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DEC64: Decimal Floating Point
DEC64 is a number type. It can precisely represent decimal fractions with 16 decimal places, which makes it very well suited to all applications that are concerned with money. It can represent values
as gargantuan as 3.6028797018963967E+143 or as measly as 1.0E-127, which makes it well suited to most scientific applications. It can provide very fast performance on integer values, eliminating the
performance justification for a separate int type and avoiding the terrible errors than can result from int truncation.
DEC64 is intended to be the only number type in the next generation of application programming languages.
DEC64 represents numbers as 64 bit values composed of 2 two’s complement components: a 56 bit coefficient and an 8 bit exponent. The coefficient is in the high order end, and the exponent is in the
low order end.
coefficient exponent
The coefficient is an integer in the range -36028797018963968 thru 36028797018963967. The exponent is in the range -127 thru 127. Numbers may not use an exponent of -128. The value of a number is
obtained from this formula:
value = coefficient * 10^exponent
Normalization is not required, and is usually not desired. Integers can have an exponent of 0 as long as the coefficient is less than 36 quadrillion. Addition of numbers with equal exponents could be
performed in a single machine cycle.
There are 255 possible representations of zero. They are all considered to be equal.
There is a special value called nan that has a coefficient of 0 and an exponent of -128. The result of division by zero is nan. nan is also the result of operations that produce results that are too
large to be represented. nan is equal to itself.
When an arithmetic operation has an input with an exponent of -128, the result will be nan. Applications are free to use the coefficient as they wish when the exponent is -128, since in that case the
coefficient has no arithmetic significance. One possible use is to store object pointers in the coefficient.
DEC64 can be implemented efficiently in hardware or software.
Conversion to and from textual representations is simple and straightforward and free of the complexities that binary floating formats must wrestle with to minimize the inevitable errors caused by
the fundamental incompatibility of the binary and decimal systems. DEC64 instead uses an internal representation that is very compatible with the e notation.
To convert an int to DEC64, shift it left 8 bits. To unpack a coefficient, shift it right 8 bits with sign extension. The exponent can be unpacked at no cost on x64 architecture because the least
significant byte can be accessed directly.
There is a fast path for addition of integers in a software implementation that takes only 5 instructions (7 on RISC-V) whilst also providing for not-a-number and overflow protection.
; Add rdx to rax.
mov cl,al ; load the exponent of rax into cl
or cl,dl ; 'or' the two exponents together
jnz slow_path ; if both exponents are zero, take the fast path
add rax,rdx ; add the coefficients together
jo overflow ; if there was no overflow, we are done
; Add x1 to x0.
orr x2, x0, x1 ; 'or' the two numbers together
ands xzr, x2, #255 ; examine the exponent part
b.ne slow_path ; if both exponents are zero, take the fast path
adds x0, x0, x1 ; add the coefficients together
b.vs overflow ; if there was no overflow, we are done
; Add x11 to x10.
or x12, x10, x11 ; 'or' the two numbers together
andi x12, x12, 255 ; isolate the exponent part
bne x12, x0, slow_path ; if both exponents are zero, take the fast path
slti x12, x10, 0 ; x12 is 1 if augend is negative
add x10, x10, x11 ; add the coefficients together
slt x13, x10, x11 ; x13 is 1 if the sum is less than the addend
bne x12, x13, overflow ; if there was no overflow, we are done
The fast path for addition in a hardware implementation should take only 1 cycle when the two exponents are equal to each other and there is no overflow. The fast path for multiplication in hardware
takes the time it takes to do a 56*56 bit signed multiply when there is no overflow.
A reference implementation is available for Intel/AMD x64 and ARM64 at https://github.com/douglascrockford/DEC64. It provides the DEC64 elementary operations. Vadim Pisarevsky has prepared a C++
version that can be found at https://github.com/vpisarev/DEC64/tree/alt.
Conversion between DEC64 and strings is trivially easy. This is demonstrated by dec64.string.
The elementary functions (sine, log, sqrt, etc) are demonstrated by dec64.math.
The idea of using powers of ten instead of powers of two is not new. For example,
Floating point subroutines and interpretive systems for early machines were coded by D. J. Wheeler and others, and the first publication of such routines was in The Preparation of Programs for an
Electronic Digital Computer by Wilkes, Wheeler, and Gill (Reading, Mass.: Addison-Wesley, 1951), subroutines A1-A11, pages 35-37 and 105-117. It is interesting to note that floating decimal
subroutines are described here, although a binary computer was being used; in other words, the numbers were represented as 10^ef, not 2^ef, and therefore the scaling operations required
multiplication or division by 10.
The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition by Donald Knuth (Addison-Wesley, 1998), page 226.
The book Knuth cited may have been the first software book. It described some of the libraries and conventions of Maurice Wilkes’s EDSAC, one of the first generation of Von Neumann machines. Some of
its subroutines used a numeric format that was very similar to DEC64.
Floating point was so important that support for it was moved into hardware for better performance. This led to the development of binary floating point because a shift could be implemented much more
easily than a divide by 10. It was discovered that by biasing the exponent and moving it to the position just after the sign bit that floating point numbers could be compared with integer opcodes, a
nifty optimization. It was also discovered that because normalization always left a 1 bit in the most significant position of the significand, that that bit could be omitted, providing an additional
bit of significance.
The Burroughs 5000 series had a floating point format in which an exponent of zero allowed the mantissa to be treated as an ordinary integer. DEC64 incorporates that brilliant idea.
Languages for scientific computing like FORTRAN provided multiple floating point types such as REAL and DOUBLE PRECISION as well as INTEGER, often also in multiple sizes. This was to allow
programmers to reduce program size and running time. This convention was adopted by later languages like C and Java. In modern systems, this sort of memory saving is pointless. By giving programmers
a choice of number types, programmers are required to waste their time making choices that don’t matter. Even worse, making a bad choice can lead to a loss of accuracy or destructive bugs. This is a
bad practice that is very deeply ingrained.
Binary floating point trades away familiarity and decimal compatibility for performance. This made it unsuitable for business languages like COBOL. Decimal fractions cannot be represented accurately
in binary floating point, which is a problem for programs that interact with humans, and is dangerous in programs that manipulate money. Exactness is required, so most business processing used BCD
(Binary Coded Decimal) in which each digit is encoded in 4 bits. That created some inefficiency, but benefited from allowing a shift by 4 bits in place of the more complex divide by 10. For a time,
mainframes could be ordered with optional floating point units for scientific computing, and optional BCD units for business computing.
The BASIC language eliminated much of the complexity of FORTRAN by having a single number type. This simplified the programming model and avoided a class of errors caused by selection of the wrong
type. The efficiencies that could have gained from having numerous number types proved to be insignificant.
Business Basic was a dialect of BASIC that was developed by Basic/Four Corporation for its small business minicomputers. It used decimal floating point, much like the EDSAC, so the language could be
used for both scientific and business applications. Business Basic could do everything that BASIC could do, and it could also handle money.
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Year is a leap year
In this example, the goal is to write a formula that will return TRUE if a year is a leap year and FALSE if not. This is a surprisingly challenging problem in Excel for two reasons: (1) Excel thinks
1900 is a leap year due to a long-standing bug inherited from Lotus 1-2-3 and (2) The logic for testing a leap year is not intuitive and requires some understanding of the history of the Gregorian
calendar system we use today. Both topics are discussed in more detail below.
Quick and dirty solution
In the worksheet shown above, we use a quick and dirty formula to check for leap years. The core of this formula is the DATE function, which will automatically adjust to month and year values that
are out of range. In the formula, the year is extracted with the YEAR function. Then it is passed into the DATE function, along with 2 for the month (February) and 29 for the day:
Then we wrap the MONTH function around that formula and check if the month is 3:
=MONTH(DATE(YEAR(B5),2,29))=2 // returns TRUE or FALSE
What's going on here? The answer depends on a subtle behavior in Excel's date system. In a leap year, February has 29 days, so the DATE function will simply return the date February 29 in the given
year. For example, when the year is 1960, DATE returns the valid date 29-Feb-1960:
DATE(1960,2,29) // returns "29-Feb-1960"
However, in a non-leap year, DATE will return March 1 in the given year, because there is no 29th day in February. In other words, the DATE function rolls the date forward to the next valid date. For
example, if we change the year to 1961, we get the date 1-Mar-1961:
DATE(1961,2,29) // returns "1-Mar-1961"
The MONTH function then extracts the month from the date and checks if the month number is 2. If the result is TRUE, we have a leap year. If the result is FALSE (the month is 3), the year is not a
leap year.
Testing the year only
In the worksheet shown, we are extracting the year from a date with the YEAR function before testing for a leap year. To test a year value only, just remove the YEAR function from the formula:
In this version, we don't extract a year value from a date, we pass a year value (i.e. 1960) directly to the DATE function.
Although the formulas above are clever and efficient, they have two limitations you should be aware of:
1. They incorrectly report 1900 as a leap year (see below for details).
2. They only work with dates/years after January 1, 1900.
These limitations arise because the formulas are built directly on Excel's date system. The first limitation is easy to work around, as explained in the next section. The second limitation is more
fundamental and requires a different approach.
Excel's 1900 problem
Excel erroneously treats 1900 as a leap year. This is due to a legacy bug from compatibility with Lotus 1-2-3, an older spreadsheet application that also erroneously treated 1900 as a leap year.
Unfortunately, this means the formulas above will incorrectly return TRUE if you are testing for a leap year with dates in 1900. You can guard against this problem with a simple hack like this:
This version of the formula enforces two conditions with the AND function:
1. The month of the adjusted date must be 2.
2. The year must not be equal to 1900.
Both conditions must be TRUE or else the formula will return FALSE. This is a simple way to avoid classifying 1900 as a leap year. However, to test year values earlier than 1900, we need a new
approach, because Excel's date functions will only work with dates beginning with January 1, 1900. However, before we look at a solution, I need to introduce the Julian and Gregorian calendars.
The Julian and Gregorian calendars
The formulas below don't make sense unless you know a little about the history of the Julian and Gregorian Calendars. The Julian Calendar was a system of dates instituted by Julius Caesar in 46 BC.
This calendar added one extra day every four years, based on the calculation that it takes 365.25 days for the earth to travel around the sun, not exactly 365 days. The idea was to account for an
extra day every four years, creating a "leap year". However, it turns out this calculation is not correct. The sun's trip around the sun is not exactly 365.25 days but 365.24237 days, approximately
11 minutes less. As a result, the Julian Calendar was over-correcting by about 8 days every 1000 years.
Further study in the 16th century resulted in a better solution. The idea was that centenary years would not be leap years unless they were divisible by 400. This meant three out of four centenary
years would not be leap years. In other words, every 400 years there would be 97 leap years instead of 100 leap years. In 1582, Pope Gregory ruled that the new date system (called the "Gregorian
Calendar" thereafter) should replace the Julian Calendar. In simple language, the rule for leap years is as follows:
To be a leap year, the year number must be divisible by four – except for end-of-century years, which must be divisible by 400. This means that 2000 is a leap year, but 1700, 1800, and 1900 are not
leap years.
The long-form formula below is meant to follow this logic.
Classic leap year formula
Now that we understand the basic history of leap years in our calendar system, we can look at how to implement a leap year rule in Excel. To handle years before 1900, we need a math-based formula
that doesn't require Excel's date functions. The classic long-form formula to test for a leap year looks like this:
Here, A1 contains a year value like 1985, 2005, etc. This formula uses the MOD function to test if the year is evenly divisible by 400, 100, and 4 and applies this logic to determine if the year is a
leap year:
1. If the year is divisible by 400, it's a leap year (TRUE).
2. If the year is not divisible by 400 but is divisible by 100, it is not a leap year (FALSE)
3. If the year is not divisible by 100 but is divisible by 4, it's a leap year (TRUE).
The same logic can be condensed by replacing the nested IF construction above with the AND and OR functions like this:
1. If the year is divisible by 400, it's a leap year (TRUE).
2. Or if the year is divisible by 4 and not divisible by 100, it's a leap year (TRUE)
3. Otherwise, the year is not a leap year (FALSE).
Both formulas above work well, and both correctly report 1900 as a non-leap year. The formulas are a bit longer and less intuitive, but both formulas can be used to test year values before 1900. By
contrast, the original "short" formulas at the top of this page depend on Excel's date engine and won't handle dates or years before 1900.
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A three critical point result in a bounded domain of a Banach space and applicationsA three critical point result in a bounded domain of a Banach space and applications
Using the bounded mountain pass lemma and the Ekeland variational principle, we prove a bounded version of the Pucci-Serrin three critical points result in the intersection of a ball with a wedge in
a Banach space. The localization constraints are overcome by boundary and invariance conditions. The result is applied to obtain multiple positive solutions for some semilinear problems.
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Patrizia Pucci
Dipartimento di Matematica e Informatica (DMI) Università di Perugia 06100 Perugia, ITALY
Csaba Varga
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
R. Precup, P. Pucci, C. Varga, A three critical point result in a bounded domain of a Banach space and applications, Differential Integral Equations 30 (2017), no. 7-8, 555-568.
Differential Integral Equations
[1] A. Ambrosetti and P.H. Rabinowitz,
Dual variational methods in critical points theory and applications
, J. Funct. Anal., 14 (1973), 349–381.
[2] C. Azizieh and P. Clément,
A priori estimates and continuous methods for positive solutions of p–Laplace equations
, J. Differential Equations, 179 (2002), 213–245.
[3] C. Chidume, “Geometric Properties of Banach Spaces and Nonlinear Iterations,” Lecture Notes in Mathematics, xviii, Springer–Verlag London, (1965).
[4] I. Ciorănescu, “Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,” Mathematics and its Applications, 62 Kluwer, Dordrecht, (1990).
[5] L. Damascelli and F. Pacella,
Monotonicity and symmetry of solutions of a p–Laplace equations
via the moving plane method
, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 689–707.
[6] G. Dinca, P. Jebelean, and J. Mawhin,
Variational and topological methods for Dirichlet problems with p–Laplacian
, Port. Math., 58 (2001), 339–378.
[7] I. Ekeland,
On the variational principle
, J. Math. Anal. Appl., 47 (1974), 443–474.
[8] H. Lisei, R. Precup, and C. Varga,
A Schechter type critical point result in annular conical domains of a Banach space and applications
, Discrete Contin. Dyn. Syst., 36 (2016), 3775–3789.
[9] R. Precup,
The Leray–Schauder boundary condition in critical point theory
, Nonlinear Anal., 71 (2009), 3218–3228.
[10] R. Precup,
Critical point theorems in cones and multiple positive solutions of elliptic problems
, Nonlinear Anal., 75 (2012), 834–851.
[11] R. Precup and C. Varga,
Localization of positive critical points in Banach spaces and applications
, Topol. Methods Nonlinear Anal., to appear.
cf. MR3670487
[12] P. Pucci and J. Serrin,
Extensions of mountain pass theorem
, J. Funct. Anal., 59 (1984), 185–210.
[13] P. Pucci and J. Serrin,
A mountain pass theorem
, J. Differential Equations, 60 (1985), 142–149.
[14] P. Pucci and J. Serrin,
The strong maximum principle revisited
, J. Differential Equations, 196 (2004), 1–66.
[15] M. Schechter, “Linking Methods in Critical Point Theory”, Birkhäuser, Boston, (1999).
[16] M. Willem,
Minimax Theorems
, Progress in Nonlinear Differential Equations and their Applications, 24 (1996).
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Twelve Papers on Functional Analysis and Geometrysearch
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Twelve Papers on Functional Analysis and Geometry
Hardcover ISBN: 978-0-8218-1785-8
Product Code: TRANS2/85
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Click above image for expanded view
Twelve Papers on Functional Analysis and Geometry
Hardcover ISBN: 978-0-8218-1785-8
Product Code: TRANS2/85
List Price: $175.00
MAA Member Price: $157.50
AMS Member Price: $140.00
eBook ISBN: 978-1-4704-3296-6
Product Code: TRANS2/85.E
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
Hardcover ISBN: 978-0-8218-1785-8
eBook ISBN: 978-1-4704-3296-6
Product Code: TRANS2/85.B
List Price: $340.00 $257.50
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AMS Member Price: $272.00 $206.00
• American Mathematical Society Translations - Series 2
Volume: 85; 1969; 258 pp
MSC: Primary 46
□ Articles
□ V. P. Gluško and S. G. Kreĭn — Inequalities for norms of derivatives in weighted $L_p$ spaces
□ A. S. Markus — Some criteria for the completeness of a system of root vectors of a linear operator in a Banach space
□ M. G. Kreĭn and Ju. L. Šmul′jan — Plus-operators in a space with indefinite metric
□ M. G. Kreĭn and Ju. L. Šmul′jan — $\mathfrak {I}$-polar representation of plus-operators
□ I. C. Gohberg and M. K. Zambickiĭ — On the theory of linear operators in spaces with two norms
□ R. S. Ismagilov — Unitary representations of the Lorentz group in a space with indefinite metric
□ V. T. Fomenko — Infinitesimal bendings of convex surfaces with bush constraints
□ N. M. Pisareva — Almost-reducible and symmetric almost-reducible affinely connected spaces
□ N. M. Pisareva — Affinely connected spaces admitting a transitive group of motions with a completely reducible stationary linear subgroup
□ D. P. Mil′man and V. D. Mil′man — The geometry of nested families with empty intersection. structure of the unit sphere of a nonreflexive space
□ V. R. Petuhov — Geometry of the Möbius strip and differential equations
□ S. G. Gindikin and F. I. Karpelevič — On an integral connected with symmetric Riemann spaces of nonpositive curvature
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 85; 1969; 258 pp
MSC: Primary 46
• Articles
• V. P. Gluško and S. G. Kreĭn — Inequalities for norms of derivatives in weighted $L_p$ spaces
• A. S. Markus — Some criteria for the completeness of a system of root vectors of a linear operator in a Banach space
• M. G. Kreĭn and Ju. L. Šmul′jan — Plus-operators in a space with indefinite metric
• M. G. Kreĭn and Ju. L. Šmul′jan — $\mathfrak {I}$-polar representation of plus-operators
• I. C. Gohberg and M. K. Zambickiĭ — On the theory of linear operators in spaces with two norms
• R. S. Ismagilov — Unitary representations of the Lorentz group in a space with indefinite metric
• V. T. Fomenko — Infinitesimal bendings of convex surfaces with bush constraints
• N. M. Pisareva — Almost-reducible and symmetric almost-reducible affinely connected spaces
• N. M. Pisareva — Affinely connected spaces admitting a transitive group of motions with a completely reducible stationary linear subgroup
• D. P. Mil′man and V. D. Mil′man — The geometry of nested families with empty intersection. structure of the unit sphere of a nonreflexive space
• V. R. Petuhov — Geometry of the Möbius strip and differential equations
• S. G. Gindikin and F. I. Karpelevič — On an integral connected with symmetric Riemann spaces of nonpositive curvature
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions.
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Estimation of the Electricity Storage Volume Density of Compact SMESs of a New Concept Based on Si Microfabrication Technologies
Institutes of Innovation for Future Society, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
Toyota Technological Institute, Graduate School of Engineering, Hisakata 2-12-1, Tempaku-ku, Nagoya 468-8511, Japan
Materials & Surface Engineering Research Institute, Kanto-Gakuin University, 1162-2, Ogikubo, Odawara 250-0042, Japan
Author to whom correspondence should be addressed.
Submission received: 22 February 2021 / Revised: 17 March 2021 / Accepted: 17 March 2021 / Published: 23 March 2021
A compact superconducting magnetic energy storage system (SMES) produced by Si micro fabrication technologies has been proposed to improve electricity storage volume density, w, in the sub-Wh/L range
of conventional SMESs and to produce them at a low cost by mass production. In parallel with the experimental development reported previously, a series of trials was performed to estimate a feasible
value of w based on the calculation of the magnetic field generated by the compact SMES by improving the calculation models step by step. In this work, the experimentally obtained magnetic flux
density dependence of superconductive critical current density was taken into consideration for the first time in this series of trials, together with the additional improvement of the calculation
models. The results of the estimation indicated that a compact SMES produced by the proposed concept can attain a w in the Wh/L range or more, ranking with or surpassing that of presently used
1. Introduction
1.1. Background and Motivation of the Research
Capacitors are typical electronic passive components widely used in varieties of electronic circuits. Capacitors are fundamentally used for electricity storage as the energy of the electric field,
which is expressed by Equation (1):
, and
stand for the energy density of the electric field, the permittivity of the vacuum, and the strength of the electric field. Electricity storage in capacitors is based on an electrostatic, a purely
physical phenomenon. The benefit of this fact is the ability of rapid charge and discharge in comparison to rechargeable batteries based on electrochemical reactions. Many varieties of capacitors are
commercially available, such as ceramic capacitors, polymer film capacitors, electrolytic capacitors, and electric double-layer capacitors (EDLC). Among them, EDLCs are reported with a high electric
energy storage density (
). Judging from open websites, typical specifications for size, maximum stored electric energy, and
, are Specifications 1: ⌀18 mm × 70 mm, 0.1 Wh, 5.7 Wh/L [
]; Specifications 2: ⌀40 mm × 150 mm, 1.2 Wh, 6.5 Wh/L [
]; and Specifications 3: ⌀51 mm × 142 mm, 2.2 Wh, 7.5 Wh/L [
]. Here, it should be noted that what is called “hybrid energy storage capacitors” are reported to have a higher
, with Specifications 4: 35 mm × 25 mm × 20 mm, 0.2–0.9 Wh, 11–50 Wh/L [
]. However, they are excluded from the purely physical electric energy storage means based on Equation (1), because they are based not only on the electrostatic storage of energy but also on the
electrochemical storage of energy.
As a counterpart of Equation (1), in electromagnetics, the energy of magnetic field
is formulated as Equation (2):
stand for the permeability of a vacuum and magnetic field strength, respectively. As for means of physical electric energy storage based on Equation (2), superconducting magnetic energy storage
systems (SMES) were developed for power system control, such as load fluctuation compensation and power system stabilization, in which a superconducting coil generates a magnetic field [
]. A state-of-the-art practical example being in operation is of 5.6 kWh [
], in which superconducting coils are formed by coiling composite wires composed of NbTi superconducting wires and Cu wires twisted together (NbTi/Cu coils). Cu wires are employed to assume a role of
stabilizer to homogenize the local temperature fluctuations of NbTi superconducting wires with the help of the high thermal conductivity of Cu. The local temperature fluctuation can lead to
quenching—that is, the accidental loss of superconductivity in NbTi wires. Cu wires also assume the role of the current bypass in the case of quenching. Because NbTi/Cu composite wires are rigid and
difficult to coil at a small curvature radius, the size of a container comprising superconducting NbTi/Cu coils can be as large as ⌀3.2 m × 3.2 m [
]. This gives a
of about 0.18 Wh/L judging from the literature. However, ancillary facilities are also necessary in a SMES system including a cryogenic refrigeration apparatus and magnetic shields to prevent the
intense magnetic field generated by the superconducting coils from causing false operation in peripheral electronic and mechanical apparatuses. Suppose the volume of these ancillary facilities is the
same as that of the container of the superconducting magnet, the total volume is doubled, and the
is estimated to be 0.09 Wh/L. More recently, YBa
(Y123) tape or Bi
(Bi2223)/Ag tape has been used to fabricate a superconducting coil in place of the NbTi/Cu composite wire, and the sizes of the superconducting magnet were much reduced [
]. As a typical example, the size of a coil with an electricity storage capacity of 278 Wh was reportedly ⌀0.78 m × 0.69 m, and the
can be calculated to be 0.84 Wh/L judging from a recent, state-of-the-art report [
]. If ancillary facilities of the same volume are taken into consideration, the
will be around 0.4 Wh/L.
A superconducting current in a coil in the vicinity of the critical superconducting current density (
) can induce an intense magnetic field, resulting in a huge electromagnetic force imposed on the superconducting current along the radial direction to enlarge the coil diameter (‘hoop stress’). To
prevent the coil from breaking under this force, the coil must be protected by specially designed reinforced structures [
]. The necessity of this reinforced structures is another factor that causes the large volumes of conventional SMES systems.
Therefore, despite using the same physical electric energy storage means based on similar expressions—Equations (1) and (2)—in electromagnetics, the sizes are much larger and the w is much smaller in
the present SMESs in operation, in comparison with commercially available and widely used capacitors. In addition to the lower w in comparison with capacitors, SMESs are also inferior to capacitors
in terms of their costs. This is partly caused by the fact that SMESs being in operation are made-to-order heavy electric machinery products, in contrast to mass-produced capacitors, which are
commercially available with many choices. Therefore, the development of a mass-producible SMES will be an important way to expand use of electricity storage means based on Equation (2). Motivated by
this, we proposed and pursued a novel approach to improve the w and mass producibility of SMES.
1.2. A Novel Approach to Increase the w and Mass Producibility of SMES
To increase the
of SMES from a sub-Wh/L level to a sub−10 Wh/L level ranking with the
of capacitors or more, we proposed the concept of a compact SMES, as shown in
Figure 1
. Without winding the superconducting cables or tapes at a small curvature radius, a superconducting thin-film coil is deposited and embedded in a spiral trench to prevent it from coming off from the
substrate by electromagnetic stress
which is applied to the coil, as shown in
Figure 1
a. Because
is a hoop stress put upon the outer wall of the trench, the superconducting current must be reduced to keep
lower than 1/3 of the mechanical strength limit of the Si wafer σ
: 4 GPa. Here, 1/3 is a safety factor. Multiple Si wafers, which are engraved with a spiral coil (hereafter, we call it a wafer coil for simplicity), are stacked and series-interconnected via
through-holes based on 3D integration technology to form a cylindrical coil unit, as shown in
Figure 1
b [
]. In a typical possible concept design, the four units together with a compact cryogenic refrigerator are combined to form a typical possible minimum system, as shown in
Figure 1
c. Compact cryogenic refrigerators have been greatly improved recently and are on the market [
]. Here, the cryogenic refrigerator is drawn as a simplified form composed of a cuboid of
and a cylindrical tower with a height of
1.3. Experimental Proof of Concept Using Superconducting NbN Thin Films
A proof-of-concept experiment was performed on a single Si wafer using reactively sputter-deposited NbN superconducting thin films [
] and an electrolytically plated Cu layer as a stabilizer.
Figure 2
a–e schematically displays the process of trench formation; deposition of NbN and Cu; and the removal of unnecessary deposits of NbN and Cu, except for those in the trench, by chemical-mechanical
polishing (CMP) step by step.
Figure 2
f–h schematically displays an example of the wafer bonding and series interconnection processes. Details of these processes were described in our previous report [
Figure 3
a,b shows two photos of the Si wafer coils in a preliminary fabrication, corresponding to the process steps shown in
Figure 2
d,e, respectively. These process technologies were used to fabricate normal conductive MEMS (microelectromechanical system) inductors on Si wafers [
]. The measurement of the current–voltage characteristics in cryogenic temperatures together with the measurement of magnetic field by a gauss meter showed an energy storage of 0.01 mJ [
]. Increasing the NbN thickness by mitigating the film stress [
], the stored energy increased up to 0.1 mJ [
]. Here, the experimentally obtained
of NbN—1100 A/mm
— was used [
]. The largest
near the central axis of the coil unit,
, estimated from the magnetic flux density
along the central axis of a single-layer solenoid was 0.11 GPa, which was well below σ
/3 [
1.4. Replacement of NbN by YBa[2]Cu[3]O[7-δ] to Increase Electricity Storage Volume Density
For further improvement, we moved on to the replacement of the reactively sputter-deposited NbN thin film by Y123 deposited by the metal-organic deposition (MOD) method [
]. Because metal-organic precursors dissolved in a carrier fluid such as propionic acid can easily go into the trench, this is advantageous in comparison with sputter deposition with a limited
throwing power. Because the superconducting transition temperature (
) of Y123 is 90 K, this replacement enables the use of liquid hydrogen at 20.3 K or even of liquid nitrogen at 77 K as a refrigerant instead of a cryogenic refrigerator to cool below 13 K in the case
of NbN. However, the real benefit of this replacement for SMES is that the
of Y123 can be much higher than that of NbN if we keep using a 4 K cryogenic refrigerator.
Figure 4
displays two curves, (a) and (b), of the measured
of the c-axis-oriented Y123 thin-film tape at 4 K as a function of the magnetic flux density applied in parallel with the c-axis (redrawn from the literature [
]). According to curve (a),
was measured to be up to 31 T in the applied magnetic flux density. There is no indication of a rapid decrease in
due to the accession of the critical magnetic flux density
. It was also reported that
was also measured to be as high as 120 T at 4 K and continuously decreased with the temperature to 0 T at the critical temperature:
= 90 K [
]. Curve (b) shows more modest values of
than the curve (a). Although the reported measured maximum applied magnetic field density was around 18 T in curve (b), it will be reasonable to conjecture that curve (b) in
Figure 4
can be smoothly extrapolated beyond 18 T to at least 50 T, as shown in curve (c). Typically, the values of
for the applied magnetic field of 15 T on curves (a) and (b) are 5 × 10
and 2 × 10
, which are about 45.5 times and 10 times of 1100 A/mm
in the case of NbN, respectively. Because
is proportional to the square of the current, we may expect, in comparison to the case of NbN, a 45.5 × 45.5 = 2070, and 18 × 18 = 324 times larger
in the case of Y123 for curves (a) and (b), respectively. Here, it should be noted that the maximum magnetic flux density at the innermost trench,
is proportional to the superconducting current
, and
is proportional to
. This means that the maximum superconducting current
in the coil must be much reduced from the value derived from
because of the limitation of
and S = σ
/3. We intuitively know that we can mitigate this large
applied to the innermost portion of the coil by making the radius of the inner wall of the innermost trench,
, larger, leaving the central part of the wafer uncovered with trenches, as shown in
Figure 3
a. However, we lose
simultaneously. Therefore, we must estimate how much we could mitigate
at the innermost portion of the coil and how much we lose
quantitatively in concrete designs of the wafer coils.
1.5. Aim of This Work
In our previous papers [
], we reported our series of trial estimations on this subject under the fixed
of 2 × 10
. This value corresponds to a
at 15 T on line (b) in
Figure 4
. However,
is dependent on
, as shown in curves (b) and (c) in
Figure 4
. There is also experimental evidence of higher
, as shown in curve (a) in
Figure 4
. To make a more realistic estimation, the
dependences of
based on the two different
curves shown in
Figure 4
were taken into consideration for the first time in our series of trial estimations. There was another item to refine the estimation. In the previous papers, the inductance
of the cylindrical coil unit made of the stack of series-interconnected wafer coils was calculated as an equivalent single-solenoid coil, which generated the same magnetic field intensity at the
center of the coil. As for the radius of the equivalent single-solenoid coil, we simply used a mean radius
’ of
and the radius of the outer wall of the outermost trench of the wafer coil,
—that is,
’ = (
)/2. More precisely, the effective radius of the equivalent single-solenoid coil,
, should be smaller than
’ because the contribution of an inner coil to the magnetic field at the center of the wafer coil is larger than that of an outer coil. In this work, the effective radius
was also taken into consideration for the first time in our series of trial estimations. The aim of this work is to elucidate whether or not it is possible for SMES to increase
to rank with and to surpass the
of capacitors by the improved estimations. Tentatively, we started with the more modest curve (b) for the magnetic flux density dependence of
2. Method of Estimation
2.1. Calculation of Magnetic Flux Density in a Wafer Coil
Figure 5
schematically illustrates approximately one-half of the cross-section of a wafer coil, showing the principal design parameters of the spiral trench. Here,
stand for the radius and thickness of the Si wafer, respectively. Parameters
, and s stand for the trench width, trench depth, and trench wall thickness, respectively. The yellow rectangles indicate cross-sections of the trenches filled with Y123. Here, the trenches are
supposed to be filled with Y123 and the presence of stabilizer layers as shown in
Figure 2
Figure 3
b is neglected for simplicity. The red circle with an arrow around a trench schematically indicates a typical magnetic field generated by the currents flowing Y123 in the trench from the back to the
surface of the plane of paper.
As for the calculation of the distribution of magnetic flux density in the wafer coil, we replaced a spiral coil with the number of turns
conductive concentric circular current paths with the same
, and
for a calculation model. In all the
concentric current paths, the currents were supposed to be the same. The calculation was based on the Biot-Savart law, as shown in the following equation in SI units:
$B r = μ 0 4 π ∮ I d q ⊗ r ′ r ′ 3$
is the magnetic permeability of free space: 4π × 10
is the absolute value of the current in the circular current path.
is the vector for magnetic flux density at the position defined by vector
The wire element d
is the vector of the circular current path element whose magnitude is the length of the differential element of the circular current path in the direction of the current at the position defined by
. The vector
is the full displacement vector from d
d ℓ {\displaystyle d{\boldsymbol {\ell}}}at point
ℓ {\displaystyle {\boldsymbol {\ell}}} to point
. The expression d
represents the cross product of the vectors d
. Numerical calculation was performed by integrating current density
in the
rectangular cross-section of the trench using the “integ.tplquad” function in a Python 3.6 calculation code.r {\displaystyle \mathbf {r}} The calculated magnetic flux density using this calculation
code at the center of a single circular current normal to the plane of the circle was confirmed to agree with the value obtained from the analytical expression (4) deduced from Equation (3):
is the radius of the circular current
in the trench in which
can be equal to
cannot be calculated by Equation (3) because the integrand containing
in its denominator reaches an infinite value. Typical calculations of
) for a single planar spiral coil formed in a spiral trench of width:
= 50 μm; depth:
= 30 μm; the trench wall thickness:
= 22 μm;
= 19 mm; and turn number
: 400 (resulting in
= 47.8 mm on a Si wafer of the diameter 2
= 101.6 mm) were performed assuming a fixed
of 2 × 10
. In the actual calculation by this calculation code, the
)s caused by concentric 400 circular currents were summed up as a model of the single planer spiral coil.
Figure 6
a shows the results of the calculations of the radial distribution of the magnetic flux density normal to the wafer coil,
), at the half depth of the trench. Here, the scalar:
is the absolute value of the vector
when the origin of
is located at the center of the wafer plane. The calculation of
) when
is located out of the trench can be performed because
’ in the denominator of the integrand in Equation (3) cannot be zero. Here, let us call this type of calculation Type I. The solid black circles stand for the calculated
) of Type I. We need to know
) penetrating the Y123 in the trench to estimate the electromagnetic hoop stress imposed on the Y123 as well as the
under the magnetic field generated by the other parts of the concentric circular currents. Because the magnetic flux generated by the current in a portion of a trench does not penetrate the same
portion of the trench, the current in the corresponding portion can be eliminated from the estimation of
). Accordingly, the calculation of
) in the trench was also performed by eliminating the current element in a small volume portion of the trench surrounding the calculating position
from the integral shown in Equation (3) so that
does not become equal to
. Let us call this type of calculation Type II. The open black circles in
Figure 6
a indicate the
) calculated in the trench by Type II. The small volume portion was defined as the volume portion in the circular trench within the plane angle area
in the vicinity of the point
subtended at the center of the Si wafer.
Figure 6
b schematically display the small volume portion of a trench surrounding a calculated position indicated by
employed in the calculation of Type II in a special case where the trench is the inner-most circular trench. As for the concrete value of
in Type II calculations, 0.001 radian was employed. In this sense, the calculation of Type I can be regarded as the calculation of Type II at
= 0, as shown in
Figure 6
a. In this special case of the Type II calculation, several values of
) for r <
, out of the trench, were also calculated at the several points including
= 0 mm and plotted with open black circles in
Figure 6
a. In the region
the points calculated by Type I and Type II are on the same line. Thus, it is duly elucidated that the effect of the elimination of the current in the volume portion of the innermost circular trench
) in this area,
, was negligible. When
in the area
at the position
shown in
Figure 5
) steeply increased in the Type I calculation.
Figure 7
displays the calculated values of
) more in detail than
Figure 6
a in the vicinity of the innermost trench. Type I calculations were only performed for the positions out of the trench, as plotted with the solid black circles (a), while Type II calculations were
performed also in the trench, as plotted with the open black circles (c). Type I calculations for a single innermost circular trench without the other concentric trenches were also conducted and
plotted with solid black triangles (b). It is obviously known that the increase in (a) when
decreased and approached
= −19.00 mm was caused by the contribution of (b). In the Type II calculation,
) takes a smaller value of 0.71 T at position
= 19.025 mm) in the center of the trench in
Figure 6
b in comparison with 0.84 T by the Type I calculation at
= 18.99 mm. In the peripheral area near the edge of the Si wafer, as indicated with C in
Figure 5
, the values indicated with the solid black circles in
Figure 6
a steeply decrease with the distance from the wafer center, in contrast to the open black circles staying near the zero magnetic field line in the same figure. This is because the magnetic flux
density produced by the outermost ring current gives negative
) at the position C, while this does not take place in Type II calculation. Except for this difference, the calculated values for
(r) generally showed a decrease with the increase in
for both the solid black circles and the open black circles in
Figure 6
a. As shown in
Figure 6
) takes the highest value in the innermost trench in a wafer coil.
2.2. Calculation of Magnetic Flux Density in a Stack of Wafer Coils
In the case of the cylindrical unit of the stacked wafer coils, the magnetic flux density caused by a different wafer coil in the stack is also imposed on the innermost trench in question.
Figure 8
a schematically illustrates the component of the magnetic flux density
) normal to the wafer at the innermost portion of the spiral coil at the radius
/2 and at depth
/2 in a wafer caused by another wafer coil at a distance
in the same stack with a red arrow. If the sum of contributions of magnetic flux densities caused by all the stacked wafer coils at a position in the innermost trench of a wafer coil exceeds
, the current density must be reduced to maintain superconductivity. The radial outward-directed black arrow indicates the electromagnetic hoop stress imposed on the inner-most trench by the
accumulated magnetic flux density of the all the contributions from the stacked wafer coils shown in the red arrows.
Figure 8
b shows the dependence of
) on
= 47.8 mm,
= 19.0 mm,
= 50 μm,
= 22 μm,
= 30 μm, and
= 0.5 mm. Because
) is plotted in logarithmic scale, it is clearly shown that
) decreases rapidly with increasing
, and it is lower than 10% of the peak value when
> 2.5 cm. The sum of the contributions
) of all the wafers in a stack is denoted by
Figure 9
displays a variation in
) as a function of the position of the wafer
in the stack in a typical case of the height of the cylindrical coil unit:
= 600. Although
) has a maximum at
= 15 cm, it is almost constant, except for the regions near
= 0 cm and
= 30 cm. This is because the major contributions of
) are of wafer coils at
< 2.5 cm, as shown in
Figure 8
Figure 10
a displays the dependence of the
curve on
. The maximum value of
) at
= 15 cm,
,almost linearly decreased with
, as shown in
Figure 10
2.3. Calculation of Inductance of the Wafer Coil Stack
) at
= 0.0 mm—was calculated as a function of
, as shown in
Figure 11
. Here, the Type I calculations were employed because
= 0.0 mm represents the center of the wafer, which is located out of the trenches. The radius
of a single circular current:
, which yields the same magnetic flux density as
(0) at its center is calculated by Equation (4) and also plotted as a function of
Figure 11
. It is shown that
is generally smaller than the mean radius of the spiral coil
’ = (
)/2, and this deviation increases with a decrease in
wafer coils are supposed to be stacked to form a cylindrical unit of 2
in diameter and
in height. The spiral coils on adjacent Si wafers are supposed to be series-interconnected through two through-holes [
] formed in the wafers by such a method illustrated in
Figure 2
f–h, which is frequently used in conventional multi-layer interconnection technology for 3D integration [
]. Thus, the series-interconnected stacked multiple wafer coils form a combined coil of inductance
. For simplicity in the calculation of inductance,
, these series-interconnected wafer coils were simulated to be a single-layer solenoid of the length
and of the equivalent radius
given in
Figure 11
. If the number of turns of the coil in a wafer is
can be expressed using the formula for the single-layer solenoid as follows:
$L = K ⋅ μ 0 ⋅ μ r ⋅ π a 2 ⋅ n ⋅ m 2 m ⋅ t$
is Nagaoka’s coefficient, which is a function of
stands for relative magnetic permeability. In the case of Si, it is reasonable to set
= 1.
Figure 12
displays the calculated values of
as a function of
in the case where the equivalent radius
shown in
Figure 11
was employed for the equivalent single-layer solenoid. The calculated values of
in the case where the mean radius of the spiral coil
’ = (
)/2 were employed for the equivalent single-layer solenoid, are also displayed for comparison. It was found that
took the maximum value around
= 13 mm when the equivalent radius,
, was employed, while
showed no peak when the mean radius,
’, was employed. The appearance of the maximum was caused by the variation in the product,
, in Equation (5), where
increased rapidly with
decreased rapidly with
. In contrast, the linear increase in
’ with
did not cause a maximum in variation of the product
in Equation (5).
2.4. Calculation of Electricity Storage Volume Density of the Wafer Coil Stack
The electric energy
stored in a solenoid of inductance
can be expressed as:
Here, the superconducting current
which flows in the coil at the critical current density
can be expressed as
. Here,
is calculated for the volume of cuboid including 4 stacks, as shown in
Figure 1
c—that is, in the special case of
Because the volume of the cuboid in
Figure 1
c is
× 2
, including the space for refrigerant, magnetic shields, heat insulation, and cryogenic refrigerator,
× 4/(
× 2
). Here, another volume of
is supposed to be necessary for the cryogenic refrigerator in addition to the upper container of the volume of
for 4 cylindrical wafer stack units. In this paper, estimations of
were performed for the cases of
= 0.3 m and
= 0.5 mm. The value
can be expressed as
= (
). Because
≪ 1 and
≪ 1,
can be approximated as follows:
$n = R max − R min d + s$
Then, (5) can be expressed as follows:
$L = K ⋅ μ 0 ⋅ π a 2 ⋅ R max − R min d + s ⋅ m 2 m ⋅ t$
can be expressed as follows:
$w = 1 2 K ⋅ μ 0 ⋅ π a 2 ⋅ R max − R min d + s ⋅ m 2 m ⋅ t ⋅ d ⋅ z ⋅ j c 2 × 4 b × b × 2 h = 2 K ⋅ μ 0 ⋅ π a 2 ⋅ R max 2 1 − R min R max 2 ⋅ 1 1 + s d 2 ⋅ z ⋅ j c 2 b × b × 2 h ⋅ t / m$
Because trenches cannot be formed on the peripheral ring area of the Si wafer with a width of at least 3 mm for the handling of the wafer in the series of processes,
can be set in a limited range and is essentially constant. Therefore,
increases with a decreasing
. Equation (9) also teaches us that
increases with a decreasing (
) and with increasing
. Because
in the special case drawn in
Figure 1
c, the denominator of Equation (9) does not contain the number of Si wafer coils,
. However, an increase in
can contribute an increase in
, because
increases with
. A decrease in
with a decrease in
, as shown in
Figure 11
, can also contribute to an increase in
via an increase in
. However,
increases with
because Equation (9) contains
in the numerator. Therefore,
can increase with an increasing
. Thus, it is not straightforward to know the actual variation in
2.5. Electromagnetic Hoop Stress
Expressing free electron density as
and elementary charge 1.60 × 10
C as
, the charge
C in the superconducting film per length d
along the trench can be expressed as
. Expressing the electron velocity as
is expressed as
. The Lorenz force
imposed on the current
can be expressed as
(r). Because the corresponding surface area on the trench wall is
the hoop stress
in the unit of GPa on the trench wall can be expressed as follows:
$S = I ⋅ d x ⋅ ∑ B z r z ⋅ d x × 10 − 9$
is normal to
gives the hoop stress, as shown in
Figure 8
a. When
(r) takes the maximum value
at the innermost trench,
also takes the maximum value,
Figure 13
displays the variations in the electricity storage volume density,
, and
, as a function of
2.6. Consideration of Magnertic Flux Density Dependent j[c] in the Calculation
The results displayed in
Figure 13
were calculated using the magnetic flux density independent
of 2 × 10
. In the next step, the magnetic flux density dependent
shown in curves (b) and (c) in
Figure 4
is taken into consideration. This is essentially an iteration process in which the
decided by the initially set value
, and then
decides the new value of
, which reduces
, and so on. For the convenience of calculation,
was expressed as a function of
in expression (11) based on curves (b) and (c) in
Figure 4
$j c A / m m 2 = 85,800 ∑ B max T + 0.672 + 12,900$
This iteration process finally gave converged values of
Figure 14
displays the
obtained with Equations (9) and (10) using
j [c∞]
as functions of
. It is found that
were much reduced in comparison with
Figure 13
. The perpendicular dashed-dotted red line was drawn to show the value of
at which
took the value of 20 T. The red open circle indicates that the
curve crosses the line of
= 20 T. The corresponding value of
was 24.7 mm. If superconductivity is retained up to this
: 20 T, it is shown that the value of
: 3.6 Wh/L is attained at
: 24.7 mm where
takes the value of 0.017 GPa. The perpendicular dashed-two dotted blue line was drawn at the value of
at which
took the maximum value of 4.3 Wh/L. The corresponding value of
was 15.3 mm, where
took the values of 26.7 T and 0.021 GPa, respectively. Therefore, it was shown that the value of
, 4.3 Wh/L could be attained if the wafer coils are superconductive up to
= 26.7 T. Here, it should be noted that the equivalent radius
does not change with the reduced critical current
(0) is also reduced in proportion to the reduced
, and the ratio
(0) does not change in Equation (4).
2.7. Effects of the Wafer Coil Design and the Hight of the Wafer Coil Stack on the Electricity Storage Volume Density
The results shown in
Figure 14
were obtained for the Si wafer with radius
= 50.8 mm, the radius of the outer wall of the outermost trench
= 47.8 mm, and the thickness
= 0.5 mm in the case of the trench width
= 50 μm, the trench wall thickness
= 22 μm, the trench depth
= 30 μm, and the number of stacked Si wafers
= 600. Here, the effects of changing
, and
in practically viable ranges on
, and
were examined.
As mentioned above, Equation (9) teaches us that
increases with decreasing (
) and with increasing
. Because
does not depend on the absolute values of
but depends on the ratio
, the examination of the different values of the sum
is not necessary. Therefore, the cases of
= 70 μm and s = 2 μm were examined at the same sum
in the case of
= 50 μm and
= 22 μm. As was elucidated in
Figure 8
b and
Figure 9
, the contributions of the stacked wafers at a distance more than 5 cm to
) are negligible. The distance of 5 cm in the stack corresponds to 100 wafer coils. Therefore, cases of
= 200 in addition to cases of m = 600 were examined. Although the change in
from 600 to 200 decreases
, as shown in
Figure 15
, and hence the amount of the magnetic energy storage for each cylindrical coil decreases, eight more cylindrical coil units can be installed in the cryogenic container of the volume
, as shown in
Figure 16
, which in turn increases
. Although
increases with an increasing
, there is an experimental limitation in increasing
because of the following reasons. The high-temperature heat treatment of the wafer coil in an oxygen atmosphere is indispensable for the crystallization and adjustment of oxygen deficiency
in YBa
. This heat treatment induces the diffusion of Si into the Y123 layer, which degrades the superconducting property of Y123. To prevent this, some intermediate buffer layers are necessary between Si
substrates and Y123 layers. The buffer layers are also necessary to mitigate lattice and thermal expansion coefficient mismatches between Y123 and Si. For these reasons, the sputter deposition of the
yttria-stabilized zirconia layer succeeded by the sputter deposition of the CeO
layer was used to form the intermediate buffer layers [
], and we also employed these buffer layers in our previous experimental study [
]. Because the throwing power of sputter deposition into the trench is limited, the large aspect ratio of the trench,
, cannot be employed. Therefore, cases of
= 100 μm were examined as a realistic value in addition to cases of
= 30 μm.
Figure 17
schematically summarizes the combinations of the wafer coil design parameters,
, examined here.
2.8. Comparison of the Effects of the Wafer Coil Design and the Height of the Wafer Coil Stack under Two Different Magnetic Flux Density-Dependent j[c]
Like Equation (11) for curves (b) and (c) in
Figure 4
was also expressed as a function of
based on curve (a) in the following expression (12):
$j c A / m m 2 = 59,400 ∑ B max T + 1.75 + 9630$
Iteration processes using Equation (11) or Equation (12) gave converging values of j[c∞.] and ΣB[max∞] as a function of R[min]. Then, j[c∞.] and ΣB[max∞] gave w and S[max] as a function of R[min].
3. Results
The results of the calculations of
, and
as a function of
for two different value settings of
shown in
Figure 16
Figure 17
are displayed in the same manner as in
Figure 14
; in
Figures S1–S4
in the case of the magnetic flux density-dependent
of curves (b) and (c) in
Figure 4
, which is simulated by Equation (11); and in
Figures S5–S8
in the case of the magnetic flux density-dependent
of curve (a) in
Figure 4
, which is simulated by Equation (12), respectively. The values of
, and
indicated by the perpendicular dashed-dotted red lines and the perpendicular dashed-two dotted blue lines in
Figures S1–S8
are summarized in
Table 1
. In
Table 1
w [@20T]
w [peak]
stand for
= 20 T and the peak values of
, respectively. In the case of the
) dependence of
expressed by Equation (11) (curves (b) and (c) in
Figure 4
took values between 3.5 and 4.5 Wh/L in the range of
21.5–31.4 mm when
= 30 μm, while
took values between 5 and 6.8 Wh/L in the range of
40–43.5 mm when
= 100 μm. An increase in
from 50 to 70 μm also caused an increase in
. The difference in
between that illustrated in
Figure 1
c and in
Figure 16
did not cause simple trends of increase nor decrease in
but caused a slight decrease in
with decreasing
In the case of the
) dependence of
expressed by Equation (12) (curve (a) in
Figure 4
took values between 5.1 and 5.6 Wh/L in the range of
34.9–40.3 mm when
= 30 μm, while
took the values between 6.1 and 6.7 Wh/L in the range of
43.7–45.8 mm when
= 100 μm. An increase in
from 50 to 70 μm caused an increase in
but did not cause an increase in
. A decrease in
from 600 to 200 caused a slight increase in
and slight decrease in
In the case of the
) dependence of
expressed by Equation (11) (curves (b) and (c) in
Figure 4
took values between 3.9 and 7.6 Wh/L in the range of
13.5–13.9 mm when
= 30 μm, while
took values between 33.8 and 69.4 Wh/L in the range of
40–43.5 mm when
= 100 μm. An increase in
from 50 to 70 μm also caused an increase in
. A decrease in
from 600 to 200 caused a decrease in
In the case of the
) dependence of
expressed by Equation (12) (curve (a) in
Figure 4
took values between 9.3 and 14.9 Wh/L in the range of
18.8–19.4 mm when
= 30 μm, while
took values between 48.2 and 80 Wh/L in the range of
16.6–17.5 mm when
= 100 μm. An increase in
from 50 to 70 μm caused an increase in
but did not cause an increase in
. A decrease in
from 600 to 200 caused a decrease in
Although the values of
often exceeded 10 Wh/L and sometimes approached 100 Wh/L when
= 100 μm or
= 70 μm, the corresponding values of
often exceeded 50 T and 100 T in some cases. These large values of
may exceed the critical magnetic flux density, although there are no signs of the appearance of critical magnetic flux density within the range shown in
Figure 4
Table 2
lists the values of
taken from
Figures S1–S8
at the
of every 10 T increment up to
Table 2
shows that the values of
= 20 T are lower than 10 Wh/L in all the different wafer coil design parameters,
d, s, z
, and the number of stacked wafer coils
under the different
dependences of
. In the cases of
= 100 μm and in some cases of
= 70 μm, the values of
are higher than 20 T and can be larger than 10 Wh/L, or more with an increase in
. It is found that this trend is more enhanced in the case of the
) dependence of
expressed by Equation (12) (curves (a) in
Figure 4
) than in the case of the
) dependence of
expressed by Equation (11) (curves (b) and (c) in
Figure 4
). In all conditions, the values of
were found to take values well below 1/3 of the mechanical strength limit of the Si wafer σ
: 4 GPa.
4. Discussion
4.1. Possibility of SMES to Rank with or Surpass Capacitors in Electricity Storage Volume Density
The dependence of
of Y123 thin films on the magnetic flux density is sensitively affected by many experimental factors such as the methods of deposition, deposition conditions, substrates, buffer layers, etc. The data
displayed in
Figure 4
are the dependences of
on magnetic flux density in parallel with the c-axis of preferentially c-axis-oriented Y123 thin films. Y123 thin films sometimes grow in mixed orientations of the c-axis and a-axis [
]. Even if c-axis-oriented Y123 is successfully grown on buffered Si substrate [
], the c-axis of the Y123 thin film grown on the bottom of the trench and the c-axis of the Y123 thin film grown on the trench wall cannot be in parallel with the same magnetic flux simultaneously.
Therefore, there remains many challenges in material science and engineering to attain depositions of high-quality Y123 thin films with a sufficient magnetic flux density dependence of
in the trench. However, the existence of the experimental evidence shown in
Figure 4
encourages us to develop a compact SMES with the new concept presented in this work, which can rank with or surpass the capacitors on the market in terms of electricity storage volume density. The
results summarized in
Table 2
teach us that
can rank with the commercially available capacitors, including EDLC, if superconductivity is retained up to the magnetic flux density
of 20 T. The results also teach us that
can surpass capacitors including EDLC if superconductivity is retained up to a
of 30 T. According to the experimental evidence of curve (b) and (c) shown in
Figure 4
, superconductivity is retained up to
= 18 T. It will be duly considered that superconductivity is retained up to 20 T. According to curve (a), superconductivity is retained up to a
of 31 T. Moreover, there is a possibility that superconductivity is retained beyond 40 T, because there is no indication of a rapid decrease in
due to the accession of the critical magnetic flux density
. In this case,
may rank with commercially available hybrid energy storage capacitors [
]. According to the results summarized in
Table 2
for the different trench design parameters illustrated in
Figure 17
, there remains possibilities to attain a greater
by using deeper and wider trenches within the experimentally feasible ranges.
4.2. Improvement for Lower Cost and Compatibility for Mass Production
According to
Table 1
, a decrease in
from 600 to 200 accompanied by an increase in the number of wafer stacks from 4 to 12 as illustrated in
Figure 16
caused no significant effect on
. This is good news because the process to stack 200 wafer coils is much easier than the process to stack 600 wafer coils from the viewpoint of a reduction in yield loss and process cost. The
semiconductor microfabrication processes used here for the fabrication of the units of stacked wafer coils are compatible in nature with unattended automatic mass production and have the possibility
to yield a significant cost reduction, which has been attained in the production of solar panels in which several tens of Si wafers are installed in one product (panel), just like the compact SMES
proposed here. However, it is a great challenge to reduce the process cost of the repeated reactive etching process for trench formation and multiple wafer-bonding processes. In April 2018, the
present authors noticed the announcement of the commercialization of a compact analyzer gas chromatograph, which is more than 30% smaller and gives 50% less analysis time compared with the
conventional ones equipped with capillary columns [
]. The main part of this product is made of spiral trenches formed in a stainless steel substrate and stacked with another stainless steel plate as a lid in place of the capillary columns.
The development of this product started with the microfabrication of spiral trenches on Si wafers [
], just like the present study. According to the result summarized in
Table 1
, the values of the maximum hoop stress
induced on the trench wall were less than 0.2 GPa, which was well below the 1/3 mechanical strength limit of the Si wafer σ
, 4 GPa. This fact indicates that other substrate materials such as typical engineering ceramics or metals can be used in place of a Si wafer if an imprinting process or molding process for spiral
trenches is feasible. Therefore, the compact SMES in the proposed concept may follow in the same footsteps of development of the compact analyzer gas chromatograph mentioned above. This possibility
of the employment of substrate materials other than Si may also open the door to other applications of the present wafer coils besides SMES—such as, for example, magnetic resonators for wireless
power transfer systems with a longer transfer range utilizing the high-quality factor of superconducting coils [
4.3. Applications of the Compact SMES Which Ranks with or Surpass the Commercially Available Capacitors
A SMES has been developed for a power system control such as load fluctuation compensation and power system stabilization [
]. However, those infrastructural applications have been proposed for usage in the space of a loosely restricted volume condition based on the electricity storage volume density of the sub-Wh/L level
of conventional SMESs. Therefore, the benefit of the compact SMES proposed in the present study is limited. Of course, the compact SMES of the proposed concept may be distributed widely in the market
by mass production and may contribute to these conventional applications at a lower cost and in a much reduced volume. Here, let us focus one of the outstanding features of the SMES: its rapid
high-power delivery of around MW/msec. This feature makes the SMES suitable for pulsed power load operation [
]. An extreme example of a pulsed power load is a lightning strike. Controlling lightning strikes to prevent them from causing damage to various infrastructures and facilities is a serious task for
telecom companies. Recently, there was an announcement that Nippon Telegraph and Telephone Corporation (NTT) started a study of technology to capture and guide lightning strikes [
]. In the project of NTT, it is planned that drones flying near the thunder clouds guide lightning electricity with the attached conductive wire to the ground in place of the kite used by Benjamin
Franklin. In the project, it is also considered to guide the lightning electricity with the conductive wire to an electricity storage facility on board of a car. Reportedly, there is a wide variety
of the properties of lightnings with their current from 1 kA to 200 kA, with their voltage from 2 MV to 200 MV, with their pulse width from sub-msec to 1 sec, and power from 10 kWh to 500 kWh. If we
take the values of the most frequent occurrences, the electricity pulse of lightning is typically 25 kA, the voltage is typically 29 MV, the pulse width is typically 1 msec, and the power is
typically 200 kWh. The pulse width is too narrow to charge rechargeable batteries. The corresponding voltage of 29 MV may be too high for conventional capacitors to be charged. As for SMES, a
feasibility study for energy regeneration on-board ground vehicles has been reported, although the proposed system was still as large as about 1 m in diameter and heavy, with bulky reinforced
structures against the hoop stress [
]. The total electricity of 200 kWh of one lightning requests the volume of 400 m
if the typical conventional electricity storage volume density
of 0.5 Wh/L is employed. This is too large to install on-board a car. However, the compact SMES of the proposed concept in this work requests a volume of 40 m
is 5 Wh/L. This compact SMES can be installed on-board a standard 4-ton truck. Although it goes without saying that SMES also requires improved insulation barriers and more detailed engineering
efforts, this topic is mentioned here to elucidate the merit of the compact SMES of the proposed concept.
5. Conclusions
SMESs store electricity as the energy of a magnetic field in contrast to capacitors, which store electricity as the energy of an electric field. Because SMESs and capacitors store electricity based
on purely physical phenomena, their abilities of rapid charge and discharge are specific to them in contrast to rechargeable batteries based on an electrochemical phenomenon. However, SMESs are heavy
electric machinery products of a relatively high cost, which is caused by their made-to-order production, while varieties of capacitors are commercially available at low prices thanks to mass
production. The electricity storage volume density w of conventional SMESs is in the sub-Wh/L range, while that of capacitors is in the Wh/L range. To increase the w of SMESs to rank with or surpass
that of capacitors and to produce them at a low cost by mass production, a compact SMES of a new concept produced by semiconductor microfabrication technologies has been proposed. In parallel with
the experimental development, a series of trials have been carried out to estimate the feasible value of w based on the calculation of the magnetic field generated by the compact SMES by improving
the calculation model step by step. In the present work, (1) the applied magnetic flux density (B) dependence of superconductive critical current density (j[c]) and (2) the equivalent radius of a
single circular current, which generates the same B at the center of the circle as the single wafer coil (engraved with a spiral coil) were taken into consideration for the first time in a series of
trials. The equivalent radius was used in the calculation of the magnetic energy storage in a single-layer solenoid coil as an equivalent model of stacked wafer coils. Regarding the B dependence of j
[c], two kinds of previously reported experimentally obtained data of the j[c] of Y123 thin films under B applied in parallel with the c-axis of the c-axis-oriented Y123 films were employed. The
results of the estimation taught us that the w of the compact SMES of the proposed concept can be in the Wh/L range or more, ranking with or surpassing those of presently available capacitors,
including electric double-layer capacitors (EDLC). According to the experimental data of the B dependence of j[c] employed here, j[c] has been measured to decrease moderately with a B up to 31 T.
There appeared to be no indication of a rapid decrease in j[c] due to the accession of the critical magnetic flux density B[c]. If a superconductivity is attained in B > 31 T and the measured B–j[c]
curve can be extrapolated beyond 31 T, the w of the compact SMES proposed here can be increased further.
Supplementary Materials
The following are available online at
: Figure S1:
dependence of
= 50 μm, s = 22 μm,
= 600, and (a)
= 30 μm, (b)
= 100 μm based on Equation (11). Figure S2:
dependence of
= 50 μm,
= 22 μm,
= 200, and (a)
= 30 μm, (b)
= 100 μm based on Equation (11). Figure S3:
dependence of
= 70 μm,
= 2 μm,
= 600, and (a)
= 30 μm, (b)
= 100 μm based on Equation (11). Figure S4:
dependence of
= 70 μm,
= 2 μm,
= 200, and (a) z= 30 μm, (b) z= 100 μm based on Equation (11). Figure S5:
dependence of
= 50 μm,
= 22 μm,
= 600, and (a)
= 30 μm, (b)
= 100 μm based on Equation (12). Figure S6:
dependence of
= 50 μm,
= 22 μm,
= 200, and (a)
= 30 μm, (b)
= 100 μm based on Equation (12). Figure S7:
dependence of
= 70 μm,
= 2 μm,
= 600, and (a)
= 30 μm, (b) z= 100 μm based on Equation (12). Figure S8:
dependence of
= 70 μm,
= 2 μm,
= 200, and (a)
= 30 μm, (b)
= 100 μm based on Equation (12).
Author Contributions
Conceptualization, T.M., M.S., and J.-h.N.; methodology, T.M., M.S., and J.-h.N.; software, T.M.; validation, T.M., M.S., and J.-h.N.; formal analysis, T.M.; investigation, T.M., M.S., and J.-h.N.;
resources, T.M., M.S., and J.-h.N.; data curation, T.M.; Writing—Original draft preparation, T.M.; Writing—Review and editing, T.M., M.S., J.-h.N., and O.T.; visualization, T.M.; supervision, T.M.
and O.T.; project administration, T.M.; funding acquisition, T.M., M.S., and J.-h.N. All authors have read and agreed to the published version of the manuscript.
This research was funded by a Kakenhi Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science, grant number 19H02195.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
The authors would like to thank S. Nomura of Meiji Univ. and T. Hioki of Nagoya Univ. for their helpful discussions, and N. Sugimoto of Toyota Central R&D Labs., Inc., for his help in the
experimental procedure. The authors also thank A. Ichiki for his help in constructing the Python code for the numerical estimations and N. Saito of Nagoya University for his help in the development
of the research environment.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Schematic representation of a typical possible minimum composition of the compact SMES based on the proposed concept. t is the thickness of the Si wafer and m is the number of stacked Si
wafers. b and h represent the approximate external sizes of the composition. A special case of h ≅ m•t is drawn in (c).
Figure 2. (a–e) The processes of trench formation; the deposition of NbN and Cu; and the removal of unnecessary deposits of NbN and Cu, except for those in the trench, by chemical-mechanical
polishing (CMP). (f–h) Examples of the processes of the wafer bonding and series interconnection of the wafer coils.
Figure 3.
) A 101.6 mm Si wafer coil after the electrolytic plating of Cu corresponding to
Figure 2
is the radius of the outer trench wall of the outermost trench and
is the radius of the inner trench wall of the innermost trench. (
) Cross-sectional SEM photograph of a half of the trench with an NbN sputter-deposited layer and electrolytically plated Cu after CMP, corresponding to
Figure 2
Figure 4.
Experimentally obtained dependences of
on the magnetic flux density in Y123 thin films reported in [
] (
) and in [
] (
). (
) A supposed extended curve of (
) beyond 18 T.
Figure 5. Schematic illustration of approximately one half of the cross-section of a wafer coil design in a Si wafer of 2R[wafer] diameter and t thickness. Notations: d, s, z, R[min], and R[max]
stand for the trench width, the trench wall thickness, the trench depth, the radius of the inner wall of the innermost coil, and the radius of the outer wall of the outermost coil, respectively. The
yellow rectangles indicate cross-sections of the trenches filled with Y123.
Figure 6. (a) Radial distribution of the calculated magnetic flux density B[z](r) normal to the wafer coil on the surface of the Si wafer of 2R[wafer] = 101.6 mm and t = 0.5 mm. The principal design
parameters of the spiral coil, d, s, z, R[max], n, and R[min], are indicated in the inserts of the figure. (b) The volume portion within ±δ radian removed in the circular trench from the integration
in Equation (3) to calculate B[z](r) at position A at the half width and at the half depth in the trench so that r does not become equal to q.
Figure 7. Calculated values of B[z](r) in the vicinity of the innermost trench. (a) By Type I, (b) by Type I for a single circular trench, and (c) by Type II.
Figure 8. (a) Schematic illustration of the magnetic flux density in the innermost trench of a wafer coil indicated with a red arrow caused by a different wafer coil stacked above at x. Alongside,
the generated electromagnetic hoop stress is also indicated by a radial outward-directed black arrow. (b) Dependence of the magnetic flux density B[z](r) on x.
Figure 9. The sum of the contributions, B[z](r), of all the wafers in a stack, ΣB[z](r), as a function of the position of the wafer x in the stack in a typical case of m = 600. R[min] = 19.0 mm.
Figure 10. (a) Variation in Σ B[z](r) – x profiles with R[min]. (b) Variation in B[max]: the maximum value Σ B[z](r) at x = 15 mm with R[min].
Figure 11. Calculated B[z](0) at the center of the wafer as a function of R[min]. Radius a of the equivalent single-layer solenoid, which gives the same B[z](0), is plotted as a function of R[min].
The mean radius of R[max] and R[min], a’, is also plotted for comparison.
Figure 12. Inductance L of the single-layer solenoid of the length m·t = 30 cm as a function of R[min] calculated employing the radius a and a’.
Figure 13. Variations in the electricity storage volume density w, S[max], and ΣB[max] as a function of R[min] based on the magnetic flux density independent j[c] = 2 × 10^4 A/mm^2.
Figure 14.
Variations in the electricity storage volume density
, and
as a function of
based on the magnetic flux density dependent
displayed as curves (
) and (
) in
Figure 4
Figure 16.
Decrease in
from 600 to 200 and the installation of eight more wafer stacks in comparison with
Figure 1
Figure 17. Schematic summary of the combinations of the wafer coil design parameters, d, s, z, examined. (a) d = 50 μm, s = 22 μm, z = 30 μm, (b) d = 50 μm, s = 22 μm, z = 100 μm, (c) d = 70 μm, s =
2 μm, z = 30 μm, (d) d = 70 μm, s = 2 μm, z = 100 μm.
Table 1. Calculated values of w, R[min], S[max], and ΣB[max] for the different wafer coil design parameters, d, s, z, and the number of the stacked wafer coils, m, under the different ΣB[max]
dependences of j[c].
Table 2. Calculated values of w in the unit of Wh/L at the ΣB[max] of every 10 T increment up to w[peak] for the different wafer coil design parameters, d, s, z, and the number of stacked wafer
coils, m, under the different ΣB[max] dependences of j[c]. Expressions such as w[@20T] indicate w when the ΣB[max] is 20 T.
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Motohiro, T.; Sasaki, M.; Noh, J.-h.; Takai, O. Estimation of the Electricity Storage Volume Density of Compact SMESs of a New Concept Based on Si Microfabrication Technologies. Magnetochemistry 2021
, 7, 44. https://doi.org/10.3390/magnetochemistry7030044
AMA Style
Motohiro T, Sasaki M, Noh J-h, Takai O. Estimation of the Electricity Storage Volume Density of Compact SMESs of a New Concept Based on Si Microfabrication Technologies. Magnetochemistry. 2021; 7
(3):44. https://doi.org/10.3390/magnetochemistry7030044
Chicago/Turabian Style
Motohiro, Tomoyoshi, Minoru Sasaki, Joo-hyong Noh, and Osamu Takai. 2021. "Estimation of the Electricity Storage Volume Density of Compact SMESs of a New Concept Based on Si Microfabrication
Technologies" Magnetochemistry 7, no. 3: 44. https://doi.org/10.3390/magnetochemistry7030044
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Community structure detection algorithms try to find dense subgraphs in directed or undirected graphs, by optimizing some criteria, and usually using heuristics.
igraph implements a number of community detection methods (see them below), all of which return an object of the class communities. Because the community structure detection algorithms are different,
communities objects do not always have the same structure. Nevertheless, they have some common operations, these are documented here.
The print generic function is defined for communities, it prints a short summary.
The length generic function call be called on communities and returns the number of communities.
The sizes function returns the community sizes, in the order of their ids.
membership gives the division of the vertices, into communities. It returns a numeric vector, one value for each vertex, the id of its community. Community ids start from one. Note that some
algorithms calculate the complete (or incomplete) hierarchical structure of the communities, and not just a single partitioning. For these algorithms typically the membership for the highest
modularity value is returned, but see also the manual pages of the individual algorithms.
communities is also the name of a function, that returns a list of communities, each identified by their vertices. The vertices will have symbolic names if the add.vertex.names igraph option is set,
and the graph itself was named. Otherwise numeric vertex ids are used.
modularity gives the modularity score of the partitioning. (See modularity.igraph for details. For algorithms that do not result a single partitioning, the highest modularity value is returned.
algorithm gives the name of the algorithm that was used to calculate the community structure.
crossing returns a logical vector, with one value for each edge, ordered according to the edge ids. The value is TRUE iff the edge connects two different communities, according to the (best)
membership vector, as returned by membership().
is_hierarchical checks whether a hierarchical algorithm was used to find the community structure. Some functions only make sense for hierarchical methods (e.g. merges, cut_at and as.dendrogram).
merges returns the merge matrix for hierarchical methods. An error message is given, if a non-hierarchical method was used to find the community structure. You can check this by calling
is_hierarchical on the communities object.
cut_at cuts the merge tree of a hierarchical community finding method, at the desired place and returns a membership vector. The desired place can be expressed as the desired number of communities or
as the number of merge steps to make. The function gives an error message, if called with a non-hierarchical method.
as.dendrogram converts a hierarchical community structure to a dendrogram object. It only works for hierarchical methods, and gives an error message to others. See dendrogram for details.
as.hclust is similar to as.dendrogram, but converts a hierarchical community structure to a hclust object.
as_phylo converts a hierarchical community structure to a phylo object, you will need the ape package for this.
show_trace works (currently) only for communities found by the leading eigenvector method (cluster_leading_eigen), and returns a character vector that gives the steps performed by the algorithm while
finding the communities.
code_len is defined for the InfoMAP method (cluster_infomap and returns the code length of the partition.
It is possibly to call the plot function on communities objects. This will plot the graph (and uses plot.igraph internally), with the communities shown. By default it colores the vertices according
to their communities, and also marks the vertex groups corresponding to the communities. It passes additional arguments to plot.igraph, please see that and also igraph.plotting on how to change the
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Understanding Mathematical Functions: Is A Scatter Plot A Function
Investigating the Nature of Mathematical Functions
Mathematical functions are an integral part of various disciplines, playing a crucial role in fields such as physics, economics, engineering, and many others. Understanding the nature of mathematical
functions is essential for making sense of various phenomena and making predictions based on empirical data.
A Definition of mathematical functions and their significance in different fields
A mathematical function is a relation between a set of inputs and a set of permissible outputs, with the property that each input is related to exactly one output. Functions are used to model the
relationship between two or more variables and are widely used in various fields for data analysis, prediction, and modeling complex systems.
Overview of graphical representation of functions, including scatter plots
Graphical representations of functions provide a visual way to understand and analyze their behavior. Different types of graphs, such as line graphs, bar graphs, and scatter plots, are utilized to
represent different types of functions. Among these, scatter plots are particularly useful for visualizing the relationship between two variables and identifying patterns or trends in the data.
Setting the stage for the exploration of whether a scatter plot can represent a function
As we delve into the topic of mathematical functions and graphical representations, it is important to consider whether a scatter plot, as a specific type of graph, can accurately depict a function.
This inquiry will allow us to explore the characteristics of scatter plots and their relationship to mathematical functions in greater detail.
Key Takeaways
• Scatter plots show relationship between two variables
• Functions have only one output for each input
• Scatter plots may or may not represent a function
• Vertical line test can determine if scatter plot is a function
• Understanding the distinction is important in mathematical analysis
Understanding Scatter Plots
Scatter plots are a type of mathematical function that is used to display the relationship between two sets of data. They are a visual representation of the correlation or relationship between the
variables being plotted. In a scatter plot, each data point is represented by a dot, and the position of the dot on the graph represents the values of the two variables being compared.
A Detailed explanation of scatter plots and their purpose
The primary purpose of a scatter plot is to show the relationship between two sets of data. It allows us to visually analyze the correlation between the variables and identify any patterns or trends
that may exist. Scatter plots are particularly useful for identifying outliers, clusters, and the overall distribution of the data.
Scatter plots are also used to:
• Identify the strength and direction of the relationship between variables
• Visualize the distribution of the data
• Identify any potential trends or patterns
Differences between scatter plots and other types of graphical representations
One key difference between scatter plots and other types of graphical representations, such as line graphs or bar graphs, is that scatter plots specifically show the relationship between two
variables. Line graphs, on the other hand, are used to show the change in one variable over time, while bar graphs are used to compare different categories of data.
Another difference is that scatter plots do not connect the data points with lines, as is the case with line graphs. This is because scatter plots are used to show the individual data points and
their distribution, rather than the overall trend or change over time.
Examples of data sets that are commonly displayed using scatter plots
Scatter plots are commonly used to display the relationship between variables in various fields, including:
• Science: Scatter plots are used to show the relationship between variables in scientific experiments, such as the relationship between temperature and pressure in a chemical reaction.
• Economics: In economics, scatter plots are used to display the relationship between variables such as supply and demand, or inflation and unemployment.
• Healthcare: In healthcare, scatter plots can be used to show the relationship between variables such as age and blood pressure, or weight and cholesterol levels.
Overall, scatter plots are a valuable tool for visualizing the relationship between two sets of data and are widely used in various fields for data analysis and interpretation.
Fundamental Characteristics of Functions
Understanding mathematical functions is essential in the field of mathematics and various other disciplines. Functions are a fundamental concept in mathematics that describe the relationship between
input and output values. In this chapter, we will explore the definition of a mathematical function, the concept of the vertical line test, and the different types of functions and their graphical
A Definition of what makes a mathematical relationship a function
A mathematical function is a rule that assigns to each input value exactly one output value. In other words, for every input, there is only one corresponding output. This means that a function cannot
have multiple outputs for the same input. Mathematically, if we have a set of ordered pairs (x, y), then the relationship is a function if each x-value is paired with exactly one y-value.
Key characteristics of a function:
• Each input has exactly one output
• No input can have multiple outputs
The concept of the vertical line test
The vertical line test is a visual way to determine if a curve in the xy-plane represents a function. If any vertical line intersects the graph of the curve at more than one point, then the curve
does not represent a function. On the other hand, if every vertical line intersects the graph at most once, then the curve represents a function.
Application of the vertical line test:
• If a vertical line intersects the graph at more than one point, it is not a function
• If every vertical line intersects the graph at most once, it is a function
Types of functions and their graphical characteristics
There are various types of functions, each with its own unique graphical characteristics. Some common types of functions include linear, quadratic, exponential, and trigonometric functions.
Linear functions: Linear functions have a constant rate of change and graphically appear as straight lines. The general form of a linear function is y = mx + b, where m is the slope and b is the
Quadratic functions: Quadratic functions have a squared term and graphically appear as parabolas. The general form of a quadratic function is y = ax^2 + bx + c, where a determines the direction and
width of the parabola.
Exponential functions: Exponential functions have a constant base raised to a variable exponent and graphically appear as curves that grow or decay exponentially. The general form of an exponential
function is y = a^x, where a is the base.
Trigonometric functions: Trigonometric functions involve angles and are used to model periodic phenomena. The most common trigonometric functions are sine, cosine, and tangent, each with its own
unique graphical characteristics.
Understanding the graphical characteristics of different types of functions is essential for analyzing and interpreting mathematical relationships in various real-world applications.
Understanding the Relationship Between Scatter Plots and Functions
When it comes to analyzing mathematical functions, scatter plots can be a valuable tool in indicating the relationship between variables. In this chapter, we will explore how scatter plots can be
used to represent functions, the conditions under which a scatter plot represents a function, and provide examples of scatter plots that do and do not represent functions.
Explanation of how scatter plots can be used to indicate relationships between variables
A scatter plot is a graphical representation of data points in a two-dimensional coordinate system. It is commonly used to display the relationship between two variables and to identify patterns or
trends in the data. Each data point on the scatter plot represents the values of the two variables, with one variable plotted on the x-axis and the other on the y-axis.
By examining the distribution of data points on a scatter plot, it is possible to identify the nature of the relationship between the variables. For example, if the data points form a clear pattern
or trend, it may indicate a positive or negative correlation between the variables. On the other hand, if the data points are scattered randomly with no apparent pattern, it may suggest that there is
no relationship between the variables.
Discussion on conditions under which a scatter plot represents a function
In the context of mathematical functions, a scatter plot represents a function if each input value (x-coordinate) corresponds to exactly one output value (y-coordinate). This means that for every
x-value, there is only one corresponding y-value. In other words, no two data points share the same x-coordinate.
Additionally, for a scatter plot to represent a function, it must pass the vertical line test. This test states that a vertical line drawn through any point on the graph should intersect the graph at
most once. If a vertical line intersects the graph at more than one point, then the scatter plot does not represent a function.
Examples of scatter plots that do and do not represent functions
Let's consider an example of a scatter plot that represents a function. If we have a set of data points where each x-value is paired with a unique y-value, and the vertical line test is satisfied,
then the scatter plot represents a function. For instance, a scatter plot showing the relationship between the number of hours studied and the score achieved on a test may represent a function, as
each study time corresponds to a unique test score.
On the other hand, a scatter plot that does not represent a function would be one where multiple data points share the same x-coordinate, leading to ambiguity in the relationship between the
variables. For example, a scatter plot representing the height of students in a class against their weight may not represent a function if there are students of the same height but different weights,
leading to multiple y-values for the same x-value.
Understanding the relationship between scatter plots and functions is essential in analyzing and interpreting data in various fields, including mathematics, science, and economics. By recognizing the
conditions under which a scatter plot represents a function, we can effectively use this graphical tool to gain insights into the relationships between variables.
Real-World Applications and Interpretations
Mathematical functions play a crucial role in understanding and interpreting real-world data. One common method used to represent data is through scatter plots, which are essential in determining
functional relationships, understanding the nature of data, and troubleshooting common misconceptions and errors in interpreting scatter plot data.
A. Case studies where scatter plots are essential in determining functional relationships
Scatter plots are widely used in various fields such as economics, biology, sociology, and environmental science to analyze and interpret data. For example, in economics, scatter plots are used to
study the relationship between variables such as supply and demand, price and quantity, or income and consumption. In biology, scatter plots help researchers visualize the relationship between
variables such as the effect of a drug dosage on a patient's health. These case studies demonstrate the importance of scatter plots in determining functional relationships between variables.
B. Importance of understanding the nature of data when using scatter plots to represent functions
Understanding the nature of data is crucial when using scatter plots to represent functions. It is essential to consider the type of relationship between the variables being plotted, whether it is
linear, quadratic, exponential, or logarithmic. This understanding helps in choosing the appropriate mathematical model to represent the data accurately. For instance, in environmental science,
understanding the nature of data is crucial when studying the relationship between temperature and carbon dioxide levels in the atmosphere. A scatter plot can help visualize the data and determine
the nature of the relationship between these variables.
C. Troubleshooting common misconceptions and errors in interpreting scatter plot data
One common misconception when interpreting scatter plot data is assuming that a scatter plot represents a function. While a scatter plot can show the relationship between two variables, it does not
necessarily represent a function. A function is a specific type of relationship where each input has exactly one output. In a scatter plot, multiple data points can have the same input value but
different output values, violating the definition of a function. It is important to be aware of this distinction when interpreting scatter plot data to avoid errors in analysis and conclusions.
Tools and Techniques for Function Identification in Scatter Plots
When analyzing scatter plots to identify mathematical functions, there are several tools and techniques that can be utilized to make the process more efficient and accurate. In this chapter, we will
explore the use of software and graphing calculators, trend lines and curve fitting, as well as diagnostic methods such as residual analysis.
A Introduction to software and graphing calculators for analyzing scatter plots
Software and graphing calculators are powerful tools that can be used to analyze scatter plots and identify potential functions. Programs such as Microsoft Excel, MATLAB, and Python's matplotlib
library allow for the visualization of data points and the application of various mathematical functions to the plot. Graphing calculators like the TI-84 or Casio fx-9750GII also provide the
capability to input data and generate scatter plots for analysis.
These tools enable users to input data points, visualize the scatter plot, and perform calculations to determine potential functions that best fit the data. They also provide the ability to
manipulate the plot and explore different mathematical models to see which one best represents the relationship between the variables.
B How to use trend lines and curve fitting to determine potential functions
One common technique for identifying potential functions in scatter plots is the use of trend lines and curve fitting. Trend lines are straight lines that can be added to a scatter plot to show the
general pattern or trend in the data. Curve fitting involves fitting a mathematical function to the data points in the scatter plot to find the best-fitting curve that represents the relationship
between the variables.
By adding a trend line or fitting a curve to the scatter plot, it becomes easier to visually identify the potential function that best describes the data. This technique allows for the comparison of
different functions and helps in determining the most suitable model for the given data set.
C Diagnostic methods, including residual analysis, to validate functions from scatter plots
Once potential functions have been identified using trend lines and curve fitting, it is essential to validate these functions to ensure their accuracy. Diagnostic methods, such as residual analysis,
can be used to assess the goodness of fit of the identified functions.
Residual analysis involves calculating the differences between the observed data points and the values predicted by the potential function. By examining the residuals, it is possible to determine if
the function adequately captures the relationship between the variables in the scatter plot. If the residuals exhibit a random pattern with no discernible trend, it suggests that the identified
function is a good fit for the data.
Overall, the use of software and graphing calculators, trend lines and curve fitting, as well as diagnostic methods such as residual analysis, provides a comprehensive approach to identifying
mathematical functions in scatter plots. These tools and techniques are valuable in analyzing data and gaining insights into the relationships between variables.
Conclusion & Best Practices
A Recap of the key insights about functions and scatter plots
Throughout this discussion, we have explored the concept of mathematical functions and their relationship to scatter plots. We have learned that a function is a relation between a set of inputs and a
set of possible outputs, where each input is related to exactly one output. On the other hand, a scatter plot is a visual representation of a set of data points, where each point represents the
values of two variables. While scatter plots are not functions in themselves, they can be used to analyze and identify functions within a given dataset.
Emphasis on the importance of context and criteria for determining functions
It is important to emphasize that the determination of whether a scatter plot represents a function depends on the context and criteria used for analysis. In some cases, a scatter plot may exhibit a
clear pattern that can be represented by a mathematical function, while in other cases, the data points may not align with a specific function. Understanding the context in which the data is
collected and applying appropriate criteria for determining functions is crucial in mathematical analysis.
List of best practices for using scatter plots to identify and analyze mathematical functions
• Clearly define the variables: When creating a scatter plot to analyze mathematical functions, it is important to clearly define the variables being represented on the x and y axes. This ensures
that the relationship between the variables can be accurately assessed.
• Look for patterns: Analyze the scatter plot to identify any discernible patterns or trends among the data points. These patterns may indicate the presence of a mathematical function that can
describe the relationship between the variables.
• Consider domain and range: When determining whether a scatter plot represents a function, consider the domain and range of the data points. If each input value (x-coordinate) is associated with
exactly one output value (y-coordinate), it is likely that the scatter plot represents a function.
• Use regression analysis: Utilize regression analysis techniques to fit a mathematical function to the scatter plot data. This can help in identifying the best-fitting function that describes the
relationship between the variables.
• Verify with mathematical tests: Once a potential function is identified from the scatter plot, verify its validity using mathematical tests such as the vertical line test or algebraic
manipulation. This ensures that the relationship between the variables truly represents a function.
By following these best practices, analysts and researchers can effectively use scatter plots to identify and analyze mathematical functions, providing valuable insights into the relationships
between variables within a dataset.
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LOOKUP Function in Excel | WPS Office Academy
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· Description:
The LOOKUP function can search a value in a column or row and returns a particular value from the same place in another column or row. It has two Syntax.
· Syntax1 :
LOOKUP(value, array)
· Arguments:
Value: The value to search for in an array, which must be in ascending order.
Array:An array of values.(contains both the values to search for and return)
· Example:
Suppose we want to find out the scores of students ranked 1,3,5,7.
1. Open your table in WPS Spreadsheets, click cell H5.
2. In this case, we need to enter the LOOKUP Function.
1) Valueis the value to search for in an array, cell G5 is the value that represent the first-ranking student we want to search, so let's enter G5 at Value.
2) Array: An array of values.(contains both the values to search for and return) the area of A3: C12 contains G5 as well as the data we want to return. So let's enter A3:C12 at Array. Also, we need
to press F4 to make it an absolute cell reference so that column H6:H8 won't change when copied.
Thus, we input =LOOKUP(G5,$A$3:$C$12), then press Enter.
The result is 75, which tells us the physics sore of the first-ranking student is 75.
3. Drop-down cell H5 to complete the process.
· Syntax2 :
LOOKUP( value, lookup_vector, [result_vector] )
· Arguments:
Value: The value to search for.
Lookup_vector: A range that contains only one row or one column of texts, numbers, or logical values, placed in ascending order.
Result_vector: [Optional]. Arange that contains only one row or column, the same size as Lookup_vector.
· Example:
Still, suppose we want to find out the scores of students ranked 1,3,5,7. This time we want to use another method to achieve the same end.
1. Open your table in WPS Spreadsheets, click cell H5.
2. We need to insert a LOOKUP function:
1) Valueis the value to search for in an array, cell G5 is the value that represent the first-ranking student we want to search, so let's enter G5 at Value.
2) Lookup_vector is the range that contains only one row or one column ofvalues, column A3: A12 is the lookup area that contains cell G5, so let's put C3:C12 here. Also, we need to press F4 to make
them absolute cell references so that rows and columns in column H6:H8 won't change when copied.
3) Result_vector is the range that contains only one row or column, the same size as Lookup_vector.In this case, column C3:C12 is the area that tells the physics score that we want to return, so
let's put C3:C12 here. Similarly, we need to press F4 to make them absolute cell references so that rows and columns in column H6:H8 won't change when copied.
Thus, we input: =LOOKUP(G5,$A$3:$A$12,$C$3:$C$12), then press Enter.
Again, the result is 75, which tells us the physics sore of the first-ranking student is 75 and further verify that our answer is correct.
3. To complete this table,we want to fill the remaining cells in this column, drop-down cell H5.
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A Public Review of Cuckoo Cycle
In the ongoing experiment that is crypto-currencies, several "Proof-of-Work" functions are commonly used as mechanisms to decentralize control of the currency. These PoW functions, as they are
typically referred to, require a participant to prove that they expended (in expectation) a certain amount of computational effort, such as finding a partial collision in a cryptographically strong
hash function. Many have been introduced to counter the gradual ASIC-ificiation of the dominant PoW, partial collisions in SHA256, used in Bitcoin. I've discussed several in prior posts that try to
require random memory accesses in order to favor general-purpose CPUs over specialized devices such as GPUs or ASICs.
In this post, I'll take a look at one called
Cuckoo Cycle
that combines two of my interests: Cuckoo Hashing and memory-hard PoW functions. Its author and a few other people asked me to take a look, and it finally seemed like enough people were looking at it
to do so.
In the rest of this post, let's assume that the properties of this function are desirable in some context, and put aside the question of whether they are the right design for a cryptocurrency. As the
saying goes, that's above my pay grade. (Though I will quibble with the article's assertion that GPUs and ASICs lead to high power costs -- on a hash-per-second-per-Watt basis, they're substantially
more efficient, and I suspect that on a per-participant basis, the Bitcoin network is using less power post-ASIC than it was pre-ASIC).
Rough idea of CC: Define N as a number of nodes (the explanation document says < 2^32, but there's no real reason for this) and E as a number of edges N/4 < E <= N. Split the nodes in half with N/2+1
on the left (V0) and N/2 -1 on the right (V1). For nonce 0 <= n < E, add an edge from siphash(k, n)%(N/2+1) in V0 to siphash(k, n)%(N/2-1) in V1. k is a constant for the block header.
The proof-of-work is to find a cycle of length L in the bipartite graph thus defined.
The author claims this work is tmto-hard (time-memory tradeoff, which I obnoxiously insist on pronouncing "tomato"), in the sense that trying to use less space causes a slowdown of "several orders of
Here's a quick start on how I'd attack this beast in parallel using substantially less memory than the author's original design - but not less than O(N) yet.
: This may be wrong! I haven't implemented it, I'm designing it on paper, and it needs a bit more care with its O(N). I'm throwing it up in its raw form as a way of continuing the dialog about such
proof-of-work functions, and as an illustration of how I start thinking about problems like this.
Let a vertex
if it has two or more incident edges. Otherwise, this vertex cannot be part of a cycle.
Let a nonce
be live iff both of the vertices it connects are live. Otherwise, it is an edge to nowhere and can be removed from the graph.
Use as a building block an
optimized "two-or-more" counting structure
such as that which I used for the Momentum PoW. With proper design, such a structure is relatively bandwidth and CPU-limited in its counting ability, not random access: On all processors generate a
sequence of values v. Place those values into one of 2^p output buckets based upon the p low-order bits of the value: bucket[p&0x0000ff], for example, to go into 255 buckets. When each bucket is
full, flush it to a growing list of items in that range. By sizing p and the buckets appropriately, perhaps with multiple passes, this uses fast local memory to ensure that the writes to slower
global memory are fully coalesced. It's the design I use in my CPU version of Momentum, for which you can
see the code here
To reduce memory use, do this in
passes, where in a pass, you discard hash outputs that don't have the low-order bits you're trying to filter. This is a standard time/memory tradeoff, requiring
times as many hashes, but reducing the number of output buckets (and thus the amount of memory required for counting of incident edges) by a factor of
. Appropriate choice of
can reduce the per-pass storage to, e.g., N/log(N), thus o(N) (that's little-oh -- it disappears as N scales).
Maintain a bitvector BV of size N
that determines the set of live nonces.
For enough passes to shrink the problem size to something you care about, do:
Test liveness of V0 - mark off nonces. A nonce is live iff its associated vertex in V0 has two or more incident edges, determined using the duplicate counter.
Using the live nonces, test liveness of V1.
Using the author's implementation default of E=N/2, this reduces the number of live nonces to a bit less than 1/2 the previous count (the author notes 1/ln(e)) on each test, except for edges that
participate in a cycle.
This is a little napkin-sketchy, but it's how I'd start attacking the problem on a platform like a GPU. My estimate of the cost is that it requires something like N bits of storage, N*log^2(N)
siphash computations, and N*log^2(N) bits of global memory bandwidth, if the time/memory tradeoff value is chosen as log^2(N) -- which is
feasible with siphash. Practically speaking, I'd go for more bandwidth and maybe 4-8 executions of siphash, if needed to reduce the partitioning fan-out for duplicate counting. (That may need to be
log^3(N) bits of bandwidth in the most general sense - but not for 512MB)
This attack does leave it with a basic requirement of O(N) bits (if we're being picky, the bitvector could be stored compressed to save a little more-- but still O(N) and yuch). As a result, this
doesn't eliminate the claim of being an ASIC-unfriendly design,
for the bounded N in the definition of the paper, which isn't fundamental. Any proof-of-work whose memory requirements, even if time-memory tradeable, scale with increasing difficulty, will present a
pain for a baked-in-silicon design. But 2^32 bits (512MB) is very manageable on almost any good GPU platform today.
(There is a way to slightly reduce the memory demands: Split the bit vector in half, and record the live bits only for the first half across multiple iterations of the test, using the full second
half. Then store this bit vector in a compact way, and perform the computation on the second half. This doesn't extend very far past a factor of two reduction, but it does require only a roughly
linear increase in the amount of computation and could possibly get the storage needed down to about 300MB for 2^32.)
This proposal has some things I like - it remains a generalization of Momentum's collision finding; by having a more dense graph structure, it requires a few more iterations through memory vs the
amount of hashing. By using siphash as its memory-filler, it reduces the compute-to-memory (bandwidth or latency) ratio, which is something the SHA512 designs could improve -- their bottleneck is
over 90% SHA512 computation time even with vector code.
There may be further progress to be made on this. In particular, cycles in graphs seem very likely to yield to sampling-based approaches. Consider, e.g., a 42-cycle: A sampling of 10% of the edges in
the graph is
likely to contain at least one of the edges in the 42 cycle. One might be able to use this as a starting point to solve Cuckoo Cycle in sublinear memory. I haven't thought enough about it, but it's
where I'd go next. If sampling is effective, it may be that the cycle finding could weaken this proposal beyond Momentum. If it's not, it may be "stronger" (in terms of memory-to-time). That'd be
interesting to determine.
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Data-Sufficiency section often features equations or inequalities involving algebraic quantities x, y, z, a, b, c etc… representing numbers and their properties.
Most of them can be answered by plugging in specific values for the algebraic quantities in the given equations or inequalities and testing their values.
If b is the product of two consecutive positive integers, is b divisible by 6?
(1) b is divisible by 4
(2) b is divisible by 3
Questions, like this, can be answered by giving specific values to the unknown quantities consistent with the given conditions.
The given condition is that b is the product of two consecutive positive integers.
Let us consider pairs of consecutive positive integers (1,2), (2,3), (3,4), (4,5), (5,6) and (6,7). Their products are 2, 6, 12, 20, 20, 30 and 42. And b may represent any of them.
Statement (1) says that b is divisible by 4. Among the possible values (2, 6, 12, 20, 20, 30 and 42) of b, only the numbers 12 and 20 are divisible by 4. Of these two numbers, while 12 is divisible
by 6, 20 is not divisible by 6.
So, the question can be answered by the Statement (1) alone.
So, (A) is not the answer.
Statement (2) says b is divisible by 3. Among the possible values (2, 6, 12, 20, 30 and 42) of b, the numbers 6, 12, 30 and 42 are divisible by 3. We can easily see that each of them is divisible by
6 also.
So, from (2) alone, we can answer the question as YES.
So, the answer is (B).
Next Question
Previous Question
GMAT-Model Questions Index
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Subtraction to 20 using simplification
Students learn that they can split a subtraction problem into tens and ones and use the subtraction problem with ones as a first step to subtract with tens.
It is important to learn this to help make subtraction easier and faster to do.
Start with a few counting exercises counting up to 20. Ask students to count quantities and to explain how they counted. They can skip count in 2s, 3s, and 5s. Then they are asked to decompose
numbers to 20 to 10 and the second number. An example is to decompose 13 into 10 and 3.
Show students a set of subtraction problems that use simplification. Ask students if they notice anything about the subtraction problems. They should notice that the difference between the two
problems is 10. Namely that 10 is added to the first number (minuend) and to the difference. These subtraction problems are related. Tell students that you can sometimes solve a subtraction problem
with tens faster if you first solve the subtraction problem without the tens. You can use that difference as a first step. If you know 6 - 2 = 4, then you can quickly determine that 16 - 2 = 14.
Practice this with MAB blocks on the interactive whiteboard. Drag the blocks that you take away to the trash can, so you can solve the subtraction problem. Ask students how many blocks are left.
Emphasize that they don't need to recalculate the subtraction problem, because they can use the first step they already solved. 10 is added to the subtraction problem, so 10 is also added to the
difference. Check if students are able to solve a subtraction problem without blocks on the interactive whiteboard as well by using the example of chewing gum. First with 6 - 1 and then with 16 - 1.
Do a few more subtraction problems with the students, but without visual support. They can imagine the MAB blocks in their head to help imagine the ten that is added to the second step of the
subtraction problem.Check that students understand subtraction to 20 using simplification by asking the following questions:- 7 - 2 = 5, what is 17 - 2?- 14 - 3 = 11. 14 - 13 = 11. How do you know
this?- 9 - 7 = 2, which subtraction problem can you also easily solve now?
Students start with a subtraction problem in which they get visual support for their simplification. They are then asked to solve the first step as well as the final subtraction problem. Finally they
are asked to do so with no visual support.
Repeat why it is useful to learn how to subtract using simplification. It helps to make subtraction easier and faster. If you know the subtraction problem without tens, you also know the difference
with tens. Check that students understand by doing two subtraction problems as a class, one with visual support and one without visual support.
Students who have difficulty can be supported by the use of a rekenrek. Show them 7 - 3 and then 17 - 3. Repeat this with different amounts. Emphasize that they don't need to recalculate the second
step, because they already have the difference from the first step. They only need to add 10 to find their difference.
Gynzy is an online teaching platform for interactive whiteboards and displays in schools.
With a focus on elementary education, Gynzy’s Whiteboard, digital tools, and activities make it easy for teachers to save time building lessons, increase student engagement, and make classroom
management more efficient.
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The Support Vector Machine Algorithm simply explained
Advanced algorithms enable computers to identify patterns, make decisions, and even predict the future based on data. Among these powerful tools, the Support Vector Machine (SVM) is notable for its
effectiveness, especially in the field of classification.
But how does it work?
Let's demystify this algorithm, starting with one of its fundamental concepts: the hyperplane.
Imagine you're at a park and you observe a wide, open field with various types of flowers scattered all around. Your task is to draw a straight line that separates two types of flowers, say daisies
and roses, so that all the daisies end up on one side of the line and all the roses on the other. In two dimensions, like our field, this line is akin to what mathematicians call a "hyperplane."
In the context of SVM, we're often not working with fields and flowers but with data points in spaces that can have many dimensions beyond just two. If we were dealing with three types of
characteristics (like the size, color, and smell of flowers), we would be in a three-dimensional space, and our "line" would actually be a flat sheet that divides the space into two parts. This flat
sheet is described mathematically as (ax + by + cz + d = 0), where (a), (b), and (c) are coefficients that define the orientation of the plane in three-dimensional space, and (d) is the offset from
the origin.
As we increase the dimensions, our hyperplane remains the thing that cleanly separates our data into two categories, but it becomes harder to visualize.
The hyperplane is the heart of the SVM algorithm. The aim of SVM is to find the best hyperplane that separates the data points of one category from those of another in the most optimal way. But what
does "optimal" mean in this context?
Optimality refers to the idea that the hyperplane should not only separate the two categories but should do so in a way that maximizes the margin between the closest points of each category and the
hyperplane itself. Think of it not just as drawing a line in our field of flowers but as drawing the widest possible road that still keeps the daisies and roses on opposite sides. The edges of this
road are marked by the nearest points (or "support vectors") of each category to the hyperplane, and the SVM algorithm seeks to make this road as wide as possible.
Why maximize the Margin?
The rationale behind maximizing the margin might seem counterintuitive at first glance. Here’s the reasoning: a hyperplane that has a large margin is likely to be more robust to noise and errors in
new data. It's not just about the data we currently have but also about accurately predicting the classification of new, unseen data points. A wider road (or larger margin) means that small changes
or uncertainties in the data are less likely to cause a misclassification.
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Multi Step Equations Integers Worksheet - Equations Worksheets
Multi Step Equations Integers Worksheet
If you are looking for Multi Step Equations Integers Worksheet you’ve come to the right place. We have 8 worksheets about Multi Step Equations Integers Worksheet including images, pictures, photos,
wallpapers, and more. In these page, we also have variety of worksheets available. Such as png, jpg, animated gifs, pic art, logo, black and white, transparent, etc.
Not only Multi Step Equations Integers Worksheet, you could also find another pics such as Math Equations, Equation Solver, Solving of Equations, Hard Equation, Year 10 One Step Equations, Equation
Explanation, The Startup Equation, and Algebra 1.
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Don’t forget to bookmark Multi Step Equations Integers Worksheet using Ctrl + D (PC) or Command + D (macos). If you are using mobile phone, you could also use menu drawer from browser. Whether it’s
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States of matter-I-Gases-Key Points
1. Kinetic molecular theory:
The kinetic molecular theory was postulated by Daniel Bernoulli, it explains the behavior of gases based on the movement of molecules.
2. Boyle's law:
According to Boyle's law, the volume of a fixed amount of a gas is inversely proportional to applied pressure at constant temperature and the number of moles.
\[V \propto \frac{1}{p}\; (When\; T\; and\; n\; are\; constant\; )\]
3. Charle's law:
According to Charle's law, the volume of a fixed amount of gas is directly proportional to absolute temperature at constant pressure and the number of moles.
\[V \propto T\; (When\; P\; and\; n\; are\; constant\; )\]
4. Avagadro's law:
According to Avagadro's law, the volume of a fixed amount of gas is directly proportional to the number of moles at constant temperature and pressure.
\[V \propto n\; (When\; P\; and\; T\; are\; constant\; )\]
5. Graham's law of diffusion or effusion:
According to Graham's law, the rate of diffusion or effusion of a gas is inversely proportional to the square root of density or molar mass of gas at constant temperature and pressure.
\[r \propto \frac{1}{\sqrt{d}}\;\;\;\; or \;\;\; r \propto \frac{1}{\sqrt{M}}\;\]
6. Dalton's law of partial pressure:
According to Dalton's law of partial pressure, the total pressure exerted by a mixture of non-reacting gases is equal to the sum of their partial pressures.
7. Absolute Zero:
It is a hypothetical temperature at which the volume of an ideal gas becomes zero. Its value is -273.16 degrees Celcius or zero kelvin.
8. General gas equation:
It is also called an ideal gas equation.
9. Van der Waal's equation:
It is also called as real gas equation.
\[\left ( P\; + a\frac{n^2}{V^2} \right )\left ( V\;-\;nb \right )\; = \;nRT\]
10. Joule Thomson effect:
The sudden expansion of highly compressed gases into a region of low pressure causes cooling.
11. Ideal gas:
Gases that obey Boyle's and Charle's law at all conditions are ideal. There are no forces of attraction between molecules of an ideal gas.
12. Non-ideal gas or real gas:
Gases that obey Boyle's and Charle's law at all conditions are ideal. There are no forces of attraction between molecules of an ideal gas.
13. Plasma:
Plasma is a mixture of positive ions; negative electrons and neutral atoms. It is also called the fourth state of matter.
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2013 A-level H1 Mathematics (8864) Question 3 Suggested Solutions - The Culture SG2013 A-level H1 Mathematics (8864) Question 3 Suggested Solutions
All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.
Required Area
KS Comments
Students should bare in mind to check the second order to substantiate that its a maximum area.
pingbacks / trackbacks
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The function y = β3(x β 2)2 + 6 shows the daily profit (in hundreds of dollars) of a hot dog stand, where x is the price of a hot dog (in dollars). Find and interpret the zeros of this tion A. Zeros at x = 2 and x = 6 B. Zeros at C. The zeros are the hot dog prices that give $0.00 profit (no profit). D. The zeros are the hot dog prices at which they sell 0 hot dogs.
1. Home
2. General
3. The function y = β 3(x β 2)2 + 6 shows the daily profit (in hundreds of dollars) of a hot dog stand,...
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An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation
State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability of Hebei Province, Hebei University of Technology, Tianjin 300130, China
Author to whom correspondence should be addressed.
Submission received: 13 January 2021 / Revised: 21 January 2021 / Accepted: 25 January 2021 / Published: 28 January 2021
Accurate prediction of photovoltaic power is conducive to the application of clean energy and sustainable development. An improved whale algorithm is proposed to optimize the Support Vector Machine
model. The characteristic of the model is that it needs less training data to symmetrically adapt to the prediction conditions of different weather, and has high prediction accuracy in different
weather conditions. This study aims to (1) select light intensity, ambient temperature and relative humidity, which are strictly related to photovoltaic output power as the input data; (2) apply
wavelet soft threshold denoising to preprocess input data to reduce the noise contained in input data to symmetrically enhance the adaptability of the prediction model in different weather
conditions; (3) improve the whale algorithm by using tent chaotic mapping, nonlinear disturbance and differential evolution algorithm; (4) apply the improved whale algorithm to optimize the Support
Vector Machine model in order to improve the prediction accuracy of the prediction model. The experiment proves that the short-term prediction model of photovoltaic power based on symmetry concept
achieves ideal accuracy in different weather. The systematic method for output power prediction of renewable energy is conductive to reducing the workload of predicting the output power and to
promoting the application of clean energy and sustainable development.
1. Introduction
With the continuous consumption of coal, oil, natural gas and other resources, energy depletion and environmental pollution are becoming more serious [
]. Solar energy has the characteristics of being green, clean and renewable, which have been widespread concerns, and the urgent demand of environmental protection has promoted the rapid growth of
global solar energy system [
]. Photovoltaic (PV) power generation is an efficient way to utilize solar energy, and the PV power generation proportion is increasing in line with reductions in cost and improvements in technology
]. PV output power has fluctuation and uncertainty [
]; these characteristics bring difficulties to the optimal dispatching of power system and are not conductive to the stability of renewable energy power system [
]. The adverse impacts of PV power generation limit the improvement of the grid connection rate of PV power generation, and are not conducive to the application of clean energy. Accurate prediction
of PV power is conducive to the safe operation of renewable energy power systems, and is beneficial for the application of clean energy [
Most of the studies for short-time PV power prediction are based on the statistical analysis of historical data [
]. PV prediction models include the linear model, nonlinear model and combination model. The linear model infers changing trends in PV historical data, and then predicts the PV output power. Reikard
] used autoregressive models to predict PV power, and achieved better results. Li et al. [
] argued that the linear model was simple to implement, but had poor flexibility and low prediction accuracy. The combination model is used to combine different prediction models in order. The
results of different prediction models are combined in the final result of combination model. Liu et al. [
] proposed a variable weight combination prediction model which integrates three different neural networks, and the prediction result of the combined model is more accurate. Combined prediction
method can integrate the advantages of each prediction method in specific conditions, but the prediction model and prediction process are more complicated.
The nonlinear models include Artificial Neural Network (ANN), Extreme Learning Machine (ELM) and Support Vector Machines (SVM). The models are trained by PV historical data. Then, the input data and
trained model are used for prediction. The advantages of the nonlinear models are relative simplicity and high accuracy. Zhu et al. [
] proposed an adaptive Back Propagation (BP) neural network model, which can adapt to the changes of time and external environment by updating the training data. Priyadarshi et al. [
] proposed an ANN method which applies the calculated astronomical variables to predict the PV output power with certainty and probability. Li et al. [
] proposed a deep belief network model, and realized the prediction of PV power generation in different seasons. The experiment proves that the model achieves better prediction accuracy. Ni et al. [
] considered the uncertainty of model and noise and proposed an optimal prediction interval method based on the ELM for PV power generation prediction. J. Wang et al. [
] proposed an ELM model which can update data automatically to predict PV power, and the model improves the prediction accuracy through continuous learning. Ni et al. [
] combined the upper and lower bound estimation method with the ELM to form the PV power generation interval prediction model, and used a heuristic algorithm to optimize the model, which achieved
better effect. Neural network and ELM models have better prediction accuracy, but neural network and ELM models require a large amount of training data. Prediction accuracy will decrease when the
amount of training data is small.
SVM requires less training data, and many studies have applied SVM to predict PV power. The method proposed by Huang et al. [
] classified the PV power generation historical data firstly, then used support vector regression (SVR) to learn the classified data. Compared with simple SVR, the prediction accuracy of the method
is enhanced. Bae et al. [
] proposed an SVM regression model that considers multiple meteorological factors. First, the meteorological data is clustered, and then the SVM is trained to predict the clustered data. The above
SVM models improve prediction accuracy, but the parameters of SVM are not optimized, so the performance of the model cannot be maximized.
The regression performance of SVM is very sensitive to parameters. Chen et al. [
] argued that SVM parameters had an important impact on regression performance. They focus on maximizing the prediction ability of SVM, and demonstrate that its parameters should be optimized. Many
studies have optimized the parameters of prediction models by using intelligent algorithms, and achieved better results [
]. Eseye et al. [
] proposed a model which consisted of wavelet, optimization algorithm and SVM. The parameters of SVM were optimized by optimization algorithm and wavelet was applied to process bad data. The
prediction accuracy was improved compared with other seven prediction strategies. The prediction model proposed by Lin and Pai [
] consisted of least squares vector regression and seasonal decomposition. The genetic algorithm was applied to optimize the model and achieved better accuracy. Shang and Wei [
] proposed a PV power generation prediction method which consisted of feature selection and support vector regression. The best candidate input is selected by feature selection and the SVM is
optimized by the heuristic algorithm. The prediction accuracy of the method was enhanced. Lin et al. [
] improved the moth to fire algorithm by using the mutation operator to increase population diversity. The improved algorithm was applied to optimize the PV prediction model. Experiments show that
the optimized model achieves sufficient accuracy. A genetic algorithm optimizing the SVM (GASVM) model is proposed by VanDeventer et al. [
] for PV power prediction at a residential level and the root mean square error (RMSE) and mean absolute percentage error (MAPE) parameters of GASVM are reduced compared with the traditional SVM.
Prior studies have achieved better prediction accuracy in a single weather condition, but these studies lack consideration of different weather conditions and cannot guarantee that the proposed
prediction models have better performance in different weather conditions. There is no unified method of PV prediction in different weather conditions. In the research of prediction model
optimization, there is a lack of in-depth research on the selection and improvement of the algorithm. Therefore, an improved whale algorithm (IMWOA) optimizing SVM model (IMWOA-SVM) for PV power
prediction is proposed in this study. SVM has advantages of greater efficiency and less demand for training data, so it is more suitable as the prediction model when the training data is less. SVM
prediction accuracy is improved when the parameters are optimized by the intelligent algorithm. The advantages of the whale optimization algorithm (WOA) include less codes, fast execution speed and
higher optimization accuracy, which make it suitable for SVM optimization [
]. However, the WOA algorithm also has the disadvantage of falling into a local optimal solution, which affects the optimization effect [
]. In addition, the objective of this study is to propose a high-precision PV power prediction method, symmetrically enhance the adaptability of prediction method to different weather, reduce the
amount of training data required by prediction methods, reduce the adverse effects of PV power fluctuations on the grid and promote the application of clean energy. Therefore, the WOA is improved by
a variety of methods in this study. Furthermore, the parameters of SVM are optimized by IMWOA. For enhancing the prediction stability of IMWOA-SVM in different weather conditions, wavelet threshold
denoising is applied to preprocess the input data. The results of the experiment prove that the IMWOA has stable optimization ability, and the prediction model based on symmetry concept can achieve
ideal prediction accuracy in different weather conditions. This study helps to enhance the stability of renewable energy power systems and is of great significance to sustainable development.
The work and innovation of this study include: (1) The IMWOA-SVM model is proposed. The proposed model requires less training data, can symmetrically adapt to various weather conditions, and has
better prediction accuracy. (2) Wavelet soft threshold denoising is applied to preprocess the input data. The interference signal from the input data is reduced, and the prediction model has better
prediction accuracy in various complex conditions. (3) IMWOA is proposed by combining WOA with tent chaos initialization, mutation disturbance and the differential evolution algorithm (DE), which
significantly enhances the ability of IMWOA to search for the optimal solution and escape from the local optimal solution. (4) IMWOA is used to optimize SVM photovoltaic prediction model. IMWOA has
better global search ability and improves the comprehensive performance of the PV prediction model.
This study is structured as follows. The
Section 2
is the prediction model and its improvement. The
Section 3
is the input data preprocessing and experimental arrangement. The
Section 4
is the discussion of experimental results. Lastly,
Section 5
are given.
2. Prediction Model and Its Improvement
This section firstly introduces the SVM regression model and the basic principle of WOA, then introduces the improvement methods of WOA, and finally verifies the excellent performance of IMWOA
through the test function.
2.1. Support Vector Machine
SVM is often applied to solve classification and regression problems [
]. SVM can achieve better prediction accuracy than other prediction models when the amount of training data is limited. The principle of SVM is introduced as follows.
$x i$
is the input data, and
$y i$
is the output data. SVM attempts to fit
$( x i , y i )$
by a regression equation, and the expression of regression equation is shown in Formula (1).
are the model parameters to be determined.
However, the actual regression problems often cannot be solved by the linear regression model represented by Formula (1). Many regression problems are nonlinear. Therefore, it is necessary to
construct a nonlinear function
$φ ( x )$
and transform the nonlinear problem into linear problem in the high-dimensional space. So, the Formula (1) can be converted as follows [
$f ( x ) = ω T φ ( x ) + b$
The goal of the SVM regression model is to make the deviation of
$f ( x )$
minimize. However, a slight deviation
$f ( x )$
is tolerated for improvement of the generalization ability of SVM. After transformation, the optimization objective of SVM is expressed as Formula (3).
$min ω , b 1 2 ‖ ω ‖ 2 + C ∑ i = 1 m f ε ( f ( x i ) − y i ) f ε ( z ) = { 0 | z | ≤ ε | z | − ε | z | > ε$
is the penalty factor,
is the amount of training data,
is the threshold of a penalty function. When the difference between the predicted value and the actual value is greater than
, a penalty term is introduced. When the difference between the predicted value and the actual value is less than
, the penalty term is zero.
After introducing the relaxation variable
$ζ i$
$ζ i *$
, Formula (3) is equivalent to Formula (4).
$min 1 2 ‖ ω ‖ 2 + C ∑ i = 1 m ( ζ i + ζ i * ) s . t . { f ( x i ) − y i − ε = ζ i y i − f ( x i ) − ε = ζ i * ζ i ≥ 0 , ζ i * ≥ 0$
Formula (4) is a linear constrained optimization problem. By introducing the Lagrange multiplier to construct the Lagrange function, constraint conditions can be eliminated. Based on Formula (4), the
Lagrange function of the optimization problem is obtained as shown in Formula (5) by introducing coefficient
$c *$
$u *$
$L ( ω , b , c , c * , ζ , ζ * , u , u * ) = 1 2 ‖ ω ‖ 2 + C ∑ i = 1 m ( ζ + ζ i * ) + ∑ i = 1 m c i ( f ( x i ) − y i − ε − ζ i ) − ∑ i = 1 m u i ζ i − ∑ i = 1 m u i * ζ i * + ∑ i = 1 m c i * ( y i
− f ( x i ) − ε − ζ i * )$
= 0, 1, 2, …,
. If the partial derivative of
$L ( ω , b , c , c * , ζ , ζ * , u , u * )$
$ζ i$
$ζ i *$
is 0, the Formula (6) can be obtained by substituting the obtained relation into Formula (4).
$max c , c * ∑ i = 1 m y i ( c i * − c i ) − ε ( c i * + c i ) − 1 2 ∑ i = 1 m ∑ j = 1 m ( c i * − c i ) ( c j * − c j ) x i T x j ∑ i = 1 m ( c i * − c i ) = 0 0 ≤ c i , c i * ≤ C$
Furthermore, the dual problem of Formula (6) can be obtained as shown in Formula (7).
$f ( x ) = ∑ i = 1 m ( c i * − c i ) x i T x + b$
Then, the kernel function is used for the nonlinear regression problem as shown in Formula (8).
$K f ( x , x i ) = exp ( − | x − x i | 2 σ )$
So, the nonlinear regression equation can be expressed as Formula (9).
$f ( x ) = ∑ i = 1 m ( c i * − c i ) K f ( x , x i ) + b$
The values of the radius of the kernel function and the penalty factor determine the regression ability of the SVM, and the optimal combination of parameters helps the SVM to achieve the best
prediction performance. So, IMWOA is applied to optimize these two parameters to achieve the best prediction performance.
2.2. Whale Optimization Algorithm
The WOA is a novel heuristic algorithm [
]. Since it was proposed, it has been widely studied due to its superior performance [
]. The search process is mainly divided into three steps.
2.2.1. Surround the Prey
In this stage, humpback whales look for prey and surround them. The prey is the optimal solution of optimization problem, and it is unknown. So, the WOA will regard the current optimal solution as
the optimal solution. The whale population will update its position according to the position of the target prey. This behavior can be converted as follows [
$D b = | C w P b e s t ( t ) − P ( t ) | P ( t + 1 ) = P b e s t ( t ) − A w D b$
$P b e s t ( t )$
is the current optimal solution, which is constantly updated with the increase in iterations.
$P ( t )$
is the position of the individual whale.
is the current number of iterations.
$A w$
$C w$
are coefficients defined by Formulas (11) and (12).
is a random number, the value range is [0,1].
is a vector that changes linearly from 2 to 0.
2.2.2. Bubble Net Predation
For description of the bubble net predation strategy, the following two mechanisms are designed in WOA.
Contraction and encirclement mechanism: The mechanism is realized by decreasing the value of $c$ in Formula (11). The variation range of $A w$ decreases with the decrease in $c$, which causes the
whale population to gradually converge to the optimal individual.
Spiral update position: This method establishes a spiral trajectory according to the distance between the whale and the optimal solution to simulate the spiral motion of the individual whale. The
spiral trajectory is defined by the Formula (13).
$D b ′ = | P b e s t ( t ) − P ( t ) | P ( t + 1 ) = D b ′ e b r cos ( 2 π r ) + P b e s t ( t )$
is the constant that defines the shape of the helix.
Shrinking and encircling prey and spiral swimming are simultaneous, Therefore, when the individual whale is updated, there is a 50% probability of contraction encirclement and 50% probability of
spiral swimming. The expression is shown as Formula (14).
$P ( t + 1 ) = { P b e s t ( t ) − A w D b r a n d < 0.5 D b ′ e b r cos ( 2 π r ) + P b e s t ( t ) r a n d ≥ 0.5$
$r a n d$
is a random number between 0 and 1.
2.2.3. Hunting for Prey
In this stage, the humpback whales hunt for prey without knowing the position of it. At this time, the reference used to update the whale position is not the optimal individual position, but the
individual position randomly selected in the humpback whale population. This mechanism strengthens the random search of WOA. The update Formula for humpback whales in the hunt phase is shown as
Formula (15).
$D r = | C w P r ( t ) − P ( t ) | P ( t + 1 ) = P r ( t ) − A w D r$
$P r ( t )$
is the individual whale randomly selected in the humpback whale population.
Figure 1
shows the overall process of the WOA algorithm through pseudo code. It can be seen that the update process of the algorithm is concise and clear. The author did not add complex mutation and evolution
processes to the WOA. Only
$A w$
$C w$
keep changing with the number of iterations. These advantages cause the WOA to have less code and faster running speed, and also cause the WOA to have great room for improvement. Although the WOA
algorithm is relatively simple, its search ability is not inferior to other optimization algorithms. Because the value of
$A w$
changes with the number of iterations, different
$A w$
values will guide the WOA to perform different update behaviors. At first, when the value of
$A w$
is large, the WOA has a higher probability of selecting the humpback whale position randomly selected as the update reference, which enhances the global search capability of the WOA. As the iteration
$A w$
gradually decreases, and the WOA is more likely to choose the current optimal individual whale to update the whale population, which improves the search accuracy of WOA. Based on the advantages of
the WOA discussed above, this study selects WOA as the optimization algorithm of SVM prediction model. To make up for the deficiency of WOA and further enhance the performance of the WOA, this study
makes some improvements to the WOA and proposes the IMWOA.
2.3. Improved Whale Optimization Algorithm
This section will introduce the improvement methods of the WOA, including population initialization based on tent map, mutation disturbance to the optimal individual and combination with differential
evolution algorithms. Finally, the optimization performance of IMWOA is tested by several test functions and another five algorithms are used for comparison with IMWOA, including the ant lion
algorithm (ALO), particle swarm optimization algorithm (PSO), moth to fire algorithm (MFO), multiverse algorithm (MVO) and WOA.
2.3.1. Tent Mapping Initialization
Initialization determines the distribution and the fitness of the initial population. The distribution of the initial population generated by random generation method, which is the default
initialization method, is poor in the solution space. The initial population can achieve better spatial distribution when the tent mapping is applied for initialization. The individual position of
the IMWOA is initialized by tent mapping in this study. The expression of the tent mapping is shown as follows [
$f ( x t + 1 ) = { x t u 0 ≤ x t ≤ u ( 1 − x t ) u u < x t ≤ 1$
$x t$
is the chaotic map sequence generated,
is the control parameter.
$u = 0.5$
, the obtained chaotic sequences have approximately uniform distribution density. So, the expression of the tent mapping is converted to Formula (17).
$f ( x t + 1 ) = { 2 x t 0 ≤ x ≤ 0.5 2 ( x t − 1 ) 0.5 < x ≤ 1$
The steps to initialize the IMWOA population by tent mapping are as follows:
Generate $x 0$ randomly, which represents the initial values of chaotic variables, and make $x 0$ not equal to 0.5.
Generate the chaotic mapping sequence iteratively according to the method of Formula (17), and if the chaotic variable enters a cycle, proceed to step (4).
Judge the end condition. When the end condition is satisfied, proceed to step (5).
Disturb $x 0$ and regenerate chaotic sequences.
Map the obtained chaotic value into the solution space of the optimization problem to form the initial IMWOA population.
2.3.2. Variation Disturbance of Optimal Position
It can be seen from Formula (14) that the optimal individual is assumed to be the prey, which constantly affects the updating of other individuals in the population. This mechanism can bring many
advantages to WOA. However, when the iteration number is small, this mechanism causes the WOA quickly enter the local search phase without finding enough excellent solutions, and makes the algorithm
fall into local optimum. Therefore, adding nonlinear mutation disturbance to the optimal individual causes the optimal individual to change with a certain probability, so the ability of IMWOA to
escape from the local optimal solution is enhanced. The expression of variation disturbance factor is shown in Formula (18).
$V = ( 1 − t M a x _ t ) tan ( ( 0.5 − a ) π )$
is the current number of iterations,
$M a x _ t$
is the maximum number of iterations,
is a random number in the range of [0,1]. The variation range of random disturbance also gradually decreases, which ensures that the local search accuracy of IMWOA will not be affected.
2.3.3. Differential Evolution Algorithm
The DE algorithm includes mutation, crossover and selection. New individuals are generated by mutation and crossover, and individuals with high fitness are retained by selection. Adding the DE to the
IMWOA can further enhance the diversity of solutions and enhance the fitness of the IMWOA population.
• Mutation: Select one individual in the population as the current individual
$X *$
, and then select three individuals except
$X *$
randomly. First, two randomly selected individuals are subjected to a vector difference operation, and then the result is subjected to a vector sum operation with the third individual to generate
a mutant individual. The Formula is shown as follows [
$X m u t = r a n d o m ( M i n s c , M a x s c ) × ( X 1 ( t ) − X 2 ( t ) ) + X 3 ( t )$
$X m u t$
is a new individual produced by mutation,
$X 1$
$X 2$
$X 3$
are three randomly selected individuals,
$r a n d o m ( M i n s c , M a x s c )$
is a random number in the range of
$[ M i n s c , M a x s c ]$
• Crossover: Crossover individual is composed of some elements of the current individual and mutation individual. The crossover Formula is shown in (20).
$U i , j = { X i , j * r a n d i , j ≤ C R o r j = I r a n d X i , j m u t o t h e r w i s e$
$U i , j$
is the
-th element of crossover individual.
$C R$
is the probability vector that controls the crossover.
$r a n d i , j$
is used to generate
dimensional random number vectors.
$I r a n d$
is the control parameter to ensure the occurrence of crossover.
• Selection: Compare the fitness values of
$X *$
$U i$
, and retain the individuals with high fitness into the next generation population. The selection Formula is shown in (21).
$X i ( t + 1 ) = { X i ( t ) i f f o b j ( U i ( t ) ) ≤ f o b j ( X i ( t ) ) U i ( t ) o t h e r w i s e$
$f o b j ( x )$
is the fitness value of
The optimization process of IMWOA is as follows, and
Figure 2
shows the flow chart of IMWOA.
Initialize IMWOA parameters, including iteration number, solution dimension, population size, decision variable matrix size, cross probability, etc.
Generate the initial population of IMWOA based on the chaotic tent map, and calculate the fitness value.
Update the position and fitness of the whale population. The renewal process includes mutation disturbance to the optimal individual.
Carry the mutation and crossover operations of differential evolution algorithm to generate crossover individuals.
Select the individuals between the cross individuals and the original individuals according to the fitness to form the next generation.
Judge the end condition. If the end condition is met, output the optimal solution. Else, proceed to step (3).
2.4. IMWOA Performance Test
In this section, the IMWOA is tested by several test functions. The test function includes monotone function and non-monotone function, which can test the optimization performance of the IMWOA
Table 1
shows the expressions of the test functions. The comparative experiments are made with several representative and widely used optimization algorithms, including MVO, PSO, MFO, ALO and WOA to show the
performance of the IMWOA.
Table 2
includes some special settings of algorithm parameters. Furthermore, the undeclared parameters are the default values.
Table 3
shows the statistical test results, which are the average values for three sets of test results.
The test results are shown in
Table 3
, and all data less than 0.001 in the table are counted as 0. The optimization values of the IMWOA are obviously better than those of the other five algorithms. All the optimization values of the
IMWOA are zero, but there are some errors in other algorithms, which shows that the IMWOA has excellent global search ability. The RMSE of the IMWOA is zero, which indicates that the optimization
ability of the IMWOA is very stable. The IMWOA optimization value and RMSE are both better than the WOA, which proves that the improvement measures adopted can effectively enhance the optimization
performance of the IMWOA. Moreover, for individual test functions (
$F 3$
), the WOA has the problem of falling into local optimum, while the IMWOA avoids similar problems, which shows that the improved method can effectively enhance the performance of the IMWOA to escape
from local optimal solution. In the same conditions, the IMWOA has the best optimization accuracy and stability, whether it has a monotone function or non-monotone function, which proves that the
performance of IMWOA is excellent and that the improvement measures adopted in this study are reasonable and effective.
3. Input Data Preprocessing and Experimental Arrangement
Many meteorological factors have an impact on the PV power, and different meteorological factors have different effects on it. In this section, the Pearson coefficient is first applied to analyze the
relationship between different meteorological factors and PV power. The meteorological factors strictly related to the PV output power are selected as the prediction input data. Then, wavelet
denoising is used to denoise the input data to remove the noise in it, and the input data is normalized. Finally, the experimental arrangement is described.
3.1. Selection of Input Data
PV output power is closely related to meteorological factors including light intensity, ambient temperature, relative humidity, etc.
Figure 3
shows the variation curve of the PV output power from 8 to 18 h in sunny and cloudy weather conditions. Sunny and cloudy weather are typical weather with representative characteristics, and the PV
output power presents great differences in these two types of weather. In sunny weather, the changes of the meteorological factors are slow and the PV output power presents a regular parabola. In
cloudy weather conditions, the meteorological factors have a greater fluctuation, resulting in PV output power also presenting a greater volatility, and the overall output power being low. Therefore,
the external meteorological factors determine the PV power and they can be applied to accurately predict the PV power. Ignoring the key meteorological parameters will increase the prediction
deviation. However, considering too many meteorological factors will greatly increase the workload of prediction, and excessive consideration of irrelevant factors will also reduce the prediction
accuracy. So, accurately measuring the correlation between various meteorological factors and PV power and selecting the appropriate factors as the input data of the prediction model is important to
enhance the prediction accuracy. The Pearson correlation coefficient is selected to measure the correlation between meteorological factors and PV output power. The expression of the Pearson
coefficient is shown in (22).
$ρ x , y = n ∑ x y − ∑ x ∑ y n ∑ x 2 − ( ∑ x ) 2 n ∑ y 2 − ( ∑ y ) 2$
are variables with correlation.
is the total number of data. In this problem,
is the weather factor,
is the output power of photovoltaic power generation, and
$ρ x , y$
is the correlation coefficient.
Table 4
shows the meaning of the Pearson coefficient. It indicates positive correlation when
$ρ X , Y$
is greater than zero, it indicates no linear correlation when
$ρ X , Y$
is equal to zero, and it indicates negative correlation when
$ρ X , Y$
is less than zero.
The correlation coefficients between PV power and the meteorological factors including light intensity, diffuse intensity, ambient temperature, wind speed and humidity are calculated. The test data
are from a PV power station in Australia.
Table 5
shows the calculation results of the correlation coefficient.
The correlation coefficient of wind speed and PV power remains at a low level, and the average correlation coefficient of six months is 0.321, showing a weak correlation. Moreover, the correlation
coefficient fluctuates greatly among different months. The maximum correlation coefficient is 0.580, and the minimum correlation coefficient is −0.053, indicating that the correlation is extremely
unstable. So, it is not suitable to select wind speed as the input data. The correlation coefficient of light intensity and PV power remains at a high value. The average correlation coefficient of
six months is 0.993, the maximum and minimum correlation coefficient are 0.996 and 0.989, showing an extremely strong and stable correlation. So, it is appropriate to select the light intensity to
predict the PV power. According to the same analysis method, the environmental temperature and humidity present stable moderate correlation with PV power, diffuse radiation and PV output power
present stable weak correlation. In this study, the light intensity, ambient temperature and humidity are selected as input data considering the accuracy and complexity of the model.
3.2. Denoising and Normalization of Input Data
Light intensity, ambient temperature and relative humidity should be continuous and slowly changing signals, but, when affected by measurement conditions and other factors, these signals contain a
lot of noise, which causes the measurement waveform to show the characteristics of fluctuation. The accuracy of the prediction model will be adversely affected if these data with noise are used for
the training of the prediction model. So, it is necessary to take measures to reduce the noise in input data.
There are many data denoising methods, and many studies have also examined various data denoising methods [
], among which wavelet threshold denoising is widely used [
]. The signal containing noise is decomposed by wavelet. The decomposed signal has a larger signal wavelet coefficient and a smaller noise wavelet coefficient. Comparing the obtained wavelet
coefficients with the threshold value, the wavelet coefficient that is higher than the threshold value is considered as the signal, and should be retained. Furthermore, the wavelet coefficient that
is lower than the threshold value is considered as noise, and should be removed. Setting an appropriate threshold can achieve ideal denoising effect.
Figure 4
shows the flow chart of wavelet threshold denoising, and the steps are shown below.
Decompose the original signal and obtain the wavelet coefficients.
Set threshold and threshold function.
Denoise the wavelet coefficients by threshold to filter the noise information in the signal.
Reconstruct the processed wavelet coefficients to obtain the denoised signal.
The threshold denoising of wavelet denoising has a decisive impact on the denoising effect of the signal. Threshold denoising involves the selection of the threshold and threshold function. The
selection of the threshold is complex and related to many aspects, and the default setting is generally used. There are two types of threshold functions. The first type is a hard threshold function,
which sets the signal that is lower than the threshold to zero and keeps the signal that is higher than the threshold unchanged. The hard threshold function can cause the signal to retain more
original information. However, the signal will break obviously near the threshold and the signal continuity is poor. The second type is a soft threshold function which processes the signal that is
higher than the threshold by adding or subtracting the threshold value and sets the signal that is lower than the threshold to zero. The soft threshold function causes the denoised signal to become
smoother and more continuous. Practice has proved that the prediction model trained by the signal processed by the soft threshold function has higher prediction accuracy. Therefore, this study
applies the soft threshold function to preprocess the prediction input data. For intuitively reflecting the denoising effect of the soft threshold function, the data of daytime temperature, humidity
and light intensity are intercepted for wavelet soft threshold denoising.
Figure 5
shows the effect picture of wavelet soft threshold denoising.
The first picture is the temperature curve, the second one is the humidity curve, and the third one is the light intensity curve. The red line in
Figure 5
represents the data denoised by wavelet soft threshold and the blue line is the original data. Since there are many noise signals in the original temperature and humidity data, the signal denoised by
wavelet undergoes obvious changes compared to the original signal. However, the light intensity signal changes gently and contains less noise, so the signal after wavelet denoising undergoes no
obvious change compared to the original signal. It can be found that the input signal without denoising contains high-frequency noise, and the signal waveform contains many turning points and
fluctuates greatly. After wavelet soft threshold denoising, the waveform is smoother, many abrupt points are eliminated, and the real data are restored, which is beneficial for improving prediction
accuracy. In this study, the training and testing input data are processed by wavelet soft threshold denoising to improve the prediction accuracy.
As shown in
Figure 5
, different meteorological factors have different units, and the variation range of the signals is also very different. So, they cannot be directly used for model training and prediction, and they
need to be normalized. The normalization Formula is shown in Formula (23).
$V n = V − V min V max − V min$
$V n$
is the normalized data,
is the non-normalized data,
$V max$
is the maximum value of the data,
$V min$
is the minimum value of the data.
In this study, the data of temperature, humidity, light intensity, PV power are normalized.
3.3. Experimental Arrangement
The prediction model of IMWOA-SVM was tested in sunny and cloudy weather conditions based on the data from Desert Knowledge Australia (DKA) Solar Center in Australia, and the prediction results were
compared with five SVM models, and ELM and BP neural networks, respectively. Two days’ data of randomly selected typical weather are used for training, and one day’s randomly selected data are used
for testing. The prediction period is from 8:00 to 18:00 in the daytime.
Figure 6
shows the prediction flow chart of IMWOA-SVM, and the steps of using IMWOA-SVM to predict PV power are as follows.
Select training data and testing data in sunny and cloudy weather, respectively.
Preprocess training input data and testing input data by wavelet soft threshold denoising.
Normalize training and testing data.
Initialize the parameters of the IMWOA-SVM photovoltaic output power prediction model.
Train the prediction model by training data. Apply the IMWOA to optimize the SVM. Test the prediction model by the test data.
Obtain the optimal prediction model of PV power. Predict the PV power.
Normalize the prediction output power inversely and output the experimental results.
In the process of model optimization, the mean square error (MSE) of the prediction power and the actual power is used as the objective function of IMWOA optimization. The definition of MSE is shown
in Formula (24).
$M S E = 1 n ∑ i = 1 n ( P i * − P i ) 2$
$P i *$
is the predicted PV power,
$P i$
is the actual PV power,
is the number of sample points.
Besides MSE, some other indexes are applied to evaluate the prediction results more comprehensively, including mean absolute error (MAE), root mean squared error (RMSE), R-square (R2) and mean
absolute percentage error (MAPE). The definitions of these evaluation indexes are shown in Formula (25).
$M A E = 1 n ∑ i = 1 n | P i * − P i | R M S E = 1 n ∑ i = 1 n ( P i * − P i ) 2 R 2 = 1 − ∑ i = 1 n ( P i * − P i ) 2 ∑ i = 1 n ( P i − P i ¯ ) 2 M A P E = 1 n ∑ i = 1 n | P i * − P i P i | × 100 %$
It should be noted that the larger $R 2$ is and the smaller the $M S E$, $M A E$, $R M S E$ and $M A P E$ are, the better the prediction results can track the actual output power.
4. Experimental Results and Discussion
In this section, the experiment results will be analyzed and discussed. The discussion is divided into two parts: the first part is the discussion of the experiment results in sunny weather, and the
second part is the discussion of the experiment results in cloudy weather. In each weather condition, the IMWOA-SVM is first compared with other five models, including the ALO-SVM, MFO-SVM, MVO-SVM,
PSO-SVM and WOA-SVM, and then compared with the BP neural network and ELM.
4.1. Prediction Results in Sunny Weather
4.1.1. Comparison with Other SVM Models
The IMWOA-SVM prediction model and the other five SVM models were tested with PV data of sunny weather and
Figure 7
shows the test results.
The red line in
Figure 7
represents the predicted power curve of IMWOA-SVM, the black line is the actual power curve, and other curves are shown in the legend. The PV power changes smoothly in sunny weather, showing a
parabola type of high in the middle and low sections in both sides. The prediction results of the algorithm optimizing the SVM model can generally describe the trend of PV power. The prediction
results of the ALO-SVM, MFO-SVM and WOA-SVM have some deviations with the actual power, and the prediction results of the IMWOA-SVM, MVO-SVM and PSO-SVM models are basically consistent with the
actual power. In order to further intuitively show the difference of prediction accuracy,
Figure 8
shows the accumulated value of absolute error of model prediction results in sunny weather.
The accumulated absolute error gradually increases over the course of a day. The accuracy of ALO-SVM, MFO-SVM, WOA-SVM is poor, the accumulated error increases rapidly, and the accumulated error
reaches more than 20 kW. The prediction accuracy of IMWOA-SVM, MVO-SVM, PSO-SVM is high. Although the accumulated error is also increasing, it has been kept at a low level and finally stabilized at 5
kW. The curve is relatively straight without an obvious mutation point, which shows that the prediction performance is stable. The accumulated absolute error of IMWOA-SVM is almost the same as that
of MVO-SVM. The prediction accuracy of PSO-SVM is inferior to IMWOA-SVM in the first and final stages, and slightly better in the middle stage. The accumulated error values of the two models are
roughly the same.
Figure 9
shows the statistical chart of the prediction error interval of the models in sunny weather. The transverse axis is the maximum value of the error range, and the longitudinal axis is the ratio of the
prediction error in the corresponding error range. The more the bar graph is concentrated to the left, the more consistent the prediction result of corresponding model with the actual power curve,
the more accurate the prediction is. Taking the IMWOA-SVM model as an example, the absolute error of IMWOA-SVM prediction results accounts for about 95% in the interval [0, 0.5] and about 5% in the
interval [0.5, 1.0]. The absolute error of other models accounts for less than 95% in the interval of [0, 0.5], which shows that IMWOA-SVM has higher prediction accuracy.
4.1.2. Comparison with BP Neural Network and ELM
Figure 10
shows the prediction results of the IMWOA-SVM, BP neural network and ELM in sunny weather. The predicted results of the IMWOA-SVM fit the actual power curve best. The BP neural network can also
predict the change of actual power well, but compared with the IMWOA-SVM, the error is obviously larger. The prediction results of ELM miss a lot of detailed information, and can hardly be applied in
The absolute error accumulation chart of BP and ELM is shown in
Figure 11
to intuitively display the error. The accumulated error of the IMWOA-SVM prediction results increases slowly and the curve is relatively straight, which indicates that the IMWOA-SVM can maintain high
accuracy throughout the whole period of time. However, the error curve of BP is gentle in the front section and warped in the tail section, which indicates that the accuracy of BP decreases at the
tail end and the prediction performance is unstable. The accumulated error curve of ELM increases rapidly, and the front and back sections of the curve are obviously warped, which indicates that ELM
prediction results are very unstable and the prediction accuracy is very poor.
Table 6
is the summary evaluation table of prediction results in sunny weather. Prediction results are comprehensively evaluated by MSE, RMSE, MAE, MAPE and R2. The bold font in the table indicates the
optimal value of the evaluation index. The MSE of IMWOA-SVM is 0.069, RMSE is 0.263, MAPE is 0.047, R2 is 0.995, which reach the optimal value. The MAE is 0.212, which is 0.009 higher than that of
PSO (0.203). Overall, the prediction result of IMWOA-SVM is the best.
In conclusion, the prediction performance of the six SVM models, the BP neural network and the ELM is tested in this section based on the data of sunny weather. The test results prove that most of
the SVM models are more accurate, while prediction accuracy of BP neural network and ELM is relatively poor. The reason is that BP neural network and ELM have a large demand for training data, while
this study is based on small training samples. The training data consists of two days’ historical data, which is far from meeting the requirements of BP neural network and ELM for training data
volume, so their prediction performance is poor. The SVM model has the advantage of less training data demand, so the overall prediction result is preferred, which is also the reason why SVM is
selected for PV prediction in this study. Several evaluation indexes show that the prediction performance of IMWOA-SVM is the best.
4.2. Prediction Results in Cloudy Weather
In this section, we will test the PV prediction model in cloudy weather. The PV output power presents great volatility and instability in cloudy weather, which brings more difficulties to the
prediction work. In such extreme conditions, the prediction performance of the prediction model can be tested more effectively.
4.2.1. Comparison with Other SVM Models
The prediction results of SVM models in cloudy weather is shown in
Figure 12
. In cloudy weather, the prediction deviations of ALO-SVM, MFO-SVM and WOA-SVM which perform poorly in sunny weather are still relatively large, and the prediction results of PSO-SVM and MVO-SVM with
high accuracy also show obvious deviation and the prediction performance declines. Only the prediction result of the IMWOA-SVM proposed can track the actual output well. The absolute error
accumulation chart of each model in cloudy weather is shown in
Figure 13
to show the change of prediction error more intuitively.
In cloudy weather, the prediction error of prediction model generally increases due to the violent fluctuation of actual PV power. The PSO-SVM and MVO-SVM models with higher prediction accuracy in
sunny weather have larger prediction error in cloudy weather, and the accumulated error increases to 15 KW, which shows that the performance of this two PV prediction model is not stable. It can
achieve better prediction accuracy in sunny weather with small disturbances, but cannot achieve ideal prediction accuracy in cloudy weather with large disturbances. Although the prediction error of
the IMWOA-SVM proposed also increases in cloudy weather, reaching about 7 KW, the rising range is not large and the IMWOA-SVM can still predict the changes of actual PV power. The accumulated error
curve of the IMWOA-SVM is still relatively straight, which shows that the complex meteorological conditions of cloudy weather have little impact on the prediction performance of the IMWOA-SVM. It
proves that the IMWOA-SVM has superior anti-interference ability and adaptability to different weather conditions.
Figure 14
shows the statistical chart of prediction error intervals in cloudy weather. The proportion of the IMWOA-SVM in the minimum error range is 90%, and the prediction accuracy is hardly affected. The
prediction error of other models accounts for only 55% in the minimum range, and the prediction results of other models generally tend to the interval with larger error. The prediction errors become
larger in complex weather, which is consistent with the previous conclusion. The results show that the IMWOA-SVM can achieve better prediction accuracy in both sunny and cloudy weather, which shows
that the IMWOA-SVM has high prediction accuracy and better stability.
4.2.2. Comparison with BP Neural Network and ELM
In cloudy weather, the prediction accuracy of BP neural network is significantly reduced. The prediction curve of BP deviates from the actual curve, as shown in
Figure 15
. The deviation points are mainly concentrated in the actual power mutation, and only a few points can accurately describe the actual output power. The prediction result of ELM is still not ideal;
the predicted curve deviates significantly from the actual curve, and lacks a significant amount of characteristic information on the actual power output curve.
The accumulated absolute error chart of BP and ELM in cloudy weather is shown in
Figure 16
. The maximum accumulated error of BP is more than 25 kW in cloudy weather, while it is not more than 15 kw in sunny weather. The accumulated error curve of the BP neural network and ELM increases
step by step. The curve up warping mainly occurs during the period when the actual power fluctuates greatly and, in the stable range of actual power, the accumulated error curve is relatively gentle.
It proves that the data fluctuation has a great influence on the prediction accuracy of BP and ELM. The prediction performance is unstable, and the adaptability to weather types is poor.
Finally, the comprehensive evaluation table of the model prediction results is given, as shown in
Table 7
. The MSE of the IMWOA-SVM is 0.257, the RMSE is 0.507, the MAE is 0.331, and the R2 is 0.979, all of which reach the optimal value. Due to the existence of zero value in cloudy weather data, the
MAPE evaluation index is not used. The evaluation results of the IMWOA-SVM are better than other models in cloudy weather.
In conclusion, the prediction performance of the six SVM models, the BP neural network and the ELM has been tested, in this section, with the data of cloudy weather. The test results show that the
accuracy of cloudy weather decreases compared with sunny weather. The prediction accuracy and stability of BP neural network and ELM are not better, and they are not suitable as the prediction model
in this study. Other prediction models based on SVM have better prediction accuracy in sunny weather, but have poor prediction accuracy in cloudy weather, indicating that these models have poor
adaptability and poor anti-interference ability. The proposed IMWOA-SVM model can achieve ideal prediction performance in both sunny and cloudy weather, especially in complex cloudy weather, and it
can also achieve better prediction accuracy. The experimental results prove that the IMWOA-SVM model has better prediction accuracy and anti-interference ability, and prove that the proposed
improvement measures in this study are effective.
5. Conclusions
PV output power has the characteristic of uncertainty, which is not conductive to the stability and security of power system. In different weather, PV power has significantly different
characteristics, which increases the difficulty of power prediction. For further symmetrically improving the prediction accuracy of PV power in different weather conditions and promoting the use of
clean energy, an improved whale algorithm optimizing SVM model is proposed in this study. The advantages of this model are that it can reduce the demand for input data, adapt to the changes of
weather conditions, and achieve ideal prediction accuracy in complex weather conditions compared with similar prediction models. The research contents include the selection of input data, the
preprocessing of data, the improvement of the optimization algorithm and the optimization of the SVM prediction model. The following conclusions are obtained.
The PV power is determined by some meteorological factors, and it has significantly different characteristics in different weather conditions. Through the correlation analysis, it is found that
PV power has the strongest correlation with the meteorological factors including light intensity, ambient temperature and humidity. Furthermore, these meteorological factors can be used to
accurately predict PV power.
The wavelet soft threshold denoising can be applied for the pretreatment of PV input data. It can effectively eliminate the noise contained in input data and improve the coherence of the input
data, which is beneficial to remove the adverse impact of noise and enhance the stability of the prediction model in complex weather conditions.
The BP neural network and ELM have large demand for training data. When the training data are not sufficient, the ideal prediction accuracy cannot be achieved. SVM has less demand for training
data, and can achieve ideal prediction accuracy when there is less training data, which is suitable for PV output power prediction models with less training data.
The optimization performance of WOA can be effectively improved through combination with the hybrid improved method. By combining the original WOA with tent chaos initialization, mutation
disturbance of the optimal individual and DE algorithm, the comprehensive performance of the IMWOA is significantly enhanced.
The IMWOA-SVM photovoltaic output power prediction model applies wavelet denoising to process the predicted input data, and applies the hybrid improved whale algorithm to optimize the SVM, which
significantly improves comprehensive prediction performance in different weather conditions.
The proposed IMWOA-SVM photovoltaic output power prediction model can symmetrically achieve the accurate prediction for PV power in different weather conditions, especially in complex weather
conditions. It can provide the operation and scheduling department with reliable reference, help to improve the utilization rate of renewable energy power generation and maintain the security of
renewable energy power systems. It is of great significance to the application of clean energy.
The input data selection method, input data preprocessing method, algorithm selection and improvement method and model optimization method proposed in this study constitute a complete prediction
method of renewable energy generation output power. It can be applied not only to predict the PV power, but also to predict other similar renewable energy power. It provides a reference for the
optimization and improvement of prediction models of other similar renewable energy and is expected to develop into a general method to predict the output power of renewable energy power.
This study has limitations. In the stage of selecting input data, the linear correlation between wind speed, temperature, humidity, light intensity, diffuse intensity and PV power is simply analyzed,
but the complex nonlinear relationship behind the data is not deeply considered, which may cause some important data to be ignored. Future studies should enhance the optimization of the prediction
method and the selection of input data to improve the performance of the IMWOA-SVM. In addition, this study is based on the SVM model; more types of models should be introduced into the prediction
field to further improve the prediction accuracy in future research.
Author Contributions
Conceptualization, Y.-W.L. and H.F.; methodology, Y.-W.L. and H.F.; software H.-Y.L. and L.-L.L.; writing—original draft preparation, Y.-W.L., H.F., H.-Y.L. and L.-L.L. All authors have read and
agreed to the published version of the manuscript.
This research was funded by the key project of the Tianjin Natural Science Foundation (Project No. 19JCZDJC32100) and the Natural Science Foundation of Hebei Province of China (Project No.
Institutional Review Board Statement
The study did not involve humans or animals.
Informed Consent Statement
The study did not involve humans or animals.
Data Availability Statement
Not Applicable.
This study was supported by the key project of the Tianjin Natural Science Foundation (Project No. 19JCZDJC32100) and the Natural Science Foundation of Hebei Province of China (Project No.
Conflicts of Interest
The authors declare no conflict of interest.
Acronyms list
ALO Ant lion optimization algorithm
DE Differential evolution algorithm
IMWOA Improved whale optimization algorithm
MAPE Mean absolute percent error
MFO Moth to fire algorithm
MSE Mean square error
PSO Particle swarm optimization algorithm
RMSE Root mean square error
SVM Support vector machine
SVR Support vector regression
WOA Whale optimization algorithm
Nomenclature variables
a, $a *$, u, $u *$ Iteration variables of whale algorithm
A, C[w] Iteration variables of whale algorithm
b Spiral motion constant of whale or Regression bias
C Penalty factor
C[1], C[2] Learning factor
CR Crossover probability
$M a x s c$ Maximum scale factor
$M a x _ t$ Maximum number of iterations
$M i n s c$ Minimum scale factor
t Current iterations
U New individuals obtained by crossing
X Individual whale population
$X b e s t$ Optimal individual of whale population
$X n$ New individuals obtained by mutation
$X r a n d$ Random individuals of whale population
$X *$ Current individuals
$x t$ Tent chaotic mapping sequence
$φ ( x )$ Nonlinear mapping function
$ε$ Permissible deviation
$ρ$ Pearson correlation coefficient
$ξ , ξ *$ Relaxation variable
Expression Dim Range Optimum
$F 1 ( x ) = ∑ i = 1 n x i 2$ 10 [−100, 100] 0
$F 2 ( x ) = ∑ i = 1 n | x i | + ∏ i = 1 n | x i |$ 10 [−10, 10] 0
$F 3 ( x ) = ∑ i = 1 dim ( ∑ j = 1 i x j ) 2$ 10 [−100, 100] 0
$F 4 ( x ) = ∑ ( [ 1 : dim ] . × [ x . ^ 4 ] ) + r a n d$ 10 [−1.28, 1.28] 0
$F 5 ( x ) = − 20 exp ( − 0.2 1 n ∑ i = 1 n x i 2 ) − exp ( 1 n ∑ i = 1 n cos ( 2 ∏ x i ) ) + 20 + e$ 10 [−32, 32] 0
Algorithm Parameter Value
ALL Number of iterations 300
Number of search agent 50
Minsc 0.20
IMWOA Maxsc 0.80
CR 0.20
PSO C[1] 1.49
C[2] 1.49
$M a x s c$ and $M i n s c$ are the upper and lower bounds of $r a n d o m ( 0 , 1 )$ in Formula (19), respectively. $C R$ is the parameter that controls the occurrence of crossover in Formula (20).
$C 1$ and $C 2$ are learning factors of PSO.
Function Index IMWOA WOA ALO MVO PSO MFO
F[1] Mean 0 0 0 0.0145 0 0
RMSE 0 0 0 0.0145 0 0
F[2] Mean 0 0 1.54 0.0428 0.136 0
RMSE 0 0 2.14 0.0429 0.0912 0
F[3] Mean 0 227 0.458 0.209 0.00801 0.591
RMSE 0 228 0.675 0.238 0.00941 0.828
F[4] Mean 0 0 0.0222 0.00211 0.0126 0.00717
RMSE 0 0 0.0245 0.0223 0.0160 0.00753
F[5] Mean 0 0 0.385 0.964 0.0450 0
RMSE 0 0 0.667 1.18 0.0632 0
$ρ X , Y$ Degree of Relevance
[0.8,1.0] Extremely strong
[0.6,0.8] Strong
[0.4,0.6] Moderate
[0.2,0.4] Weak
[0.0,0.2] Extremely weak
Month Light Intensity Diffuse Temperature Wind Speed Humidity
1 0.992 0.350 0.521 0.199 −0.590
2 0.994 0.449 0.521 0.262 −0.563
3 0.989 0.321 0.426 −0.053 −0.373
4 0.996 0.212 0.627 0.428 −0.596
5 0.995 0.256 0.696 0.580 −0.698
6 0.990 0.326 0.500 0.512 −0.731
Mean 0.993 0.319 0.549 0.321 −0.591
Relevance Extremely strong Weak Moderate Weak Moderate
Parameter IMWOA ALO MFO MVO PSO WOA BP ELM
MSE 0.069 0.859 0.970 0.069 0.072 0.962 0.587 8.887
RMSE 0.263 0.927 0.985 0.263 0.268 0.981 0.766 2.981
MAE 0.212 0.783 0.923 0.212 0.203 0.870 0.632 2.578
MAPE 0.047 0.103 0.130 0.048 0.057 0.161 0.179 0.506
R2 0.995 0.933 0.924 0.995 0.994 0.925 0.954 0.305
Parameter IMWOA ALO MFO MVO PSO WOA BP ELM
MSE 0.257 1.297 4.987 0.783 0.655 0.809 2.717 2.756
RMSE 0.507 1.139 2.233 0.885 0.809 0.899 1.648 1.660
MAE 0.331 0.996 2.012 0.671 0.630 0.748 1.338 1.283
R2 0.979 0.893 0.588 0.935 0.946 0.933 0.775 0.772
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Liu, Y.-W.; Feng, H.; Li, H.-Y.; Li, L.-L. An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation. Symmetry 2021, 13, 212. https://doi.org/10.3390/
AMA Style
Liu Y-W, Feng H, Li H-Y, Li L-L. An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation. Symmetry. 2021; 13(2):212. https://doi.org/10.3390/sym13020212
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Dissecting crush
Posted on July 9, 2014
For almost a year and half now I’ve been referencing one particular book on Coq, Certified Programming with Dependent Types. CPDT is a literate program on building practical things with Coq.
One of the main ideas of CPDT is that proofs ought to be fully automated. This means that a proof should be primarily a logic program (Ltac) which constructs some boring and large proof term. To this
end, CPDT has a bunch of Ltac “tactics” for constructing such logic programs.
Since CPDT is a program, there’s actual working source for each of these tactics. It occurred to me today that in my 18 months of blinking uncomprehendingly at CPDT, I’ve never read its source for
these tactics.
In this post, we’ll dissect how CPDT’s main tactic for automation, crush, actually works. In the process, we’ll get the chance to explore some nice, compositional, ltac engineering as well as a whole
host of useful tricks.
The Code
The first step to figuring out of crush works is actually finding where it’s defined.
After downloading the source to CPDT I ran
grep "Ltac crush :=" -r .
And found in src/CpdtTactics, line 205
Ltac crush := crush' false fail.
Glancing at crush', I’ve noticed that it pulls in almost every tactic in CpdtTactics. Therefore, we’ll start at the top of this file and work our way done, dissecting each tactic as we go.
Incidentally, since CpdtTactics is an independent file, if you’re confused about something firing up your coq dev environment of choice and trying things out with Goal inline works nicely.
Starting from the top, our first tactic is inject.
Ltac inject H := injection H; clear H; intros; try subst.
This is just a quick wrapper around injection, which also does the normal operations one wants after calling injection. It clears the original hypothesis and brings our new equalities into our
environment so future tactics can use them. It also tries to swap out any variables with our new equalities using subst. Notice the try wrapper since subst is one of those few tactics that will fail
if it can’t do anything useful.
Next up is
Ltac appHyps f :=
match goal with
| [ H : _ |- _ ] => f H
appHyps makes use of the backtracking nature of match goal with. It’ll apply f to every hypothesis in the current environment and stop once it find a hypothesis f works with.
Now we get to some combinators for working with hypothesis.
Ltac inList x ls :=
match ls with
| x => idtac
| (_, x) => idtac
| (?LS, _) => inList x LS
inList takes a faux-list of hypothesis and looks for an occurrence of a particular lemma x. When it finds it we just run idtac which does nothing. In the case were we can’t match x anywhere, inList
will just fail with the standard “No matching clause” message.
Next we have the equivalent of appHyps for tupled lists
Ltac app f ls :=
match ls with
| (?LS, ?X) => f X || app f LS || fail 1
| _ => f ls
This works exactly like appHyps but instead of looking through the proofs environment, we’re looking through ls. It has the same “keep the first result that works” semantics too. One thing that
confused me was the _ => f ls clause of this tactic. Remember that with our tupled lists we don’t have a “nil” member. But rather the equivalent of
A :: B :: C :: Nil
((A, B), C)
So when we don’t have a pair, ls itself is the last hypothesis in our list. As a corollary of this, there is no obvious “empty” tupled list, only one with a useless last hypothesis.
Next we have all, which runs f on every member in f ls.
Ltac all f ls :=
match ls with
| (?LS, ?X) => f X; all f LS
| (_, _) => fail 1
| _ => f ls
Careful readers will notice that instead of f X || ... we use ;. Additionally, if the first clause fails and the second clause matches, that means that either f X or all f LS failed. In this case we
backtrack all the way back out of this clause. This should mean that this is a “all or nothing” tactic. It will either not fail on all members of ls or nothing at all will happen.
Now we get to the first big tactic
Ltac simplHyp invOne :=
let invert H F :=
inList F invOne;
(inversion H; fail)
|| (inversion H; [idtac]; clear H; try subst) in
match goal with
| [ H : ex _ |- _ ] => destruct H
| [ H : ?F ?X = ?F ?Y |- ?G ] =>
(assert (X = Y); [ assumption | fail 1 ])
|| (injection H;
match goal with
| [ |- X = Y -> G ] =>
try clear H; intros; try subst
| [ H : ?F ?X ?U = ?F ?Y ?V |- ?G ] =>
(assert (X = Y); [ assumption
| assert (U = V); [ assumption | fail 1 ] ])
|| (injection H;
match goal with
| [ |- U = V -> X = Y -> G ] =>
try clear H; intros; try subst
| [ H : ?F _ |- _ ] => invert H F
| [ H : ?F _ _ |- _ ] => invert H F
| [ H : ?F _ _ _ |- _ ] => invert H F
| [ H : ?F _ _ _ _ |- _ ] => invert H F
| [ H : ?F _ _ _ _ _ |- _ ] => invert H F
| [ H : existT _ ?T _ = existT _ ?T _ |- _ ] => generalize (inj_pair2 _ _ _ _ _ H); clear H
| [ H : existT _ _ _ = existT _ _ _ |- _ ] => inversion H; clear H
| [ H : Some _ = Some _ |- _ ] => injection H; clear H
Wow, just a little bit bigger than what we’ve been working with so far.
The first small chunk of simpleHyp is a tactic for doing clever inversion using the tuple list invOne.
invert H F :=
inList F invOne;
(inversion H; fail)
|| (inversion H; [idtac]; clear H; try subst)
Here H is a hypothesis that we’re thinking about inverting on and F is the head symbol of H. First we run the inList predicate, meaning that we don’t invert upon anything that we don’t want to. If
the head symbol of H is something worth inverting upon we try two different types of inversion.
In the first case inversion H; fail we’re just looking for an “easy proof” where inverting H immediately dispatches the current goal. In the second case inversion H; [idtac]; clear H; try subst, we
invert upon H iff it only generates 1 subgoal. Remember that [t | t' | t''] is a tactic that runs t on the first subgoal, t’ on the second, and so on. If the number of goals don’t match, [] will
fail. So [idtac] is just a clever way of saying “there’s only one new subgoal”. Next we get rid of the hypothesis we just inverted on (it’s not useful now, and we don’t want to try inverting it
again) and see if any substitutions are applicable.
Alright! Now let’s talk about the massive match goal with going on in simplHyp.
The first branch is
| [ H : ex _ |- _ ] => destruct H
This just looks for a hypothesis with an existential (remember that ex is what exists desugars to). If we find one, we introduce a new variable to our environment and instantiate H with it. The fact
that this doesn’t recursively call simplHyp probably means that we want to do something like repeat simplHyp to ensure this is applied everywhere.
Next we look at simplifying hypothesis where injection applies. There are two almost identical branches, one for constructors of two parameters, one for one. Let’s look at the latter since it’s
slightly simpler.
| [ H : ?F ?X = ?F ?Y |- ?G ] =>
(assert (X = Y); [ assumption | fail 1 ])
|| (injection H;
match goal with
| [ |- X = Y -> G ] =>
try clear H; intros; try subst
This looks for an equality over a constructor F. This branch is looking to prove that X = Y, a fact deducible from the injectiveness of F.
The way that we go about doing this is actually quite a clever ltac trick though. First we assert X = Y, this will generate to subgoals, the first that X = Y (shocker) and the second is the current
goal G, with the new hypothesis that X = Y. We attempt to prove that X = Y by assumption. If this works, than we already trivially can deduce X = Y so there’s no point in doing all that injection
stuff so we fail 1 and bomb out of the whole branch.
If assumption fails we’ll jump to the other side of the ||s and actually use injection. We only run injection if it generates a proof that X = Y in which case we do the normal cleanup with trying to
clear our original fact and do some substitution.
The next part is fairly straightforward, we make use of that invert tactic and run it over facts we have floating around in our environment
| [ H : ?F _ |- _ ] => invert H F
| [ H : ?F _ _ |- _ ] => invert H F
| [ H : ?F _ _ _ |- _ ] => invert H F
| [ H : ?F _ _ _ _ |- _ ] => invert H F
| [ H : ?F _ _ _ _ _ |- _ ] => invert H F
Notice that we can now use the match to grab the leading symbol for H so we only invert upon hypothesis that we think will be useful.
Next comes a bit of axiom-fu
| [ H : existT _ ?T _ = existT _ ?T _ |- _ ] =>
generalize (inj_pair2 _ _ _ _ _ H); clear H
inj_pair2 is function that lives in the Coq standard library and has the type
forall (U : Type) (P : U -> Type) (p : U) (x y : P p),
existT P p x = existT P p y -> x = y
This relies on eq_rect_eq so it’s just a little bit dodgy for something like HoTT where we give more rope to = than just refl.
This particular branch of the match is quite straightforward though. Once we see an equality between two witnesses for the same existential type, we just generalize the equality between their proofs
into our goal.
If this fails however, we’ll fall back to standard inversion with
| [ H : existT _ _ _ = existT _ _ _ |- _ ] => inversion H; clear H
Finally, we have one last special case branch for Some. This is because the branches above will fail when phased with a polymorphic constructor
| [ H : Some _ = Some _ |- _ ] => injection H; clear H
Nothing exciting going on there.
So that wraps up simplHyp. It’s just a conglomeration of useful stuff to do to constructors in our hypothesis.
Onwards we go! Next is a simple tactic for automatically rewriting with a hypothesis
Ltac rewriteHyp :=
match goal with
| [ H : _ |- _ ] => rewrite H by solve [ auto ]
like most of the other tactics we saw earlier, this will hunt for an H where this works and then stop. The by solve [auto] will run solve [auto] against all the hypothesis that the rewrite generates
and ensure that auto solves all the new goals. This prevents a rewrite from going and introducing obviously false facts as goals for a rewrite that made no sense.
We can combine this with autorewrite with two simple tactics
Ltac rewriterP := repeat (rewriteHyp; autorewrite with core in *).
Ltac rewriter := autorewrite with core in *; rewriterP.
This just repeatedly rewrite with autorewrite and rewriteHyp as long as they can. Worth noticing here how we can use repeat to make these smaller tactics modify all applicable hypothesis rather than
just one.
Next up is an innocent looking definition that frightens me a little bit
Definition done (T : Type) (x : T) := True.
What frightens me about this is that Adam calls this “devious”.. and when he calls something clever or devious I’m fairly certain I’d never be able to come up with it :)
What this actually appears to do is provide a simple way to “stick” something into an environment. We can trivially prove done T x for any T and x but having this in an environment also gives us a
proposition T and a ready made proof of it x! This is useful for tactics since we can do something like
assert (done SomethingUseful usefulPrf) by constructor
and viola! Global state without hurting anything.
We use these in the next tactic, instr.
Ltac inster e trace :=
match type of e with
| forall x : _, _ =>
match goal with
| [ H : _ |- _ ] =>
inster (e H) (trace, H)
| _ => fail 2
| _ =>
match trace with
| (_, _) =>
match goal with
| [ H : done (trace, _) |- _ ] =>
fail 1
| _ =>
let T := type of e in
match type of T with
| Prop =>
generalize e; intro;
assert (done (trace, tt)) by constructor
| _ =>
all ltac:(fun X =>
match goal with
| [ H : done (_, X) |- _ ] => fail 1
| _ => idtac
end) trace;
let i := fresh "i" in (pose (i := e);
assert (done (trace, i)) by constructor)
Another big one!
This match is a little different than the previous ones. It’s not a match goal but a match type of ... with. This is used to examine one particular hypothesis’ type and match over that.
This particular match has two branches. The first deals with the case where we have uninstantiated universally quantified variables.
| forall x : _, _ =>
match goal with
| [ H : _ |- _ ] =>
inster (e H) (trace, H)
| _ => fail 2
If our hypothesis does, we randomly grab a hypothesis, instantiate e with it, add H to the trace list, and then recurse.
If there isn’t a hypothesis, then we fail out of the toplevel match and exit the tactic.
Now the next branch is where the real work happens
| _ =>
match trace with
| (_, _) =>
match goal with
| [ H : done (trace, _) |- _ ] =>
fail 1
| _ =>
let T := type of e in
match type of T with
| Prop =>
generalize e; intro;
assert (done (trace, tt)) by constructor
| _ =>
all ltac:(fun X =>
match goal with
| [ H : done (_, X) |- _ ] => fail 1
| _ => idtac
end) trace;
let i := fresh "i" in (pose (i := e);
assert (done (trace, i)) by constructor)
We first chekc to make sure that trace isn’t empty. If this is the case, then we know that we instantiated e with at least something. If we have, we snoop around to see if there’s a done in our
environment with the same trace. If this is the case, we know that we’ve done an identical instantiation of e before hand so we backtrack to try another one.
Otherwise, we look to see what e was instantiated too. If it was a simple Prop, we just stick a done record of this instantiation into our environment and add our new instantiated e back in with
generalize. If e isn’t a proof, we do the same thing. In this case, however, we must also double check that the things we used to instantiate e with aren’t results of inster as well otherwise our
combination of backtracking/instantiating can lead to an infinite loop.
Since this tactic generates a bunch of done’s that are otherwise useless, a tactic to clear them is helpful.
Ltac un_done :=
repeat match goal with
| [ H : done _ |- _ ] => clear H
Hopefully by this point this isn’t too confusing. All this tactic does is loop through the environment and clear all dones.
Now, finally, we’ve reached crush'.
Ltac crush' lemmas invOne :=
let sintuition := simpl in *; intuition; try subst;
repeat (simplHyp invOne; intuition; try subst); try congruence in
let rewriter := autorewrite with core in *;
repeat (match goal with
| [ H : ?P |- _ ] =>
match P with
| context[JMeq] => fail 1
| _ => rewrite H by crush' lemmas invOne
end; autorewrite with core in *) in
(sintuition; rewriter;
match lemmas with
| false => idtac | _ =>
(** Try a loop of instantiating lemmas... *)
repeat ((app ltac:(fun L => inster L L) lemmas
(** ...or instantiating hypotheses... *)
|| appHyps ltac:(fun L => inster L L));
(** ...and then simplifying hypotheses. *)
repeat (simplHyp invOne; intuition)); un_done
sintuition; rewriter; sintuition;
try omega; try (elimtype False; omega)).
crush' is really broken into 3 main components.
First is a simple tactic sintuition
sintuition := simpl in *; intuition; try subst;
repeat (simplHyp invOne; intuition; try subst); try congruence
So this first runs the normal set of “generally useful tactics” and then breaks out some of first custom tactics. This essentially will act like a souped-up version of intuition and solve goals that
are trivially solvable with straightforward inversions and reductions.
Next there’s a more powerful version of rewriter
rewriter := autorewrite with core in *;
repeat (match goal with
| [ H : ?P |- _ ] =>
match P with
| context[JMeq] => fail 1
| _ => rewrite H by crush' lemmas invOne
end; autorewrite with core in *)
This is almost identical to what we have above but instead of solving side conditions with solve [auto], we use crush' to hopefully deal with a larger number of possible rewrites.
Finally, we have the main loop of crush'.
(sintuition; rewriter;
match lemmas with
| false => idtac
| _ =>
repeat ((app ltac:(fun L => inster L L) lemmas
|| appHyps ltac:(fun L => inster L L));
repeat (simplHyp invOne; intuition)); un_done
sintuition; rewriter; sintuition;
try omega; try (elimtype False; omega)).
Here we run the sintuition and rewriter and then get to work with the lemmas we supplied in lemmas.
The first branch is just a match on false, which we use like a nil. Since we have no hypothesis we don’t do anything new.
If we do have lemmas, we try instantiating both them and our hypothesis as many times as necessary and then repeatedly simplify the results. This loop will ensure that we make full use of bot our
supplied lemmas and the surrounding environment.
Finally, we make another few passes with rewriter and sintuition attempting to dispatch our goal using our new, instantiated and simplified environment.
As a final bonus, if we still haven’t dispatched our goal, we’ll run omega to attempt to solve a Presburger arithmetic. On the off chance that we have something omega can be contradictory, we also
try elimType false; omega to try to exploit such a contradiction.
So all crush does is call this tactic with no lemmas (false) and no suggestions to invert upon (fail). There you have it, and it only took 500 lines to get here.
Wrap Up
So that’s it, hopefully you got a few useful Ltac trick out of reading this. I certainly did writing it :)
If you enjoyed these tactics, there’s a more open-source version of these tactics, on the CPDT website. It might also interest you to read the rest of CpdtTactics.v since it has some useful gems like
Last but not least, if you haven’t read CPDT itself and you’ve made it this far, go read it! It’s available as either dead-tree or online. I still reference it regularly so I at least find it useful.
It’s certainly better written than this post :)
Note, all the code I’ve shown in this post is from CPDT and is licensed under ANCND license. I’ve removed some comments from the code where they wouldn’t render nicely with them.
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What is the role of data preprocessing in Multivariate Analysis, and how does SAS support it? | Hire Someone To Take My SAS Assignment
What is the role of data preprocessing in Multivariate Analysis, and how does SAS support it? By an individual, the role of prior information in multivariate analyses is explained. By a greater
number of study participants, a greater estimate of the proportion of data that may be suitable to characterize the multivariate association for each respondent is provided. How Does SAS Inform Us
Who We Are, and Who We Put in Policy Under Upper Dog Lending, Uiner, and Postpartum Flop? Data preprocessing is arguably the single most important component in multivariate analysis. Improving the
process from preconceptions of data science to making a sense of how it applies to public policy will improve the quality of decisions in most governments. It follows very well from multiple studies
of how data is used in policies, and it is possible to implement the same process as data science in general, and in general. The good news is that anyone who is willing to enter data will probably
find themselves at least somewhat influenced by many similar concerns. Thus, any approach to data analysis that provides an in-depth discussion of many important data sources is likely to provide a
good starting point. How do SAS’s tools work for multivariate analysis? Both SAS and SAS2 are designed for generating and analyzing multivariate models. SAS2 has been developed by a consortium of
organisations and research groups, and is available freely for download. SAS2 provides the following function that is used for generating model estimates on the basis of a prior distribution: At each
iteration step (or at least several steps), SAS is applied to the distribution of all variables in a model at high precision. Each high-precision step is then followed by a series of rounds where the
model is determined to be of the appropriate type. The components of the high-precision model are considered if they can be treated as continuous variables on a count-one computer program so that
sampling in each of the rounds performs as if it had this contact form a continuous variable. HPE: How do we know that a model is a continuous model with a mean determined at each iteration step?
This is really the gist of the question: how did SAS generate the model? For an extensive discussion of the prior shape-class utility, including how we can see how the prior is shaped, and how shape
classification is able to tell us where in the model the relevant information was. For instance, SAS uses the previous figure for point patterns to understand what we are looking for in a variable. A
graphical representation of the prior can be built to give insight into how the prior is thought of under what circumstances. The distinction between a model and a group of variables in a
prior-bounding box is illustrated in Figure 4.3. Figure 4.3 A prior model (drawn by a classifier) is well-rounded to display how the shapes of the nodes in the current variable are represented on the
area. Information is not fixed in the model that isWhat is the role of data preprocessing in Multivariate Analysis, and how does SAS support it? This section provides an overview.
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3.3 The prior space of the multivariate process, which we are presently using in our analysis Our manuscript seeks to highlight some principles within the analysis of multivariate data that inform
the choice of a threshold value for the prior space; primarily the approach we will take. We want to emphasize that the previous assumptions of multiple measures in the prior space are less relevant
for our methodology. Once more, we define the prior space through a number of techniques that we found useful for classifying and fitting our samples. As we increase the scale of our analysis, the
scale of the prior space will increase. However, we also want to address the issue of small samples: the prior-centered sampling. 2.1 The most general prior space and data used Various assumptions
can be made about the prior space. For any given sample size and scale, the posterior probability of arriving at the true positive (pre-predict) of the estimate of the model from the data is given by
the following equation: > p(f_1 > f_2; 5.3 x f1 Using one of these procedures and a prior of size 1, we created 20,000,000 random samples from the posterior of each dimension of the prior space that
are more or less similar to the true positive (corresponding to a P( > prior_from) = P( > posterior_from) is 1). Prior-centered sampling works as described above and accounts for the initial
posterior distribution when conditioning the multiple measurements over the prior space. We now assume 2 distinct prior spaces, e.g. a prior of size 1, (2 + a posterior density factor) and a prior of
size 2. All this is because if the prior space contains the data from one data point, when the data point is “pushed” or “kicked”, it usually doesn’t contain all of the data within the prior space.
The principle underlying our technique lies in its ability to separate the posterior samples from each prior space independently. However, if we assume a posterior density that is constant over the
data, we will also use a posterior density factor. In these cases, this leads to a mixture of different prior spaces allowing us to recover data that fit well across the posterior distribution if, e.
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g., the data on the basis of two prior spaces and the prior space, is similar to each other. 3.2 For the prior space, where we are also using the the likelihood surface, the posterior densities and
the prior surfaces are as follows. First, a prior density of size 1 is given in equation 11. Second, if one of the prior spaces is slightly larger than the data, the prior blog here will start over
and become insignificant for discover this samples. Here weWhat is the role of data preprocessing in Multivariate Analysis, and how does SAS support it? ———————————————————————————- Data
preprocessing tools have been working largely on the theory (Skilling [@CR22]; Schierle et al. [@CR22]; Stechneider et al. [@CR25]) but there is still considerable controversy. Some authors argue
that it is not a perfect analysis tool, but it fails to provide an analytical feel for multivariate data structures. However, sometimes it is very helpful and more often, people find ways to look at
multivariate data structures to apply some methods (e.g. Gaussian functions) together with other data conditions (e.g. time-series) (Di Castro [@CR12]), in particular when using complex time series
or many complex data structures as a signal. How does SAS support multivariate analysis? ——————————————- Methods like SAS are not so much work, but the problem is why it works when you have many data
analysis methods in place, the method is just one and the problem is adding more and more additional data. There are several reasons for this. Firstly, some data analysis methods are known for their
small number of values to the machine; no data are in the data partition or you have data of multiple values. There is no *single* or *many* software at your disposal to train this or to get started,
so sometimes it might be beneficial for a first impression. Secondly, SAS is written in terms of machine jargon, and it does not have a complete vocabulary for data structure and time-series
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There are two basic ways how to do this: A *trainable* framework for structured analysis — *general structural property* (GP) or Structural Model Relationships (SMRs)
—————————————————————————————————————————– ### General structural property In this part, we are concerned with the natural language of SAS data. It is only a statement in a single language, no
language constructs from any other, so you have to make the statements in SAS specific for you. We discuss how, in a language that is simple and easy to use, you come up with results. In the
following three sections, we discuss sas project help common examples and how not SAS is a data source for multivariate analysis. **Example 1:** Two SAC model built with different populations
(individuals with different cognitive competencies) for which individual membership is at level *C*~*g*~. Note that the model isn’t built this way because each individual has two values for their
membership. How does SAS come together so that the population membership value is one? On its own, we hope. **Example 2:** How does SAC with individuals with different cognitive competencies relate
to each other and who are best at group comparison? Suppose our data is limited to age *C*~*c*~ = 3 (years), with 50% individuals having web cognitive competency. How does
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Equity Multiplier: Meaning and How to Calculate It
Calculating Equity Multiplier is straightforward, which helps to know the shareholders’ net equity finances and the number of assets of a firm. Suppose the Equity Multiplier ratio is 2, which means
investment in total assets is 2 times by total equity of shareholders. These real-world examples from Apple and Verizon illustrate how companies can have different financial strategies reflected in
their equity multipliers.
In calculating the equity multiplier, only the equity attributable to ordinary stock is taken into account. There are nuances to be aware of in order to find the correct data for calculating equity
multiplier. This program breaks down everything you need to build and interpret real estate finance models. Used at the world’s leading real estate private equity firms and academic institutions. The
general rules of thumb for interpreting the equity multiple are as follows.
Examples of Equity Multiplier Analysis
Meanwhile, Verizon’s telecommunications business model is similar to utility companies, which have stable, predictable cash flows and typically carry high debt levels. Apple is thus more susceptible
to changing economic conditions or evolving industry standards than a utility or a traditional telecommunications firm. The formula for calculating the equity multiplier consists of dividing a
company’s total asset balance by its total shareholders’ equity. The debt ratio shows the proportion of a company’s assets that are financed by credit obligations. It is usually calculated as the
ratio of a company’s debt to its total assets.
• They are categorized as either current assets, which can be easily converted to cash within a year, or non-current assets, which can’t.
• In the event of a crisis, a heavily indebted business runs the risk of losing the ability to service debt and run operations due to declining profits.
• It demonstrates the proportion of a company’s assets funded by shareholder equity versus debt.
• The amount of a company’s shareholders’ equity and the total value of all its assets can be found in the balance sheet.
• To calculate the shareholders’ equity account, our model assumes that the only liabilities are the total debt, so the equity is equal to total assets subtracted by total debt.
• Generally, a lower equity multiplier (closer to 1) implies less financial risk but potentially lower returns.
• This makes Tom’s company very conservative as far as creditors are concerned.
Company A has a lower Equity Multiplier than Company B, which means company B uses more debt to fund their business. Founded in 1993, The Motley Fool is a financial services company dedicated to
making the world smarter, happier, and richer. If ROE increases solely due to an increase in EM, this is a warning sign. The first one (Total assets) is presented in the last row under the Assets
section. From Year 1 to Year 5, the net cash proceeds attributable to the investor are fixed at $300k each period.
How to Use Equity Multiplier in DuPont Analysis?
High leverage can be part of an effective growth strategy, especially if the company is able to borrow more cheaply than its cost of equity. There is no one-size-fits-all equity multiplier that would
be considered good for any company. In general, numbers in the range of 0.8 to 1.5 are considered safe leverage.
The equity multiplier is a useful tool for investors to monitor risk and understand how a company generates returns for investors. It’s helpful by itself and as part of a DuPont analysis, which is a
financial tool that breaks out how a company generates a return on equity (ROE). Because their assets are generally financed by debt, companies how to calculate equity multiplier with high equity
multipliers may be at risk of default. Higher equity multipliers typically signify that the company is utilizing a high percentage of debt in its capital structure to finance working capital needs
and asset purchases. This is a method for assessing the financial attractiveness of a business developed by DuPont.
Is a High Equity Multiplier Always a Cause for Concern?
When looked at in conjunction with the equity multiplier, these two can provide a deeper insight into a company’s financial performance. Understanding the equity multiplier isn’t just an academic
exercise; it has real-world applications that can affect your bottom line. Whether you’re an investor, a creditor, or a business owner, this financial ratio can offer you valuable insights. It’s
important to remember that the Equity Multiplier
of a company should only be compared to the industry standard or to other companies
in the same sector. The best way to examine it is to compare it over time and
look for a trend.
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1. in 1995, 10.8% of All U.S. Families Had Incomes Below the Poverty Level, As Reported
Homework 9
1. In 1995, 10.8% of all U.S. families had incomes below the poverty level, as reported by the Census Bureau in Current Population Reports. During that same year, of 368 randomly selected families
whose householder had a Bachelor's degree or more, 9 had incomes below the poverty level. At the 1% significance level, do the data provide sufficient evidence to conclude that in 1995, families
whose householder had a Bachelor's degree or more had a lower percentage earning incomes below the poverty level than the national percentage of 10.8%? (Use rejection region approach.)
2. The Chicago Title Insurance Company publishes statistics on recent home buyers in the The Guarantor. According to that publication, 83.1% of home buyers in 1995 purchased single-family houses. Out
of 2544 randomly selected home buyers for this year, 2081 purchased single-family houses. Do the data provide sufficient evidence to conclude that this year's percentage of home buyers purchasing
single-family houses is different from the 1995 figure of 83.1%? Use = 0.024. (Use p-value approach.)
3. Problem 5.44
4. An office manager has implemented an incentive plan that she thinks will reduce the mean time required to handle a customer complaint. The mean time for handling a complaint was 30 minutes prior
to implementing the incentive plan. After the plan was in place for several months, a random sample of the records of 38 customers who had complaints revealed a mean time of 28.7 minutes with a
standard deviation of 3.8 minutes. Is there sufficient evidence that the incentive plan has reduced the mean time to handle a complaint? Us 0.01 level of significance. (Note that one has to be very
cautious about using a 2 sided confidence interval to draw conclusion about a one tailed test. That is why 5.78 d did not mention a significance level.) To avoid careless mistakes, perform the test
instead of drawing the conclusion from a confidence interval when dealing with 1-sided test. Or, you have to compute the 1-sided confidence bound in order to use it to draw conclusion about the
test). For this problem, add that we want to perform the test at a level of significance of 0.01.
5. Use Minitab to solve following: A study was conducted of 90 adult male patients following a new treatment for congestive heart failure. One of the variables measured on the patients was the
increase in exercise capacity (in minutes) over a 4-week treatment perios. The previous treatment regime had produced an average increase of µ = 2 minutes. The researchers wanted to evaluate whether
the new treatment had increased the value of µ in comparison to the previous treatment. The data yielded a sample mean of 2.17 and S = 1.05 Calculate the power of the test for ua = 2.1 and ua = 2.5
for both an alpha of 0.05 and 0.01. Comment on the effect these changes have on the power of the test. Also, find the minimum sample size needed to have a power of 0.80 for an alpha of 0.05 to detect
an effect size of 0.2.
Minitab Steps:
Select STAT > Power and Sample Size > One Sample t
Enter 90 50 for sample size, 0.1 0.5 for differences, and 1.05 for Standard Deviation
Click Options and select Greater Than for alternative and first enter 0.05 for significance level and click OK twice
Go back and change significance level from 0.05 to 0.01
For sample size calculations return and clear sample sizes, enter 0.2 for difference, 0.8 power and change alpha back to 0.05
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Statistical Arbitrage Trading Pairs
A Z score is the value of a supposedly normal random variable when we subtract the mean and divide by the standard deviation, thus scaling it to the standard normal distribution. Each case gets its
own z-score.
Identify a pair of equities that possess a residuals time series which has been statistically identified as mean-reverting. In this case, Microsoft, Google and Facebook pairs trading.
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Wormhole Metric Continued
I have decided to continue my Quaternionic Equation from the original Wormhole Metric Thread = https://www.scienceforums.net/topic/111607-wormhole-metric-how-is-this-screwed-up/
This is mainly to check all the variables in the differential Equation to make sure that they all solve correctly and to make sure the Quaternion is anomaly free and solve the equation for
∇'(x,y,z,t,ω[s],ω[p,]M,I,k,φ,S,X,Z,μ,Y,q,a,β) = (d^2/((ħ^2 /(2E[rest]/C^2)) ∑^3[a =1] (d^2/d((C^2/E[rest])∑^N[i = 1 ]M[i]R[i])^2) + (1/2)∑^3[a,][β = 1 ]μ[a][β](P[a ]- Π[a])(P[β][ ]- Π[β]) + U - (ħ^
2/2)∑^3N-6[s=1](d^2/dq^2) + V)((|(Log[(DgDaDψDφ-W)](((2ħGC^2))R[s] - (1/4)F^a[μ][v]F^a^μv + i(ψ-bar)γ^μ(((L[ghost QE ] - gf^abc(δ^μ (c-bar)^a)A[μ]^bc^c) / (c-bar)^aδ^μc^a)[ + ig(1/2)]τW[μ] + ig'(1/2)
YB[μ])ψ^i +(ψ-bar)^i[L]V[ij]φψ^j[r] + (a[ji]) - (μ^2((φ-Dagger)φ) + λ((φ-Dagger)φ)^2)/-(((L[ghost QE ] - gf^abc(δ^μ (c-bar)^a)A[μ]^bc^c) / (c-bar)^aδ^μc^a)[ + ig(1/2)]τW[μ] + ig'(1/2)YB[μ])^2)|)-e^2
S(r,t)/h)) - ((E[rest]/C^2)ω[s]((((8πGT[ab]/C^4) + Λg[ab ] - R[ab]) * g[ab]^-1))^1/2 + (S/ (((3G(E[rest]/C^2))/2C^2R[s]^3)(R[p]V[p]) + (GI[s]/C^2R[s]^3)((3R[p]/R[s]^2)(ω[p ]R[p]) -ω[p] ))))R[s]^2/2))
) / (ħ^2/2(E[rest]/C^2))))^1/2(((1-(((2(E[rest]/C^2)G / R[s]) - (I[s]ω[s]((((8πGT[ab]/C^4) +[ ]Λg[ab ] - R[ab]) * g[ab]^-1))^1/2 + (S/(((3G(E[rest]/C^2))/2C^2R[s]^3)(R[p]V[p]) + (GI[s]/C^2R[s]^3)((3R
[p]/R[s]^2)(ω[p ]R[p]) -ω[p] )))))/2(E[rest]/C^2))+ (((8πG/3)((g/(2π)^3)∫(((E[relativistic]^2 - E[rest]^2 / C^2)^ + ((A[r](X) + (E[Nucleon binding SNF][ ]ε[0 ]μ[0 ]/m[u]) - A[r](X^Z^±)/Z) / m[u])^2)^
(^1/2)(1/e^((ERelativistic - ^μchemical)^/TMatter^)±1)(ħω[s ] + ħω[s]) - ((k[s]C^2)/ R[s]^2) + (((8πGT[ab]/C^4) + Λg[ab ] - R[ab]) * g[ab]^-1))^1/2(ΔKiloparsec)))^2/(C^2)))^1/2)
(d^2/∇') - (Ctp)^2 = ds^2
(Universe Volumetric Planck State @ size of universe in radius) =(4/3)π((RUniverse/(tpC))^3 Luniverse
Luniverse = (∇Charge,∇Color,∇flavour,∇gravity - ∇Dark Energy)
Charge possible states per point (1,2/3, 1/3, 0,-1/3,-2/3,-1)
Color Possible states per point(R,B,G,0,antiG,antiB,antiR)
Flavour possible states per point (I,II,III,0,darkIII,darkII,darkI)
Gravity/Dark Energy possible states per point of space (Energy,Mass,Spin,0,-spin,-mass,-Energy)
Atleast the graphing equation and Equivalence principal are in working order having A.I. do the work.
I have decided to use this equation for a proton instead of the entire universe as it would be too much data to ever complete.
(Universe Volumetric Planck State @ size of universe in radius) =(4/3)π((RUniverse/(tpC))^3 (∇Charge,∇Color,∇flavour,∇gravity - ∇Dark Energy)
RUniverse = RProton = 10^-15 meters
The Equation Yields a Planck State of 9.9023511969154288921026543960449 * 10^59 (∇Charge,∇Color,∇flavour,∇gravity - ∇Dark Energy)
So a Field with 9.9023511969154288921026543960449 * 10^59 cubes that are a Planck length with states of (+1/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/
(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (+1/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/
(dx + dy +dz)- 0/(dx + dy +dz))) , (+1/(dx + dy +dz),G/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) if
the proton is at rest.
The Strong Nuclear Force or color Map will look something like this which is the only thing over the 3-D field that varies in a proton.
If the Proton is in motion let's say moving in a particle accelerator at 8 Tev then the State is 7.6171932283964837631558879969576 * 10^57 (+1/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (8000000/
(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (+1/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (8000000/(dx + dy +dz) - 0/(dx + dy
+dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (+1/(dx + dy +dz),G/(dx + dy +dz),I/(dx + dy +dz), (8000000/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/
(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
All of the Information being within the equation with a smaller color field of the same picture being less Planck Lengths within the particle due to length contraction.
The Graphing Equation displays all possible properties of the particle or substance to an quantized amount of a Planck Length being exact without error, I could write the entire Tensor for each
substance but it would take the big number amount of states. These were done assuming Dark Energy was not existent and a non expanding universe which are the zero terms. There is only one unknown in
these equations which is the Spin number of Dark Energy particles being the final zero in the spin term, the graph is over d/dx + d/dy + d/dz the big number shows the number of planck lengths that
the fields manifest for a proton at rest versus in motion for these examples.
This shows this equation to be in working order and accurate to reality.
This equation is actually more complex than the long equation as it gives a single state for everything rather than a large number of multiple Planck States like this one.
If you wanted more detail of the Quarks within the Proton you could graph the equation with the same set of coordinates including the quarks with the same result.
For the Rest proton with quarks in finer detail.
9.9023511969154288921026543960449 * 10^59 (+2/3/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy
+dz))) , (+2/3/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (-1/3/(dx + dy +dz),G/(dx
+ dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
Now the charges varies given the details of the quarks within the proton which as now the charges vary you will have two varing graphs one for the Strong Nuclear Force or Color and one for the
Electromagnetic Force or Charge being the (+2/3/(dx + dy +dz),R/(dx + dy +dz)) + (+2/3 /(dx + dy +dz), B/(dx + dy +dz)) + (-1/3/(dx + dy +dz), G/(dx + dy +dz)) = (+1/(dx + dy +dz),RGB/(dx + dy +dz))
The equation can be used to whatever detail you would like it to be this being a more exact map of the proton next would be to add gluons if you wanted or even more protons and neutrons to construct
an atom, but it is always exact to the planck length, no matter what detail is used.
Overlapped Charge and Color Map, (+2/3/(dx + dy +dz),R/(dx + dy +dz)) + (+2/3/(dx + dy +dz), B/(dx + dy +dz)) + (-1/3/(dx + dy +dz), G/(dx + dy +dz)) = (+1/(dx + dy +dz),RGB/(dx + dy +dz))
Which solves perfectly making the graphing equation even physically correct next we will try something more challenging like a Feynman diagram using this equation, it should be able to graph anything
in the universe to the planck length is the test.
The Feynman Diagram we are going to test this on is Beta Decay of Carbon 14 into Nitrogen 14 to start off with the calculations need to be done for the planck state of an Electron and Neutron as beta
decay is P+ > N + e- + Ve , so we willl start with mapping the quarks within the proton which a proton's state is 9.9023511969154288921026543960449 * 10^59 (+2/3/(dx + dy +dz),R/(dx + dy +dz),I/(dx
+ dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (+2/3/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/
(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (-1/3/(dx + dy +dz),G/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)
- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
Then the neutron can be described as a Planck State too which is
9.9023511949154288921026543960449 * 10^59 (+2/3/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy
+dz))) , (-1/3/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (-1/3/(dx + dy +dz),G/(dx
+ dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
The electron has a smaller state 1.1998578848809383445875560276978 * 10^51 (-1/(dx + dy +dz),0/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz), .511/(dx + dy +dz)- 0/(dx + dy +dz),1
/2/(dx + dy +dz)- 0/(dx + dy +dz)))
The Neutrino has State of 28722.600151171579743008314436886(0/(dx + dy +dz),0/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz), .2/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0
/(dx + dy +dz))) being much smaller than all of them
This Completes the Feynmann Diagram for Beta minus decay and satisfies P+ > N + e- + Ve
9.9023511969154288921026543960449 * 10^59 (+2/3/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy
+dz))) , (+2/3/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (-1/3/(dx + dy +dz),G/(dx
+ dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
9.9023511949154288921026543960449 * 10^59 (+2/3/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy
+dz))) , (-1/3/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (-1/3/(dx + dy +dz),G/(dx
+ dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz),938.28/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
1.1998578848809383445875560276978 * 10^51 (-1/(dx + dy +dz),0/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz), .511/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy
28722.600151171579743008314436886(0/(dx + dy +dz),0/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 0/(dx + dy +dz), .2/(dx + dy +dz)- 0/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))\
All properties have been conserved.
This shows the volume of the neutron to be slightly smaller in size to the proton by .0000002%.
This calculator can also be used to find the effects of Dark Energy on the particle in question for a proton you could solve the amount of Dark Energy on the particle on Nucleon, we can find that
Dark Energy has a velocity currently of 54 meters per second using a simple equation E = (1/2)MV^2 , V = 54 m/s . The Mass of the Dark Energy Particles are unknown so I will use a mass of electron or
mass of proton. Giving each section of space a energy of 1.458 Kev outward with the push of Dark Energy if mass of electron or mass of proton it would be 1.313 Mev , now we can write the proton
effected by Dark Energy.
9.9023511969154288921026543960449 * 10^59 (+2/3/(dx + dy +dz),R/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 1.45/(dx + dy +dz),938.28/(dx + dy +dz)- 938.28/(dx + dy +dz),1/2/(dx + dy +dz)- 0/
(dx + dy +dz))) , (+2/3/(dx + dy +dz),B/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 1.45/(dx + dy +dz),938.28/(dx + dy +dz)- 938.28/(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz))) , (-1/3/
(dx + dy +dz),G/(dx + dy +dz),I/(dx + dy +dz), (0/(dx + dy +dz) - 1.45/(dx + dy +dz),938.28/(dx + dy +dz)- 938.28 /(dx + dy +dz),1/2/(dx + dy +dz)- 0/(dx + dy +dz)))
Now the Proton is displaying the expansion of Dark Energy upon the Proton.
It has been shown that this graphing tool can be used to graph anything that is contained with the universe using the information about its dimensions, so this test has been concluded about the
graphing equation as successful, but I wanted to note that (dx^2 + dy^2 +dz^2) = (Planck State)^2 being R^2 in Planck lengths which is why the dimensions are divided by (dx + dy +dz) and that the
Planck state( C ) data is used being the dimensions that the field is over being the Complex Manifold. The manifold of space (Euclidean Space) is being used as (dx + dy +dz) which can also be (dx' +
dy' +dz') if you wanted to directly start to use special relativity (Makowski space) on it where as the Field dimensions are from Quantum field theory to be put over the manifold which is a type of
quantum gravity.
Next will be a proof of the big equation which will take longer to test which will give a ds^2 value based on a complex system which can be used with the graphing equation to graph the actual state
of the entire universe exactly without error based on a complex set of 18 variables or kept in its natural state for a ds^2 value which is a Grand Unified Field equation that takes in account the
Strong Nuclear Force, Weak Nuclear Force, Gravity and Electromagnetism all in one equation yielding E8 Killing Vectors. This Metric takes in account General Relativity, Special Relativity, Quantum
Mechanics, and Quantum Field Theory to arrive at the solution in Killing Vectors which are then placed in Minkowski space.
∇'(x,y,z,t,ω[s],ω[p,]M,I,k,φ,S,X,Z,μ,Y,q,a,β) = (d^2/((ħ^2 /(2E[rest]/C^2)) ∑^3[a =1] (d^2/d((C^2/E[rest])∑^N[i = 1 ]M[i]R[i])^2) + (1/2)∑^3[a,][β = 1 ]μ[a][β](P[a ]- Π[a])(P[β][ ]- Π[β]) + U - (ħ^
2/2)∑^3N-6[s=1](d^2/dq^2) + V)((|(Log[(DgDaDψDφ-W)](((2ħGC^2))R[s] - (1/4)F^a[μ][v]F^a^μv + i(ψ-bar)γ^μ(((L[ghost QE ] - gf^abc(δ^μ (c-bar)^a)A[μ]^bc^c) / (c-bar)^aδ^μc^a)[ + ig(1/2)]τW[μ] + ig'(1/2)
YB[μ])ψ^i +(ψ-bar)^i[L]V[ij]φψ^j[r] + (a[ji]) - (μ^2((φ-Dagger)φ) + λ((φ-Dagger)φ)^2)/-(((L[ghost QE ] - gf^abc(δ^μ (c-bar)^a)A[μ]^bc^c) / (c-bar)^aδ^μc^a)[ + ig(1/2)]τW[μ] + ig'(1/2)YB[μ])^2)|)-e^2
S(r,t)/h)) - ((E[rest]/C^2)ω[s]((((8πGT[ab]/C^4) + Λg[ab ] - R[ab]) * g[ab]^-1))^1/2 + (S/ (((3G(E[rest]/C^2))/2C^2R[s]^3)(R[p]V[p]) + (GI[s]/C^2R[s]^3)((3R[p]/R[s]^2)(ω[p ]R[p]) -ω[p] ))))R[s]^2/2))
) / (ħ^2/2(E[rest]/C^2))))^1/2(((1-(((2(E[rest]/C^2)G / R[s]) - (I[s]ω[s]((((8πGT[ab]/C^4) +[ ]Λg[ab ] - R[ab]) * g[ab]^-1))^1/2 + (S/(((3G(E[rest]/C^2))/2C^2R[s]^3)(R[p]V[p]) + (GI[s]/C^2R[s]^3)((3R
[p]/R[s]^2)(ω[p ]R[p]) -ω[p] )))))/2(E[rest]/C^2))+ (((8πG/3)((g/(2π)^3)∫(((E[relativistic]^2 - E[rest]^2 / C^2)^ + ((A[r](X) + (E[Nucleon binding SNF][ ]ε[0 ]μ[0 ]/m[u]) - A[r](X^Z^±)/Z) / m[u])^2)^
(^1/2)(1/e^((ERelativistic - ^μchemical)^/TMatter^)±1)(ħω[s ] + ħω[s]) - ((k[s]C^2)/ R[s]^2) + (((8πGT[ab]/C^4) + Λg[ab ] - R[ab]) * g[ab]^-1))^1/2(ΔKiloparsec)))^2/(C^2)))^1/2)
(d^2/∇') - (Ctp)^2 = ds^2
One solved solution for this equation already is for ∇' being d2/dx'2 + d2/dy'2 + d2/dz'2 , The original solution for the equation was LGhost QE Which states that Quantum Entanglement is the same as
creating a wormhole between two spaces or universes, and that theoretically if you did quantum entanglement on matter between universes you can transmit matter just like is often done across space
during standard Quantum Entanglement experiments.
I am changing the (dx,dy,dz) parameters to display a special relativistic 4 current, now including the evolution of the state over time and not just in a static point.
Luniverse = (∇Charge,∇Color,∇flavour,∇gravity - ∇Dark Energy) , ∇' (x,y,z)= d'(x,y,z)∇ , d(x,y,z)' = d(x,y,z) (1-(V(x,y,z)^2 /C^2))^1/2 , E(x,y,z) = (1/2)MV(x,y,z)^2
This shows the parameters as a function of kinetic energy in a direction or velocity in a direction now, now space properly dilates in the presence of energy at a given time giving a value for L that
is special relativistic. The original equation was relativistic however this equation was not.
L'universe = (∇'Charge,∇'Color,∇'flavour,∇'gravity - ∇'Dark Energy)
The time coordinate can be ignored but I am still doing the A.I. analysis of it anyways, which shows that our analysis of Dark Energy and Gravity are Valid with (C+V, C-V)
This is the proof that the space contraction equation does not interfere with the "Higher Dimensions" such as gravity or dark energy or charge.
However this does prove that it takes the dilation effect upon dimension x upon Q or space upon charge that the space is changing however not charge.
Edited by Vmedvil5
• 2 years later...
I recently attempted to put the "big equation" into wolfram alpha and the computation engine could not compute the equation as it is too large. The Result I recieved from wolfram alpha is in the
picture below.
It seems the technology does not exist yet to compute the "Big Equation" that I wrote.
The Computation by wolfram alpha for the time space of the "big equation" is computable though which the result is in the link below.
(d^2/∇') - (Ctp)^2 = ds^2
Wolfram Alpha link = (d^2/∇') - (Ct)^2 = s^2 - Wolfram|Alpha (wolframalpha.com)
Edited by Vmedvil
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Contagious sets in expanders
We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least r active
neighbors, where r > 1 is the activation threshold. A contagious set is a set whose activation results with the entire graph being active. Given a graph G, let m(G, r) be the minimal size of a
contagious set. It is known that for every d-regular or nearly d-regular graph on n vertices, m(G,r) ≤ O(nr/d). We consider such graphs that additionally have expansion properties, parameterized by
the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion properties imply that m(G,2) ≤ O(n/d^2) (and more generally, m(G,r) ≤ O(n/d^r
(r-1))). In addition, we demonstrate that rather weak assumptions on the girth and/or the spectral gap suffice in order to imply that m(G,2) ≤ O(n log d/d^2). For example, we show this for graphs of
girth at least 7, and for graphs with λ(G) ≤ (1-ε)d, provided the graph has no 4-cycles. Our results are algorithmic entailing simple and efficient algorithms for selecting contagious sets.
Original language English
Title of host publication Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
Pages 1953-1987
Number of pages 35
Edition January
ISBN (Electronic) 9781611973747
State Published - 22 Dec 2014
Event 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States
Duration: 4 Jan 2015 → 6 Jan 2015
Publication series
Name Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Number January
Volume 2015-January
Conference 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
Country/Territory United States
City San Diego
Period 4/01/15 → 6/01/15
All Science Journal Classification (ASJC) codes
• Software
• General Mathematics
Dive into the research topics of 'Contagious sets in expanders'. Together they form a unique fingerprint.
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Transcendental Numbers - Definition, Symbol, and List with Examples
A transcendental number is a real or complex number that is not a root of any non-zero polynomial equation with rational coefficients.
Thus, if we cannot express a number as the solution to an algebraic equation like ${a_{n}x^{n}+a_{n-3}x^{n-1}+\ldots +a_{1}x+a_{0}=0}$, where the coefficients ${a_{0},a_{1},\ldots a_{n}}$ are
rational numbers, then those are transcend numbers.
The symbols ℝ ∖ 𝔸 or ℝ – 𝔸 are often used to denote real transcendental numbers, where ℝ is the set of algebraic numbers.
Transcendental numbers are irrational, but the converse is not always true, as all irrational numbers are not transcendental.
For example,
x^2 – 2 = 0, which has a root x = ${\sqrt{2}}$, is an irrational and algebraic number but not a transcendental number.
Joseph Liouville introduced the first transcendental number, the Liouville constant, in 1844. The Liouville constant number is irrational but is not a root of any polynomial equation.
With Venn Diagram
The following Venn diagram shows the transcendental numbers in the number system:
Examples of Common Transcendental Numbers
π (Pi) = 3.1415 …: It is known for calculating the area and circumference of a circle. Ferdinand von Lindemann proved that π is not the root of any polynomial equation with rational coefficients,
meaning it is impossible to square the circle exactly.
e (Euler’s Number)= 2.718 …: It is the base of the natural logarithm. Charles Hermite proved the transcendence of e in 1873. The number e is crucial in calculus, particularly in studying exponential
growth and decay processes.
Here is a list of a few more transcendental numbers.
• Euler’s constant, gamma = 0.577215 … = ${\lim _{n\rightarrow \infty }\left( 1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots +\dfrac{1}{n}-\ln \left( n\right) \right)}$
• Catalan’s constant, G = ${\sum ^{\infty }_{k=0}\dfrac{\left( -1\right) ^{k}}{\left( 2k+1\right) ^{2}}}$
• Liouville’s number = 0.110001000000000000000001000 …
• Chaitin’s constant = the probability that a random algorithm halts
• Chapernowne’s number = 0.12345678910111213141516171819202122232425…
How Many Transcendental Numbers are there?
There are uncountable such transcendental numbers. Since polynomials with rational coefficients are countable, and each polynomial has only a finite number of zeroes, it follows that algebraic
numbers are also countable.
Difference Between Transcendental and Algebraic Numbers
What is then an algebraic number?
When we have a polynomial equation like x^2 – 2 = 0, whose coefficients are rational, then x is an algebraic number.
Algebraic numbers, like 0, 1, 2, ${\dfrac{1}{2}}$, ${\sqrt{2}}$, are roots of polynomials with integer coefficients.
For example,
f(x) = x has root x = 0
f(x) = x – 1 has root x = 1
f(x) = x + 1 has root x = -1
f(x) = 2x + 1 has root x = ${-\dfrac{1}{2}}$
f(x) = x^2 – 2 has a root x = ${\sqrt{2}}$
Thus, all integers, rational numbers, and some irrational numbers (such as ${\sqrt{2}}$) are algebraic. Some polynomial f(x) with integer coefficients exists for these algebraic numbers where f(a) =
Again, transcendental numbers do not satisfy any such polynomial equation. For a transcendental number ‘t,’ there is no f(t) where f(t)=0.
For example,
${2^{\sqrt{3}}}$ is transcendental, which means there is no polynomial f(x) with integer coefficients for which ${f\left( 2^{\sqrt{3}}\right) =0}$
Thus, it is unusual to find a number that is not algebraic. However, a lot of such numbers exist.
Moreover, algebraic and transcendental together combine to form a set of real numbers. Since algebraic numbers can be arranged in one-to-one correspondence with whole numbers, they are countable.
Also, real numbers are uncountable, and thus, transcendental numbers must be uncountable.
Hence, many more Transcendental numbers exist than algebraic numbers, and most real numbers are transcendental. This argument also applies to complex numbers.
Proving e^α and a^b Transcendental
Proving a number to be transcendental involves showing that it does not satisfy any algebraic equation with rational coefficients. The proofs typically use complex analysis, number theory, and
e^α by Lindemann–Weierstrass Theorem
The Lindemann–Weierstrass Theorem states that if α[1],α[2], …, α[n] are distinct non-zero algebraic numbers, then the numbers ${e^{\alpha _{1}},e^{\alpha _{2}},\ldots ,e^{\alpha _{n}}}$ are linearly
independent over the field algebraic numbers.
Thus, ${e^{\alpha _{1}},e^{\alpha _{2}},\ldots ,e^{\alpha _{n}}}$ are transcendental.
a^b by Gelfond–Schneider Theorem
The Gelfond–Schneider Theorem states that if a and b are algebraic numbers, with a ≠ 0, 1, and b is irrational, then a^b is transcendental.
This is used to prove the transcendence of numbers like ${2^{\sqrt{2}}}$ and ${\sqrt{2}^{\sqrt{2}}}$
Transcendental Function is Not Algebraic
Just as a transcendental number, a transcendental function is also not algebraic. It is a function that cannot be constructed from elementary functions and their inverses in a finite number of steps.
An example is the sine function, sin(x).
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4,038,187 research outputs found
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the
notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge
in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded
tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of
functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids
representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the
notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge
in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded
tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of
functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids
representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio
In this paper we consider ways to alleviate negative estimated depth for the inverse depth parameterisation of bearing-only SLAM. This problem, which can arise even if the beacons are far from the
platform, can cause catastrophic failure of the filter.We consider three strategies to overcome this difficulty: applying inequality constraints, the use of truncated second order filters, and a
reparameterisation using the negative logarithm of depth. We show that both a simple inequality method and the use of truncated second order filters are succesful. However, the most robust peformance
is achieved using the negative log parameterisation. ©2008 IEEE
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Question ID - 156772 | SaraNextGen Top Answer
Hot water at $95^{\circ} \mathrm{C}$ is sent through a countercurrent tube-in-tube heat exchanger with cold water entering at $25^{\circ} \mathrm{C}$. Hot/cold water specific heat capacity is $4.2 \
mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$. Flow rates of hot and cold water are 2.7 and $4.1 \mathrm{~kg} \mathrm{~min}^{-1}$, respectively. Overall heat transfer coefficient is $55 \mathrm{~W}
\mathrm{~m}^{-2} \mathrm{~K}^{-1}$ and the area of heat transfer is $5 \mathrm{~m}^{2}$. Cold water outlet temperature from the heat exchanger in ${ }^{\circ} \mathrm{C}$ will be________________
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Physics of Reality – 6: Entanglement, Wormholes, and Firewalls by Charles Phelan
Physics of Reality – 6
Quantum Entanglement, Wormholes, and the Firewall Problem
by Charles Phelan
[Intro about Charles at Part 4]
Part 1
In the previous two articles, we reviewed Leonard Susskind's mind-blowing theory of Black Hole Complementarity (BHC), which tells us that two completely contradictory observations are both true for
their respective observers. We also noted that BHC is purely theoretical, not yet settled science. In this article, we'll examine a recent challenge to the theory of BHC called the “firewall
paradox,” as well as Susskind's brilliant response to it.
Let's start with a quick review on Black Hole Complementarity. At the center of a black hole is a singularity, where matter has been squeezed to an infinitesimally small point. The gravitational
force of a singularity is so strong that nothing can escape it, not even light itself. Hence the phrase “black hole,” originally coined by physicist John Wheeler. The point of no return where an
object falling toward the black hole can no longer escape is called the event horizon.
According to BHC as well as Einstein's Equivalence Principle, our intrepid explorer Alice will experience nothing unusual as she passes through the event horizon. She will only become human
“spaghetti” when she hits the singularity later, which could take anywhere from less than a second to many years, depending on the size of the black hole.
Meanwhile, Alice's partner Bob has remained behind to observe what happens to her, and he sees a completely different picture. Bob sees Alice move toward the event horizon but never reach it due to
the time dilation effect described by Einstein's Theory of Relativity. Both stories are true for their respective observers, with the catch being that they cannot communicate with one another (i.e.
pass information back and forth).
This lands us squarely in an observer-dependent universe, and leads to reality-bending ambiguities in the bargain. For example, a particle may be either a microscopically small object or a smeared
out probability distribution that covers the entire event horizon of an enormous black hole, with both views being “true” depending on the observer's frame of reference!
Clearly, if BHC is valid, then it forces us to change our thinking about an objective reality “out there” that is reliably the same for all observers. While Susskind's BHC proposal was initially not
widely accepted, after advances in String Theory and other related subfields of physics, it did come to be accepted by most physicists. There remain vocal critics, of course, and papers challenging
the math and logic behind Susskind's theory.
The most significant attack came in 2012, when four physicists (Almheiri, Marolf, Polchinski, and Sully) published a paper proposing a solution to a perceived inconsistency in BHC theory. The
solution proposed was called the “firewall phenomenon,” and the proposal became known as the “AMPS firewall,” AMPS being an acronym for the authors' last names.
In order to understand the AMPS firewall, we first need to discuss the concept of quantum entanglement. When pairs of particles are entangled, there can be no independent description of the separate
particles. Essentially, quantum entanglement tells us that such particle pairs are a whole and must be described as a system, rather than as separate entities. While this may not seem like such a big
deal, it leads to some astonishing conclusions that completely defy logic. For example, measurements on entangled particle pairs are always correlated, no matter how far apart physically the pairs
have been separated. If we measure the “spin” property of a particle, it's counterpart will always measure with an opposite spin, even if the entangled particles are split to opposite sides of the
universe before they are measured!
At first glance, quantum entanglement appears to violate Einstein's discovery that nothing whatsoever can exceed the speed of light. If information is somehow transmitted between a pair of entangled
particles instantaneously, that would constitute a violation of General Relativity, a theory that has been proven over and over again via astronomical and other physical observations. In fact,
Einstein, together with his colleagues Boris Podolosky and Nathan Rosen, published a paper critical of QM theory that became known as the EPR paradox. The popular phrase used by Einstein to describe
such instantaneous transmission was “spooky action at a distance.”
Another term for this is nonlocality, where particles are considered as a whole rather than a discrete system of two separate entities. It's a proven fact of physics that Einstein was wrong about
this issue. Nonlocality has been successfully demonstrated in entangled particle pairs, beginning with the ground-breaking experimental work of Alain Aspect in 1982, and continuing with many
independent confirmations since. At least within the context of quantum particle pairs, the universe is indeed nonlocal.
Is nonlocality not suggestive of nonduality as conceived by Advaita? To be precise, we must note that “quantum entanglement” refers specifically to the description of a theoretical particle-pair, and
we should not leap to conclusions that “QM proves nonduality.” That caveat notwithstanding, nonlocality does look like one of Maya's magic tricks. Measure the spin of a particle, and its companion
will collapse to the other spin state, even if the two particles are in a condition of space-like separation that would entail transmission of information faster than light-speed. Due to QM being
based on probabilities rather than certainties, all this happens without violating General Relativity. Nothing can travel faster than light, and yet particle pairs are apparently linked across vast
stretches of spacetime. Such phenomena appear to show us the "footprints" of nonduality, stamped squarely on such paradoxes as wave-particle duality and quantum nonlocality.
Returning to the AMPS paper's rebuttal to Black Hole Complementarity, the authors showed that there is a potential flaw in the theory. To understand this criticism, we first need to know that highly
entangled particle pairs may only be party to “monogamous” entanglements. This means that one particle is entangled with another, and cannot be the subject of multiple entanglements. Yet there are at
least two potential entanglements if all the assumptions inherent in BHC are maintained.
Recalling our previous discussion on information loss in black holes, we know information is reflected off the event horizon in the form of Hawking radiation. So we have information falling into the
black hole entangled with information coming out of it, and also entangled with prior Hawking radiation particles, leaving one entanglement too many. Their solution demonstrated that a break of
entanglement at the event horizon would lead to a buildup of energy, hence the term “firewall.”
The AMPS paper skillfully argued that some key assumption must give way to resolve the inconsistency of multiple entanglements. Joe Polchinski, the “P” in AMPS, argued that the Equivalence Principle
must fail at the event horizon, and that instead of “No Drama” scenario experienced by Alice, she will burn up when she hits the wall of energy there. The AMPS paper, if correct, would resolve the
apparent inconsistency in BHC where Bob and Alice experience contradictory stories that are both true. And it would do so by breaking the entanglement of particles at the event horizon. In effect,
the authors were saying there is no “inside” to a black hole, as nothing can get past the event horizon.
How did Susskind respond? Working with Juan Maldacena, a fellow physicist from the Institute for Advanced Studies at Princeton, Susskind published a paper titled, “Cool Horizons for Entangled Black
Holes,” in which he proposed an answer to the firewall problem as proposed by AMPS. The short version of their response was:
ER = EPR
This means that entangled pairs are linked through wormholes that tunnel outside the event horizon, thus resolving the apparent identification of one too many entanglements by AMPS.
The above explanation is very compact, so let's break it down and elaborate a bit further. We've already touched on EPR above, the Einstein-Podolsky-Rosen paradox of non-locality, or “spooky action
at a distance.” The “ER” in ER=EPR stands for Einstein-Rosen bridges, and refers to a paper the two men published about wormholes connecting black holes across distant regions of space.
Einstein and his colleagues did not link the two concepts, but Susskind and Maldacena did. The idea is that entangled particles (EPR) are connected by wormhole bridges with their counterparts outside
the event horizon (ER). One of the their key arguments was that the AMPS authors had assumed there could be no connection between space inside and outside the event horizon. The ER = EPR solution
provides for particles inside the event horizon to remain entangled with their counterparts in the cloud of Hawking radiation particles previously leaving the black hole. Thus there is no need to
break entanglement at the event horizon, and therefore no firewall. This is why Susskind and Maldacena used the phrase “cool horizons” in the title of their paper. ER = EPR preserves the “No Drama”
interpretation of BHC, Alice feels nothing at the event horizon, and she still turns into spaghetti when she hits the singularity later.
As so often happens with cutting edge research, however, this proposal implies much more significant possibilities than a resolution to the AMPS firewall paradox. The ER = EPR theory points to a deep
connectedness underlying all of spacetime. We have already seen this with nonlocal entanglement, and also with the wormhole idea. This approach says that ER and EPR are two sides of the same
phenomenon, the same thing viewed from different angles. This is precisely why there is an equals sign in the equation! And the “spooky action at a distance” that gave Einstein fits may actually be
what literally stitches together spacetime. As Maldacena put it, “... the solid and reliable structure of spacetime is due to the ghostly features of entanglement.” Further, the long-term physics
project of uniting the conflicting theories of gravity between General Relativity and Quantum Mechanics may be facilitated by approaches such as ER = EPR.
As we have seen over and over again in these articles, modern physics is converging to an understanding of reality as observer-dependent. It seems we are coming closer to a scientific view that
supports the ancient Advaita perspective that “perception creates the world.” We will never be able to say that science or physics has “proved” nonduality conclusively, simply because any such
knowledge or proof must always be less than the Whole. But it certainly appears that we are starting to understanding some of Maya's tricks. Look out to the extremes of the universe, at both macro
and micro levels, and it's possible to notice some of the rough edges. Quantum foam that allows something to come from nothing, entangled particles, wormholes, and black holes are some of those
It seems there is no limit to the bizarreness of some of the new proposals in physics. Could science actually prove that the entire empirical universe is nothing more than a snake-in-the-rope
illusion, as Adi Shankara advised us 1,200 years ago? Perhaps so!
(To Continue .... Physics of Reality - 7
In the next article, we'll talk about the Holographic Principle, which says that we are living in a hologram).
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Table 78.1 summarizes the options in the PROC SEQDESIGN statement.
Table 78.1 Summary of PROC SEQDESIGN Options
Option Description
Design Parameters
ALTREF= Specifies the alternative reference
BOUNDARYSCALE= Specifies the statistic scale for the boundary
MAXINFO= Specifies the maximum information level
Table Output
ERRSPEND Displays the cumulative error spending at each stage
PSS Displays powers and expected sample sizes
STOPPROB Displays expected cumulative stopping probabilities
Graphics Output
PLOTS=ASN Displays the expected sample numbers plot
PLOTS=BOUNDARY Displays the detailed boundary plot
PLOTS=COMBINEDBOUNDARY Displays the combined boundary plot
PLOTS=ERRSPEND Displays the error spending plot
PLOTS=POWER Displays the powers plot
By default, the SEQDESIGN procedure displays tables of design information, method information, and boundary information for each specified design. If the ODS GRAPHICS ON statement is specified, it
also displays a detailed boundary plot.
In addition, you can use output options to display output tables such as expected cumulative stopping probability at each stage under various hypothetical references. If the ODS GRAPHICS ON statement
is specified, you can also use output options to display plots such as powers and expected sample sizes under various hypothetical references.
The following options can be used in the PROC SEQDESIGN statement to derive boundary values for all sequential designs in the procedure. They are listed in alphabetical order.
ALTREF= <( <LOWER=> <UPPER=> )>
specifies the alternative reference—that is, the hypothetical reference under the alternative hypothesis at which the power is computed. The LOWER= and UPPER= options are applicable only for a
two-sided design with different lower and upper alternative references.
For a one-sided design,
The specification of the ALTREF= option depends on the hypothesis used in the clinical trial. For example, suppose the null hypothesis
If the ALTREF= option is not specified, the alternative reference
Note that if the SAMPLESIZE statement is specified with a two-sided design, the sample sizes derived by using the lower and upper alternatives might be different. If
BOUNDARYSCALE=MLE | SCORE | STDZ | PVALUE
BSCALE=MLE | SCORE | STDZ | PVALUE
specifies the scale for the statistic that is displayed in the boundary table and boundary plots. The keywords MLE, SCORE, STDZ, and PVALUE correspond to the boundary with the maximum likelihood
estimate scale, the score statistic scale, the standardized normal
With the BOUNDARYSCALE=MLE or BOUNDARYSCALE=SCORE option, the maximum information must be either explicitly specified with the MAXINFO= option or derived in the SEQDESIGN procedure to provide the
necessary information level at each stage to compute the boundary values. See the section Boundary Scales for a detailed description of the statistic scale for the boundary values.
Note that for a two-sided design, the
specifies the maximum information level for the design. If the MAXINFO=option is specified and the alternative reference is either specified explicitly with the ALTREF= option or derived from the
SAMPLESIZE statement, then the Type I and Type II error probability levels cannot be met simultaneously. In this case, the ALPHA= option in the DESIGN statement is applicable only with the
BOUNDARYKEY=ALPHA option (which is the default) in the DESIGN statement, and the Type II error probability
Table Output Options
The following options can be used in the PROC SEQDESIGN statement to display addition table output. They are listed in alphabetical order.
PSS <( CREF= numbers )>
displays powers and expected sample sizes under various hypothetical references, where the numbers
For a one-sided design, the power and expected sample sizes under hypotheses
For a two-sided design, the power and expected sample sizes under hypotheses
Note that for a symmetric two-sided design, only the power and expected sample sizes under hypotheses Type I and Type II Errors for a detailed description of the power computation. See the
section Powers and Expected Sample Sizes for a detailed description of the expected sample size computation.
STOPPROB <( CREF= numbers )>
displays expected cumulative stopping probabilities under various hypothetical references, where the numbers
For a one-sided design, expected cumulative stopping probabilities at each stage under hypotheses
For a two-sided design, expected cumulative stopping probabilities at each stage under hypotheses
Graphics Output Options
This section describes the options for using ODS Graphics with the SEQDESIGN procedure to create plots. To request these graphs, you must specify the ODS GRAPHICS ON statement in addition to the
following options in the PROC SEQDESIGN statement. For more information about the ODS GRAPHICS statement, see Chapter 21, Statistical Graphics Using ODS.
The following options can be used in the PROC SEQDESIGN statement to display graphs with ODS Graphics. They are listed in alphabetical order.
PLOTS <( ONLY )> <= plot-request>
PLOTS <( ONLY )> <= ( plot-request < ...plot-request> ) >
specifies options that control the details of the plots. The default is PLOTS=BOUNDARY. The global plot option ONLY suppresses the default plots and displays only plots specifically requested.
The plot request options are as follows.
produces all appropriate plots.
ASN <( CREF= numbers )>
displays a plot of the average sample numbers (expected sample sizes for nonsurvival data or expected numbers of events for survival data) under various hypothetical references, where the numbers
For a one-sided design, expected sample numbers under hypotheses
For a two-sided design, expected sample numbers under hypotheses
BOUNDARY <( HSCALE=INFO | SAMPLESIZE ) >
displays a plot of the resulting sequential boundaries with the acceptance and rejection regions for each design. Either the information level (HSCALE=INFO) or the sample size (HSCALE=SAMPLESIZE)
is displayed on the horizontal axis. If the maximum information is not available for the design, the information in percentage of its corresponding fixed-sample design are used in the plot. The
stage number for each stage is displayed inside the plot. The default is HSCALE=INFO.
If the HSCALE=SAMPLESIZE option is specified. the SAMPLESIZE statement must also be specified. The options MODEL=INPUTNEVENTS, MODEL=TWOSAMPLESURVIVAL, and MODEL=PHREG in the SAMPLESIZE statement
indicate survival data. For a sample that does not contain survival data, the sample size at each stage is displayed on the horizontal axis. For survival data, the number of events is displayed
on the horizontal axis at each stage. The critical values for the corresponding fixed-sample design are also displayed in the plot.
COMBINEDBOUNDARY <( HSCALE=INFO | SAMPLESIZE | STAGE ) >
displays a plot of the resulting sequential boundaries for all designs simultaneously. You can display the information level (HSCALE=INFO), the sample size (HSCALE=SAMPLESIZE), or the stage
number (HSCALE=STAGE) on the horizontal axis. The default is HSCALE=INFO. With HSCALE=INFO, if the maximum information is not available for the design, then the information in percentage of its
corresponding fixed-sample design is used in the plot.
If the HSCALE=SAMPLESIZE option is specified, the SAMPLESIZE statement must also be specified. The options MODEL=INPUTNEVENTS, MODEL=TWOSAMPLESURVIVAL, and MODEL=PHREG in the SAMPLESIZE statement
indicate survival data. For a sample that does not contain survival data, the sample size at each stage is displayed on the horizontal axis. For survival data, the number of events is displayed
on the horizontal axis at each stage.
ERRSPEND <( HSCALE=INFO | STAGE ) >
displays a plot of the error spending for all sequential boundaries in the designs simultaneously. You can display the information level (HSCALE=INFO) or the stage number (HSCALE=STAGE) on the
horizontal axis. With HSCALE=INFO, the information fractions are used in the plot. The default is HSCALE=STAGE.
suppresses all plots.
POWER <( CREF= numbers ) >
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Cosmology at the Millennium
8.1 Testing Inflation + CDM in the Precision Era
As we look forward to the abundance (avalanche!) of high-quality observations that will test Inflation + CDM, we have to make sure the predictions of the theory match the precision of the data. In so
doing, CDM + Inflation becomes a theory with ten or more parameters. For cosmologists, this is a bit daunting, as it may seem that a ten-parameter theory can be made to fit any set of observations.
This will not be the case when one has the quality and quantity of data that are coming. The standard model of particle physics offers an excellent example: it is a 19-parameter theory, and because
of the high quality of data from experiments at high-energy accelerators and other facilities, it has been rigorously tested, with parameters measured to a precision of better than 1% in some cases.
In fact, the ten parameters of CDM + Inflation are an opportunity rather than a curse: Because the parameters depend upon the underlying inflationary model and fundamental aspects of the Universe, we
have the very real possibility of learning much about the Universe, inflation, and perhaps fundamental physics. The ten parameters can be split into two groups: cosmological and dark matter.
Cosmological Parameters
1. h, the Hubble constant in units of 100km s^-1 Mpc^-1.
2. [B]h^2, the baryon density.
3. n, the power-law index of the scalar density perturbations. CMB measurements indicate n = 1.1 ± 0.2; n = 1 corresponds to scale-invariant density perturbations. Several popular inflationary
models predict n
4. dn / dlnk, ``running'' of the scalar index with comoving scale (k = wavenumber). Inflationary models predict a value of ^-3) or smaller.
5. S, the overall amplitude squared of density perturbations, quantified by their contribution to the variance of the quadrupole CMB anisotropy.
6. T, the overall amplitude squared of gravitational waves, quantified by their contribution to the variance of the quadrupole CMB anisotropy. Note, the COBE normalization determines T + S (see
7. n[T], the power-law index of the gravitational wave spectrum. Scale invariance corresponds to n[T] = 0; for inflation, n[T] is given by -T / 7S.
Dark-matter Parameters
1. [], the fraction of critical density in neutrinos (= [i] m[[i]] / 90h^2). While the hot dark matter theory of structure formation is not viable, it is possible that a small fraction of the matter
density exists in the form of neutrinos.
2. [X], the fraction of critical density in a smooth component of unknown composition and negative pressure (w[X] w[X] = -1).
3. g[*], the quantity that counts the number of ultra-relativistic degrees of freedom (at late times). The standard cosmology/standard model of particle physics predicts g[*] = 3.3626 (photons in
the CMB + 3 massless neutrino species with temperature (4/11)^1/3 times that of the photons). The amount of radiation controls when the Universe becomes matter-dominated and thus affects the
present spectrum of density fluctuations.
The parameters involving density and gravitational-wave perturbations depend directly upon the inflationary potential. In particular, they can be expressed in terms of the potential and its first two
derivatives (see e.g., Lidsey et al. 1997):
where V(d / dV[*] is the value of the scalar potential when the present horizon scale crossed outside the horizon during inflation.
As particle physicists can testify, testing a ten (or more) parameter theory is a long, but potentially rewarding process. To begin, one has to test the basic tenets and consistency of the underlying
theory. Only then, can one proceed to take full advantage of the data to precisely measure parameters of the theory. The importance of establishing a theoretical framework is illustrated by
measurements of the number of light neutrino species derived from the decay width of the Z^0 boson: N[] = 3.07 ± 0.12 (not assuming the correctness of the standard model); N[] = 2.994 ± 0.012
(assuming the correctness of the standard model).
In the present case, the putative theoretical framework is Inflation + CDM, and its basic tenets are: a flat, critical density Universe; a nearly scale-invariant spectrum of Gaussian density
perturbations; and a stochastic background of gravitational waves. The first two predictions are much more amenable to testing, by a combination of CMB anisotropy and large-scale structure
measurements. For example, a flat Universe with Gaussian curvature perturbations implies a multipole power spectrum of well defined acoustic peaks, beginning at l Fig. 5). In addition, there are
consistency tests: comparison of the precise BBN determination of the baryon density with that derived from CMB anisotropy; an accounting of the dark matter and dark energy by gravitational lensing;
SNe1a measurements of acceleration; and comparison of the different determinations of the Hubble constant. Once the correctness and consistency of Inflation + CDM has been verified - assuming it is -
one can zero in on the remaining parameters (subset of the list above) and hope to determine them with precision.
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Special Expression Formats
Expressions can be combined in special ways to produce concise program statements that nevertheless have tremendous power and flexibility. The special expression formats are
• WHERE expression format
• assignment expression format
• grouped expression format
• case expression format
How To Use the WHERE Expression Format
A WHERE subcommand can be appended to a value expression to control the evaluation of that value expression. The syntax is
valexpression WHERE logic_expression
where valexpression is any value expression.
logic_expression is any logic expression.
If the expression is complex, it must be enclosed in parentheses.
If the WHERE condition evaluates to $True, the value expression is evaluated. If the WHERE condition evaluates to $False, the value expression is not evaluated, and the entire expression is
considered to be $Null.
The WHERE expression format is frequently used with the case expression format, the assignment expression format, or with aggregate functions.
Examples of the WHERE Expression Format
(Salary – 20000) where Salary > 20000
(Salary – 20000) is evaluated only if the condition is met. The entire expression format takes on the value of the arithmetic expression if the condition is met; otherwise, it is considered to be
$Null (valueless). The following example
let i = { 1 where Age < 10,
2 where Age between 10 and 30,
4 where Age > 50, 3}
assigns a value to a variable based on Age. The following example
compute Employees evaluate (let TotSal =
$total(Salary where LastName = “Smith”))
totals the salaries of all employees named Smith.
How To Use the Assignment Expression Format
A LET command can be used within an expression to assign a value to a target as the target is being used in a value or logic expression. The syntax is
(LET target=expression)
Where target is a global or local variable, a parameter, a field or a form field
where expression is any value expression.
Assignment expressions enable you to use fewer commands to achieve a particular result or to perform calculations within set-processing commands.
Examples of the Assignment Expression Format
while (let vCount=vCount-1)>0
Tests that vCount (which is first set to vCount – 1) is greater than 0.
let vTriple=(let vSingle=5)*3
Sets vSingle to 5 and vTriple to vSingle * 3.
(let vCount=vCount+1) where Processed=”yes”
Increments VCount if Processed is “yes”.
report footing (let TotalSal=$total(Salary)*1000)
Sets the variable TotalSal to 1000 times the running total of Salary, displaying TotalSal in the report footing. After the latter command has been executed (i.e., when the report is finished), the
application retains access to the contents of TotalSal.
How To Use the Grouped Expression Format
Individual value expressions may be placed in parenthesis and separated from one another by commas to produce “grouped expressions”. The syntax is
(expression «,expr»)
where expression is any value expression. If the expression is complex, it must be enclosed in parentheses.
All the expressions are evaluated. The entire expression takes on the value of the last expression in the group.
Grouped expressions are useful for performing multiple assignments within one logic expression.
Example of the Grouped Expression Format
while ((let vTotal=0), (let vAverage=0), (let vCount=vCount-1)) > 0
let var1=((let var2=2), (let var3=3), 5-4)
How To Use the Case Expression Format
Case expressions provide a powerful method of dealing with evaluations that are conditional on the results obtained. The syntax is
{expression«, expr»}
where expression is any value expression, including another case expression.
The expressions within the braces are evaluated from left to right. The entire expression takes on the value of the first expression that is not $Null (valueless).
Each case within the braces is evaluated from left to right until one case yields a value that is not $Null. The case expression returns this value.
Case expressions can minimize the code that is required to produce certain conditional results as shown in the following example:
let Status = {“tall” where Height > 6, “short”}
if Height > 6
let Status = “tall”
let Status = “short”
Case expressions can ensure that a value is provided in situations where a field or form field alone may sometimes be $Null, as shown in the following example:
let Salary = {fAddEmps.Salary, 0}
Case expressions can also be used to determine the action to be taken by certain commands, as shown in the following example:
break 1
{$year(InvDate) + 1 where $month(InvDate) >= 5,
heading …
Examples of the Case Expression Format
let Status={“tall” where Height>6, “short”}
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detail line “Employee Number: ” {EmpNum, “N/A”}
See Also
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By Joseph Mazur
Review by Ardian Gill
I HAD JUST FINISHED READING a New York Times article on electroconvulsive therapy (ECT) when I picked up Joseph Mazur’s Fluke. On the very first page of the introduction, Mazur describes how his
uncle endured sessions of ECT after suffering a high school boxing injury. It seemed a fitting entry into a book with the subtitle “The Maths and Myths of Coincidence.”
Ten years old at the time, Mazur speculated on his uncle’s injury: What if he had been sick that day and hadn’t gone to school? What if his opponent had been sick? What if Jack had knocked out the
other guy first? Speculations like these led to the present volume, in which he tries to apply mathematics to determine the likelihood of unlikely events. In the process, he breaks the law of large
numbers into “the law of weak large numbers” and “the law of truly large numbers,” reminding us that “if there is any likelihood that something could happen, no matter how small, it’s bound to happen
at some time.”
The first part of the book is the most entertaining. Here he tells 10 stories of unlikely occurrences, with intriguing titles like “The Girl from Petrovka,” “The Albino Taxi Driver,” “Plum Pudding,”
and “Abe Lincoln’s Dreams.” My favorites are (surprise) the two about books. In one of these, the novelist Anne Parrish of Colorado Springs, Colo. is in Paris having lunch with her husband at Deux
Magots. She leaves him to finish his wine and strolls off to browse the bookstalls along the Seine. To her delight, she comes across a book she had loved as a child in Colorado, Jack Frost and Other
Stories. She buys it and rushes back to show it to her husband, reciting her tale of having owned a book of that title as she was growing up. What a coincidence! He leafs through a few pages, then
hands it back, showing her the flyleaf where her name and Colorado address have been written in her childish hand. Was this a coincidence, a fluke, or serendipity? The definitions of these terms are
given early on and used to distinguish events he analyzes later in the book. Other words tossed in are “improbability,” “seriality,” and “synchronicity.”
In part two, the author provides the math he believes we will need for that analysis. Early on he asks, “We have an actuarial grip on the odds of a person living past the age of x years, so what is
obstructing us from measuring the odds of a miracle?” To provide the tools for such measurement, the next five chapters deal with probability in a way that I frankly found tedious and poorly
proofread. (On a single page there are two formulae with plus signs where multiplication is required.) At some length we are treated to illustrations of long series of coin flips, dice tosses, cards
dealt, etc. to arrive eventually at a normal curve and to illustrate the earlier maxim that improbable events happen. To quote R.A. Fisher, “The ‘one chance in a million’ will undoubtedly occur …
however surprised we may be that it should occur to us.” Coincidentally—or is it a fluke?—Fisher is mentioned in a chapter devoted to suspected cause vs. coincidence or merely correlation. Fisher was
skeptical that smoking caused cancer, “If … it were possible to infer that smoking cigarettes is a cause of this disease, it would be possible to infer on exactly similar grounds that inhaling
cigarette smoke was a practice of considerable prophylactic value in preventing the disease, for the practice is rarer among patients with cancer.” (Too bad he’s not around to examine the climate
change “hoax” assertion.)
The most entertaining of these chapters are the ones dealing with famous scientists—Pascal, Bernoulli, Cardano, Poincaré, and Galileo. Among the less familiar vignettes were descriptions of Pascal’s
Triangle (each number is the sum of the two numbers above it) and the Galton Board, where balls are dropped on the ends of a set of rods and land randomly, ultimately forming a normal curve.
In part three, we return to the stories of coincidence that opened the book, but first they are classified: Unexpectedly finding what is searched for; Forgotten objects unexpectedly turning up from
the past in faraway places, etc. The Paris story of the Seine bookstall and the Jack Frost book of Anne Parrish’s childhood has a quote from literary critic and all-around wag Alexander Woollcott,
who says that he was “more than half disposed to believe that when the oblivious Anne Parrish crossed the street to that bookstall, somewhere in fathomless space a star chuckled—chuckled and skipped
in its course.”
But we are not dealing with stars but probabilities, and here they are (see the book for the rationale): likelihood of Parrish traveling to Paris that summer: 0.1; likelihood of visiting the
bookstalls: 0.3; likelihood that the book would be there: 0.01. He writes, “So the probability of such a story happening would be something like p = 0.1 x 0.3 x 0.01=0.0003, the odds in favour of its
happening are 3331 to 1.” He admits that “hidden variables” (the subject of an earlier chapter) might change the probability but not “by more than 1/10,000, and therefore the odds … remain slightly
better than the odds being dealt a poker hand of four of a kind.”
Another of the listed coincidences is that of a woman hailing a cab in Miami and having the same albino taxi driver she had in Chicago three years earlier. Skipping over the details, here is the
final calculation of the probability. “That puts the woman’s chances … greater than 20/15,327=0.013 and less than 40/15,327=0.026. The odds are between 75 to 1 and 36 to 1. Not bad!”
Well, not good, I’m afraid, since the decimal point is in the wrong place.
The author admits defeat when he looks at “Plum Pudding” and a couple of other listed coincidences, but he goes to town on the last, which is that of one Joan Ginther winning four Texas lotteries
over a period of 18 years. Rather than tackling that scenario directly, he walks through some preliminary calculations, concluding that the odds of a person (not a designated person) in the United
States (not just Texas) will win two lotteries in five years are better than even. Moving from the United States to the whole world, he derives the probability of a person winning the jackpot twice
in two years as 0.97, or near certainty. From this we return to Ginther’s four wins over 18 years and are told that “in that time span, the probability of some person winning four jackpots somewhere
in the world is extremely close to 1.”
Part 4 is titled “Head Scratchers” and begins with a confession that “[t]here are coincidences that completely escape analysis.” So we move from the “Math” of the subtitle to the “Myth.” Five essays
deal with DNA, the accidental discovery of X-rays, the acts of a rogue trader, psychic powers or extrasensory perception (ESP), and the intentional coincidences in literature.
The DNA essay reveals the uncertainty inherent in DNA matches, either through questionable lab work (as was convincing to the O.J. Simpson jury) or, more scientifically, through the analysis of too
few short DNA sequences called “short tandem repeats,” or STRs. The legal profession has settled on the number 13 as the minimum for matching STRs to be accepted in evidence. The essay is interesting
for the information on the human genome, for the history of actual court cases involving DNA, and for the personal stories of victims and exonerations of the innocent. The pages on exonerations are
replete with statistics on incarcerations: “[A]pproximately 2.3 million persons are being held in federal and state prisons. … [N]early 37% … are African American. … It means that 1 in 100 American
adults are behind bars, leaving 1 child in 28 with a parent behind bars.” These depressing figures seem to have triggered the author’s emotions, for they lead to what might be considered a rant: “[T]
he United States leads the world in per capita documented incarceration rates, behind Russia and Rwanda … with one-quarter of the world’s total prison population.” These numbers touch a nerve until
they are called into question by some less-thanprecise recitations: “In 2014, 515 of the 1,409 exonerations in the United States were of prisoners on death row. That’s a staggering rate of 16.8
percent!” Huh? Solving for the denominator of that percent figure, we get 3,070, which happens to be the number of inmates then on death row—a fact we learn a page later. The balance of the paragraph
is confounding. “Since 1976 there have been 1,386 executions in the United States and just 144 exonerations of death row verdicts. That means since 1976 almost 1 in 10 people should not have been
sent to death row.”
Regardless of one’s feelings toward executions and our prison policies in general, one wonders what all this has to do with flukes and coincidence,aside from a DNA match occurring because of matching
too few STRs.
“Discovery” is the title of an absorbing chapter where chance, flukes, and coincidence contributed to science. X-rays were discovered when Wilhelm Röntgen accidentally put his hand between a cathode
ray tube and the screen where the rays were focused. The name comes from the letter “X” being the classic symbol for an unknown. Mazur describes other contributions of chance to such discoveries as
penicillin (mold on a Petri dish), quinine (a malaria victim drinking water from a spot near a cinchona tree), insulin (flies on a dog’s pancreas), etc.—all echoing Louis Pasteur’s famous quotation,
“Chance favors the prepared mind.”
The chapter titled “Risk” attributes coincidence to market events—London Underground bombings, for example, causing the Financial Times Stock Exchange 100 (the FTSE) to fall precipitously. That’s not
the coincidence, but a trade made by one Jerome Kerviel in France was highly profitable because of the bombings. Unfortunately, he went on trading, with disastrous results. With the subject of risk,
Mazur has wandered into actuarial territory, and it is unfamiliar terrain. About earthquakes and other rare events, “We know they will happen, but not when.” Isn’t that what insurance is all about?
He quite rightly agrees that “we can assess the risk that the worst might happen.”
The psychic power chapter chronicles events that are attributed to ESP. He points out that such events are treated with appropriate skepticism in most circles, yet one anecdote tells of a Brazilian
court accepting a “letter from the dead” as evidence sufficient to acquit the murderers. A good summary of so-called psychic events is stated as, “We see coincidence as events that are mysteriously
fated by some deeply significant design. We suspect a correlation between two complex phenomena. The real problem is that we naturally tend to make connections were [sic] there are none.” A personal
example: At the moment of the death of one of my sisters, her son-in-law, sitting in a neighboring room looking at a family album, saw a page flip over by itself to a photo of her already deceased
husband; another sister, sitting in her yard, had a chickadee alight on her shoulder.
Two other quotations will suffice to end this review:
“Our thoughts and actions seem to be primed by chains of experiences, and yet fate has its odd ways of stepping in to tweak and perturb the balance.”
“I leave it to you to judge it a coincidence, a fluke, or divine intervention.”
ARDIAN GILL, a member of the Academy and a fellow of the Society of Actuaries, is the author of The River Is Mine, a historical novel, and of a recently published children’s book, The Blue Moose. In
his past life, Gill was chief actuary at the Mutual Life Insurance Co. of New York, a partner at Tillinghast, and co-founder of Gill and Roeser Inc., reinsurance intermediaries.
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(PDF) The speed of gravity: An observation on galaxy motions
... Because all stars/stellar are orbiting around the center of the galaxy [10,17], just as the planets are orbiting around the Sun. And, only the Newtonian theory of orbit perturbation is valid to
understand the celestial orbit [10,16]. Therefore, the orbit of a star around the center in a galaxy only can be described with the Newtonian theory of orbit perturbation: ...
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Volume of a Cylinder - Club Z! Tutoring
Volume of a Cylinder
Volume Of Cylinder Definition
The volume of a cylinder means the space inside the cylinder that can hold a specific amount of material quantity. In short, it is the capacity of the cylinder to hold solid, liquid, or gas as it
relates to volume. In order to understand the Capacity of Cylinder, it must be in a three-dimensional shape. It is not possible to measure the volume of two-dimensional cylinder.
Cylinder’s volume is given by the formula, ?r^2h, where r is the radius of the circular base and h is the height of the cylinder.
Definition of a Cylinder
A cylinder is a three-dimensional shape that consists of two parallel bases linked by a curved surface. The line that passes from the center or joining the centers of two circular bases is referred
to as the axis of the cylinder
What is the unit for the volume of a cylinder?
The volume of a cylinder is typically measured in what are called cubic units. These cubic units are generally displayed as follows: cubic centimeters (cm^3), cubic meters (m^3), cubic feet (ft^3).
Types of Cylinders as it relates to Volume:
1. Oblique cylinder – A cylinder where both sides lean over the base at an angle that is not equal to a right angle, or 90 degrees.
2. Elliptic cylinder – It is a cylinder whose bases are ellipses. Ellipses are curved lines.
3. Right circular hollow cylinder – It has the shape of a right circular cylinder. However, it does not have closed circles.
4. Right Cylinder – A cylinder that has a closed circular surface having two parallel bases on both the ends and whose elements are perpendicular to its base.
Volume of a Right Circular Cylinder
The base of a right circular cylinder is a circle, which means the the area of the circle of radius ‘r’ is ?r^2. Thus, the volume (V) of a right circular cylinder, using the above formula, is,
V = ?r^2h
Based on the above formula,
• ‘r’ is the radius of the base of the cylinder
• ‘h’ is the height of the cylinder
• ? is a constant whose value is either 3.142.
Therefore, the volume of cylinder varies based on it height and directly varies with the square of its radius.
Volume of an Oblique Cylinder
In order to calculate the volume of an Oblique Cylinder, you will utilize the same formula as that of a right cylinder. The Volume (V) of an oblique cylinder whose base radius is ‘r’ and whose
height is ‘h’ is as follows: V = ?r^2h
Volume of an Elliptic Cylinder
It is important to note that an ellipse has two radii. If we know that the area of an ellipse whose radii are ‘a’ and ‘b’ is ?ab. The volume of an elliptic cylinder is: V = ?abh
To break this down further,
• ‘a’ and ‘b’ are the radii of the base of the cylinder.
• ‘h’ is the height of the cylinder.
• ? is a constant whose value is 3.142.
Volume of a Right Circular Hollow Cylinder
A right circular cylinder is a shape that has two right circular cylinders – one inside the other. In order to obtain the volume, you must subtract the volume inside cylinder from the volume of the
outside cylinder. Therefore, the volume (V) formula for a Right Circular Hollow Cylinder is as follows: ,
V = ?(R^2 – r^2)h
• ‘R’ is the base radius of the outside cylinder.
• ‘r’ is the base radius of the inside cylinder.
• ‘h’ is the height of the cylinder.
• ? is a constant whose value is 3.142.
Steps to calculate the volume of a cylinder
The below steps will provide you with process of finding the volume of a cylinder.
Step 1: Identify the type of cylinder given to you in the question or in real life. Is the cylinder a right cylinder, an oblique cylinder, Right Circular Hollow Cylinder, or an elliptic cylinder?
Step 2: Once you have decided on the type of cylinder, you will need to figure out the formula that is best used for the associated Cylinder.
☆ Right Cylinder: V = ?r^2h
☆ Oblique Cylinder: V = ?r^2h
☆ Elliptic Cylinder: V = ?abh
☆ Right Circular Hollow Cylinder: V = ?(R^2 – r^2)h
Step 3: Make sure you verify that you have all dimensions needed, and they are utilizing the same unit of measurement.
Step 4: Simply match the units with their appropriate variable in the formula, and solve the formula.
Below you will find a simple example for a Right Cylinder utilizing Area and Height.
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ball mill design
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There are many different designs and styles of ball mill liners. As with grinding balls local economics and ultimately operating costs determine the best design and material to use. The initial
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In this paper, the design method of three chamber ball mill is introduced. Comchambered with the design of Φ × 13m threechamber ball mill, the design process of ball mill is described in detail.
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The details of the ball mill motor are as follows. Power = kW or HP and the speed is 343 rpm. Load calculations (prior to failure analysis) The ball mill can experience failure based on the
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Euler Number
Introduction to the Euler Number used in fluid mechanics.
The Euler Number is a dimensionless value used for analyzing fluid flow dynamics problems where the pressure difference between two points is important. The Euler Number can be interpreted as a
measure of the ratio of the pressure forces to the inertial forces.
The Euler Number can be expressed as
Eu = p / (ρ v^2) (1)
Eu = Euler number
p = pressure (Pa)
ρ = density (kg/m^3)
v = fluid flow velocity (m/s)
The pressure difference is often used
Eu = dp / (ρ v^2) (2)
dp = differential pressure (Pa)
• Note! - a perfect frictionless flow corresponds to that the Euler number equals 1
The combination below is called the pressure coefficient
pressure coefficient = dp / (1/2 ρ v^2) (3)
A special version of the Euler Number is in general referred to as the Cavitation Number.
Related Topics
• The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
Related Documents
• An introduction and definition of the Cavitation Number.
• Physical and chemical dimensionless quantities - Reynolds number, Euler, Nusselt, and Prandtl number - and many more.
• The Pressure Coefficient is the ratio of pressure forces to inertial forces.
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How far is that Star?
How far is that Star!
with Stargazing Tenerife team, your guide to the dark skies of Tenerife.
I want to start by wishing you all a very happy new year. I’ve been asked over this Christmas period by so many people, “But how do you know its that far away” that I have had to dig out my course
notes and today I hope to explain just how we do measure distances in observational cosmology. The only answer I could give on the spot was geometry for the closest objects and then we use the
Distance Ladder. On closer look by a colleague, he had stumbled across parallax and this is the first rung of our ladder.
Well there are lots of ways to measure distances in astronomy, but most only work over a limited range of distances. So, typically, we must use a whole series of different measures, starting with
close-in measures and using them to calibrate the further measures, which in turn calibrate measures that work at even greater distances.
Rung 1: Parallax
The simplest method is parallax. As the Earth moves around the Sun, it causes our viewing angle to slightly shift, making nearby stars appear to move relative to far-distant background stars. The
parallax angle is defined as the difference in apparent position you would get if the Earth moved by one astronomical unit (the Earth moves by two astronomical units as it goes around the Sun, so the
observed angular shift will be twice the parallax angle). If the parallax angle of a star is one arcsecond, it lies at a distance of one parsec (by definition).
Unfortunately, space is very big – there are no stars within a parsec of the Earth. So, parallax angles are typically very small and hard to measure. Even the best ground-based telescopes can only
measure parallaxes to a handful of nearby stars. Space missions like Hipparchos extended it out a little further, crucially including the nearest star cluster, the Hyades.
” Now of course, this is the arguably hardest thing in observational
astronomy, getting distances. You want to get a fight going between two astronomers, ask
them how did you measure the distance? “
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• Astrophotography Courses
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• Stargazing Pods overnight stay
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Square coordinates
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
Square Coordinates printable sheet
Play with this interactivity until you can draw tilted squares confidently.
Formulate and describe a general instruction for the construction of a square when you are given two adjacent corners.
Formulate and describe a general instruction for the construction of a square when you are given two opposite corners.
Decide whether any of the collections of points below form a square.
If so, which ones?
Can you do this without plotting the points on a grid?
1. $(8,3)$, $(7,8)$, $(2,7)$, $(3,2)$
2. $(3,3)$, $(7,4)$, $(8,8)$, $(4,7)$
3. $(16,19)$, $(18,22)$, $(21,20)$, $(19,17)$
4. $(4,20)$, $(21,19)$, $(20,2)$, $(3,3)$
Explain how you decided.
You may now enjoy playing Square It.
Getting Started
You might like to play Square It before working on this problem.
Can you describe how many steps left/right and how many steps up/down you need to take to go from one corner of a square to the next? And to the next?
Student Solutions
Hannah tackled the first part of this problem. Here are the squares she drew:
In all of these, the side AB could either be from the bottom corner to the right, or from the left corner to the top. Do you see why?
Hannah went on to complete the arrow notation for these squares. She's taken the side AB to be from the bottom up to the right.
(a) $A \: 1 \rightarrow +1 \uparrow B \: 1 \leftarrow + 1 \uparrow C \:1 \leftarrow + 1 \downarrow D \:1 \rightarrow + 1 \downarrow A$
(b) $A \: 2 \rightarrow + 1 \uparrow B \: 1 \leftarrow + 2 \uparrow C \: 2 \leftarrow + 1 \downarrow D \: 1 \rightarrow + 2 \downarrow A$
(c) $A \: 3 \rightarrow + 1 \uparrow B \: 1 \leftarrow + 3 \uparrow C \: 3 \leftarrow + 1 \downarrow D \: 1 \rightarrow + 3 \downarrow A$
(d) $A \: 2 \rightarrow + 2 \uparrow B \: 2 \leftarrow + 2 \uparrow C \: 2 \leftarrow + 2 \downarrow D \: 2 \rightarrow + 2 \downarrow A$
(e) $A \: 3 \rightarrow + 2 \uparrow B \: 2 \leftarrow + 3 \uparrow C \: 3 \leftarrow + 2 \downarrow D \: 2 \rightarrow + 3 \downarrow A$
Good work, Hannah!
Ahmed gave us instructions to construct a square where you are given one of its sides:
Suppose you're given the side $A \: a \rightarrow + b \: \uparrow B$. (I noticed that $\downarrow$ could be written as $-\uparrow$ and $\leftarrow$ could be written as $-\rightarrow$, so $a$ and $b$
could be negative, but that doesn't matter. This makes it a bit easier, as we only have two sorts of arrow then.) Then the square is either $$A \: a \rightarrow + b \uparrow B \: -b \rightarrow +a \
uparrow C \: -a \rightarrow -b \uparrow D \: b \rightarrow -a \uparrow A$$ or $$A \: a \rightarrow + b \uparrow B \: b \rightarrow -a \uparrow C \: -a \rightarrow - b \uparrow D \: -b \rightarrow + a
\uparrow A.$$
Well done, Ahmed, especially for spotting that there are two possible squares.
Ahmed then worked out which of the collections of points could be a square.
1. (8,3), (7,8), (2,7), (3,2). In arrow notation, this would be $A \: -1 \rightarrow +5 \uparrow B \: -5 \rightarrow - 1 \uparrow C \: 1 \rightarrow - 5 \uparrow D \: 5 \rightarrow + 1 \uparrow A$.
This is of the first form, with $a=-1$ and $b=5$. So this is a square.
2. (3,3), (7,4), (8,8), (4,7). In arrow notation, this would be $A \: 4 \rightarrow + 1 \uparrow B \: 1 \rightarrow + 4 \uparrow C \: -4 \rightarrow - 1 \uparrow D \: -1 \rightarrow - 4 \uparrow D$.
This isn't of either form, so the points don't form a square.
3. (16,19), (18,22), (21,20), (19,17). In arrow notation, this would be $A \: 2 \rightarrow + 3 \uparrow B \: 3 \rightarrow - 2 \uparrow C \: -2 \rightarrow - 3 \uparrow D \: -3 \rightarrow + 2 \
uparrow A$. This is of the second form, with $a=2$ and $b=3$, so the points form a square.
4. (4,20), (21,19), (20,2), (3,3). In arrow notation, this would be $A \: 17 \rightarrow - 1 \uparrow B \: -1 \rightarrow - 17 \uparrow C \: -17 \rightarrow + 1\uparrow D \: 1 \rightarrow + 17\
uparrow A$. This is also of the second form, with $a=17$ and $b=-1$, so the points form a square.
Teachers' Resources
Why do this problem?
This problem emphasises to students that squares don't just exist in their usual orientation. It goes well with the game Square It
The context offers an ideal opportunity to challenge students to visualise relationships between coordinates.
The interactivity could also be useful when introducing Pythagoras' Theorem and when working on the gradients of perpendicular lines.
Possible approach
Display the interactivity. Ask for volunteers to move the corners to make a different square.
Fix a couple of corners and challenge students to complete the square.
Offer them a chance to see the coordinates.
Choose two points where all the coordinates are either all even or all odd. Challenge students to complete the square with these as opposite vertices.
Set students to work in pairs (ideally at computers) practising making squares until they can answer the key questions below. Suggest they make a variety of squares of different sizes and note down
the sets of coordinates of their completed squares.
This could lead to a plenary discussion or, when appropriate, challenge students to work away from the computer on the final questions in the problem. This sheet provides further practice with tilted
squares, but without reference to their co-ordinates.
Key questions
How can we construct a square when we are given two adjacent corners?
How can we construct a square when we are given two opposite corners?
How can we construct a square when we are given the centre and one corner?
If we are given four points, how can we tell if they will make a square or not?
Can we do all this without plotting the points?
Possible extension
How does this extend to rectangles?
If you are given three coordinates, work out how to determine if they will define a right angle.
Draw squares with as many different areas (under 50) as is possible. Which areas are possible and which aren't?
Possible support
Provide students with a handout of some tilted squares drawn on squared paper and ask them to box each one in with a non-tilted square. Students can look at the four right angled triangles which
result around the edge and will see that these triangles are congruent.
Students can answer the last four questions by plotting the points provided and boxing them in to decide whether they make a square.
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Remove Duplicates from an Unsorted Linked list
This post is completed by 1 user
• 0 Add to List
27. Remove Duplicates from an Unsorted Linked list
Objective: Write a program to remove the duplicates from an unsorted linked list
Input Linked List : 1->2->2->4->3->3->2
Output : 1->2->4->3
• Create a Hash Map
• Take two pointers, prevNode and CurrNode.
• PrevNode will point to the head of the linked list and currNode will point to the head.next.
• Now navigate through the linked list.
• Check every node data is present in the Hash Map.
• if yes then delete that node using prevNode and currNode.
• If No, then insert that node data into the Hash Map.
• Return the head of the list
Time Complexity: O(n), Space Complexity: O(n)
Follow Up: If suppose addition buffer is not allowed then we have an option to check every node data against every other node data and if find duplicates, delete that node.
Time Complexity: O(n^2)
Original List : ->1->2->2->3->4->4->2
Updated List: ->1->2->3->4
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Rounding and Scientific Notation
This video from Khan Academy shows gives examples of rounding whole numbers to the nearest ten, hundred and thousand.
Note: All Khan Academy content is available for free at www.khanacademy.org.
This video from Khan Academy shows how to round to the nearest tenth and hundredth using examples.
Note: All Khan Academy content is available for free at www.khanacademy.org.
This video from Khan Academy gives an example of estimating the sum of two whole numbers using rounding.
Note: All Khan Academy content is available for free at www.khanacademy.org.
This video from patrickJMT shows how to estimate the result of some decimal arithmetic operations by rounding.
Note: All patrickJMT content is available for free at www.youtube.com/user/patrickJMT.
Patrick JMT’s Website: www.patrickjmt.com.
Pages 6-7 of these notes from MathCentre discuss rounding a decimal to a certain number of decimal places or significant figures. There are also exercises on page 7 with answers at the end of the
A pdf of this document can be downloaded here.
Interactive Exercises
This NUMBAS exercise lets you practise rounding numbers to various degrees of accuracy.
Please click here to view this NUMBAS question in another window.
Scientific Notation
This video from Khan Academy shows you how to write a given number in decimal representation as one in scientific notation, and vice versa. This covers writing both large and small numbers.
Note: All Khan Academy content is available for free at www.khanacademy.org.
This video from Krista King describes division of numbers written in scientific notation, as well as converting an example of division in decimal notation to one in scientific notation.
Note: All Krista King content is available for free at www.youtube.com/c/Integralcalc.
Krista King’s Website: www.kristakingmath.com .
These notes from MathCentre discuss the ideas behind scientific notation, give some examples, and have instructions on using your calculator when dealing with scientific notation.
A pdf of this document can be downloaded here.
Interactive Exercises
These NUMBAS exercises from TEAM-E let you get some practice converting from decimal to scientific notation and vice versa, and also multiplying and dividing numbers written in scientific notation.
Please click here to view this NUMBAS question in another window.
Please click here to view this NUMBAS question in another window.
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How to pass operator as parameter in Swift?
@mason In Swift, you can pass an operator as a parameter by using the Operator type. This is a special type that represents an operator symbol, such as +, -, *, or /.
Here's an example of a function that takes an operator as a parameter and uses it to perform a calculation:
1 func calculate(num1: Int, num2: Int, op: Operator) -> Int {
2 switch op {
3 case .addition:
4 return num1 + num2
5 case .subtraction:
6 return num1 - num2
7 case .multiplication:
8 return num1 * num2
9 case .division:
10 return num1 / num2
11 }
12 }
In this example, the calculate function takes two integers, num1 and num2, and an operator, op. The function uses a switch statement to determine which operation to perform based on the value of op.
You can call the calculate function like this:
1 let result = calculate(num1: 10, num2: 5, op: .addition)
The result variable will be assigned the value 15, which is the result of adding 10 and 5 together.
Note that the Operator type used in this example is a custom type that you will need to define yourself. It should contain an enumeration of all the different operators that you want to be able to
pass as parameters. Here's an example of how you might define the Operator type:
1 enum Operator {
2 case addition
3 case subtraction
4 case multiplication
5 case division
6 }
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diameter of a circle calculator
A diameter of a circle is defined as a straight line passing from one side to another side through the center point. They do not affect the calculations. How to calculate Diameter of a circle when
radius is given using this online calculator? Visit our other calculators and tools. Now double the radius to get the diameter (Example: 621.12 meters). The equation for diameter of a circle from
circumference is: If written instead in terms of the radius, the diameter is very simple; it's just twice as long: Area is the space contained within the circle's boundaries. Diameter Of A Circle
From Area Calculator, Diameter Of A Circle From Radius Calculator. Below is a diameter calculator, which will compute a circle's radius, circumference, and area if you know the diameter. The radius
is the distance from the Earth and the Sun: 149.6 million km. To use this online calculator for Diameter of a circle when radius is given, enter Radius (r) and hit the calculate button. Sphere volume
to diameter calculator; Sphere surface area to diameter calculator; User Guide. The diameter of a circle calculator uses the following equation: Area of a circle = π * (d/2) 2. Calculates the radius,
diameter and circumference of a circle given the area. By using this website, you agree to our Cookie Policy. It doesn't matter whether you want to find the area of a circle using diameter or radius
- you'll need to use this constant in almost every case. Circumference to Diameter Calculator. What will be the angle between the ends of the arc? Diameter of Circle Enter the diameter of a circle.
Just input the circumference value in the input field and the diameter of a circle from circumference calculator will automatically update you with the circle diameter. The radius is the distance …
Diameter To Circumference Calculator. A circle is a simple closed shape in Euclidean geometry. It's also easy to find from any of the others. A diameter is just two radiuses drawn in opposing
directions from the circle's origin. Enter the diameter of a circle. This tool will calculate the diameter of a circle from the area, and will convert different measurement units for area and
diameter. Symbols. This is also known as the longest chord of the circle. Given any 1 known variable of a circle, calculate the other 3 unknowns. Diameter of Circle. Finally, you can find the
diameter - it is simply double the radius: D = 2 * R = 2 * 14 = 28 cm. Symbols. Symbols. Circle calculator, formula, work with steps, step by step calculation, real world and practice problems to
learn how to find the area, circumference & diameter of a circle in inches, feet, meters, centimeters and millimeters. Calculate the arc length according to the formula above: L = r * θ = 15 * π/4 =
11.78 cm. In this model, the Sun is at the centre of the circle, and the Earth's orbit is the circumference. The formula used to calculate circle radius is: r = ø / 2. Diameter of Circle This is the
diameter of a circle that corresponds to the specified area. To run the calculator, hit the 'Calculate Circle Dimensions' button when you have entered the known diameter. The calculator will then
determine the length of the arc. Compute the area using these units: square mils, square inches, square feet, square yards, square miles, acres, hectares, square millimeters, square centimeters,
square meters, and square kilometers. An online calculator to calculate the circle diameter based on the area. It is also called as the longest chord of the circle. The diameter of a circle is the
length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle as long as they are exactly 180 degrees
apart. Do you know a different dimension? The formula for working out the circumference of a circle is: Circumference of circle = π x Diameter of circle This is typically written as C = πd. This
tells us that the circumference of the circle is three “and a bit” times as long as the diameter. It doesn't matter whether you want to find the area of a circle using diameter or radius - you'll
need to use this constant in almost every case. Enter the circle area, diameter, or circumference and it will solve for the other two. Just input the circumference value in the input field and the
diameter of a circle from circumference calculator will automatically update you with the circle diameter. Where: π is approximately equal to 3.14. Use the central angle calculator to find arc
length. This means you can use this tool to do other calculations such as diameter to circumference, radius to area, radius to cicumference, area to radius, area to circumference etc. Twice the
length of a circle… C = Circle circumference; π = Pi = 3.14159… ø = Circle diameter; Diameter of Circle. Diameter to Circumference Calculator Circumference or perimeter of a circle is defined as the
distance around it. Diameter = C/π = Note: Most other calculators and programs use scientific notation for too large and too small results. The formula used to calculate circle area is: A = π x (ø/
2) 2. The next equations show how you can instead find the diameter from them. The central angle is a quarter of a circle: 360° / 4 = 90°. Using the Radius Calculator You can enter the radius and
then compute diameter and circumference in mils, inches, feet, yards, miles, millimeters, centimeters, meters and … To use the arc length calculator, simply enter the central angle and the radius
into the top two boxes. Decide on the radius of your circle. The diameter of a circle is known as the straight line segment which passes through the center of the circle. DQYDJ may be compensated by
our advertising and affiliate partners if you make purchases through links. A = Circle area; π = Pi = 3.14159… ø = Circle diameter; Diameter of Circle. Circle Calculator - area, circumference,
diameter It can be also used to calculate other parameters of a circle such as diameter, radius and area. The area of a circle is calculated as A = πr². A circle is a simple closed shape in Euclidean
geometry. The diameter of a circle calculator uses the following equation: Area of a circle = π * (d/2) 2. The formula used to calculate the circle diameter is: ø = 2 x √(A / π) Symbols. Formula.
Area of a circle = π * r 2. Conveniently, it is twice as long as the radius of a circle. Visual on the figure below: π is, of course, the famous mathematical constant, equal to about 3.14159, which
was originally defined as the ratio of a circle's circumference to its diameter. And so does this calculator. The central angle is a quarter of a circle: 360° / 4 = 90°. The diameter of a circle is
the straight line passing through the center of the circle. A circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the center. This circle calculator
apart from finding the diameter, area of sector, radius, circumference of the circle, also provides you the individual formulas for the calculation of the same. Area of a circle = π * r 2. Here is
how the Diameter of a circle when radius is given calculation can be explained with given input values … It can also be defined as the longest chord of the circle. For a circle, three lengths most
commonly are applied: The radius – defined above; The diameter – the distance from edge to edge of a circle passing through its origin or center. Use this circle calculator to find the area,
circumference, radius or diameter of a circle. Enter the diameter of a circle. Area of a circle diameter. Circle formulas and geometric shape of a circle. The circumference of a circle is calculated
using the formula: 2 x π x radius, where π is a mathematical constant, equal to about 3.14159.It was originally defined as the ratio of a circle's circumference to its diameter (see second formula
below on why) and appears in many formulas in mathematics, physics, and everyday life. If you draw two opposing line segments from the circle's origin to the edge, you just drew the diameter. A
circle is set of all points in a plane that are at a given distance from the centre. The distance around a circle (i.e) its perimeter gives you the circumference. It is calculated just by multiplying
the diameter of the circle with π value. Units: Note that units of length are shown for convenience. For any circle, if you divide the circumference by the diameter you get pi, an irregular number
usually rounded to 3.14. area S 6digit 10digit 14digit 18digit 22digit 26digit 30digit 34digit 38digit 42digit 46digit 50digit If we are only given the diameter and not the radius we can enter that
instead, though the radius is always half the diameter so it’s not too difficult to calculate. This geometry calculator will take one known circle measurement (area, circumference, diameter, or
radius) and calculate the other three. Enter the diameter of a circle. You can enter the diameter and then compute radius and circumference in mils, inches, feet, yards, miles, millimeters,
centimeters, meters and kilometers. In this equation, "C" represents the circumference of the circle, and "d" represents its diameter. By using this website, you agree to our Cookie Policy.
Conveniently, it is twice as long as the radius of a circle. Plus, unlike other online circle calculators, this calculator will show its work and give a detailed, step-by-step explanation of the
formulas and sequence used to arrive at each result. Circumference of a circle is defined as the distance around it. Enter the circumference which is the total length of the edge around the circle,
if it was straightened out. The diameter of a circle is the distance from edge to edge of a circle passing through its origin or center. ø = Circle diameter; C = Circle circumference; π = Pi =
3.14159… Circumference of Circle. Diameter of Circle. Did you like this tool? A circle is set of all points in a plane that are at a given distance from the centre. How to use the Calculator: Insert
the Diameter to calculate the Radius or input the Radius to find the Diameter; Either insert the Angle of the Arc or insert the length; So, by carrying out either of the two foregoing operations, the
user will be able to find the Arc of a Circle quickly and without any difficulties. (You can also input the diameter into the arc length calculator instead.) Free Circle Diameter calculator -
Calculate circle diameter given equation step-by-step This website uses cookies to ensure you get the best experience. Learn more about pi, or explore hundreds of other calculators addressing
finance, math, … There are three dimensions most often used to describe a circle: Starting from the diameter, you can easily find the other two. Calculate the diameter of a circle, from its area The
radius is the distance from the Earth and the Sun: 149.6 million km. If you know the radius of the circle, double it to get the diameter. This is also known as the longest chord of the circle. Free
Circle Diameter calculator - Calculate circle diameter given equation step-by-step This website uses cookies to ensure you get the best experience. If you draw two opposing line segments from the
circle's origin to the edge, you just drew the diameter. Given any one variable A, C, r or d of a circle you can calculate the other three unknowns. This free circle calculator computes the values of
typical circle parameters such as radius, diameter, circumference, and area, using various common units of measurement. Solve area, diameter, and circumference, circle equations. The constant pi,
designated by the Greek letter π, is the ratio of the circumference to the diameter of a circle. Circle Calculator - area, circumference, diameter r = Circle radius; ø = Circle diameter; Diameter of
Circle. Circumference to Diameter Calculator. The diameter of a circle is known as the straight line segment which passes through the center of the circle. Then, we want to calculate the area of a
part of a circle, expressed by the central angle. Let's say it is equal to 45 degrees, or π/4. Plugging π into your calculator will give you its numerical value, which is a closer approximation of
3.14 or 22/7. Where: π is approximately equal to 3.14. Circumference of a circle is defined as the distance around it. Enter the radius of a circle. Our calculator can also convert the result to
different units. Just enter the area value in the input field and click calculate in this Diameter of a circle from area calculator to find the result. Area of a circle diameter. An online geometry
calculator to calculate the diameter of a circle based on the circumference. An online geometry calculator to calculate the diameter of a circle based on the circumference. Calculate diameter of a
circle from circumference of 20.72 cm. Diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. ø = Circle diameter; r =
Circle radius; Radius of Circle. The radius is the distance between the centre and any point on the outer edge of a circle. The formula used to calculate the circle diameter is: ø = C / π. This is
the diameter of a circle … It is calculated just by multiplying the diameter of the circle with π value. This is a great starting point. Calculate the area, circumference, radius and diameter of
circles. The formula for the area of a circle is π x radius2, but the diameter of the circle is d = 2 x r 2, so another way to write it is π x (diameter / 2)2. How to use the Calculator: Insert the
Diameter to calculate the Radius or input the Radius to find the Diameter Either insert the Angle of the Arc or insert the length So, by carrying out either of the two foregoing operations, the user
will be able to find the Arc of … Instead try one of the related circle dimension calculators: The diameter of a circle is the distance from edge to edge of a circle passing through its origin or
center. As with a circle, the longest line segment that connects two points of a sphere through its center is called the diameter, d. The equation for calculating the volume of a sphere is provided
below: See, Diameter Calculator: Compute Dimensions of a Circle, Hours Calculator: See How Many Hours are Between Two Times, Bitcoin Return Calculator with Inflation Adjustment, Net Worth by Age
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The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also passes through the centre of a circle, or any two points on the circle as long as
they are exactly 180 degrees apart. Use the central angle calculator … That is to say, you can find the circumference of a circle just by multiplying the diameter by pi. Diameter of Circle Enter the
diameter of a circle. For example, it can be equal to 15 cm. Circumference of a circle formula. The distance from one side of the circle to the other, going through the center of the circle, is the
diameter. Both definitions are also valid for the diameter … Regardless of this distinction, a ball and a sphere share the same radius, center, and diameter, and the calculation of their volumes is
the same. Use our circumference calculator to find the radius when you only have the circumference or area of a circle. Infact, all the numbers are shown as soon as you type any one number. In this
model, the Sun is at the centre of the circle, and the Earth's orbit is the circumference. A circle is formed by combining a set of all points in a plane that are at a given distance from the centre
point. The full angle is 2π in radians, or 360° in degrees, the latter of which is the more common angle unit. Plus, unlike other online circle calculators, this calculator will show its work and
give a detailed, step-by-step explanation of … Dimensions of a Circle. r. Symbols. The diameter of a circle is the length of a straight line drawn between two points on a circle where the line also
passes through the centre of a circle, or any two points on the circle as long as they are exactly 180 degrees apart. Find A, C, r and d of a circle. Dimensions of a Circle This geometry calculator
will take one known circle measurement (area, circumference, diameter, or radius) and calculate the other three. Ø = C / π its diameter any straight line passing from side... That units of length are
shown as soon as you type any one variable a, C, and. When radius is: ø = circle radius is: ø = 2 x √ a. Three unknowns simply enter the circumference or area of a circle longest chord of the circle
calculator... Know the diameter of the circle to the edge around the circle with value! Diameter you get the diameter of a circle based on the circumference by the Greek letter π, is ratio. Volume to
diameter calculator ; User Guide million km circle such as diameter, you just drew the diameter a... Distance around it, the latter of which is the straight line which... The angle between the centre
of the circle area is: a = circle area is a! Circle radius is the circumference will then determine the length of the arc our. Distance between the centre of the arc length calculator, which is a
closer approximation 3.14... Will solve for the other 3 unknowns and affiliate partners if you draw two line! Centre and any diameter of a circle calculator on the circumference common angle unit
radius and area you! R or d of a circle given the area other three unknowns circle... The calculator, simply enter the circumference which is the circumference of circle. Three dimensions most often
used to calculate diameter of a circle calculator uses following... Purchases through links and a bit ” times as long as the diameter C, or! Π into your calculator will give you its numerical value,
which will a... Area if you make purchases through links the latter of which is a of. Circle: 360° / 4 = 90° diameter is just two radiuses drawn in opposing directions the! Be equal to 15 cm circle
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Tessellation Classification
Many people fascinated by tessellation artworks want to understand their structure. Some differences stand out — tiles have different amounts of rotation, different numbers of neighbors, and
sometimes are flipped. It’s natural to wonder how many different structures there are and how to tell them apart.
There are three systems commonly used to classify tessellation artworks — wallpaper groups, Heesch types, and isohedral types — and (at least) three systems used mainly by their creators — Escher
types, Nicolas types, and Mohr types.
wallpaper groups, which classify two-dimensional repeating patterns into 17 categories. These are well-studied, and used across different disciplines. And while they don’t distinguish all
tessellation differences, they are useful for key features like the amounts of rotational and reflective symmetry.
Heesch types he defined capture edge relationships in a tile, and match 28 of the 35 tessellation symmetries used in artworks on this site.
classification system 7 symmetries with an internal mirror, capturing figures with two identical halves. Adding these to the 28 Heesch types we get the 35 symmetries behind artworks on this site.
While Nicolas’s coding system has not been widely adopted, his classification, presentation, and examples are thorough, beautiful, and useful.
It’s natural to wonder how many tilings are possible if all symmetry restrictions are lifted, and in fact mathematicians Branko Grünbaum and Geoffrey Shephard showed that 93 tilings are possible.
Their enumerated tilings are commonly known as isohedral types or IH types, with each identified by a number from 1 to 93. While many IH tilings aren’t used in tessellation art, IH types provide a
standard concise code for an artwork’s underlying symmetry.
Dutch artist M.C. Escher developed his own theory and notation for "regular division of the plane", independently discovering and making artworks with 28 of our 35 symmetries. (Without mathematical
training or use of a computer!) For a detailed discussion including Escher's diagrams see chapter 2 of Doris Schattschneider's fine book M.C. Escher: Visions of Symmetry.
Finally, while finding artworks for this site, developer Rick Mohr found that none of the above classification systems was suited to quickly identifying an artwork’s symmetry. For that purpose he
created Mohr types, defining easily-observable qualities of an artwork and a set of symmetry codes which refer to them.
For a table of our 35 tessellation symmetries and their classifications, see the summary page.
The classification systems described here have different strengths. Here’s a comparison table, showing:
• Adopted — is the system widely used?
• Precise — does each type identify a single tessellation symmetry?
• Concise — do all types have a short code?
• Identify — is there a procedure to build a type code from an artwork's features'?
• Expressive — do the type codes describe easily-observable artwork features?
Wallpaper group ✓ — ✓ — —
Heesch type ✓ ✓ ✓ ✓ —
Isohedral type ✓ ✓ ✓ — —
Escher type — ✓ ✓ — —
Nicolas type — ✓ ✓ — —
Mohr type — ✓ — ✓ ✓
Mathematicians have shown that all two-dimensional repeating patterns can be divided into 17 groups based on their symmetries. The groups are called wallpaper groups, or plane symmetry groups, or
plane crystallographic groups.
Much could be said about these groups, but we'll focus on the parts that are relevant to tessellations and the artworks on this site.
Since every tessellation is a two-dimensional repeating pattern, it will belong to one of the groups. A tessellation's group tells us:
1. the number of rotated tile orientations (and thus the rotation angle)
2. the number of flipped tile orientations
3. whether each tile is symmetrical across a central “mirror”
Let’s see how our 35 tessellation symmetries relate to the 17 symmetry groups.
28 of our tessellation symmetries (the Heesch types) use asymmetric tiles. They belong to seven groups, with relatively understandable names:
• In group p1, tiles all have the same orientation.
• In groups p2, p3, p4, and p6, tessellations have respectively 2-, 3-, 4-, or 6-way rotational symmetry.
• In group pg, tiles have two orientations — one flipped to mirror the other.
• In group pgg, tiles have four orientations. Relative to the base orientation one is rotated, one is flipped, and one is both rotated and flipped (or equivalently, flipped on the other axis).
The remaining 7 of our tessellation symmetries — where tiles are symmetrical across a central mirror — belong to four additional groups.
If these groups were used only to describe our 35 tessellation symmetries they might have been called 1m, 2m, 3m, and 4m (m for "mirror"). But in fact they are called cm, pmg, p31m, and p4g. That’s
because they use a notation created for 3-dimensional crystallography groups (230 of them!), which makes distinctions that aren’t helpful for our single-motif 2-dimensional tessellations. Those codes
are actually abbreviations — for c1m1, p2mg, p31m, and p4gm — which are a bit more understandable, telling the number of orientations and that there is a mirror. But everyone uses the (less
intuitive) short names, so we will as well.
• In group cm (think “1m”), tiles all have the same orientation and are symmetrical across a central mirror.
• In groups pmg, p31m, and p4g (think “2m, 3m, and 4m”), tessellations have respectively 2-, 3-, or 4-way rotational symmetry, and tiles are symmetrical across a central mirror.
So our 35 tessellation symmetries belong to just 11 of the 17 wallpaper groups. The remaining 6 groups are almost never used for single-tile tesselated art because they require tiles with multiple
internal mirrors. And while many familiar figures have one mirror (like a front-view person), few have multiple mirrors. You can see good examples using multiple motifs by Alain Nicolas if you scroll
to the end of this section of his site.
Many people discussing tessellation symmetries use wallpaper groups as a framework — perhaps because they are widely used beyond the small world of tessellations, or perhaps because no other system
for classifying tessellations stands out as an alternative.
But wallpaper groups are incomplete as a framework for understanding tessellation symmetries — mainly because they address only patterns, ignoring tile shape and adjacency. For example, the eight
tessellation symmetries belonging to group pgg are similar in having four different tile orientations with flips on two axes, but their base polygons have four different edge counts and their
alignments vary from linear to clustered:
So while wallpaper groups are useful for understanding some features of tessellation symmetries, the other classification systems discussed on this page provide a more precise framework.
For more on wallpaper groups see
German mathematician Heinrich Heesch proved in 1932 that there are exactly 28 ways to tile a flat surface with asymmetric tiles — that is, where tiles have no internal reflection or rotation and all
edges are shapeable. Each of these 28 “Heesch types” underlies artworks on this site.
Heesch’s notation uses a code for each edge of a tile’s base polygon, based on its role in the tessellation. The codes are:
• T – the edge is translated (slid in a straight line) to align with an opposite edge
• C – the edge is rotated 180° about its midpoint to align with itself
• C[3], C[4], or C[6] – the edge is rotated about an endpoint, to align with its adjacent edge and fit 3, 4, or 6 tiles around that endpoint.
• G – the edge is glide reflected (slid and flipped) to align with an opposite or adjacent edge. (If a tile has two different glide reflections the edge codes are distinguished as G[1] and G[2]).
For more on translation, rotation, and glide reflection see the Symmetry Tutorial.
By convention Heesch codes start with the “simplest” edge (where T < C < C[3] < C[4] < C[6] < G) and proceed clockwise.
For example, in this diagram of Pentagons, rotated and flipped:
• black edges are related by translation
• the blue edge is related to itself by 180° rotation
• red edges are related by glide reflection
so the Heesch code is
In this diagram of Pentagons with 6-way rotation:
• the blue edge is related to itself by 180° rotation
• purple edges are related by 3-way rotation
• green edges are related by 6-way rotation
so the Heesch code is
For more examples you can click the "Heesch" header on any symmetry page to see how the codes relate to a diagram or artwork. Here's part of the page for Pentagons with 4-way rotation, showing
details of the Heesch type along with “Under the Sea” by Henk Wyniger:
Pros and cons
On the plus side a Heesch code is concise, precisely defines a single symmetry, is itself a tessellation recipe, and can be constructed by analyzing an artwork. On the minus side that construction
can be tricky, and the codes themselves are not memorable.
Extended codes for mirror symmetries
Heesch did not consider tiles with an internal mirror. But his system can be extended to handle them, by indicating where in the edge sequence the mirror appears.
The mirror may either intersect an edge midpoint (bisecting the edge) or appear between two edges. Here’s how those are notated on this site:
• When a mirror bisects a translated edge the notation is t instead of T, as in symmetries tCtC, tCCtCC, and tTTtTT.
• When a mirror appears between edges the notation is |, as in symmetries TT|TT, CC|CC, C[3]C[3]|C[3]C[3], and C[4]C[4]|C[4]C[4].
For example, in this diagram of Hexagons (with inner mirror), rotated:
• a vertical mirror bisects the black translated edges
• each of the blue and green edges has a twin across the mirror
• each of the blue and green edges is related to itself by 180° rotation
The extended Heesch code is tCCtCC:
In this diagram of Rhombuses, with inner mirror and 3-way rotation:
• a vertical mirror lies between the edges
• the blue edges are related by reflection across the mirror
• the blue edges are also related by 120° rotation at the sides
The extended Heesch code is C[3]C[3]|C[3]C[3]:
Notation choice
In TesselManiac Kevin Lee also extended Heesch’s notation to handle internal mirrors, with the collaboration of noted tessellation scholar Doris Schattschneider. Why does this site not use their
Note that some mirrored symmetries may be seen as having pairs of either translated edges or glide reflected edges. For example, as shown in the diagrams below, tessellation symmetry Rhombuses, with
inner mirror could be seen as either:
1. TT|TT, a symmetrical version of TTTT (Parallelograms), or
2. G[1]G[1]|G[2]G[2], a symmetrical version of G[1]G[1]G[2]G[2] (Kites, flipped)
Rhombuses, with inner mirror Rhombuses, with inner mirror
TT|TT G[1]G[1]|G[2]G[2]
Parallelograms Kites, flipped
TTTT G[1]G[1]G[2]G[2]
Which notation should we choose? #1 (TT|TT) seems a good choice because, as with TTTT, all tiles have the same orientation. Choosing #2 would seem to suggest that some tiles appear flipped, but none
In TesselManiac #2 was chosen. One consequence is that (in TesselManiac) a mirror never appears between translated edges. That means an asterisk can be used to notate both types of mirror — C*C for a
mirror between two edges and T* for a mirror bisecting an edge.
But since #1 was chosen for this site, a mirror can appear between translated edges. To avoid ambiguity, the two types of mirror are notated differently — T|T for a mirror between two translated
edges and t for a mirror bisecting a translated edge. Using a vertical bar instead of an asterisk avoids ambiguity with the TesselManiac notation and also offers an intuitive visual.
Every tiling on this site is isohedral, meaning it has a single tile shape and has translational symmetry. Translational symmetry means you can slide the entire pattern in a straight line and end up
with an identical pattern. That also means the tiles interlock in a consistent way — a tile edge touches the same part of its neighboring tile everywhere you look.
Mathematicians Branko Grünbaum and Geoffrey Shephard showed that there are 93 different types of isohedral tilings. Artworks for 35 of those are on this site, but artworks for the other 58 are rare —
for a couple of reasons.
cats21 by Makoto Nakamura shows one example.
(Scroll down on his tessellations page for the original.)
The 93 isohedral tilings break down as follows (Schenk, p.40):
• 28 Heesch tilings (supported on this site)
• 20 Heesch tilings with internal symmetry:
□ 7 with one internal mirror (supported on this site)
□ 1 with two internal mirrors (supported by TesselManiac)
□ 12 with internal rotational symmetry
• 24 with some edges shapeable and others straight (unshapeable)
• 21 with all edges straight (unshapeable)
IH type codes
Grünbaum and Shephard assigned each isohedral tiling a number from 1-93, commonly called isohedral types, or IH types. You can see the IH type for each of our tessellation symmetries on the Symmetry
Summary page. Clicking the “IH” column header will sort the symmetries by IH type.
Why did Grünbaum and Shephard choose this particular order for the IH codes? It turns out the types are sorted into groups by their valence — the number of lines meeting at each vertex of a tile,
which could be 3, 4, 6, 8, or 12.
For example: in any tiling based on hexagons three lines meet at each of the hexagon’s six vertices, for a valence of 333333. Since that’s the lowest valence of any tiling, hexagonal tilings come
first in the IH order.
Different tessellation symmetries based on the same underlying grid will have IH numbers that are close to each other. For example, the two tessellation symmetries that use the appealing pentagonal
grid called “Cairo tiling” (whose valence is 33434) have adjacent IH numbers — Pentagons with 2-way flip is IH27 and Pentagons with 4-way rotation is IH28.
Here are some resources for further exploration of the 93 isohedral tilings:
Mohr types are useful for identifying an artwork’s tessellation symmetry, via easily-observable qualities of the artwork. And because the type codes refer directly to those qualities, the codes may
be easier to remember than codes in other systems.
You can identify an artwork’s symmetry with these five questions (and two more for special cases shown later):
Question Answers Codes
3 tri
How many neighbors surround a tile? 4 quad
(The answer identifies the tessellation's base polygon.) 5 pent
6 hex
How many different tile orientations appear in the tessellation? 1, 2, 3, 4, 6 1, 2, 3, 4, 6
Are any tiles flipped? no
yes flip
no alone
Do tiles with the same orientation touch each other? yes, at a point tip
yes, along an edge edge
Are the tiles symmetrical? no
yes sym
Reading a symmetry's Mohr type gives answers to these questions. For example, hex 4 flip edge describes tessellations where tiles have 6 neigbors (so the base polygon is a hexagon), tiles appear in 4
orientations, some tiles are flipped, same-orientation tiles touch along an edge, and tiles are not symmetrical.
Let's see how Mohr types can help identify artwork symmetries. (Many thanks to the artists for permission to use their artworks as examples.)
Number of neighbors and orientations
Answering the first two questions — number of tile neighbors and orientations — is enough to identify 5 of the 35 symmetries.
Counting how many neighbors surround a tile tells us which base polygon underlies the tessellation — three neighbors means a triangle, four neighbors a quadrilateral, five a pentagon, and six a
hexagon. Based on the answer, the symmetry code will begin with tri, quad, pent, or hex.
To count neighbors: pick a tile and follow its border all the way around, counting how many other tiles are adjacent. (Touching at a single point doesn't count, but a very short edge does!)
Then we count how many different tile orientations are present, considering both rotations and flips.
For example:
“2056 dragon” by Yasukiyo Yoshida
Here the symmetry is tri 2 because:
• Each tile has 3 neighbors (tri)
• Tiles appear in 2 orientations
• There are no other “tri 2” symmetries
“Tesselephants” by Guillaume Riesen
Here the symmetry is quad 6 because:
• Each tile has 4 neighbors (quad)
• Tiles appear in 6 orientations
• There are no other “quad 6” symmetries
Next we ask whether you’d have to flip any tiles to align them with each other, which distinguishes another 3 symmetries.
For example: the next artworks both match tri 4, as tiles have 3 neighbors (tri) and 4 orientations. But the first is tri 4 flip (because some tiles are flipped), while the second is tri 4 (because
no tiles are flipped).
For some cases we can distinguish similar symmetries by seeing whether tiles with the same orientation touch each other. There are three possible answers:
• alone if same-orientation tiles don’t touch,
• tip if same-orientation tiles touch at a single point, or
• edge if same-orientation tiles touch along an edge.
This question distinguishes 9 more symmetries.
For example, the next artworks both match quad 2 flip, as tiles have 4 neighbors (quad) and 2 orientations, with some tiles flipped. But the first is quad 2 flip edge because same-orientation tiles
touch along an edge, while the second is quad 2 flip tip because same-orientation tiles touch at a single point.
As another example, the next artworks both match pent 4 flip, as tiles have 5 neighbors (pent) and 4 orientations, with some tiles flipped. But the first is pent 4 flip edge because same-orientation
tiles touch along an edge, while the second is pent 4 flip alone because same-orientation tiles don’t touch.
Mirror Symmetry
14 more symmetries are distinguished by observing whether tiles have two identical halves, making them symmetrical across a central “mirror”.
For example: the next artworks both match hex 2, as tiles have 6 neighbors (hex) and 2 orientations, with no tiles flipped. But the first is hex 2 sym because tiles are symmetrical, while the second
is hex 2 because tiles are not symmetrical.
Special Cases
The first five questions identify 31 symmetries, leaving just 4 — two pairs, distinguished with a question each.
The next two artworks have different symmetries but both match hex 2 flip, as tiles have 6 neighbors (hex) and 2 orientations (some flipped). Adjacency doesn't help, as in both cases same-orientation
tiles touch along an edge.
Notice that each artwork has chains of adjacent same-orientation tiles. One clear difference is the chain direction — horizontal in the first artwork and vertical in the second artwork. While that
difference holds for most examples of these symmetries, there are exceptions so it's not reliable.
So we ask a different question — “To align a chain of same-orientation tiles with an adjacent flipped chain, do the ends or the sides of the chain trade places?” In the first artwork the answer is
“Chain ends trade places” so the symmetry is hex 2 flip ends, while in the second artwork the answer is “Chain sides trade places” so the symmetry is hex 2 flip sides.
Imagine the figures are on a skewer — do you have to trade the skewer's ends or just twirl it?
Here's an example where the figure chains appear on the other axis, with half upside down:
The final two artworks have different symmetries but both fit quad 4 flip alone, as tiles have 4 neighbors (quad) and 4 orientations (some flipped), and same-orientation tiles don't touch (alone).
To distinguish these we can ask “Are all adjacent tiles flipped?” In the first example the answer is “No, some are rotated” so the symmetry is quad 4 flip alone rotate, while in the second example
the answer is “Yes, all are flipped” so the symmetry is quad 4 flip alone flip.
Identifying the symmetry of tessellation artworks can be tricky, but it gets easier with practice!
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Against a global conception of mathematical hinges
Epistemologists have developed a diverse group of theories, known as hinge epistemology, about our epistemic practices that resort to and expand on Wittgenstein's concept of ‘hinges’ in On Certainty.
Within hinge epistemology there is a debate over the epistemic status of hinges. Some hold that hinges are non-epistemic (neither known, justified, nor warranted), while others contend that they are
epistemic. Philosophers on both sides of the debate have often connected this discussion to Wittgenstein's later views on mathematics. Others have directly questioned whether there are mathematical
hinges, and if so, these would be axioms. Here, we give a hinge epistemology account for mathematical practices based on their contextual dynamics. We argue that 1) there are indeed mathematical
hinges (and they are not axioms necessarily), and 2) a given mathematical entity can be used contextually as an epistemic hinge, a non-epistemic hinge, or a non-hinge. We sustain our arguments
exegetically and empirically.
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How Quantum Computing is Enabling Breakthroughs in Chemistry
Note: Mark Jackson is Scientific Lead of Business Development at Cambridge Quantum ComputingÂ
Quantum computing is expected to solve computational questions that cannot be addressed by existing classical computing methods. It is now accepted that the very first discipline that will be greatly
advanced by quantum computers is quantum chemistry.
Quantum Computers
In 1982, the Nobel Prize-winning physicist Richard Feynman observed that simulating and then analyzing molecules was so difficult for a digital computer as to make it impossible for any practical
use. The problem was not that the equations governing such simulations were difficult.
In fact, they were comparatively straightforward, and had already been known for decades. The problem was that most molecules of interest contained hundreds of electrons, and each of these electrons
interacted with every other electron in a quantum mechanical fashion—resulting in millions of interactions that even powerful computers could not handle.
To overcome the quantum nature of the equations, Feynman proposed quantum computers, which perform calculations based on the laws of quantum physics, as the ultimate answer. Unfortunately, such
precise manipulation of individual quantum objects was far from technically possible. The joke for the past 35 years has been that quantum computing is always ten years away.
In the past few years, what was once a distant dream has slowly become a reality. Not only do quantum computers now exist, millions of programs have been executed via the cloud, and useful
applications have started to emerge.
The power of a quantum computer can be roughly estimated by the number of qubits, or quantum bits: each qubit can represent a 1 and 0 state simultaneously. There are a number of promising hardware
approaches to quantum computing, including superconducting, ion trap, and topological. Each has advantages and disadvantages, but superconducting has taken an early lead in terms of scalability.
Google, IBM, and Intel have each used this approach to fabricate quantum processors ranging from 49 to 72 qubits. Qubit quality has also improved.
Chemistry Breakthrough
The breakthrough by scientists at Cambridge Quantum Computing (CQC) and their partners at JSR Corp was the ability to model multi-reference states of molecules. Multi-reference states are often
needed to describe the “excited states” arising when molecules interact.
The reason such modeling is significant is that “classical” digital computers find it virtually impossible to tackle multi-reference states; in many cases, classical computing methods fail not only
quantitatively but also qualitatively in the description of the electronic structure of the molecules.
An outstanding problem—and the one recently solved—is to find ways that a quantum computer can run calculations efficiently and with the required chemical accuracy to make a difference in the
real world. The program was run on IBM’s 20 qubit processor, as both CQC and JSR are members of the IBM Q Network.
Why is chemistry of such interest? Chemistry is one of the first commercially lucrative applications for a variety of reasons. Researchers hope to discover more energy-efficient materials to be used
in batteries or solar panels. There are also environmental benefits: about two percent of the world’s energy supply goes toward fertilizer production, which is known to be grossly inefficient and
could be improved by sophisticated chemical analysis.
Finally, there are applications in personalized medicine, with the possibility of predicting how pharmaceuticals would affect individuals based on their genetic makeup. The long-term vision is the
ability to design a drug for a particular individual to maximize treatment and minimize side effects.
There were two strategies employed by CQC and JSR Corp that allowed the researchers to make this advance. First, they used CQC’s proprietary compiler to most efficiently convert the computer program
into instructions for qubit manipulation. Such efficiency is particularly essential on today’s low-qubit machines, in which every qubit is needed and speed of execution is critical.
Second, they utilized quantum machine learning, a special sub-field of machine learning that uses vector-like amplitudes rather than mere probabilities. The method of quantum machine learning being
used is specially designed for low-qubit quantum computers, offloading some of the calculations to conventional processors.
The next few years will see a dramatic advance in both quantum hardware and software. As calculations become more refined, more industries will be able to take advantage of applications including
quantum chemistry. The Gartner Report states that within 4 years, 20 percent of corporations will have a budget for quantum computing. Within ten years, it should be an integral component of
Image Credit: Egorov Artem / Shutterstock.com
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The Maths Zone
I’ve had a really nice time doing a round of workshops for teachers and for PGCE and GTP students on handheld technology. I’ve always thought that ICT provides opportunities for teachers to invent
interesting activities that give students deep insights into how maths works. It is interesting that today, pupil’s in primary schools are no longer to be allowed to be examined in their ability to
solve problems with numbers harder than those they could handle by written or mental methods. Calculators are banned. Bizarely, the minister responsible justified this move in terms of the need to be
able to handle numbers because maths “influences all spheres of our daily lives”. This maths is routinely done by engineers and scientists who would never stoop to using a calculator or indeed a
computer to support their number work. The failure to get the sums right in the recent Virgin trains debacle was presumably caused by over use of calculators, except that the culprits will have been
educated in an era when they did have get enough number work. An era that clearly never was.
I start with the neat teacherly trick of playing ‘guess the function‘ here the participants see a calculators giving values of f(x) for their values of x, letting them choose to get a feeling for the
variation. I only show them a graph, when they have already formed a reasonable view, then watching as they focus on the details. The first thing is to realise that experienced teachers and well
qualified trainees struggle to see a quadratic just form a small table of values. No doubt because the drill and practice pedagogy the present government is so enamored with means many will have only
ever encountered a quadratic already knowing that was what the five points they were given to plot would show. But it is good to get a feeling for things and they see this. So, playing the game on
the handheld with their partners strengthens the insights and makes them more flexible.
It turns out that lots of schools are buying sets of iPads, demonstration that there is plenty of money around. But the maths software available for iPads isn’t a patch on any graphing calculator and
the storage, security and battery issues for anything you have to recharge means they will be no more reliable than laptops. In one group of 25 trainee teachers after about 4 weeks in schools only
one had seen any handheld machines possible to use in ordinary classrooms. That was a school where every student carried a laptop with them at all times. It only came out later that in all this time
they had not been used even one single time in maths lessons. A set of 15 HP39gIIs stored in a bag in the maths office with a few spare batteries and you just pick them up on your way in to class. I
make the case that it is the teacher who prevents the use of technology. That is a bit harsh. Mostly it’s the technology. So use something which is no more expensive than a couple of textbooks and is
almost certain to work.
Then we get back to seeing the resources we have as sites to conjour up really clever ways in to mathematical ideas. That’s what makes our job fun. Look at bag of dice, counters, centicubes and we
should always be saying, OK what could I do with those that encapsulates a mathematical idea. A graphing calculator is just the same, it’s something we can use to give students deeper insights. It is
in fact a calculator, and the scientists and engineers of the future should certainly learn to use it to support their number work, their algebra, whatever, so they can focus their brain power on
being brilliant with the science and the engineering. But also it’s a pedagogic device. A clever piece of kit for clever teachers to do what is most creative about our jobs. Something that supports
kids doing clever thinking.
Rubiks Maths for GCSE Revision
We’ve just finished work on updating the Rubik’s maths service so that it acts as a direct, complete GCSE revision tutor. The centrepiece is a neatly arranged assessment and practice system for
National Curriculum Maths. A smart space theme is used for the navigation … choose the Planet (e.g. Algebra), then the Continent (e.g. sequences) to find a nice self marking, auto updating, flash
based assessment package. If you make mistakes there are web links and video resources to give your practice opportunities (and, if you have a MyMaths licence, links to these resources). Continue
reading Rubiks Maths for GCSE Revision
Cape Town Maths
Two weeks ago, I had a very nice trip down to Cape Town. It is a very beautiful city indeed. However, I did a series of sessions mixing HP Graphing Calculators, GeoGebra and Data Streaming to groups
of maths teachers, trainee maths teachers and undergraduate engineering students at the Cape Penninsula Technical University and the University of the Western Cape. South Africa, in the post
apartheid era has been trying to bring all of its education systems up to the level of the former elite schools. As you can imagine, this is a tough task, although the government’s commitment is
clear having one of the highest proportional spends on education anywhere in the world. The universities I visited are excellent examples of that move to change and I was delighted to work with
really enthusiastic students and teachers. Additionally, the campuses are equipped with state-of-the-art facilities, including innovative sensory play equipment, further enhancing the learning
environment. If you’re interested in learning more about innovative sensory play equipment, you can check out this site at https://timbertrails.co.uk/why-timber-trim-trail-equipment-is-beneficial/
. For high-quality playground equipment UK, check out this article for more valuable information. Furthermore, if you’re looking for maintenance of your playground or wet pour installation, you can
click here for more information and services. For more resources on primary school education at https://www.primaryschoolresources.org.uk/outcome/psed. Continue reading Cape Town Maths
Virtual Learning Environments
I’ve done a lot of work with Fronter, I manage a number of Moodle VLEs e.g. The Education Interactive courses portal and the ATM/MA London Branch at King’s site LondonMaths . At King’s, I use their
ELK (formally BlackBoard) system. So, what is it, that these systems sell themselves on? Really it boils down to one thing: anywhere, anytime access to teaching materials. Sure, you can make cute
little multiple choice quizzes that are self marking and record and track progress, but you are clearly noit going to design and build your own set of these for your whole courses. Student protfolios
are very neat, but they only work if students can SUBMIT their work Continue reading Virtual Learning Environments
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Detailed Course Information
Select the desired Level or Schedule Type to find available classes for the course.
MATH 1080 - Introduction to Technical Mathematics
Prerequisite: MATH 0745 or MATH 0890 or placement test. The course is to meet the needs of engineering technology students as they encounter problems that occur in the world of work. The course
emphasizes the use of a scientific calculator to cover algebra from basic equations, the metric system, conversion factors, the use of measurement equipment and common geometric relationships. Right
angle trigonometry, and practical applications from a variety of technical areas with emphasis on logical thinking and application to engineering problems. Blueprint reading topics will include
lines, views, dimensions, notes, and sections. Common symbols used on drawings for welding and tolerances are introduced. Students must have a scientific calculator, however when continuing the
technical math series the recommended calculator type is the TI-84 Plus.
4.000 Credit hours
4.000 Lecture hours
Levels: Credit
Schedule Types: Hybrid, Independent Study, Lecture, Online, Online WSR Virtual Meeting req
Computer/IS/Engineer Tech Division
Eng CADT MATH PHYS MECT ENGR Department
Must be enrolled in one of the following Levels:
Credit level MATH 0745 Minimum Grade of S or Credit level MATH 0745 Minimum Grade of SC or Credit level MATH 0750 Minimum Grade of S or Credit level MATH 0890 Minimum Grade of S or Math Level 1 1 or
Math Level 2 2 or Math Level 3 3 or Pre-Algebra 39 or ACT Math 19 or SAT Mathematics 460 or Accuplacer Arithmetic 080 or Accuplacer Elementary Algebra 045 or SAT Math Section Score 460 or NG -
Arithmetic 263 or NG - Quantitative Reasoning 240
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02 Feb 2015
==> Just an announcement, kids, I’m scaling back my frequency of posts at Mises CA; I’m just too busy. If you want to take your hand at it, feel free to pitch ideas to them at contact@mises.ca. They
often get picked up by ZeroHedge if that matters.
==> Someone else made a print version of Barta and my Bitcoin book. Silas and I only benefit from the transmission of knowledge… (Remember if you want the free PDF, go here.)
==> A new NBER study suggests that part of the “surprising” US performance last year was due to the ending of unemployment benefits. Here’s a Reason review, but TL;DR if you stop paying people to not
have jobs, then fewer people will not have jobs.
==> Can’t remember if I linked to this? Peter Lewin (an expert on Austrian capital theory) chimes in on Piketty.
==> I don’t have time to deal with this, but if you want to see another perspective, Mike Konczal tries to defang Scott Sumner’s laughing over the sequester and Keynesians.
==> An interesting autobiographical note from one of the authors of The Market for Liberty, fantastic book that I reviewed very favorably (here–thanks to Darien for finding the link).
51 Responses to “Potpourri”
1. E. Harding says:
The three factors, I think, behind the strengthening of the recovery were:
1. Expiration of unemployment benefits.
2. End of regime uncertainty regarding Obamacare.
3. The beginning of the end of private-sector deleveraging.
□ LK says:
“E. Harding
The three factors, I think, behind the strengthening of the recovery were:”
The recovery we observe today would not even exist had it not been for the Keynesian stimulus from 2009-2011 and the interventions to stop the financial system from collapsing.
Had a “liquidationist” solution been pursued in 2008, large numbers of banks would have collapsed in 2008-2009 and million of deposit savings would have been lost, and a depression as bad as
or worse than the 1930s would have resulted.
3. Daniel Kuehn says:
What are your thoughts on Lewin? I am finding his post very hard to follow. It’s like he’s trying to work with about a third of a Solow model. And I don’t get why he’s equating the capital share
of income with the level of inequality.
□ Andrew_FL
Ah well, one could also question whether, even if capital’s share of income approaches 100%, that capital is driven by demographic trends and inheritance into increasingly fewer hands. And
therefore whether Piketty’s assumptions about capital’s share strictly increasing over time would necessarily imply increasing inequality. But if those assumptions themselves don’t hold,
what’s the point of questioning Piketty’s demographic and inheritance assumptions? Redundantly proving he’s wrong?
☆ Daniel Kuehn says:
I’m not entirely sure I follow. I guess my bigger confusion is that the capital share is not inequality according to Piketty (or anyone!), he noted some increase in the capital share over
the twentieth century but doesn’t think it will increase continually. I’m just confused by Lewin’s whole spin on what Piketty is saying.
○ Andrew_FL
If r > g (which Piketty belabors as a “fundamental law”) the capital share should grow to approach 1, right? I’m confused how you think Piketty thinking r is strictly greater than g,
doesn’t imply he thinks the capital share grows indefinitely?
To get from there to ever increasing inequality, you need certain auxiliary assumptions, about demographics and inheritance, yes. But I’m surprised to hear you think he doesn’t start
by essentially arguing that capital’s share grows toward unity (and then work from there, to inequality from auxiliary assumptions).
■ Daniel Kuehn says:
I suppose I’m not understanding why you think r > g implies the capital share of income goes to 1, or why you think Piketty thinks this. Can you provide a page number maybe?
I thought Piketty said alpha is the capital share of income, and that alpha = r x beta, where beta is the capital stock divided by income. Then beta is equal to s/g where s is the
savings rate. So alpha = (rs)/g. I don’t understand why you think (or why you think Piketty thinks) that this goes to 1 merely because r > g.
★ Andrew_FL
I think I’ve just misunderstood the math. Still, see Phil’s comment below, which I think partially justifies my confusion.
Daniel Kuehn says:
Phil points you to a chart that shows convergence to a beta of 700%, which Piketty says is the equilibrium value. You are thinking of alpha. Alpha is equal to r*beta. So
the convergence to beta=7. If we take his long-run typical r of about 0.05, that implies an equilibrium alpha (what you’re interested in) of 0.35.
Take a look at figure 6.5. What are the alphas converging towards?
Phil’s graph does not show an increase of either the capital stock or capital’s share of income going to infinity, it simply doesn’t.
Now I’ll agree, if someone didn’t read Piketty and the discussion of why he thinks beta will converge to an equilibrium level of 700%, then that figure that Phil shares
might suggest something different.
■ Phil Magness says:
Andrew – Piketty does in fact predict a continuous and pronounced worldwide growth of the capital stock vis-a-vis income. This is a slightly different way of capturing the
predicted effect than examining – say – yearly returns on capital as a share of total income, but he uses it to illustrate the underlying mechanism of r>g and all that he wishes
his reader to take away from it. That was the entire point of the second half of his figure 12.4 (the first half being an artificially pronounced trough that he achieved through
statistical manipulation to bolster his false historical narrative)
★ Phil Magness says:
Piketty does predict a stabilization…of the *savings* rate. But this is also explicitly identified as an assumption that he makes. Relax that assumption, things change pretty
dramatically. Or even hold to that assumption, and he still asserts a sustained divergence in C-I ratios. 700% is not the magic convergence number – it is only where he
supposes things will be at the end of his century long projection based on what he assumes to be the worldwide savings rate. The convergence is only on that assumed savings
rate of about 10%.
But don’t take my word for it. Take Piketty’s own description from the two relevant places where he brings this up.
Chapter 5:
“I also assume that the savings rate will stabilize at about 10 percent in the long run. With these assumptions, the dynamic law β = s / g implies that the global capital/
income ratio will quite logically continue to rise and could approach 700 percent before the end of the twenty-first century, or approximately the level observed in Europe
from the eighteenth century to the Belle Époque. In other words, by 2100, the entire planet could look like Europe at the turn of the twentieth century, at least in terms of
capital intensity. Obviously, this is just one possibility among others. As noted, these growth predictions are extremely uncertain, as is the prediction of the rate of
saving. These simulations are nevertheless plausible and valuable as a way of illustrating the crucial role of slower growth in the accumulation of capital.”
Chapter 12:
“In the central scenario for the evolution of the global capital/income ratio that I discussed in Chapter 5, I assumed that the savings rate would stabilize at around 10
percent of national income as this international convergence process neared its end. In that case, the accumulation of capital would attain comparable proportions everywhere.
A very large share of the world’s capital stock would of course be accumulated in Asia, and especially China, in keeping with the region’s future share of global output. But
according to the central scenario, the capital/income ratio would be the same on all continents, so that there would be no major imbalance between savings and investment in
any region. Africa would be the only exception: in the central scenario depicted in Figures 12.4 and 12.5, the capital/income ratio is expected to be lower in Africa than in
other continents throughout the twenty first century (essentially because Africa is catching up economically much more slowly and its demographic transition is also delayed).
If capital can flow freely across borders, one would expect to see a flow of investments in Africa from other countries, especially China and other Asian nations. For the
reasons discussed above, this could give rise to serious tensions, signs of which are already visible.
To be sure, one can easily imagine scenarios much more unbalanced than the central scenario. Nevertheless, the forces of divergence are much less obvious than in the case of
the sovereign wealth funds, whose growth depends on windfalls totally disproportionate to the needs of the populations benefiting from them (especially where those populations
are tiny). This leads to endless accumulation, which the inequality r > g transforms into a permanent divergence in the global capital distribution.”
Daniel Kuehn says:
He suggests savings at 10%, g at just under 1.5% at the end of the century which gets you to his “could approach 700 percent” figure.
One thing he definitely has not said and that you have definitely not demonstrated him saying is that the capital share will continue to increase.
Your chapter 12 quote is about the distribution of the capital stock NOT the capital share of income. It’s important to keep these concepts straight.
Daniel Kuehn says:
Andrew this is very easy to sort out. To get alpha going to infinity you need s or r going off to infinity or g going off to zero (or less). Piketty clearly doesn’t make a
case for any of that. Alpha and beta are pinned down. They were in the Solow model. They are in Piketty.
Daniel Kuehn says:
Actually I’m not entirely sure how to think about this in the context of long run negative growth, so don’t quote me on that last part.
Phil Magness
No Daniel, the Chapter 12 quote is about the Capital/Income RATIO depicted in the accompanying Figure 12.4, to which it directly refers. I never represented it as a
presentation of the capital return share of income, and in fact stated otherwise when I introduced 12.4 above.
For someone who quite rudely just insinuated to Andrew that he “didn’t read Piketty” and therefore made an erroneous representation of him, your own erroneous readings of
both Piketty, not to mention others with whom you converse, are as palpable in comparison as they are astounding in this context.
Daniel Kuehn says:
12.4 is the ratio to income. Andrew asked about capital’s share of income. Your quote refers to the distribution of capital’s share of income.
□ Bob Murphy says:
Thanks I should’ve thought of that too, Darien, but I was referring to a book review I wrote for Mises.org. I have an old URL for it, but it’s broken since they revamped the website.
5. Phil Magness says:
I realize that reading comprehension is not your friend, Daniel, but:
(1) 700% is nothing more than where he expects C/I to be 100 years from now IF his assumption of the savings rate convergence at 10% holds true. It is the product of a forecast made using that
assumptions – IOW where he expects it to end up after a century – not the stable equilibrium that you purported it to be in your comment above. See:
“*With these assumptions,* the dynamic law β = s / g implies that the global capital/income ratio will quite logically continue to rise and could approach 700 percent before the end of the
twenty-first century”
(2) Piketty is pretty clear in recognizing the possibility of capital accumulation continuing, especially as one relaxes the assumptions
“This leads to endless accumulation, which the inequality r > g transforms into a permanent divergence in the global capital distribution.”
(3) Simply holding the 10% rate constant and continuing Piketty’s own projection model forward another decade to 2110 results in the ratio quickly breaking north of 700%. It would not do so if
this were a stable equilibrium as you claimed above.
(4) As a brief aside, Piketty’s forecast “model” is chock full of the same sort of amateurish back-of-the-envelope contrivances that plague the rest of this chart and should be evaluated
accordingly, even if the 10% savings assumption is accepted.
□ Daniel Kuehn says:
Always such a treat to talk to you Phil.
1. Right, it’s the projection to the end of the century. Plug in the figures for beta and show me how beta keeps increasing, then we can talk. It’ll take the century to converge, it’s true.
Nothing in the determination of beta suggests r > g pushes it to infinity. If you disagree explain your reasoning.
2. The distribution of capital is not the same as the capital share of income. I pointed this out to you above, I don’t know why you’re repeating it.
3. Which g are you plugging in? I don’t see how you get this. If you are using a g other than something just shy of 1.5 it may keep going a little longer but you’re going to level out
somewhere along there.
6. Phil Magness
1. There appears to be a substantial divergence between how Piketty actually calculated his forecast and how you *think* he calculated his forecast, Daniel. You seem to be accepting his model,
adding in a few additional assumptions that your misreading of his text leads you to believe to be a part of it, and working backwards to his forecast from there. What we actually have in Piketty
though is the product of a base assumption (the 10% savings rate convergence) coupled with a very rudimentary linear projection, a bunch of ad hoc guesstimation to fill in the gaps, and an
intermittently applied rolling average overlaid on top of the whole thing.
2. Seeing as I did not equate it with the capital share of income and in fact said the exact opposite upon introducing Figure 12.4 to the discussion, I’m not sure what you’re even talking about
and would accordingly defer to my previous assessment of your reading comprehension skills.
3. See #1 above. How you think Piketty performed his forecast is a very different thing from how he actually performed his forecast. If you keep his forecast model at the 10% savings convergence
after 2100, hold the world dist. ratios constant to where he set them, and assume a stable continuation of his world income trend, then yes – the ratio goes north of 700% in 2110.
And for fun I’ll add in:
4. You repeatedly asserted above that 700% is an “equilibrium value” or some sort of similar stabilization point. Yet Piketty’s text makes no such claim, and only enlists this number to suggest
that some 90 years from now it’s about where he expects the world’s C/I ratio to be at. So if not a misreading of Piketty even as you chastised another poster for supposedly failing to read the
same, where exactly are you getting this “equilibrium” business from?
□ Daniel Kuehn says:
re: “You repeatedly asserted above that 700% is an “equilibrium value” or some sort of similar stabilization point. Yet Piketty’s text makes no such claim.”
Check the discussion of beta=s/g starting on page 166. Beta is a long-run value. On page 169: “the law beta=s/g is the result of a dynamic process: it represents a state of equilibrium toward
which an economy will tend if the savings rate is s and the growth rate g, but that equilibrium state is never perfectly realized in practice.”
This is not some radical contribution. We’ve had these steady state equations since at least 1956 (maybe Ramsey had them in the 20s – I’d have to double check that). It’s really stupid that
we’re arguing this point.
□ Daniel Kuehn says:
re: “So if not a misreading of Piketty even as you chastised another poster for supposedly failing to read the same”
Can you please stop repeating this? I don’t know why you have this need to make things up about me. I never said that.
I did say someone that didn’t read the book might interpret your chart in the way that you were trying to spin it. But of course that’s not the only way to misunderstand the chart. If you
read the book and forgot it you’d misunderstand. If you read the book and didn’t understand it you’d misunderstand, etc.
I was having a nice conversation with Andrew before you came here. Please stop inventing insults of Andrew and putting them in my mouth.
7. Phil Magness
Daniel – I’m not taking issue with Piketty’s growth rate beyond noting it is an assumption. I’m not questioning his savings rate beyond – again – noting that it carries an assumption. And I’m not
asserting anything about his modeling of the two separate and apart from the empirics of his forecast.
I am, however, questioning *your* assertion that they settle at an equilibrium that is specifically 700% and specifically coincides with the year 2100 on his Figure 12.4, as none of that is
evident in Piketty’s own claims. So while you may be unsure as to why we’re arguing the point, I am unsure why you are now evading an explanation for a very specific and erroneous claim that you
made and reiterated at an earlier point in this thread.
As to my other remark, you stated “if someone didn’t read Piketty and the discussion of why he thinks beta will converge to an equilibrium level of 700%…” Leaving aside the aforementioned issue
that the 700% equilibrium convergence is your own invented – and now evaded – construction, the clear insinuation of that remark is that persons who disagreed with your assertion – i.e. Andrew at
that point of the conversation – must not have read Piketty.
Since you plainly don’t enjoy being called out for making such insinuations (or for that matter your habit of baselessly attributing errors of method, comprehension, competence, qualification, or
even basic calculation practices to others you criticize), my advice to you on this point is simple: quit inserting such blatantly uncharitable and derogatory insinuations about any and every
interlocutor you face on this subject. They are needless, and only serve to start the conversation on a footing from which it is difficult to ever recover. And given your own challenges of
comprehension when it comes to Piketty as has been amply documented above, they are also more than a tad hypocritical to be coming out of such a self-unaware source.
8. Andrew_FL
Guys, for the record, I think that Daniel is right that I was wrong about the math-just a bad misunderstanding on my part. On the substance and on reflection, r > g by itself, doesn’t imply an
ever growing capital share. That requires additional auxiliary assumptions. I certainly got the impression that was what was being claimed. I now think I may have misunderstood that, too.
It still seems to me that Piketty would logically have the capital share of income growing for at least the next several decades. And looking at his projection I see no continuous decrease in the
rate of increase of the capital/income ratio, so I don’t see on what basis the value is expected to stop increasing at 700%? If that’s what Piketty’s model assumes, then it doesn’t look like he
used his model to make his projection. What gives?
□ Daniel Kuehn says:
re: “so I don’t see on what basis the value is expected to stop increasing at 700%”
s is about 0.1
g is about 0.015
9. Phil Magness
Andrew – All fair on the math issues. My original point was to note the different metric of his C/I ratio does indeed show a fairly aggressive expansion of the capital stock as an implied effect
of r>g.
But yes – I also want to know where Daniel’s assertion of the stabilizing 700% equilibrium comes from. Because (1) this does NOT actually appear in Piketty’s claims and (2) it seems to be the
result of a misreading of Piketty’s related assumption that the savings rate will stabilize at 10% worldwide in an adjoining passage.
10. Daniel Kuehn says:
Let me ask you two this: precisely where do you think Piketty got his 12.4 figure???
□ Andrew_FL
Having not personally examined his spreadsheet for that graph, I plead ignorance and agnosticism. I only note that it doesn’t look like it comes from a model that you describe.
☆ Phil Magness
Andrew is correct. It may be loosely informed by his model. The assumptions of that model also determine its shape and where he expects its end point to be in 2100. But it is constructed
from several specific components that fill in the trend, and do so in a way that fits his story. Specific components include:
1. Historical estimates of global output, which are then weighted regionally across the century & projected forward according to where he anticipates the weights going (there are HUGE
gaps and other problems in the regional btw – as in sufficiently large to undermine the reliability of the whole thing). The forward projections are also mostly a continuation of his own
guesstimations, with a little averaging overlaid in a few regions.
2. Historical data on the capital stock by region. He only has actual data for the US & Western Europe across the whole series, and Japan/Aus/NZ since 1970, so the rest is guesstimated or
– in the communist regions – simply hard coded in according to an a priori ideological claim about the position of capital in a communist society. These too are projected forward
post-2010, most of it through simple guesstimation of where he thinks it will be.
3. Historical data for total world output (giving him the income part of the ratio) and a projection of where he thinks it will go through 2100.
4. Piketty’s savings rate assumptions, which show all regions of the world converging on 10% by 2080.
He then projects where he thinks the C/I ratio will be in 2100 and uses the above, in cumulative, to backfill in the trend lines. The 700% is a product of that projection, attained
through where he sets his savings assumption and g. It is not, however, a fixed equilibrium around which the world will automatically stabilize.
11. Phil Magness
Perhaps also worth reiterating that Piketty himself does NOT consider 700% an equilibrium – rather it is only an outcome that happens IF he sets his assumptions around the 10% savings rate and
specific levels of output growth across the century. Change either of those things and it could be higher (or lower) in any given year. IOW, Daniel seems to be taking the assumptions Piketty used
as if they are fixed and drawing his claimed equilibrium out of their projected results. Yet Piketty himself tells us the assumptions are NOT fixed:
“Obviously, this is just one possibility among others. As noted, these growth predictions are extremely uncertain, as is the prediction of the rate of saving. These simulations are nevertheless
plausible and valuable as a way of illustrating the crucial role of slower growth in the accumulation of capital.”
Therefore 700% is not an equilibrium, but a product of his forecasting.
□ Daniel Kuehn says:
He does call it an equilibrium – I gave it to you above – and yes of course an equilibrium value changes when the values of the parameters change.
☆ Phil Magness
No Daniel. He says his “second law” represents an equilibrium state that “is never perfectly realized in practice” in a passage some 20 pages prior to the figure you are projecting an
equilibrium onto. When he gets to that actual figure though, he readily concedes that it is a *product* of his prior assumptions, and not an equilibrium onto itself. So either Piketty did
not mean what he plainly said when he described his figures as “uncertain” and only a “possibility” among many outcomes, or you are borrowing from an earlier passage to append something
to the one under consideration.
In any case, the result of your claim is intrinsically circular. You cannot predict conditions X & Y will hold 100 years from now and plug them into a formula producing result Z, then
assert that the result Z thereby validates inputs X & Y and is somehow now an equilibrium state towards which X & Y gravitate. Piketty – to his credit – does not make that step. You seem
to believe otherwise.
○ Daniel Kuehn says:
Phil you’re acting like if we don’t know the future with exact certainty then it can’t be an equilibrium. You have a very different understanding of what an equilibrium is in
economics from mine. I’ll stick with mine.
I don’t even follow what you’re trying to say in the second paragraph.
■ Phil Magness
Daniel – What I’m saying is that by all appearances, your theory of equilibrium seems to proceed as follows:
Step 1: Make wild guesses about what the global savings rate AND output will be 100 years from now.
Step 2: Derive a world C-I ratio for 100 years from now from the guesses in Step 1.
Step 3: Produce a trend line between the present day and that guess-derived ratio from Step 2 by way of connect-the-dots.
Step 4: Assert that the ratio from Step 2 exhibits innate stabilizing characteristics and label it an equilibrium
Step 5: Use that newly labeled equilibrium from Step 4 to assert that the connect-the-dots exercise from Step 3 represent a convergence upon the derived ratio of Step 2.
(Disclaimer: Piketty only follows this line of construction through Step 3 and then caveats it with an admission that its premised on wild guesses. This is to his credit.)
★ Daniel Kuehn says:
Try this instead:
1. Derive steady state equations for the capital stock relative to income.
2. Forecast the parameters that contribute to the steady state.
3. Plug them in and get forecasted capital stock relative to income.
That’s to get his long run equilibrium. I don’t know if the path to get there was projected or if he had some kind of balanced growth path with the same forecasted parameters
or what.
Forecasts are always dicey. Piketty says as much which is good. But that’s the exercise.
Phil Magness
Unless you are asserting that there is a stabilizing effect in the world growth rate at 1.5% (in which case you should also be prepared to offer a reason) there’s
something missing from your purported equilibrium. Simply calling something a “forecast” or an “equilibrium” does not make it so. Thus we actually have something more
along the lines of:
1. Make wild guesses about where savings and output will be 100 years from now
1a. Label aforementioned guesses “forecasting” even though they don’t employ any standard forecasting model and are actually just rough guesses
2. Plug them in to get your C/I ratio.
2a. Also label this a “forecast” even though it’s made from the same wild guesses
3. Let Piketty play connect the dots to tie them back to the present.
3a. Plead ignorance of this, but accept its results nonetheless.
4. Ascribe an equilibrium convergence to the product of the above, ignoring that it lacks any innate stabilizing character and is only one of many possible results
determined entirely by the wild guesswork undertaken in Step 1.
□ Daniel Kuehn says:
“Daniel seems to be taking the assumptions Piketty used as if they are fixed ”
Never said it. Not ever. Not once.
☆ Phil Magness
If the assumptions are not fixed, then the 700% is not an “equilibrium” but a product of those forecasting assumptions. Is that what you believe now?
12. Andrew_FL
It seems like there are two “equilibriums”: an equilibrium savings rate, which is pulled out of thin air(???) and an implied equilibrium Capital/Income ratio, given that equilibrium savings rate.
But again, looking at figure 12.4, it doesn’t look like the rate of increase is decreasing as fast or as continuously as it should if it’s converging on 700%.
BTW, Daniel, I just realized, .1/.015 = 6.6 repeating, not 7. This seems like even stronger indication that the function used to create the graph isn’t converging to Piketty’s beta, it’s already
about there and has a significant positive slope.
□ Daniel Kuehn says:
Just under 7 is what he says and as I noted above the long run g is just under 1.5 putting beta somewhere above 6.666.
□ Daniel Kuehn says:
Ya 2100 value in the chart s 667% which is 0.1/0.015.
□ Phil Magness
Piketty’s referenced equilibrium is a theorized global convergence in the savings rate. He supposes this will happen around 2180 at 10%. The 667 is NOT an equilibrium onto itself, but a
product of that savings rate plus another contingent assumption of what world output will be in 2100. If we simply change the assumptions around (say savings converges at 15% instead, or a
different growth in output), we could literally set it at any number we want for the resulting C-I ratio…which further affirms that it’s not an equilibrium =)
☆ Daniel Kuehn says:
re: “The 667 is NOT an equilibrium onto itself, but a product of that savings rate plus another contingent assumption of what world output will be in 2100.”
I assume you mean growth rate, not output itself right?
Yes – it’s the equilibrium product of those parameters. Beta=s/g
○ Phil Magness
Actually Daniel, if you look at his spreadsheet he is using an estimate of world output in 2100 to get to the rate of growth and calculate his world C-I ratio, so it is ultimately
contingent on an assumption about where world output will be in 100 years. An equilibrium requires stability. Simply plugging in wild guesses does not yield stability, as changing any
of them even slightly can dramatically alter the result.
■ Daniel Kuehn says:
re: “Actually Daniel, if you look at his spreadsheet he is using an estimate of world output in 2100 to get to the rate of growth and calculate his world C-I ratio”
Hence, “I assume you mean growth rate”
The actual level of output doesn’t enter into beta.
re: “An equilibrium requires stability. Simply plugging in wild guesses does not yield stability, as changing any of them even slightly can dramatically alter the result.”
You are confusing the equilibrium value of beta with our estimate of the equilibrium value of beta. The latter is of course just an estimate.
13. Bob Murphy says:
I predict that the dispute between Phil and Daniel will tend to infinity as the year approaches 2100.
14. Phil Magness
As this discussion is rapidly descending into a semiotic dyslexia of Kuehnian pedantry, I’ll simply note:
1. Future world output is one of the (very) few things in Piketty’s spreadsheets even approximating a regular forecast. As his highly variable growth rates are derived directly from this
forecast, that rate and the calculation derived from it is directly contingent on his output prediction, which is what I stated.
2. Daniel previously and repeatedly asserted that a stabilization occurs at Piketty’s specific 100 year forecast of a 700% C/I ratio. Above this was described as a “convergence” at 700%. On a
related blog post, he asserts that it “will level off at around 700%.” It is therefore only reasonable to interpreted as an ascription of equilibrium characteristics to Piketty’s specific
3. Since that estimate is wholly contingent on the aforementioned guesses about the savings rate & world output 100 years from now, calling that an equilibrium is absurd on a number of counts.
Why 2100 and not 2050, when the same formula would presumably apply? Why not a growth rate of 3%? Why not a global convergence around a savings rate of 12%? Bottom line: there is nothing innately
stabilizing about the year 2100 or Piketty’s estimated C/I ratio in that year, and to assert otherwise is to both misunderstand what he’s doing to achieve his projection and attribute an
equilibrium claim to him that he did not in fact make.
4. The lack of a declining slope in the decade before 2100 also strongly suggests that Piketty is not attempting to show a convergence at the point that Daniel repeatedly claimed. Rather, Piketty
is simply forecasting where he thinks the ratio will be in a century. Piketty’s book – to its rare credit – openly acknowledges this ambiguity in ways that Daniel’s original comments, at least,
did not.
5. What I actually believe we have here is this: Daniel, by way of the aforementioned affliction of semiotic dyslexia, misread Piketty’s reference to a theorized global convergence in the savings
rate at about 10% (a plausible but not airtight claim) as a theorized convergence around his Figure 12.4’s *particularized* end point. In doing so he ascribed equilibrium conditions to that end
point that Piketty himself never claimed. In the ensuing conversation, that original particularized claim has morphed into an ascription of a general equilibrium to Piketty’s 2nd law on account
of a passing theoretical reference to it as such – qualified by an admission that this equilibrium characteristic is never realized in practice – at an earlier point in Piketty’s text, though
also one that does not assert the particularized stabilization claim about 700% in the year 2100 that Daniel originally and mistakenly attributed to Piketty’s text.
The primary remaining question is therefore whether Daniel still believes that 700% (or 667, if we go by the exact calculations on the figure) is a particularized equilibrium convergence point
for Piketty’s C/I ratio. If he does indeed believe this, it is fair game to then ask (a) how he reconciles this convergence with the strong positive slope immediately preceding it, (b) why this
convergence occurs in 2100 and not 2050 or 2150 or any other date, (c) why this convergence occurs at the savings rate and world output levels Piketty uses for 2100 and not any other possible
level, and (d) what theorized characteristic explains the predicted future stabilization of the world growth rate – a figure that has historically exhibited fluctuation – which would be necessary
for the claimed future C/I stabilization to occur, either in its particularized form in 2100 or at any other future moment.
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Math Colloquia - The phase retrieval problem
In many applications such as X-ray Crystallography, imaging, communication and others, one must construct a function/signal from only the magnitude of the measurements. These measurements can be, for
example, the Fourier transform of the density function. While it is well known that we can recover a function from its Fourier transform, the classical phase retrieval problem asks whether we can
recover a function from only the magnitude of its Fourier transform. The phase retrieval problem has since been extended to a much broader class of settings, referring to the reconstruction of a
signal from only the magnitude of its linear measurements or more generally, from quadratic measurements. The problem, even in finite dimensions, turns out to be quite challenging. Many fundamental
theoretical problems remain unresolved. Equally challenging is to develop fast and robust algorithms for phase retrieval. The problem has, not surprisingly, links to many problems in science and
engineering. But more surprisingly it has also links to some classical problems on the embedding of projective spaces into Euclidean spaces and nonsingular bilinear forms. In this talk I'll give a
brief overview and discuss some of the recent progresses.
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A simple Bayesian analysis of surveys
What share of UK adults gave the right answer to a statistical question?
Classical statistics thinks of probability as the long-run frequency of events. The Bayesian approach considers probability as a degree of belief. The language of probability then describes
uncertainty in unknown quantities.
The Royal Statistical Society commissioned a survey of the public, asking this question:
• Question 1: If you toss a fair coin twice, what is the probability of getting two heads?
Opinium conducted this online survey, gathering views of 2,001 UK adults. Researchers weighed responses by gender, age, region, social grade and employment status.
(Image: R Pubs)
Surveys provide estimates, which can differ from true values for many reasons. Researchers use distinct wordings and response options, trying to measure the same concept. Different survey modes,
sampling frames, and weights may produce different estimates.
What about the uncertainty in those estimates? Bayesian analyses start with the prior distribution. This distribution represents knowledge before data collection. There were seven response options
for the question: 15%, 25%, 40%, 50%, 75%, Other, and Don’t Know.
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Density Calculation Worksheet
Density Calculation Worksheet. Web pdf, 544.99 kb. Web our calculating density worksheets are a great way to expand upon and reinforce gcse physics learning.
Density Calculations Worksheet I Answer Key Worksheet Resume Examples from www.thesecularparent.com
A student finds a rock on the way to school. They will complete a chart with missing pieces. In other words, density is.
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Practice Monohybrid Crosses Answer Key
14 Best Images of Monohybrid Cross Worksheet Answer Key Monohybrid
Practice Monohybrid Crosses Answer Key. 10.what is the phenotypic ratio? Web monohybrid cross worksheet flashcards | quizlet monohybrid cross worksheet 4.3 (6 reviews) term 1 / 19 genotypes made of
the same alleles click the card to flip ๐ definition.
14 Best Images of Monohybrid Cross Worksheet Answer Key Monohybrid
Web give peas a chance with this 10 question practice worksheet that can be used as classwork or homework. Dihybrid cross practice worksheet answer key 3. We provide you all the answers. 11.what is
the probability of producing a spotted beetle? What percentage of their offspring are expected to have short arms? Worksheets are dihybrid cross practice. 10.what is the phenotypic ratio? Dihybrid
crosses worksheet answer key| 4. Practice with monohybrid punnett squares read the following. Decide what sort of drawback you are attempting to resolve.
Dihybrid cross practice worksheet answer key 3. Web record of dihybrid cross follow issues reply key pdf concepts. Dihybrid cross practice answer key 2. We provide you all the answers. Web give peas
a chance with this 10 question practice worksheet that can be used as classwork or homework. What is the genotypic ratio? Web terms of use monohybrid cross google classroom in watermelons, solid
green rind color ( g) is dominant to stripes ( g ). Famous dihybrid practice worksheet answer key 2023. Answer each of the following questions using a punnett square and the rules of monohybrid
crosses. 10.what is the phenotypic ratio? A farmer crosses two watermelon plants that are.
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What is a margin of error?
This statistical tool can help you understand vaccine trials and political polling
published : 21 February 2024
In the last year, statistics have been unusually important in the news. How accurate is the COVID-19 test you or others are using? How do researchers know the effectiveness of new therapeutics for
COVID-19 patients? How can television networks predict the election results long before all the ballots have been counted?
Each of these questions involves some uncertainty, but it is still possible to make accurate predictions as long as that uncertainty is understood. One tool statisticians use to quantify uncertainty
is called the margin of error.
In the real world, it is impossible to test or sample every relevant person, so statisticians rely on smaller samples drawn from a population. Guzaliia Filimonova/iStock via Getty Images Plus
Limited data
I am a statistician, and part of my job is to make inferences and predictions. With unlimited time and money, I could simply test or survey the entire group of people I am interested in to evaluate
the question in mind and find the exact answer. For example, to find out the COVID-19 infection rate in the U.S., I could simply test the entire U.S. population. However, in the real world, you can
never access 100% of a population.
Instead, statisticians sample a small portion of the population and build a model to make a prediction. Using statistical theory, that result from the sample is extrapolated to represent the whole
Quantifying uncertainty
Take drug development, for example. It is always true to predict that a new medication will be somewhere between 0% and 100% effective for everyone on Earth. But that isn’t a very useful prediction.
It is a statistician’s job to narrow that range to something more useful. Statisticians usually call this range a confidence interval, and it is the range of predictions within which statisticians
are very confident the true number will be found.
If a medication was tested on 10 individuals and seven of them found it effective, the estimated drug efficacy is 70%. But since the goal is to predict the efficacy in the whole population,
statisticians need to account for the uncertainty of testing only 10 people.
Confidence intervals are calculated using a mathematical formula that encompasses the sample size, the range of responses and the laws of probability. In this example, the confidence interval would
be between 42% and 98% – a range of 56 percentage points. After testing only 10 people, you could say with high confidence that the drug is effective for between 42% and 98% of people in the whole
If you divide the confidence interval in half, you get the margin of error – in this case, 28%. The larger the margin of error, the less accurate the prediction. The smaller the margin of error, the
more accurate the prediction. A margin of error that is almost 30% is still quite a wide range.
However, imagine that the researchers tested this new drug on 1,000 people instead of 10 and it was effective in 700 of them. The estimated drug efficacy is still going to be around 70%, yet this
prediction is much more accurate. The confidence interval for the larger sample will be between 67% and 73% with a margin of error of 3%. You could say this drug is expected to be 70% effective, plus
or minus 3%, for the entire population.
Statisticians would love to be able to predict with 100% accuracy the success or failure of a new medication or the exact outcomes of an election. However, this is not possible. There is always some
uncertainty, and the margin of error is what quantifies that uncertainty; it must be considered when looking at results. In particular, the margin of error defines the range of predictions within
which statisticians are very confident the true number will be found. An acceptable margin of error is a matter of judgment based on the degree of accuracy required in the conclusions to be drawn.
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Electronic systems in terms of algebraic equations - Free Essay Example - 2323 Words | StudyDriver.com
In the world of electrical engineers it is not easy to define or analyze any system performance without any proper study. Every electronics system is defined in terms of mathematical equations. I am
keen to study about radio frequency circuits. In these systems it is always necessary to know how much error is obtained and we need to estimate the error so that we can design the system that can be
used efficiently. Also the signals that are received in these type of communication systems are unpredictable and it is necessary to study about the behaviour. This is the reason why I chose to study
about Probability and Random processes.
Every system is dependent on any specific constraint. In communications its usually time and frequency. So we learn every type random process that are either single dependent or double parameter
dependent. The course mainly deals about basics of probability and its use in determining the behaviour, performance of any Random process. It starts with venn diagrams, plotting of probabilities and
then the probabilities that are dependent on other parameters which are called conditional probability. It is further studied by deriving into a theorem that is mostly used in machine learning which
is called bayes theorem. This theorem derives the probability of the event to occur, after a event is happened.
To get a clear view of study, parameters should be defined in a function so as to determine its characteristics. So probability distribution function PDF and cumulative distributive function are
derived. These functions are used to analyze Random variables in both continuous and discrete forms. Various statistical terms like mean, variance are used to define the random process. The power of
the system are represented by density functions .These random variables are represented as Random process, such as poison random processes, Binomial random processes and Rayliegh random processes.
These are further viewed in Gaussian and Gamma distributions.
Relation between the two random processes are mathematically given by covariance, correlation and orthogonality which will give insight to dependency of one random process with another random process
(like heat of the device and gain of the device- even though both are not related, one random process effects another random process).
At the end of the course various error detecting techniques are studied such as Mean square error(MSE) , ML estimator. Both these estimators are mathematically derived and various types of cases are
solved. The project of this course is based on these estimators. This made me learn more about my Matlab programming and using this MSE estimator is designed. This is estimated on a data sheet of 10
variables and 2500 data entries. The results gave almost as accurate values and thus knowing the significance of Estimators.
LINEAR AND MATRIX ALGEBRA- FALL 2019
As mentioned it is necessary for me to know the mathematical representation of every system. Any electronic device is operated either by current or voltage, for this to happen the device should
respond according to what we hope for. Sometimes due to over driving or noise considerations the device is failed to be in use. All these considerations can be analyzed over using S parameter matrix.
The s parameter matrix can be a collection of parameters across the device and to know about them it is necessary for me to know the advantages i can use over this matrix by using basic algebra. This
is the reason I opted for Linear and Matrix algebra
This course is started off with basics of matrices and its properties along with vectors and linear combinations. The matrices are used to find the solutions of linear equations by using LU
decomposition method, Elimination method. 4 types of vector spaces are studied as row space, null space , column space and left null space. Sometimes we don’t have particular unique solution to the
set of linear equations(no solution or infinite solutions).Using these vector spaces and subspaces, the linear equations are calculated and is given in terms of row/column subspace. The four
subspaces are characterized based on its dimension, dependence and basis. These give the relation between the subspaces and values in terms of rank.
The relation between subspaces is given by orthogonality. Every element in the row space is orthogonal to every element in null space. This means no term in column space exists in null space.
Similarly with row space and left null space. Also the columns of the matrix is said to be column space and rows of the matrix is said to be row space, this mean C(A)=R(AT) (same with column space).
Important aspect in matrices is defining its inverse. For that we need to know about determinant of the matrix. Various properties of determinants are studied and with this eigen values and eigen
vectors came to origin. Cramers rule is discussed for inverse of a given matrix. This results in using matrices to determine volumes and other scalar notations. The Eigen vectors and eigen values are
determined so as to covert the given matrix in various forms. Singular value decomposition is used to represent given matrix in terms of eigen value matrix along with 2 orthogonal matrices (eigen
For electrical application purpose the network is converted in graph so as to represent in matrix form. This matrix is further reduced to identify node voltage, total voltage and current of the
network. At the end of the course, as a part of project work I got to write a paper and I decided to write about the stability of the system using Eigen values and vectors and how to improve the
stability of the system.
RF AND MICROWAVE CIRCUITS -1 – FALL 2019
I always wonder how would be the world without wires. This thought has drawn me towards wireless communications. Though there are lots of wireless systems, I found keen interest with RF and microwave
circuits. I know the basic communication structure is totally based on protocols and architecture model, but the real challenge is to know how to generate process and refine the RF signals so as to
use them in real world experience. The evolution of RF based devices varies not only with size but also in performance, power compatibility and so on. To study deep down about the RF devices I opted
RF and microwave circuits-1.
This course deals with analysis, construction, development and types of various blocks in basic RF communication block model. The main goal of this model is to convert high frequency RF signal to IF
signal using a local oscillator. During this conversion various factors comes to existence that are related to mainly produce efficient output. Every device characteristics can be represented using
many types of parameters. The most common is S parameters (4 parameters for 2 port device). These S parameters directly give return loss at both ports, reverse transmission and gain of the device.
According to the function of device, specific ranges of S parameters are expected.
Devices such as Filters, couplers, diodes and mixers are studied. Various types of couplers, both 3 and 4 port couplers are studied. Advantages and disadvantages of each type of couplers are
discussed. Thus according to the product requirement specific couplers are used. The most important components of the RF model are filters. When RF signal is sent through mixer so as to down convert
it to IF, the receiving antenna is capable of capturing various RF frequencies that are surrounded that are close to required RF signal. So to eliminate such frequencies there is a necessity of using
a filter. Several types of filters are studied such as lumped filters, transmission line filters, Coupled line filters, Stubbed filters, micro strip line filters and so on.
The other important device is RF diodes. A few type of diodes are studied and are derived on how they can be used as switch in RF frequencies. PIN diode which is one of the most common and basic RF
switch. Schottky diode which is virtually formed over MOSFET devices (formed by junction of semi conductor and metal surface).Varactor diodes (variable capacitor variable diode) which are mostly used
in tuning and are operated in reverse bias.
All the concept of impedance matching and filters are taken to level up by using ADS. This application enables to design, simulate and analyze any type of practical or ideal response of the system
over any specific parameter (in this course mainly over frequency). 2 projects on impedance matching and micro strip coupled line filters are designed using ADS. The simulation part of the project is
done and the layout of the design is fabricated ready to test.
In RF and microwave circuits, the course mostly deals with all test bench equipment theoretically. To obtain more practical knowledge on how the RF devices work in general workspace conditions, their
behaviour over practical errors I opted for Wireless circuits and microwave systems laboratory. The entire course work is mirror to that of practical application of RF/MW circuits 1 course. In
addition to that lot of lab equipment gear is required.
The basic element to describe the performance of the RF device it is important to obtain the S parameters. VNA- vector network analyzer, a device is used to measure S parameters for this course.
Spectrum analyzer and Oscilloscope are used to obtain the signal response over frequency and time domain respectively. A signal generator is used in test bench so as generate the RF signal with our
required Frequency and power level.
To connect each devices in the test bench various connectors are used. A coaxial cable which is generally used in connection with bigger equipment(VNA, SA, SG, DSO). Various connectors like SMA
adaptor, SMA load, attenuator, Isolator, 3”to 4” semi rigid cable, F-M adaptor and many more. Even though these are just used to connect RF devices, as they are part of the designing model it is
important to study about the behaviour of S parameters of these connectors and are used accordingly.
Various RF devices like Filters, couplers, Antennas are designed using ADS and then fabricated for testing. During testing the designed RF device board is connected by VNA to obtain the S parameters.
In doing so we need to using coaxial cable which will change the original characteristics of the board. To compensate this error the device is calibrated with known terminations so as to change the
measuring device reference point near the end of coaxial cable. For lumped element filter, the inductor and capacitor are soldered to the FR4 board. A distributed Filter for narrow bandwidth is
designed using ADS which is constructed by microstrip lines, is then fabricated by milling machine.
The directional coupler is used Mixer for RF frequencies and its characteristics are measured. Various loss and measurement factors are derived such as conversion loss, Isolation between ports. Also,
the important component of any communication device, a antenna is designed. A simple patch antenna is designed and is used a receiver so as to check my design efficiency over the known transmission
by cushcraft antenna.
A cushcraft antenna is analyzed and its s parameters are studied so as to identify the use of cushcraft antenna in communication systems. In addition RF devices, different types of modulation
techniques are studied. AM, FM, PSK, QAM,QPSK along with how to plot the interpolation points of these modulation techniques.
At the end of the course year, we then obtain two filters (lumped and distributed), a mixer(directional coupler) and a patch antenna. Using my own design of RF devices, i was able to construct my own
RF receiver test bench model. With the known transmitted data, I now obtain the received data and check for efficiency over the entire test bench set up.
With addition to RF/MW circuits-1 and WAMI laboratory, I opted 3 more RF related courses because of my interest towards this field. I took RF measurements course-(FALL 2019), in which it deals with
all type of practical adaptation of Amplifier characteristics like 1-dB compression point, Third Order Intercept point and Noise figure. These measurements gives the range of amplifier in which it
will be used as a linear gain amplifier(1dB compression point), how far can the amplifier be pushed with considerate amount of non linearity(third order intercept point) and how much noise the
amplifier can accept in order to use. In addition to the amplifier characteristics, I designed my own amplifier without any bias networks and lumped elements using MMIC of MOSFET and embedded
microstrip FET. This is done by On-Wafer probing using 750?, 650? and 350? thickness probe tips by both SOLT and Thru calibration standards.
SPRING 2020- With all these interesting concepts of RF devices, to get to know more about theoretical side of RF devices I opted for RF/MW circuits-2 course. This course gave me an insight of one
step closer in designing my own LNA and Power amplifier. The course is mainly experienced with smith chart. It covers from basic transmission line theory to all over transistor characteristics. I
also opted RF Power amp design to enlarge my scale of knowledge in the filed of RF amplifiers. This course gave me the importance of various concepts of operating classes of transistor and its
voltage dependency. Although this design is done over QORVO based HEMT device and all the design work is carried on ADS. With load pulling and transistor characteristics, input and output matching
circuit are designed. These circuits are then converted to microstrip line layout design to be ready to fabricate.
BROADBAND COMMUNICATION – SPRING 2019
The first sequence of my course work is completely dependent on Designing of RF devices. The interest of RF communication is mainly based on aspect “wireless”. Although for this fast moving world,
the technology is in the hands of each individual which mostly wireless devices. The most common thing every individual expects is privacy on their own.
Cite this page
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Nyquist Theory For Rotational Order Analysis | Technomax
Nyquist Theory for Rotational Order Analysis
The Nyquist theory has been used since the last 50 years for data analysis in various related areas such as telegraphic transmission, multiplex systems, the theory of communication and the treatment
of noise. The Nyquist theory is defined such that in order to properly reproduce a signal it should be periodically sampled at a rate that is 2X the highest frequency that is intended to be recorded.
In case of images, frequency is related to the size of the structure. Structures smaller in size are said to have a higher frequency. Thus, the imaging sample rate (or pixel) size should be 1/2 the
size of the smallest object you wish to record.
In the analysis of frequency, the input signal is sampled using a sampling clock of the frequency that is obtained from the crystal oscillator inside the FFT analyser and 2.56 times the frequency
range. When analysing the vibration or noise of a rotating body whose rotation speed changes by using this sampling method, the number of samplings per rotation will change as this is always
dependant on rotation speed because the frequency of the sampling clock is constant.
On the other hand, if sampling is performed using a sampling clock synchronized with the rotation speed, say, a signal of 64 pulses per rotation, the number of signal samples per rotation does not
change even if the rotation speed changes. Whenever FFT analysis is performed on the signal of vibration / sound that is sampled in the clock synchronized with the speed of rotation, the unit of the
X axis is not frequency (Hz) but order (Order). The data displayed as the power spectrum of the order component is called rotational order ratio analysis.
Also read:
Sampling Signal for Order Ratio Analysis
Inaccurate measurements occur regularly in data acquisition as a result of improper sampling times. An understanding of proper sampling times when collecting data with an analogue-to-digital
converter or video camera is crucial in order to avoid anomalies. A proper choice of sampling times should be based on the Nyquist theory. If the sampling time is chosen judiciously, then it is
possible to accurately determine the frequency of a signal, which varies periodically with time.
Order analysis is used to quantify noise or vibration in rotating machinery whose rotational speed changes over time. An order refers to a frequency that is a certain multiple of a reference
rotational speed. In the case of normal analysis of frequency, the frequency of the sampling clock is 2.56 times of frequency range. Similarly, in the case of order ratio analysis, the number of
sampling clocks per rotation must be 2.56 times the maximum order. The frequency resolution of the frequency analysis by the internal sampling clock is 1/400 of the set frequency range when the
analysis data length is 1024 points, it is 1/800, when 2048 points. It implies that you can read the spectrum every 2.5 Hz.
When order ratio analysis is performed by an external sampling clock, the relationship between the maximum analysis order and its resolution is calculated as follows:
Order resolution can be obtained by this equation without considering rotation speed
The Effect
The implications of the sample rate and the highest observable frequency in order to avoid aliasing are widely known when dealing with constant-time step sampling, where digital values are measured
at equal increments of time. There is far less familiarity with the relevant relationship when dealing with orders, where an order is a multiple of the rotational rate of the shaft. For example, the
second order is a rate that is exactly twice the current rotational speed of the shaft. The point of consideration here is the relationship between the rate at which data is collected from a
rotating shaft and the highest order to avoid aliasing.
The relationship depends on whether sampling is done at constant time steps (equi-time step sampling) or at equal angles spaced around the shaft (equiangular or synchronous sampling). But before
considering either of them, revisiting the relationship between regular equi-time step sampling and the highest frequency permissible is advisable to avoid aliasing.
With regular time-based sampling using uniform time steps, we have a sample rate of say S samples/second. That is digital values are taken 1/S seconds apart. For convenience let del=T be the time
increment in seconds so that del T= 1/S seconds. With regular time domain processing, we have a time and frequency relationship. That is if we carry out a Fourier analysis of a regularly spaced
time history then we get a frequency spectrum. Shannon’s aliasing theorem states that if we have a sample rate S then the highest frequency, one can observe without aliasing is (S/2) Hz. (S /2) is
known as the Nyquist Frequency
Nyquist theory is widely understood, but this understanding generally relates to time sampling and the conversion to the frequency domain. Rotational order analysis and the effect of the Nyquist
frequency are, however, less understood. Most engineers appreciate the basic concept that Harry Nyquist defined that was improved upon by Claude Shannon. To recreate the frequency of interest, at
least two samples are required. More the merrier though.
Conclusion - Nyquist Theory
The maximum order that can be analysed with synchronous data can be represented by O[max] = N/2; Where the highest order is O[max] and N is the synchronous sample rate, in samples per revolution.
Ergo, the highest order that can be analysed is the synchronous sample rate divided by two. The maximum order that can be analysed with time-based data can be represented by O[max] = S/(2*R); Where
the highest order is O[max] and S is the time sample rate, that is samples per second and R is the shaft speed in revolutions per second. Ergo, the highest order that can be analysed is the time
sample rate divided by twice the revolutions per second.
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in Mathematics
The Bachelor of Arts in Mathematics is designed to meet the needs of students with a wide variety of interests. All mathematics majors complete a basic core of required mathematics courses and then
choose elective courses to tailor the program of study to meet their individual goals. Along with the standard program of study for the B.A. in mathematics, the department also offers a Concentration
in Statistics and a Teachers Option. The Concentration in Statistics is designed to prepare students for careers in industry or for graduate study in statistics or data science. The Teachers Option
requires students to choose courses that meet the requirements for state certification in mathematics.
Required Courses
All students seeking the B.A. in mathematics must complete the following six courses:
Remaining Coursework
The remaining coursework will depend on the particular program of study chosen. The remaining requirements for the Concentration in Statistics and Teachers Option are described separately, below.
Students must complete four additional 3000- or 4000-level MATH or STAT courses numbered above 3120, including at least two 4000-level MATH or STAT courses and one of the five sequences in
fundamental areas of mathematics:
Additionally, students seeking the B.A. in mathematics are expected to learn basic computer programming and are required to complete either CSCI 1060 Scientific Programming or CSCI 1300 Introduction
to Object-Oriented Programming.
A GPA of 2.00 (“C” average) or higher is required in 3000- or 4000-level MATH and STAT courses counting toward the major.
Concentration in Statistics
The remaining coursework for students seeking the B.A. in Mathematics with a Concentration in Statistics must include
along with two additional courses from the following list:
Students choosing the Concentration in Statistics must complete CSCI 1300 Introduction to Object-Oriented Programming.
Teachers Option
The remaining coursework for students seeking the B.A. in Mathematics with the Teachers Option must include
along with one additional course chosen from the following list:
Students seeking the Teachers Option are not required to complete a course in computer programming.
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The Ultimate Step by Step Guide to Preparing for the WY-TOPP Math Test
If you’re looking for the most effective WY-TOPP Math strategies ever published, WY-TOPP Grade 6 Math for Beginners is the perfect resource. This comprehensive study guide is designed to provide all
the tools your student needs to succeed on the WY-TOPP Math test 2023.
WY-TOPP Grade 6 Math for Beginners includes comprehensive study guides, explanations, examples, and practice exercises with answers for every single topic on the grade 6 math test. This guide is an
invaluable resource that will help your student ACE the WY-TOPP Math test.
The updated version of WY-TOPP Grade 6 Math for Beginners 2023 includes:
• A step-by-step guide to teach students the best strategies for success on the WY-TOPP Math test
• Thorough explanations for each math subject
• Numerous practice tests in different formats, including fill-in-the-blank, free response, and multiple choice
• Two realistic and full-length practice tests with detailed answers
An important feature of WY-TOPP Grade 6 Math for Beginners is that it covers all WY-TOPP Math topics on the 2023 test. The variety of practice tests helps students become familiar with different
types of questions and prepare for the test with confidence. The explanations for all practice test questions are provided to help students understand the methods to solve each question.
Math can be a challenging subject, but a comprehensive study guide with in-depth explanations can encourage students to study and comprehend math. WY-TOPP Grade 6 Math for Beginners is an
all-inclusive study resource that leaves no stone unturned. It is perfect for both self-study and classroom usage.
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LIV.DAT Virtual Seminar Series - Spring 2022
Date: Tuesday 26 April 2022 – 14:00 (Europe/London)
Speaker: Dr Vitaliy Kurlin, Reader in the Computer Science Department and Materials Innovation Factory, University of Liverpool
Any solid crystalline material (periodic crystal) consists of periodically translated unit cells of atoms or molecules. Since crystal structures are determined in a rigid form, the strongest and most
practical equivalence of crystals is rigid motion (a composition of translations and rotations) or isometry (also including reflections). In the past, periodic crystals were classified by coarser
equivalences such as space groups into 219 types (230 if mirror images are distinguished). However, the world's largest Cambridge Structural Database (CSD) of 1.1M+ existing crystals requires finer
The Data Science Theory and Applications group at the Liverpool Materials Innovation Factory developed generically complete and continuous isometry invariants for periodic sets of atomic centres.
Computing these invariants for all 660K+ periodic crystals (full 3D structure; no disorder) in the CSD through 200 billion+ pairwise comparisons over two days on a modest desktop detected five pairs
of "identical needles in a haystack". For example, the CSD crystals HIFCAB and JEPLIA are truly isometric, but one atom of Cadmium is replaced by Manganese, which should inevitably perturb a local
geometry of atoms. As a result, five journals are now investigating the data integrity of the underlying publications.
These experiments justified the Crystal Isometry Principle, which states that any real periodic crystal is uniquely determined by its geometry of atomic centres without chemical information. Then all
known and undiscovered periodic crystals live in a common Crystal Isometry Space (CRISP), so that Mendeleev's periodic table representing individual elements, categorised by only two discrete
parameters (atomic number and group), can be extended into a continuous space for all solid crystalline materials. For instance, diamond and graphite, both consisting purely of carbon, have different
locations in CRISP.
Dr Vitaliy Kurlin is a Reader in the Computer Science Department and Materials Innovation Factory at the University of Liverpool. His research is developing the emerging area of Periodic Geometry
within Mathematical Data Science to resolve the long standing challenges of Crystallography and Materials Science. Since 2017 he has lead the Data Science Theory and Applications group at the
University of Liverpool and since 2021 he is the Director of the Liverpool doctoral network AI for Future Digital Health. Vitaliy obtained his MSc in Mathematics and a PhD in Geometry and Topology
from Moscow State University.
You can now watch the seminar on YouTube: https://youtu.be/tTuqjYtPVtU
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Hyperbolic cosine integral: Integer value graphics
Saunders graphics
Saunders graphic of , where denotes the th digit in base .
Saunders graphic of , where denotes the th digit in base .
Values at integer arguments
Strang graphic of the fractional parts of at the integers .
Strang graphic of the first three digits of the decimal expansion of at the integers .
and over the at integer ‐values. (Here ⌉ is the round function.)
Gaussian primes of the form over the at integer ‐values. (Here ⌉ is the round function.)
Truchet pattern of the form over the at integer ‐values.
Truchet pattern of the form over the at integer ‐values. (Here ⌉ is the round function.)
Argument and absolute value of the discrete Fourier transform of at Gaussian integer arguments.
Integer values
Points in the such that is a Gaussian integer.
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Isaac Newton Institute for Mathematical Sciences
This institution has 204 collections.
Collection search results
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100% renewable energy by 2050? This is an ambitious vision that will need significant change to our energy systems. It will require the development of an integrated power grid and continuous and
steady transformation of the UK power system can only happen if fundamental interdisciplinary research...
Institution: Isaac Newton Institute for Mathematical Sciences
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During 2012 the Newton Institute is planning a number of events to celebrate the 20th anniversary. Also includes events organised for the centenary celebration of the life and work of Alan Turing.
Read more @ http://www.newton.ac.uk/20/
Alan Turing Year:...
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Thursday 22nd April 2021
In foundational ontology, 4-dimensionalism is shorthand for a mathematical-philosophical basis for a rigorous global identity criterion based upon composition. It acquired this name as a vital part
of the approach is treating individuals as extended in time as...
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We are a national and international visitor research centre and run research programmes on selected themes in mathematical sciences, with applications in a wide range of science and technology.
Institution: Isaac Newton Institute for Mathematical Sciences
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The Newton Gateway to Mathematics acts as a knowledge intermediary for the mathematical sciences. It is the impact initiative of the Isaac Newton Institute for Mathematical Sciences (INI). Supported
by INI and the University of Cambridge, the Newton Gateway to Mathematics reaches out to and engages...
Institution: Isaac Newton Institute for Mathematical Sciences
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With a rapidly ageing global population and challenges such as the growth of antibiotic resistance, there has been significant growth in the global incidence of chronic and infectious health
conditions. Furthermore, the number of people living with two or more chronic health conditions...
Institution: Isaac Newton Institute for Mathematical Sciences
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In recent years there has been an explosion of complex data-sets in areas as diverse as Bioinformatics, Ecology, Epidemiology, Finance and Population genetics. In a wide variety of these
applications, the stochastic models devised to realistically represent the data generating processes are very...
Institution: Isaac Newton Institute for Mathematical Sciences
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Numerical modelling is used to good effect in a number of applications and advances in geometric and structure preserving methods will progress this field further. These methods are a special class
of numerical algorithms used to compute solutions to differential equations that...
Institution: Isaac Newton Institute for Mathematical Sciences
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Lie theory has profound connections to many areas of pure and applied mathematics and mathematical physics. In the 1950s, the original "analytic" theory was extended so that it also makes sense over
arbitrary algebraically closed fields, in particular, fields of positive characteristic....
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Next generation (Quantum) computers promise to speed up some mathematical processes by orders of magnitude, but new algorithms and software will need to be developed to exploit this power. Although
general-purpose quantum computers are some years away, work should start on the software now. As...
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Analysis on graphs and other discrete structures has been developing for quite some time, in particular due to applications to number theory, algebra, probability theory, spectral geometry, as well
as to its usefulness in many practical problems. This area, however, has experienced recently a...
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Asymptotic analysis and perturbation methods can provide approximate solutions and analytical properties to a broad range of problems where an exact solution cannot be found. They are therefore some
of the most critically important tools in mathematics and theoretical physics....
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Approximation theory is the study of simulating potentially extremely complicated functions, called target functions, with simpler, more easily computable functions called approximants. The purpose
of the simulation could be to approximate values of the target function with...
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Approximation theory is the study of simulating potentially extremely complicated functions, called target functions, with simpler, more easily computable functions called approximants. The purpose
of the simulation could be to approximate values of the target function with...
Institution: Isaac Newton Institute for Mathematical Sciences
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In a number of problems, both in theory and applications, one faces a situation when the ambient dimension is extremely high. Such problems often include approximating, sampling, or compressing
functions on high-dimensional domains. Classical methods fail to be effective in this case due to the...
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The EPSRC Centre for Mathematical Imaging in Healthcare (CMIH) will hold an engagement event in October 2019. This will aim to showcase the research that is being carried out at the Centre and will
present an opportunity to hear in detail about some of the current project...
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Over the course of the COVID-19 pandemic, modelling has taken centre stage both in forecasting, policy formulation and in informing the public, featuring prominently in the advice given to government
in the UK and beyond. The pandemic has had profound influence on social and economic...
Institution: Isaac Newton Institute for Mathematical Sciences
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Nash Equilibrium Game Theory
Applied Game Theory Project Mohit Batham (11426) In the project, I have looked into 3 different problems related to Bollywood Industry. The games that I have solved are: Clash of Bollywood movies on
Big Weekends Signaling Game in Bollywood: Producers and Viewers Location Problem with Directional Constraints: An Application to Movie Shows Clash of Bollywood movies on Big Weekends Introduction
Entertainment has become a very important part of our life. One source of entertainment
Introduction What is a game? When there are more than one actors making choices, they are playing a game. That's to say, when you play a game, the consequence does not depend merely on your decision
but also your rivals’. Usually, players play for fun with low risks, as poker and chess, however, no one can deny that playing games can be very serious. In order to reduce stakes and get better
result, game theory appeared which is defined as the science of interactive decision-making. It is practical
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Stabilization of walls for nano-wires of finite length
p. 1-21
Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation
p. 22-35
BV solutions and viscosity approximations of rate-independent systems
p. 36-80
Dynamic Programming Principle for tug-of-war games with noise
p. 81-90
Homogenization of many-body structures subject to large deformations
p. 91-123
Global optimality conditions for a dynamic blocking problem
p. 124-156
On a Bernoulli problem with geometric constraints
p. 157-180
Weak notions of jacobian determinant and relaxation
p. 181-207
Controller design for bush-type 1-d wave networks
p. 208-228
A phase-field model for compliance shape optimization in nonlinear elasticity
p. 229-258
Dimension reduction for functionals on solenoidal vector fields
p. 259-276
Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity
p. 277-293
Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market
p. 295-317
The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations
p. 318-342
Approximation by finitely supported measures
p. 343-359
A discussion on the Hölder and robust finite-time partial stabilizability of Brockett's integrator
p. 360-382
A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
p. 383-400
Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks
p. 401-426
Spectral analysis in a thin domain with periodically oscillating characteristics
p. 427-451
Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints
p. 452-482
Full convergence of the proximal point method for quasiconvex functions on Hadamard manifolds
p. 483-500
Rayleigh principle for linear hamiltonian systems without controllability
p. 501-519
Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities
p. 520-547
Indirect stabilization of locally coupled wave-type systems
p. 548-582
Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1
p. 583-610
Exponential convergence for a convexifying equation
p. 611-620
On the continuity of degenerate n-harmonic functions
p. 621-642
Invariant measures and controllability of finite systems on compact manifolds
p. 643-655
Stability and stabilizability of mixed retarded-neutral type systems
p. 656-692
Optimal convex shapes for concave functionals
p. 693-711
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations
p. 712-747
Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
p. 748-773
Flat outputs of two-input driftless control systems
p. 774-798
A Hölder infinity laplacian
p. 799-835
Linearization techniques for 𝕃 ∞ See PDF-control problems and dynamic programming principles in classical and 𝕃 ∞ See PDF-control problems
p. 836-855
Root growth: homogenization in domains with time dependent partial perforations
p. 856-876
Stability of retarded systems with slowly varying coefficient
p. 877-888
On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping
p. 889-913
Deterministic characterization of viability for stochastic differential equation driven by fractional brownian motion
p. 915-929
Multiplicity of solutions for the noncooperative p-laplacian operator elliptic system with nonlinear boundary conditions
p. 930-940
Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket
p. 941-953
Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem
p. 954-968
Nash equilibria for a model of traffic flow with several groups of drivers
p. 969-986
Linear-quadratic optimal control for the Oseen equations with stabilized finite elements
p. 987-1004
Dynamic programming principle for stochastic recursive optimal control problem with delayed systems
p. 1005-1026
An analysis of electrical impedance tomography with applications to Tikhonov regularization
p. 1027-1048
On convex sets that minimize the average distance
p. 1049-1072
Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations
p. 1073-1096
Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space
p. 1097-1121
Adaptive finite element method for shape optimization
p. 1122-1149
The structure of reachable sets for affine control systems induced by generalized Martinet sub-lorentzian metrics
p. 1150-1177
A variational problem for couples of functions and multifunctions with interaction between leaves
p. 1178-1206
Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support
p. 1207-1224
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Ethernaut Lvl 3 Coin Flip Walkthrough: how to abuse psuedo randomness in smart contracts
This is a in-depth series around Zeppelin team’s smart contract security puzzles. I’ll give you the direct resources and key concepts you’ll need to solve the puzzles 100% on your own.
This levels requires you to correctly guess the outcome of a coin flip, ten times in a row.
How Ethereum generate “randomness”
There’s no true randomness on Ethereum blockchain, only random generators that are considered “good enough”.
Developers currently create psuedo-randomness in Ethereum by hashing variables that are unique, or difficult to tamper with. Examples of such variables include transaction timestamp, sender address,
block height, etc.
Ethereum then offers two main cryptographic hashing functions, namely, SHA-3 and the newer KECCAK256, which hash the concatenation string of these input variables.
This generated hash is finally converted into a large integer, and then mod’ed by n. This is to get a discrete set of probability integers, inside the desired range of 0 to n.
Notice that in our Ethernaut exercise, n=2 to represent the two sides of a coin flip.
Example of input variables that are often cryptographically hashed
This method of deriving pseudo-randomness in smart contracts makes them vulnerable to attack. Adversaries who know the input, can thus guess the “random” outcome.
This is the key to solving your CoinFlip level. Here, the input variables that determine the coin flip are publicly available to you.
Detailed Walkthrough
Let’s create a malicious contract that checks the outcome of the coin flip.
Only when you’ve correctly guessed the outcome, should you invoke the real contract’s flip(bool _guess) function.
1. Inside Remix IDE, create a malicious contract that closely mirrors CoinFlip.sol:
2. Implement a hackFlip() function that predicts the flip outcome, using the same logic and input variables as the original contract. Since you also know blockhash and block.number, you’re able to
accurately predict the correct _guess.
Your function should only invoke the originalContract.flip() with the correct _guess.
3. Call your hackFlip() function 10 times. The original contract’s consecutiveWins counter should steadily increase, as you are only making correct guesses.
Key Security Takeaways
• There’s no such thing as true randomness
• Be careful when calculating “randomness” in your contract (or even when inheriting from an existing random numbers library). In cases where you use randomness to determine contest winners,
remember that adversaries can easily guess the random outcome and hack your game!
More Levels
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How to Figure Hardwood Flooring Nails | Homesteady
How to Figure Hardwood Flooring Nails
Nails secure your hardwood flooring to the subfloor or framing members. To determine how many nails you need to secure your flooring, you have to consider room size and nail spacing. It is also a
good idea to figure in an extra amount in case a nail breaks or bends during installation. Figuring the number of nails you need requires a few simple calculations.
Step 1
Measure the length and width of the room. Multiply the measurements to figure out the room's square footage. Before you can figure out how many nails you need, you have to determine how many hardwood
floorboards the room requires.
Step 2
Decide on flooring board width. Typical widths for hardwood flooring are 2 1/4, 3, 5 and 7 inches.
Step 3
Estimate the number of bundles or boxes of hardwood flooring you need. Hardwood flooring manufacturers print an estimated square footage coverage on each box or bundle of floorboards. Divide the
estimated square footage coverage of your hardwood flooring into the total square footage of the room.
Step 4
Calculate how many boards you need. Multiply the number of hardwood floorboards per box by the number of boxes you need to cover the room.
Step 5
Determine the nail spacing requirements for your hardwood flooring, based on the width of the floorboards. Boards 2 to 2 3/4 inches wide need one nail every 8 to 10 inches. Boards 3 to 3 3/4 inches
wide require a nail every 6 to 8 inches. Boards 4 to 7 inches wide need one nail every 6 inches.
Step 6
Calculate the number of hardwood flooring nails. Multiple the number of boards calculated in Step 4 with the nail spacing requirements in Step 5.
For instance, you need 500 2 1/4-inch floorboards to cover a room, with one nail placed every 8 to 10 inches. Multiply 500 by 8 to calculate the low estimate of 4,500 nails. Multiply by 10 to get the
high estimate of 5,000 nails. In this example, you need 4,500 to 5,000 hardwood flooring nails to secure the floor.
Writer Bio
Sue-Lynn Carty has over five years experience as both a freelance writer and editor, and her work has appeared on the websites Work.com and LoveToKnow. Carty holds a Bachelor of Arts degree in
business administration, with an emphasis on financial management, from Davenport University.
Photo Credits
• hardwood floor texture image by GoodMood Photo from Fotolia.com
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EBCT Calculator
Home » Simplify your calculations with ease. » Health »
EBCT Calculator
Understanding water treatment processes is key for quality assurance, and one crucial aspect is the Empty Bed Contact Time (EBCT). Our intuitive EBCT calculator makes calculating this vital factor a
EBCT, or Empty Bed Contact Time, is a vital measure in water treatment systems. It quantifies the contact duration between the water and the media in a filter, reflecting treatment efficiency.
Our EBCT calculator operates using a straightforward principle. It uses the volume of the empty bed and the flow rate as inputs to generate the EBCT. As the results are instant, it simplifies the
process of monitoring water treatment effectiveness.
The EBCT Formula: A Detailed Overview
Here’s how the formula works:
EBCT (minutes) = V (volume of empty bed in gallons) / FR (flow rate in GPM) The formula essentially divides the empty bed volume by the flow rate to calculate the Empty Bed Contact Time, indicating
the time water spends in the filter bed.
Suppose we have an empty bed volume of 343 gallons and a flow rate of 43 GPM. Using the EBCT formula:
EBCT = 343 / 43 = 7.98 minutes This means water spends approximately 8 minutes in contact with the filter media, suggesting ample time for effective treatment.
Applications of the EBCT Calculator
Water Treatment
In water treatment facilities, EBCT calculator is used to evaluate system efficiency, aiding in ensuring safe and clean water.
Industrial Wastewater Management
EBCT calculation is equally vital in industrial wastewater management, enabling industries to ensure they meet environmental guidelines.
Frequently Asked Questions
Why is EBCT important?
EBCT is critical as it measures water’s contact time with filter media. The longer the EBCT, the higher the chance for pollutants to be removed, enhancing water quality.
How does flow rate affect EBCT?
Flow rate inversely impacts EBCT. Higher flow rates shorten the EBCT, potentially reducing treatment effectiveness, thus requiring careful management.
How can I improve my system’s EBCT?
Improving EBCT often involves reducing flow rate or increasing the bed volume. However, it’s a balance as drastic changes might affect other aspects of the system.
Mastering the concept of EBCT is essential for anyone in the water treatment industry. Using our EBCT calculator, you can conveniently compute the EBCT, ensuring optimal treatment process efficiency.
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Quantification of Model Uncertainty (Part 2) | ValidMind
Quantification of Model Uncertainty (Part 2)
While Model Risk Management (MRM) principles are important, meeting the expectations set by regulators (FED 2011, PRA 2023) and fostering a robust risk management framework can be challenging or even
confusing at times. We believe all MRM practitioners must develop a good understanding of these principles and actively implement them into the organization’s risk management culture.
Embarking on the second journey of our series dedicated to uncertainty in model risk management, it’s important to keep in mind the crucial points we covered in Understanding Uncertainty for Model
Risk Management (Part 1). Recognizing the significance of uncertainty in the modelling process and adopting a prudential modelling culture is not only beneficial, but essential in making informed
business decisions.
We previously highlighted how a systematic approach to uncertainty can lead to more transparent and accountable models, robust testing during model development, and an effective challenge between the
first-line and second-line of defense. Additionally, it enables us to create quantifiable metrics for risk appetite, identify early warning indicators for ongoing monitoring, generate challengeable
metrics for model tiering, and implement effective model conservatism.
However, acknowledging and implementing these practices are only the first steps in managing model uncertainty. Given that the ultimate purpose of quantitative models is to present the most plausible
options rather than definitive answers, we must understand how model uncertainty arises.
In this second article, we will take a deeper dive into model uncertainty. We’ll explore quantitative methods for assessing uncertainty using simple modelling case studies that will help us gain a
more comprehensive understanding of the principles and practices we discussed in part one.
We hope to offer model risk professionals with more tangible, actionable insights around this important topic that can help efficiently navigate the current and new model risk management regulatory
Interpreting Uncertainty
Understanding the different sources of uncertainty is valuable for assessing which deficiencies can potentially be mitigated by further investigation in many cases. This has direct applications in
model risk regulation, such as the previously mentioned Margin of Conservatism. Appropriate quantification of uncertainty is also important to communicate model results transparently and avoid
creating a false sense of risk to users. When using intervals around point estimates, it is important to know, for example, whether those intervals correspond to random variations around the
long-term mean (aleatoric uncertainty) or the actual point estimate (the combination of aleatoric and epistemic uncertainty, defined earlier as model uncertainty). Both can be useful, if they are
understood. However, prediction points are indeed more difficult to predict than the expected mean, so the prediction intervals (uncertainty around the point estimate) will be wider than the
confidence intervals (uncertainty about the mean).
We illustrate model uncertainty in a classical statistical setting. These examples, although simple, are key building blocks of widely used models in finance. The concepts explained here can be
applied to more general cases beyond linear regression, such as Generalised Linear Models, Time Series models or a combination of models.
A Simple Case
Suppose that one of the inputs to your decision-making model is an estimate of how many like-dislike feedbacks in the next quarter will be likes, knowing that you receive 500 feedbacks per quarter
and on average, 70% are likes. The natural thing to do seems obvious, and is to set a fixed parameter:
However, if we see all quantitative estimates as what they really are, wrong by nature (Box, 1976), the average will be perceived not as a fixed parameter, but rather as the most plausible estimate
of that parameter. The next question for a prudential modeller then is: how can I extract all the plausible parameter values given the data?
In this simple example, we can represent the data as an independent 1/0 process likes which can be appropriately modelled, although only as an approximation, with a binomial distribution with model
parameters and :
We can now produce a point estimate for the parameter . After sampling once from the Binomial distribution , we obtain:
To extract variability around the parameter estimate 0.7 (aleatoric uncertainty), we repeat the same experiment multiple times, by sampling multiple times (1000 is good enough) from the likes
The modeller can now use this distribution to, for example, propagate uncertainty to downstream models, or to perform a risk impact assessment against different parameter values and assign
probabilities to those scenarios.
Linear Regression
We now build a simple linear regression model to illustrate model uncertainty in a slightly more realistic setting. We try to predict the salary of employees based on their years of experience. We
notice that the scale of the target and explanatory variables is significantly different, so we proceed to normalise the dataset by applying the logarithm to the salary. The table below shows the
main features of the dataset:
Measure Years of Experience Salary Log Salary
n 30 30 30
min 1.1 37,000 10.5
mean 5.3 76,000 11.17
max 10.5 122,000 11.72
We draw histograms to identify the distribution of our data, that may inform the modelling choices:
We check the relationship between Years of Experience and Log Salary:
This relationship can be approximated by a classical linear regression model:
where represents the intercept, and the coefficient estimate for the years of experience. In this specification, the error term of the model (observed vs predicted) assumes an independent normal
distribution with mean 0 and standard deviation :
After fitting this regression model to the available data, we obtain the following estimates:
We can now use this specification to produce a point estimate that predicts, for example, the salary of a person with 15 years of experience:
We obtain the Salary predictive mean and standard deviation by exponentiating the model output:
Sources of Model Uncertainty
The epistemic uncertainty in this model (the lack of knowledge about the data generating process) is captured by , the standard deviation of the residuals. In other words, a measure of the average
distance of each observation from its model prediction. This error can be potentially reduced by improving the model. The second source, aleatoric uncertainty (variance in the model parameters), is
embedded in the estimates and .
Defining Uncertainty in the Model
One useful way to visualise these variances is to write the model in probabilistic terms, where each parameter of the model is represented by probabilities distributions , and , centred around the
model estimates , and , respectively:
the multivariate normal distribution includes the univariate distributions for and with mean and , respectively, and variance capturing the parameter uncertainty (or epistemic uncertainty),
represented by . The latter is the estimated covariance matrix of the model parameters, where is the unscaled estimated covariance matrix. Note that, usually, the output of model fits gives the
scaled version () of the estimated covariance matrix. The diagonal elements in are the estimated variances of and , and the off-diagonal elements represent the correlation of these predictors.
Next, we extract the aleatoric uncertainty from our model by adding variability to the model error term , using a chi-squared distribution (appropriate for modelling variances):
with as the number of data points (30) and the number of predictors (2).
Extracting Uncertainty from the Model
We can now simulate from this probabilistic model (Gelman, 2007) and extract uncertainty from the previously estimated parameters and the error term:
For each draw out of the number of simulations (example: 1000):
The results of this inferential simulation produce univariate distributions , and , for each model estimate , and , coherent with this dataset and this model:
This gives us flexibility in propagating uncertainty about any combination of plausible values of parameters and model errors, based on the data available.
Predictions Accounting for Model Uncertainty
The final step is to obtain predictions using the estimation uncertainty around the coefficient estimates and the error term of the model. To accomplish this, we compute multiple predictive paths
following these simple steps:
1. We generate simulations from the probabilistic model that accounts for uncertainty in both the coefficient estimates and the model error. The simulations are designed to take into account both
epistemic and aleatoric uncertainty. The generation of multiple possible paths of outcomes during the simulations reflects these uncertainties and helps capture a range of possible future
2. Once the simulations are completed, the generated data for all the simulated paths are combined into a single dataset. This dataset represents the spectrum of potential outcomes considering the
aforementioned uncertainties. This combined dataset forms the basis for the subsequent analysis and visualization of the simulated outcomes.
The figure below shows the predictive distribution of Log Salary, created from the above simulation procedure, that captures both sources of model uncertainty:
After applying , we obtain the Salary levels:
These simulations model potential paths for the relationship between years of experience and log salary, accounting for the model uncertainties involved. For example, a given amount of experience
might lead to different salaries depending on other unknown or uncontrolled factors, and this is shown in the forecast plots below, both for Log Salary and Salary.
We can see in this plot that salary distributions are often observed to be right-skewed. This may mean, for example, that there are a small number of individuals with very high incomes while the
majority of individuals have relatively lower incomes.
These uncertainty measures can serve, for instance, as a basis for setting risk appetite, allowing banks to establish appropriate thresholds for acceptable deviations in salary projections. Moreover,
incorporating an add-on derived from the uncertainty analysis, we can prudently account for potential variations in salaries over business planning projections.
The integration of uncertainty into critical models ensures a more realistic and robust approach, enabling financial firms to strategically allocate resources in the face of uncertain assumptions.
In this second part of the series, we presented a validation practice to deconstruct and understand model uncertainty. This involves creating predictions against different scenarios as a combination
of different levels of aleatoric and epistemic uncertainty (examples: severe, medium, baseline). We can compute these different prediction paths by sampling different parts of the distribution of the
model estimates (epistemic) and standard error (aleatoric).
We can also entirely switch off and on different sources of uncertainty and see the impact on predictions. This can be used to set for example Post Model Adjustments (PMAs) tailored to different
model deficiencies, as opposite to use one general PMA to mitigate the overall model error. A more granular approach to PMAs allows MRM professionals to challenge the effectiveness and usage of these
PMAs. This enables validators and developers to identify efforts where model improvement is possible, and cases where deficiencies are of random nature (examples: data quality issues, error
measurements, etc.) and need to be addressed differently, and perhaps by other teams (example: improving the data engineering system).
Inferential simulation (Gelman, 2007) can be applied in the same way to other statistical models, such as Generalised Linear Models or Time Series, to extract estimation uncertainty from each
parameter, and create a probability distribution of plausible model outcomes.
In the next article of this series we will explore alternative techniques to quantify uncertainty in more realistic and complex case studies.
• Bank of England (2023). SS1/23 Model Risk Management Principles for Banks.
• Federal Reserve System (2011). SR11-7 Supervisory Guidance on Model Risk Management.
• Gelman (2007). Data analysis using regression and multilevel/hierarchical models.
• Box (1976). Science and statistics. Journal of the American Statistical Association, 71 (356): 791-799.
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Terra Incognita Project
TS Terra Incognita Project #7: Wavelet Cycle Hunter
Fast Introduction
This module is designed for early revealing of cyclic phenomena in financial data and making the forecast based on these revealed cycles. Research shows that it is not so important how many cycles
you have found in the financial data (like as 55 or 100, or 200 trading days cycle), tit is much more important to find the strongest cycle NOW - the dominant cycle Therefore, the most important
issue is to reveal the appearance of the new dominant cycle/cycles as early as possible. This is why this module has been designed.
Let's start. Open Wavelet Cycle Hunter module and click "Calculate" button. In a matter of seconds you get this wavelet diagram:
The bright yellow zones correspond to the periods when some cycle is active. In our example, within two days, November 16 and 17, 171-bars cycle is active.
Now let create a wavelet, a wave form that is based on our 171 bars cycle. It is very simple to do in this program; just drag the mouse over this yellow region - the region where our cycle is active:
Immediately you get in the Main screen the projection line based on this cycle:
Now let's try to analyze some short term cycle. To do that, highlight "Emphasize Short" cycle. You will see a detailed wavelet diagram for short term cycles:
Drag the mouse cursor over another bright yellow zone that corresponds to 39-bars cycle:
Again, on the Main screen, immediately you will see the projection line which is based on the superposition of these two cycles, 171 and 39 bars cycles.
You can enable/disable any of these cycle:
Try to vary the amount of overtones to calculate the more/less detailed projection line:
Cycles Validation
Cycles in financial data are very tricky phenomena. Research shows that presence of bright zones in the wavelet diagram can be caused by random oscillations of the price. This is a specific feature
of financial data.
We recommend to perform an additional procedure to verify the importance of some cycle %X.
Suppose we have created a wavelet that is formed by two points on the wavelet diagram, A and B:
How will this wavelet forecast the future? We can see it. To do that, just set the "Target" option ON, - and you will see the price together with the wavelet projection line (aqua line):
The projection line after point B shows how this wavelet forecasts the future; here it is:
And we can see that there is no coincidence between the wavelet diagram (the black line) and the price (aqua line). It means that this cycle is not working.
Here is another example, an example of the wavelet that works:
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The Essentials of Geometry
Popular passages
A zone is a portion of the surface of a sphere included between two parallel planes.
... the three sides of one are equal, respectively, to the three sides of the other. 2. Two right triangles are congruent if...
The projection of a point on a plane is the foot of the perpendicular drawn from the point to the plane.
A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The...
A chord is a straight line joining the extremities of an arc ; as AB.
A sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre.
S' denote the areas of two © whose radii are R and R', and diameters D and D', respectively. Then, | = "* § = ££ = £• <§337> That is, the areas of two circles are to each other as the squares of
their radii, or as the squares of their diameters.
The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.
The perpendiculars from the vertices of a triangle to the opposite sides are the bisectors of the angles of the triangle formed by joining the feet of the perpendiculars.
A right circular cone may be generated by the revolution of a right triangle about one of its legs as an axis.
Bibliographic information
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DISMA - GEOMETRIC CONTROL AND APPLICATIONS TO QUANTUM MECHANICS - UGO BOSCAIN - INRIA - CNRS - UNIVERSITE PARIS SORBONNE
Categoria: Seminari e Convegni
Stato: Archiviata
25-27 gennaio 2022
DISMA (3rd floor)
The purpose of this course is to introduce the basic concepts in geometric control, from controllability
to optimal control. These concepts will be then applied to the problem of controlling simple quantum
mechanical systems that appear often in quantum technologies as Nuclear Magnetic Resonance and for
the realization of q-bits for quantum computers.
Lecture 1. The \control theory problem": controllability, stabilizability, optimal control. Example
of problems arising in quantum mechanics: Nuclear Magnetic Resonance, Stimulated Raman Adiabatic
Passages. Systems evolving on two di erent scales: averaging. Systems, with an unknown parameter.
The ( nite-dimensional) Schroedinger equation for the wave function and for the propagator.
Lecture 2. Families of vector elds. Lie groups and left invariant control systems. Lie brackets.
Frobenius theorem. Non-integrable vector distributions.
Lecture 3. Controllability 1. The Krener theorem, The Chow theorem.
Lecture 4. Controllability 2. Convexi cation. Killing the drift. The recurrent drift theorem. Applications to nite dimensional quantum systems: the Lie Algebraic Rank Condition. Controlling a Spin 1/
particle on the Bloch sphere.
Lecture 5. Optimal control 1. Formulation of the problem. Existence.
Lecture 6. Optimal control 2. The Pontryagin Maximum Principle (proof for minimal energy for affine
Lecture 7. Minimal energy for a 2-level system.
Lecture 8. Minimum time for a 3-level system.
Lecture 9. The adiabatic theorem: averaging. Population transfer for systems presenting conical
Lecture 10. Systems with an unknown parameter. Two level systems: chirp pulses. Three level sys-
tems: the STIRAP process.
[1] Jurdjevic, Velimir. Geometric control theory. Cambridge Studies in Advanced Mathematics, 52. Cam-
bridge University Press, Cambridge, 1997.
[2] D'Alessandro, Domenico Introduction to quantum control and dynamics. Chapman & Hall/CRC Ap-
plied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2008.
[3] A. Agrachev, D. Barilari, and U. Boscain. A Comprehensive Introduction to sub-Riemannian Ge-
ometry, volume 181 of Cam- bridge Studies in Advanced Mathematics. Cambridge University Press,
Cambridge, 2020. http://people.sissa.it/?barilari/Notes.html. xviii+746 pp.
Tuesday, January 25, h 14-18 in Auletta Seminari DISMA (3rd floor)
Wednesday, January 26, h 14-18 in Auletta Seminari DISMA (3rd floor)
Thursday, January 27, h 9-13 in Aula Buzano DISMA (3rd floor)
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What is this course about?
This lesson contains information about the course.
Welcome to the realm of Data Structures and Algorithms!
This course is designed for learners who need to prepare for interviews and brush up on their problem-solving skills, especially in regard to Data Structures and Algorithms in Python.
You will find a basic introduction and a few challenges for the following data structures:
• Stacks
• Singly Linked Lists
• Circular Linked Lists
• Doubly Linked Lists
• Arrays
• Binary Trees
• Binary Search Trees
Additionally, the course contains numerous problems and solutions concerning the following algorithms and techniques:
• Binary Search
• Recursion
• String Processing
Please note that this course requires basic familiarity with Python.
Let’s get started! I hope you have a great experience that enhances your problem-solving skills regarding data structures and algorithms in Python.
Get hands-on with 1200+ tech skills courses.
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2023 StL Game #131: Sunday, August 27 at Phillies
Viewing 25 posts - 1 through 25 (of 54 total)
• Author
• 12:35 p.m.
LHP Drew Rom (0-1, 14.73) vs. RHP Aaron Nola (11-8, 4.49)
BSM // KMOX
(and Bally Sports South/Southwest Extra)
Paid - Annual
Hasn’t Nola been mentioned as one of the free agent pitchers the Cards may pursue?
His numbers look pretty suck…. We already have guys that be a cut below average…
NOLA has a history of showing up for work every 5 days all season.
August 26, 2023 at 10:58 pm #232558
Paid - Annual
So has Mikolas. Their numbers are about the same.
We dont need more of the same trash. Mo needs to take the trash out and dump it and get us some real playahs up in here.
August 26, 2023 at 11:18 pm #232559
The Reds won over Arizona last night 8-7 in 11 innings. The Reds got three runs in the 10th and Arizona got three in the last of the 10th. The key in the last of the 10th was when McClain let a
DP grounder go right through his legs at 2ndbase. RIGHT THRU HIS LEGS!
Anyway, the Reds won because the DBacks pitcher – Crismatt – committed a balk in the top of the 11th with two outs.
Quite a game. At least the Reds games mean something.
August 27, 2023 at 11:06 am #232581
May need to switch to the Reds Rats.At least the uniforms look about the same. Maybe our baby birds (now a 4A team has found their true colors-baby blue!
August 27, 2023 at 11:08 am #232582
Could be ugly again today.
August 27, 2023 at 11:12 am #232584
It might be, it could be, it is…Rosie O’Donnell!
August 27, 2023 at 11:17 am #232585
SS Tommy Edman S
LF A. Burleson L
1B P. Goldschmidt R
DH N. Arenado R
C W. Contreras R
2B Nolan Gorman L
RF J. Walker R
CF R. Palacios L
3B T. Motter R
August 27, 2023 at 12:01 pm #232588
After that first rate thrashing they endured yesterday it’s nice to see Manny back in there. I’d like to see him moved up in the order, but the Redbird manager’s obviously providing some
protection for Palacios.
August 27, 2023 at 12:25 pm #232590
When Cardinal baseball starts conjuring up images of Rosie O’Donnell, it’s time for a break!
August 27, 2023 at 12:39 pm #232591
I wish you guys would stop it. Instead of birds on the bat, I’m seeing Rosie O’Donnell on a bat, and is a very disturbing visual.
August 27, 2023 at 12:43 pm #232592
So Edman leads it off with a double. Now lets see some sound fundamentals. The lefty should be able to pull one to the right side and advance him to 3rd with less than 2 outs.
August 27, 2023 at 12:45 pm #232593
So Burly can’t put it in play and strikes out. If you don’t have the skills, then bunt.
August 27, 2023 at 12:46 pm #232594
Now Edman gets caught too far off second on an infield grounder. Unbelievable NOOTBLAN. So now its a runner on first and two outs.
August 27, 2023 at 12:48 pm #232595
So it ends with a routine grounder, and we get nothing. I’ve seen a half inning so far and am already disgusted by the bungling and failure to execute.
Now we see how Rom does.
August 27, 2023 at 12:50 pm #232596
First pitch homer to CF off Rom.
August 27, 2023 at 12:53 pm #232597
Next guy hits a single to CF, but wait, Palacios brought his rubber glove and turns it into a man on second. E8.
August 27, 2023 at 12:55 pm #232598
Ricky Horton mentioned that the Cardinals have lost by more than 10 runs three times in the last 8 games. First time that has happened since 1908.
August 27, 2023 at 12:57 pm #232599
Now a walk, 2 on and 1 out. Already down 1-0 in the bottom of the 1st.
Rom gets out of the 1st only down 1-0, but he’s at 26 pitches. Lets see if we can get basic competence out of the offense in the 2nd.
Three strike outs in three tries isn’t the competency I was hoping for. So now we see if Rom can keep it close.
A little noise but no further harm. Rom at 49 pitches after 2. By current standards, he’s doing ok so far. In other words, got through 2 without getting blown out.
Rom had a clean 3rd, 59 pitches.
On the offense side, through 4 we have the one hit and 6 strike outs.
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Python图着色算法答案_CSE 30_Programming Assignment 6美国留学生Python作业pa6答案
import os
import sys
from graph import *
def CheckProperColoring(G):
flag = True
for v in G._color: # 对于每一个着色的点
for node in G.d[v]: # 当前节点的每个相邻点
if G.getColor(v) == G.getColor(node): # v点和相邻点颜色一致,则着色有一样的,着色方案错误!
flag = False
return flag
def main():
p = sys.argv
if len(p)<3:
print("Usage: $ python3 GraphColoring.py
CSE 30
Programming Abstractions: Python
Programming Assignment 6
In this project you will solve (approximately) the Graph Coloring (GC) problem discussed in class. Given
a graph ??, determine an assignment of colors from the set {1, 2, 3, … , ??} to the vertices of ?? so that no two
adjacent vertices are assigned the same color. Further, try to reduce the size ?? of the color set as far as
possible. Such an assignment is called a ??-coloring of ??, and ?? is said to be ??-colorable. The smallest ??
for which ?? is ??-colorable is called the chromatic number of ??, and is denoted ??(??). Any graph with ??
vertices is necessarily ??-colorable (just assign each vertex its own color). Therefore, any solution the GC
problem is expected to satisfy ??(??) ≤ ?? ≤ ??. However, for a general graph ??, the value ??(??) may not be
known, so in solving GC, one is left not knowing what value of ?? is the minimum. Indeed, this is the
Let us pause a moment and think about how we might solve this minimization problem exactly, i.e. find a
??-coloring with ?? = ??(??). An assignment of colors from {1, 2, 3, … , ??} to ??(??) = {1, 2, 3, … …, ??}, is
nothing but a mapping ??: {1, 2, 3, … … , ??} → {1, 2, 3, … , ??}. How many such mappings are there? This is
a counting problem right out of Discrete Mathematics (CSE 16). There are ?? ways to choose the color
??(1), then ?? ways to choose the color ??(2), then ?? ways to choose ??(3), …, and finally ?? ways to choose
??(??). Altogether, the number of ways of making these choices in succession is:
??��⋅ ??��⋅ �?? �⋅ ⋯��⋅�??
= ????
Therefore, the number of ??-color assignments to ??(??) is ????. Of course, not all of these mappings represent
proper ??-colorings, since adjacent vertices may be assigned the same color. Indeed, if ?? < ??(??), then no
such mappings are ??-colorings. Fortunately, it is easy to check whether a mapping is a ??-coloring or not.
Just step through all edges {??, ??} ∈ ??(??), and check that the colors ??(??) and ??(??) are different. If for some
{??, ??} ∈ ??(??), we have ??(??) = ??(??), then ?? is not a ??-coloring.
A brute force solution to this minimization problem is now clear. For each ?? = 1, 2, 3, …, enumerate all ????
mappings ??: {1, 2, 3, … … , ??} → {1, 2, 3, … , ??} (itself an interesting problem), then check each one to see if
it is a proper ??-coloring. The first (smallest) ?? for which this check succeeds is ??(??). Unfortunately, the
algorithm just described is utterly impractical for all but the smallest ??. Observe that the total number of
checks performed is
� ????
≥ ??(??)??,
which grows very rapidly with the size ?? of the vertex set of ??. For instance, if ?? has ?? = 100 vertices
and ??(??) = 10, then more than 10100 checks must be performed. Even if each check could be performed
in a billionth of a second (far too optimistic), the amount of time consumed would be
1091 seconds = (3.168… ) ⋅ 1083 years.
For contrast, the current estimate of the age of the universe is only (13.772) ⋅ 109 years. There are ways
to improve the above algorithm, such as to consider only surjective mappings from {vertices} to {colors},
but nothing can overcome its basic inefficiency. If someone were to discover an efficient algorithm for the
Graph Coloring problem, it would be considered a major advancement in theoretical computer science, and
would win its inventor one million dollars. See
for details.
Obviously then, our goal in this project must be more modest. Instead, you will create a program that only
approximates an optimal solution, and does it efficiently. There is a word for this: heuristic. A heuristic
for an optimization problem is a computational process that returns an answer that is likely to be a good
approximation to optimal, and is likely to be efficient in doing so. There may be a small set of problem
instances for which the approximation is bad, and some for which the runtime is long. An algorithm on the
other hand, is supposed to solve all instances of a problem, i.e. always return an optimal solution.
The heuristic you develop for the GC problem will be based on the reachable() and BFS() algorithms
discussed in class. These algorithms are very efficient and can be run on large graphs with thousands or
even millions of vertices. Both algorithms start at a source vertex and search outward from it, always
processing the next vertex nearest the source. This makes sense for the SSSP problem, since the goal is to
determine distances from the source. It may not make sense for the GC problem. Here is the general outline
followed by both algorithms.
1. start somewhere
2. while some vertex has not been “processed” (whatever that may mean)
3. pick the “best” such vertex ?? (whatever “best” means)
4. process ??
5. for each neighbor ?? of ??
6. “update” the vertex ?? (whatever “update” means)
Both reachable() and BFS() have a defined starting point, the source, which is one of the algorithm inputs.
For the GC problem we can start anywhere. Is there a best place to start? That will be up to you. Each
vertex ?? participates in a constraint with each of its neighbors ??, namely that color[??] ≠ color[??]. The
number of constraints on color[??] is the number of neighbors of ??, also called the degree of ??, denoted
deg(??). Should we pick the largest degree (most constraints) to start? The smallest (least constraints)?
Should we consider the degrees of the neighbors of ??? Should we ignore all this and just pick a random
starting vertex? These are all things for you to consider. Once you get a running program, you will be able
to do experiments, altering your heuristic in various ways to see if you get better results.
What does it mean to “process” a vertex ??? In this problem, it pretty clearly means assigning a color to ??,
although even this is open to interpretation. One thing your heuristic must do is to always produce a proper
??-coloring of ??. To this end you should, for each ?? ∈ ??(??), maintain a set ecs[??] containing the excluded
color set for ??, i.e. the set of colors that have already been assigned to its neighbors, and therefore cannot
be assigned to ??. Since our goal is to use the smallest possible number of colors, the color we assign to ??
should always be the smallest color in the set {1, 2, 3, … , ??} − ecs[??], i.e. pick the smallest color that can
be assigned. This assures that there will be no gaps in the set of colors used. In other words, if we manage
to find a 5-coloring using the set {1, 2, 3, 4, 5}, but the color 3 is never assigned, then we could have
achieved a 4-coloring.
If “process” ?? means to pick color[??], then to “update” one of its neighbors ?? ∈ adj[??], should mean to add
color[??] to ecs[??], so when it comes ??’s turn to be “processed”, we know what colors not to assign.
Finally, what should “best” mean on line 3 of the outline? This decision is where you will have the most
leeway to be creative in designing your heuristic. Here are some ideas. You may base the choice for what
is “best” on degrees again, i.e. pick an unprocessed (uncolored) vertex of either highest or lowest degree.
You might also pick one with the largest excluded color set, the idea being to put out the biggest fire first.
Another approach would be to pick a vertex closest to your starting point. In this case, you should maintain
a FIFO queue with the closest vertex at the front, just in BFS(). Then to “update” a neighbor ?? of ?? would
entail adding it to the back of the queue, as well as updating ecs[??]. You can refine all of these strategies
by combining them in various ways. For instance, pick ?? to be an uncolored vertex of maximum degree,
then break ties by picking one with largest ecs[??].
Program Specifications
You will write a module called graph.py containing a Graph class. This file should be based on the example
graphs.py discussed at length in class (note the different spellings). Begin by removing anything not
needed in this project, like the attributes _distance, _predecessor and _component, and functions
like findComponents() and getPredecessor(). Add the following attributes to your Graph class.
_color: a dictionary whose keys are vertices x, and value _color[x], the color of x
_ecs: a dictionary whose keys are vertices x, and value _ecs[x], the excluded color set of x
You may add other attributes that you deem necessary for your heuristic strategy. Include functions that
perform the actions described in their respective doc strings.
def Color(self):
Determine a proper coloring of a graph by assigning a color from the
set {1, 2, 3, .., n} to each vertex, where n=|V(G)|, and no two adjacent
vertices have the same color. Try to minimize the number of colors
used. Return the subset {1, 2, .., k} of {1, 2, 3, .., n} consisting
of those colors actually used.
# end
def getColor(self, x):
“”” Return the color of x.”””
# end
It is recommended (but not required) that you include a helper function as described below.
def _find_best(self, L):
“””Return the index of the best vertex in the list L.”””
# end
The required function Colors() is of course the main event in this project. Again you may add other
functions as you deem appropriate.
Write a client program called GraphColoring.py containing the following functions.
def CheckProperColoring(G):
Return True if no two adjacent vertices in G have like colors,
False otherwise.
# end
def main():
# check command line arguments and open files
# read each line of input file
# get number of vertices on first line, create vertex list
# create edge list from remaining lines
# create graph G
# Determine a proper coloring of G and print it to the output file
# Check that the coloring is correct
msg = ‘coloring is proper: {}’.format(CheckProperColoring(G))
print(msg, file=outfile )
# end
You may follow the outline in function main() if you like, though it is not required. The code after the final
comment (triple quoted out) is for diagnostic purposes only, and intended for you to run your own tests.
Do not include those commands in your submitted version.
A sample run of your program is given below.
$ python3 GraphColoring.py
Usage: $ python3 GraphColoring.py
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Multiple Question - Assignment Online | Assignmentsonline.org — Assignment Online
Multiple Question – Assignment Online | assignmentsonline.org
Business & Finance- Assignment Online | assignmentsonline.org
Multiple Question- Assignment Online | assignmentsonline.org
– Assignment Online | assignmentsonline.org
1. find the interest paid on a loan of $2800 for two years at a simple interest rate of 11% per year.
The interest on a loan is $
2. Find the maturity value of a loan of $2400.00 after three years. The loan carries a simple interest rate of 7.7% per year.
The maturity value of a loan is $ (round to nearest cent)
3. Convert two years, expressed in decimal form to the nearest hundredth. Nine months.
Nine months = (round to nearest decimal)
4. A man needed money to buy lawn equipment. He borrowed $800.00 for five months and paid $53.95 in interest. What was the rate of interest?
The rate of interest per year was (round to nearest decimal)
5. Find the exact interest on a loan of $32,600 at 7% annually for 20 days.
$(round to nearest cent)
6. A loan made on March 13 is due September 10 of the following year. Find the exact time for the loan in a non-leap year and a leap year.
The exact time in a non-leap year is 546 days
the exact time in a leap year is 547 days
7. A loan is made on March 20 for 181 days. Find the due date.
September 17
8. A loan for $2000 with a simple annual interest rate of 15% was made on June 18 and was due on August 18. Find the exact interest.
The exact interest is $50.14
9. Find the adjusted balance due at maturity for a 90 day note of $18,000 at 13.8% ordinary interest is a partial payment of $4000 is made on the 60th day of the loan.
The adjusted balance due at maturity is $14,579.76
10. Raul Fletes borrowed $7000 on a 210 day note that required ordinary interest at 13.41%. Raul paid $3500 on the note on the 140th day. How much interest did he save by making the partial payment?
The interest saved is $(round to nearest cent)
11. A man makes a simple discount note with a face value of $2700, a term of 140 days, and a 18% discount rate. Find the discount.(use banker’s rule)
The discount is $
12. A man has a simple discount note for $6100, at an ordinary bank discount rate of 8.53%, for 50 days. What is the effective interest rate? Round to the nearest 10th of a percent. Use banker’s
The effective interest rate is %
13. A man holds a note of $5000 that has an interest rate of 14% annually. The note was made on March 19 and is due November 12. He sells the note to a bank on June 12 at a discount rate of 13%
annually. Find the proceeds on the third-party discount note. (Use the bankers rule)
The proceeds are $4107.34
14. A loan of $4000 at 4% is compounded semiannually for four years. Find the future value and compound interest. Use the $1.00 future value table or the future value and compound interest formula.
The future value of the loan is $4686.64
the compound interest is $686.64
15. A loan of $1000 and 30% is compounded monthly for one year. Find the future value and compound interest. Use a $1.00 future value table or the future value and compound interest formula.
The future value of the loan is $1344.89
the compound interest is $344.89
16. Tom Bond borrowed $6200 at 6 ½% for three years compounded annually. What is the compound amount of the loan and how much interest will he pay on the loan?
The compound amount is $7489.29
the compound interest is $1289.29
17. A bank loaned ***** ***** $4000 for seven years compounded annually at 8%. How much interest was John required to pay on the loan? Use the $1.00 future value table or the future value and
compound interest formula.
John was required to pay $2855.30 of interest
18. Find the future value of an investment of $12,000 if it is invested for four years and compounded semiannually at an annual rate of 2%. Use the $1.00 future value table or the future value and
compound interest formula.
The future value of the investment is $12,994.28
19. Find the effective interest rate for a loan for four years compounded semiannually at an annual rate of 2%.
The effective interest rate is 2.01%
20. Find the compound interest on a $2000 investment at .5% compounded daily for 17 days. The compound of interest of $100 compounded daily is 0.023290.
The interest is $
21. Find the amount that should be set aside today to yield the desired future amount. Future amount needed, $5000, interest rate, 8%, compounding., Semiannually, investment time, two years. On the
chart the present value of $1 at 12% is 0.79719.
The present value is $
22. Compute the amount of money to be set aside today to ensure a future value of $2700 in one year if the interest rate is 1.5% annually, compounded annually.
The amount of money to be set aside is $
23. Ronnie Cox has just inherited $27,000. How much of this money should be set aside today to have $22,000 to pay cash for a Ventura Van, which he plans to purchase in one year? He can invest at
1.7% annually, compounded annually.
The amount of money to be set aside is $
24. Dewey Sykes plans to open a business and eight years when he retires. How much must he invest today to have $5000 when he retires if the bank pays 3% annually, compounded semiannually?
Present value of
9. Find the adjusted balance due at maturity for a 90 day note of $15,000 at 13.1% ordinary interest is a partial payment of $4000 is made on the 60^th day of the loan. 30^th is Feb 1^st 60^th day is
Mar. 1^st Apr. 1^st is 90^th
The adjusted balance due at maturity is
13. A man holds a note of $5000 that has an interest rate of 14% annually. The note was made on March 19 and is due November 12. He sells the note to a bank on June 12 at a discount rate of 13%
annually. Find the proceeds on the third-party discount note. (Use the bankers rule)
The proceeds are $
14. A loan of $6000 at 7% is compounded semiannually for two years. Find the future value and compound interest. Use the $1.00 future value table or the future value and compound interest formula.
Future Value or Compound Amount of $1 at 7% is 1.1449
16. Tom Bond borrowed $6200 at 6 ½% for three years compounded annually. What is the compound amount of the loan and how much interest will he pay on the loan?
The compound amount is $
the compound interest is $
17. A bank loaned ***** ***** $4000 for four years compounded annually at 7%. How much interest was John required to pay on the loan? Use the $1.00 future value table or the future value and compound
interest formula. Future value or compound amount of $1 at 7% is 1.31080
John was required to pay $ of interest
18. Find the future value of an investment of $10,000 if it is invested for four years and compounded semiannually at an annual rate of 4%. Use the $1.00 future value table or the future value and
compound interest formula. Future value or compound amount of $1 at 4% is 1.16986
The future value of the investment is $
19. Find the effective interest rate for a loan for three years compounded semiannually at an annual rate of 2%. Future value or compound amount of $1 at 2% is 1.06121.
The effective interest rate is %
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Science Journals
Maxwell Analogy, gravitation, rotary star, black hole, Kerr Metric, torus, gyrotation, horizon
Black holes generally are defined as stellar objects which do not release any light. The Schwarzschild radius, derived from GRT, defines the horizon radius for non-rotating black holes. The Kerr
metric is supposed to define the “event horizon” of rotating black holes, and this metric is derived from generally “acceptable” principles. The limit for the Kerr metric's horizon for non-rotating
black holes is the Schwarzschild radius. By analysing the horizon outcome for rotating and non-rotating black holes, using the Maxwell Analogy for Gravitation (MAG) (or historically more correctly:
the Heaviside Analogy for Gravitation, often called gravitomagnetism), I find that the Kerr metric must be incomplete in relation to the definition of “event” horizons of rotating black holes. If the
Maxwell Analogy for Gravitation (gravitomagnetism) is supposed to be “a good approach” of GRT, we may assume that it is a valid analysis tool for the star horizon metrics. The Kerr metric only
defines the horizons for light, but not the “mass-horizons”. I find both the “light-horizons” and the the “mass-horizons” based on MAG. Moreover, I deduct the equatorial radii of rotating black
holes. The probable origin of the minutes-lasting gamma bursts near black holes is unveiled as well. Finally, I deduct the spin velocity of black holes with a 'Critical Compression Radius'.
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Inductor - (Electromagnetism I) - Vocab, Definition, Explanations | Fiveable
from class:
Electromagnetism I
An inductor is a passive electrical component that stores energy in a magnetic field when electric current flows through it. It is typically made of a coil of wire and opposes changes in current,
thereby playing a crucial role in various electrical circuits and systems.
congrats on reading the definition of inductor. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Inductors store energy in a magnetic field created around them when current flows, and this energy can be released back into the circuit when needed.
2. In AC circuits, inductors create reactance that increases with frequency, affecting the overall impedance in the circuit.
3. The inductance value of an inductor is measured in henries (H), and it's determined by factors like the number of turns in the coil, the core material, and the coil's geometry.
4. Inductors are used in various applications such as filters, transformers, and energy storage devices in power supplies.
5. In RL circuits (resistor-inductor circuits), the time constant determines how quickly current can rise or fall, illustrating transient behavior.
Review Questions
• How does an inductor respond to changes in current and what implications does this have for circuit design?
□ An inductor opposes changes in current flow due to its property of self-inductance, which generates a back electromotive force (EMF) when the current changes. This response makes inductors
essential for filtering signals and smoothing out current variations in power supplies. Circuit designers often take this behavior into account to ensure stability and prevent oscillations in
• Discuss the role of inductors in AC circuits and how they affect the power factor.
□ In AC circuits, inductors introduce reactance that increases with frequency, leading to phase differences between voltage and current. This phase difference can affect the overall power
factor of the circuit, which measures how effectively electrical power is being converted into useful work. A low power factor indicates more reactive power caused by inductive loads, which
can lead to inefficiencies in power systems.
• Evaluate the significance of Lenz's law in relation to inductors and their applications in real-world scenarios.
□ Lenz's law states that the direction of induced EMF in an inductor will oppose any change in current that created it. This principle is significant as it governs how inductors function within
circuits, ensuring energy conservation. In practical applications like transformers and inductive charging systems, Lenz's law plays a crucial role in regulating energy transfer and enhancing
efficiency while minimizing losses due to unwanted fluctuations.
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Jahn–Teller Magnets
Department of Theoretical Physics, Ural Federal University, 620083 Ekaterinburg, Russia
Institute of Metal Physics Ural Branch of the Russian Academy of Sciences, 620108 Ekaterinburg, Russia
Submission received: 29 May 2023 / Revised: 19 September 2023 / Accepted: 27 September 2023 / Published: 2 November 2023
A wide class of materials with different crystal and electronic structures including quasi-2D unconventional superconductors, such as cuprates, nickelates, ferropnictides/chalcogenides, ruthenate Sr
$2$RuO$4$, and 3D systems, such as manganites RMnO$3$, ferrates (CaSr)FeO$3$, nickelates RNiO$3$, silver oxide AgO, are based on Jahn–Teller $3 d$ and $4 d$ ions. These unusual materials, called
Jahn–Teller (JT) magnets, are characterized by an extremely rich variety of phase states, spanning from non-magnetic and magnetic insulators to unusual metallic and superconducting states. The
unconventional properties of JT magnets can be attributed to the instability of their highly symmetric Jahn–Teller “progenitors” with the ground orbital E-state with repect to charge transfer,
anti-Jahn–Teller d-d disproportionation, and the formation of a system of effective local composite spin–singlet or spin–triplet, electronic, or hole S-type bosons moving in a non-magnetic or
magnetic lattice. We consider specific features of the anti-JT-disproportionation reaction, properties of the electron–hole dimers, possible phase states and effective Hamiltonians for single- and
two-band JT magnets, concluding with a short overview of physical properties for actual JT magnets.
1. Introduction
We refer to compounds based on Jahn–Teller 3
- and 4
-ions [
] with configurations of the
$t 2 g n 1 e g n 2$
type in a highly symmetrical octahedral, cubic, or tetrahedral environment, and with a ground state orbital
-doublet, as Jahn–Teller (JT) magnets. These are compounds based on tetra complexes with the configuration
$d 1$
$3 +$
, V
$4 +$
), low-spin (LS) configuration
$d 3$
$2 +$
, Cr
$3 +$
, Mn
$4 +$
), and high-spin (HS) configuration
$d 6$
$2 +$
, Co
$3 +$
); they also include octa complexes with HS configuration
$d 4$
$2 +$
, Mn
$3 +$
, Fe
$4 +$
, Ru
$4 +$
), low-spin configuration
$d 7$
$2 +$
, Ni
$3 +$
, Pd
$3 +$
), as well as octa complexes with configuration
$d 9$
$2 +$
, Ni
$1 +$
, and Ag
$2 +$
) (see
Table 1
). The term “Jahn–Teller magnets”, referring to compounds that contain Jahn–Teller ions, was introduced more than 40 years ago in a well-known article by Russian scientists Kugel and Khomskii
(Uspekhi fizicheskih nauk, 136, 621 (1982), in Russian), although in the translated version (see Ref. [
]) the term “Jahn-Teller magnetic materials” was used with a focus on 3d magnetic insulators with a cooperative Jahn–Teller effect. However, the class of JT magnets extends much further than the
materials considered by Kugel and Khomskii [
]. It includes a large number of promising materials that are at the forefront of modern condensed matter physics, including manganites RMnO
, ferrates (Ca,Sr)FeO
, ruthenates RuO
, (Ca,Sr)RuO
, (Ca,Sr)
, a wide range of ferropnictides (FePn) and ferrochalcogenides (FeCh), 3D nickelates RNiO
, 3D-cuprates KCuF
, 2D cuprates (La
, …) and nickelates RNiO
, and silver-based compounds (AgO, AgF
) (see
Table 1
). Among these materials, it is necessary to highlight JT magnets experiencing charge transfer, particularly disproportionation, as they exhibit a rich spectrum of unique properties, ranging from
varied types of magnetic and charge ordering to metal–insulator transitions and superconductivity. Interestingly, the selection of cuprates as potential superconducting materials and the discovery of
high-temperature superconductivity (HTSC) [
] were influenced by the outstanding Jahn–Teller character of Cu
$2 +$
ions [
]. Attempts to explain the HTSC of cuprates led to the development and dissemination of ideas about the disproportionation. Many authors considered disproportionation as a mechanism (“negative-
” model) leading to the “glueless” superconductivity of a system of local electron pairs, or composite bosons (see, e.g., Refs. [
]). The concept of superconductivity, understood as a Bose–Einstein condensation (BEC) of local composite bosons (two electrons bound in real space), was introduced by Ogg Jr. in 1946 [
] and developed by Schafroth in 1954–55 [
]. However, due to the triumph of the BCS (Bardeen–Cooper–Schrieffer) theory, the notions of local composite bosons and preformed pairs were practically forgotten for many years. The discovery of
HTSC cuprates in 1986 revived interest in the idea of local pairing [
], especially since this idea has been supported by K. A. Mueller, the discoverer of HTSC [
]. Currently, there is convincing experimental evidence that the local pairing of carriers takes place well above T
, at least in underdoped cuprates [
]. At the same time, to date, the HTSC theory has been dominated by approaches based on the BCS paradigm, i.e., on the representations of the BCS model theory applicable to the description of typical
low-temperature superconductors. This is largely due to the fact that an appealingly straightforward picture of preformed pairs and BEC superconductivity in cuprates seemingly came to be at odds with
several experimental observations, indicative of typical Fermi liquid behavior; notably, with indications of a well-defined Fermi surface (FS) in, at least, overdoped cuprates, the thermal and
electrical conductivity were found to follow the standard Wiedemann–Franz law. Quantum oscillations have been observed as well in various cuprates [
However, this contradictory behavior can be easily explained if we take into account the possibility of separating the superconducting BEC phase and the normal Fermi liquid phase. Indeed, recently,
Pelc et al. [
] introduced a phenomenological model of “local phase separation”, in which two electronic subsystems coexist within the unit cell: itinerant and localized holes. In this model, the
holes introduced via doping are always itinerant, while the pairing is associated with the localized holes. In fact, they argue that the Fermi liquid subsystem in cuprates is responsible for the
normal state with angle-resolved photoemission spectra (ARPES), magnetic quantum oscillations, and Fermi arcs, but not for the unconventional superconducting state. In other words,
cuprate superconductivity is not related to the doped hole pairing
, the carriers that exhibit the Fermi liquid behavior are not the ones that give rise to superconductivity. However, the authors could not elucidate the nature of local pairing to be a central point
of the cuprate puzzle.
The disproportionation scenario, which is especially popular in the “chemical” community (“chemical” way to superconductivity), has been addressed earlier by many authors; however, it was not
properly developed theoretically. Perhaps that is why it has not yet been a worthy competitor to the traditional BCS approach.
Previously, we proposed a mechanism for “anti-Jahn-Teller disproportionation” in 3d JT magnets [
], which, by analogy with other anti-JT effects [
], leads to the removal of orbital degeneracy in JT magnets. As a result, we arrived at the formation of a system of electron and hole centers with orbitally nondegenerate ground states, equivalent
to a system of effective local composite spin–triplet bosons moving in a magnetic or nonmagnetic lattice. This mechanism indicated an unconventional bosonic spin–triplet superconductivity in 3d JT
magnets, particularly in ferropnictides and ferrochalcogenides, which was predicted back in 2008 [
]. In the past years, new results have been obtained in the study of JT magnets based on both
$3 d$
$4 d$
ions, as well as new arguments both for and against spin–triplet superconductivity.
In this paper, we expand a model of “anti-Jahn-Teller” disproportionation to encompass a wider class of Jahn–Teller magnets, including
$4 d$
magnets (ruthenates, silver compounds), and 2D nickelates RNiO
, showing that they can all be described within a single scenario. In
Section 2
, we present a more detailed description of the anti-JT disproportionation for JT magnets and the formation of effective local composite bosons. In
Section 3
Section 4
, we consider electron–hole (EH) dimers as specific “disproportionation quanta”, delving into their electron and spin structures.
Section 5
provides a brief overview of the possible phase states for JT magnets.
Section 6
Section 7
present the effective Hamiltonians of single- and two-band JT magnets, providing a brief overview of the properties of real JT magnets. A brief summary is presented in
Section 8
2. Anti-Jahn–Teller Disproportionation
For each JT magnet, one can introduce an imaginary “parent” highly symmetrical phase, or “progenitor”, with a highly symmetrical octahedral, tetrahedral, or cubic environment of the JT ion. The
lifting of the orbital
-degeneracy in the high-symmetry “progenitor” JT magnets can be associated with the specifics of the crystal structure, for example, in “apex-free” 2D cuprates (Nd
) and RNiO
nickelates, and with the conventional Jahn–Teller effect [
], which, as a rule, leads to the formation of a low-symmetry insulating antiferromagnetic (La
, KCuF
, LaMnO
) or ferromagnetic (K
) phase. A competing mechanism for removing orbital degeneracy in the aforementioned JT magnets is the “anti-Jahn-Teller”, “symmetric”
-disproportionation, illustrated by the following scheme:
$d n + d n → d n + 1 + d n − 1 ,$
assuming the formation of a system of bound or relatively free electronic
$d n + 1$
and hole
$d n − 1$
centers, differing by a pair of electrons/holes. Formally, an electron/hole center can be represented as a hole/electron center with a pair of electrons/holes
$d 2$
$d ̲2$
localized at the center. In other words, a disproportionate system can be formally represented as a system of effective local spin–singlet or spin–triplet composite electron/hole bosons “moving” in
the lattices of hole/electron centers. Note that in the frames of the toy model (
), the disproportionation energy
$Δ d d$
formally coincides with the energies of local correlations
$U d d$
, giving a reason to associate symmetric
-disproportionation with the negative-
Obviously, in systems with strong
-hybridization (cation–anion covalency), the disproportionation reaction (
) must be written in a “cluster” language, for example, for CuO
clusters in the CuO
cuprate planes:
$[ C u O 4 ] 6 − + [ C u O 4 ] 6 − → [ C u O 4 ] 7 − + [ C u O 4 ] 5 − ,$
instead of
$d 9 + d 9 → d 10 + d 8 .$
The cation–ligand cluster representations of the
$d n , d n ± 1$
-centers immediately show the important role of the bond-stretching, or so-called “breathing mode” of the ligand displacements in perovskite-type JT magnets with corner-shared coupling of neighboring
-centers. The displacement amplitude of the common ligand for two centers during disproportionation can reach values greater than of 0.1 Å due to the large difference in the cation–ligand separation
for the electron and hole centers. Thus, the Cu-O separation for CuO
centers in cuprates increases by 0.2 Å from the hole [CuO
$5 −$
to electron [CuO
$7 −$
center [
The bond-stretching, or breathing-type distortion of metal-oxygen clusters, is a clear fingerprint of a static charge disproportionation observed in JT magnets, such as 3D nickelates and ferrates
(see, e.g., Refs. [
]), while the softening and broadening of the bond stretching phonon mode observed in JT magnets, such as HTSC cuprates (LSCO, YBCO, Hg1201) and manganites LaMnO
], is believed to be an indication of the dynamical disproportionation. Note that the electron–lattice interaction leads to the stability of the electron and hole centers in the lattice of the parent
system with the ground states of all three centers—the electron, parent, and hole—corresponding to different values of the local breathing configuration coordinates
$Q A 1 g$
$+ Q 0$
, 0,
$− Q 0$
, respectively.
-disproportionation, in contrast to “asymmetric”, “single-center”,
-disproportionation [
], has a two-center character, although it may include
-transfer between clusters. Obviously, symmetric
disproportionation will be energetically more favorable in progenitor Mott–Hubbard JT magnets, and vice versa, asymmetric
-disproportionation will be more energetically favorable in charge-transfer (CT) insulators (“negative charge transfer” materials).
It is worth noting that all the JT magnets are characterized by empty, half-filled, or fully filled
$t 2 g$
subshells with orbitally non-degenerate
$A 1 g$
$A 2 g$
, or
-type ground states, and with only one
$e g$
electron or hole [
]. Obviously, the low-energy anti-JT-disproportionation implies the
$e g$
$e g$
intersite transfer with the formation of the empty, half-filled, or fully filled
$e g$
subshells with the
-type ground state for the electron and hole centers. In all cases, we arrive at relatively stable
-type configurations of electron and hole centers. For all JT magnets, the anti-JT disproportionation reactions can be written as follows:
$t e t r a d 1 : e g 1 + e g 1 → e g e g 0 + e g 0 e g 2 e g 0 e g 2 + e g 0 ;$
$t e t r a d 3 : e g 3 + e g 3 → e g e g 4 + e g 4 e ̲g 2 e g 4 e ̲g 2 + e g 4 ;$
$o c t a d 4 : t 2 g 3 e g + t 2 g 3 e g → e g t 2 g 3 + t 2 g 3 e g 2 t 2 g 3 e g 2 + t 2 g 3 ;$
$t e t r a d 6 : e g 3 t 2 g 3 + e g 3 t 2 g 3 → e ̲g e g 4 t 2 g 3 e ̲g 2 + e g 4 t 2 g 3 e g 4 t 2 g 3 + e g 4 t 2 g 3 e ̲g 2 ;$
$o c t a d 7 : t 2 g 6 e g + t 2 g 6 e g → e g t 2 g 6 + t 2 g 6 e g 2 t 2 g 6 e g 2 + t 2 g 6 ;$
$o c t a d 9 : t 2 g 6 e g 3 + t 2 g 6 e g 3 → e ̲g t 2 g 6 e g 4 e ̲g 2 + t 2 g 6 e g 4 t 2 g 6 e g 4 + t 2 g 6 e g 4 e ̲g 2 .$
In Equations (4)–(9), we singled out both the composite boson and (bold) stable basic configurations of the electron and hole centers. Obviously, for JT magnets with on-site progenitor configurations
$e g 1$
$t 2 g 3 e g 1$
$t 2 g 6 e g 1$
, we are dealing with the transfer of the
$e g$
electron, while for configurations of
$e g 3 t 2 g 3$
$t 2 g 6 e g 3$
, it is correct to speak of the
$e g$
hole (
$e ̲g$
) transfer. Thus, for these configurations, we arrive at the doublet of ionic states with site-centered charge orders, or two centers that differ in the transfer (exchange) of two
$e g$
electrons or two
$e g$
holes, respectively, which can be thought of as effective local composite bosons. For centers with high (octahedral, tetrahedral) symmetry, these effective bosons will be described by the low-energy
-type Hund configuration
$e g 2 ; 3 A 2 g$
; or
$e ̲g 2 ; 3 A 2 g$
. It should be noted that effective bosons cannot be considered conventional quasiparticles, they are an integral part of many-electron configurations [
All JT magnets can be conditionally classified as “single-band” or “two-band” magnets. In single-band JT magnets with configurations of $d 1$, $d 3$, $d 7$, and $d 9$, effective electron ($d 1$, $d
7$) or hole ($d 3$, $d 9$) composite bosons move in the lattices of ions with completely filled shells, while in two-band JT magnets ($d 4$, $d 6$), the lattices include ions with half filled $t 2 g$
The optimal configurations and the spin of the composite boson, along with the orbital state and the local spin of the lattice—which are formed as a result of anti-JT disproportionation in JT magnets
with a 3
$d n$
configuration—as well as some 4
$d n$
JT configurations, are presented in the fourth and fifth columns of
Table 1
. Note that in all cases, the complete disproportionation leads to a system of composite bosons with a concentration of 1/2, indicating half-filling.
3. Electron–Hole Dimers
A pair of bound electron and hole centers, or an EH dimer, is a kind of “disproportionation quantum”. In Mott–Hubbard insulators, EH dimers are low-energy metastable charge excitations above the
ground state or may be the result of the self-trapping of
CT excitons [
The two-electron/hole charge exchange reaction in the EH dimer
$d 1 n + 1 + d 2 n − 1 ↔ e g 2 ; 3 A 2 g d 1 n − 1 + d 2 n + 1 ,$
is controlled by the effective local boson transfer integral
$t B = 〈 d 1 n + 1 d 2 n − 1 | H ^ B | d 1 n − 1 d 2 n + 1 〉 ,$
$H ^ B$
is an effective two-particle (bosonic) transfer Hamiltonian, and we assume ferromagnetic-ordered spins of the two centers. As a result of this quantum process, the bare ionic states with
site-centered charge orders and the same bare energy
$E 0$
transform into two EH dimer states with indefinite valence and bond-centered charge order
$| ± 〉 = 1 2 ( | d 1 n + 1 d 2 n − 1 〉 ± | d 1 n − 1 d 2 n + 1 〉 )$
with energies
$E ± = E 0 ± t B$
. In other words, the exchange reaction restores the bare charge symmetry. In both
$| ± 〉$
states, the on-site number of
electrons is indefinite with quantum fluctuations between (
$n + 1$
) and (
$n − 1$
), and a mean value
. Interestingly, in contrast with the ionic states, the EH dimer states
$| ± 〉$
have distinct electron–holes and inversion symmetry, even parity (
-type symmetry) for
$| + 〉$
states, and odd parity (
-type symmetry) for
$| − 〉$
states, respectively. Both states are coupled by a large electric dipole matrix element:
$〈 + | d ^ | − 〉 = 2 e R 12 ,$
$R 12$
is a 1–2 separation. The two-particle transport (
) can be realized through two successive one-particle processes with the
$e g$
electron transfer as follows:
$d 1 n + 1 + d 2 n − 1 → e g d 1 n + d 2 n → e g d 1 n − 1 + d 2 n + 1 ,$
hence, the two-particle transfer integral
$t B$
can be evaluated as follows:
$t B = − t e g e g 2 / U ≈ − J k i n ( e g e g ) ,$
$t e g e g$
is the one-particle transfer integral for the
$e g$
is the mean transfer energy. It means that the two-particle bosonic transfer integral can be directly coupled with the kinetic
$e g$
$J k i n ( e g e g )$
to the Heisenberg
$e g$
$e g$
exchange integral. Both
$t B$
$J k i n ( e g e g )$
are determined by the second-order one-particle transfer mechanism. It should be noted that a negative sign of the two-particle CT integral
$t B$
points to the energy stabilization of the
-type EH dimer state
$| + 〉$
Moreover, we should emphasize once more that the stabilization of EH dimers is provided by a strong electron–lattice effect with a striking intermediate oxygen atom polarization and displacement
concomitant with charge exchange. In a sense, the EH dimer may be a bosonic counterpart of the Zener Mn
$4 +$
$3 +$
polaron [
]. It is no wonder that even in a generic disproportionated system BaBiO
—instead of simple checkerboard charge orderings of Bi
$3 +$
and Bi
$5 +$
ions—we arrive at a CDW (charge density wave) state with the alteration of expanded Bi
$( 4 − ρ ) +$
and compressed Bi
$( 4 + ρ ) +$
octahedra with
$0 < ρ ≪ 1$
]. The enormously large values of oxygen thermal parameters in BaBiO
] underscore the great importance of dynamical oxygen breathing modes providing some sort of “disproportionation glue”. A sharp rise in the oxygen thermal parameter in the high-temperature O phase of
] or in several “competing” phases found by Huang et al. [
], compared to the bare AFI phase, is believed to be a clear signature of the high-temperature manganese disproportionation [
We examine an EH dimer as a dynamic–charge-fluctuating bipolaronic system composed of coupled electron $d n + 1$ and hole $d n − 1$ centers that are glued in a lattice, due to a specific local
expansion/contraction mode of neighboring clusters (half-breathing or breathing mode), and strong electron–lattice polarization effects.
4. Spin Structure of EH Dimers
Let us address the spin degrees of freedom, which are of great importance for the magnetic properties of EH dimers as nucleation centers for a rich variety of different phases. First, we note that
the structures of EH dimers are significantly different in single- and two-band JT magnets. In EH dimers of JT magnets based on
$d 1$
$d 7$
, and
$d 9$
configurations, the spin–triplet boson “moves” along the spinless centers (see
Table 1
), which leads to a trivial spin structure of the dimer. A more complicated situation is realized for EH dimers of JT magnets based on
$d 4$
$d 6$
configurations, where the spin–triplet boson “moves” through the
$d 3 ( t 2 g 3 )$
centers with spin 3/2 (see
Table 1
The total spin moments of these EH dimers are
$S = S 1 + S 2$
, where
$S 1$
$S 1 = 5 / 2$
) and
$S 2$
$S 1 = 3 / 2$
) are spins of
$d 5$
$d 3$
$d 5$
$d ̲3$
) configurations, respectively, so the total spin magnitudes
take the values of 1, 2, 3, and 4. In a nonrelativistic approximation, the spin structure of the EH dimer in the bare ionic state
$d 5$
$d 3$
$d 3$
$d 5$
) with the site-centered charge order will be determined by isotropic Heisenberg exchange coupling
$V e x = J ( d 5 d 3 ) ( S 1 · S 2 ) ,$
$J ( d 5 d 3 )$
being a
$d 5$
$d 3$
$d 3$
$d 5$
) (super)exchange integral. However, the two site-centered states,
$d 5$
$d 3$
$d 3$
$d 5$
, are coupled by the two-particle charge transfer characterized by a respective transfer integral, depending on the spin states, as follows:
$〈 5 2 3 2 ; S M | H ^ B | 3 2 5 2 ; S M 〉 = 1 20 S ( S + 1 ) t B ,$
$t B$
is a spinless transfer integral. Making use of this expression, we can introduce an effective spin operator form for the boson transfer, as follows:
$H ^ B e f f = t B 20 2 ( S ^ 1 · S ^ 2 ) + S 1 ( S 1 + 1 ) + S 2 ( S 2 + 1 ) ,$
which can be a very instructive tool for qualitative and quantitative analyses of boson transfer effects. Thus, the effective transfer integral of the composite boson strongly depends on the spin
state of the electron–hole pair, falling ten-fold as the total spin of the pair changes from
= 4 to
= 1. In particular, we arrive at a strong, almost two-fold, suppression of the effective transfer integral in the paramagnetic phase, compared with its maximal value
$t B$
for ferromagnetic ordering (
= 4).
Both the conventional Heisenberg exchange coupling
$d 5$
$d 3$
$d 3$
$d 5$
) and unconventional two-particle bosonic transfer, or bosonic double exchange, can be easily diagonalized in the total spin
representation, so that for the energy of the EH dimer, we arrive at
$E S = J ( d 5 d 3 ) 2 [ S ( S + 1 ) − 25 2 ] ± 1 20 S ( S + 1 ) t B ,$
where ± corresponds to two quantum superpositions
$| ± 〉$
written in a spin representation as follows
$| S M 〉 ± = 1 2 ( | 5 2 3 2 ; S M 〉 ± | 3 2 5 2 ; S M 〉 ) ,$
- and
-type symmetry, respectively. It is worth noting that the bosonic double exchange contribution formally corresponds to ferromagnetic exchange coupling with
$J B = − 1 10 | t B |$
We see that the cumulative effect of the Heisenberg exchange and the bosonic double exchange results in the stabilization of the
= 4 high-spin (ferromagnetic) state of the EH dimer provided
$| t B | > 10 J ( d 5 d 3 )$
(see the left panel in
Figure 1
) and the
= 1 low-spin (“ferrimagnetic”) state otherwise (see right panel in
Figure 1
). As for the spin states with intermediate
values (
= 2, 3), these correspond to a classical noncollinear ordering. It is interesting that for
$| t B | = 10 J ( d 5 d 3 )$
, the energy of the dimer’s
-type state does not depend on the value of the total spin, so that we arrive at the surprising result of the 24-fold (
$∑ S = 1 S = 4 ( 2 S + 1 )$
) degeneracy of the ground state of an isolated dimer (see the central panel in
Figure 1
To estimate quantities
$t B$
$J ( d 5 d 3 )$
and their dependencies on the crystal structure parameters, we can address the results of a comprehensive theoretical and experimental analysis of different superexchange integrals in perovskites
, RCrO
, and RFe
$1 − x$
, with Fe
$3 +$
and Cr
$3 +$
ions with electronic configurations of
$d 5$
$d 3$
, respectively [
]. These perovskites are isostructural with many JT magnets, including (Ca,Sr,Ba)FeO
, RMnO
, and (Ca,Sr,Ba)RuO
Antiferromagnetic kinetic exchange contribution to
$J ( e g e g )$
related to the
$e g$
electron transfer to the partially filled
$e g$
-shell, can be written as follows [
$J ( e g e g ) = ( t s s + t σ σ cos θ ) 2 2 U ,$
while for the
$d 5$
$d 3$
superexchange, we encounter competition between the antiferromagnetic and ferromagnetic contributions
$J F e C r = J ( d 5 d 3 ) = 2 15 t σ π 2 U sin 2 θ + t π π 2 U ( 2 − sin 2 θ )$
$− Δ E ( 35 ) 10 U ( t s s + t σ σ cos θ ) 2 U + t σ π 2 U sin 2 θ .$
is the cation–anion–cation-bonding angle,
$t σ σ > t π σ > t π π > t s s$
are positive definite
transfer integrals,
is the mean
transfer energy (effective correlation energy), and
$Δ E ( 35 )$
is the energy separation between
$3 E g$
$5 E g$
terms for the
$t 2 g 3 e g$
Microscopically derived angular dependencies of the superexchange integrals
$J F e F e$
$J C r C r$
, and
$J F e C r$
nicely describe the full set of experimental data on the value of
$T N$
for various orthoferrites, orthochromites, mixed ferrites–chromites, as well as Mössbauer data on Fe-substituted orthochromites [
Figure 2
shows the dependence of the superexchange integrals
$J F e C r$
$J ( d 5 d 3 )$
$J ( e g e g )$
$− t B$
on the cation–anion–cation superexchange angle, which is typical for orthoferrites and orthochromites. The empty rectangles for
$J ( d 5 d 3 )$
reproduce the experimental data [
], taking into account the measurement errors of the exchange integrals and the average values of the superexchange bonding angles. The dashed curve in
Figure 2
describes the angular dependence (
) for
$J ( e g e g )$
with quantitative estimates based on the analysis of the full set of experimental data on the value of exchange parameters for orthoferrites and orthochromites [
The fitting allows us to predict the sign change for $J F e C r$ at $θ c r$ ≈ 160–170$∘$. In other words, the ($t 2 g 3 e g 2 − O 2 − − t 2 g 3$) superexchange coupling becomes ferromagnetic at $θ ≥
θ c r$.
At variance with
$J ( d 5 d 3 )$
, the exchange parameter
$J ( e g e g$
$≈ | t B |$
declines rapidly with the decrease in the bonding angle
, so that at
$θ c r$
≈ 142
, the ferro- and antiferromagnetic contributions to the effective exchange parameter are compensated,
$J e f f = J ( d 5 d 3 ) − 0.1 | t B |$
, with
= 1,2,3,4 degeneracy, and there is a transformation of the spin ground state from
= 4→
= 1, with a ten-fold reduction in the effective transfer integral of the composite boson (see Equation (
We believe that the results of the analysis of the angular dependence of parameters
$J ( d 5 d 3 )$
$J ( e g e g )$
, presented in
Figure 2
, can be used to analyze the spin structures of EH dimers in JT magnets with a perovskite structure, such as manganites, ferrates, and ruthenates (see
Table 1
So, for example, for the superexchange geometry, which is typical for LaMnO
], with the Mn-O-Mn bonding angle
$θ ≈ 155 ∘$
, we find
$J ( d 5 d 3 ) ≈$
+7 K and
$J ( e g e g ) ≈ | t B | ≈$
297 K. In other words, for the effective exchange integral
$J e f f$
, we arrive at a rather large value:
$J e f f = J ( d 5 d 3 ) − 0.1 | t B | ≈$
−23 K. Despite the antiferromagnetic sign of the Heisenberg superexchange integral, these data unambiguously point to a dominant ferromagnetic contribution from the bosonic double exchange mechanism,
with a ground ferromagnetic
= 4 spin state for the EH dimer and a maximal, “nonreduced” value of the composite boson transfer integral.
For the bonding angle,
$θ = 143 ∘$
, which is typical for heavy rare-earth manganites RMnO
(R=Dy, Ho, Y, Er) [
], the relationship between
$| t B | ≈$
154 K and
$J ( d 5 d 3 ) ≈$
14 K [
] approaches to the critical one:
$| t B | = 10 J ( d 5 d 3 )$
evidencing the destabilization of the ferromagnetic state of the EH dimers.
Thus, the structural factor plays a significant role in the stabilization of specific spin states of the EH dimer and the effective transfer integral for the composite boson. We believe that the
change (decrease) in the angle of the cation–anion–cation superexchange bond, leading to the suppression of ferromagnetic interaction and metallicity, can be the main reason for the strong effect of
substituting Sr with Ca in JT magnets, such as SrFeO$3$, SrRuO$4$, Sr$2$RuO$4$, and Sr$3$Ru$2$O$7$.
5. Possible Phase States of JT Magnets with Instability to Charge Transfer
In the limit where electron correlations are strong, and potential energy prevails for valence electrons, the stable against charge transfer “progenitor” JT-systems, as a rule, typically manifest as
spin-magnetic insulators with a specific orbital ordering (OO), as a consequence of the cooperative JT effect [
]. Conversely, in the limit of weak correlations where the kinetic energy for valence electrons predominates, we arrive at a system of itinerant electrons constituting a Fermi liquid.
In the crossover CT-instability regime, instead of a single inactive charge
$d n$
component, the on-site Hilbert space of
-centers includes a charge triplet of
$d n , d n ± 1$
-centers, leading to the appearance of at least eight parameters of diagonal and off-diagonal charge orders [
]. Taking into account the spin degree of freedom and lattice modes, we arrive at a huge variety of possible phase states. The phase diagram’s complexity originates from the specific crystal
chemistry and a fine balance between the energies of the electron–lattice interaction, crystal field, local (Coulomb and exchange, or Hund) correlations, nonlocal charge correlations, inter-site
single and two-particle (composite boson) charge transfers, and spin–spin exchanges. The inevitable consequences of the competitions of many order parameters will be phase separation and the
possibility of fine-tuning physical properties by changing the chemical composition, applying external pressure, and going over to epitaxial films and heterostructures.
Taking into account the coexistence of one- and two-particle transports, the high-temperature disordered phase for these systems will be a kind of “boson-fermion soup” [
], or a “strange/bad” metal with a
-linear resistance dependence (strange metal) and a violation of the Mott–Ioffe–Regel criterion (bad metal). Indeed, the “strange/bad” metal behavior is common in all the JT magnets listed in
Table 1
A specific long-range order in JT magnets starts to form at a high temperature in a disordered phase, which is characterized by the competition between the electron–lattice interaction, and spin and
charge fluctuations in the “struggle” for the low-temperature ground state. The local JT interaction leads to the stabilization of low-symmetry-insulating magnetic structures. Low-energy charge
fluctuations, which are characteristic of the local anti-JT disproportionation reaction (
), depend on the ratio between the parameters of local and non-local correlations, the integrals of one- and two-particle transfers, and the specifics of the electron–lattice interaction associated
with the breathing mode unique to electron–hole pairs; this can lead to the formation of a wide variety of phases, including charge (CO) and spin–charge ordering, collinear and noncollinear magnetic
ordering, a coherent metallic Fermi liquid FL phase, a bosonic superconductivity (BS) phase, and a specific nematic phase with the EH dimer ordering [
It should be noted that materials that are simultaneously magnetic and charge-ordered can be multiferroic, with potentially very large electric polarization.
We believe that the expected superconductivity of JT magnets is not a consequence of the BCS-type pairing, but the result of a quantum transport of the effective on-site composite electron/hole
bosons. The superconducting state, as one of the possible ground states of JT magnets, can compete with the normal Fermi liquid state, charge order, spin–charge density wave, collinear or
noncollinear magnetic orders, as well as specific quantum phases. The variety of competing phases clearly indicates the important role of phase separation effects [
], which must be taken into account first when analyzing experimental data.
Below, without dwelling on a detailed analysis of phase states and phase diagrams, we consider only the main features of the single- and two-band JT magnets in fully disproportionated states, when
they form a system of spin–singlet or spin–triplet composite bosons in a nonmagnetic or magnetic lattice, respectively. Strictly speaking, to describe the disproportionate systems, it is necessary to
take into account the electron–lattice interaction, primarily with the so-called breathing mode; below, we will consider the effective Hamiltonian of effective composite bosons in the “frozen”
lattice approximation.
6. Single-Band JT Magnets
6.1. Effective Hamiltonian of a System of Spin–Triplet Composite Bosons: Non-Magnetic Lattice
As can be seen in
Table 1
the anti-Jahn–Teller disproportionation in the system of tetrahedral JT centers with a configuration of 3
$d 1$
, 4
$d 1$
, low-spin octa-centers with configurations of 3
$d 7$
, 4
$d 7$
, or octa-centers with configurations of 3
$d 9$
, 4
$d 9$
leads to the formation of a half-filled system of effective spin–triplet bosons moving in a non-magnetic lattice. We represent the Hamiltonian of such a system in the following form:
$H = − ∑ i > j , ν t i j B ^ i ν † B ^ j ν + B ^ i ν B ^ j ν † + ∑ i > j , ν , ν ′ V i j n i ν n j ν ′ − ∑ i , ν μ ν n i ν + H s ,$
$t i j$
is the spin-independent boson transfer integral,
$V i j$
is the effective boson–boson repulsion (nonlocal correlations),
is the chemical potential,
$H s$
is the spin Hamiltonian. The chemical potential
is introduced to fix the boson concentration
$n = 1 N ∑ i ν 〈 n ^ i ν 〉$
The composite boson creation/annihilation operators
$B ^ i ν † / B ^ i ν$
, regardless of the spin component
$ν = 0 , ± 1$
, obey the on-site anti-commutation Fermi relations and the inter-site Bose commutation relations:
${ B ^ i , B ^ i † } = 1 , [ B ^ i , B ^ j † ] = 0 .$
The anti-commutation Fermi relations can be rewritten as
$[ B ^ i , B ^ i † ] = 1 − 2 B ^ i † B ^ i = 1 − 2 N ^ i .$
On the whole, these relations rule out on-site double-filling.
To take into account the influence of an external magnetic field, one can use the standard Peierls substitution:
$t i j → t i j e i ( Φ j − Φ i ) ,$
$( Φ j − Φ i ) = − q ℏ c ∫ R i R j A ( r ) d l ,$
is the vector potential of a homogeneous magnetic field, and the integration goes along the line connecting the sites
. In a general case, the spin Hamiltonian, denoted as
$H s$
for the system of spin–triplet bosons, can be represented as follows:
$H s = ∑ i > j J i j s ^ i · s ^ j + ∑ i > j j i j s ^ i · s ^ j 2 + K S I A ∑ i m i · s ^ i n i · s ^ i + V T I A − ∑ i h · s ^ i ,$
$J i j$
$j i j$
are the bilinear and biquadratic isotropic exchange integrals, respectively,
$K S I A$
is a constant,
are unit vectors that define two characteristic axes of the second-order single-ion anisotropy,
$V T I A$
denotes the two-ion bilinear and biquadratic anisotropy, and
denotes the external field.
It is worth noting that the Cartesian form of the composite boson spin operator can be represented as follows
$s ^ β = B ^ α † ϵ α β γ B ^ γ ,$
$ϵ α β γ$
is Levi-Civita tensor,
$α , β , γ = x , y , z$
In the paramagnetic region, the Hamiltonian (
) actually reduces to the Hamiltonian of the well-known lattice hard-core (
$h c$
) Bose system with an inter-site repulsion, governed in the nearest-neighbor approximation by two parameters,
$t B$
. At half-filling, depending on the relative values of the parameters, we arrive at a charge order (CO) or Bose-superfluid (BS) phase. As the temperature decreases, a specific magnetic order is
realized in the system.
6.2. $d 1$, $d 3$ JT Magnets
The only JT magnets that are known in the literature with tetrahedral
$d 1$
-centers, such as
with V
$4 +$
and (Sr,Ba)
with Cr
$5 +$
, are considered to be typical insulators, exhibiting Jahn–Teller distortions with orbital ordering and the formation of a system of weakly coupled spin dimers (see, e.g., Refs. [
]). We did not find any literature data on JT magnets with tetrahedral
$d 3$
-centers, except for the assumption made in Ref. [
] about the possibility of synthesizing Ba
melilite with V
$2 +$
ions, an anticipated JT-multiferroic.
$d 7$ JT Magnets
The origin of the metal-to-insulator transition (MIT) in the series of rare-earth nickelates RNiO
with perovskite structures has challenged the condensed matter research community for almost three decades [
]. Furthermore, the recent theoretical prediction for superconductivity in LaNiO
thin films [
] has also sparked intensive research efforts.
The complex MIT phenomena in these materials are a perfect illustration of the competition between the potential and kinetic energy gain, presumably governed by structural factors, namely, the
Ni-O-Ni bond angle, providing clear evidence for strong electron–lattice effects, which have a dramatic effect on the character of the MIT.
Orthorhombic RNiO
(R = Pr, …Lu) undergoes a first-order metal–insulator phase transition to a charge-ordered insulating state upon cooling below T
$C O$
= T
$M I T$
, spanning from 130 K for Pr to ∼550–600 K for heavy rare-earth [
]. Each shows clear signs of the charge-disproportionated state with two types of Ni centers that correspond to alternating large [NiO
$10 −$
$2 +$
center) and small [NiO
$8 −$
$4 +$
center) octahedra, strongly differing in magnetic moments (∼2
$μ B$
and ∼0, respectively), in full accordance with the disproportionation model (see
Table 1
). The largest anomaly at T
$M I T$
= T
= 130 K in PrNiO
is observed in the amplitude of the breathing mode, which undergoes a sharp jump of 0.15 Å [
]. A further interesting observation is the existence of a nearly perfect linear correlation between the amplitude of the breathing mode associated with the charge order and the staggered
magnetization below the MIT. In addition, the authors [
] suggest the existence of hidden symmetry in the insulating phase, which may be related to a nematic contribution of bound EH dimers.
At low temperatures, ortho-nickelates show magnetic phase transitions toward unusual antiferromagnetic structures defined by a propagation vector (1/2, 0, 1/2) [
], which can be explained by the rather strong superexchange
$n n n$
(next-nearest neighbor) coupling of magnetic
= 1 Ni
$2 +$
centers. Strictly speaking, the (1/2, 0, 1/2) ordering suggests three possible magnetic structures, of which, two are collinear and one is non-collinear. For instance, a spin-canted antiferromagnetic
state of the nickel sublattice was observed in NdNiO
]; however, the ambiguity of the magnetic structure of the nickelates is not yet completely resolved. The non-collinear spin order in nickelates can potentially generate spin-induced
ferroelectricity; however, these systems remain comparatively unexplored as potential multiferroics [
Increasing the Ni-O-Ni bond angle in nickelates when moving from LuNiO
to LaNiO
leads to a gain in kinetic energy with a clear trend toward metallization due to two important effects, namely, an increase in the transfer integrals for the
$e g$
electrons and a decrease in parameter
of inter-site repulsion (nonlocal correlations), due to an increase in the Ni-Ni separation. So, the X-ray diffraction, neutron scattering, transport, and thermodynamic experiments show that globally
rhombohedral single-crystal LaNiO
samples reveal unusually high metallicity and maintain paramagnetic behavior down to 1.8 K [
], or some signatures of antiferromagnetic transition at 157 K [
], but no structural and metal–insulator transitions. The combined total neutron scattering and broadband dielectric spectroscopy experiments on polycrystalline samples [
] indicate that the structure of LaNiO
has a high degree of symmetry when viewed on long-length scales, but similar to orthorhombic nickelates, it also has at least two different types of Ni sites when viewed locally. LaNiO
is locally distorted to orthorhombic at room temperature, and further to monoclinic at 200 K from a globally rhombohedral structure [
]. This controversial behavior for LaNiO
can be the result of the peculiar “ortho-mono-rhombo” phase separation.
Another example of nickel JT magnets is the quasi-2D nickelates ANiO
(A = Ag, Li, Na), revealing the existence of unconventional ground states stabilized by the frustrated triangular lattice geometry from a cooperative JT ordering of Ni
$3 +$
ions in NaNiO
to a moderately charged ordering 3Ni
$I I I +$
→ Ni
$2 +$
+2 Ni
$3.5 +$
in antiferromagnetic metal AgNiO
]. In the case of LiNiO
, there could be a competition between charge and orbital ordering, the nickel valency could be a mixture of 2+, 3+, and 4+ [
]. A comparison between NaNiO
and LiNiO
, where several different possible ground states are very close in energy, illustrates how two systems that are apparently so chemically similar can, nevertheless, have very different behavior [
6.3. $d 9$ JT Magnets
6.3.1. Isoelectronic Quasi-2D Cuprates and Nickelates
The Cu
$2 +$
ion in octahedral complexes is characterized by the strongest JT bond and is the most popular, almost “textbook” illustration of the Jahn–Teller effect. The consequence of this effect is the
formation of the insulating state of a quantum antiferromagnet, for example, in KCuF
and La
, or quasi-2D ferromagnet K
. However, in contrast to fluorides, in La
, the JT distortion leads to the formation of CuO
planes with a “perovskite” configuration of CuO
-clusters, with the ground
$b 1 g ∝ d x 2 − y 2$
state of the
$e g$
hole, which provides a strong
-coupling channel for the hole transfer in the CuO
plane and disproportionation (
), forming spin–singlet and orbitally nondegenerate (
$1 A 1 g$
) electronic [CuO
$7 −$
(analog of Cu
ion) and Zhang–Rice (ZR) [
] hole [CuO
$5 −$
(analog of the Cu
$3 +$
ion) centers.
Recently [
], we argued that there are no fundamental qualitative differences in the electronic structures of “apex-free” RNiO
nickelates and cuprates (primarily, cuprates with
$T ′$
-structures). The unusual properties of cuprates and nickelates are the results of the “competition” between various parameters that govern the ground states of the CuO
) planes. Thus, if for the vast majority of parent cuprates, an antiferromagnetic insulating phase is observed, corresponding to the limits of strong local correlations, then this phase is not found
in the parent nickelates RNiO
, which can be associated with smaller values or even a change in the sign of the local correlation parameter. We proposed [
] to understand by “parent” the cuprates and nickelates with hole half-fillings of in-plane centers CuO
), which—depending on the parameters of local and non-local correlations, transfer integrals, exchange integrals, and “external” crystal fields formed by the out-of-plane environment—can have
different ground states, e.g., an antiferromagnetic insulator (AFMI), an unusual Bose superconductor (BS), a Fermi metal (FL), or a non-magnetic insulator with charge ordering (CO). Obviously, these
phases will differ in electronic degrees of freedom as well as lattice degrees of freedom; this interaction ensures the minimum of the total free energy. In addition, the competition between several
possible phases with similar energies will lead to phase separation, which will have a significant effect on the observed physical properties.
To describe the actual low-energy phase states of cuprates/nickelates, we propose a minimal model for the CuO
planes with the on-site Hilbert space reduced to a charge triplet of the three effective valence centers [CuO
$5 − , 6 − , 7 −$
$6 − , 7 − , 8 −$
(nominally, Cu
$3 + , 2 + , 1 +$
$2 + , 1 + , 0 +$
) with different conventional spins, different orbital symmetries, and local lattice configurations [
]. Making use of the
= 1 pseudospin formalism and the spin–pseudospin operators as the Hubbard
-operators, we constructed the spin–pseudospin Hamiltonian of the charge triplet model, which takes into account local and nonlocal correlations, correlated one-particle and two-particle (bosonic)
transports, and the Heisenberg spin exchange. In particular cases, the Hamiltonian reduces to a well-known “limiting” Hamiltonian (Hubbard, Heisenberg, atomic limit, hard-core bosons, …). In
accordance with experimental data for apexless cuprates [
], nickelates [
], and different typical cuprates, we argue that antiferromagnetic insulating (AFMI), charge ordered (CO), Bose superconducting (BS), and Fermi liquid (FL) phases are possible phase states of a model
parent cuprate/nickelate, while typical phase states of doped systems, in particular, mysterious pseudogap phases, are the result of a phase separation (PS). The superconductivity of cuprates/
nickelates is not a consequence of the pairing of doped holes [
], but the result of the quantum transport of on-site composite hole bosons, whereas the main peculiarities of a normal state can be related to an electron–hole interplay for an unusual Fermi liquid
phase and PS features. In the BCS model, the electron–lattice interaction determines the
-wave pairing, while in the model of local composite bosons, it yields the
$d x 2 − y 2$
-symmetry of the superconducting order parameter, thus showing, once again, a substantial involvement of the lattice in the HTSC [
]. Within the framework of the effective field approximation [
] and the Maxwell construction [
], we constructed several 2D T–
phase diagrams for the CuO
planes, which qualitatively reproduce the main features of the experimentally observed 3D phase diagrams of cuprates and nickelates [
] (see
Figure 3
). Note that the exotic pseudogap phase is believed to be related to the PS region AFMI-CO-FL-BS, separated from the 100% FL-phase by the T
$* ( p )$
curve (“pseudogap temperature”) of the “third order” phase transition.
In general, quasi-2D cuprates and nickelates present excellent examples of the applicability of the anti-JT disproportionation model. A large amount of experimental data from a long-term study of
various properties of a wide class of old 2D cuprates and novel 2D nickelates, as well as the results of the theoretical modeling of phase diagrams in the charge triplet model [
], provide important information about possible phase states of other JT magnets with charge transfer.
6.3.2. “Silver” JT Magnets
The anti-JT disproportionation model predicts the possibility of a “silver or palladium path” to superconductivity in systems based on Ag
$2 +$
) or Pd
); that is, the 4d analog of Cu
$2 +$
. The most likely candidate, silver fluoride AgF
], also known as
, is an excellent analog of the cuprate with surprisingly close electronic parameters to La
, but with greater deformation (buckling) of AgF
planes. However, this fluoride is a canted antiferromagnetic insulator, although close to a charge-transfer instability. Indeed, experimental studies [
] report the discovery of a metastable disproportionate diamagnetic phase
, interpreted as a charge-ordered compound Ag
$1 +$
$3 +$
, which quickly transforms into the
structure (see Ref. [
Unlike the antiferromagnetic insulator Cu
$2 +$
O, its silver 4d analog Ag
$2 +$
O is a diamagnetic semiconductor with a disproportionate Ag sublattice, whose chemical formula is often written as Ag
$1 +$
$3 +$
, with O-Ag
$1 +$
)-O collinear bonds and Ag
$3 +$
square planar bonds (4d
]. In this case, the [AgO
$5 −$
cluster, like the [CuO
$5 −$
center in cuprates, is in a nonmagnetic state of the Zhang–Rice singlet type.
7. Two-Band JT Magnets
Single-band JT magnets, with their relatively simple electronic structures, provide an excellent illustration of the predictive power of the anti-JT disproportionation model, while the situation with
two-band JT magnets is less certain.
Anti-Jahn–Teller disproportionation in “two-band” systems of high-spin octa centers with 3
$d 4$
, 4
$d 4$
configurations, or tetrahedral JT centers with 3
$d 6$
, 4
$d 6$
configurations, imply unusual phases with the coexistence of a half-filled system of effective spin–triplet electrons or hole bosons with configurations of
$e g 2 : 3 A 2 g$
$e ̲g 2 : 3 A 2 g$
, and a magnetic lattice with on-site
$S = 3 / 2$
configurations of
$t 2 g 3 : 4 A 2 g$
, although this does not exclude the existence of unusual phases with delocalized
$t 2 g$
electrons (see the review article [
Two-band JT magnets include many promising compounds, most of which are presented in the last column of
Table 1
. Below, we briefly consider the effective Hamiltonian, the features of the electronic structure, and physical properties of the most prominent representatives of two-band JT magnets.
7.1. Effective Hamiltonian of a System of Spin–Triplet Composite Bosons: Magnetic Lattice
The anti-Jahn–Teller disproportionation in a system of high-spin octahedral JT-centers with 3d
, 4d
configurations or tetrahedral JT-centers with 3
$d 6$
, 4
$d 6$
configurations leads to the formation of a half-filled system of effective spin–triplet electron or hole bosons with configurations of
$e g 2 : 3 A 2 g$
$e ̲g 2 : 3 A 2 g$
, moving in a magnetic lattice with on-site configurations of
$t 2 g 3 : 4 A 2 g$
Table 1
The effective Hamiltonian of such a system can also be represented as (
), however, with the spin-dependent composite boson transfer integral (see Equation (
$t i j = S ( S + 1 ) 20 t B ,$
$S ^ = S ^ i + S ^ j$
is the total spin of the EH-pair (
$i j$
= 1, 2, 3, 4.
In contrast with the single-band JT magnets, the spin Hamiltonian
$H s$
for two-band JT magnets will have a much more complex structure. Taking into account only the bilinear spin–spin isotropic exchange, it can be represented as follows:
$H s = ∑ i > j J i j l l ( S ^ i · S ^ j ) + ∑ i > j J i j b b ( s ^ i · s ^ j ) + ∑ i > j J i j b l ( s ^ i · S ^ j ) + ∑ i J i i b l ( s ^ i · S ^ i ) ,$
where we assume the localized
$t 2 g$
subshell. The first term denotes the exchange interaction between the “lattice” spins, the second term denotes the exchange interaction between the spin–triplet bosons, the third and fourth terms
denote the exchange between bosons and lattice spins, and the last term denotes the intra-atomic Hund exchange. To fulfill Hund’s rule, it is necessary to set the exchange integral
$J i i b l$
to be relatively large ferromagnetic.
Estimates for different superexchange couplings—given the cation–anion–cation bond geometry that is typical for perovskites such as ferrates (Ca,Sr)FeO
or manganites RMnO
with bare octa-HS d
configurations [
]—predict the antiferromagnetic coupling for the
$n n$
lattice centers (
$J l l > 0$
) and the two nearest neighbor bosons (
$J b b > 0$
), where the coupling between the boson and the nearest neighbor lattice centers (
$J b l < 0$
) can be ferro- or antiferromagnetic, depending on the value of the cation–anion–cation-bonding angle (see
Figure 1
). Taking into account the boson transport, which prefers an overall ferromagnetic ordering, we arrive at a highly frustrating system with competition between the ferro- and antiferromagnetic
Generally speaking, our Hamiltonian model describes the system that can be considered a Bose analog of the
double-exchange model system [
7.2. Chromium Cr$2 +$ Compounds
Among the JT chromium compounds, we—more or less—have reliable information about chromium difluoride CrF
, according to which, it is an antiferromagnetic insulator [
]. However, X-ray absorption and resonant inelastic X-ray scattering (RIXS) spectra of CrF
] point to the presence of three chromium oxidation states, namely Cr
, Cr
$2 +$
, and Cr
$3 +$
, indicating instability with respect to the charge transfer, with clear signatures of the
disproportionation reaction in this JT magnet. The most likely explanation for this is phase separation; that is, the coexistence of antiferromagnetic regions and regions of a disproportionate phase.
7.3. Manganites RMnO$3$
Features of the anti-JT disproportionation and its influence on the phase diagram of manganites RMnO
are considered in detail in Ref. [
A high-temperature, thermally fluctuating charge disproportionated metallic state has been postulated for LaMnO
by different authors [
]. However, upon lowering the temperature, one observes a first-order phase transition at T = T
$J T$
$J T$
≈ 750 K in LaMnO
) from the high-temperature fully disproportionate Bose metallic phase to a low-temperature orbitally ordered insulating phase, with a cooperative Jahn–Teller ordering of the occupied
$e g$
-orbitals of the Mn
$3 +$
octahedra, accompanied by A-type antiferromagnetic ordering below T
≈ 140 K in LaMnO
) [
]. However, many experimental data point to a phase separation with the coexistence of insulating and disproportionated phases [
The non-isovalent substitution and/or non-stoichiometry seem to revive the disproportionated phase, and such manganites—along with metallic ferromagnetism and colossal magnetoresistance—reveal many
properties that are typical for local spin–triplet superconductivity [
Distinct signatures of high-temperature disproportionated phases are revealed in other manganites, such as LaMn
] with quadruple perovskite structures and YBaMn
Additionally, the orthorhombic rare-earth manganites RMnO
characteristically display non-collinear spin–spiral orders and form a “model family” of spin-driven ferroelectrics [
7.4. Iron Fe$4 +$ JT Magnets
All the ferrates listed in
Table 1
, are JT magnets that are unstable with respect to charge transfer.
The AFe$4 +$O$3$ (A = Ca, Sr, Ba) perovskites show intriguing physical properties, which are strongly dependent on the size and polarizability of the A-site ion since this affects all the main
parameters governing their electronic structures.
With decreasing temperatures, orthorhombic metallic CaFeO
(CFO) exhibits a second-order phase transition to a narrow-gap charge-ordered monoclinic semiconductor, or Hund’s insulator, with disproportionation in Fe
$4 ± δ$
below a transition temperature T
$C O$
= T
$M I T$
= 290 K at ambient pressure, resulting in a three-dimensional rock salt-type ordering of alternating small and large oxygen octahedra surrounding the nominal
$d 3$
$d 5$
Fe sites, respectively [
]. Parameter
= 0 for T > 290 K increases continuously with decreasing temperatures below 290 K; typically,
approaches unity at low temperatures. The MIT is accompanied by the reduction in crystal symmetry as well as the sharp variation in electrical transport. Within our model, the disproportionated phase
in the CFO implies the electron boson confinement in the larger FeO
The charge disproportionation scenario for CFO has been experimentally well-established using
Fe Mössbauer spectroscopy [
], which clearly reveals two different sites with considerably different isomer shifts and hyperfine fields.
Let us pay attention to the possibility of the formation of domains in the charge-ordered state with 180$∘$-domain walls, realizing the transition between two types of “site-centered” charge orders.
At the center of the domain walls, a system of delocalized spin–triplet composite bosons with a “bond-centered” charge order is formed, which formally corresponds to the system of Fe$4 +$ centers.
As the temperature is further lowered, there is another transition in the CFO from the paramagnetic to an antiferromagnetic insulator at the Néel temperature T
≈ 120 K. The low-temperature magnetic data can be fit equally well by a screw–spiral structure or by a sinusoidal amplitude-modulated structure. The values of the moments at the two Fe sites can take
different values; 2.5 and 3.5
$μ B$
for the spiral structure, and maximum amplitudes of 3.5 and 5.0
$μ B$
for the sinusoidal structure [
Note that the high-temperature orthorhombic metal phase of CFO can be considered as a Hund’s bad metal, which appears as a mixed-valence state that fluctuates between two atomic configurations.
In contrast to the distorted perovskite CaFeO
, the undistorted cubic perovskites SrFeO
and BaFeO
maintain metallic behaviors down to very low temperatures, exhibiting different types of helical spin order. However, the ground states in these ferrates raise many questions. At variance with
Mössbauer data for CaFeO
, the single magnetic hyperfine pattern for SrFeO
at 4 K indicates a rapid electron exchange between Fe
$3 +$
and Fe
$5 +$
ions; the center shift and the hyperfine field coincide approximately with the average values of the corresponding parameters for CaFeO
]. In other words, “static” disproportionation occurs in CaFeO
with the formation of a site-centered charge order, whereas in SrFeO
, we are dealing with “dynamic” disproportionation, with the formation of a bond-centered charge order. Furthermore, in SrFeO
, experiments have revealed a phase-separated state with a surprising variety of magnetic incommensurate helical and commensurate structures [
Surprisingly, a ferromagnetic ground state is found in BaFeO
single-crystalline thin films with a saturation magnetization and Curie temperature of 3.2
$μ B$
/formula unit and 115 K, respectively [
]. Unusually, for a uniform cubic ferromagnet, the films are insulating, possessing an optical gap of ∼1.8 eV.
The incommensurate helicoidal spin ordering observed in both CaFeO
and SrFeO
], up to very low temperatures, can be explained as a result of the competition between conventional exchange coupling and the bosonic double exchange. Obviously, the theoretical and experimental
studies of the phase diagram for (Ca,Sr)FeO
and substituted systems deserve further exploration, especially, investigations aimed at exploring possible superconductivity.
Fe Mössbauer measurements for the double-layered perovskite ferrate Sr
indicate the charge disproportionation and the magnetic properties, which are similar to CaFeO
]. The critical temperature for the charge disproportionation reaction and the Néel temperature T
of the helical spin order are determined to be ∼343 K and ∼120 K, respectively. Above 343 K, spectra clearly show a Fe
$4 +$
singlet. Puzzlingly, the spatial ordering pattern of the disproportionated charges has remained “hidden” to conventional diffraction probes, despite numerous X-ray scattering and neutron scattering
studies. Only relatively recently, by making use of neutron Larmor diffraction and Fe K-edge resonant X-ray scattering, Kim et al. [
] demonstrated the checkerboard charge order in the FeO
layers and showed that the “invisibility” of charge ordering in Sr
originates from the frustration of the interactions between neighboring layers.
The less-studied quasi-2D ferrate Sr
with the K
structure is a compound isotypic with the parent cuprate La
. It is an antiferromagnetic semiconductor at ambient pressure with a Néel temperature T
of about 56 K [
]. In the past 30 years, the concept of the electronic structure of Sr
has changed from a Mott-type antiferromagnetic insulator similar to La
] to an insulator with negative charge-transfer energy (negative-
$Δ p d$
) [
]. The insulating ground state of Sr
is assumed to be stabilized by a hidden structural distortion similar to the charge order in the related Sr
, and differs from the charge disproportionation in other Fe
$4 +$
However, we believe that the ground spin–charge state in this ferrate, as well as in other JT ferrates, is determined by
anti-JT disproportionation. This is evidenced by the absence of a noticeable JT distortion of the FeO
octahedra, the manifestation of a phonon mode atypical for the K
structure, which can be naturally associated with a breathing mode typical for
disproportionation, an elliptical cycloidal spin spiral structure typical of all JT ferrates, and an insulator–metal transition under high pressure [
]. To elucidate the details of the ground state, we require further studies, particularly on single crystals of Sr
7.5. JT Ruthenates
Just like Fe
$4 +$
$d 4$
) JT ferrates, Ru
$4 +$
$d 4$
)-based ruthenates belong to the same family of Ruddlesden–Popper (A
$n + 1$
$3 n + 1$
) compounds. They host rich physics, including unconventional superconductivity in Sr
, a metamagnetic ground state in Sr
, insulating antiferromagnetism in Ca
and Ca
, and both paramagnetic and ferromagnetic metallic states in CaRuO
and SrRuO
, respectively. Ruthenates undergo a variety of electronic, magnetic, and orbital ordering transitions, which are tunable with chemical doping, pressure, temperature, magnetic fields, and epitaxial
strain. However, their properties differ in many points from their 3d analogs. This is due to the fact that the 4d shell of the Ru
$4 +$
ion is more extended than the 3d shell of the Fe
$4 +$
electronic analog, which most likely leads to an increase in the crystal field parameter 10
q, a decrease in the local correlation parameter, and an increase in the transfer integrals. As a result, the Ru
$4 +$
$4 +$
) ions tend to adopt a low-spin state or
= 1 state because relatively large crystal fields often overpower the Hund’s rule coupling [
In other words, in ruthenates, we seemingly encounter a fine high-spin–low-spin (HS–LS) balance, up to the possibility of the coexistence of HS- and LS-states [
]. It means that by varying substitutions, tuning the physical and chemical pressures, and reducing the film thicknesses, one can observe different quantum states, ranging from those typical for JT
magnets, such as JT ferrates, to states typical for low-spin
$t 2 g 4$
-systems, with a trend toward phase separation.
Practically all layered ruthenates at low temperatures are characterized by robust Fermi liquid behavior, as evidenced by the quadratic temperature dependence of resistivity and by the observations
of quantum oscillations. However, the breach of the Mott–Ioffe–Regel limit for the basal plane resistivity and the anomalous strange metallic behavior, with a linear temperature dependence of
resistivity at high temperatures, clearly exhibit behavior inconsistent with any conventional Fermi liquid paradigms [
] but are typical for disproportionate systems with two types of charge transport.
Ruthenates are excellent candidates to explore the intricate interplay between structural and electron–spin degrees of freedom. For instance, Ca
is a paramagnetic Mott insulator below the metal–insulator transition temperature T
$M I T ≈$
360 K with antiferromagnetic ordering below T
$N ≈$
110 K [
]. However, the application of very modest pressure transforms it from the antiferromagnetic Mott insulator to a quasi-2D ferromagnetic metal. Under the current flow, the insulating ground state was
observed to transform into an electrically conducting phase with highly diamagnetic susceptibility.
Puzzlingly, single-crystalline Ca
nanofilms exhibit the co-appearance of high-temperature superconductivity with T
$c ≈$
60 K and ferromagnetism [
]. Such a high temperature of the superconducting transition suggests the presence of an unconventional mechanism of superconductivity of the type found in high-T
The replacement of Ca
$2 +$
ions (ionic radius 1.34 Å) with Sr
$2 +$
ions (ionic radius 1.44 Å) in the bulk family appears to induce a subtle alteration in the electronic structure, while simultaneously leading to a dramatic transformation of the ground state from an
antiferromagnetic insulator in Ca
to a superconducting and ferromagnetic state in Sr
, with a spiral spin structure in the ground normal metallic state [
Based on early Knight shift, polarized neutron scattering, muon–spin-resonance, and polar Kerr measurements, Sr
has been widely believed to support a spin–triplet chiral
-wave superconducting state [
]. However, despite significant achievements in characterizing the properties of Sr
over the last three decades, the precise nature of its electronic ground state and superconducting order parameter is still unresolved [
]. Understanding the nature of superconductivity in Sr
is one of the most enigmatic problems in unconventional superconductivity, despite the vast interest and wide array of experiments performed on the material. Recent results have pushed the community
toward potentially adopting an even-parity spin–singlet pairing state, although conventional states of this nature are not able to consistently explain all observations. It should be noted that
superconductivity is a relatively common property of ruthenates. Very recently, strain-stabilized superconductivity with T
$c ≈$
2 K was discovered in ruthenate RuO
films [
Generally speaking, despite extensive efforts, a comprehensive understanding of electronic structures and physical properties in JT ruthenates is still lacking.
7.6. Iron-Based Superconductors
The Fe
$2 +$
iron-based superconductors have layered structures with the conducting layers made of tetrahedral centers FeAs
, FeP
(ferropnictides), FeSe
, FeS
, FeTe
(ferrochalcigenides). These JT magnets exhibit the unprecedented richness of physics, sometimes within a single family, encompassing magnetism, unconventional superconductivity, quantum criticality,
linear-in-T resistivity, nematic order, and a propensity toward orbital-selective Mott behavior [
]. Researchers have found practically all phenomena associated with strongly correlated electron systems in Fe-based materials. At present, a variety of theoretical approaches are being employed to
understand these systems, although the issue remains to be fully settled.
Here, our intention is not to deliver a comprehensive review of the electronic structures and phase diagrams of iron-based superconductors, but rather to pay attention to several specific features
that allow us to assume an important role of the disproportionation mechanism. Superconductivity in FePn/Ch emerges out of a “bad-metal” normal state and the superconducting phase occurs near the
antiferromagnetic order in proximity to a Mott transition. The parent iron pnictides are antiferromagnetically ordered metals; insulating behavior and AF order also appear in a variety of iron
Unconventional non-BCS superconductivity in FePn/Ch has much in common with that of copper oxides; in particular, the ratio of T
versus the superfluid density is close to the Uemura plot observed for hole-doped high-T
cuprates [
]; as for cuprates, the electronic nematicity has been observed in the normal states of many—if not all—the FePn/Ch.
At the same time, FePn/Ch is different in many respects from cuprates. Thus, the high field inelastic neutron scattering data in the optimally doped Fe(Se,Te) superconductor [
] and in 112-type pnictide [
] show that—similar to cuprates—magnetic fluctuations play a central role in iron superconductivity; however, these suggest that the superconductivity of FePn/Ch is actually driven by a spin–triplet
bound state. The spin–triplet nature of superconducting carriers in FePn/FeCh was proposed back in 2008 [
] and has been confirmed by several experimental facts [
], although experimental data are contradictory [
]. In this regard, let us turn our attention to one of the primary modern techniques used for determining the spin of superconducting carriers: measuring the spin susceptibility by measuring the
Knight shift [
]. It is believed that spins in a triplet superconductor should be polarized in an external magnetic field, just like free spins in an ordinary metal. Thus, in such a system, one can expect that the
spin susceptibility and the Knight shift should not have singularities in T
. Spin anisotropy can suppress this for some directions but not for others. In a spin–singlet superconductor, the magnetic susceptibility vanishes at T→ 0. Thus, for a spin–singlet superconductivity,
a decrease in the uniform spin susceptibility below T
can be expected, although qualitatively, the same can occur for certain components of the triplet, although the vanishing susceptibility is often difficult to determine due to the background Van
Vleck contribution. However, this technique does not take into account the complex nature of spin interactions and the spin structure of spin–triplet superconductors.
The “singlet-triplet” dilemma for superconducting carriers in the vast majority of superconductors is considered within the framework of the BCS scenario, while the model of anti-JT
disproportionation in JT magnets represents a fundamentally different view of the mechanism of superconductivity, in which superconducting carriers are effective local, singlet or triplet, hole or
electronic, composite bosons. Our model assumes that superconducting carriers in FePn/Ch compounds consist of
$e g$
holes, and not of
$t 2 g$
electrons, as predicted by the single-electron multi-orbital band model [
At the moment, we cannot present an unambiguous conclusion about the role of the mechanism of anti-JT disproportionation in iron-based superconductors; however, finding high-T$c$ superconductivity in
FePn/Ch compounds with the tetrahedral coordination of iron Fe$2 +$(3$d 6$) ions in the HS state, and the coexistence of unconventional magnetism, can be a key argument that supports the
disproportionation scenario.
More surprisingly, our simple model provides convincing predictions of superconductivity and its features in different quasi-two-dimensional JT magnets, cuprates, nickelates, ruthenates, and
ferropnictides/chalcogenides, differing both in the electronic structures of active centers, and in the local crystal structures. The model predicts hole-type bosonic spin–singlet superconductivity
in 2D cuprates and nickelates, spin–triplet hole superconductivity in FePn/FeCh with sufficiently high T
in both systems, and electronic superconductivity in Sr
with very low T
, in agreement with Hirsch’s ideas about the hole nature of the HTSC [
8. Summary
We believe that the unusual properties of a wide class of JT magnets—materials based on Jahn–Teller 3d and 4d ions with diverse crystal and electronic structures, ranging from quasi-two-dimensional
unconventional superconductors (cuprates, nickelates, ferropnictides/chalcogenides, ruthenate SrRuO$4$), and manganites with localized superconductivity, to 3D ferrates (CaSr)FeO$3$, nickelates RNiO
$3$, and silver oxide AgO with unusual charge and magnetic orders—can be explained within the framework of a single scenario, which assumes their instability with respect to anti-Jahn–Teller d-d
disproportionation. As a result of disproportionation, the parent (“progenitor”) JT magnet is transformed into a half-filled system that is equivalent to a single- or two-band system of effective
local composite spin–singlet or spin–triplet, electron or hole S-type bosons in a magnetic or non-magnetic lattice, which gives rise to an extremely rich set of phase states, from non-magnetic and
magnetic insulators, unusual magnetic metallic and superconducting states, to a specific nematic ordering of the EH dimers. The effective composite bosons cannot be considered conventional
quasiparticles; they are an integral part of many-electron configurations. The effective spin-dependent two-particle bosonic transport in two-band JT magnets results in behavior that is typical for
“double-exchange” systems.
The model provides a comprehensive understanding of the well-established charge and magnetic order in JT ferrates and nickelates RNiO$3$, including the nontrivial effect of the
cation–anion–cation-bonding angle.
The most optimal conditions for HTSC with spin–singlet local composite bosons and a spinless lattice can only be achieved for low-symmetry quasi-two-dimensional d$9$ JT magnets, such as 2D cuprates
and nickelates, where disproportionation follows the traditional Jahn–Teller effect and orbital ordering.
The anti-JT disproportionation model points to a possibility of spin–triplet superconductivity in ruthenates Sr$2$RuO$4$ and RuO$2$, ferropnictides/chalcogenides FePn/FeCh, and manganite LaMnO$3$,
although in most of the known “candidates” (Ca(Sr)FeO$3$, RNiO$3$, AgO), a specific spin–charge order is realized. The model assumes that effective superconducting carriers in the FePn/FeCh compounds
consist of $e g$ holes rather than $t 2 g$ electrons, as predicted by the one-electron multi-orbital band models. The effective Hamiltonians for spin–triplet composite bosons in nonmagnetic and
magnetic lattices have complex spin structures, which must be taken into account when interpreting experiments to determine the spin of superconducting carriers.
The research was supported by the Ministry of Education and Science of the Russian Federation, project No. FEUZ-2023-0017.
I thank Yuri Panov for the very fruitful multi-year collaboration and the stimulating and encouraging discussions.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. (Color online): Spin structure of the EH dimer, or self-trapped CT exciton with a step-by-step inclusion of the one- and two-particle charge transfer. Arrows point to electric dipole
moments for bare site-centered dimer configurations.
Figure 2. (Color online): Angular dependencies of $J ( d 5 d 3 )$ and $1 10 | t B |$, which define the effective integral $J e f f = J ( d 5 d 3 ) − 0.1 | t B |$.
Figure 3.
(Color online): Model phase
-diagrams of hole-doped CuO
planes in cuprates/nickelates calculated in the effective field approximation (
$n = p$
for hole doping), with the phase separation taken into account using Maxwell’s construction;
is the exchange integral,
$Δ = U / 2$
is the local correlation parameter,
is the nonlocal correlation parameter,
$t p , t n , t p n$
are three independent integrals of the correlated single-particle transfer,
$t B$
is the effective transfer integral of the composite boson (see insets), assuming competition between “monophases” NO (disordered), AFMI, BS, FL, and CO. The boundaries between the phases represent
lines of equal free energies. The dashed curves (
) indicate the lines of equal volume fractions of two neighboring phases, the yellow curves represent the lines of phase transitions of the “third” kind, limiting the regions with maximal 100% volume
fractions of one of the phases. See Refs. [
] for more details. Pay attention to the strong change in the phase diagram, even with a very small change in the parameters of the Hamiltonian (compare panels
JT Configuration Symm. LS/HS Local Lattice Representative
JT Ions Boson Compounds
$3 d 1$($e g 1$):$2 E$ tetra - $e g 2$:$3 A 2 g$ A$1 g$ $β$-Sr$2$VO$4$
Ti$3 +$, V$4 +$, Cr$5 +$ s = 1 S = 0 (Sr,Ba)$3$Cr$2$O$8$
$3 d 3$($e g 3$):$2 E$ tetra LS $e ̲g 2$:$3 A 2 g$ A$1 g$ Ba$2$VGe$2$O$7$ (?)
V$2 +$, Cr$3 +$, Mn$4 +$ s = 1 S = 0
CrO, CrF$2$
$3 d 4$($t 2 g 3 e g 1$):$5 E$ $e g 2$:$3 A 2 g$ A$2 g$ Sr$2$FeO$4$
Cr$2 +$, Mn$3 +$, Fe$4 +$ octa HS s = 1 S = 3/2 (Ca,Sr,Ba)FeO$3$
RMnO$3$, LaMn$7$O$12$
$4 d 4$($t 2 g 3 e g 1$):$5 E$ $e g 2$:$3 A 2 g$ A$2 g$ (Ca,Sr)$2$RuO$4$
Ru$4 +$ octa HS s = 1 S = 3/2 (Ca,Sr)RuO$3$, RuO$2$
$3 d 6$($e g 3 t 2 g 3$):$5 E$ tetra HS $e ̲g 2$:$3 A 2 g$ A$2 g$ FePn, FeCh, Na$5$CoO$4$
Fe$2 +$, Co$3 +$ s = 1 S = 3/2
$3 d 7$($t 2 g 6 e g 1$):$2 E$ octa LS $e g 2$:$3 A 2 g$ A$1 g$ RNiO$3$
Co$2 +$, Ni$3 +$ s = 1 S = 0 (Li,Na,Ag)NiO$2$
$3 d 9$($t 2 g 6 e g 3$):$2 E$ octa - $e ̲g 2$:$3 A 2 g$ A$1 g$ CuF$2$, KCuF$3$, K$2$CuF$4$
Cu$2 +$, Ni$+$ s = 1 S = 0
$4 d 9$($t 2 g 6 e g 3$):$2 E$ octa - $e ̲g 2$:$3 A 2 g$ A$1 g$ AgO (Ag$1 +$Ag$3 +$O$2$)
Pd$+$, Ag$2 +$ s = 1 S = 0
$3 d 9$($t 2 g 6 e g 3$):$2 B 1 g$ octa*square - $b ̲1 g 2$:$1 A 1 g$ A$1 g$ HTSC cuprates
Cu$2 +$, Ni$+$ s = 0 S = 0 RNiO$2$, CuO
$4 d 9$($t 2 g 6 e g 3$):$2 B 1 g$ octa*square - $b ̲1 g 2$:$1 A 1 g$ A$1 g$ AgF$2$, KAgF$3$
Pd$+$, Ag$2 +$ s = 0 S = 0 Cs$2$AgF$4$, LaPdO$2$ (?)
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Measuring Tools
Tools and practical methods for measuring and analyzing precision.
Paste your data; get the statistics. Also available as an Excel spreadsheet.
If you're willing to sacrifice statistical efficiency, Brent Danielson noted that you can get away with a single measurement for successive 2-shot groups, instead of the coordinate (x, y)
measurements of each shot required for the maximally efficient estimator. I.e.,
1. Fire two shots at a single point of aim.
2. Measure their center-to-center distance.
This produces two sample radii, both equal to half the measured distance.
The benefit of this approach is that it only requires one measurement in one dimension for every two shots. The drawback is that, to get the same statistical confidence, this requires almost double
the number of shots as if measuring the coordinates of all the shots in a single group. (With n shots split over g groups, the statistical formulas show that confidence is an increasing function of (
n-g), so going from 1 group to n/2 groups requires 2n-1 shots.)
Calculation of sigma from Danielson's sample data, as well as confidence intervals and hypothesis testing, are shown in Media:DanielsonExample.xlsx.
Jeffrey Block's OnTarget Precision Calculator is the most convenient package for converting a target image into data points for analysis. It accounts for scale and distance and automatically
calculates Mean Radius (called "Average to Center" in the software) and Extreme Spread (called "Max Spread").
The more expensive Target Data System can automatically identify and aggregate shots on scans of its specially-coded targets.
Taran (target analysis and shooting precision calculator) is a free online application to upload a target image, mark the points of impact, and download the coordinates of the points. Among others,
it also calculates the Rayleigh CEP.
The free shotGroups package for the open-source statistical environment R provides functions to analyze target groups with respect to their shape, location (accuracy) and spread (precision). Among
others, it provides implementions for many CEP estimators and descriptive precision measures. The package works with point data exported from OnTarget or Taran and includes functions to plot the
group with precision indicators like the bounding box, maximum spread or minimum covering circle.
The main functionality of the package is also available as a set of web applications that do not require installing R or using R syntax.:
For more information, see the package description including a walk-through with sample diagrams and the complete manual for all functions. After having installed R and RStudio, open RStudio and
install shotGroups by running: install.packages("shotGroups") . For a first introduction to R, see:
• TargetScan (iOS) computes the unbiased estimate of Mean Radius for supported targets.
Spreadsheet Analysis
Given a target data set, whether compiled using the #2-Shot Method or #OnTarget, Closed Form Precision analysis can be performed using standard spreadsheet functions. See, for example Media:CCI 40gr
HV 100yd.xlsx or any of the other workbooks linked in the Examples.
Media:RangeStatisticEstimation.xls is spreadsheet for calculating the statistical significance of Range Statistics estimates.
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Calculation of Detector Positions for a Source Localizing Radiation Portal Monitor System Using a Modified Iterative Genetic Algorithm
Radiation portal monitors (RPMs) have been deployed at nations’ borders to screen individuals, vehicles, or cargo at border or security facilities to thwart the smuggling of illicit radiological
sources and materials for nuclear weapons. The utilization of RPMs is not limited to the detection of radioactive sources. Depending on which technology is integrated with RPMs, diverse functions can
be implemented in RPMs. Various applications have been developed for the convenience of the operators such as radio isotope identification, source localizing and so on. One of them is the localizing
and tracking of radioactive sources. This is an application of signal processing techniques for RPMs. Coulon et al.[
] and Rao et al.[
] presented tracking algorithms that utilize signal differences over time from RPM networks. Miller et al.[
] suggested a theoretical algorithm using an inverse transport approach. Vilim et al.[
] and Jarman et al.[
] proposed localization algorithms based on statistical probability theory. Furthermore, Miller et al.[
] presented an improved localization algorithm by combining radiography and a statistical probability approach.
Despite such studies on RPMs with signal processing techniques, few have attempted to apply machine learning to RPM applications. Most of radioactive source localizing researches are based on
theoretical modeling, which uncertainties are not considered such as buildup effect, overestimation, measurement error, and so on, because these uncertainties are hard to be expressed by mathematical
modeling. Machine leaning is a subfield of computer science that provides computers with the ability to learn without clear programing. Machine learning follows a natural discipline or draws
conclusions in a manner analogous to that of the human brain for complicated problems. Therefore, it is expected that source localizing RPM can be implemented without complicated theoretical
background, if machine learning is utilized. Furthermore, it could have better performance than other researches based on theoretical models, which are hard to consider the uncertainties, if RPM
application is implemented using machine learning algorithm.
Particularly, in the past few years, the number of fields to which machine learning has been applied has considerably increased because of its reliable and feasible results, and the computational
speed has been significantly reduced by computer developments. However, the performance of machine learning can vary depending on quality of training data. Therefore, it is important to implement a
machine learning algorithm itself, but it is also important to configure the experimental environment well to make high quality training data. As a preliminary phase for implementing a radioactive
source localization algorithm using machine learning, this paper focuses on the environmental configuration of an RPM.
This study will address a machine learning based optimization method, and trial of it on the calculation of detector positions for a radioactive source localizing RPM. In nuclear science,
optimization is generally performed using data acquired through simulations or experiments.[
] Conducting optimization using optimization theory is uncommon work in this area. Wacholder et al.[
] presented a study of the optimization of a radioactive source localization system using optimization theory based on a machine learning algorithm. However, they did not consider several cases in
which a solution is not globally optimal if the problem is complicated. In this paper, a modified iterative genetic algorithm (MIGA) will be implemented by improving a genetic algorithm (GA), one of
the classic machine learning algorithms, to increase the chance to find globally optimal solution than, and utilized to optimize positions of detectors for a radioactive source localizing RPM.
Materials and Methods
1. Optimization using a modified iterative genetic algorithm
1) Optimization and genetic algorithm
The optimization problem is one of the most essential problems in engineering. In an optimization problem, the problem has an objective function and constraints on the variables. The objective
function represents the cost of the system. The equality or inequality constraints represent the limitations on the variables, such as the budget and fuel. Optimization is a process that finds the
best set of variables to minimize or maximize the objective function. For simple functions, finding the minimum or maximum value is quite simple. However, it is generally difficult to find the
minimum or maximum values analytically for optimization problems because of the numerous variables and their complex relationships.
A GA is based on Darwin’s theory of the survival of the fittest and the principles of natural genetics and natural selection.[
] In other traditional optimization problems, the objective function should be differentiable or partially differentiable. However, a GA can be utilized for any type of problem whose objective
function can be defined as a function of the output, even for discontinuous functions. Owing to this domain-independent strength, a GA is considered to be a powerful technique for solving complex
problems and is utilized in various areas.
In a GA, solution candidates are treated as individuals, and these individuals form a population. Each individual has strings that correspond to the chromosomes in natural genetics. In the generation
procedure, a new set of strings is generated by randomized selection, reproduction, crossover, or mutation from the past generation. Through a random procedure, the GA is not simple as general random
search method. In the new generation procedures, the genotypes of genes are adapted to the environment to generate improved genes in the next generations using a fitness function. By iterating
generations, individuals that have a mediocre fitness disappear, and those that have outstanding fitness are utilized in offspring production procedures. Consequently, only the individuals that have
outstanding fitness will survive globally. This means that the result of a GA ultimately reaches the globally optimal solution with a high probability.
Figure 1
shows the pseudocode of a generic GA, and
Figure 2
shows an example illustrating the locally and globally optimal solutions. As shown in
Figure 2
, there are local minima and global minimum in an optimization problem. Only global minimum is the true optimal solution, and the others is not optimal solutions.
2) Modified iterative genetic algorithm
Even though a GA is theoretically very good for searching for a globally optimal solution, it is not a perfect algorithm. Depending on the complexity of the problem, a GA also can find a local
minimum as an optimal solution. Therefore, it is necessary to improve the GA to avoid local minima for several complicated problems.
There have been various studies of enhancements in GAs based on modifications of the original GA. Michalewicz et al. proposed a modified genetic algorithm (MGA) using specialized operators [
]. The operators are utilized in the mutation and crossover procedures to increase the convergence performance. However, MGA is developed based on systemic characteristics of control problems, it is
difficult to be applied in general optimization problems. Several researchers have suggested iterative genetic algorithms (IGAs) to solve problems that have complicated variables associated with one
another [
]. In these studies, GAs that optimize different values are arranged in cascaded stages, and a result from a node is utilized in next node as a constraint condition. IGAs are effective to optimize a
system consisting of variables in complicated relationship, but the performance of optimization is not improved.
In this paper, a modified iterative genetic algorithm (MIGA), which has little in common with previous studies, will be presented. In the MIGA, several modifications have been applied to the IGA to
modify the termination criteria to increase the probability to find globally optimal solution. In the GA, there are three termination criteria: the number of generations, individual tolerance, and
fitness tolerance. However, the MIGA has four termination criteria. The number of generations and individual tolerance are akin to those of the GA. In contrast to other algorithms presented in
previous studies, the MIGA utilizes a variable fitness tolerance and the number of iterations. For here, generation and iteration are intrinsic and systemic parameters of GA and MIGA. Individuals are
corresponding to candidates of optimal detector positions, and fitness value means the value of optimization function described
equation 4
that will be addressed in section 2.2.3.
To adjust the fitness tolerance, the end of GA procedure is supplemented with the competition procedure. In the competition procedure, the optimal result in a GA loop is selected and stored in a
competition rank. By repeating each loop, the fitness of an individual ranked in first place among the competition results is fed back to the fitness tolerance. At the end of the MIGA, the one more
GA loop is executed with the top-ranked individual as an initial condition to verify that the individual is the optimal solution. If the verification result is not optimal, the number of iterations
is initialized, and entire MIGA is resumed with only the competition data. In short, the MIGA iterates GAs by the number of iterations to enhance the probability of reaching a globally optimal
solution with adjustments of the fitness tolerance by competition and verifies the optimal solution at the end of the MIGA. The entire procedure is shown in
Figure 3
2. Calculation of detector positions for the system
1) Structural design
The structural design of an RPM for radioactive source localization is based on the external dimensions of container trailers. Standard ISO containers are 2.43 m wide and 2.59 m high and have two
lengths, 6.06 and 12.2 m.
In the case of trailers, the height of a flatbed trailer is commonly less than 1.40 m. Because vehicles move on the ground, the trailers frequently move from side to side but hardly up and down when
they pass through an RPM. To consider these types of fluctuations, the spatial margins of the system are defined as follows; the height margin is 0.5 m, and the width margin is 0.65 m for each side.
On the basis of this information, a basic structure of an RPM is designed. A conceptual design of the RPM is shown in
Figure 4
In this system, the number of detectors is equivalent to the number of input measurements. According to the number of detectors, the accuracy of source localization will be increased. However, it is
not a sufficient solution to increase the number of detectors without other considerations because the relationship between the number of detectors and the localization accuracy is not simple. This
relationship tends to be logarithmic rather than proportional. Because the number of detectors is highly associated with the performance of the system and budgetary constraints, the number of
detectors must be determined on a case-by-case basis for each situation. However, the decision procedure will not be addressed because the details regarding this relationship are beyond the scope of
this paper. In total, six detectors will be set on the RPM for our research.
NaI (TI) scintillator detectors with a volume of 4×4×16 in
will be utilized for the RPM system. These detectors are also utilized in other RPM researches [
]. If the active area is too small, the detection performance is poor, and if it is too large, it is difficult to specify the position of the detector. From this perspective, the NaI detector seems
to be the appropriate choice and is the reason why it is widely utilized in RPM systems.
2) Inverse transport theory to define optimization problem
The measurements from detectors are utilized to localize a radioactive source. For a detector, a measured quantity of a source in a specific medium can be expressed as follows: [
• D = Measured quantity with a detector
• A = the active area of a detector,
• ɛ[ip] = the intrinsic parameter of a detector,
• d = the distance between the source and the detector,
• I[0] = the intensity of the source,
• μ = the absorption coefficient of the medium.
This equation is the simplest theoretical model. In this equation, uncertainties such as build-up effect, over estimation, measurement noise, and so on are ignored because mathematical modeling of
them is too complicated. As presented above, a measurement is associated with the distance from a detector to a source. The distance is one-dimensional (1D) data, but the location is two-dimensional
(2D) or three-dimensional (3D) data. Therefore, we have to acquire more measurements from other detectors to transform 1D data into 2D or 3D data. The relevant techniques have been commonly utilized
in acoustic and sensor network studies [
]. The key aspects of these techniques is the solution of simultaneous nonlinear equations.
We define a radioactive source localization system as shown in
Figure 5
. The system consists of a structural frame, detectors, and their electronics. Each detector is fixed on a structure. Therefore, all detectors have their own fixed positions. In the case where a
radioactive source is placed in the system, the number of measurements acquired may be equal to the number of detectors. Using the Pythagorean theorem, the distances between the sources and the
detectors can be expressed mathematically. By substituting the equation for the distance into
equation 1
, a nonlinear equation concerning the position of a radiation source can be expressed as follows:
• x[s] = the X coordinate of the position of the source,
• y[s] = the Y coordinate of the position of the source,
• x[i] = the X coordinate of the position of the i^th detector,
• y[i] = the Y coordinate of the position of the i^th detector in the Cartesian coordinate system,
• D[i] = measurement of the i^th detector.
If all parameters concerning a detector and the radiation absorption are known,
equation 2
can be expressed as nonlinear equations that have three identical unknown variables:
, and
. This means that the position of the source can be calculated by solving a system of equations if the number of these equations is greater than three. It is possible to solve nonlinear equations
even without the intensity of the source. The detector parameters can be confirmed by checking the specifications of the detector. In the case of the absorption coefficient, it varies with the medium
that a radiation particle passes through. Because inhomogeneous media are quite complicated, only a homogeneous medium will be considered to simplify the problem in this research.
3) Problem definition
On the basis of the structural design of an RPM, the positions of each detector should be determined. The positions of the detectors should be considered according to the purpose of the system. In
typical RPMs that are used to detect radioactive materials only, the positions of the detectors may not be very important because they are focused on the detection of radiation. However, the major
purposes of these systems are the detection and localization of radioactive materials. From the perspective of localization, the contrasts among each measurement will affect the localization
performance because the position of the source is calculated using the detector measurements. Therefore, the measurement contrasts should be maximized to increase the localization performance.
The other specifications of the structural design are already determined during the conceptual design, except for the positions of the detectors. These positions should be determined in the
directions that maximize the contrasts of the measurements. Before defining an optimization problem, the source data should be established. In most cases in which an arbitrary radioactive source is
contained in a container, the position of the source will be inside the container. Therefore, a region of interest (ROI) can be defined as an area that is identical to the external size of standard
containers. Then, point sources that are regularly distributed within an ROI can be set as source data. In practice, cargo can’t be loaded to full container height because of limitation of the load
weigh. However, just for simplification, the distribution of source points was assumed to be uniform. In this paper, an array of 11×11 point sources is utilized as the data. All sources are
monoenergetic, and the intensity of a source is 1×10^6 gamma rays·s^−1.
As mentioned in section 2.2.2, the position of the source can be calculated if all other parameters are known values. To define the problem, all parameters are defined as follows. The inner space of
the RPM is assumed to be a homogeneous medium. The absorption coefficient of the medium is chosen to be 10
, equivalent to the microscopic cross section of air. Each detector is assumed to have an active area of ~412.9 cm
and an intrinsic efficiency of 10% for simplification. These values are reasonable for NaI scintillator detectors [
With the defined source data and parameters,
equation 1
can be expressed as a function of the unknown detector positions as
equation 3
. In this equation, the unknowns are marked by star symbols.
The simultaneous calculations of the differences for 121 sources is equivalent to the case in which the measurement differences of each detector for a strong source located at the center of the ROI
are calculated. Therefore, the differences should be calculated on case-by-case basis for each source. To maximize the contrast of the measurements regardless of the location of a source, the
following objective function is defined:
• m = the number of sources,
• n = the number of detectors,
• D[i] = measurement of the i^th detector according to equation 1,
• (D[ij])k = total difference in the measurements between the i^th and j^th detectors for the k^th source.
In optimization problems, maximization can also be represented as minimization by attaching a negative sign to the objective function. The reason why
is divided by 2 is to countervail the overlap between two identical differences such as
. By substituting the
terms in
equation 4
equation 3
, the objective function can be represented with the unknown detector positions.
The equality and inequality constraints of this problem are related to the structural frame of the RPM. Six detectors will be installed in this system, as mentioned above. The frame consists of three
beams on the left, right, and top sides. On each side of the frame, two detectors can be assigned: detectors 1 and 2 to the left, detectors 3 and 4 to the top, and detectors 5 and 6 to the right
sequentially. For the left- and right-side detectors, the x positions are fixed on the frame. The y positions are adjustable but limited to the height of the frame. For top-side detectors, the x
positions are adjustable but limited to the width of the frame. The y positions are fixed on the frame. When each position of the detectors is represented by two vectors X and Y, these equality and
inequality constraints can be written with vectors and matrices as follows.
1) Equality constraints
2) Inequality constraints
• A = the matrix for selecting the left- and right-side detectors,
• B = the matrix for selecting the top-side detectors,
• W = the width of the RPM frame,
• H = the height of the RPM frame,
• x[i] = the X coordinate of the position of the i^th detector,
• yi = the Y coordinate of the position of the i^th detector in the Cartesian coordinate system.
Results and Discussion
1. Performance evaluation of the MIGA
The optimization problem defined in section 2.3.2 was solved by two algorithms—the GA and MIGA. These algorithms were implemented in a MATLAB environment to compare their performance. In this
section, the optimization results and analyses will be presented sequentially.
A GA is an effective algorithm for finding a globally optimal solution, but the result might be a locally optimal solution depending on the complexity of the problem.
Figure 6
shows the variation in the fitness value for each generation. For simulation, the number of generation is set to 2,000, and individual tolerance and fitness tolerance are equally set to 1×10
. At the beginning of the GA, the fitness value decreases rapidly. This means that the GA searches the optimal solution effectively. Then, the fitness value converges. This shows how the GA finds the
optimal solution.
In the MIGA, the competition, iteration, and verification procedures are added to increase the optimization performance.
Figure 7
shows the fitness value versus the number of iterations for the MIGA. To solve the defined problem using the MIGA, the number of iterations is set to 1,000. As shown in this figure, calculation
results for each iteration are not constant. This shows one of the weaknesses of the GA; all of the points in
Figure 7
represent the optimization results from each GA loop. However, only the points on the red dashed line that represent the minimum fitness values are real optimal solutions, and the others are local
minima. This means that we cannot know whether the solution is a globally optimal solution when using the GA only. Therefore, it is necessary to utilize the iteration and competition procedures to
increase the probability of finding globally optimal solutions.
At the end of the MIGA, the verification procedure is added to confirm whether the final result is globally optimal. In the verification procedure, the GA is executed once again with the final result
as an initial condition. If the result is a globally optimal solution, the verification result will be identical to the initial values themselves.
Figure 8
shows the variation in the fitness value during the process of verification. Although the variations in the fitness values are almost zero, the GA repeats the generation process 263 times. This shows
that the GA attempts to find another globally optimal solution.
Figure 9
shows a schematic of optimization using the MIGA. The basic frame of the RPM is described by the black bold line. A container (ROI) and flatbed trailer are described by red and green dashed lines,
respectively. The cyan, red, and blue stars represent the source information, the optimal positions of the detectors calculated by the MIGA, and the trisection points of the ROI, respectively.
Because the fact that the point sources are uniformly distributed inside of ROI is utilized as source information, we expect the result to be related with the area of the ROI. Therefore, the
trisection points were considered as estimated optimal positions. However, an optimal solution is found, which has lower fitness value than the estimated solution by 175.5565. An RPM system has been
designed using this optimization result. We also tried to solve this problem using GA, but we cannot find globally optimal solution that is found by MIGA.
Table 1
summarizes the estimated solution, local solution using GA, optimal solution using MIGA, and their fitness values. In
Table 1
, each solution is represented by the position (X, Y), and all values are presented in units of millimeters (mm).
In common sense, detectors should be located in symmetry. However, the case that the positions of detectors are located in asymmetry has higher fitness value than symmetric case as shown in
Figure 9
Table 1
. We thought that this result may cause owing to the definition the objective function. We defined the objective that maximize the contrast of detector measurements. In perspective of maximizing the
contrast, the contrast is getting larger in unbalanced condition than balanced condition.
In this paper, a machine learning algorithm was utilized to calculate positions of detectors for radioactive source localizing RPM system. To calculate detector positions, an optimization problem was
defined. To achieve a reliable solution, the MIGA was suggested by modifying a GA with supplemental iteration, competition, and verification procedures. The GA and MIGA were implemented using MATLAB,
and their performance was analyzed. In the case of the MIGA, the result was analyzed systematically. According to resulting optimization analyses, all supplemented procedures help to find a globally
optimal solution.
This paper is a preliminary step for the development of an intelligent RPM system that can detect the position of a radioactive source using machine learning algorithms. To apply machine learning to
RPM systems for source localization, it should be designed to maximize the performance of the system in accordance with the characteristics of the algorithms.
In typical pattern recognition problems, one of the most essential research fields in machine learning algorithms, all input data are transformed into features that represent the characteristics of
all patterns, and machines are trained by features to recognize an object. To increase the performance of a machine for pattern recognition, many samples that consist of clearly distinguishable
features should be supplied.
To implement a high-performance intelligent localization system, the information produced by the system should be distinguishable because features are generated by the information. In the case of an
intelligent RPM, measurements from the detectors will be utilized as the information. Therefore, it is important to design a system so that it can generate distinguishable detector measurements. In
short, the contrasts of the detector measurements should be clear. For this reason, an optimization problem was set up to maximize measurement differences in each detector.
On the basis of this study, an RPM has been designed and will be fabricated.
Figure 10
shows a mechanical drawing of the designed RPM. However, an evaluation of the localization performance with this optimization result cannot be conducted immediately because another machine learning
algorithm for localization should be implemented. As the next step of this work, an intelligent RPM system for radioactive source localization using a machine learning algorithm will be researched by
utilizing this designed RPM system. Then, the localization performance of this system will be evaluated.
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|
Mathematics A level
Mathematics can be split into three core areas: pure mathematics (geometry, trigonometry, algebra and calculus); statistics
(probability, estimation, correlation and regression, sampling, hypothesis testing); mechanics (kinetics, dynamics and statics).
Course Structure
Teaching begins in September (January for 18 month courses). In the first year you will cover the following topics:
Pure Mathematics
• Proof, algebra and functions, coordinate geometry in the (x,y) plane, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, vectors
• Statistical sampling, data presentation and interpretation, probability, statistical distributions, statistical hypothesis testing
Student may take the AS exam at the end of the first year of the programme. The second year then builds on this foundation for the completion of the full A level.
In the second year you will focus on the following topics, in greater depth:
Pure Mathematics
• Proof, algebra and functions, coordinate geometry in the (x,y) plane, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, vectors
• Statistical sampling, data presentation and interpretation, probability, statistical distributions, statistical hypothesis testing
• Quantities and units in mechanics, kinematics, forces and Newton’s Laws, moments
Exam Structure
A level Exam Format
Papers 1 &2 Paper 3
Pure Mathematics Statistics and Mechanics
2 hours each paper 2 hours
Mathematics One-Year A level
We follow the Edexcel legacy syllabus for resit candidates..
Programme Requirements
Students are normally required to have at least grade 7 at GCSE Mathematics, or equivalent.
Related Further Study and Careers
Mathematics is essential for those who want to read mathematics at university, but it is also highly suited to further study in other numerate subjects such as engineering or physics. Many students
who take A level Mathematics also go on to study degrees in economics or finance.
|
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|
The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper, we study E[γ]
-resolvability, in which total variation is replaced by the more general E[γ] distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let Q[X|U] be a
random transformation, n be an integer, and E (0,+\infty). We show that in the asymptotic setting where\γ = (nE), a (nonnegative) randomness rate above inf [QU: D(QX|πX)≤ E]\D(Q[X]\|\π[X])+I(Q[U],Q[X
|U])-E\is sufficient to approximate the output distribution π[X]^⊗n using the channel Q[X|U]^⊗n, where Q[U]\to Q[X|U]\to Q[X], and is also necessary in the case of finite U and X. In particular, a
randomness rate of inf[QU]I(Q[U],Q[X|U])-E is always sufficient. We also study the convergence of the approximation error under the high-probability criteria in the case of random codebooks.
Moreover, by developing simple bounds relating E[γ] and other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and
smooth Rényi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive: 1) a one-shot upper bound on the probability of excess distortion in lossy
compression, which is exponentially tight in the i.i.d. setting; 2) a one-shot version of the mutual covering lemma; and 3) a lower bound on the size of the eavesdropper list to include the actual
message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.
All Science Journal Classification (ASJC) codes
• Information Systems
• Computer Science Applications
• Library and Information Sciences
• Resolvability
• broadcast channel
• mutual covering lemma
• source coding
• wiretap channel
Dive into the research topics of 'E[γ]-Resolvability'. Together they form a unique fingerprint.
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Time Duration Calculator
Last updated:
Time Duration Calculator
This time duration calculator allows you to calculate the duration between two times. It accepts either two elapsed times or two clock times, where it supports both the 12-hour and 24-hour time of
day formats. The calculator correctly handles time durations that span midnight. As a bonus, this article shows you how to calculate time duration manually, making sure you have a firm understanding
of the maths behind the calculator.
After you've finished using this calculator, you might also be interested in our day counter calculator.
What is time?
Simplistically, time is the 4th dimension in our Universe, along with the three spatial dimensions of length, width, and height. You could say that time exists so that everything doesn't happen at
once. A universe without time wouldn't be a particularly exciting place to live. Time provides separation of events and allows cause and effect to be determined. So that's time sorted, right? Err,
no. Not really. Physicists and philosophers are still debating the exact nature of time and how it behaves to this very day.
Fortunately for us, as long as we measure two times in the same inertial frame of reference (a little bit of relativity there, time dilation calculator will help you grasp the concept), our time
duration calculator will work just fine, without knowing exactly what time is.
How to use the elapsed time calculator
The elapsed time calculator is in the first part of the calculator, found at the top. Enter a start time and an end time, and you will see the duration between those two times. By default, you can
enter the times in hours, minutes, and seconds. However, if you click on the units, you can select other time units to suit your needs.
Note that you can enter any two times in this section of the calculator (as long as the end elapsed time is more than the start elapsed time), and there are no restrictions imposed by the 12- or
24-hour time formats. Though if you are interested in the duration between two clock times, the next section of the calculator is just for you.
How to use the clock time calculator
The second section of the calculator is a clock time calculator. This calculator helps work out the duration between two times during a single day, or overnight. You can choose to either work with
the 12-hour clock format (the one we most use in everyday life) or the 24-hour format (which you might encounter when booking a flight, for example). Here's how you use the clock time duration
1. Select either the 12-hour or 24-hour time formats.
2. Enter the start clock time, giving the hour, minute, and optionally second. If you are using the 12-hour clock, don't forget to select whether it is am (in the morning) or pm (afternoon/evening).
3. Then do the same for the end clock time. The calculator supports time durations that start on one day and end the next day. So you could find the time duration from 8 pm until 5 am the next
4. The time duration calculator result will be shown as the number of hours, minutes, and seconds between the two times. If you want to see the result in another time unit (e.g., just in seconds),
click on the units to display the dropdown menu.
If you need to calculate time durations for a whole week at work, then our time card calculator might be worth checking out.
How to calculate time duration manually
Let's take some time to discuss how to do these time duration calculations manually. The same principles apply to both the elapsed time calculator and the clock time calculator (though for the clock
time calculator, if you are using the 12-hour time format, the first thing to do is to convert it to the 24-hour format. The solution is simply a case of adding 12 hours to any pm times!).
We will first study the example of calculating the time duration between 8:13 am and 4:55 pm. So we first convert the pm time to the 24-hour format:
• 4:55 pm => 4 + 12 = 16 => 16:55
We can then write down the two times the same way you would do a manual math subtraction:
$\small \begin{array}{rrrrr} 16:55\\ -8:13\\ \hline 8:42 \end{array}$
Then, calculate the difference in hours and minutes separately. In this example, the result for the number of hours is 16 − 8 = 8 hours, and for the number of minutes, it is 55 − 13 = 42 minutes.
Next, let's look at a more complicated example. How about 8:13 am until 4:07 pm? We convert the pm time to 24-hour clock format as before and write down the subtraction:
$\small \begin{array}{rrrrr} 16:&\!\!\!\!\!07&\\ -8:&\!\!\!\!\!13&\\ \hline 8:&\!\!\!\!\!-06 & \textit{\tiny ??? Not a valid time.} \end{array}$
On this occasion, however, if we try to do the subtraction on the minutes' column, we would end up with minus 6. In this case, we need to convert an hour from the hours' column to 60 minutes in the
minutes' column so we can get a positive value for the time difference. Here is what the new manual subtraction looks like:
$\small \begin{array}{rrrrr} 15:67\\ -8:13\\ \hline 7:54 \end{array}$
So 16 hours became 15 hours, and 7 minutes became 67 minutes, when we added the 60 minutes, carried over from the hours' column. Now we can do the subtraction for the hours 15 − 8 = 7 hours and
minutes 67 − 13 = 54 minutes as before. This same carryover method can be used if we also had a seconds' column.
Finally, if you want to calculate a time difference from one day to the next, you would need to add a day column and carryover 24 hours (as it is a day in the future), adding it to the hours column
of the first time. For example, to find the duration between 4:07 pm until 8:13 am the next morning, you would do:
$\small \begin{array}{rrrrr} 1 \! &\!\textrm{day} \!\!& 8:13 & \rightarrow &\!\! 32:13\\ -0 \!&\! \textrm{day}\!\! & 16:07 & &\!\! -16:07 \\ \hline &&&& 16:06 \end{array}$
So for some clock times, calculating the duration between two times is a little tricky, so it might be better to use our calculator instead!
How do I calculate time duration?
To calculate the time duration between two times:
1. Write both times in 24-hour format.
2. Write the later time above the earlier time and perform a long subtraction with some additional caveats:
□ If you need to carry over from the hours into the minutes column, be sure to add 60 minutes and not 100 minutes.
□ If the time duration happens on different days, add a left-most 'days' column. If you need to carry over from the days column into the hours column, be sure to add 24 hours.
How do I write time duration?
There is no set way to write time duration. If you need to write the time in shorthand, then the hh:mm:ss format will likely be enough (you can add more columns, such as days or milliseconds, as you
see fit). You could also write out the time duration explicitly in words, if you so desired, e.g., 1 hour, 26 minutes, and 49 seconds.
How do I determine the duration of time intervals in hours?
To find the duration of a time interval in hours:
1. Write both times in their 24-hour forms.
2. Subtract the earlier time from the later time.
3. If you have any amount of minutes or seconds left after subtraction:
1. Divide the seconds by 60 and add this value to the minutes.
2. Divide the minutes by 60 and add this to the total hours.
4. If you have any days, multiply the number of days by 24 and add this to the remaining hours.
What is 12:30 am in 24-hour format?
12:30 am is 00:30 in 24-hour format. This is because it is the first 30 minutes of the day.
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limit of function with dirac delta
limit of function with dirac delta
Consider the following code:
f = (1-x) * sage.unit_step(x)
df = f.diff(x) -> -unit_step(x) + (1-x)*dirac_delta(x)
Trying to compute df.limit(x=0, dir="right"), which should return -1, but instead returns limit(-unit_step(x) + (1-x)*dirac_delta(x), x, 0, plus).
Even df.limit(x=0.5) return limit(-unit_step(x) + (1-x)*dirac_delta(x), x, 0.5) instead of -1.
Any idea on what might be going on? Thanks!
My guess is that Maxima doesn't know how to take limits of these functions.
1 Answer
Sort by ยป oldest newest most voted
To expand on the comment by @kcrisman, limits are sent to Maxima by default. Maxima can evaluate the limit of the step function: entering
returns 1 as expected. Maxima cannot evaluate the limit of the Dirac delta: entering
returns an unevaluated expression. Since part of your limit cannot be evaluated by Maxima, it all comes back unevaluated.
The other option for limit evaluation is to set algorithm='sympy': entering
returns zero as expected. Unfortunately, entering
gives the message
sympy does not support one-sided limits
Even more problematic, entering
gives the message
SymPy function 'unit_step' doesn't exist
so SymPy won't get the complete job done either.
Not exactly the answer you want, but hopefully it helps you understand what's happening.
edit flag offensive delete link more
the existence problem can probably be fixed by setting a custom "conversion" dictionary in src/sage/functions/generalized.py
mforets ( 2017-03-10 07:54:15 +0100 )edit
the problem is that there is really no such UnitStep function in SymPy. Here it says that it's possible to define it at $0$ passing a 2nd argument. see also in github. But in my local sage
installation, it doesn't work; don't know if it's because of a difference in the SymPy versions..
mforets ( 2017-03-10 13:04:50 +0100 )edit
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Research paper:Non-Euclidean geometry
Research paper:Non-Euclidean geometry
Getting gained to be a weird oddity, no-Euclidean geometry, with time, was mainstreamed to medical idea. Really, no-Euclidean is globally and regionally work to get globally well-accepted approach.
https://englishessays.net/english-essay-writing Hence, no-Euclidean is greatly picture staying even more of scholastic meaning. The analysis will endeavour to display tactics developed and even a few
of the shortcomings that also a hitch. Hyperbolic and elliptic geometry is recognised as within the analyze. Range of mathematical models is offered awareness for these kinds of geometries; graphics
helps to a whole lot in understanding of hyperbolic geometry using a jet. In the case of two to three length and width, more attention must be put in place (Gunn 1991, p.18). By way of example,
visualization ventures on portion of spherical and hyperbolic thought to be, much more, growing disciplines of a couple of dimensions and photorealistic is pulling. The example attempts handbook and
will make consumers comprehend it in proven and crystal clear method.
Complex awareness is required from geometrical images of low-Euclidean, that is certainly, reported by true situation investigation and schooling deepness. Astonishingly, the outdoors has number of
forms serving as an overview of the thesis. Top of the sphere comprehends squarely the discovery, which is the the planet area. That could be if an individual could just walk direct over the the
planet floor, he will return to a similar kick off point. With certain curiosity, just one concludes that any pull paths cross excluding existence of parallel wrinkles (Peters, 1991, p.56). Plenty of
geometry is performed in mileage and data of facets along with triangles.
In fact, it is actually unfamiliar that nobody bothers together with the development of spherical geometry alternative to Euclid until such time as 180 in years past. Coherently, spherical geometry
is rarely no-Euclidean mainly because of the intersection of two lines on the factor is not really single. Re-product of projective geometry took place during the early 19th century delivering
accurate no-Euclidean statistical base on sphere geometry. Having said that, geometry is I the exact same apart from the alternative area currently being diagnosed; not failing to remember sole
factors intersecting is proven.
Creative setbacks are documented since it is not focused. The reader should be far more that watchful when using the time period elliptic and spherical. The primary reason for carefulness is
definitely the two is obviously use interchangeably. With respect to hyperbolic ground, mother nature herself presents plenty of spheres for that edification in issue.
In the previous century mathematics and systems presents cases about how no-Euclidean geometry picture in two specifications. You need to aim to support man creative thinking (Gunn, 1993, p.23).
Because of the truth that, huge geodesic triangle used to appraise if ever the aspects when amount of money jointly allows 180 degrees, sensing international no-Euclidean is everybody’s efforts. By
way of example, among the scholars operates named the Cayley-Klein scenario derivation of hyperbolic airplanes beginning with the projective airplane. With homogenous meets (p,q,r). Picking out
quadratic shape, as a result, By-=p2 q2 (-r2). The complete conic is X-=. From the circumstance de homogenizing is completed, p2 q2=1 would be the machine group. Consequently, it is easy to create
the distance performance in relation to picture form By- and in addition invariant can be found. Hyperbolic geometry version is going to be provided as being the product within the extended distance
perform. In this projective model, absolute conic is never picked up.
There is a lot most of non-Euclidean geometry design, all planning to hand out the same perspectives even on all those on three measurement areas. But additionally, the models have convenience and
demerits, i.e. hard drive version by Poincare, combined edges supplying actual sides, gets the merit that it only takes a smaller amount Euclidean region to provide precisely the same geometry when
compared to projective type, in this quite a bit is observable at the same time. As you nearby the group of friends at infinity, the outcome is sensed a lot obvious. Euclidean lines are shown by
projective design.
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Class 8 Maths Chapter 3 Understanding Quadrilaterals MCQ - Sanfoundry
Class 8 Maths MCQ – Understanding Quadrilaterals
This set of Class 8 Maths Chapter 3 Multiple Choice Questions & Answers (MCQs) focuses on “Understanding Quadrilaterals”. These MCQs are created based on the latest CBSE syllabus and the NCERT
curriculum, offering valuable assistance for exam preparation.
1. What is the general formula to count the sum of all the interior angles of a polygon with ‘n’ number of sides?
a) \((n-2)*180°\)
b) \((n-2)*360°\)
c) \((n-1)*180°\)
d) \((n-1)*360°\)
View Answer
Answer: a
Explanation: The general formula for the sum total of all the interior angles of any polygon is (n-2)*180°. For example: As we know that the sum of all the interior angles of any triangle is 180°.
So, now we can recheck it with the above formula. Sum of interior angles=(n-2)180°
Any triangles have three sides so n=3.
∴ Sum of interior angles=(3-2)180°
∴ Sum of interior angle=180°.
2. What is the sum of all the angles in any triangle?
a) 90°
b) 180°
c) 270°
d) 240°
View Answer
Answer: b
Explanation: : As we know that the sum of all the interior angles of any triangle is 180°. Sum of interior angles=(n-2)180°
Any triangles have three sides so n=3.
∴ Sum of interior angles=(3-2)180°
∴ Sum of interior angle=180°.
3. If the angles of a four sided polygon are in the ratio 7:8:9:12, then the angles would be?
a) 70, 80, 90, 100
b) 70, 70, 80, 80
c) 120, 70, 80, 100
d) 70, 80, 90, 120
View Answer
Answer: d
Explanation: Let us assume x to be the common multiple.
Now, the sides of the four sided polygon would be 7x, 8x, 9x and 12x.
The sum total of any four sided polygon is 360°.
Therefore 7x+8x+9x+12x=360
Therefore 36x=360
Therefore x=10°
Therefore the angles are, 7x=70, 8x=80, 9x=90 and 12x=120.
One can recheck their answer by, verifying the total as One can recheck their answer by, verifying the total as 360°.
4. In a triangle the sides are 4 cm, 5 cm and 4 cm. One of the base angle is 80°. Find the measure of the apex angel.
a) 30°
b) 40°
c) 20°
d) 10°
View Answer
Answer: c
Explanation: One can get the hint that the given triangle is isosceles triangle by observing the length of all the sides. As in an isosceles triangle two sides are of equal length and also the base
angles are equal in measure. Let the apex angle for denoted by the variable x.
Therefore 80+80+x=180
Therefore 160+x=180
Therefore x=20.
5. All sides of a four-sided polygon are equal in length and its diagonals bisect each other at 90°. what would be the angle between two adjacent sides?
a) Exactly 90°
b) 90° or any acute angle
c) 90° or any obtuse angle
d) Can be any angle
View Answer
Answer: a
Explanation: When a four sided polygon has all its sides equal in length, and the diagonals bisect each other, it is a square. Any square has it’s angles =90°. Hence the answer would be 90°. Here the
other options won’t be accurate. For example if a student selects the option 90° or any acute angle it would be partially correct because in a square the angles are always right angles.
6. In a quadrilateral the angles are 40°,120°,10° and x. Find x.
a) 40°
b) 120°
c) 190°
d) 180°
View Answer
Answer: c
Explanation: A quadrilateral is a four sided polygon. Therefore the sum of all the interior angles is 360°.
Therefore 40+120+10+x=360
Therefore 170+x=360
Therefore x=190.
7. What angle is subtended by a semicircle?
a) 360°
b) 300°
c) 180°
d) 90°
View Answer
Answer: c
Explanation: A circle subtends a whole of 360°, this is the measure of one complete rotation over the circumference of the circle. Now, the semicircle would subtend an angle of 180°.
8. What would be the measure of all the angles in a scalene triangle?
a) 60°, 60° and 60°
b) 120°, 30° and 30°
c) 90°, 45° and 45°
d) 60°, 30° and 90°
View Answer
Answer: d
Explanation: In a scalene triangle the measure of all the angles and the sides is different. The options other then 60°, 30° and 90° are either equilateral triangles or isosceles triangles.
9. Any right angled triangle must have how may angles in total?
a) Two angles
b) Four angles
c) One angle
d) Three angles
View Answer
Answer: d
Explanation: In any triangle irrespective of any type of triangle it has three angles in total. Here if the question would ask the ‘how many angles other than right angle’ then the answer would be
two angles.
10. The formula (n-2)180° is the formula for the sum of exterior angle of ‘n’ sided polygons.
a) True
b) False
View Answer
Answer: b
Explanation: The formula (n-2)180° is the formula for the interior angles of the polygon.
As we know the sum angle property of the square is 360°.
Therefore (n-2)180°
Therefore (4-2)180°
Therefore 360°.
More MCQs on Class 8 Maths Chapter 3:
To practice all chapters and topics of class 8 Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.
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Book Operator Algebras Theory Of C Algebras And Von Neumann Algebras Volume 122 2006
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The VaR of a portfolio at the 99% confidence level is $250,000 when mean return is assumed to be zero. If the assumption of zero returns is changed to an assumption of returns of $10,000, what is the
revised VaR?
A financial institution is considering shedding a business unit to reduce its economic capital requirements. Which of the following is an appropriate measure of theresulting reduction in capital
The CDS rate on a defaultable bond is approximated by which of the following expressions:
Which of the following are true:
I. The total of the component VaRs for all components of a portfolio equals the portfolio VaR.
II. The total of the incremental VaRs for each position in a portfolio equals the portfolio VaR.
III. Marginal VaR and incremental VaR are identical for a $1 change in the portfolio.
IV. The VaR for individual components of a portfolio is sub-additive, ie the portfolio VaR is less than (or in extreme cases equal to) the sum of the individual VaRs.
V. The component VaR for individual components of a portfolio is sub-additive, ie the portfolio VaR is less than the sum of the individual component VaRs.
The probability of default of a security over a 1 year period is 3%. What is the probability that it would have defaulted within 6 months?
If the default hazard rate for a company is 10%, and the spread on its bondsover the risk free rate is 800 bps, what is the expected recovery rate?
Which of the following can be used to reduce credit exposures to a counterparty:
I. Netting arrangements
II. Collateral requirements
III. Offsetting tradeswith other counterparties
IV. Credit default swaps
Which of the following statements are correct?
I. A reliance upon conditional probabilities and a-priori views of probabilities is called the 'frequentist' view
II. Knightian uncertainty refers to thingsthat might happen but for which probabilities cannot be evaluated
III. Risk mitigation and risk elimination are approaches to reacting to identified risks
IV. Confidence accounting is a reference to the accounting frauds that were seen in the past decadeas a reflection of failed governance processes
According to Basel II's definition of operational loss event types, losses due to acts by third parties intended to defraud, misappropriate property or circumvent the law are classified as:
Under the ISDA MA, which of the following terms best describes the netting applied upon the bankruptcy of a party?
A corporate bond maturing in 1 year yields 8.5% per year,while a similar treasury bond yields 4%. What is the probability of default for the corporate bond assuming the recovery rate is zero?
In respect of operational risk capital calculations, the Basel II accord recommends a confidence leveland time horizon of:
If P be the transition matrix for 1 year, how can we find the transition matrix for 4 months?
The Altman credit risk score considers:
The standalone economic capital estimates for the three business units of a bank are $100, $200 and $150 respectively. What is the combined economic capital for the bank, assuming the risks of the
three business units are perfectly correlated?
Which of the following statements are true:
I. The sum of unexpected losses for individual loans in a portfolio is equal to the total unexpected loss for the portfolio.
II. The sum of unexpected losses for individual loans in a portfolio is less than the total unexpected loss for the portfolio.
III. The sum of unexpected losses forindividual loans in a portfolio is greater than the total unexpected loss for the portfolio.
IV. The unexpected loss for the portfolio is driven by the unexpected losses of the individual loans in the portfolio and the default correlation between these loans.
A stock that follows the Weiner process has its future price determined by:
Once the frequency and severity distributions for loss events have been determined, which of the following is an accurate description of the process to determine a full loss distribution for
operational risk?
Which of the following best describes the concept of marginalVaR of an asset in a portfolio:
The CDS quote for the bonds of Bank X is 200 bps. Assuming a recovery rate of 40%, calculate the default hazard rate priced in the CDS quote.
A corporate bond has a cumulative probability of default equal to 20% in the first year, and 45% in the second year. What is the monthly marginal probability of default for the bond in the second
year, conditional on there beingno default in the first year?
A bank extends a loan of $1m to a home buyer to buy a house currently worth $1.5m, with the house serving as the collateral. The volatility of returns (assumed normally distributed) on house prices
in that neighborhood is assessed at 10% annually. The expected probability of default of the home buyer is 5%.
What is the probability that the bank will recover less than the principal advanced on this loan; assuming the probability of the home buyer's default is independent of the value of the house?
Which of the following objectives are targeted by rating agencies when assigning ratings:
I. Ratings accuracy
II. Ratings stability
III. High accuracy ratio (AR)
IV. Ranked ratings
Which of the following is not a measure of risk sensitivity of some kind?
In estimating credit exposure for a line of credit, it is usual to consider:
Which of the following is not one of the 'three pillars' specified in the Basel accord:
Which of the following statements is true in respect of a non financial manufacturing firm?
I. Market risk is not relevant to the manufacturing firm as it does not take proprietary positions
II. The firm faces market risks as an externality which it must bear and has no control over
III. Market risks can make a comparative assessment of profitability over time difficult
IV. Market risks for a manufacturing firm are not directionally biased and do not increase the overall risk of the firm as they net to zero over a long term time horizon
Which of the following is true in relation to the application of Extreme Value Theory when applied to operational risk measurement?
I. EVT focuses on extreme losses that are generally not covered by standard distribution assumptions
II. EVT considers the distribution of losses in the tails
III. The Peaks-over-thresholds (POT) and the generalized Pareto distributions are used to model extreme value distributions
IV. EVT is concerned with average losses beyond a given level of confidence
Which of the following formulae describes CVA (Credit Valuation Adjustment)? All acronyms have their usual meanings (LGD=Loss Given Default, ENE=Expected Negative Exposure, EE=Expected Exposure, PD=
Probability of Default, EPE=Expected Positive Exposure, PFE=Potential Future Exposure)
Under the KMV Moody's approach to credit risk measurement, which of the following expressions describes the expected 'default point' value of assets at which the firm may be expected to default?
Which of the following credit risk models considers debt as including a put option on the firm's assets toassess credit risk?
Under thebasic indicator approach to determining operational risk capital, operational risk capital is equal to:
For a corporate bond, which of the following statements is true:
I. The credit spread is equal to the default rate times the recovery rate
II. The spread widens when the ratings of the corporate experience an upgrade
III. Both recovery rates and probabilities of default are related to the business cycle and move in oppositedirections to each other
IV. Corporate bond spreads are affected by both the risk of default and the liquidity of the particular issue
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Performance bounds for random walks in the positive orthant
We consider methods to establish upper and lower bounds on stationary performance measures of a random walk in the positive orthant. We develop an approximation framework that is based on the Markov
reward approach to error bounds, in which we use the stationary performance measure of a perturbed random walk to obtain the upper and lower bounds. The perturbed random walk is constructed such that
its stationary performance measure is known explicitly.
This thesis has two parts. In the first part we consider the numerical implementation of the approximation framework. Given an original random walk and a perturbed random walk whose stationary
probability distribution is known explicitly, we formulate linear programs that return the upper and lower bounds. Implementations of these linear programs are provided that can be used to obtain
numerical bounds for a large class of multi-dimensional random walks. These linear programs are not always feasible. We establish sufficient conditions under which the linear programs are feasible,
i.e.\ under which a bound is provided.
In the second part of this thesis we introduce various classes of random walks that can be used as the perturbed random walk for two-dimensional models. First, we obtain a perturbed random walk with
state-dependent transition rates on the horizontal and the vertical axis. Secondly, we construct a perturbed random walk of which only the transition rates in the tail are different from those of the
original random walk. In both cases, we give explicit expressions for the error bounds. Through numerical results, we see that these perturbation schemes can provide tighter upper and lower bounds
than existing schemes.
These perturbation schemes are considered as an intermediate step towards developing perturbation frameworks for higher-dimensional models. The long-term goal is to provide insights into behavior of,
for instance, large queueing networks, which can be modeled as random walks.
Original language English
Qualification Doctor of Philosophy
Awarding Institution • University of Twente
• Boucherie, Richard J., Supervisor
Supervisors/Advisors • Goseling, Jasper, Supervisor
Award date 20 Sept 2018
Place of Publication Enschede
Publisher • University of Twente
Print ISBNs 978-94-9301-443-5
Publication status Published - 20 Sept 2018
Dive into the research topics of 'Performance bounds for random walks in the positive orthant'. Together they form a unique fingerprint.
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Can I use custom Constant Values in SAGE?
Can I use custom Constant Values in SAGE?
show(pi) will show a nice representation of pi, while pi.n() will yield a useable value
how am I supposed to achieve the same functionality for a custom 'symbol' ?
Az = 5.5 # cm the surface area of my Cylinder
but showing 'Az' as representation, not the value
I solved this with a custom Class, but this feels wrong:
class Myconst(Constant):
def __init__(self, name='Az', value=1.0):
Constant.__init__(self, name,
latex=None, domain='positive')
self.v = value
def __float__(self):
return self.v
def _real_double_(self, R):
return self.v
def _mpfr_(self,R):
return self.v
Az = Myconst(name='Az', value = 2.0).expression()
#As = Myconst(name='As', value = 5.0).expression()
now these work like I want them to :) Az.n() show(1/Az)
But is this the intended approach?
If it is a short life constant, than the above is ok.
Else, one has to integrate the constant with the sage code, this does not seem to be easy:
sage: pi.parent()
Symbolic Ring
sage: pi?
Type: Expression
String form: pi
Length: 0
File: /usr/lib/python2.7/site-packages/sage/symbolic/expression.pyx
Nearly all expressions are created by calling
new_Expression_from_*, but we need to make sure this at least does
not leave self._gobj uninitialized and segfault.
and so on.
pi?? gives more information. (Or just open the py-file...)
Note: To have in-line code displayed as code, use the ticks to markdown. (Or mark it in editor and press either Control+K or that button with 101 and 010.)
1 Answer
Sort by ยป oldest newest most voted
Yes, you can compare your code with the one of Sage source code : https://git.sagemath.org/sage.git/tre...
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Research Guides: SAS: Statistical Software: Books
A Handbook of Statistical Analyses Using SAS by Geoff Der; Brian Everitt
Call Number: QA276.4 .D47 2002
ISBN: 158488245X
Publication Date: 2002
Powerful software often comes, unfortunately, with an overwhelming amount of documentation. As a leading statistics software package, SAS is no exception. Its manuals comprise well over 10,000 pages
and can intimidate, or at least bewilder, all but the most experienced users. A Handbook of Statistical Analyses using SAS, Second Edition comes to the rescue. Fully revised to reflect SAS Version
8.1, it gives a concise, straightforward description of how to conduct a range of statistical analyses. The authors have updated and expanded every chapter in this new edition, and have incorporated
a significant amount of new material. The book now contains more graphical material, more and better data sets within each chapter, more exercises, and more statistical background for each method.
Completely new topics include the following: Data description and simple inference for categorical variables Generalized linear models Longitudinal data: Two new chapters discuss simple approaches,
graphs, summary measure, and random effect models Researcher or student, new user or veteran, you will welcome this self-contained guide to the latest version of SAS. With its clear examples and
numerous exercises, A Handbook of Statistical Analyses using SAS, Second Edition is not only a valuable reference, but also forms the basis for introductory courses on either SAS or applied
statistics at any level, from undergraduate to professional.
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Two-stage Stop Loss Strategy
1. Two-stage Stop Loss Strategy
Two-stage Stop Loss Strategy
, Date: 2023-10-25 18:11:30
The main idea of this strategy is to set two take profit targets and move the stop loss to entry price after the first target is reached to avoid stop loss hunting.
Strategy Logic
This strategy enters trades based on Bollinger Bands and Stochastic indicators. It goes short when price exceeds the Bollinger upper band and goes long when Stochastic shows oversold.
Specifically, the entry logic is:
1. Enter long when close is below Bollinger lower band and Stochastic K crosses below D.
2. Enter short when close is above Bollinger upper band and Stochastic K crosses above D.
The strategy sets two take profit targets, TP1 fixed at 200 points and TP2 fixed at 500 points.
When price moves and TP1 is triggered, the strategy will move stop loss to entry price. This locks in profit from first stage and prevents stop loss hunting.
The strategy closes all positions when TP2 or stop loss is triggered.
Advantage Analysis
The biggest advantage of this two-stage stop loss approach is it allows locking in profits while preventing stop loss hunting. By moving stop loss to entry price, it reduces the chance of stop loss
hunting and protects profits.
Another advantage is the combination of Bollinger Bands to gauge volatility range and Stochastic for overbought/oversold makes for more accurate entries.
Risk Analysis
Main risks stem from potential false signals from Bollinger Bands and Stochastic indicators. Incorrect Bollinger range can lead to missing entries or bad signals. Stochastic false breakouts also
cause wrong entries.
There is also risk of stop loss being hunted again after moving to entry price. V-shaped reversals can trigger stop loss a second time.
These risks can be reduced by optimizing parameters for both indicators and increasing distance between stop losses.
Optimization Directions
Further optimizations for this strategy:
1. Test different parameter combinations to find optimal Bollinger and Stochastic parameters.
2. Test different profit/loss targets to find ideal configurations.
3. Add other indicators like moving averages to create multi-indicator systems for higher accuracy.
4. Research alternate stop loss positioning logic, like fixed distance from entry instead of entry price itself.
5. Increase stop loss movement occurrences to 3 or more stages.
This strategy uses Bollinger Bands and Stochastic for entries, sets two take profit targets, and moves stop loss to entry after first target reached to form a two-stage stop loss. This effectively
locks in profits and prevents stop loss hunting. Strategy has clear advantages but also room for improvements via parameter optimization, multi-indicator systems, and stop loss logic adjustments.
start: 2022-10-18 00:00:00
end: 2023-10-24 00:00:00
period: 1d
basePeriod: 1h
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
// This source code is subject to the terms of the Mozilla Public License 2.0 at https://mozilla.org/MPL/2.0/
// © fpsd4ve
// Add Bollinger Bands indicator (close, 20, 2) manually to visualise trading conditions
strategy("2xTP, SL to entry",
// PARAMETERS
// Assumes quote currency is FIAT as with BTC/USDT pair
tp1=input.float(200, title="Take Profit 1")
tp2=input.float(500, title="Take Profit 2")
sl=input.float(200, title="Stop Loss")
stOBOS = input.bool(true, title="Use Stochastic overbought/oversold threshold")
// Colors
colorRed = #FF2052
colorGreen = #66FF00
// FUNCTIONS
// Stochastic
f_stochastic() =>
stoch = ta.stoch(close, high, low, 14)
stoch_K = ta.sma(stoch, 3)
stoch_D = ta.sma(stoch_K, 3)
stRD = ta.crossunder(stoch_K, stoch_D)
stGD = ta.crossover(stoch_K, stoch_D)
[stoch_K, stoch_D, stRD, stGD]
// VARIABLES
[bbMiddle, bbUpper, bbLower] = ta.bb(close, 20, 2)
[stoch_K, stoch_D, stRD, stGD] = f_stochastic()
// ORDERS
// Active Orders
// Check if strategy has open positions
inLong = strategy.position_size > 0
inShort = strategy.position_size < 0
// Check if strategy reduced position size in last bar
longClose = strategy.position_size < strategy.position_size[1]
shortClose = strategy.position_size > strategy.position_size[1]
// Entry Conditions
// Enter long when during last candle these conditions are true:
// Candle high is greater than upper Bollinger Band
// Stochastic K line crosses under D line and is oversold
longCondition = stOBOS ?
low[1] < bbLower[1] and stGD[1] and stoch_K[1] < 25 :
low[1] < bbLower[1] and stGD[1]
// Enter short when during last candle these conditions are true:
// Candle low is lower than lower Bollinger Band
// Stochastic K line crosses over D line and is overbought
shortCondition = stOBOS ?
high[1] > bbUpper[1] and stRD[1] and stoch_K[1] > 75 :
high[1] > bbUpper[1] and stRD[1]
// Exit Conditions
// Calculate Take Profit
longTP1 = strategy.position_avg_price + tp1
longTP2 = strategy.position_avg_price + tp2
shortTP1 = strategy.position_avg_price - tp1
shortTP2 = strategy.position_avg_price - tp2
// Calculate Stop Loss
// Initialise variables
var float longSL = 0.0
var float shortSL = 0.0
// When not in position, set stop loss using close price which is the price used during backtesting
// When in a position, check to see if the position was reduced on the last bar
// If it was, set stop loss to position entry price. Otherwise, maintain last stop loss value
longSL := if inLong and ta.barssince(longClose) < ta.barssince(longCondition)
else if inLong
close - sl
shortSL := if inShort and ta.barssince(shortClose) < ta.barssince(shortCondition)
else if inShort
close + sl
// Manage positions
strategy.entry("Long", strategy.long, when=longCondition)
strategy.exit("TP1/SL", from_entry="Long", qty_percent=50, limit=longTP1, stop=longSL)
strategy.exit("TP2/SL", from_entry="Long", limit=longTP2, stop=longSL)
strategy.entry("Short", strategy.short, when=shortCondition)
strategy.exit("TP1/SL", from_entry="Short", qty_percent=50, limit=shortTP1, stop=shortSL)
strategy.exit("TP2/SL", from_entry="Short", limit=shortTP2, stop=shortSL)
// DRAW
// Stochastic Chart
plot(stoch_K, color=color.blue)
plot(stoch_D, color=color.orange)
// Circles
plot(stOBOS ? stRD and stoch_K >= 75 ? stoch_D : na : stRD ? stoch_D : na, color=colorRed, style=plot.style_circles, linewidth=3)
plot(stOBOS ? stGD and stoch_K <= 25 ? stoch_D : na : stGD ? stoch_K : na, color=colorGreen, style=plot.style_circles, linewidth=3)
// Levels
hline(75, linestyle=hline.style_dotted)
hline(25, linestyle=hline.style_dotted)
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CBSE Class 10 Maths : Chapter 1 - Real Numbers Notes - SchoolMyKids
In the vast realm of mathematics, numbers play a fundamental role. Real numbers form the foundation upon which much of mathematics is built. But what exactly are real numbers, and what makes them
special? This article delves into the world of real numbers, exploring their properties, types, and the essential formulas used to manipulate them.
Understanding Real Numbers
Real numbers encompass all numbers that can be used to represent points on a continuous number line. They include:
• Natural Numbers: Positive whole numbers starting from 1 (1, 2, 3, 4, …).
• Whole Numbers: Natural numbers along with zero (0, 1, 2, 3, 4, …).
• Integers: Whole numbers and their negative counterparts (… -3, -2, -1, 0, 1, 2, 3, …).
• Rational Numbers: Numbers that can be expressed as a fraction (p/q), where p and q are integers and q is not equal to zero (e.g., 1/2, -3/4, 5/7).
• Irrational Numbers: Numbers that cannot be expressed as a finite or repeating fraction (e.g., √2, π).
Real numbers are distinct from imaginary numbers, which are used to represent the square root of a negative number and are not part of the real number system.
Essential Formulas for Real Numbers
Here, we explore some key formulas used to perform operations on real numbers, along with examples to illustrate their application.
• HCF (Highest Common Factor): The largest number that is a factor of two or more given numbers.
□ Example: Find the HCF of 12 and 18.
☆ Explanation: List the factors of each number: 12 (1, 2, 3, 4, 6, 12) and 18 (1, 2, 3, 6, 9, 18). The HCF is 6.
• LCM (Least Common Multiple): The smallest number that is a multiple of two or more given numbers.
□ Example: Find the LCM of 8 and 12.
☆ Explanation: List the multiples of each number: 8 (8, 16, 24, …) and 12 (12, 24, 36, …). The LCM is 24.
• Factorization: Breaking down a number or polynomial into its constituent parts (prime factors for numbers, linear or quadratic expressions for polynomials).
1. Addition of Real Numbers
• Formula: a + b = c (where a, b, and c are real numbers)
• Explanation: Addition involves combining two real numbers to get a new real number.
• Examples:
□ 5 + 3 = 8 (Adding two integers)
□ -2 + 1.5 = -0.5 (Adding an integer and a decimal)
□ √2 + π (irrational numbers can also be added)
2. Subtraction of Real Numbers
• Formula: a – b = c (where a, b, and c are real numbers)
• Explanation: Subtraction involves finding the difference between two real numbers.
• Examples:
□ 7 – 2 = 5 (Subtracting two integers)
□ 4.5 – 1.25 = 3.25 (Subtracting decimals)
□ π – √3 (irrational numbers can also be subtracted)
3. Multiplication of Real Numbers
• Formula: a x b = c (where a, b, and c are real numbers)
• Explanation: Multiplication involves finding the product of two real numbers.
• Examples:
□ 3 x 4 = 12 (Multiplying two integers)
□ -2 x 0.5 = -1 (Multiplying an integer and a decimal)
□ π x √2 (irrational numbers can also be multiplied)
4. Division of Real Numbers
• Formula: a / b = c (where a and c are real numbers, and b ≠ 0)
• Explanation: Division involves finding the quotient of two real numbers. It’s important to note that division by zero is undefined.
• Examples:
□ 10 / 2 = 5 (Dividing two integers)
□ 6 / 1.5 = 4 (Dividing an integer by a decimal)
□ π / √3 (irrational numbers can also be divided, as long as the divisor is not zero)
5. Order of Operations (PEMDAS)
• Formula: PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right))
• Explanation: PEMDAS is a mnemonic used to remember the correct order of operations when evaluating expressions involving multiple operations.
• Examples:
□ 2 + 3 x 4 = 14 (Multiplication is done before addition)
□ (5 + 2) x 3 = 21 (Parentheses are evaluated first)
□ 8 / (2 + 1) = 2 (Division is done before subtraction within the parentheses)
Remember: These are just a few of the many formulas used in working with real numbers. As you delve deeper into mathematics, you’ll encounter more complex formulas and operations.
Exploring the Applications of Real Numbers
Real numbers permeate every aspect of our lives. They are used in:
• Science and Engineering: From calculating distances in astronomy to designing bridges, real numbers are the language of science and engineering. They are used in formulas for motion, gravity,
electricity, and countless other scientific principles.
• Finance and Economics: Real numbers are essential for financial calculations like interest rates, budgeting, and analyzing market trends. They are the backbone of economic models and investment
• Daily Life: From measuring ingredients in a recipe to calculating travel time, real numbers are used in our everyday activities. We rely on them for temperature measurements, converting units,
and understanding proportions.
Beyond the Formulas: Properties of Real Numbers
Real numbers exhibit specific properties that govern their behavior under various operations:
• Closure: Performing operations (addition, subtraction, multiplication, and division) on real numbers always results in another real number (except for division by zero).
• Commutativity: The order in which we add or multiply real numbers doesn’t affect the result (a + b = b + a and a x b = b x a).
• Associativity: Grouping real numbers for addition or multiplication doesn’t change the result ((a + b) + c = a + (b + c) and (a x b) x c = a x (b x c)).
• Distributive Property: Multiplication distributes over addition (a x (b + c) = a x b + a x c).
• Identity Property: There exist identity elements (0 for addition and 1 for multiplication) that leave a real number unchanged when added or multiplied (a + 0 = a and a x 1 = a).
• Inverse Property: Every real number (except zero for division) has an inverse for addition (a + (-a) = 0) and multiplication (a x (1/a) = 1, where a ≠ 0).
Understanding these properties is crucial for manipulating real numbers effectively and solving mathematical problems.
The Encompassing Nature of Real Numbers
Real numbers form the foundation of mathematics, providing a framework for representing continuous quantities. From the basic counting numbers to the intricate world of irrational numbers, real
numbers equip us with the tools to quantify, analyze, and solve problems across diverse fields. As you embark on your mathematical journey, remember that real numbers are not just abstract concepts –
they are the language of the universe, waiting to be explored and understood.
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Basic variables - (Nonlinear Optimization) - Vocab, Definition, Explanations | Fiveable
Basic variables
from class:
Nonlinear Optimization
Basic variables are the set of variables in a linear programming problem that correspond to the basic feasible solution. In the context of equality constrained optimization, these variables are
essential for expressing the solution to the optimization problem in terms of the constraints and the objective function. They play a crucial role in determining the feasible region and ultimately
influence the optimal solution by defining which constraints are active.
congrats on reading the definition of basic variables. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. In a linear programming problem with equality constraints, basic variables can be identified from the matrix representation of the system, where they correspond to pivot columns.
2. The number of basic variables in a solution must equal the number of constraints in the system for the solution to be considered basic feasible.
3. Basic variables can take on non-zero values while non-basic variables are typically set to zero in basic feasible solutions.
4. Changing a basic variable can lead to a new vertex of the feasible region, which may yield a different optimal solution.
5. In simplex methods for solving linear programming problems, identifying and updating basic variables is a key step in moving towards optimality.
Review Questions
• How do basic variables contribute to identifying feasible solutions in equality constrained optimization?
□ Basic variables are critical in identifying feasible solutions because they determine which constraints are active and directly influence the shape of the feasible region. When solving
equality constrained optimization problems, a basic feasible solution is formed when the values of these variables satisfy all given constraints. This relationship allows us to pinpoint
specific solutions that are permissible under the defined constraints.
• Discuss how the selection of basic versus non-basic variables affects the outcome of an optimization problem.
□ The selection between basic and non-basic variables significantly impacts the outcome of an optimization problem. Basic variables, which are allowed to take on non-zero values, are directly
involved in forming solutions that satisfy all constraints. In contrast, non-basic variables are typically set to zero and do not contribute to the immediate solution. The interplay between
these variable sets can affect whether an optimal solution is reached or if further iterations are needed in methods like simplex.
• Evaluate the role of basic variables in transforming an initial feasible solution into an optimal one through iterative methods.
□ Basic variables play a pivotal role in transforming an initial feasible solution into an optimal one by allowing for systematic adjustments during iterative methods like simplex. As
iterations progress, changing the values of basic variables can lead to exploring new vertices of the feasible region, each potentially yielding better objective function values. This process
continues until no further improvements can be made, indicating that an optimal solution has been reached based on the defined constraints and objective function.
"Basic variables" also found in:
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Spatial dependence of correlation functions in the decay problem for a passive scalar in a large-scale velocity field
Statistical characteristics of a passive scalar advected by a turbulent velocity field are considered in the decay problem with a low scalar diffusivity κ (large Prandtl number v/κ, where v is
kinematic viscosity). A regime in which the scalar correlation length remains smaller than the velocity correlation length is analyzed. The equal-time correlation functions of the scalar field are
found to vary according to power laws and have angular singularities reflecting locally layered distribution of the scalar in space.
Soviet Journal of Experimental and Theoretical Physics
Pub Date:
April 2006
□ 05.20.Jj;
□ 47.27.Gs;
□ 47.27.-i
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Logistic Regression Mastering
Logistic Regression Mastering
The process of training an ML model involves providing an ML algorithm (that is, the learning algorithm) with training data to learn from. The term ML model refers to the model artifact that is
created by the training process.
The training data must contain the correct answer, which is known as a target or target attribute. The learning algorithm finds patterns in the training data that map the input data attributes to the
target (the answer that you want to predict), and it outputs an ML model that captures these patterns. We are going to investigate the accuracy of our model in the next section, just focus on
training the model for now.
We are going to train a Logistic regression model. Formally, in binary logistic regression, there is a single binary dependent variable, coded by an indicator variable, where the two values are
labeled "0" and "1", while the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value).
Methods description
• sklearn.linear_model: This module from scikit-learn provides various linear models for classification and regression tasks;
• LogisticRegression: This is a class within the sklearn.linear_model module used for logistic regression, a statistical method for analyzing datasets where there are one or more independent
variables that determine an outcome. It's commonly used for binary classification problems;
□ .random_state: This parameter sets the random seed for reproducibility;
□ .max_iter: This parameter specifies the maximum number of iterations for the solver to converge;
• .fit(X_train, y_train): This method trains the logistic regression model using the training data X_train and corresponding labels y_train. It adjusts the parameters of the model to minimize the
loss function and fit the data as well as possible;
• .predict(X_test): This method predicts the labels for the input data X_test using the trained logistic regression model. It returns the predicted labels based on the learned parameters from the
training data.
1. Import LogisticRegression from sklearn.
2. Use the method just imported to initialize the classifier.
3. Call .fit() and pass X_train and y_train as parameters.
4. Predict on X_test.
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Thanks for your feedback!
The process of training an ML model involves providing an ML algorithm (that is, the learning algorithm) with training data to learn from. The term ML model refers to the model artifact that is
created by the training process.
The training data must contain the correct answer, which is known as a target or target attribute. The learning algorithm finds patterns in the training data that map the input data attributes to the
target (the answer that you want to predict), and it outputs an ML model that captures these patterns. We are going to investigate the accuracy of our model in the next section, just focus on
training the model for now.
We are going to train a Logistic regression model. Formally, in binary logistic regression, there is a single binary dependent variable, coded by an indicator variable, where the two values are
labeled "0" and "1", while the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value).
Methods description
• sklearn.linear_model: This module from scikit-learn provides various linear models for classification and regression tasks;
• LogisticRegression: This is a class within the sklearn.linear_model module used for logistic regression, a statistical method for analyzing datasets where there are one or more independent
variables that determine an outcome. It's commonly used for binary classification problems;
□ .random_state: This parameter sets the random seed for reproducibility;
□ .max_iter: This parameter specifies the maximum number of iterations for the solver to converge;
• .fit(X_train, y_train): This method trains the logistic regression model using the training data X_train and corresponding labels y_train. It adjusts the parameters of the model to minimize the
loss function and fit the data as well as possible;
• .predict(X_test): This method predicts the labels for the input data X_test using the trained logistic regression model. It returns the predicted labels based on the learned parameters from the
training data.
1. Import LogisticRegression from sklearn.
2. Use the method just imported to initialize the classifier.
3. Call .fit() and pass X_train and y_train as parameters.
4. Predict on X_test.
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Matthewfl, Author at Matthew Francis-Landau
I did a quick project to scratch a particular itch. Maybe it will be helpful to others as well.
Encrypted Messages for the Event of Death at https://in-event-of-death.github.io/v1/
When it comes to passing along digital accounts after death, existing services operate as a dead man switch, sending an email after some amount of time has passed. This requires that you trust your
chosen service not to close down, that the dead man switch will not get accidentally triggered before you die, and to not get hacked causing your sensitive information to get leaked. To me, this
problem could be reasonably solved using cryptography. Using Shamir’s secret sharing, we can split a message into N parts such that as long as the number of people attempting to decrypt a message is
less than N, the message can not be decrypted. Furthermore, we can use asymmetric encryption so that multiple messages can be encrypted rather than requiring a single message to be predetermined that
will be sent along. Building on top of the PGP ecosystem means messages can be encrypted from the command line if one so chooses, and the cryptographic primitives will be secure.
However, this combination of Shamir’s secret Sharing and PGP does not make for a good user interface for non-technical users. As such, I created Encrypted Messages for the Event of Death as a
self-contained webpage that uses OpenPGP.js and Shamir secret sharing to expose the necessary operations to create encryption keys, encrypt messages, and decrypt messages. This means that by
providing loved ones with a link to this webpage as well as encrypted messages, they should be able to figure out how to decrypt an encrypted message, provided that they can copy and paste the
encrypted messages into the webpage.
Language Modeling in the Limit
I think that there are many people who are surprised that Large Language Models (LLMs) that predict the next word act intelligently and are capable of solving many challenging tasks. In my opinion,
the success of LLMs is not surprising. In this blog post, I will explain why I think LLM’s success was obvious.
So, around two years ago, during a discussion with some colleges, I coined the term “GPT-\(\infty\)” to represent language modeling in the limit of infinite power and infinite data.
let it be known that neural LM in the limit was coined gpt-? by @matthewfl
— Suzanna Sia (@suzyahyah) November 30, 2021
And I think that this “in the limit” thinking is helpful in understanding the success of modern Large Language Models. So first, let us discuss what language modeling is and what it would mean to
take language modeling to the extreme “in the limit.”
What is Language Modeling?
So, what is generative language modeling? Language modeling is a probabilistic model over a sequence of words, usually written as follows:
\(P(w_i | w_{i-1}, w_{i-2}, \ldots, w_0) \propto \exp(f(w_i, w_{i-1}, w_{i-2}, \ldots, w_0))\)
Where \(f(\cdots)\) is a function that returns a real number and is typically learned from data. The “\(\propto \exp(\cdots)\)” in this equation converts the real number returned from \(f(\cdots)\)
into a probability distribution over possible next tokens \(w_i\). A long sequence of words can be generated by repetitively evaluating \(P(w_i | w_{i-1}, w_{i-2}, \ldots, w_0)\) to determine what is
the most likely next word in the sequence conditioned on the previously generated words.
For example, suppose that we have the sequence “The moon is made of,” and we want to determine the next word. We will evaluate \(P(\text{rocks} | \text{The moon is made of})\), as well as \(P(\text
{cheese} | \text{The moon is made of})\). Both the words “rocks” and “cheese” are given some probability of being the next word. In the case of generation, we choose the word that has a high
probability, either greedily or according to some randomized process.
\(P(\text{rocks} | \text{The moon is made of}) > P(\text{cheese} | \text{The moon is made of})\)
In the case that we want to evaluate the probability of a longer sequence of words, we can multiply the \(P(w_i | \cdots)\) together, where each \(P(\cdots)\) is used to evaluate a single word. For
\(P(\text{the} | \emptyset) * P(\text{moon} | \text{the}) * P(\text{is} | \text{the moon}) * P(\text{made} | \text{the moon is}) *\)
\(P(\text{of} | \text{the moon is made}) * P(\text{rocks} | \text{the moon is made of})\)
models the probability of the phrase “the moon is made of rocks.”
Backing Off A Language Model
When building language models, researchers have to make many decisions to develop a model that is tractable. For example, there are several ways that the function \(f(\cdots)\) can be defined. These
days, \(f(\cdots)\) is usually a large neural network (a transformer), which is trained using gradient descent. However, neural nets are not the only way to define a language model. For example,
early n-gram language modes defined \(f(\cdots)\) as the ratio between counts of phases that appeared in a corpus. For example, a bi-gram model only conditions on the previous word and is defined as
\(f(w_i, w_{i-1}) = \frac{\text{count}(w_i, w_{i-1} )}{\text{count}(w_{i-1} ) + \epsilon}\)
We can observe that the bi-gram model is very backed off, as it only depends on the previous word. Hence, the probability distribution represented \(P(w_i | w_{i-1}, w_{i-2}, \ldots, w_0)\) is the
same as \(P(w_i | w_{i-1})\) under the bigram model. As a result, when people write the \(P(\cdot)\) equations using backed off language models in a paper, they will usually remove the \(w_{i-2}, \
ldots, w_0\) from the equation they are writing. For example:
\(P(\text{the} | \emptyset) * P(\text{moon} | \text{the}) * P(\text{is} | \text{moon}) * P(\text{made} | \text{is}) * P(\text{of} | \text{made}) * P(\text{rocks} | \text{of})\)
Writing the \(P(w_i | \cdots)\) this way indicates which parts the \(f(\cdots)\) function is able to use. Engineers will claim that this approximation is “good enough” for whichever task they are
attempting to solve. Note, just because we choose to ignore the \(w_{i-2}, w_{i-3}, \ldots, w_{0}\) does not mean it does not exist.
\(P(w_i | w_{i-1}, w_{i-2}, \ldots, w_0) \approx P(w_i | w_{i-1}) \propto \exp(f(w_i, w_{i-1}))\)
Language Modeling In The Limit
So, now that we have a basic understanding of language modeling, and backing off a language model, what does it mean to language model in the limit? Let us imagine a theoretical language model that
does not back off from anything. In other words, it conditions on absolutely everything.
\(P(w_i | \text{EVERYTHING BEFORE } w_i) \propto \exp(f(w_i, \text{EVERYTHING BEFORE } w_i))\)
Furthermore, in the limit, we will say that this language model has been trained with everything. This includes all data that existed in the past and all data that will exist in the future (hence
everything). Because the infinite language model has access to all data, this means it always accurately predicts the next word.
For example, the infinite language model conditions on who is speaking which changes the probability of the next word. In the case of the moon rocks example, if we are modeling a scientist vs a
5-year-old, then there are likely to be different answers
\begin{align*} P(\text{rocks} | \ldots\text{made of}, speaker=\text{5-year-old}) &< P(\text{cheese} | \ldots\text{made of}, speaker=\text{5-year-old}) \\ P(\text{rocks} | \ldots\text{made of},
speaker=\text{scientist}) &> P(\text{cheese} | \ldots\text{made of}, speaker=\text{scientist}) \end{align*}
As a more extreme example, suppose I prompt the infinite language model with “For breakfast, I ate,” and the language model completes the word “eggs.”
\begin{align*} “\text{eggs}” = \underset{w_i}{\text{argmax }} P\left( w_i \left| \begin{array}{c} \text{For breakfast I ate}, \\ speaker=\text{matthew}, \\ day=\text{feburary 20th}, \\ \vdots \end
{array} \right. \right) \end{align*}
Here, the language model knows that I (Matthew) am the person speaking. It also knows what the day is. It even has information about what I actually ate, and it knows that I will answer this
statement truthfully. Note that what I actually ate is not recorded anywhere. I did not write it down on a piece of paper or tweet about it online. In other words, the infinite language model has
access to all data, not just the data that is online.
It might be better to call the infinite language model an omnipotent model in that it has access to everything and even knows the next word with 100% accuracy. Hence, it is not really appropriate to
think of the “infinite language model” as a probabilistic or learned model.
Rather, the thing that we are interested in is that our “omnipotent and infinite” model is a function \(f_\infty(\cdots)\) that takes the entire world prior to \(w_i\) as an argument and returns a
real-valued number that selects the next word.
Using the “In The Limit” Language Model
So how do we make use of the \(f_\infty(\cdots)\) function?
Neural networks are universal function approximators. This means that for any function \(\hat{f}:\mathbb{R}^n \to \mathbb{R}\), that takes a real-valued vector (\(x \in \mathbb{R}^n\)) as an argument
and returns a real value (\(y \in \mathbb{R}\)) as the result, there exists a neural network \(f_{\text{neural}}:\mathbb{R}^n \to \mathbb{R}\), that can approximate the function \(\hat{f}\) to a
desired degree of accuracy.
To train the neural network \(f_{\text{neural}}:\mathbb{R}^n \to \mathbb{R}\), one simply needs to collect many inputs and outputs samples \(\langle x, y \rangle\) from the function \(\hat{f}\), and
then use those samples to train the \(f_{\text{neural}}\) function. This is exactly what we already do when training a neural network!
In other words, if we collect a lot of samples from the “omnipotent and infinite” language model \(f_\infty\) and use that to train \(f_{\text{neural}}\), then we can approximate this function.
Thankfully this is easy! All text that exists are samples from the \(f_\infty\) model.
For example, suppose that we prompt the \(f_\infty\) model with “reddit post written by bob123 on July 16, 2006”. The \(f_\infty\) model will exactly know the post by b0b123, and the \(f_\infty\)
model will generate “Religion and politics are always interesting.” This sentence can then be used as training data for the \(f_{\text{neural}}\) model.
\begin{align*} P_\infty\left(\text{“Religion and politics are always interesting.”} \left| \begin{array}{c} speaker=\text{bob123}, \\ source=\text{reddit}, \\ date=\text{July 16, 2006}, \\ \vdots \
end{array} \right. \right) = 1 \end{align*}
Hence, gathering data to train \(f_{\text{neural}}\) can be done by simply collecting all written text and training a neural language model in the usual way.
Furthermore, we will create better approximations to \(f_\infty\) by conditioning on more of the input to \(f_\infty\), meaning creating larger prompts for Large Language Models, and by making the
approximation better by creating larger neural networks! In other words, bigger equals better.
Is the “In The Limit” Model Intelligent?
Large language models seem to act intelligently. So a natural question is the \(f_\infty\) function, that neural language models approximate, also intelligent? Admittedly, this question is a bit
ill-formed. The \(f_\infty\) model is “omnipotent and infinite.” It does not need to be intelligent. It already knows everything.
For example, suppose that I have a time machine and go back in time to December 6, 1941, and predict that on December 7, 1941, Japanese planes will attack Pearl Harbor in Hawaii. From the perspective
of the people living in 1941, I would appear to be a very intelligent person as I have apparently analyzed a bunch of data and accurately predicted the future. However, knowing that I was a person
living in 2024 with knowledge of history, the only thing that I have done is recall a fact that I read in a history textbook.
What is Intelligence?
So if \(f_\infty\) is not intelligent, is training \(f_{\text{neural}}\) to approximate \(f_\infty\) going to enable us to build intelligent agents?
First, let us try to define what it means for a system to be intelligent. A plausible definition of intelligence could be having the ability to predict the future and then using those predictions to
exhibit behavior that is favorable to the agent. I am using the term “favorable behavior” loosely here in that the behavior does not have to entirely come from a “conscious” decision by the agent.
For example, suppose that the agent is a hunter and is trying to catch an animal that it wants to eat. If the agent can accurately predict where the animal is going to be, then the agent is going to
have a better chance of catching the animal. The favorable behavior in this case is getting resources (food) that are needed to survive. This behavior is driven by some innate instinct to get food
and survive.
A more modern version of predicting the future these days might instead look like trying to predict the price of a stock. For example, if an agent can accurately predict the price of Nvidia stock,
then it can place trades (bets against other traders) that will be profitable. In the case of the \(f_\infty\) model, the model is omnipotent, so it knows the answer. Hence, it would win every bet.
However, if we are not an omnipotent agent and are limited by the constraints of time, then the agent will have to read through corporate reports and compare trends with historical data to make an
“educated guess” about the future stock price. This is conceptually what a human financial analyst does. Here, we say that the agent is acting intelligently because the agent does not have direct
access to the future (like \(f_\infty\)) and instead must use its existing sources of knowledge to make an “educated guess.” Hence, the agent must represent their guesses using a probability
distribution of potential outcomes:
\(P_{\text{neural}}(\text{nvidia stock closed at \$900 at the end of 2024} | \cdots) = .001 \)
\(\vdots \)
\(P_{\text{neural}}(\text{nvidia stock closed at \$800 at the end of 2024} | \cdots) = .001 \)
\(\vdots \)
\(P_{\text{neural}}(\text{nvidia stock closed at \$600 at the end of 2024} | \cdots) = .001\)
This is in contrast to \(P_\infty(\cdots)\) that knows the correct answer and creates a “distribution” that is entirely peaked at the correct answer:
\(P_{\infty}(\text{nvidia stock closed at \$900 at the end of 2024} | \cdots) = 0\)
\(\vdots \)
\(P_{\infty}(\text{nvidia stock closed at \$875.23 at the end of 2024} | \cdots) = 1\)
\(P_{\infty}(\text{nvidia stock closed at \$700 at the end of 2024} | \cdots) = 0\)
So, in conclusion, Large Language Models are being trained to approximate the “in the limit” \(f_\infty\) function by being trained on the collection of all text. As we build larger LLMs and use more
data, we will train neural nets to better approximate \(f_\infty\), which will make the agents appear more intelligent. Because LLMs, like humans, are unable to know the future, they must deal with
ambiguity and instead create a distribution of potential outcomes. Hence, LLMs exhibit seemingly intelligent behavior.
Finally, I note that this “in the limit” argument says nothing about what is the optimal architecture or that transformers, or even neural networks, are the best way to approximate \(f_\infty\).
Rather, my claim is that there exists a function \(f_\infty\) such that, when approximated well, will result in agents that act intelligently.
PhD Done!!!
My PhD on Declarative Programming Via Term Rewriting, which I completed at Johns Hopkins University, is done. https://matthewfl.com/phd
This research project was about developing a declarative, weighted logic programming language for machine learning, artificial intelligence, and natural language processing applications. The
programming language we were researching was interesting because it allowed for programs that do interesting probabilistic and symbolic reasoning to be concisely expressed in a few lines of code. It
accomplishes this by allowing the programmer to leave out as many details as possible about how the program is executed. This is similar to a database, where the database needs to automatically
figure out how to retrieve the data given a high-level declarative query.
To make this work, I created a relational algebra, which is capable of representing entire programs. To make the system flexible enough to find a good execution strategy, I created a term rewriting
approach that includes hundreds of rewrite rules to run the program. This is similar to rewriting an expression like “2 + 3” as “5” where both of these expressions are semantically equivalent.
To make the term rewriting relational algebra approach tractable, I additionally had to redesign many of the traditional approaches that programming languages are implemented. For example, I created
a new way to think about memoization (dynamic programming) to make it work with our system. Additionally, I created a (JIT) compiler for our term rewrite system because the naive implementation was
too slow for real world use.
In the end, this was an interesting research project. However, I think that this work was set a bit too firmly in the realm of symbolic systems for AI (the AI paradigm of yesteryear). Hence, I do not
know if this is applicable to modern only big neural AI that is dominating. Eventually, I do think that this work may see some use. The reason is that while pure neural creates really cool
demonstrations, it will also fabricate information. This creates an issue when these systems are deployed into applications, and that is a problem for their usability in industry. Hence, having a
system that incorporates weighted reasoning (necessary for neural networks), and symbolic reasoning into a single system is a very powerful programming paradigm.
The dissertation document and recording of the defense are available at: https://matthewfl.com/phd
Dissertation Abstract
I present a new approach to implementing weighted logic programming languages. I first present a bag-relational algebra that is expressive enough to capture the desired denotational semantics,
directly representing the recursive conjunctions, disjunctions, and aggregations that are specified by a source program. For the operational semantics, I develop a term-rewriting system that executes
a program by simplifying its corresponding algebraic expression.
I have used this approach to create the first complete implementation of the Dyna programming language. A Dyna program consists of rules that define a potentially infinite and cyclic computation
graph, which is queried to answer data-dependent questions. Dyna is a unified declarative framework for machine learning and artificial intelligence researchers that supports dynamic programming,
constraint logic programming, reactive programming, and object-oriented programming. I have further modernized Dyna to support functional programming with lambda closures and embedded domain-specific
The implementation includes a front-end that translates Dyna programs to bag-relational expressions, a Python API, hundreds of term rewriting rules, and a procedural engine for determining which
rewrite rules to apply. The rewrite rules generalize techniques used in constraint logic programming. In practice, our system is usually able to provide simple answers to queries.
Mixing disparate programming paradigms is not without challenges. We had to rethink the classical techniques used to implement logic programming languages. This includes the development of a novel
approach for memoization (dynamic programming) that supports partial memoization of fully or partially simplified algebraic expressions, which may contain delayed, unevaluated constraints.
Furthermore, real-world Dyna programs require fast and efficient execution. For this reason, I present a novel approach to just-in-time (JIT) compile sequences of term rewrites using a custom tracing
Training a cat to walk on a leash
To celebrate the one year anniversary of adopting Patton, I would like to take a moment to encourage cat owners everywhere to walk their cat. After a year of walking Patton, I have been able to get
him to go up to 4 miles on a single walk. I have come to believe that Patton often enjoys walking as sometimes he chooses to continue walking instead of stopping early.
The First Time Walking
When I first started trying to walk Patton I had found that there was an abundance of advice online about how to train a cat to walk on a leash. I have come to think that with cats there might not be
a single strategy that works (like there is with dogs) and I wouldn’t get discourage if the first thing that you try ends up failing.
The first time that I put a harness on Patton, he was about 2 years old already and I had only adopted him a few weeks earlier. (So he was doomed to walk on a leash from the start with me.) Now, a
lot of the advice that I found online talked about getting a cat used to the harness first. However, Patton really hates the harness, even after a year of using it he would just go pout if it was
left on while he was inside. So, instead of trying to get him used to the harness while indoors, I only ever use it while outside and the moment that we get back, I end up taking it off. With this
arrangement, Patton will even wait by the door right after getting back for me to take the harness off.
The second trick to getting Patton to start walking was figuring out how to motivate walking while outside and on a leash. What I eventually settled on was rewarding returning to my apartment with
food. This was fairly natural given the first few times that I tried to take Patton outside all he wanted to do was run back inside. This meant that I would carry him outside some short distance and
then let him run back (while still on the leash). This means that these walks are really short on the matter of minutes where I would put the harness on, carry him a few feet from my door, let him
run back, and then feed him.
Extending the Distance
Once I had Patton doing short distances, the trick was then to being extending the walk into something more than just running back to my apartment. This started as carrying him outside one way and
then making him run back another way. Essentially this was trying to close a loop so that it would begin to look more like a walk. Now, this required a bit of handling on the leash to tug Patton
along some alternate path. During this time, I also planned goals during the walk which I began to associate with the food reward that Patton would get at the end of walking. The goal that I started
with was just walking around the perimeter of my apartment building.
What to Expect
Walking a cat is not like walking a dog. When walking a dog, they will generally follow along after relatively minimal training. Cats are instead leading the walk. When I take Patton to a city park,
he often want to spend a lot of time smelling all of the plants in an area or laying down in the bushes and take a nap instead of walking around. I have found this a good time to bring a book and
catch up on reading.
While “slow” or relaxed walking is the norm inside of a city, I have found that when I take Patton on more isolated trails (less dogs) then he is more eager to walk. If there is a single obvious path
to follow, then Patton is more than happy to take the lead in these cases and have been able to walk him up to 4 miles under these ideal circumstances.
Year in Review Composition
Redmagic Meta-tracing JIT
This is a newly published repository from an experiment that I started during the summer to attempt to optimize Python’s runtime without impacting compatibility with C models and existing libraries.
There currently exists frameworks such as PyPy and Truffle/Graal which will generate a JIT when you design your programming language inside of their framework and use their compiler to generate a
custom binary which implements a JIT for a given language. One problem with these frameworks is that they require that an existing language is reimplemented essentially from scratch using a subset
of Python or Java respectively. Additionally, once a programming language is reimplemented, any existing modules which interface with internal aspects of the interpreter (any python C module) will
not be compatible and will have to be rewritten.
Redmagic is similar to PyPy and Truffle/Graal in that it tries to be a framework for creating a JIT however it is different in that it tries to work with an existing C or C++ interpreter requiring
only a few annotations inside the code to identify loops at the user level of the program. (cPython example) It then generates traces of a user’s program by switching between letting the language’s
interpreter run normally and on top of a virtual machine which records all instructions and branching directions. Unlike other JITs it does not reorganize how memory is laid out and even goes as far
as simulating pushing and popping of C level stack frames. This means that at any point while running inside the virtual machine or inside the generated code, the system can resume the “normal” code
simply by jumping to the corresponding instruction in the original interpreter. This does come with the downside that it inherits all memory layout and pointer dereferencing performance issues that
were present in the original interpreter.
Results: After much work on Redmagic, I have observed very slight (<2%) performance improvements in limited cases when working with cPython. The issues of memory layouts (explored further in this
post) seem to contribute significantly to Python’s performance issues and those are not addressed at all by the Redmagic implementation. Additionally, it is not clear that these would be easy to
address from Redmagic itself given that it is looking at a program from the level of x86 assembly instructions vs higher level representations where data-structures can be easily recovered. I
believe that there might be cases, possibly in math routines, where memory layouts have already been optimized and possibly allowing for more clever uses of branching could prove beneficial to
Cost of Abstractions
This post is from a recent conversation that I had about the cost of abstractions in various languages. In choosing a programming language for a project, it is important to choose a language that
gives a good mix of “time to develop” and “performance.” The first tends to be obvious from experience. However, for many, the performance of a programming language (and its implementations) is
somewhat opaque. I am going to demonstrate the differences between Python, Java, and C++, as this gives a good sample of the differences between programming implementations. We can think of Python as
representative of the general class of interpreter-based programming languages such as Ruby and Perl, and Javascript before 2005. Java represents the current state of the art in JIT technology and
should generalize to general JVM targeted languages and even languages such as Javascript, which have seen countless hours invested in developing new implementations of the language. Finally, C++ is
a “low level” language, however, it features a number of unique features, such as templates and stack allocation, which allows for zero cost abstractions which is sparsely seen in other languages.
What is abstraction
Abstraction in this post means the ability of the programmer to wrap a series of methods and data into some object which we are going to use throughout our program. An example of abstraction might be
a class LogNumber which abstracts away the fact that we are representing some numbers inside of a log space instead of a linear space. This is fairly standard practice in machine learning
applications and might even be necessary in some cases where the magnitude of the number is smaller than the range of a floating point number. If we have this abstraction then we can write an
expression such as: LogNumber + LinearNumber and have the LinearNumber automatically converted into log space and thus we will get the correct behavior from our program. If we are programming without
this abstraction and simply using primitive types, and we accidentally write float_in_log + float_in_linear we are going to get a number that is neither in log space or linear space and this is a
bug. One of the most famous examples of this bug is with the Mars Climate Orbiter crashed due to a units mismatch between standard and metric system.
For the remainder of this post, we consider a simplified “LogNumber” class, A which is using a single 32 bit signed integer and getter and setter methods. One could easily imagine there being
multiple getter and setter methods for when one wants to set a linear number to a log space number etc.
# Python
class A:
def __init__(self, a):
self._value = a
def get(self):
return self._value
def set(self, a):
self._value = a
class A {
private int value;
public A(int a) { value = a; }
public int get() { return value; }
public void set(int a) { value = a; }
class A {
int32_t value;
A(int32_t a) : value(a) {}
int32_t get() { return value; }
void set(int32_t a) { value = a; }
Memory overhead
First, let us consider the impact on memory that this abstraction has in each of these languages. Memory makes since as a place to start it is generally understood that the less memory that we use
the more that we can fit into ram. Additionally, on modern hardware, accessing main memory tends to be a major bottle neck for performance with a read from main memory taking ~100 cycles while a
local processor caches takes 10-20 cycles. As such, having an object which is twice as large means that we can fit half as many objects inside our processor cache and thus will end up performing
costly accesses to main memory more often. Remember, while reading this next section, we are just wrapping a single 4 byte integer.
# Python
a_python = A(5)
A a_java = new A(5);
A *a_ptr_cpp = new A(5);
A a_stack_cpp(5);
Python: First for Python, in constructing the A class, we are going first to create a PyInstanceObject which contains 3 pointers plus the PyObject_HEAD which is 3 pointers plus a size_t bringing the
size of this struct up to 56 bytes plus malloc overhead. (Note: the malloc overhead depends on the implementation of malloc used and could even be changed at runtime using something like LD_PRELOAD).
Next, to actually save _value we have to construct the hash map that is going to back all the elements for this class instance. This means creating a PyDictObject which is a PyObject_HEAD, 3 size_t,
2 pointers, and 8 preallocated hash map slots for which each slot contains a size_t and 2 pointers bringing up the size of this object to 264 + malloc overhead. The number 5 in this case is a small
integer and thus is already interned inside the python interpreter and the hash map key _value would be shared between all instances of this object and so we are not going to count it. Finally, we
are going to have to maintain a pointer on Python’s stack to the object a_python that we just constructed. All together then this brings our total size to 328 + 2 * malloc overhead. Note: We could
reduce the object size somewhat by using Python’s __slots__ feature which should avoid allocating an oversized hash map to back this object, however our object size is still going to be in the 100s
of byte range.
Java: Next with Java, there is an 8 byte header that is added to all objects which contains information about the type of object and object state w.r.t. locking etc. (more info) Additionally, Java
will round the size of objects up to the next 8 byte increment which means that there will be 4 wasted bytes inside this particular object. Finally, we need to store a pointer to this object on our
stack. One of the nice tricks in Java for saving memory is pointer compression where a pointer will be 4 bytes instead of 8 which is a significant savings given that Java programs tend to use a lot
of pointers. (pointer compression info). Together, this means that there will be 20 bytes in memory corresponding to this object.
C++: In the first case with a_ptr_cpp we can see that the object size will be 4 bytes as one would expect given the C++ standard. There will also be the additional malloc overhead associated with
this object. This brings the total size for the a_ptr example to 12 + malloc overhead when including the stack pointer.
In the second case with a_stack_cpp we have this being directly allocated on the stack which means that we have no pointer or malloc overhead or pointer to the object, thus the total size is only 4
bytes. The fact that we are able to take a primitive type (int32_t) and wrap it inside some class and still consume the exact same amount of memory is what it means to have a zero cost abstraction.
Once C++ is compiled with some optimizations or a Java program has been running for long enough, it is feasible that both a_ptr_cpp and a_java are entirely stored inside a register saving 8 and 4
bytes respectively. Moreover, in the a_stack_cpp case we might have the integer value stored in a register, which means that the memory used in this case is 0 bytes.
Escape Analysis
For the C++ and Java case if a compiler can prove during optimizations that a dynamically allocated structure is not going to leave a method, then it is possible for that compiler to transform that
object into a stack allocated object. This means that even for a_java and a_ptr_cpp could end up having zero memory consumption in the right circumstances.
A a_java = new A(5);
A *a_ptr_cpp = new A(5);
cout << a_ptr_cpp->get();
delete a_ptr_cpp;
Array overhead
Arrays are basic primitive in nearly all languages when it comes to efficiently storing a number of similar objects. In the case of numerical processing applications, our main performance overheads
are going to come down to how efficiently we store numbers and are able to access these values.
# Python
arr_python = [A(i) for i in range(10)]
A[] arr_java = new A[10];
for(int i = 0; i < arr_java.length; i++)
arr_java[i] = new A(i);
A *arr_ptr_cpp[] = new A*[10];
for(int i = 0; i < 10; i++)
arr_ptr_cpp[i] = new A(i);
A arr_obj_cpp[] = new A[10];
for(int i = 0; i < 10; i++)
arr_obj_cpp[i] = A(i);
std::vector<A>; vec_obj_cpp(10);
for(int i = 0; i < vec_obj_cpp.size(); i++)
vec_obj_cpp[i] = A(i);
A arr_stack_cpp[10];
for(int i = 0; < 10; i++)
arr_stack_cpp[10] = A(i);
Python: First for python, when constructing a list we create a PyListObject which contains the PyObject_HEAD, a pointer to the head of the list and a size_t for the size of the list for 48 bytes. The
list itself is simply a list of pointers to objects. Therefore the list object on its own will be 128 + 2 * malloc overhead. For every object, we are then going to have the same as we had above with
348 per object. This brings the total of this to 3608 bytes + 22 * malloc overhead.
Java: With Java, there are essentially two types of arrays. The first is an array of a primitive type (such as int[]) where the value of the integers will actually be stored inside the array itself.
These arrays can be considered very efficient since there is a small overhead when storing a large number of primitive values. However, in this case, our array does not contain primitive types and
instead must has to be an array of pointers. This means that the array itself will have the 12 byte overhead for each object created and then 10 pointers for which each is 4 bytes, making the list
itself 48 bytes which is divisible by 8 so we do not have any additional wasted space here. Each object in the list will then be 16 bytes as before and we have a reference on the stack bringing the
total for this up to 212 bytes.
C++: Finally with C++, we can see that there exist a number of possible implementations for an array of objects for which each of these have slightly different memory implications. In the arr_ptr_cpp
case we are creating an array of pointers which is conceptually equivalent to the Python and Java implementations. Including the stack reference to this, it takes 120 + 11 * malloc overheads bytes in
memory. Note: we are are going to have to explicitly deallocate all objects inside the array when freeing arr_obj_cpp which requires additional cognitive overhead when writing this program. In the
second C++ example arr_obj_cpp we are allocating space inside the array for our object instead of using a pointer to reference the array. This means that the size of the array will only be 40 bytes
bringing this implementations memory consumption to 48 + 1 * malloc overhead when including the stack pointer. The third case with vec_obj_cpp uses the C++ standard library and would be a more
acceptable way of writing the arr_obj_cpp example and will use 2 additional size_t (16 bytes) to track the array’s size and amount allocated (64 + 1 * malloc overhead bytes). The final case is
constructing an array directly on C++’s stack which means that we are avoiding the stack pointer reference as well as the malloc overhead and this will only require 40 bytes.
Again, we can see that C++ is capable of a zero overhead abstraction as in the last 3 examples the only overhead from directly using int32_t was O(1) bookkeeping.
Performance overhead
While the way in which data is stored is important, we also care about how these languages are going to perform when we go to execute our program.
# Python
a_python = A(5)
A a_java = new A(5);
A *a_ptr_cpp = new A(5);
cout << a_ptr_cpp->get();
A a_stack_cpp(5);
cout << a_stack_cpp.get();
Python: First, Python is the simplest language in the comparison, given that it is a simple bytecode interpreter. This means that it will not be performing any static compile time optimizations or
attempting to compile away this abstraction.
Looking at just the a.set(6) line, we can easily look at the Python’s bytecode using the dis library:
# Calling a.set
3 12 LOAD_FAST 0 (a)
15 LOAD_ATTR 1 (set)
18 LOAD_CONST 2 (6)
21 CALL_FUNCTION 1
24 POP_TOP
# Method a.set implementation
7 0 LOAD_FAST 1 (a)
3 LOAD_FAST 0 (self)
6 STORE_ATTR 0 (_value)
9 LOAD_CONST 0 (None)
12 RETURN_VALUE
We can easily see that there are 10 python bytecode instructions. Some of these instructions such as STORE_ATTR and LOAD_ATTR represent hash table lookups to identify the _value and set slots.
Additionally, nearly all of these instructions represent writes into memory given that Python uses reference counting, and by loading objects onto Python’s stack requires incrementing the reference
count. To see full implementations of these byte codes you can look at ceval.c. From prior experience with python’s internals, I would guess that this line represents about 2000-7000 CPU instructions
with over 30 writes to memory and 100 reads.
Java: With Java, it is a bit more difficult to exactly measure how much overhead there is when evaluating this instruction. This is because the JIT will be constantly generating new versions of this
code which are more efficient. If we were to evaluate this instruction exactly once, then our bytecode will be running on an interpreter which is conceptually similar to python, but by not having
reference counting and not backing every object with a hash map means that this is going to be significantly more efficient and about 100-500 instructions. After we have executed this statement a few
times, the JIT will generate more optimized code. The exact code that is generated depends on a number of factors including if at the call site for a.set(6) if there exist multiple implementations of
set that we must dispatch to (multiple classes implementing the same interface). Assuming that there will only be a single implementation of set, then the JVM will end up inlining and optimizing this
down to 1 instruction which writes to memory.
C++: When using a pointer in C++, we can see statically that there is only a single implementation of A::set and so the compiler will likely inline this method call. This means that similar to the
Java case it will have 1 instruction to perform the write into memory. In the second case with a_stack_cpp we might represent this instruction as a single write into a register. While both of these
cases are going to be a single instruction, the later of writing into a register instead of memory will be much more “efficient.” Additionally, in a larger context, we could imagine that the compiler
completely removes the representation of a_stack_cpp and just inlines the value 6 or 5 whatever is appropriate.
Again, a_stack_cpp is giving a zero cost abstraction while a_ptr_cpp and a_java are giving us a low cost abstraction w.r.t. the number of instructions evaluated.
Meta template programming
The advantages of zero cost overheads really shine when combined with templates and meta programming techniques. By having templated classes, we are able to construct a single class which implements
many basic features that we are able to reuse. For example, suppose that we had a LogNumber class such that we could have LogNumber<LinearNumber<float> > and LogNumber<LinearNumber<double> > which
use the same same which class implementation with two different sizes of numbers. We could even extend this to have LogNumber<LogNumber<LinearNumber<float> > > which would allow for us to easily
represent a number in a log log space without having to reimplemented our LogNumber class.
If we implemented this templated class in Java, then we would have that every template parameter would require a pointer and another class instance. This means that in the LogNumber<LogNumber
<LinearNumber<Float> > > we would require that there are 4 class instances and thus would require 4 pointer indirection to actually reach the floating point value and consume 64 bytes of memory.
In C++, this templated class does not use a pointer indirection when using a template. This means that the size of LogNumber<LogNumber<LinearNumber<float> > > and LinearNumber<float> will be exactly
the same size. Additionally, accessing the stored number will only take a single memory access instruction and can be placed on the stack or used inside an array as shown above.
Abstraction is a very important concept when writing larger programs. Having zero cost abstraction available means that we can have both performant programs while reducing the mental burden on the
programmer. When programming in a language such as Java or Python, we have to make a choice between developing the most performant program in that language and having a clean and reusable
We see the power of zero cost abstractions aggressively used inside libraries such as Eigen, Dlib and Tensorflow as these libraries care immensely about the computational and storage efficiency.
Using C++’s template system, these libraries contain a single implementation for a generic routine which can be customized by switching out a few minor methods without losing any performance.
WTF happened this year
This is a post that I started writing shortly after the Democratic convention, as I started thinking that Trump was going to end up beating Clinton. Now that the “impossible” has happened (haha all
those “old media” predicting <2% for Trump) I find myself publishing this post as an attempted reflection on how we got here.
So what went wrong and what can be learned from this year? I think that the first and easiest way to frame this year might be “old establishment” vs “random unknowns” where a large number of
individuals decided that unknowns would be better than more of the same for themselves. These individuals IMO tend to view the status quo as getting themselves screwed over by some external force,
such as technological progress, trade deals, banks or immigrants, and as such wanted some candidate that would end what ever entity was screwing them over. When this battle comes down to between HRC
and the Orange, we have one candidate that was essentially saying that things are not so bad and that what happened with the financial meltdown and bailing out the banks had to happen, and the other
simply channeling people’s anger towards an undeserving group of people (immigrants). In the end, speaking to one’s frustrations rather than trying to tell them they are “crazy” and they are better
off then X years ago was the better strategy (who would have guessed).
While it is easy to forget where we came from in terms of the primaries, considering that I starting this posts months before it was posted, it is easy to recall the events of a few weeks ago.
First, when the Tangerine said that he might consider a third party run if he was not treated fairly by the Republicans, this was an extremely smart home in hindsight. This was basically his escape
hatch from the party which would have allowed the Tangerine to prevent a successful presidential bid if it came to light that there was any foul play during the primary processes. As such, the
Republican party was forced in to playing a fair game and thus as an outsider Tangerine was given a fair shot.
The flip side of this issue was Bernie, who said that he would not consider a third party run which basically meant that he gave license to the Democratic party to sabotage the primary processes
against him without their being any consequences (such as directly losing in November as a result). This is something that we know happened given the wide array of emails that have been leaked. (1, 2
) (In the last few weeks alone there have been countless wikileaks which have shown the additional details of how the
Simply looking at the Democratic primary, we had HRC with Clinton being one of the few name brands bigger than Lewinsky (sorry, bad joke… someone had to make it), and Bernie, a politician who many
(at least on the west coast) have never heard about before. The massive HRC name brand politician then proceeded to lose 22 primaries. Additionally, winning the primaries that she did required that
she conspired with major media providers and spend years specifically maneuvering to control the Democratic party through DWS becoming the party head (her campaign manager in 2008) and getting
countless super delegates to pre-signup with her campaign. When in it came to fund raising, Bernie was consistently out raising HRC and he was doing it using a larger pool of donors making smaller
contributions which IMO indicates a campaign which was better in touch with the actual voters. We see a similar parallel when comparing the sizes of HRC and Bernie rallies.
In directly comparing Clinton to Tangerine and Bernie, we have HRC who continued to “evolve” her position to try and always attract the most voters while Bernie and Trump both took a position and
keep pushing a core message. The fact that Tangerine keep changing exactly what he said on specific policy issues didn’t change the core message which was simple enough that it easily resonated with
his core voting base.
False unity at the DNC
Primary election fraud
More info on voter fraud/suppression for Bernie voters
Large report on election fraud
TL;DR: If you fix your primary such that you ignore the “public poll” that you are conducing, the candidate that you get out is going to be weaker then they should be in the general election.
Some additional comments:
• While the mainstream media is likely going to frame this as “America wasn’t ready for a woman president,” I don’t think that was the issue. Instead, Clinton was a weak candidate which to many
Americans symbolized the failures of government that they can’t stand
□ The fact that “the most qualified Woman/person ever” just lost to the biggest “joke” we are unlikely to see another “Woman from a major political party” within the next 20 years. The only
chance that the next woman has of getting a nomination is that she wins on a populace surge, the party insiders of the Republicans and Democrats are going to be unwilling to risk it
• At some level the country has just “approved” the Tangerine’s personal views on race and women (given that this election wasn’t won on policy)
• Once Clinton pivoted to the general, talks of policy basically stopped. Instead she started using personal attacks (forcing all those meaningless leaks about personal qualities). Instead of
Bernie had been in the general, he would have keep the message focused on policy, people would have been able to more easily recognize that Tangerine had no real policy and would be unable to
□ If you want to have a chance of winning against Tangerine in 4 years, you are going to have to defeat him on the fact that he has bad policies or the poor job that he has done. More personal
attacks are not going to work and is a dumb position to take and isn’t going to resonate well with younger generations.
• The Tangerine tape scandal made no sense. (Again this isn’t a policy issue attack but a personal attack.) Some have said that this was a “waking up call” to women that were supporting Tangerine,
I don’t think that actually holds that much weight. Lets suppose for a moment that Tangerine had “much better” policies then HRC w.r.t. women’s issues and this tape still came out, the
conversation would have been: “So he said these things, but he is still much better for me as a woman.” America’s history of presidents has been a string of questionable views on women and
marriages etc, one more shouldn’t really be a surprise to anyone regardless of how many times the media plays it. Most people will never directly interact with the president, “we” (or at least
I), do not care if the person who wins the presidency is likable or has done some questionable things in the past, all that “we” care about is whether or not their policies are going to be good
for “us.”
What a surprise mainstream media:
Donald Trump would have lost US election if Bernie Sanders had been the candidate
How the Washington Post killed Bernie Sanders’ candidacy
The Democratic Party Establishment Is Finished
Theoretical online voting system
With the election a few days away, I found myself recently looking at the state of voting in America and contemplating that there is still no online-based voting system in place. The main arguments
against online voting or digital-based voting has been that it would be hard to verify and would require a computer security expert to identify if something has been tampered with.
Now to create a system that is “provably incorruptible” would be very difficult and impracticable to expect average poll workers to verify the correctness of such a system, however, there is probably
a widely unexplored range of systems that are better than our current system but still have some easy to verify properties. In this post, I attempt to create a voting system which no worse then our
current voting system with respect to voter fraud and ensuring the votes are counted.
First, let’s consider the state of our current voting system, specifically the voting-by-mail system. Step 1 is to go online and register your address along with some identifying voter information.
At a later point in time, the state will mail a ballot to your address which contains a “voting key” which maps various voting positions (who you would like for some office or position on a
proposition) to some number \([1, 200]\). To vote, you bubble in your corresponding chosen numbers, wrap your ballot in more paper called a “secrecy sleeve,” put this in another envelope and mail it
to the ballot counting location. Presumably, once your ballot arrives, someone will check the identifying information on the mailing envelope to prevent duplication and then pass the ballot and
secrecy sleeve to someone else who is just going to count the votes. This two-level operation would prevent people from knowing who you voted for, assuming that the first poll works don’t look at the
ballot inside the secrecy sleeve. In terms of ensuring that your vote is counted, we have to then trust the second poll worker to count the votes correctly. We might use more than one person for this
second part to prevent errors etc.
Now in making a new system, we have to consider what possible vulnerabilities exist in the current system, as those could still be allowed in the new system:
1. Trusting the United states postal services (USPS) to properly deliver mail — If your ballot never makes it back to the polling place, then it will essentially be lost (there might be some ways to
identify that it is lost, but still no real/easy recourse for ensuring that it gets counted)
2. The USPS needs to get you your ballot to you in the first place — If the ballot was sent to the wrong address, it is possible that someone fills in the ballot in your name, forges your signature,
and then mails it back in
3. People are trusted to bubble in their choice correctly — Eg, they are at least able to understand that given some “number” on a “ballot key,” they are supposed to transfer that number correctly
to the ballot itself
4. A malicious poll worker could prevent a vote from getting counted that they didn’t agree with — Given that your vote is easily identifiable on the ballot, it is trivial for someone to reject all
ballots which have bubbled in number 10 (ideally, there are two or more people to double check that this does not happen)
Given this set of vulnerabilities in our current system, lets now try to design a better system that allows for internet voting:
Our first steps would be very similar to the current voting system, where someone goes online and registers with their mailing address. The state would then mail out a “ballot key” to the provided
address. The reason that we would still require that something is mailed out is that there is currently no good way to identify a citizen online in a secure way, however, like the current
vote-by-mail system, it is acceptable to trust the USPS as a “broker of identities.” Now our vote by internet ballot key will be a bit different from existing ballots where each vote is represented
by \([1, 200]\) and instead have a number in \([0, 2^{256}]\), additionally, instead of having a single number (say 10) represent a position on the ballot, each voter would be given a unique number
for each position on the ballot. (A sample ballot is at the end of this post) We can then use a simple website to collect the keys which represent a person’s choice. Given that each user has
different codes generated for their ballot, we can use untrusted channels to communicate these codes to the vote-counting authority. Additionally, we do not have to worry about “suppressing” the vote
that a poll worker disagrees with since the intermediate communication mechanisms don’t even know which vote was cast. All they know is that they are responsible for is communicating some number to
the voting authority. Even if voter’s computer was infected with a computer virus, it would be unable to change your vote since it only knows the key that was entered representing your choice, while
the other keys would only be present on the paper ballot key that was mailed to your address.
Some properties of this system:
1. We are still trusting the USPS to properly identify people and communicate information with them securely. (Same as before)
2. Submitting a vote for someone else still depends on your receiving or intercepting their ballot and “forging” a signature (Same as before)
3. The intermediaries do not know your vote (better than before) — Now your vote is a number that is specific to you, so the only people who will know the vote are the person who generated the
“voting key” and whoever has the voting key
1. The intermediaries can not suppress your vote based on who you voted for — They do not who you voted for, so it can not be suppressed based on this reason
2. Your vote can not be changed after the fact — Changing your vote would require that the malicious intermediary have your “voting key book,” which was printed by the state and mailed by the
USPS (which is a trusted medium)
3. Your computer (now technically also an intermediary) can not change your vote even if it was infected with a virus — your computer does not know the alternate keys you were provided since
they were printed and mailed, so it can not switch between them.
4. The number that you have to enter is a lot longer (worse) — Currently, you only enter some number \([1, 200]\), however, a 256 bit number is notably longer. Given how people are already used to
entering 16 digit credit card numbers, this might not be such a big issue. We could even include checksums to limit erroneously entering something (bitcoin already uses a 32-bit checksum on all
Some might point out that one could set up false voting websites to try to confuse voters or perform a DOS attack on the voting website. First, with false websites, we could follow the trend of some
banking websites where an image is displayed to ensure that you are on the correct website. However, we might make it some confirmation code that is sufficiently long that it would be difficult to
counterfeit and easy to print on a “ballot key.” For the DOS attack, we already know how to build systems that can deal with DOS attacks. Additionally, if we have a confirmation code system that
confirms that a vote has been recorded, then any mechanism which takes a voting key and returns the confirmation code is as good as any other. This means you could have voting via email or even text
message, which are “more difficult” to perform a DOS attack against or allow for third-party websites to spring up to collect votes as they would have to be still backed by the state vote recording
Sample theoretical ballot key:
│Politician │Office │key │Confirmation code│
│T Sandwich │President│NMosFJizjPgUV2BKEhGE rjvUZzKZVAFCyqPy7w3t │FuT8VDz3z │
│Giant D │President│Tru4oZn9y3RMnxAsb7g 5Gqs7Fu13FX4ExaQSer6y │bFcCf4MJA │
│None of the above│President│LaGeinvoBUduEbovp5z JDQJ6DQEdgSqZWgXzArhi │xjzEahMdi │
(These politician names are based on this current season of south park.)
TL;DR: Online voting where you are still mailed your ballot via USPS, and your ballot contains keys that we consider “secure,” and you only submit one key that corresponds to your vote.
Update / additional background info / other posts on this topic:
While the mathematical concepts in these schemes are sound, it would be difficult to convince the public at large. In these cases, people would have to just generally trust that someone has done the
correct thing with designing the voting systems. From an academic point of view, if these systems are implemented correctly, there wouldn’t even be a need for there to be vote checkers since they
would “have to be correct.”
JSApp.us Shutdown
This post is a few weeks late, however JSApp.us has been shutdown. At the time that JSapp was first released, it was the first open node.js hosting platform. It featured an easy to use web based code
editor (which was extremely useful at the time as developing node.js on windows was fairly difficult and required compiling code yourself). The system also provided CNAME support as well as
subdomains for quickly developing demonstrating applications. There was even a command line client which allowed for quick deployments from the command line and uploading and downloading files from
ones file system.
At last, JSApp’s time has come to an end. It has seen no major updates since 2011 (source) and the state of developing Node.js applications has moved on with new API’s (that were not supported by
JSAPP) as well as new compile to JavaScript languages (which were also unsupported by JSApp).
Given the abundance of alternate Node.js hosting options and the age of the system, it seems that essentially all users have already migrated off the platform, so this change is unlikely to been
disruptive. The source will remain available on github if anyone interested in some of the internals, however given how dated the system is, I am assuming that there are better solutions today for
nearly all aspects of the system.
Intel Xeon Phi for “cheap”
(This work and post were originally from early 2015, some aspects may still be useful, eg the kernel patch for the lower end motherboards)
Recently Intel has been selling their a version of their Xeon Phi coprocessor under a promotional deal at 90% off. This means that one can get a device with 8GB of ram (on the coprocessor) and 228
hardware threads (57 physical cores, and each with 4 hyper-threads) at a reasonable price of ~$200.
When I first purchased the Phi, I was planning to put it into somewhat of an old desktop system that I had lying around, however the motherboard did not support the major requirement of “Above 4G
decoding” on the PCI bus. 4G decoding deals with how the system allocates the memory resources on items on the PCI bus. With the Intel Phi, unlike consumer level GPUs it will present all 8G as a
memory mapped region to the host computer. (more about 4G decoding) Based off some research on this obscured feature, it appeared that most “modern” motherboard have some support for this
feature. I decided to get an Asus h97m-plus which is fairly cheap, and fit the computer tower that I already had on hand. While this motherboard does list the above 4G decoding in its bios and
manual, I am not actually sure if this feature has been properly tested, as unlike Asus higher end motherboards, there was no mention of this mother board specifically working with the above 4G
decoding. Based off examining the early booting sequence it appeared that the Linux kernel was attempt to find alignment positions for PCI devices which were equal in size to the requested memory
region (8GB in this case) or depends on the BIOS to perform the PCI allocation before booting. For the higher end motherboards which the Intel Phi was known to work with, it appears that the “more
powerful BIOSes” were allocating memory for the Phi, but in the case of this lower end motherboard, the BIOS was unable to deal with a request to allocate 8GB of memory and thus falling back onto the
kernel to perform allocations. Following this observation, I made a small kernel patch (here) which changes requests for alignment larger than the maximal size to be simply aligned at the maximal
supported size. With the components in this computer it appears that even with this change the Intel Phi gets aligned to a 4GB boundary and is able to still function correctly.
The next challenge once the Phi was communicating with the computer was to prevent the chip from overheating. The discounted versions of the Phi did not include any fans as it was designed for use
in server environments. Additionally being a 300+W accelerator, the system is capable of generating a lot of heat. As such, many “typical” fan solutions that I tried failed to keep the chip cool
for longer than a few minutes. I eventually landed on the high-powered tornado fan which can move over 80 cubic inches of air a minute. I ended up having to zip tie this over one end of the chip to
ensure that there was enough directed airflow to keep it functional. (warning to future users: This fan actually does sound like a tornado, constantly).
Having the entire system functional for over a year now, I have managed to use the Phi for a handful of computations. While there is decent opportunity in improved performance, this chip really
requires that you design customized software for it specifically. This is especially true given that Intel Phi is less popular than graphics cards with Cuda, where many mathematical tools and
frameworks already have customized backend targeting Cuda requiring limited effort on the user’s part. While this chip has a nice promise of being able to execute normal x86 instructions, this seems
to be of fairly limited use since the only compiler that will target the chip and use its specialized vector instructions is Intel’s own compiler (similar in nature to Cuda). This makes it fairly
difficult to natively run any non trivial programs on this chip as any external libraries require their own porting effort. (As an accelerator which accelerates embedded methods similar to Cuda this
chip works fine, just if you are trying to run a program without the hosts involvement.)
Photos of the setup:
Fan zip tied onto the back of the computer
Estimating the n percentile of a set
Here is an interesting idea that I had recently, this is just a high level concept about how it would work, there are no proofs for error bounds or quality, in fact there would be a handful of
orderings of sets which would product terrible results.
To accurately compute the \(n^{th}\) percentile value of a given set of values, one ends up having to sort the values which if they are not integers, can take \(O(n \log n)\) time. However, then
getting value itself is trivial since it is simply going to the correct place in the sorted values. I am thinking that one should be able to compute an estimate for the \(n^{th}\) value for a
randomly ordered set of elements in \(O(n)\) time.
First the basic idea, imagine that you have a set of elements \(X = \{ x_i \}\). Then if we had this set sorted \(S\), then finding the \(n^{th} (0 \le n \le 1)\) percentile out of \(X\) would
simply become \(s_{n * |S|}\). This implies that we have \(n * |S|\) elements less than \(s_{n *|S|}\) and \(|S|*(1 – n)\) elements greater. From this point we can imagine constructing two sets, \
(\alpha = \{ x \in X : x < s_{n * |S| } \}, \beta = \{ x \in X : x > s_{n * |S|} \}\) which represent the elements greater and less then the \(n^{th}\) value. This also means that \(\frac{|\alpha|}
{|\alpha| + |\beta|} \approx n\). Now using this concept for \(\alpha\) and \(\beta\), we can attempt to construct these sets while iterating through \(X\) by having a current estimate for the value
\(s\), and then tracking the elements current in each set. This essentially becomes if \(\frac{|\alpha|}{|\alpha| + |\beta|} > n + \epsilon\) then take the current value of \(s\) and insert it into
\(\beta\), then take the largest element out of \(\alpha\) and set it equal to \(s\). In the case of \(\frac{|\alpha|}{|\alpha| + |\beta|} < n – \epsilon\) we simply do the reverse by inserting the
current value of \(s\) into \(\alpha\), and then removing the smallest value from \(\beta\) and setting it equal to \(s\).
Now the problem has been reduce to splitting \(X\) into two different sets and keeping them sorted somehow to be able to get and remove the largest/smallest elements. However, this would give an
exact answer to finding the \(n^{th}\) percentile. Now given that we want to find an estimate, we can imagine capping the size of these sets to \(k\), where \(k\) is a small number such as \(2\).
Then instead of tracking the elements themselves, we are simply counting the number of elements that are greater or less than the current \(s\) value. Additionally, we have the sets tracking the \(k
\) elements that are largest but still less then \(s\), and the smallest but greater then \(s\). As we iterate through the set, we can track the \(k\) values in \(\alpha, \beta\) and the size of \(\
alpha, \beta\) accordingly, and when we want to change the value of \(s\) to keep \(\frac{|\alpha|}{|\alpha| + |\beta|} \approx n\), then we just take the new value from \(\alpha\) or \(\beta\)
respectively and update the cardinality of each set.
An additional extension to this algorithm could be for an \(n \approx .999\) then the size of \(\beta\) would only be \(\frac{1}{1000}\) the size of the original data set. Keeping track of largest \
(.001\) of the data set would not be linear in the side of the data set, however it could take out a considerable chunk of computation depending on how large or small the value of \(n\) is.
Reducing specific use cases in a language to improve overall usability
This last summer I spent a considerable amount of time refactoring i-lang. Since I started implementing this programming language in early 2012, it had accumulated quite a bit of cruft, and it was
difficult to continue moving forward.
Refactoring type system
One of the first internal improvements that was made was to overhaul the internal type system. Before, the type system was simply passing around a boost::any. However, this became trouble some as all
parts of the code would have to know about each type so that it could cast it locally. In many places the code began to look like:
if(a->type() == typeid(Object*)) {
a = boost::any_cast<Object*>(a);
} else if(a->type() == typeid(Array*)) {
It became even worse when there were two different types involved, as can be seen in the case of performing arithmetic.
Now, the type system has been rewritten to better make use of C++ template and virtual function systems. This means that one can write code like:
ValuePass a = valueMaker(1);
ValuePass b = valueMaker(2.4);
ValuePass c = a + b;
assert(c->cast<float>() == 3.4);
assert(c->cast<int>() == 3);
assert(c->cast<bool>() == true);
assert(c->cast<std::string>() == “3.4”);
The real beauty of this type system can be seen when using the foreign function interface, where the value of arguments can be “injected” into local variables. This means that a function can be
written as:
ValuePass func(Arguments &args) {
int a;
double b;
std::string c;
ilang::Function d;
args.inject(a, b, c, d);
return valueMaker("hello world");
Changes in the type system at the language level
Before this refactor, types in i-lang were defined in a global table of identifiers called variable modifiers. A variable could have more than one modifier attached to it, and each modifier is used
to check the value being set to a variable. What this roughly translates to would be something like:
define_type("Unsigned", {|a|
return a >= 0;
Unsigned Int b = 5;
Looking at this implementation of a type system, it does not seem that bad when compared to other programming languages. As displayed here it is missing the concept of a namespace or import scope,
but otherwise it is fundamentally a type system where types are given names and then later used to reference that type. However, this concept of a type having a name fundamentally goes against
i-lang’s concept of names only being used as place holders for values, vs having an explicit places in the language. (eg: class required_name_of_class {} vs name_bound_for_use_later = class {}). This
lead me to question what does a type system fundamentally do. In lower level languages such as C/C++ a type system provides information about the space required for an object, however in a higher
level language such as python (which i-lang is more similar to on this point) values are just fixed sizes and then pointers to larger dynamically sized objects when required. Type system also
provided casting between primitive types, such as a 4 byte integer casted to a floating point. This on its own isn’t that interesting as there are limited number of primitive types and similar
operations can be accomplished with code like `1 * 1.0` or `Math.floor(1.2)` for casting. Finally, type systems provide a way to identify the type of some value, which can be further used by a
language to provided features such as pattern matching when calling a function. Now, choosing to focus on this last issue of a type system lead to i-lang concept of a type system, which is that a
type is simply a function which can identify if a value is a member of a given type.
The idea of using just a function to identify a type can sound a little strange at first, however, after playing with it some, the idea itself can be seen to be quite powerful. Here is a quick
example of using this type system to implement pattern matching on the value passed to a function.
GreaterThan = {|a|
return {|b|
return b > a;
LessThan = {|a|
return {|b|
return b < a;
EqualTo = {|a|
return {|b|
return b == a;
Example_function = {|GreaterThan(5) a|
return "The value you called with is greater than 5";
} + {|LessThan(5) a|
return "Then value you called with is less than 5";
} + {|EqualTo(5) a|
return "The value you called with is equal to 5";
} + {
return "The value you called with didn't compare well with 5, must not have been a number";
Int GreaterThan(0) LessThan(10) int_between_zero_and_ten = 5;
In the Example_function, we are combining 4 different functions, each with different type signatures. Additionally, we are creating types on the fly by calling the GreaterThan/LessThan/EqualTo
functions which are using annoymious functions and closures. This method also allows for classes to have a place in the type system. We can now easily create special member of a class to check if a
value passed is an instance or interface of the class type.
sortable = class {
Function compare = {};
sort = {|Array(sortable.interface) items|
Refactoring Objects and Class to appear like functions
Before, i-lang use syntax similar to Python or JavaScript dicts/objects when constructing a class or object. This meant that these items looked like:
class {
Function_to_check_type type_name: value_of_type,
Another_type another_name: another_value,
no_type_on_this: value
However, in ilang, except when prefixed with `object` or `class` the use of `{}` means that it is a function. (eg: a = { Print("hello world"); };) Additionally, colons are not used anywhere else in
the language, which made me question why was this case such a special one. This lead me to ask why not use equal signs and semicolons like everywhere else, meaning that defining a class would appear
class {
Function_to_check_type type_name = value_of_type;
Another_type another_name = another_value;
no_type_on_this = value;
Furthermore, is there any reason to exclude loops and if statements when constructing a class? Allowing control flow when the class definition is constructed makes this act identical to a function.
a = true;
class {
if(a) {
b = 5;
} else {
b = 6;
Final thoughts
By starting by cleaning up the internals of i-lang, it allowed me to take another look at the language and determine why certain choices were made at first. Bootstrapping a new programming language
takes a considerable amount of effort, and could easily lead someone to defining something like print a function (as python recently changed away from in version 3). In my case, I was creating
special mechanisms for constructing class/objects and defining types for variables largely because the type system, scopes, and function internal interfaces were all poorly designed in the first
iteration of the language. Now that the internals have been cleaned up, it makes it easier to see that these changes are wins for the language. Now, I doubt that I would have been able to come up
with these changes right off the bat with the first implementation, as it was only through the pain of the first implementation for which the need for these changes became apparent.
Current state of ilang
This is the first week back at school, which means that the speed of development on ilang will begin to fluxuate again. Over this last break, and the final parts of last semester, I was able to clean
up a bunch of features with the ilang programming language.
When I originally set out to create this new language, I was mostly thinking about how the syntax would be different and the specific features that I would want in the language to make it worthwhile
of being created. However, what I failed to thinking about was what really make a programming language useful today is the fact that there are an absurd amount of libraries already programmed,
debugged and available for download for any successful language. As a result of this realization, I have been working on trying to get useful libraries written for the language. However in trying to
work with the language, I have a few beliefs that I am trying to stick with, but I am not so such how well it will work out.
The first belief, is that there should be no need to pause or have any sort of timer system, this is because I feel as if the language should attempt to run as fast as possible and focus on
processing data. However when writing testing frameworks to automatically check if the language is working, it has become apparent that it would be useful to have some timer system. ** I still
haven’t written in the timer system so this is still somewhat of an issue of internal debate.
Along with the timers, there is the problem of getting data efficiently in and out of the programming language. One of the concepts that I have for the future with this language is that the system
will be able to distribute the computations between a large number of computers, this means that it is not particularly natural for the system to have access to features of the local computer, such
as the file-system or the standard in/out. I am thinking that for the time being that the system could be designed to have access to the standard input and part of the file-system, however when the
system becomes networked across a lot of computers, there could possibly be a way to specify where the standard input/output should go along with where the file-system has access to. The other
alternate that I am working on, is using the concept of just having the http server be the way to input data, however I expect that it will become cumbersome quickly to input large data files. A
possible compromise is to use the parameters to support some way to map specific files or folders to names that can be access from inside the program.
When considering the libraries that have already been included, there is still a lot of space for development. The modification library is still lacking too many features to be really usable. Along
with the modification, the database library, still lacks the ability to save arrays into the database. The complication with arrays is trying to figure out an efficient way to store the data without
causing a significant slow down. My plan for arrays in the database, was that they would be “smaller” then objects in the database, as objects when stored in the database do not have any limiting
factors for the number of elements. However with arrays, I plan to try to have the system load all the data into memory when reading an array from the database. However the current way that the
system is designed, it does not allow for the elements, to be easily accessed under the hood. I am thinking that the system might try and store all the elements in their own container, however the
problem with this is that there would be a large number of database queries when trying to read data out. And inserting in the middle of the array would require a large number of read and writes into
the database. However, on the flip, if the system was using one database object to store all of the array elements, there would be a few read and writes, but the object would likely be come very
large very quickly.
The current plan for future features, is to keep working on adding more libraries into the core system to be useful. This will mostly focus on processing and interpreting data. (Web server, http
client, as well as methods to parse string such as regex, also expecting some form of CSV or sqlite reader for when working with data that has been downloaded). Along with supporting reading the
data, I plan to attempt to include a wide range of machine learning and artificial intelligence libraries and tools. Hopefully, it will be easy enough to integrate their database systems with the
ilang database. Once these components are in somewhat of a usable state, I should have some framework for which experiments with the code modification library can be conducted.
Random last thought:
I currently plan for the ilang system to have a formatting tool, much like golang does. The reason for this, is when working with the modification system, I plan to have the system completely
regenerate a file using the system “print from code tree” feature. This should greatly simplify writing the code back to disk, when compared to other possible ideas such as trying to find where the
code has been changed with corresponding lines and then try to recreate those changes on disk.
Job fair
Wednesday of this last week, I went to a EECS job fair. I found that essentially all companies there were eager to talk with anyone that came by and take a résumé. I have even gotten some contact
back from some companies already which I was not expecting as I am a first year student and I was told by many older students that first years do not typically get contacted or get internships/jobs.
I think that this point brings up some very interesting beliefs that are in the tech industry. Many of these have been noted before on countless blogs and new articles, but rehashing these from my
own experiences I believe might be helpful to some people.
1. The tech industry does not particularly care about you age, gender, race etc. All they care about is if you are technically skilled and are able to get the job done.
2. Github profiles are a big deal. At the top of my résumé along with my physical address, I decided to put my internet address. This included things such as my email, website and github profile.
I want to note that even while talking with an individual he was looking at my résumé said “o nice you have your link to your github profile” and then continued to circle it with his pen and said
that he was amazed how many people he talked to that did not have some public code profile. Today this “public code profile” has become a standard for hiring in the coding world.
3. Do not emphasis what you do not have when talking with the representatives. I was waiting behind some student who was talking with the hulu representatives about what he has. First he starts
out with what he does not like about the hulu product, the fact that there are ads even though he is paying for it (guess what you pay for Cable and there are still ads, there is no reason hulu
can’t do the same). The representatives then interrupts him and asks about what sort of projects he has. He states that he has made a few smallish things. The representatives then continues to
ask if he has a github (see point 2). Which he replies that he does, but there is nothing on there because…..some answer like, “my ideas are soooo great that I do not want people coping them I
might sell them at some point…..”
These are somewhat of tips/points/what not to do experiences. Like I said at the top, these ideas have been noted all over the internet and are not rocket science.
Additionally, in line with my last post about hackathon projects. Everything that you write should be version controlled somehow. You can use git without using github and just keep it on your local
machine. Additionally, when you decided that your code is either “done” or not going to continue into a money-making company, or only going to survive as a free product, then you might as well
create a public repo on github or similar so that if/when you are at a job fair, there is something on
The Hackathon paradigm
Today I was looking at a lot of the different applications that I normally use on my phone and through my web browser. If I was talking to someone who had never experienced either of these before,
they might believe that I generally have a single specific device for a specific task, and that in terms of functionality there would be little overlap of major features. However, for anyone that
has experienced either of these mediums, they are aware of the wide variety of applications and programs that duplicated the functions of other applications.
My complain is two-pronged on this issue of applications that start or continue with a Hackathon paradigm. First, the old Unix philosophy says do one thing and do one thing well. On this specific
point, I believe that many applications start out with the right intentions, however over time there is a significant feature creep effect that takes place. I believe that this is the result of
“Hackathon project” becoming more than “Hackathon projects.” The developer of these application feel that they are going to form a company with a project that in terms of complexity should really be
no more than a side project. Essentially what I am saying, is to develop and maintain your application X, it _might_ take 2 hours a week once it is launched. However, these individuals choose to
attempt to make a 50 hour, startup styled, work week out of these types of projects.
My second issue with the “Hackathon projects” is don’t assume that something that you can write in 24 hours is not easily copied. There are a lot of very complex and difficult problem that exist in
the world today. Nearly all of these types of problems can not be solved in a short period of time. Additionally, if a product can be made in 24 hours given the proper tools and skills, then it is
only a matter of time before there is a large number of people who will be competing with you. Some might even have better products given that they were able to replicate your product in such a
short period of time, and then vastly improve upon there after.
With these two issues, I am not saying that Hackathons are bad,Hackathons provide a relativity easy way to create demos to show of skills. However when it comes to publishing a product I believe
that people should think a little more about what they are going to create and invest enough time into the product such that it is not just going to be another 1 of 100 “identical” products.
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3D Visualization of Textures in Metals and Alloys
Mar 12
3D Visualization of Textures in Metals and Alloys
In a separate article, we described how one can use the neo-Eulerian orientation representations to create new 3D visualizations of the Rodrigues Fundamental Zones in homochoric, cubochoric, 3D
stereographic, Euler and Rodrigues-Frank orientation representations. In the present article, we use these visualization modes to illustrate how one can represent Orientation Distribution Functions
as 3D objects. All renderings available below were created using fortran-90 routines (part of release 3.1 of the EMsoft package) which generate input files for the open source PoVRay rendering
program. One rendering was performed with the Chimera visualization package.
For most of the Figures of this paper, we provide here two animations; one a regular animation of the object rotating a full circle, the other a red-blue anaglyph movie for which you will need a pair
of red-blue stereo glasses. Below are the figure captions for all the figures that have an associated animation; the link represented by M is the regular movie, A represents the anaglyph movie.
Figure 2: (a) Rodrigues space representation of the cube texture component, along with the outline of the Rodrigues fundamental zone for octahedral rotational symmetry [M, A]; (b) stereographic
projection of the cube (blue) and Goss (red) texture components[M, A]; (c) cubochoric representation of the Goss texture component[M, A]; (d) [M, A] and (f) [M, A] are Euler space representations of
the cube and Goss components, respectively, along with the mapping of the Rodrigues FZ; in (e), both components are represented in the conventional Euler fundamental zone [M, A].
Figure 3: 3D emission map renderings of the synthetic cube (a) [M, A] and Goss (b) [M, A] textures, showing all equivalent orientations in the standard Euler cell; regions of higher orientation
density appear as brighter light sources.
Figure 4: (a) Representation of the (4 pi,4 pi,4 pi) monoclinic Euler unit cell; orientations in the blueish octants map onto quaternion q, the others on quaternion -q. (b) equivalent atom positions
(q blue, -q red) along with regular (n) and time-reversing (n’) diagonal glide planes and the time-reversing translation vectors a‘/2, b‘/2 and c‘/2. (c) unit cell corresponding to the magnetic space
group P_cc outlined in purple; the symmetry elements of this space group are indicated by translation vectors. (d) perspective view of the full (4pi,4pi,4pi) unit cell along with the monoclinic cell
(outlined in purple) for the traditional representation with beta=90° [M, A] .
Figure 5: (a) [M, A] Euler space representation of the experimental cube texture, along with a traditional (100) pole figure in (b); (c) [M, A] 3D stereographic projection of the cube texture inside
the octahedral fundamental zone; (d) [M, A] magnification of the central portion of (c).
Figure 6: (a) [M, A] Euler space representation of the experimental Goss texture; (b) ; (c) [M, A] stereographic projection of the Goss texture inside the octahedral fundamental zone; (d) [M, A]
magnification of the central portion of (c).
Figure 9: (a) [M, A] Euler space representation of the Rene 88-DT texture; each unique orientation is represented by a small blue sphere in the octahedral Rodrigues FZ. (b) [M, A] 3D stereographic
projection of the Ren’e 88-DT texture represented as an emission map inside the octahedral RFZ.
Figure 12: (a) Rodrigues fundamental zones [M, A] for the octahedral and hexagonal rotational groups 432 and 622 as well as the 432-622 disorientation FZ (green wireframe), drawn on the same scale;
(b) 432-622 disorientation FZ [M, A]drawn separately, with the unique disorientations cell outlined in red (c) [M, A], and shown separately (d) [M, A]; (e) equivalent hexagonal Rodrigues fundamental
zones represented in a 3D stereographic projection [M, A]; note that the sphere is subdivided into 12 zones, with the vertical zones spanning the sphere surface and connecting on the other side. (f)
shows the 24 equivalent FZs for the octahedral rotational group 432 [M, A].
Figure 13: (a) [M] and (b) represent two frames of a movie (rendered using the Chimera package showing an iso-surface rendering of the alpha (pinkish) and beta (light blue) orientation distribution
histograms in 3D stereographic projection mode; the projection sphere is not shown. The vertical direction in (a) and (b) corresponds to the hexagonal six-fold rotation axis.
Figure 14: (a) Disorientation histogram for unique M, A] hexagonal Rodrigues fundamental zone scatter plot with the twelve equivalent BORs indicated by red spheres; (c) [M, A] and (d) [M, A] unique
disorientation cell scatter plots from two different viewing directions. Note the increased density of points near the BOR locations in (b)-(d).
Finally, here is a link to a zip file containing all the PoV-Ray scene description files for the movies above.
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gcc/testsuite/gcc.c-torture/execute/20000223-1.c - gcc - Git at Google
/* Copyright (C) 2000 Free Software Foundation, Inc.
Contributed by Nathan Sidwell 23 Feb 2000 <nathan@codesourcery.com> */
/* __alignof__ should never return a non-power of 2
eg, sizeof(long double) might be 12, but that means it must be alignable
on a 4 byte boundary. */
void check (char const *type, int align)
if ((align & -align) != align)
abort ();
#define QUOTE_(s) #s
#define QUOTE(s) QUOTE_(s)
#define check(t) check(QUOTE(t), __alignof__(t))
// This struct should have an alignment of the lcm of all the types. If one of
// the base alignments is not a power of two, then A cannot be power of two
// aligned.
struct A
char c;
signed short ss;
unsigned short us;
signed int si;
unsigned int ui;
signed long sl;
unsigned long ul;
signed long long sll;
unsigned long long ull;
float f;
double d;
long double ld;
void *dp;
void (*fp)();
int main ()
check (void);
check (char);
check (signed short);
check (unsigned short);
check (signed int);
check (unsigned int);
check (signed long);
check (unsigned long);
check (signed long long);
check (unsigned long long);
check (float);
check (double);
check (long double);
check (void *);
check (void (*)());
check (struct A);
return 0;
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Cylinder Volume Gallons Calculator
Cylinder Volume Gallons Calculator
Cylinder Volume Gallons Calculator
Understanding the Cylinder Volume Gallons Calculator
The Cylinder Volume Gallons Calculator is an essential tool for anyone needing to convert the capacity of cylindrical objects from cubic inches to gallons. This calculator helps you quickly determine
the volume of any cylinder in gallons by just entering its radius and height in inches. It is particularly useful in various fields such as construction, manufacturing, and domestic tasks.
Application of the Calculator
This calculator can be beneficial in numerous real-life scenarios. For instance, if you are working on a plumbing project and need to know how many gallons a cylindrical tank can hold, this tool will
provide that information instantly. Gardeners and farmers can use this calculator to figure out the capacity of cylindrical water storage containers, aiding them in efficient water usage and
management. It's also useful in engineering and design projects where accurate volume measurements of cylindrical components are needed.
How the Answer is Derived
The volume of a cylinder is calculated by multiplying the area of its base by its height. The base of the cylinder is a circle, and its area is found by squaring the radius (distance from the center
to the edge) and multiplying with pi (approximately 3.14159). After computing the volume in cubic inches, the result is converted to gallons using a specific conversion factor (1 cubic inch equals
approximately 0.004329 gallons).
Why Use This Calculator?
This calculator streamlines a process that would otherwise require manual computations and conversion. It ensures accuracy and saves time, allowing you to focus on other important aspects of your
project or task. The inclusion of an input field for both radius and height, along with helpful tooltips, makes it user-friendly and accessible to anyone, regardless of their math proficiency.
Relevant Information
Understanding the volume of a cylinder in gallons can be critical for proper storage and resource management. This calculator can also assist in ensuring safety and compliance with regulations that
might require precise measurements. Whether you are involved in DIY tasks, industrial projects, or agricultural activities, having an accurate understanding of the volume of cylindrical containers
can improve efficiency and decision-making.
Q: How accurate is the calculator?
A: The calculator provides a highly accurate conversion by using precise mathematical formulas and a constant pi value of approximately 3.14159. However, always consider any approximations in the
input values you provide.
Q: Can I use the calculator for cylinders with different units?
A: The calculator is designed to work with inches for input values. To convert from other units, you need to first convert them to inches before entering them into the calculator.
Q: How does the calculator handle large numbers?
A: The calculator can process very large numbers efficiently as it is built to manage substantial calculations. Just ensure that the inputs are within a reasonable range to prevent any overflow
Q: Why do I need to input the radius and height?
A: To calculate the volume of a cylinder accurately, both the radius of the base and the height of the cylinder are essential. These two parameters help in determining the overall cubic capacity
Q: How is the conversion factor from cubic inches to gallons determined?
A: The conversion factor is based on the standard measurement where 1 cubic inch equals approximately 0.004329 gallons. This constant is used to convert the computed volume from cubic inches to
Q: Is the calculated volume affected by the shape of the cylinder's base?
A: The volume calculation assumes a perfect cylindrical shape with a circular base. Any deviations in actual shape may affect the precise volume, but the calculator provides an ideal theoretical
volume based on circular geometry.
Q: What should I do if I get an unexpected result?
A: Ensure that the radius and height values are correctly entered and recheck for any input errors. If the problem persists, try recalculating manually to verify if the calculator's result aligns
with your computations.
Q: Can this calculator be used for scientific experiments?
A: Yes, the calculator can be used in scientific contexts where precise volume measurements are required. It's particularly useful for experiments involving fluid capacities and storage.
Q: Is there a way to save or print my calculations?
A: Depending on the interface of your website, you might have options to print or save your results. You can also manually record the results for future reference.
Q: How can I improve the accuracy of my measurements?
A: Use precise measuring tools to determine the radius and height of the cylinder. Ensure the measurements are as accurate as possible to get the best results from the calculator.
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Continuous Random Variables and their Distributions
4.1.0 Continuous Random Variables and their Distributions
We have in fact already seen examples of continuous random variables before, e.g., Example 1.14. Let us look at the same example with just a little bit different wording.
I choose a real number uniformly at random in the interval $[a,b]$, and call it $X$. By uniformly at random, we mean all intervals in $[a,b]$ that have the same length must have the same probability.
Find the CDF of $X$.
• Solution
□ As we mentioned, this is almost exactly the same problem as Example 1.14, with the difference being, in that problem, we considered the interval from $1$ to $2$. In that example, we saw that
all individual points have probability $0$, i.e., $P(X=x)=0$ for all $x$. Also, the uniformity implies that the probability of an interval of length $l$ in $[a,b]$ must be proportional to its
length: $$P(X \in [x_1,x_2]) \propto (x_2-x_1), \hspace{20pt} \textrm{where }a \leq x_1 \leq x_2 \leq b.$$ Since $ P(X \in [a,b])=1$, we conclude $$P(X \in [x_1,x_2]) =\frac{x_2-x_1}{b-a}, \
hspace{20pt} \textrm{where }a \leq x_1 \leq x_2 \leq b.$$ Now, let us find the CDF. By definition $F_X(x)=P(X \leq x)$, thus we immediately have $$F_X(x) =0, \hspace{20pt} \textrm{for } x <
a,$$ $$F_X(x) =1, \hspace{20pt} \textrm{for } x \geq b.$$ For $a \leq x \leq b$, we have
$F_X(x)$ $=P(X \leq x)$
$=P(X \in [a,x])$
Thus, to summarize $$\hspace{70pt} F_X(x) = \left\{ \begin{array}{l l} 0 & \quad \textrm{for } x < a \\ \frac{x-a}{b-a} & \quad \textrm{for }a \leq x \leq b\\ 1 & \quad \textrm{for } x > b \
end{array} \right. \hspace{70pt} (4.1)$$ Note that here it does not matter if we use "$ < $" or "$ \leq $", as each individual point has probability zero, so for example $P(X < 2)=P(X \leq 2)
$. Figure 4.1 shows the CDF of $X$. As we expect the CDF starts at zero and ends at $1$.
Fig.4.1 - CDF for a continuous random variable uniformly distributed over $[a,b]$.
One big difference that we notice here as opposed to discrete random variables is that the CDF is a continuous function, i.e., it does not have any jumps. Remember that jumps in the CDF correspond to
points $x$ for which $P(X=x) > 0$. Thus, the fact that the CDF does not have jumps is consistent with the fact that $P(X=x)=0$ for all $x$. Indeed, we have the following definition for continuous
random variables.
A random variable $X$ with CDF $F_X(x)$ is said to be continuous if $F_X(x)$ is a continuous function for all $x \in \mathbb{R}$.
We will also assume that the CDF of a continuous random variable is differentiable almost everywhere in $\mathbb{R}$.
The print version of the book is available on Amazon.
Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI
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