content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
I AM PISSED AGIAN!!!!!!!!
FOR GOD'S SAKE ...LOWER YOUR VOICE! I FIND MY PEOPLE TO BE SPEAKING IN HIGHER PITCHER THAN NECESSARY WHEN THEY HAVE TO...! My roommates(all somalis) are tired of me when i tell them to lower their
voices in the library or any where else on campus. I am growing to be sick of that sh*t, and at times, I avoid them knowing that they are too loud.
Firstly, we British wallahs would read the title of your post as, 'I am drunk again!!!!'. Secondly, how is your state of agitation relevant to the Camel Milk Debate? Is this up for intellectual
debate or are u just blowing off steam? Thirdly, keep it down, will ya?
i love shouting the odds...especially when i'm talking on the phone and in the bus. i love to see people getting annoyed all because of little me..whoopie...swells me with joy to know i have made yet
another person's day a misery...drink up mate!
it's in our blood...well in mine at least, who cares about yous
In andOut
U and u'r friends, u all needs to consult an Audiologist just to make sure that none is wrong with u'r ears or simply pluck u'r ears with wax or even better chewing gum will do u good.
other ppl actions shouldn't bother u so much....besides who made u the boss :rolleyes: ...reading ur post u were doing some shounting urself....besides.....lighten up...what they do is thier
bizznaz...just worry about urself....
Originally posted by besbaaso:
other ppl actions shouldn't bother u so much....besides who made u the boss :rolleyes: ...reading ur post u were doing some shounting urself....besides.....lighten up...what they do is thier
bizznaz...just worry about urself....
I neva said anyone or anything bothers me but he did ask for and i'm just saying yours for the asking.
As for who made me the boss,i say the supreme court, i guess u can blame it on the hanging chads and u might as well go ahead and ask for a recount.
You might think what they do is their business but sometimes it might become all somali students' business...like when your school administration sends a complaint letter to the "SOMALI STUDENT
ASSOCIATION". Some Somalis are just loud and inconsiderate.
Abdilatif-lool. Your school got mad problems :eek: yeah sometimes some somalis do misbehave-thank god school is out
sheyhem.....i wasn't writing about u.....but the person who started the topic.....chill bro...
as for being the boss.....the only person in ur jurisdiction is urself... :rolleyes:
Walahi I thought SOL had a alcoholic in it's mist. pheww!
And there I was bracing myself for SOL's version of AA.
"Hi, my name is InAndOut and I'm aN alcoholic"
Originally posted by Sue:
Walahi I thought SOL had a alcoholic in it's mist. pheww!
And there I was bracing myself for SOL's version of AA.
"Hi, my name is InAndOut and I'm aN alcoholic"
I just dont see what you are getting at with this Alcoholic thing?? why you gotta attack someone who didnt disrespect you at all??
wooooooooow i love when i see my people talking loud. that is the only way that i can't tell the different between as somalis and the rest of the african's
I'm sorry AJ I wasn't trying to attack anyone walahi. I just thought it was funny, because pissed means drunk in the UK. Sorry obviously what I think is funny is not so funny for you. I'm sorry if I
ofended anyone,I seriously did not mean too. | {"url":"https://www.somaliaonline.com/community/topic/18671-i-am-pissed-agian/","timestamp":"2024-11-15T04:28:44Z","content_type":"text/html","content_length":"277802","record_id":"<urn:uuid:1a0323a4-da94-41bc-b4b3-81f4d869a088>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00473.warc.gz"} |
Universal Hashing
Written by Mike James
Friday, 27 January 2023
Hashing is a fun idea that has lots of unexpected uses. Here we look at a novel type of hash function that makes it easy to create a family of universal hash functions. The method is based on a
random binary matrix and is very simple to implement.
Put simply you give a hash function an item of data x and it returns a number h(x).
You can use this number for all sorts of things but in general a hash function has a number of desirable properties.
• The first is that h(x), the hash value should be smaller in the sense it takes less storage than the original data x.
• The second is that if you have a a set of data items then h(x) should spread the data items out over the range of possible hash values i.e. the relationship between x and h(x) should not be too
Typically hash functions are used for storage, store x in Data[h(x)], or for security, if h(x) changes then x has been changed.
If you are using a hash function for security then you need a higher grade of hash function - a cryptographic hash. The hash function described in the example can be extended to a cryptographic hash
but in its current form it is suitable for use as a storage hash.
Notice that as a hash function "condenses" down an item of data to a smaller number of possible hash values it is not unique. That is, for any hash function there will be usually quite a lot of other
data values, y say, for which h(y)=h(x). This is often called a collision and it is perfectly natural for a hash function and all hashing algorithms have to deal with the situation in one way or
In this article the focus is on hash functions used for data storage and retrieval.
Families of hash functions
In the early days of hashing you generally just needed a single good hash function. Today things are getting increasingly complex and you often need whole families of hash functions.
Once such family is the Universal Hash. This is a set of hash functions with an interesting additional property. First we need to look at the problem that this additional property is designed to
Consider a hash storage scheme based on storing x in a location given by h(x) which ranges from 0 to N-1.
What is the worse thing that can happen?
The answer is that you are given N things to store that all map to the same hash value - i.e. you try and store everything in the same location and every access involves a collision.
You can prove quite easily that for any hash function it is possible to find a dataset that provides this worst case.
For example if the number of things you could ask to store is greater than N*N then if you attempt to store N*N data elements in the array with only N storage locations it is obvious that each
location will store N data elements i.e. there are going to be N collisions. Just pick the data mapped to the same location and you have your worst case dataset.
It is a simple consequence of mapping a big set to a small number of locations - there have to be collisions - and you can find any number just by filling the storage and keeping going.
So can you protect your hashing scheme against this attack?
Yes you can by having a family of hashing functions H that you randomly select from before starting the algorithm. In this case any worst case set of data that some one has selected has only a
probability of being worst case for the hash function selected.
If the family of hash functions is such that given x and y then the probability that h(x)=h(y) for a hash function drawn randomly from m possible hash functions is 1/m then the family is called a
universal hash family.
Universal hash families are particularly useful for algorithms that need multiple hash functions or which need the data structure to be rebuilt if too many collisions occur (look out for Cuckoo
hashing coming soon).
So we need an example of a universal hash family.
There are standard examples of universal hash functions created using the usual "multiplication mod a prime" - i.e. similar to congruential generators.
However, there is a little known method based on using a random matrix. It has lots of advantages - it's a universal family, it performs well, it's easy to understand and it's quick to compute.
The idea is very simple.
Suppose you have an input data item that you have input data with m bits and you want a hash function that produces n bits then first generate a random nxm binary matrix M. The hash function simply
consists of working out
where you consider x to be a binary vector.
For example, suppose you have a four bit input value and want to generate a three bit hash value. Then generating a random matrix gives say:
( 0 1 0 0 )
M= ( 1 0 1 1 )
( 1 1 0 1 )
and if the data value was 1011 the hash value would be computed as:
( 0 1 0 0 )(1) (0)
Mx= ( 1 0 1 1 )(0) = (1)
( 1 1 0 1 )(1) (0)
or in other word h(1011)=M(1011)=010.
If you find the math difficult to follow it might help to be reminded that in binary (or mod 2) arithmetic 1x1=1, 0x0=0 and 1x0=0, also 0+0=0, 0+1=1 and 1+1=0.
There are a number of other ways to look at the way the arithmetic is done that suggest different ways of implementing the algorithm.
The first is to notice that what you are doing is anding each row with the data column vector. That is taking the second row as an example:
( 1 0 1 1 )And (1 0 1 1) = (1 0 1 1)
and then you add up the bits in the result:
Adding up the bits in the result can also be interpreted as the parity function because the result is zero if there are an even number of ones and one if there are an odd number of ones. Notice also
that this involves m Ands and m parity determinations.
The second way of looking at the arithmetic is to notice that the multiplication picks out the columns of M corresponding to where the data has a one and then does a bitwise addition or exclusive or.
For example:
( 0 1 0 0 )(1) (0)
Mx= ( 1 0 1 1 )(0) = (1)
( 1 1 0 1 )(1) (0)
picks out the first, third and fourth columns of M and adds or exclusive ors them together:
( 0 + 0 + 0 ) (0)
( 1 + 1 + 1 ) = (1)
( 1 + 0 + 1 ) (0)
Notice that this might involve as many as m columns and probably m iterations to form the result. As m>n the first method is worth looking at more closely.
There may be other ways to interpret the arithmetic as logical operations but these two are the most useful. Which is best depends on the hardware available. For example, if the machine you are
working with has a fast way to determine the parity of a word then the first method should work well.
An implementation
To demonstrate how the idea works let's create a small C# class that implements both approaches to the random binary matrix hash.
The code is easy enough to convert into any other language.
First we need to create the random matrix. Instead of working with a bit array it makes more sense to use an Int32 for each row of the array and assume that the input data is an int32. For simplicity
we can use Int32 and just use the lower 31 bits to give a positive number range for the data.
So start a new C# project and add a new class complete with random number generator and Int32 array ready to hold the rows of the bit matrix:
class Mhash
Random R = new Random();
Int32[] hashmatrix;
Int32 M=0;
The constructor creates the random matrix with m rows and stores m for other methods to make use of:
public Mhash(Int32 m)
hashmatrix= new Int32[M];
for (int i = 0; i < M; i++)
hashmatrix[i] = R.Next();
Once the constructor is finished we can use it to create a hash object capable of taking a positive Int32 and returning an m bit hash.
The next step is to create the method that does this job:
public Int32 Hash1(Int32 x)
Int32 hash=0;
for (int i = 0; i < M; i++)
hash = hash | parity(x & hashmatrix[i]);
return hash;
The method takes each row of the matrix and ands it with the data. The result is then converted by the parity function into a 0, for even parity or a 1, for odd parity. Don't worry how parity works
for the moment. At the end of the routine we have an m bit hash to return.
If you want to write the method a little more compactly you could write the body of the for loop as:
for (int i = 0; i < M; i++)
hash <<= 1;
hash |= parity(x & hashmatrix[i]);
Now to try it out all you have to do is:
Mhash hashobj=new Mhash(8);
Int32 h1 = hashobj.Hash1(x);
and you have an 8-bit hash for any positive Int32 you care to supply.
Last Updated ( Friday, 27 January 2023 ) | {"url":"https://www.i-programmer.info/programming/72-theory/2664-universal-hashing.html","timestamp":"2024-11-06T02:17:11Z","content_type":"text/html","content_length":"36491","record_id":"<urn:uuid:11634dfb-bfca-4b3e-841b-ff4dc3c7261b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00720.warc.gz"} |
Quantum Computing: Unleashing the Power of Qubits
yuj- Top UX Design Agency in USA & India | Quantum Computing: Unleashing the Power of Qubits
Quantum computing is a revolutionary field that harnesses the principles of quantum mechanics to perform computations far beyond the capabilities of classical computers. In this article, we’ll
explore the fundamental concepts of quantum computing, from qubits to applications and limitations, shedding light on its transformative potential.
What is Physics?
Physics is the study of the fundamental laws governing the universe. It seeks to understand the behavior of matter, energy, and forces, providing the framework for exploring everything from the
smallest particles to the vast cosmos.
Branches of Physics
Physics branches into several subfields, including:
• Classical Mechanics: Describes the motion of macroscopic objects using Newtonian principles.
• Quantum Mechanics: Focuses on the behavior of particles at the atomic and subatomic levels, introducing concepts that challenge our classical intuitions.
What is Mechanics?
Mechanics is a branch of physics that focuses on motion and the forces that act on objects. It encompasses both classical and quantum mechanics, offering insights into a wide array of physical
Types of Mechanics
1. Classical Mechanics:
• Describes everyday phenomena.
• Based on Newton’s laws of motion.
• Predicts the behavior of macroscopic objects.
2. Quantum Mechanics:
• Applies to particles at the atomic and subatomic scales.
• Involves wave functions, probabilities, and uncertainty.
• Challenges our classical intuition.
The Atom and Classical Computers
The Atom: Atoms are the building blocks of matter, governed by the principles of quantum mechanics. Electrons occupy discrete energy levels around the nucleus, contributing to the unique properties
of each element.
Classical Computers: Classical computers utilize bits (0s and 1s) for computation, processing information sequentially. Physical constraints, such as the size and speed of transistors limit them.
Introducing Quantum Computers
Qubits: Quantum bits (qubits) are the fundamental units of quantum information. Unlike classical bits, qubits can exist in superpositions (0, 1, or both), enabling parallel computation.
Superposition: Qubits can represent multiple states simultaneously, allowing quantum computers to explore vast possibilities at once, greatly enhancing computational power.
Entanglement: When qubits become entangled, their states become correlated. Changes in one qubit instantly affect the other, regardless of distance, enabling powerful quantum algorithms that can
solve complex problems more efficiently.
Applications of Quantum Computing
• Cryptography: Quantum computers have the potential to break classical encryption methods (e.g., RSA). Consequently, quantum-resistant algorithms are being developed to safeguard data.
• Optimization: They can solve intricate optimization problems, such as route planning and supply chain logistics, using techniques like quantum annealing and adiabatic algorithms.
• Drug Discovery: Quantum computing can simulate molecular interactions, significantly accelerating drug development and leading to new medical breakthroughs.
Limitations and Quantum Supremacy
Limitations: Despite their promise, quantum computers face significant challenges. Qubits are fragile and susceptible to decoherence, while error correction remains an ongoing hurdle. Currently,
practical quantum computers are still in their infancy.
Quantum Supremacy: Quantum supremacy is achieved when a quantum computer performs calculations beyond the reach of classical supercomputers. Notably, Google’s Sycamore reached this milestone in 2019,
marking a pivotal moment in computing history.
Quantum computing holds immense promise, poised to revolutionize industries and reshape our understanding of computation. While significant hurdles remain, continued research and innovation in this
field are expected to yield breakthroughs that will transform technology as we know it.
Unlock the Future of Design with Us! At yuj, a top UX design agency, we leverage cutting-edge technologies like quantum computing to create innovative solutions that drive success for our clients.
Let’s embark on this journey together! Talk to us today. | {"url":"https://medium.com/yuj-designs-blogs/quantum-computing-unleashing-the-power-of-qubits-f6f23ae70136","timestamp":"2024-11-07T15:45:03Z","content_type":"text/html","content_length":"119225","record_id":"<urn:uuid:cb2ab132-bce3-4776-acb7-bd0e33abb605>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00361.warc.gz"} |
Hyperons in Finite and Infinite Nuclear Systems
Istituto Nazionale di Fisica Nucleare, Dipartimento di Fisica “Ettore Majorana”, Università di Catania, Via Santa Sofia 64, I-95123 Catania, Italy
Submission received: 6 September 2021 / Revised: 30 September 2021 / Accepted: 6 October 2021 / Published: 9 October 2021
In this work, we shortly review the role and properties of hyperons in finite and infinite nuclear systems such as hypernuclei and neutron stars. Particularly, we describe different production
mechanisms of hypernuclei, discuss some aspects of their $γ$-ray spectroscopy and their weak decay modes, and give a few strokes on their theoretical description. We reexamine also the role played by
hyperons on the properties of neutron and proto-neutron stars with a special emphasis on the well-known “hyperon puzzle”, of which we discuss some of the solutions that have been proposed to tackle
this problem. Finally, we review the role of hyperons on the cooling properties of newly born neutron stars and on the so-called r-mode instability.
1. Introduction
The presence of strange baryons, commonly known as hyperons, in hypernuclei and neutron stars permits the study of baryon interactions from an enlarged perspective, and the extension of our present
knowledge of conventional nuclear physics to the SU(3)-flavor sector [
]. Hypernuclei, bound systems composed of neutrons, protons and one or more hyperons, were first observed in 1952 by Danysz and Pniewski in a balloon-flown emulsion stack where a hyperfragment was
discovered [
]. Pion and proton beam production in emulsions and later in
He bubble chambers, where single-
hypernuclei were identified from the weak decay of the
hyperon into a proton and a
$π −$
, followed these initial cosmic-ray observations of hypernuclei. The advent of separated
$K −$
beams, which permitted the realization of counter experiments, lead to more systematic investigations of hypernuclei. A considerable amount of hypernuclear features, such as, e.g., the small
spin-orbit strength of the hyperon–nucleon (YN) interaction or the fact that the
essentially retains its identity inside the nucleus, were revealed by in-flight (
$K − , π −$
) counter experiments carried out at CERN and at Brookhaven National Laboratory (BNL). Experiments using
$( π + , K + )$
$( K s t o p p e d − , π 0 )$
reactions were conducted later at the Brookhaven AGS and KEK accelerators with higher intensities and improved energy resolution of the beams. A high-precision tool for the study of
-hypernuclear spectroscopy with resolutions of several hundred keV [
] is provided by the electromagnetic production of hypernuclei through the reaction
$( e , e ′ K + )$
carried out at the Thomas Jefferson National Laboratory (JLAB) and the Mainz Microtron Accelerator (MAMI-C). A promising new way to produce hypernuclei by using stable and unstable heavy-ion beams
was proposed a few years ago by the HypHI Collaboration at FAIR/GSI [
], and it has recently allowed the observation of the
hyperon, and the
$Λ 3$
H and
$Λ 4$
H hypernuclei in a first experiment using a
Li beam on a
C target at 2 AGeV [
]. Today, thanks to the use of high-energy accelerators and modern electronic counters, more than 40 single-
hypernuclei, and a few double-
and single-
ones have been identified. The existence of single-
hypernuclei, however, has not been experimentally confirmed yet without ambiguity, suggesting that the
N interaction is most likely repulsive.
In addition to hypernuclei, a big interest has being put in the study of hyperonic matter (nuclear matter with nucleonic and hyperonic degrees of freedom), especially in connection with the physics
of neutron star interiors [
]. The density in the interior of neutron stars is large enough to allow for the appearance of new particles with strangeness content besides the conventional nucleons and leptons by means of weak
interaction processes. Hyperons are expected to appear in neutron stars at around twice normal nuclear matter saturation density
$ρ 0 = 0.16$
$− 3$
. Neutron star properties are closely related to the underlying Equation of State (EoS) of matter at high densities. Therefore, despite the fact that hypernuclear matter is an idealized system, the
theoretical determination of its EoS is an essential step towards the understanding of those neutron star properties which can be affected by the presence of hyperons. It is well known that the
presence of hyperons softens the EoS and reduces the mass of neutron stars (see, e.g., [
]). In addition, hyperons can strongly influence also the thermal evolution and gravitational instabilities of these objects. The presence of hyperons, for instance, can modify the neutrino
emissivity of dense matter and it can allow also for additional cooling mechanisms. Furthermore, hyperons dominate the bulk viscosity of matter as soon as they appear in the neutron star interior.
Consequently, the emission of gravitational waves in hot and rapidly rotating neutron stars due to the so-called r-mode instability is affected also by their presence. Conversely, further constraints
on the YN and hyperon–hyperon (YY) interactions can be provided by comparing the theoretical predictions for these properties with astrophysical observations.
A detailed knowledge of the EoS of hypernuclear matter over a wide range of densities is required to understand better the effect of hyperons on neutron stars. However, this is a very hard task. Two
types of approaches have been traditionally used to describe baryon interactions in the nuclear medium and, to construct from them the (hyper)nuclear EoS: phenomenological and microscopic approaches.
Relativistic or non-relativistic phenomenological approaches are based on effective density-dependent interactions which contain typically a certain number of parameters that are adjusted to
reproduce (hyper)nuclear observables, and neutron star properties. Among the most commonly used ones, we can mention Skyrme-type interaction models and relativistic mean field (RMF) models. Several
authors have used density-dependent baryon–baryon interactions based on Skyrme-type forces including hyperons to derive phenomenological EoSs of hyperonic matter [
]. Properties of nuclei and the experimental data from hypernuclei are employed within this approach to fix the in-medium nucleon–nucleon (NN), YN and YY interactions. RMF models are based on
effective Lagrangian densities in which baryon–baryon interactions are described in terms of meson exchanges. A RMF description of the EoS of dense matter with hyperons is turning out today to be one
of the most popular ones (see, e.g., [
]). The parameters of RMF models are usually determined by using, in the case of the nucleons, the properties of nuclei and nuclear bulk matter, and by employing symmetry relations and hypernuclear
observables to fix the coupling constants of the hyperons with the mesons. The Quark-Meson-Coupling (QMC) model has also been employed to determine the EoS of (hyper)nuclear matter and the properties
of neutron stars [
]. In the QMC model, baryons are treated as confined non-overlapping bags of three quarks where the interaction is modeled through the exchange of mesons between quarks from different bags, which
are, in turn, modeled using the MIT bag model. The in-medium properties of hyperons have been also studied within the Non Linear Derivative (NLD) model, an alternative RMF approach which incorporates
an explicit momentum dependence of the in-medium baryon optical potentials [
]. Microscopic approaches, on the other hand, are based on realistic two-body baryon–baryon interactions that describe the scattering data in free space. These interactions have been mainly
constructed within the framework of the meson-exchange theory [
], although a new approach based on chiral perturbation theory has recently emerged as a powerful tool [
]. To obtain the EoS, one has to solve then the very complicated many-body problem. The main difficulty of this problem lies in the treatment of the repulsive core which dominates the short-range
behavior of the baryon interaction. Different microscopic many-body methods have been extensively used to study nuclear matter, however, very few of them have been extended to the hypernuclear
sector. To the best of our knowledge, the many-body methods extended to the strange sector include the Brueckner–Hartree–Fock (BHF) approximation [
] of the Brueckner–Bethe–Goldstone theory, the Hartree–Fock theory based on the soft V
$l o w k$
interactions [
] and the Dirac–Brueckner–Hartree–Fock (BHF) theory [
]. The Auxiliary Field Diffusion Monte Carlo method [
] was also extended to the hyperonic sector a few years ago. Very recently BHF calculations of hyperonic matter using YN interactions derived within SU(3) chiral effective field theory have also been
done by the Jülich–Bonn–Munich group [
] and Kohno [
This work is not intended to be an exhaustive review but rather a short introduction to the several aspects of hypernuclear physics. Here, we briefly review the properties of hyperons in hypernuclei
and neutron stars. Particularly, in
Section 2
, we shortly discuss different production mechanisms of hypernuclei, as well as some aspects of their
-ray spectroscopy, their weak decay modes, and their theoretical description. In
Section 3
, we reexamine the role played by hyperons on the properties of neutron and proto-neutron stars with a special emphasis on the so-called “hyperon puzzle” and present some of the possible solutions
that have been proposed to tackle it. We also reexamine in this section the effect of hyperons on the cooling properties of newly born neutron stars and on the development of the so-called r-mode
instability. The manuscript is finished in
Section 4
with a summary.
2. Hypernuclear Physics in a Nutshell
Compared to the NN interaction, the YN and YY ones are still poorly constrained due, mainly, to the scarce amount of YN scattering data and to the complete absence of them in the YY case. The reason
of this scarce amount of data should be traced back to the experimental difficulties associated with the short lifetime of hyperons and the low-beam intensity fluxes. In addition to the scattering
data, information on the YN and YY interactions can be obtained, using the so-called femtoscopy technique, by measuring the correlations (in momentum space) of Yp and YY pairs produced in heavy-ion
collisions [
]. The ratio of the distribution of relative momenta
$k → * = ( p → 1 − p → 2 ) / 2$
between a correlated and uncorrelated pair defines the correlation function of a given baryon pair. If the interaction of a baryon pair is attractive, then the measured correlation function will be
found to be larger than one. Conversely, if the interaction of the pair is repulsive, the correlation function will take values between zero and one. The correlation function between two baryons can
be theoretically expressed as [
$C ( k → * ) = ∫ d 3 r → | ψ ( k → * , r → ) | 2 S ( r → ) ,$
$ψ ( k → * , r → )$
is the relative wave function of the baryon pair of interest and
$S ( r → )$
is the so-called source function, and it represents the distribution of the distance
$| r → |$
at which the particles are emitted. The comparison between the theoretical correlation function and the measured one permits the testing and improvement of the existing YN and YY potentials.
Lattice QCD offers also a very powerful way to derive baryon–baryon interactions, see, e.g., [
] for reviews. A big progress in this direction has been made in the last years by the HALQCD [
] and the NPLQCD [
] collaborations. We should note, however, that the methods employed by these two collaborations are quite different. Whereas the HALQCD collaboration follows a method to extract the different
baryon–baryon potentials from the Nambu–Bethe–Salpeter wave function measured on the lattice, the NPLQCD collaboration combines calculations of correlation functions at several light-quark-mass
values with low-energy effective field theory (EFT). This second approach is particularly interesting because it allows the matching of lattice QCD results with low-energy EFT providing, in this way,
the means for first predictions in the physical quark mass limit. Results for various NN, NY and YY interaction channels at a single value of the lattice spacing and of the lattice volume have been
recently obtained by the HALQCD collaboration which managed to approach the region of physical masses [
]. Very recently, the NPLQCD collaboration has studied the interaction between two octet baryons for strangeness
$S = 0 , − 1 , − 2 , − 3$
$− 4$
at low energies using larger-than-physical quark masses corresponding to a pion mass of
$m π ∼ 450$
MeV and a kaon mass of
$m K ∼ 596$
MeV, and have extracted the corresponding values of
-wave scattering phase shifts, low-energy scattering parameters, and binding energies of two-baryon bound systems [
]. A detailed review of the last lattice QCD developments in the strangeness sector is beyond the scope of the present paper and, therefore, the interested reader is referred to the original works of
both the HALQCD and the NPLQCD collaborations for further information.
Alternative and complementary information on the YN and YY interactions can be extracted from the study of hypernuclei. The main aim of hypernuclear physics [
] is, in fact, to relate hypernuclear observables with the underlying YN and YY interactions. In this section, we briefly review the different production mechanisms of hypernuclei, discuss some
aspects of their
-ray spectroscopy and their weak decay modes, and we finish with a few strokes on their theoretical description.
2.1. Production of Hypernuclei
Several reactions can be used to produce single-
hypernuclei. One of them is the so-called
$( K − , π − )$strangeness exchange reaction
$K − + A Z → Λ A Z + π − ,$
where a neutron of the nucleus target hit by a
$K −$
is changed into a
that remains bound to the nucleus and a
$π −$
is emitted. With this reaction, it is possible to determine accurately the mass and the binding energy of the formed hypernucleus by measuring the momenta of both the incoming
$K −$
and the outgoing
$π −$
using two magnetic spectrometers with good energy resolution. Strangeness exchange reactions, initially performed at CERN, have been mainly used later at BNL in the USA, and at KEK and J-PARC in
The so-called
$( π + , K + )$associated production reaction
is another production mechanism of hypernuclei that makes use of
$π +$
beams instead of
$K −$
$π + + A Z → Λ A Z + K + .$
Here, an $s s ¯$ pair is created from the vacuum, and a $K +$ and a $Λ$ are produced in the final state. The production cross section of this reaction is smaller than that of the $( K − , π − )$ one,
but this is compensated by the fact that the intensities of the $π +$ beams are larger than those of the $K −$ ones. The production of hypernuclei by means of these reactions have been also performed
at BNL and KEK, and later at GSI in Germany.
The use of electron beams with an excellent spatial and energy resolution has allowed the
of hypernuclei by means of the
$( e , e ′ K + )$
$e − + A Z → e − + K + + Λ A ( Z − 1 ) ,$
providing in addition a high-precision tool for the study of hypernuclear spectroscopy, with energy resolutions of several hundred keV [
]. Electroproduction of hypernuclei is carried out at JLAB in the USA and the MAMI-C laboratory in Germany. These two laboratories are presently the only ones with the instrumental capabilities
required to perform these kind of experiments.
The kinematics of the elementary processes
$n ( K − , π − ) Λ$
$n ( π + , K + ) Λ$
$p ( γ , K + ) Λ$
underlying the three production mechanisms of single
-hypernuclei discussed above is shown in
Figure 1
, adapted from [
]. Note that the momentum transferred to the
is much lower in the case of the
$n ( K − , π − ) Λ$
reaction than in the other two. The lower the momentum transferred to the
is, the larger its probability of interacting with, or being bound to, the nucleus will be. In addition, the lower the momentum transferred is, the smaller the angular momentum transfer will also be
and, consequently, the
will more easily retain the quantum numbers of the nucleon that has been eliminated in the reaction. Therefore, in the case of the
$n ( π + , K + ) Λ$
$p ( γ , K + ) Λ$
reactions, since the recoil momentum of the hyperon is high, the cross sections to bound states are reduced, and the produced
has a higher probability of escaping the nucleus.
Similar reactions can be used to produce single-
hypernuclei. However, as we have already said, their existence has not been yet experimentally confirmed without ambiguity. The production of double-
hypernuclei requires a two-step mechanism in which, first, a
$Ξ −$
is created by means of reactions such as
and second, the
$Ξ −$
is captured in an atomic orbit and interacts with the nuclear core, producing two
’s by hitting one of the protons of the nuclei
This process releases an energy of about 30 MeV that, in most of the cases, is equally shared between the two
’s, leading to the escape of one or both hyperons from the nucleus.
hypernuclei are currently the best systems to investigate the properties of the baryon–baryon interaction in the strangeness
$S = − 2$
sector. The
$Λ Λ$
bond energy
$Δ B Λ Λ$
in double-
hypernuclei can be experimentally determined by measuring of the binding energies of double and single-
hypernuclei as
$Δ B Λ Λ = B Λ Λ ( Λ Λ A Z ) − 2 B Λ ( Λ A − 1 Z ) .$
A few double-
$Λ Λ 6$
$Λ Λ 10$
Be and
$Λ Λ 13$
B, have been reported in emulsion experiments. A quite large value of the
$Λ Λ$
bond energy of around 4 to 5 MeV was deduced from the subsequent analysis of these emulsion experiments. We should also note that the identification of some of these double-
hypernuclei was ambiguous. Therefore, careful attention should be paid when using the data from this old analysis to put any kind of constraint on the
$Λ Λ$
interaction. However, a new
$Λ Λ 6$
He candidate with a
$Λ Λ$
bound energy
$Δ B Λ Λ = 1.01 ± 0 . 2 − 0.11 + 0.18$
MeV (recently corrected to
$Δ B Λ Λ = 0.67 ± 0.17$
MeV) was observed without ambiguity in 2001 at KEK [
The reactions (
) and (
) can be used to produce single-
hypernuclei and, in fact, a few of them have been identified. The analysis of the experimental data from reactions such as
$K − , K +$
$Ξ − 12$
Be [
] indicates an attractive
-nucleus interaction of
$∼ − 14$
MeV. Recently, however, Friedman and Gal [
] have analyzed several
$Ξ − p → Λ Λ$
two-body capture events in
C and
Ni emulsion nuclei, concluding that the
-nuclear interaction is strongly attractive, with a
$Ξ −$
potential depth in nuclear matter
$V Ξ − ≥ 20$
MeV. We should mention here the observation [
] of a deeply bound state of the
$Ξ −$
N system with a binding energy of
$3.87 ± 0.21$
MeV [
]. This event provides the first clear evidence of a deeply bound state of this system by an attractive
N interaction. The latest experimental data obtained by the J-PARC E07 collaboration [
] indicate a value for the binding energy of the
$Ξ −$
in the
$Ξ −$
N system of
$1.27 ± 0.21$
MeV. Future
hypernuclei experiments are being planned at J-PARC.
2.2. $γ$-ray Spectroscopy of Hypernuclei
Excited states of hypernuclei can be produced when a nucleon in a
or a higher shell is replaced by a hyperon. The energy of the hypernuclear excited states can be released either by emitting nucleons or, sometimes,
-rays. The analysis of hypernuclear excited states with very good energy resolution has been possible thanks to the detection of
-ray transitions in single-
hypernuclei. The construction of large-acceptance germanium detectors, dedicated to hypernuclear
-ray spectroscopy, has overcome some of the initial technical difficulties found in the application of
-ray spectroscopy to hypernuclei. These difficulties were mostly associated with the detection efficiency of
-ray measurements and with the necessity of covering a large solid angle with
-ray detectors. Several weak points in hypernuclear
-ray spectroscopy, however, still persist such as, for instance, the fact that the observation of
-rays is mostly limited to the low excitation region, maybe up to the
-shell. The reason is that a number of single-particle
states are bound in heavy
hypernuclei with a potential depth of ∼28 MeV but the energy levels of many single-particle states are above the neutron and proton emission thresholds. Another weak point is clearly the fact that
-ray transition only measures the energy difference between two states and this single energy information is not enough to fully identify the two levels. This problem can be solved, of course, by
measuring two
-rays in coincidence. We show in
Figure 2
the energy of a
hyperon in the single-particle states
$s , p , d , f$
of several hypernuclei, deduced from emulsion,
$( K − , π − )$
$( π + , K + )$
reactions, as a function of the mass number to the power
$− 2 / 3$
. The value of ∼28 MeV extrapolated at
$A − 2 / 3 = 0$
is usually interpreted as the binding energy of a single
hyperon in infinite symmetric nuclear matter at saturation density, and it is used to fix the parameters of the majority of the models of the hyperonic EoS. Systematic spectroscopic studies of single
hypernuclei indicate that the
N interaction is attractive [
To finish this section, in
Figure 3
, we show, as an example, the level scheme and
-ray transitions of
$Λ 16$
O identified and determined by
-ray spectroscopy using the
$( K − , π − )$
reaction and the germanium detector array
at BNL [
]. The twin peaks observed confirm the hypernuclear fine structure for the
$( 1 − → 1 − )$
$( 1 − → 0 − )$
transitions in
$Λ 16$
O. We note that the small spacing between the twin peaks is due to the spin dependence of the
N interaction.
2.3. Weak Decay of Hypernuclei
The so-called mesonic weak decay
$Λ → N + π , p N ∼ 100 MeV / c$
is the main decay mode of the
hyperon in free space, where ∼60% of the times the
decays into a proton and a
$π −$
, and ∼40% into a neutron and a
$π 0$
. When the
is bound in the nucleus, however, this mode is strongly suppressed by the Pauli exclusion principle since the momentum of the outgoing nucleon (∼100 MeV/c) is smaller than the typical Fermi momentum
of a nucleon in the nucleus (∼270 MeV/c). Consequently, in hypernuclei (specially in medium and heavy ones), the dominant decay mode becomes the non-mesonic one
$Λ + N → N + N , p N ∼ 420 MeV / c$
$Λ + N + N → N + N + N , p N ∼ 340 MeV / c$
where the
interacts with one (or more) of the surrounding nucleons. The weak decay of hypernuclei has been mainly studied within the framework of meson-exchange models [
] and, more recently, using the effective field theory [
]. In [
], the interested reader can find two comprehensive reviews on the theoretical aspects of hypernuclear weak decay.
Figure 4
, we present the weak decay rate
(expressed in units of the decay rate of the
in free space) as function of the total number of particles
$A + 1$
. The figure has been adapted from the original one in [
]. The dot, dashed and solid lines show, respectively, the theoretical predictions of the mesonic
$Γ M$
, non-mesonic
$Γ N M$
and total
$Γ T$
decay rates. The curves labeled
$Γ 1$
$Γ 2$
correspond to the contributions of one-nucleon and two-nucleon induced decay modes to the non-mesonic decay rate (see Equations (
) and (
)). Experimental values of the total and non-mesonic decay rates are given, respectively, by the squares and circle marks. As it can be seen in the figure, the mesonic decay mode gets blocked as A
increases, while the non-mesonic decay increases up to a saturation value of the order of the free decay, reflecting the short-range nature of the weak
$Δ S = 1$
baryon–baryon interaction.
2.4. Theoretical Description of Hypernuclei
A simple theoretical description of a hypernucleus consists of an ordinary nucleus with a hyperon sitting in a single particle state of an effective hyperon-nucleus mean field potential. Based on
this simple description, several approaches have been followed to derive the properties of hyperons in finite nuclei. Traditionally, Woods–Saxon potentials have been used, for instance, to describe
in a shell model picture the single-particle properties of the
from medium to heavy hypernuclei [
]. To improve the overall fit of the
single-particle energies, non-localities and density-dependent effects have been included in non-relativistic Hartree–Fock calculations with Skyrme type YN interactions [
]. Relativistic mean field theory [
] and Dirac phenomenology [
] have been also employed to perform hypernuclear structure calculations. Several hypernuclear structure studies based on ab initio approaches do also exist in the literature [
]. The single-particle properties of the
in the hypernucleus are derived in these studies from effective YN G-matrices built from bare YN interactions which describe the scarce scattering data in free space. A Quantum Monte Carlo
calculation of single- and double-
hypernuclei has also been recently done using two- and three-body forces between the
and the nucleons [
]. We would like to note that the NPLQCD collaboration has been able to obtain the binding energies of the light hypernuclei including
$Λ 3$
$Λ 4$
H and
$Λ Λ 4$
He [
The quality of the description of hypernuclei in most of these approaches relies on the validity of the mean field picture. Correlations induced by the YN interaction can, however, change
substantially this picture and, therefore, should not be ignored. While many authors have extensively studied the correlations of nucleons in nuclear matter and finite nuclei, those of hyperons have
not received so much attention so far. The effect of the
correlations in nuclear matter, beyond the mean field description, was studied for the fist time by Robertson and Dickhoff [
] using the Green’s function formalism. These authors calculated the spectral function and quasi-particle parameters of the
finding results qualitatively similar to those of the nucleons. They showed that the
is, in general, less correlated than the nucleons. A few years ago, the author of the present review studied the spectral function of the
hyperon in finite nuclei [
], showing, in agreement with the work of Robertson and Dickhoff, that the
is less correlated than the nucleons, and confirming the idea that it maintains its identity inside the nucleus. The results of this study showed also that in hypernuclear production reactions, the
hyperon is formed mostly in a quasi-free state.
As an example of a theoretical calculation of hypernuclei, we briefly describe here a microscopic method that allows one to determine the single-particle bound states of a
-hyperon in finite nuclei. This method starts with the construction of all the YN
-matrices which describe the interaction between a hyperon and a nucleon in infinite nuclear matter. To this end, the coupled-channel Bethe–Goldstone equation is solved. These
-matrices are then used to obtain the YN
-matrices in finite nuclei through the following integral equation:
$G F N = G + G Q E F N − Q E G F N = G + G Q E F N − Q E G + G Q E F N − Q E G Q E F N − Q E G + · · · ,$
which expresses the finite nuclei
$G F N$
, in terms of the nuclear matter ones,
, and the difference between the finite-nucleus and the nuclear-matter propagators, written schematically as
$( Q / E ) F N − ( Q / E )$
. This difference, which accounts for the relevant intermediate particle–particle states has been shown to be quite small and, thus, in all practical calculations,
$G F N$
can be well approximated by truncating the expansion of Equation (
) up to second order in the nuclear matter
-matrices. Therefore, we have
$G F N ≈ G + G Q E F N − Q E G .$
Using then
$G F N$
as an effective YN interaction, one can obtain the
self-energy in the BHF approximation (see diagram (a) of
Figure 5
). This approximation can be split into the sum of two contributions: the one shown by diagram (b), which originates from the first-order term on the right-hand side of Equation (
), and that of diagram (c), which stands for the so-called two-particle-one-hole (2p1h) correction, where the intermediate particle–particle propagator has to be viewed as the difference of
propagators appearing in Equation (
). Solving finally the Schrödinger equation with the real part of the
self-energy, it is then possible to determine, as mentioned before, the different
single-particle bound states. Further details of this method can be found, e.g., in [
As an example of the application of this method, in
Table 1
, we show the energies of the
single-particle bound states in several hypernuclei. The results have been obtained with the NLO13 [
] and the NLO19 [
] chiral YN interaction of the Jülich–Bonn–Munich group for different values of the cutoff of the interaction including both contributions (first order term and 2p1h correction) to the
self-energy. We note that, due to technicalities of this method, it can be only applied to hypernuclei consisting of a closed-shell nuclear core plus a
sitting in a single-particle state. The values reported are to be compared with the experimental separation energies for the corresponding hypernuclei. However, since experimental data for the
particular hypernuclei we consider do not always exist, the comparison is done with the closest representative hypernuclei for which experimental information is available. As it can be seen, in
general, there is an underbinding of light hypernuclei such as
$Λ 5$
He and
$Λ 13$
C while the description of medium and heavy hypernuclei improves. Note, however, that neither the NLO13 interaction nor the NLO19 one yield a quantitative description of all medium and heavy
hypernuclei. Whereas the NLO19 interaction describes reasonably well
$Λ 17$
$Λ 41$
Ca and
$Λ 91$
Zr, it seems to overbind
$Λ 209$
Pb. For the latter, the predictions of the NLO13 interaction are more in line with the experiment. We note that the results of the calculation agree with the experimental fact that the spin-orbit
splitting of the
- and
-states is very small. The interested reader is referred to [
] for further detail of these results.
3. Hyperons and Neutron Stars
The presence of hyperons in the interior of neutron stars has been considered by many authors for more than 60 years since the seminal work of Ambartsumyan and Saakyan [
]. Different phenomenological or microscopic models have been used to describe the neutron star matter EoS. All the works show that hyperons may appear in the neutron star interior at densities
$2 − 3 ρ 0$
. The reason for their appearance is simply that, at such densities, the nucleon chemical potential is large enough to make the conversion of nucleons into hyperons energetically favorable. As a
result of this conversion, the Fermi pressure exerted by nucleons is relieved and, therefore, the EoS becomes softer. Consequently, the mass of the star, and particularly, its maximum value
$M m a x$
is reduced. How much the EoS is softened, and how much
$M m a x$
is reduced depends on the attractive or repulsive character of the YN and YY interactions. In general, attractive (repulsive) interactions lead to an earlier (later) onset and larger (smaller)
concentration of hyperons and, thus, to a stronger (more moderate) softening of the EoS and a larger (smaller) reduction of
$M m a x$
. It is well known (see, e.g., [
]), however, that hyperons equalize the effect of different nucleonic interactions through several compensation mechanisms: a stiffer nucleonic EoS will lead to an earlier onset of hyperons enhancing
in this way the softening due to their presence. Conversely, a retarded onset of a certain hyperon species will favor the appearance of other species leading also to a softer EoS. As a result,
$M m a x$
is surprisingly quite insensitive to the pure nucleonic EoS, and even to the details of the YN and YY interactions (see, e.g.,
Figure 2
in [
] or
Figure 3
in [
3.1. The Hyperon Puzzle and Some Possible Solutions
Although the presence of hyperons in neutron stars seems to be energetically unavoidable, the strong softening of the EoS associated with the onset of hyperons (notably in microscopic models) leads
to values of
$M m a x$
not compatible with observations. This controversy is known in the literature as the “hyperon puzzle”, and is currently a subject of intensive research. The discrepancy between the theory and
observations became more dramatic after the measurements in the last decade of unusually high masses of the millisecond pulsars PSR J1903 + 0327 (
$1.667 ± 0.021 M ⊙$
) [
], PSR J1614-2230 (
$1.928 ± 0.017 M ⊙$
) [
], PSR J0348 + 0432 (
$2.01 ± 0.04 M ⊙$
) [
], and the most massive one observed up to now PSR J0740 + 6620 (
$M > 2 . 14 − 0.09 + 0.10 M ⊙$
) [
], which rule out almost all currently proposed EoS with hyperons (both microscopic and phenomenological).
To solve this puzzle, a mechanism (or mechanisms) is necessary that could provide the additional repulsion needed to make the EoS stiffer and, therefore,
$M m a x$
compatible with the current observational limits. Three possible mechanisms that could provide such additional repulsion are: (i) more repulsive hyperon–hyperon interactions driven by either
repulsive vector mesons exchanges [
] or less-attractive scalar
meson exchange [
] (ii) repulsive hyperonic three-body forces [
] and (iii) a phase transition to deconfined quark matter at densities below the hyperon threshold [
]. The possible appearance of other hadronic degrees of freedom, such as the
isobar or meson condensates, that can push the onset of hyperons to higher densities, has been also considered. An interesting possibility to circumvent the problem is the so-called two-families
scenario proposed in [
], in which stars made of hadrons are stable only up to
$( 1.5 − 1.6 ) M ⊙$
while most massive compact stars are entirely made of strange quark matter. In the following, we briefly review some of these possible solutions.
3.1.1. Hyperon–Hyperon Repulsion
This possible solution has been explored mostly in the context of RMF models [
], and it is based on the well-known fact that, in meson-exchange models of nuclear forces, while the scalar meson
is responsible for the intermediate-range attraction, vector mesons generate repulsion at short distances. It has been argued that if the interaction between two hyperons, driven by vector mesons, is
repulsive enough or the attraction mediated through the exchange of the
meson is weak enough, then the EoS could be sufficiently stiff to reconcile the current high pulsar mass observations with the existence of hyperons in neutron stars. However, the strength of meson
exchanges should be modified consistently with hypernuclear data which requires, at least, the
N interaction to be attractive and suitably tuned to the hypernuclear data [
]. Such tuning is not required if the repulsive vector meson interactions act only among the hyperons through the exchange of the strange
vector meson (which couples only to hyperons). In this way, the onset of hyperons is shifted to higher densities and it is possible to obtain neutron stars with maximum masses larger than
$2 M ⊙$
and a significant amount of hyperons in their interior.
3.1.2. Hyperonic Three-Body Forces
It is well known that three-nucleon forces are fundamental to reproduce accurately the properties of few-nucleon systems as well as the empirical saturation point of symmetric nuclear matter in
non-relativistic many-body approaches. It seems natural, therefore, to suggest that three-body forces of the type NNY, NYY and YYY, involving one or more hyperons, could provide, as in the case of
three-nucleon forces, the additional repulsion needed at high densities to make the EoS stiff enough, solving in this way the hyperon puzzle. This idea was suggested even before the observation of
neutron stars with ∼
$2 M ⊙$
(see e.g., [
]), and it has been explored by a number of authors in the last years [
]. However, no general consensus has been reached yet regarding the role played by the hyperonic three-body forces in solving the hyperon puzzle. A multi-Pomeron exchange potential (MPP) model to
introduce a universal three-body repulsion among three baryons in the hyperonic matter EoS was proposed in [
]. This universal three-body repulsive potential was based on the extended soft core (ESC) baryon–baryon interaction of the Nijmegen group [
]. The strength of the MPP was determined by analyzing the nucleus–nucleus scattering with the use of a G-matrix folding potential derived from the ESC interaction complemented with the MPP and a
three-nucleon attractive part, added phenomenologically in order to reproduce the nuclear saturation properties. The results of those works [
] showed that when the MPP contribution was taken into account, universally for all baryons, neutron star radii
at a typical mass
$1.5 M ⊙$
were predicted to be around 12.3–13.1 km [
], and a maximum mass of ∼
$2.2 M ⊙$
was obtained. This result for the maximum mass is in contradiction with that reported in [
] where the case of a universal three-body repulsion was also analyzed. The authors of [
] used a model based on the BHF approach of hypernuclear matter using the Argonne V18 NN potential [
] and the Nijmegen YN soft core NSC89 one [
] supplemented with additional simple phenomenological density-dependent contact terms to establish numerically lower and upper limits to the effect of hyperonic three-body forces on the maximum mass
of neutron stars. Assuming that the strength of these forces was either smaller than or as large as the pure nucleonic ones, the results reported in [
] showed that, although the employed hyperonic three-body forces stiffened the EoS, they were, however, unable to provide the repulsion needed to make the predicted maximum masses compatible with the
recent observations. In [
], a Monte Carlo calculation of pure neutron matter with a non-vanishing
-hyperon concentration was carried out including NN, NNN,
N and NN
two- and three-body interactions. In particular, the NN
force used in this work was tuned in order to provide a reasonable description of the measured
separation energy of several hypernuclei [
]. The authors of this work concluded that, with the model they considered, the presence of hyperons in the core of neutron stars could not be satisfactorily established and, consequently, according
to these authors, there is no clear incompatibility with astrophysical observations when the
is included. However, one should note, that the presence of protons, necessary to establish the correct
-equilibrium inside neutron stars and, thus, a proper treatment of nuclear matter was neglected in their calculation. Very recently, the authors [
] have studied the effects of chiral hyperonic three-body forces on neutron stars, showing that the inclusion of a moderate repulsive NN
force leads already to an EoS stiff enough such that the resulting neutron star maximum mass is compatible with the largest currently measured massed. These authors have also examined the effect of
this NN
force on the separation energy of a
in some hypernuclei obtaining a good description of the experimental data. This is in agreement with the results of [
], where also a moderate hyperonic three-body force was found to be enough to reproduce the binding energies of single-
hypernuclei, but contrary to the results reported in [
] where a very repulsive NN
was required to describe the data. In conclusion, it seems that although hyperonic three-body forces offer an interesting microscopic solution to the hyperon puzzle, the uncertainties associated to
these forces are still too large to allow for a definite conclusion.
3.1.3. Quark Matter Phase Transition below the Hyperon Threshold
An early phase transition from hadronic matter to deconfined quark matter at densities below the hyperon threshold could provide the solution to the hyperon puzzle, as it has been suggested by
several authors. In this case, massive stars could actually be hybrid stars with a stiff quark matter core. This solution, however, leads to a new question: can quarks provide the sufficient
repulsion required to produce a
$2 M ⊙$
neutron star? In [
], the maximum mass was found to be in a relatively narrow interval, ∼
$1.4 M ⊙ < M m a x < ∼ 1.7 M ⊙$
which is incompatible with the current observational limit of
$2 M ⊙$
. To obtain values of
$M m a x$
larger than
$2 M ⊙$
, the interaction among quarks should fulfill two important and necessary conditions. First of all, it should be significantly repulsive, for example, in vector channels in order to guarantee that
the EoS is stiff enough. Second, it should be strongly attractive in certain channels, leading to color superconductivity, so that the deconfined quark-matter phase is energetically favorable with
respect to the hadronic one. Current theoretical descriptions of quark matter at high density rely on phenomenological models, which are constrained using the few available experimental information
on high density baryonic matter from heavy-ion collisions. Several models of hybrid stars have been proposed in the recent years with the necessary properties to generate a
$2 M ⊙$
(see, e.g., [
]). Very recently, Shahrbaf et al. [
] have applied, for the first time, the finite-range polynomial interpolation method of Masuda et al. [
] for constructing a transition between hadronic and quark matter phases. The predicted maximum mass of the hybrid star is about
$2.2 M ⊙$
in agreement with current observations. The observation of
$2 M ⊙$
neutron stars, on the other hand, may also help constrain better the models of hybrid and strange stars (compact stars completely made of deconfined
quark matter), and improve our present understanding of the hadron–quark phase transition.
3.1.4. $Δ$ Isobar and Kaon Condensation in Neutron Stars
An alternative possible solution to the hyperon puzzle that has been also considered is the appearance of other hadronic degrees of freedom, such as for instance the
isobar or meson condensates, that can push the onset of hyperons to higher densities. The presence of the
isobar in neutron stars was usually neglected because in former calculations its threshold density was found to be larger than the typical densities of neutron star cores. A few years ago, it has
been shown [
], however, that the onset density of the
isobar depends crucially on the slope parameter
of the nuclear symmetry energy. Using recent experimental constraints and a state-of-the-art EoS, the authors of [
] have shown that the
isobar could actually appear at densities below the hyperon threshold. However, they found that, as soon as the
appears, the EoS becomes also considerably soft and, as a consequence,
$M m a x$
is reduced to values below the current observational limits, giving rise to what they have named the “
puzzle”. Very recently, using an RMF approach, the authors of [
] found, however, that the presence of the
isobar in the neutron star interior is compatible with the observation of
$2 M ⊙$
millisecond pulsars, provided that the couplings of the
to the
meson fields are at least
$10 %$
stronger than the corresponding ones of the nucleons. The possibility that pions or kaons form Bose–Einstein condensates in the interior of neutron stars has also been extensively considered in the
literature [
]. However, as in the case of the hyperons or the
isobar, the appearance of a meson condensate induces also a strong softening of the EoS reducing the value of
$M m a x$
below the current observational limits too.
3.2. Effect of Hyperons on Proto-Neutron Stars
Thermal effects and neutrino trapping affect the properties of newly born neutron stars during the first tens of seconds after their formation. In particular, the composition and the overall
stiffness of the EoS of the star are strongly influenced by these two effects. Matter becomes more proton rich, the number of muons is significantly reduced and the onset of hyperons is shifted to
higher densities [
]. Furthermore, the number of strange particles is on average smaller, and the EoS is stiffer in comparison with the cold and neutrino-free case.
A very important consequence of the neutrino trapping in dense matter is the possibility of having metastable neutron stars and a delayed formation of a “low mass” (
$M = 1 − 2 M ⊙$
) black hole. To illustrate this, we show in
Figure 6
the gravitational mass
$M G$
of the star as a function of its baryonic mass
$M B$
(proportional to the total number of baryons in the star) obtained by the authors of [
]. When hyperons are present in the star (
Figure 6
a), the deleptonization lowers the range of gravitational masses that can be supported by the EoS from about
$1.59 M ⊙$
to about
$1.28 M ⊙$
(see dotted horizontal lines in the figure). The neutron star baryonic mass can be considered constant during the evolution from the initial proto-neutron star configuration to the final
neutrino-free one because most of the matter accretion on the forming neutron star happens in a very early stage after its birth (t < 1 s). For this particular calculation, proto-neutron stars born
with gravitational masses between
$1.28 M ⊙$
$1.59 M ⊙$
(a baryonic mass between
$1.40 M ⊙$
$1.72 M ⊙$
) will be stabilized by neutrino trapping effects long enough to carry out nucleosynthesis accompanying a Type-II supernova explosion. After neutrinos leave the star, the EoS is softened and it
cannot support anymore the star against its own gravity and the newborn star collapses then to a black hole [
]. Conversely, when only nucleons are considered to be the relevant baryonic degrees of freedom (
Figure 6
b), no metastability occurs and a black hole is unlikely to be formed during the deleptonization because the gravitational mass increases during this stage which happens at (almost) constant baryonic
mass. If a black hole were to form from a star with only nucleons, it is much more likely to form during the post-bounce accretion stage.
3.3. Hyperons and Neutron Star Cooling
The cooling of a newly born hot neutron star is driven in a first stage by the neutrino emission from the interior, and later by the emission of photons at the surface. Depending on whether the
number of involved baryons is one or two, neutrino emission reactions can be classified as fast or slow processes, respectively. The simplest possible neutrino emission reaction is the well-known
direct Urca process
$n → p + l + ν ¯ l , p + l → n + ν l .$
This is a fast mechanism which, due to momentum conservation, is only possible when the proton fraction exceeds a critical value
$x D U R C A ∼ 11 %$
$15 %$
]. Other neutrino reactions that lead to medium or slow cooling scenarios, but that are operative at any density and proton fraction, are the modified Urca processes:
$N + n → N + p + l + ν ¯ l , N + p + l → N + n + ν l ,$
the bremsstrahlung:
$N + N → N + N + ν + ν ¯ ,$
or the Cooper pair formation:
$n + n → [ n n ] + ν + ν ¯ , p + p → [ p p ] + ν + ν ¯ ,$
this last operating only when the temperature of the star drops below the critical temperature for neutron superfluidity or proton superconductivity.
The presence of hyperons can affect the cooling of neutron stars because they can modify neutrino emissivities allowing for additional cooling mechanisms such as, for example, the direct
$Y → B + l + ν ¯ l , B + l → Y + ν l ,$
and modified
$B ′ + Y → B ′ + B + l + ν ¯ l , B ′ + B + l → B ′ + Y + ν l ,$
hyperonic Urca processes. These additional neutrino emission reactions, however, can lead to temperatures at the surface of the star that are much lower than those observed, unless they are
suppressed by hyperon pairing gaps. The study of hyperon superfluidity becomes then of particular interest because it could play a key role in the thermal history of neutron stars. Nevertheless,
whereas the presence of superfluid neutrons in the inner crust of neutron stars, and that of superfluid neutrons together with superconducting protons in their quantum fluid interior are well
established and have been the subject of many studies, a quantitative estimation of the hyperon pairing has not received so much attention, and just few calculations exist in the literature [
3.4. Hyperons and R-Modes
It is well known that if a neutron star rotates with a spin frequency above the so-called Kepler frequency
$Ω K$
, i.e., the absolute maximum rotational frequency of a neutron star, then matter is ejected from the star’s equator [
]. Different types of perturbations, however, can lead to instabilities that prevent the star from reaching rotational frequencies as high as
$Ω K$
and set more stringent limits on their rotation [
]. One of these possible instabilities which is particularly interesting is the so-called r-mode instability [
]. It is a toroidal mode of oscillation whose restoring force is the Coriolis force which leads to the emission of gravitational waves in hot and rapidly rotating neutron stars through the
Chandrasekhar–Friedman–Schutz mechanism [
]. The r-mode grows with the emission of gravitational waves whereas it is damped by dissipation mechanisms. The r-mode becomes unstable when the driving time of the gravitational radiation is
shorter than the damping time associated to viscous processes. A rapidly rotating neutron star could transfer in this case a significant fraction of its angular momentum and rotational energy to the
emitted gravitational waves.
Bulk (
) and shear (
) viscosities are the main dissipation mechanisms of r- and other pulsation modes. At high temperatures (
$T > 10 9$
K), the bulk viscosity is the dominant mechanism and, therefore, it is the most important one for hot young neutron stars. It results from the variations in the pressure and the density induced by
the oscillation mode which drives the star away from
-equilibrium. The weak interaction tries then to reestablish the equilibrium with the consequent dissipation of energy. If no hyperons or other exotic components are present in the neutron star
interior, the bulk viscosity is mainly determined by the direct and modified Urca processes. However, as soon as hyperons appear, new mechanisms such as strong interaction reactions
$Y + Y ↔ N + Y , N + Ξ ↔ Y + Y , Y + Y ↔ Y + Y ,$
weak non-leptonic hyperon reactions
$N + N ↔ N + Y , N + Y ↔ Y + Y ,$
or direct and modified hyperonic Urca (see Equations (
) and (
)) contribute to the bulk viscosity and dominate it for densities above 2–3 times the nuclear saturation density. Hyperon bulk viscosity has been considered by several authors, see, e.g., [
The time dependence of an r-mode oscillation is given by
$e i ω t − t / τ ( Ω , T )$
is the frequency of the mode and
$1 τ ( Ω , T ) = − 1 τ G W ( Ω ) + 1 τ ξ ( Ω , T ) + 1 τ η ( Ω , T ) ,$
is an overall timescale which describes both its exponential growth due to the gravitational wave emission and its decay due to the viscous damping, being
$τ G W , τ ξ$
$τ η$
the time scales associated, respectively, to the gravitational wave emission and the bulk and shear viscosity dampings. If
$τ G W$
is shorter than both
$τ ξ$
$τ η$
, the mode will exponentially grow, whereas in the opposite case, it will be quickly damped away. For each star at a given temperature T, one can define a critical angular velocity
$Ω c$
as the smallest root of the equation
This equation defines the boundary of what is usually called the r-mode instability region. If the angular velocity of a neutron star is smaller than its corresponding
$Ω c$
, then the star is stable against the r-mode instability. Conversely, a star with an angular velocity larger than
$Ω c$
will develop an instability that will cause a rapid loss of angular momentum through gravitational radiation until its angular velocity falls below the critical value. As an example, on panel (a) of
Figure 7
, we show the r-mode instability region for a pure nucleonic (black solid line) and a hyperonic (red dashed line) star with
$1.27 M ⊙$
both described using the BHF theory with realistic baryon interactions. On panel (b) of the figure, we show the contributions from direct and modified nucleonic Urca processes as well as from the
weak non-leptonic process
$n + n ↔ p + Σ −$
included in the calculation of the bulk viscosity. As it is clearly seen, the r-mode instability is smaller for the hyperonic star. The reason is simply the increase of the bulk viscosity due to the
presence of hyperons which makes the damping of the r-mode more efficient.
4. Summary
In this work, we have briefly reviewed several topics related with the physics of hypernuclei and hyperons in neutron stars. In particular, we have revised the different production mechanisms of
hypernclei as well as some aspects of their $γ$-ray spectroscopy and their weak decay modes. We have also given a few strokes on their theoretical description. We have discussed also the main effects
of hyperons on the properties of neutron and proto-neutron stars with an emphasis on the well known “hyperon puzzle”, the problem of the strong softening of the EoS of dense matter due to the
appearance of hyperons which leads to maximum masses of compact stars that are not compatible with the recent observations of approximately $2 M ⊙$ millisecond pulsars. We have shortly reexamined
some of the different solutions that have been proposed to tackle this problem. The first of the mechanisms we have revised consists of the inclusion of a repulsive hyperon–hyperon interaction
through the exchange of vector mesons and it has been mainly explored in the context of RMF models. However, since presently there are not enough experimental data to constrain the YN and YY
interaction accurately, it is not clear whether the large repulsion invoked in such models is realistic. The second one requires the inclusion of repulsive hyperonic three-body forces. However, at
present, it is still an open issue whether these forces can, by themselves, solve completely the hyperon puzzle or not, although, it seems that even if they cannot provide the full answer they can
contribute to it in an important way. The third possible solution to the problem we have reviewed is to consider the possibility of a phase transition to deconfined quark matter at densities below
the hyperon threshold. However, the description of the quark phase via phenomenological models also suffers from uncertainties. Most models of hybrid stars unanimously agree that to construct massive
neutron stars, it is required both a sufficiently stiff hadronic EoS as well as a color superconducting quark phase with strong interaction among quarks to provide sufficient repulsion. We have also
briefly discussed a possible solution to the hyperon puzzle by invoking the appearance of other hadronic degrees of freedom such as the $Δ$ isobar or meson condensates that push the onset of hyperons
to higher densities. Finally, we have also revised how the presence of hyperons can affect the cooling of neutron stars and the r-mode instability window through modifications of the microscopic
input of the weak interaction rates and transport coefficients, such as the bulk viscosity, of dense matter.
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Conflicts of Interest
The author declares no conflict of interest.
1. Lenske, H.; Dhar, M.; Gaitanos, T.; Cao, X. Baryons and baryon resonances in nuclear matter. Prog. Part. Nucl. Phys. 2018, 98, 119–206. [Google Scholar] [CrossRef]
2. Danysz, M.; Pniewski, J. Delayed disintegration of a heavy nuclear fragment: I. Philos. Mag. 1953, 44, 348–350. [Google Scholar] [CrossRef]
3. Hugenford, E.V. Experimental considerations in electromagnetic production of hypernuclei. Prog. Theor. Phys. Suppl. 1994, 117, 135–149. [Google Scholar]
4. Bianchin, S.; Achenbach, P.; Ajimura, S.; Borodina, O.; Fukuda, T.; Hoffmann, J.; Kavatsyuk, M.; Koch, K.; Koike, T.; Kurz, N.; et al. The HypHI project: Hypernuclear spectroscopy with stable
heavy ion beams and rare isotope beams at GSI and Fair. Int. J. Mod. Phys. E 2009, 18, 2187–2191. [Google Scholar] [CrossRef] [Green Version]
5. Rappold, C.; Kim, E.; Nakajima, D.; Saito, T.R.; Bertini, O.; Bianchin, S.; Bozkurt, V.; Kavatsyuk, M.; Mab, Y.; Ma, F.; et al. Hypernuclear spectroscopy of products from ^6Li projectiles on a
carbon target at 2AGeV. Nucl. Phys. A 2013, 913, 170–184. [Google Scholar] [CrossRef] [Green Version]
6. Shapiro, S.L.; Teukolsky, S.A. Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Stars; Wiley and Sons: Hoboken, NJ, USA, 1983. [Google Scholar]
7. Weber, F. Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics; Institute of Physics Publishing: Bristol, UK, 1999. [Google Scholar]
8. Glendenning, N.K. Compact Stars: Nuclear Physics, Particle Physics and General Relativity, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
9. Haensel, P.; Potekin, A.Y.; Yakovlev, D.G. Neutron Stars 1: Equation of State; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
10. Rezzolla, L.; Pizzochero, P.; Jones, I.; Rea, N.; Vidaña, I. (Eds.) The Physics and Astrophysics of Neutron Stars; Springer Nature: Cham, Switzerland, 2018. [Google Scholar]
11. Balberg, S.; Gal, A. An effective equation of state for dense matter with strangeness. Nucl. Phys. A 1997, 625, 435–472. [Google Scholar] [CrossRef] [Green Version]
12. Balberg, S.; Lichtenstadt, I.; Cook, G.B. Role of hyperons in neutron stars. Astrophys. J. Suppl. Ser. 1999, 121, 515. [Google Scholar] [CrossRef]
13. Millener, D.J.; Dover, C.B.; Gal, A. Λnucleus single-particle potentials. Phys. Rev. C 1988, 38, 2700. [Google Scholar] [CrossRef]
14. Yamamoto, Y.; Bandō, H.; Žofka, J. On the Λ-hypernuclear single particle energies. Prog. Theor. Phys. 1988, 80, 757–761. [Google Scholar] [CrossRef] [Green Version]
15. Fernández, F.; López–Arias, T.; Prieto, C. Skyrme-Hartree-Fock calculation of Λ-hypernuclear states from (π^+,K^+) reactions. Z. Phys. A 1989, 334, 349–354. [Google Scholar]
16. Lanskoy, D.E.; Yamamoto, Y. Skyrme-Hartree-Fock treatment of Λ and ΛΛ hypernuclei with G-matrix motivated interactions. Phys. Rev. C 1997, 55, 2330. [Google Scholar] [CrossRef]
17. Tretyakova, T.Y.; Lanskoy, D.E. Structure of neutron-rich Λ hypernuclei. Eur. Phys. J. A 1999, 5, 391–398. [Google Scholar] [CrossRef]
18. Cugnon, J.; Lejeune, A.; Schulze, H.-J. Hypernuclei in the Skyrme-Hartree-Fock formalism with a microscopic hyperon-nucleon force. Phys. Rev. C 2000, 62, 064308. [Google Scholar] [CrossRef] [
Green Version]
19. Vidaña, I.; Polls, A.; Ramos, A.; Schulze, H.-J. Hypernuclear structure with the new Nijmegen potentials. Phys. Rev. C 2001, 64, 044301. [Google Scholar] [CrossRef] [Green Version]
20. Zhou, X.-R.; Schulze, H.-J.; Sagawa, H.; Wu, C.-X.; Zhao, E.-G. Hypernuclei in the deformed Skyrme-Hartree-Fock approach. Phys. Rev. C 2007, 76, 034312. [Google Scholar] [CrossRef]
21. Zhou, X.-R.; Polls, A.; Schulze, H.-J.; Vidaña, I. Λ hyperons and the neutron drip line. Phys. Rev. C 2008, 78, 054306. [Google Scholar] [CrossRef] [Green Version]
22. Bednarek, I.; Haensel, P.; Zdunik, J.L.; Bejger, M.; Mańka, R. Hyperons in neutron-star cores and a 2M[⊙] pulsar. Astron. Astrophys. 2012, 543, A157. [Google Scholar] [CrossRef] [Green Version]
23. Weissenborn, S.; Chatterjee, D.; Schaffner–Bielich, J. Hyperons and massive neutron stars: Vector repulsion and SU(3) symmetry. Phys. Rev. C 2012, 85, 065802. [Google Scholar] [CrossRef] [Green
24. Van Dalen, E.N.E.; Colucci, G.; Sedrakian, A. Constraining hypernuclear density functional with Λ-hypernuclei and compact stars. Phys. Lett. B 2014, 734, 383–387. [Google Scholar] [CrossRef] [
Green Version]
25. Oertel, M.; Providência, C.; Gulminelli, F.; Raduta, A.R. Hyperons in neutron star matter within relativistic mean-field models. J. Phys. G 2015, 42, 075202. [Google Scholar] [CrossRef]
26. Maslov, K.A.; Kolomeitsev, E.E.; Voskresensky, D.N. Solution of the hyperon puzzle within a relativistic mean-field model. Phys. Lett. B 2015, 748, 369–375. [Google Scholar] [CrossRef] [Green
27. Fortin, M.; Avancini, S.S.; Providência, C.; Vidaña, I. Hypernuclei and massive neutron stars. Phys. Rev. C 2017, 95, 065803. [Google Scholar] [CrossRef]
28. Pal, S.; Hanauske, M.; Zakout, I.; Stöcker, G.W. Neutron star properties in the quark-meson coupling model. Phys. Rev. C 1999, 60, 015802. [Google Scholar] [CrossRef] [Green Version]
29. Stone, J.R.; Guinchon, P.A.M.; Matevosyan, H.H.; Thomas, A.W. Cold uniform matter and neutron stars in the quark-meson-coupling model. Nucl. Phys. A 2007, 792, 341–369. [Google Scholar] [CrossRef
] [Green Version]
30. Bombaci, I.; Panda, P.K.; Providência, C.; Vidaña, I. Metastability of hadronic compact stars. Phys. Rev. D 2008, 77, 083002. [Google Scholar] [CrossRef] [Green Version]
31. Carroll, J.D.; Leinweber, D.B.; Williams, A.G.; Thomas, A.W. Phase transition from quark-meson coupling hyperonic matter to deconfined quark matter. Phys. Rev. C 2009, 79, 045810. [Google Scholar
] [CrossRef] [Green Version]
32. Miyatsu, T.; Saito, K. Effect of gluon and pion exchanges on hyperons in nuclear matter. Prog. Theor. Phys. 2009, 122, 1035–1044. [Google Scholar] [CrossRef] [Green Version]
33. Carroll, J.D. QMC and the nature of dense matter: Written in the stars? AIP Conf. Proc. 2010, 1261, 226–231. [Google Scholar]
34. Panda, P.K.; Santos, A.M.S.; Menezes, D.P.; Providência, C. Compact stars within a soft symmetry energy quark-meson-coupling model. Phys. Rev. C 2012, 85, 055802. [Google Scholar] [CrossRef] [
Green Version]
35. Stone, J.R.; Dexheimer, V.; Guichon, P.A.M.; Thomas, A.W.; Typel, S. Equation of state of hot dense hyperonic matter in the Quark-Meson-Coupling (QMC-A) model. Month. Not. R. Astron. Soc. 2019,
502, 34767. [Google Scholar]
36. Antić, S.; Stone, J.R.; Thomas, A.W. Neutron stars from crust to core within the Quark-meson-coupling model. EPJ Web Conf. 2020, 232, 03001. [Google Scholar] [CrossRef]
37. Gaitanos, T.; Kaskulov, M. Momentum dependent mean-field dynamics of compressed nuclear matter and neutron stars. Nucl. Phys. A 2013, 899, 133–169. [Google Scholar] [CrossRef] [Green Version]
38. Moustakidis, C.C.; Gaitanos, T.; Margaritis, C.; Lalazissis, G.A. Bounds on the speed of sound in dense matter and neutron star structure. Phys. Rev. C 2017, 95, 045801. [Google Scholar] [
39. Gaitanos, T.; Chorozidou, A. Momentum dependent mean-fields of (anti)hyperons. Nucl. Phys. A 2021, 1008, 122153. [Google Scholar] [CrossRef]
40. Nagels, M.M.; Rijken, T.A.; de Swart, J.J. Determination of the mixing angle, F/(F +D) ratio, and coupling constants of the scalar-meson nonet. Phys. Rev. Lett. 1973, 31, 569. [Google Scholar] [
41. Nagels, M.M.; Rijken, T.A.; de Swart, J.J. Low-energy nucleon-nucleon potential from Regge-pole theory. Phys. Rev. D 1978, 17, 768. [Google Scholar] [CrossRef]
42. Machleidt, R.; Holinde, K.; Elster, C. The bonn meson-exchange model for the nucleon-nucleon interaction. Phys. Rep. 1987, 149, 1–89. [Google Scholar] [CrossRef]
43. Holzenkamp, B.; Holinde, K.; Speth, J. A meson exchange model for the hyperon-nucleon interaction. Nucl. Phys. A 1989, 500, 485–528. [Google Scholar] [CrossRef]
44. Maesen, P.M.M.; Rijken, T.A.; de Swart, J.J. Soft-core baryon-baryon one-boson-exchange models. II. Hyperon-nucleon potential. Phys. Rev. C 1989, 40, 2226. [Google Scholar] [CrossRef]
45. Rijken, T.A.; Stoks, V.G.J.; Yamamoto, Y. Soft-core hyperon-nucleon potentials. Phys. Rev. C 1999, 59, 21. [Google Scholar] [CrossRef] [Green Version]
46. Stoks, V.G.J.; Rijken, T.A. Soft-core baryon-baryon potentials for the complete baryon octet. Phys. Rev. C 1999, 59, 3009. [Google Scholar] [CrossRef] [Green Version]
47. Haidenbauer, J.; Meissner, U.-G. Jülich hyperon-nucleon model revisited. Phys. Rev. C 2005, 72, 044005. [Google Scholar] [CrossRef] [Green Version]
48. Rijken, T.A. Extended-soft-core baryon-baryon model. I. Nucleon-nucleon scattering with the ESC04 interaction. Phys. Rev. C 2006, 73, 044007. [Google Scholar] [CrossRef] [Green Version]
49. Rijken, T.A.; Yamamoto, Y. Extended-soft-core baryon-baryon model. II. Hyperon-nucleon interaction. Phys. Rev. C 2006, 73, 044008. [Google Scholar] [CrossRef] [Green Version]
50. Rijken, T.A.; Nagels, M.M.; Yamamoto, Y. Baryon-baryon interactions. Prog. Theor. Phys. Suppl. 2010, 185, 14. [Google Scholar] [CrossRef]
51. Weinberg, S. Nuclear forces from chiral lagrangians. Phys. Lett. B 1991, 251, 288. [Google Scholar] [CrossRef]
52. Weinberg, S. Effective chiral lagrangians for nucleon-pion interactions and nuclear forces. Nucl. Phys. B 1991, 363, 3–18. [Google Scholar] [CrossRef]
53. Entem, D.R.; Machleidt, R. Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory. Phys. Rev. C 2003, 68, 041001. [Google Scholar] [CrossRef] [Green
54. Epelbaum, E.; Glöcke, W.; Meissner, U.-G. The two-nucleon system at next-to-next-to-next-to-leading order. Nucl. Phys. A 2005, 747, 362–424. [Google Scholar] [CrossRef] [Green Version]
55. Entem, D.R.; Machleidt, R.; Nosyk, Y. High-quality two-nucleon potentials up to fifth order of the chiral expansion. Phys. Rev. C 2017, 96, 024004. [Google Scholar] [CrossRef]
56. Epelbaum, E. Few-nucleon forces and systems in chiral effective field theory. Prog. Nucl. Part. Phys. 2006, 57, 654–741. [Google Scholar] [CrossRef] [Green Version]
57. Polinder, H.; Haidenbauer, J.; Meissner, U.-G. Hyperon–nucleon interactions—A chiral effective field theory approach. Nucl. Phys. A 2006, 779, 244–266. [Google Scholar] [CrossRef] [Green Version]
58. Haidenbauer, J.; Petschauer, S.; Kaiser, N.; Meissner, U.-G.; Nogga, A.; Weise, W. Hyperon–nucleon interaction at next-to-leading order in chiral effective field theory. Nucl. Phys. A 2013, 915,
24–58. [Google Scholar] [CrossRef] [Green Version]
59. Haidenbauer, J.; Meissner, U.-G.; Nogga, A. Hyperon-nucleon interaction within chiral effective field theory revisited. Eur. Phys. J. A 2020, 56, 91. [Google Scholar] [CrossRef] [Green Version]
60. Schulze, H.-J.; Baldo, M.; Lombardo, U.; Cugnon, J.; Lejeune, A. Hypernuclear matter in the Brueckner-Hartree-Fock approximation. Phys. Lett. B 1995, 355, 21–26. [Google Scholar] [CrossRef] [
Green Version]
61. Schulze, H.-J.; Baldo, M.; Lombardo, U.; Cugnon, J.; Lejeune, A. Hyperonic nuclear matter in Brueckner theory. Phys. Rev. C 1998, 57, 704. [Google Scholar] [CrossRef] [Green Version]
62. Baldo, M.; Burgio, G.F.; Schulze, H.-J. Onset of hyperon formation in neutron star matter from Brueckner theory. Phys. Rev. C 1998, 58, 3688. [Google Scholar] [CrossRef]
63. Baldo, M.; Burgio, G.F.; Schulze, H.-J. Hyperon stars in the Brueckner-Bethe-Goldstone theory. Phys. Rev. C 2000, 61, 055801. [Google Scholar] [CrossRef] [Green Version]
64. Vidaña, I.; Polls, A.; Ramos, A.; Hjorth-Jensen, M.; Stoks, V.G.J. Strange nuclear matter within Brueckner-Hartree-Fock theory. Phys. Rev. C 2000, 61, 025802. [Google Scholar] [CrossRef] [Green
65. Vidaña, I.; Polls, A.; Ramos, A.; Engvik, L.; Hjorth-Jensen, M. Hyperon-hyperon interactions and properties of neutron star matter. Phys. Rev. C 2000, 62, 035801. [Google Scholar] [CrossRef] [
Green Version]
66. Schulze, H.-J.; Polls, A.; Ramos, A.; Vidaña, I. Maximum mass of neutron stars. Phys. Rev. C 2006, 73, 058801. [Google Scholar] [CrossRef] [Green Version]
67. Schulze, H.-J.; Rijken, T. Maximum mass of hyperon stars with the Nijmegen ESC08 model. Phys. Rev. C 2011, 84, 035801. [Google Scholar] [CrossRef] [Green Version]
68. Dapo, H.; Schaefer, B.-J.; Wambach, J. Appearance of hyperons in neutron stars. Phys. Rev. C 2010, 81, 035803. [Google Scholar] [CrossRef] [Green Version]
69. Sammarruca, F. Effect of Λ hyperons on the nuclear equation of state in a Dirac- Brueckner-Hartree-Fock model. Phys. Rev. C 2009, 79, 034301. [Google Scholar] [CrossRef] [Green Version]
70. Lonardoni, D.; Pederiva, F.; Gandolfi, S. Accurate determination of the interaction between Λ hyperons and nucleons from auxiliary field diffusion Monte Carlo calculations. Phys. Rev. C 2014, 89,
014314. [Google Scholar] [CrossRef] [Green Version]
71. Petschauer, S.; Haidenbauer, J.; Kaiser, N.; Meissner, U.-G.; Weise, W. Hyperons in nuclear matter from SU(3) chiral effective field theory. Eur. Phys. J. A 2016, 52, 15. [Google Scholar] [
CrossRef] [Green Version]
72. Haidenbauer, J.; Meissner, U.-G.; Kaiser, N.; Weise, W. Lambda-nuclear interactions and hyperon puzzle in neutron stars. Eur. Phys. J. A 2017, 53, 121. [Google Scholar] [CrossRef]
73. Kohno, M. Comparative study of hyperon-nucleon interactions in a quark model and in chiral effective field theory by low-momentum equivalent interactions and G matrices. Phys. Rev. C 2010, 81,
014003. [Google Scholar] [CrossRef] [Green Version]
74. Kohno, M. Single-particle potential of the Λ hyperon in nuclear matter with chiral effective field theory NLO interactions including effects of YNN three-baryon interactions. Phys. Rev. C 2018,
97, 035206. [Google Scholar] [CrossRef] [Green Version]
75. Ohnishi, A.; Morita, K.; Miyahara, K.; Hyodo, T. Hadron–hadron correlation and interaction from heavy–ion collisions. Nucl. Phys. A 2016, 954, 294–307. [Google Scholar] [CrossRef] [Green Version]
76. Adamczewski–Musch, J.; Agakishiev, G.; Arnold, O.; Atomssa, E.T.; Behnke, C.; Berger-Chen, J.C.; Hades Collaboration. Λp interaction studied via femtoscopy in p→Nb reactions at s[NN]=3.18 GeV.
Phys. Rev. C 2016, 94, 025201. [Google Scholar] [CrossRef] [Green Version]
77. Hatsuda, T.; Morita, K.; Ohnishi, A.; Sasaki, K. pΞ^- correlation in relativistic heavy ion collisions with nucleon-hyperon Interaction from Lattice QCD. Nucl. Phys. A 2017, 967, 856–859. [Google
Scholar] [CrossRef]
78. Mihaylov, D.L.; Sarti, V.M.; Arnold, O.W.; Fabbietti, L.; Holweger, B.; Mathis, A.M. A femtoscopic correlation analysis tool using the Schrödinger equation (CATS). Eur. Phys. J. C 2018, 78, 394.
[Google Scholar] [CrossRef] [Green Version]
79. Acharya, S.; Adamová, D.; Adhya, S.P.; Adler, A.; Adolfsson, J.; Aggarwal, M.M.; A Large Ion Collider Experiment Collaboration. First observation of an attractive interaction between a proton and
a cascade baryon. Phys. Rev. Lett. 2019, 123, 112002. [Google Scholar] [CrossRef] [PubMed] [Green Version]
80. Acharya, S.; Adamová, D.; Adolfsson, J.; Aggarwal, M.M.; Rinella, G.A.; Agnello, M.; ALICE Collaboration. p-p, p-Λ, and Λ-Λ correlations studied via femtoscopy in pp reactions at s=7 TeV. Phys.
Rev. C 2019, 99, 024001. [Google Scholar] [CrossRef] [Green Version]
81. Acharya, S.; Adamová, D.; Adhya, S.P.; Adler, A.; Adolfsson, J.; Aggarwal, M.M.; Castro, A.J.; ALICE Collaboration. Study of the Λ-Λ interaction with femtoscopy correlations in pp and p-Pb
collisions at the LHC. Phys. Lett. B 2019, 797, 134822. [Google Scholar] [CrossRef]
82. Acharya, S.; Adamová, D.; Adler, A.; Adolfsson, J.; Aggarwal, M.M.; Rinella, G.A.; Casula, E.A.R.; ALICE Collaboration. Investigation of the p-Σ^0 interaction via femtoscopy in pp collisions.
Phys. Lett. B 2020, 805, 135419. [Google Scholar] [CrossRef]
83. Fabbietti, L.; Sarti, V.M.; Vázquez Doce, O.V. Hadron-hadron interactions measured by ALICE at the LHC. arXiv 2012, arXiv:2012.09806. [Google Scholar]
84. Tolós, L.; Fabbietti, L. Strangeness in nuclei and neutron stars. Prog. Part. Nucl. Phys. 2020, 112, 103770. [Google Scholar] [CrossRef] [Green Version]
85. ALICE Collaboration. Unveiling the strong interaction among hadrons at the LHC. Nature 2020, 588, 232–238. [Google Scholar] [CrossRef]
86. Pratt, S. Pion interferometry of quark-gluon plasma. Phys. Rev. D 1986, 33, 1314. [Google Scholar] [CrossRef]
87. Lisa, M.A.; Pratt, S.; Soltz, R.; Wiedemman, U. Femtoscopy in relativistic heavy ion collisions: Two decades of progress. Ann. Rev. Nucl. Part. Sci. 2005, 55, 357–402. [Google Scholar] [CrossRef]
[Green Version]
88. Beane, S.R.; Savage, M. Nucleon–nucleon interactions on the lattice. Phys Lett. B 2002, 535, 177–180. [Google Scholar] [CrossRef] [Green Version]
89. Ishii, N.; Aoki, S.; Hatsuda, T. Nuclear force from lattice QCD. Phys. Rev. Lett. 2007, 99, 022001. [Google Scholar] [CrossRef] [PubMed] [Green Version]
90. Aoki, S.; Hatsuda, T.; Ishii, N. Theoretical foundation of the nuclear force in QCD and its applications to central and tensor forces in quenched lattice QCD simulations. Prog. Theor. Phys. 2010,
123, 89–128. [Google Scholar] [CrossRef] [Green Version]
91. Beane, S.; Detmold, W.; Orginos, K.; Savage, M. Nuclear physics from lattice QCD. Prog. Part. Nucl. Phys. 2011, 66, 1–40. [Google Scholar] [CrossRef] [Green Version]
92. Aoki, S. Hadron interactions in lattice QCD. Prog. Part. Nucl. Phys. 2011, 66, 687–726. [Google Scholar] [CrossRef] [Green Version]
93. Aoki, S.; Doi, T.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Ishii, N.; Murano, K.; Nemura, H.; Sasaki, K. Lattice quantum chromodynamical approach to nuclear physics. Prog. Theor. Exp. Phys. 2012, 1,
01A105. [Google Scholar] [CrossRef]
94. Nemura, H.; Aoki, S.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Iritani, T.; Ishii, N.; Miyamoto, T.; Sasaki, K.; et al. Lambda-Nucleon and Sigma-Nucleon interactions from lattice QCD with
physical masses. arXiv 2017, arXiv:1702.00734. [Google Scholar]
95. Doi, T.; Aoki, S.; Doi, T.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Iritani, T.; Ishii, N.; Miyamoto, T.; et al. Baryon interactions from lattice QCD with physical masses—overview and S=0,
-4 sectors—. arXiv 2017, arXiv:1702.01600. [Google Scholar]
96. HALQCD Collaboration. Baryon interactions from lattice QCD with physical masses—S = -3 sector: ΞΣ and ΞΛ-ΞΣ—. PoS 2017, 256, 127. [Google Scholar]
97. HALQCD Collaboration. Baryon interactions from lattice QCD with physical masses—S = −2 sector—. arXiv 2017, arXiv:1702.06241. [Google Scholar]
98. Doi, T.; Aoki, S.; Doi, T.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Iritani, T.; Ishii, N.; Miyamoto, T.; et al. Baryon interactions from lattice QCD with physical quark masses—Nuclear
forces and ΞΞ forces—. EPJ Web. Conf. 2018, 175, 05009. [Google Scholar] [CrossRef]
99. Nemura, H.; Aoki, S.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Iritani, T.; Ishii, N.; Miyamoto, T.; Sasaki, K.; et al. Baryon interactions from lattice QCD with physical masses—strangeness
S = -1 sector—. EPJ Web. Conf. 2018, 175, 05030. [Google Scholar] [CrossRef]
100. Iritani, T.; Aoki, S.; Doi, T.; Etminan, F.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Ishii, N.; Miyamoto, T.; et al. NΩ dibaryon from lattice QCD near the physical point. Phys. Lett. B
2019, 792, 284–289. [Google Scholar] [CrossRef]
101. Iritani, T.; Aoki, S.; Doi, T.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Ishii, N.; Nemura, H.; Sasaki, K. Systematics of the HAL QCD potential at low energies in lattice QCD. Phys. Rev. D
2019, 99, 014514. [Google Scholar] [CrossRef] [Green Version]
102. Sasaki, K.; Aoki, S.; Doi, T.; Gongyo, S.; Hatsuda, T.; Ikeda, Y.; Inoue, T.; Iritani, T.; Ishiie, N.; Murano, K.; et al. ΛΛ and NΣ interactions from lattice QCD near the physical point. Nucl.
Phys. A 2020, 998, 121737. [Google Scholar] [CrossRef]
103. Beane, S.R.; Chang, E.; Cohen, S.D.; Detmold, W.; Lin, H.W.; Luu, T.C.; Orginos, K.; Parreño, A.; Savage, M.J.; Walker-Loud, A. Hyperon-nucleon interactions from Quantum Chromodynamics and the
composition of dense nuclear matter. Phys. Rev. Lett. 2012, 109, 172001. [Google Scholar] [CrossRef] [PubMed]
104. Orginos, K.; Parreno, A.; Savage, M.J.; Beane, S.R.; Chang, E.; Detmold, W. Two nucleon systems at m[π]∼450 MeV from lattice QCD. Phys. Rev. D 2015, 92, 114512. [Google Scholar] [CrossRef] [
Green Version]
105. Illa, M.; Beane, S.R.; Chang, E.; Davoudi, Z.; Detmold, W.; Murphy, D.J.; Orginos, K.; Parreño, A.; Savage, M.J.; Shanahan, P.E.; et al. Low-energy scattering and effective interactions of two
baryons at m[π]∼450 MeV from lattice quantum chromodynamics. Phys. Rev. D 2021, 103, 054508. [Google Scholar] [CrossRef]
106. Botta, E.; Bressani, T.; Garbarino, G. Strangeness nuclear physics: A critical review on selected topics. Eur. Phys. J. A 2012, 48, 41–64. [Google Scholar] [CrossRef] [Green Version]
107. Gal, A.; Hugerford, E.V.; Millener, D.J. Strangeness in nuclear physics. Rev. Mod. Phys. 2016, 88, 035004. [Google Scholar] [CrossRef]
108. Hugenford, E.V. Topics in strangeness nuclear physics. Lect. Notes Phys. 2007, 274, 1. [Google Scholar]
109. Takahashi, H.; Ahn, J.K.; Akikawa, H.; Aoki, S.; Arai, K.; Bahk, S.Y. Observation of a ΛΛ6He double hypernucleus. Phys. Rev. Lett. 2001, 87, 212502. [Google Scholar] [CrossRef] [PubMed]
110. Khaustov, P. Evidence of Ξ hypernuclear production in the ^12C(K^-,K^+)Ξ^-12Be reaction. Phys. Rev. C 2000, 61, 054603. [Google Scholar] [CrossRef] [Green Version]
111. Friedman, E.; Gal, A. Constraints on Ξ^- nuclear interactions from capture events in emulsion. Phys. Lett. B 2021, 820, 136555. [Google Scholar] [CrossRef]
112. Nakazawa, K.; Endo, Y.; Fukunaga, S.; Hoshino, K.; Hwang, S.H.; Imai, K.; Ito, H.; Itonaga, K.; Kanda, T.; Kawasaki, M.; et al. The first evidence of a deeply bound state of Ξ^--^14N system.
Prog. Theor. Exp. Phys. 2015, 2015, 033D02. [Google Scholar] [CrossRef] [Green Version]
113. Hiyima, E.; Nakazawa, K. Structure of S=-2 Hypernuclei and Hyperon–Hyperon Interactions. Ann. Rev. Nucl. Part. Sci. 2018, 68, 131–159. [Google Scholar] [CrossRef]
114. Hayakawa, S.H.; Agari, K.; Ahn, J.K.; Akaishi, T.; Akazawa, Y.; Ashikaga, S.; J-PARC E07 Collaboration. Observation of Coulomb-assisted nuclear bound state of Ξ^--^14N system. Phys. Rev. Lett.
2021, 126, 062501. [Google Scholar] [CrossRef]
115. Hashimoto, O.; Tamura, H. Spectroscopy of Λ hypernuclei. Prog. Part. Nucl. Phys. 2006, 57, 564. [Google Scholar] [CrossRef]
116. Ukai, M. γ-ray spectroscopy of Λ16O and Λ15N hypernuclei via the ^16O(K^-,π^-γ) reaction. Phys. Rev. C 2008, 77, 054315. [Google Scholar] [CrossRef]
117. Bauer, E.; Garbarino, G.; Parreño, A.; Ramos, A. Microscopic approach to the proton asymmetry in the nonmesonic weak decay of Λ hypernuclei. Phys. Rev. C 2012, 85, 024321. [Google Scholar] [
CrossRef] [Green Version]
118. Bauer, E.; Garbarino, G.; Rodríguez Peña, C.A. Nonmesonic weak decay of Λ hypernuclei: The ΛN-ΣN coupling. Phys. Rev. C 2017, 96, 044303. [Google Scholar] [CrossRef] [Green Version]
119. Parreño, A.; Bennhold, C.; Holstein, B.R. ΛN→NN weak interaction in effective-field theory. Phys. Rev. C 2004, 70, 051601. [Google Scholar] [CrossRef] [Green Version]
120. Pérez-Obiol, A.; Entem, D.R.; Juliá-Díaz, B.; Parreño, A. One-loop contributions in the effective field theory for the ΛN→NN transition. Phys. Rev. C 2013, 87, 044614. [Google Scholar] [CrossRef
] [Green Version]
121. Alberico, W.M.; Garbarino, G. Weak decay of Λ hypernuclei. Phys. Rep. 2002, 369, 1. [Google Scholar] [CrossRef] [Green Version]
122. Parreño, A. Weak decays of hypernuclei. Lect. Note Phys. 2007, 724, 141–189. [Google Scholar]
123. Alberico, W.M.; De Pace, A.; Garbarino, G.; Ramos, A. Weak decays of medium and heavy Λ hypernuclei. Phys. Rev. C 2000, 61, 044314. [Google Scholar] [CrossRef] [Green Version]
124. Bouyssy, A.; Hüfner, J. Hypernuclei with A≥12. Phys. Lett. B 1976, 27, 276. [Google Scholar] [CrossRef]
125. Bouyssy, A. Strangeness exchange reactions and hypernuclear spectroscopy. Phys. Lett. B 1979, 84, 41–45. [Google Scholar] [CrossRef]
126. Dover, C.D.; Liedking, L.; Walker, G.E. Hypernuclear physics with pions. Phys. Recv. C 1980, 22, 2073. [Google Scholar] [CrossRef]
127. Motoba, T.; Bandō, H.; Wünsch, R.; Žofka, J. Hypernuclear production by the (π^+,K^+) reaction. Phys. Rev. C 1988, 32, 1322. [Google Scholar] [CrossRef] [PubMed]
128. Boguta, J.; Bohrmann, S. Relativistic quantum field theory of a hypernuclei. Phys. Lett. B 1981, 102, 93–96. [Google Scholar] [CrossRef] [Green Version]
129. Mareš, J.; Žofka, J. On Λ-hyperon(s) in the nuclear medium. Z. Phys. A 1989, 333, 209. [Google Scholar]
130. Glendenning, N.K.; Von-Eiff, D.; Haft, M.; Lenske, H.; Weigel, M.K. Relativistic mean-field calculations of Λ and Σ hypernuclei. Phys. Rev. C 1993, 48, 889. [Google Scholar] [CrossRef] [PubMed]
131. Mareš, J.; Jennings, B.K. Relativistic description of Λ, Σ, and Ξ hypernuclei. Phys. Rev. C 1993, 49, 2472. [Google Scholar] [CrossRef]
132. Sugahara, Y.; Toki, H. Relativistic mean field theory for lambda hypernuclei and neutron stars. Prog. Theor. Phys. 1994, 92, 803–813. [Google Scholar] [CrossRef]
133. Lombard, R.J.; Marcos, S.; Mareš, J. Description of hypernuclei in the scalar derivative coupling model. Phys. Rev. C 1995, 51, 1784. [Google Scholar] [CrossRef]
134. Ma, Z.; Speth, J.; Krewald, S.; Chen, B.; Reuber, A. Hypernuclei with meson-exchange hyperon-nucleon interactions. Nucl. Phys. A 1996, 608, 305–315. [Google Scholar] [CrossRef]
135. Ineichenm, F.; Von-Eiff, D.; Weigel, M.K. A density-dependent relativistic Hartree approach for hypernuclei. J. Phys. G 1996, 22, 1421. [Google Scholar] [CrossRef]
136. Tsushima, K.; Saito, K.; Thomas, A.W. Self-consistent description of Λ hypernuclei in the quark-meson coupling model. Phys. Lett. B 1997, 411, 9–18. [Google Scholar] [CrossRef] [Green Version]
137. Tsushima, K.; Saito, K.; Haidenbauer, J.; Thomas, A.W. The quark-meson coupling model for Λ, Σ and Ξ hypernuclei. Nucl. Phys. A 1998, 630, 691–718. [Google Scholar] [CrossRef] [Green Version]
138. Brockmann, R.; Weise, W. Relativistic single particle motion and spin-orbit coupling in nuclei and hypernuclei. Nucl. Phys. A 1981, 355, 365–382. [Google Scholar] [CrossRef] [Green Version]
139. Chiapparini, M.; Gattone, A.O.; Jennings, B.K. Dirac phenomonology and the Λ-nucleus potential. Nucl. Phys. A 1991, 529, 589–597. [Google Scholar] [CrossRef]
140. Yamamoto, Y.; Bandō, H. Chapter II. baryon-baryon interactions and single-particle aspects of hypernuclei. Prog. Theor. Phys. Suppl. 1985, 81, 9–41. [Google Scholar] [CrossRef] [Green Version]
141. Yamamoto, Y.; Bandō, H. Hypernuclear properties derived from the Nijmegen soft-core OBE potential. Prog. Theor. Phys. 1990, 83, 254–264. [Google Scholar] [CrossRef] [Green Version]
142. Yamamoto, Y.; Reuber, A.; Himeno, H.; Nagata, S.; Motoba, T. Hypernuclear properties derived from the Jülich hyperon-nucleon interaction (in comparison with the Nijmegen interactions). Czec. J.
Phys. 1992, 42, 1249–1260. [Google Scholar] [CrossRef]
143. Yamamoto, Y.; Motoba, T.; Himeno, H.; Ikeda, K.; Nagata, S. Hyperon-nucleon and hyperon-hyperon interactions in nuclei. Prog. Theor. Phys. Suppl. 1994, 117, 361–389. [Google Scholar] [CrossRef]
144. Halderson, D. G-matrix calculations in finite hypernuclei. Phys. Rev. C 1993, 48, 581. [Google Scholar] [CrossRef]
145. Hjorth–Jensen, M.; Polls, A.; Ramos, A.; Müther, H. Self-energy of Λ in finite nuclei. Nucl. Phys. A 1996, 605, 458. [Google Scholar] [CrossRef] [Green Version]
146. Vidaña, I.; Polls, A.; Ramos, A.; Hjorth–Jensen, M. Hyperon properties in finite nuclei using realistic YN interactions. Nucl. Phys. A 1998, 644, 201–220. [Google Scholar] [CrossRef] [Green
147. Haidenbauer, J.; Vidaña, I. Structure of single-Λ hypernuclei with chiral hyperon-nucleon potentials. Eur. Phys. J. A 2020, 56, 55. [Google Scholar] [CrossRef] [Green Version]
148. Lonardoni, D.; Gandolfi, S.; Pederiva, F. Effects of the two-body and three-body hyperon-nucleon interactions in Λ hypernuclei. Phys. Rev. C 2013, 87, 041303(R). [Google Scholar] [CrossRef]
149. Beane, S.R.; Chang, E.; Cohen, S.D.; Detmold, W.; Lin, H.W.; Luu, T.C.; Orginos, K.; Parreño, A.; Savage, M.J.; Walker-Loud, A. Light nuclei and hypernuclei from quantum chromodynamics in the
limit of SU(3) flavor symmetry. Phys. Rev. D 2013, 87, 034506. [Google Scholar] [CrossRef] [Green Version]
150. Robertson, N.J.; Dickhoff, W.H. Correlation effects on Λ propagation in nuclear matter. Phys. Rev. C 2004, 70, 044301. [Google Scholar] [CrossRef]
151. Vidaña, I. Single-particle spectral function of the Λ hyperon in finite nuclei. Nucl. Phys. A 2017, 958, 48–70. [Google Scholar] [CrossRef] [Green Version]
152. Botta, E.; Bressani, T.; Felicello, A. On the binding energy and the charge symmetry breaking in A≤16 Λ-hypernuclei. Nucl. Phys. A 2017, 960, 165–179. [Google Scholar] [CrossRef] [Green Version]
153. Pile, P.H.; Bart, S.; Chrien, R.E.; Millener, D.J.; Sutter, R.J.; Tsoupas, N.; Peng, J.-C.; Mishra, C.S.; Hungerford, E.V.; Reidy, J.; et al. Study of hypernuclei by associated production. Phys.
Rev. Lett. 1991, 66, 2585. [Google Scholar] [CrossRef]
154. Ambartsumyan, V.A.; Saakyan, G.S. The degenerate superdense gas of elementary particles. Sov. Astron. 1960, 4, 187. [Google Scholar]
155. Champion, D.J.; Ransom, S.M.; Lazarus, P.; Camilo, F.; Bassa, C.; Kaspi, V.M.; Nice, D.J.; Freire, P.C.C.; Stairs, I.H.; van Leeuwen, J.; et al. An eccentric binary pulsar in the galatic plane.
Science 2008, 320, 1309–1312. [Google Scholar] [CrossRef] [PubMed] [Green Version]
156. Demorest, P.; Pennucci, T.; Ransom, S.M.; Roberts, M.S.E.; Hessels, J.W.T. A two-solar-mass neutron star measured using Shapiro delay. Nature 2010, 467, 1081–1083. [Google Scholar] [CrossRef]
157. Antoniadis, J.; Freire, P.C.; Wex, N.; Tauris, T.M.; Lynch, R.S.; Van Kerkwijk, M.H.; Kramer, M.; Bassa, C.; Dhillon, V.S. A massive pulsar in a compact relativistic binary. Science 2013, 340,
1233232. [Google Scholar] [CrossRef] [Green Version]
158. Cromartie, H.T.; Fonseca, E.; Ransom, S.M.; Demorest, P.B.; Arzoumanian, Z.; Blumer, H.; Brook, P.R.; DeCesar, E.M.; Dolch, T.; Ellis, J.A.; et al. Relativistic Shapiro delay measurements of an
extremely massive millisecond pulsar. Nat. Astron. 2019, 4, 72–76. [Google Scholar] [CrossRef] [Green Version]
159. Takatsuka, T.; Nishizaki, S.; Yamamoto, Y. Necessity of extra repulsion in hypernuclear systems: Suggestion from neutron stars. Eur. Phys. J. A 2002, 13, 213–215. [Google Scholar] [CrossRef]
160. Takatsuka, T.; Nishizaki, S.; Tamagaki, R. Three-body force as an extra repulsion suggested from hyperon- mixed neutron stars. Prog. Theor. Phys. Suppl. 2008, 174, 80–83. [Google Scholar] [
161. Vidaña, I.; Logoteta, D.; Providência, C.; Polls, A.; Bombaci, I. Estimation of the effect of hyperonic three-body forces on the maximum mass of neutron stars. Eur. Phys. Lett. 2011, 94, 11002.
[Google Scholar] [CrossRef] [Green Version]
162. Yamamoto, Y.; Furumotom, T.; Yasutake, B.; Rijken, T.A. Multi-Pomeron repulsion and the neutron-star mass. Phys. Rev. C 2013, 88, 022801. [Google Scholar] [CrossRef] [Green Version]
163. Yamamoto, Y.; Furumoto, T.; Yasutake, B.; Rijken, T.A. Hyperon mixing and universal many-body repulsion in neutron stars. Phys. Rev. C 2014, 90, 045805. [Google Scholar] [CrossRef] [Green
164. Lonardoni, D.; Lovato, A.; Gandolfi, S.; Pederiva, F. Hyperon puzzle: Hints from quantum Monte Carlo calculations. Phys. Rev. Lett. 2014, 114, 092301. [Google Scholar] [CrossRef] [PubMed] [Green
165. Yamamoto, Y.; Furumoto, T.; Yasutake, N.; Rijken, T.A. Hyperon-mixed neutron star with universal many-body repulsion. Eur. Phys. J. A 2016, 52, 19. [Google Scholar] [CrossRef] [Green Version]
166. Yamamoto, Y.; Togashi, H.; Tamagawa, T.; Furumoto, T.; Yasutake, N.; Rijken, T.A. Neutron-star radii based on realistic nuclear interactions. Phys. Rev. C 2017, 96, 065804. [Google Scholar] [
CrossRef] [Green Version]
167. Logoteta, D.; Vidaña, I.; Bombaci, I. Impact of chiral hyperonic three-body forces on neutron stars. Eur. Phys. J. A 2019, 55, 207. [Google Scholar] [CrossRef]
168. Burgio, G.F.; Baldo, M.; Sahu, P.K.; Schulze, H.-J. Hadron-quark phase transition in dense matter and neutron stars. Phys. Rev. C 2002, 66, 025802. [Google Scholar] [CrossRef] [Green Version]
169. Burgio, G.F.; Baldo, M.; Sahu, P.K.; Santra, A.B.; Schulze, H.-J. Maximum mass of neutron stars with a quark core. Phys. Lett. B 2002, 526, 19–26. [Google Scholar] [CrossRef] [Green Version]
170. Alford, M.; Blaschke, D.; Drago, A.; Klähn, T.; Pagliara, G.; Schaffner–Bielich, J. Quark matter in compact stars ? Nature 2007, 445, E7. [Google Scholar] [CrossRef] [PubMed]
171. Özel, F.; Psaltis, D.; Ransom, S.; Demorest, P.; Alford, M. The massive pulsar PSR J1614–2230: Linking quantum chromodynamics, gamma-ray bursts, and gravitational wave astronomy. Astrophys. J.
Lett. 2010, 724, L199. [Google Scholar] [CrossRef] [Green Version]
172. Weissenborn, S.; Sagert, I.; Pagliara, G.; Hempel, M.; Schaeffner–Bielich, J. Quark matter in massive compact stars. Astophys. J. Lett. 2011, 740, L14. [Google Scholar] [CrossRef]
173. Schramm, S.; Negreiros, R.; Stenheimer, J.; Schürhoff, T.; Dexheimer, V. Properties and stability of hybrid stars. Act. Phys. Pol. B 2012, 43, 749. [Google Scholar] [CrossRef]
174. Bonanno, L.; Sedrakian, A. Composition and stability of hybrid stars with hyperons and quark color-superconductivity. Astron. Astrophys. 2012, 539, A16. [Google Scholar] [CrossRef] [Green
175. Astowiecki, R.; Blaschke, D.; Grigorian, H.; Typel, S. Strangeness in the cores of neutron stars. Acta Phys. Polon. Suppl. 2012, 5, 535. [Google Scholar] [CrossRef]
176. Zdunik, J.L.; Haensel, P. Maximum mass of neutron stars and strange neutron-star cores. Astron. Astrophys. 2013, 551, A61. [Google Scholar] [CrossRef]
177. Klähn, T.; Blaschke, D.; Łastowiecki, D. Implications of the measurement of pulsars with two solar masses for quark matter in compact stars and heavy-ion collisions: A Nambu–Jona–Lasinio model
case study. Phys. Rev. D 2013, 88, 085001. [Google Scholar] [CrossRef] [Green Version]
178. Shahrbaf, M.; Blaschke, D.; Grunfeld, A.G.; Moshfegh, H.R. First-order phase transition from hypernuclear matter to deconfined quark matter obeying new constraints from compact stars. Phys. Rev.
C 2020, 101, 025807. [Google Scholar] [CrossRef] [Green Version]
179. Shahrbaf, M.; Blaschke, K.S. Mixed phase transition from hypernuclear matter to deconfined quark matter fulfilling mass-radius constraints of neutron stars. J. Phys. G Nucl. Part. Phys. 2020, 47
, 115201. [Google Scholar] [CrossRef]
180. Drago, A.; Lavagno, A.; Pagliara, G.; Pigato, D. The scenario of two families of compact stars. Part 1. Equations of state, mass-radius relations and binary systems. Eur. Phys. J. A 2016, 52,
40. [Google Scholar] [CrossRef] [Green Version]
181. Drago, A.; Lavagno, A.; Pagliara, G.; Pigato, D. The scenario of two families of compact stars. Part 2: Transition from hadronic to quark matter and explosive phenomena. Eur. Phys. J. A 2016, 52
, 41. [Google Scholar] [CrossRef]
182. Wiringa, R.B.; Stoks, V.G.J.; Schiavilla, R. Accurate nucleon-nucleon potential with charge-independence breaking. Phys. Rev. C 1995, 51, 38. [Google Scholar] [CrossRef] [PubMed] [Green Version]
183. Isaka, M.; Yamamoto, Y.; Rijken, T.A. Effects of a hyperonic many-body force on B[Λ] values of hypernuclei. Phys. Rev. C 2017, 95, 044308. [Google Scholar] [CrossRef] [Green Version]
184. Masuda, K.; Hatsuda, T.; Takatsuka, T. Hadron-quark crossover and massive hybrid stars with strangeness. Astrophys. J. 2013, 764, 12. [Google Scholar] [CrossRef] [Green Version]
185. Masuda, K.; Hatsuda, T.; Takatsuka, T. Hadron-quark crossover and massive hybrid stars. Prog. Theor. Exp. Phys. 2013, 7, 073D01. [Google Scholar]
186. Drago, A.; Lavagno, A.; Pagliara, G.; Pigato, D. Early appearance of Δ isobars in neutron stars. Phys. Rev. C 2014, 90, 065809. [Google Scholar] [CrossRef]
187. Ribes, P.; Ramos, A.; Tolós, L.; Gonzalez–Boquera, C.; Centelles, M. Interplay between Δ particles and hyperons in neutron stars. Astrophys. J. 2019, 883, 168. [Google Scholar] [CrossRef]
188. Kaplan, D.B.; Nelson, A.E. Strange goings on in dense nucleonic matter. Phys. Lett. B 1986, 175, 57–63. [Google Scholar] [CrossRef]
189. Kaplan, D.B.; Nelson, A.E. Erratum. Phys. Lett. B 1986, 179, 409. [Google Scholar]
190. Brown, G.E.; Lee, C.-H.; Rho, M.; Thorsson, V. From kaon-nuclear interactions to kaon condensation. Nucl. Phys. A 1994, 567, 937–956. [Google Scholar] [CrossRef] [Green Version]
191. Thorsson, V.; Prakash, M.; Lattimer, J.M. Composition, structure and evolution of neutron stars with kaon condensates. Nucl. Phys. A 1994, 572, 693–731. [Google Scholar] [CrossRef] [Green
192. Lee, C.-H. Kaon condensation in dense stellar matter. Phys. Rep. 1996, 275, 255–341. [Google Scholar] [CrossRef] [Green Version]
193. Glendenning, N.K.; Schaffner-Bielich, J. Kaon condensation and dynamical nucleons in neutron stars. Phys. Rev. Lett. 1998, 81, 4564. [Google Scholar] [CrossRef] [Green Version]
194. Keil, W.; Janka, H.-T. Hadronic phase transitions at supranuclear densities and the delayed collapse of newly formed neutron stars. Astron. Astrophys. 1996, 296, 145. [Google Scholar]
195. Bombaci, I. The maximum mass of a neutron star. Astron. Astrophys. 1996, 305, 871. [Google Scholar]
196. Prakash, M.; Bombaci, I.; Prakash, M.; Ellis, P.J.; Knorren, R.; Lattimer, J.M. Composition and structure of proto-neutron stars. Phys. Rep. 1997, 280, 1–77. [Google Scholar] [CrossRef] [Green
197. Vidaña, I.; Bombaci, I.; Polls, A.; Ramos, A. Microscopic study of neutrino trapping in hyperon stars. Astron. Astrophys. 2003, 399, 687–693. [Google Scholar] [CrossRef]
198. Burgio, G.F.; Schulze, H.-J.; Li, A. Hyperon stars at finite temperature in the Brueckner theory. Phys. Rev. C 2011, 83, 025804. [Google Scholar] [CrossRef] [Green Version]
199. Lattimer, J.M.; Pethick, C.J.; Prakash, M.; Haensel, P. Direct URCA process in neutron stars. Phys. Rev. Lett. 1991, 66, 2701. [Google Scholar] [CrossRef]
200. Balberg, S.; Barnea, N. S-wave pairing of Λ hyperons in dense matter. Phys. Rev. C 1998, 57, 409. [Google Scholar] [CrossRef] [Green Version]
201. Takatsuka, T.; Tamagaki, R. Superfluidity of Λ-hyperons admixed in neutron star cores. Prog. Theor. Phys. 1999, 102, 1043–1048. [Google Scholar] [CrossRef] [Green Version]
202. Takatsuka, T.; Nishizaki, S.; Yamamoto, Y.; Tamagaki, R. The possibility of hyperon superfluids in neutron star cores. Prog. Theor. Phys. 2000, 105, 179–184. [Google Scholar] [CrossRef] [Green
203. Takatsuka, T.; Nishizaki, S.; Yamamoto, Y.; Tamagaki, R. Superfluidity of hyperon-mixed neutron stars. Prog. Theor. Phys. Suppl. 2002, 146, 279–288. [Google Scholar] [CrossRef] [Green Version]
204. Vidaña, I.; Tolós, L. Superfluidity of Σ^- hyperons in β-stable neutron star matter. Phys. Rev. C 2004, 70, 028802. [Google Scholar] [CrossRef] [Green Version]
205. Zhou, X.-R.; Schulze, H.-J.; Pan, F.; Drayer, J.P. Strong hyperon-nucleon pairing in neutron stars. Phys. Rev. Lett. 2005, 95, 051101. [Google Scholar] [CrossRef] [PubMed] [Green Version]
206. Wag, Y.N.; Shen, H. Superfluidity of Λ-hyperons in neutron stars. Phys. Rev. C 2010, 81, 025801. [Google Scholar]
207. Lindblom, L. Estimates of the maximum angular velocity of rotating neutron stars. Astrophys. J. 1986, 303, 146–153. [Google Scholar] [CrossRef]
208. Friedman, J.L.; Ipser, J.R.; Parker, L. Rapidly rotating neutron star models. Astrophys. J. 1986, 304, 115–139. [Google Scholar] [CrossRef]
209. Lindblom, L. Critical angular velocities of rotating neutron stars. Astrophys. J. 1995, 438, 265–268. [Google Scholar] [CrossRef]
210. Anderson, N. A new class of unstable modes of rotating relativistic stars. Astrophys. J. 1998, 502, 708. [Google Scholar] [CrossRef] [Green Version]
211. Friedman, J.L.; Morsink, S.M. Axial instability of rotating relativistic stars. Astrophys. J. 1998, 502, 714. [Google Scholar] [CrossRef] [Green Version]
212. Chandrasekhar, S. Solutions of two problems in the theory of gravitational radiation. Phys. Rev. Lett. 1970, 24, 611. [Google Scholar] [CrossRef]
213. Friedman, J.L.; Schutz, B.F. Lagrangian perturbation theory of non-relativistic fluids. Astrophys. J. 1978, 221, 937–957. [Google Scholar] [CrossRef]
214. Friedman, J.L.; Schutz, B.F. Secular instability of rotating Newtonian stars. Astrophys. J. 1978, 222, 281–296. [Google Scholar] [CrossRef] [Green Version]
215. Langer, W.D.; Cameron, A.G.W. Effects of hyperons on the vibrations of neutron stars. Astrophys. Space Sci. 1969, 5, 213–253. [Google Scholar] [CrossRef]
216. Jones, P.B. Astrophysical significance of the dissipation of turbulence in a dense baryon fluid. Proc. R. Soc. Lond. A 1971, 323, 111–125. [Google Scholar]
217. Levin, Y. Runaway heating by R-modes of neutron stars in low-mass X-ray binaries. Astrophys. J. 1999, 517, 328. [Google Scholar] [CrossRef] [Green Version]
218. Jones, P.B. Comment on “gravitational radiation instability in hot young neutron stars”. Phys. Rev. Lett. 2001, 86, 1384. [Google Scholar] [CrossRef]
219. Jones, P.B. Bulk viscosity of neutron-star matter. Phys. Rev. D 2001, 64, 084003. [Google Scholar] [CrossRef]
220. Lindblom, L.; Owen, B.J. Effect of hyperon bulk viscosity on neutron star r-modes. Phys. Rev. D 2002, 65, 0653006. [Google Scholar] [CrossRef] [Green Version]
221. Haensel, P.; Levenfish, K.P.; Yakovlev, D.G. Bulk viscosity in superfluid neutron star cores. Astron. Astrophys. 2002, 381, 1080–1089. [Google Scholar] [CrossRef]
222. Van Dalen, E.N.E.; Dieperink, A.E. Bulk viscosity in neutron stars from hyperons. Phys. Rev. C 2004, 69, 025802. [Google Scholar] [CrossRef] [Green Version]
223. Chatterjee, D.; Bandyopadhyay, D. Effect of hyperon-hyperon interaction on bulk viscosity and r-mode instability in neutron stars. Phys. Rev. D 2006, 74, 023003. [Google Scholar] [CrossRef] [
Green Version]
224. Bondarescu, R.; Teukolsky, S.A.; Wasserman, I. Spin evolution of accreting neutron stars: Nonlinear development of the r-mode instability. Phys. Rev. D 2007, 76, 064019. [Google Scholar] [
CrossRef] [Green Version]
225. Chatterjee, D.; Bandyopadhyay, D. Hyperon bulk viscosity in the presence of antikaon condensate. Astrophys. J. 2008, 680, 686. [Google Scholar] [CrossRef]
226. Gusakov, M.E.; Kantor, E.M. Bulk viscosity of superfluid hyperon stars. Phys. Rev. D 2008, 78, 083006. [Google Scholar] [CrossRef] [Green Version]
227. Sinha, M.; Bandyopadhyay, D. Hyperon bulk viscosity in strong magnetic fields. Phys. Rev. D 2009, 79, 123001. [Google Scholar] [CrossRef] [Green Version]
228. Patruno, A. The accreting millisecond X-ray pulsar IGR J00291 + 5934: Evidence for a long timescale Spin evolution. Astrophys. J. 2010, 722, 909. [Google Scholar] [CrossRef] [Green Version]
229. Jha, T.K.; Mishra, H.; Sreekanth, V. Bulk viscosity in a hyperonic star and r-mode instability. Phys. Rev. C 2010, 82, 025803. [Google Scholar] [CrossRef] [Green Version]
Figure 1.
Momentum transferred to the
as a function of the incident particle momentum for the elementary process
$n ( K − , π − ) Λ$
$n ( π + , K + ) Λ$
$p ( γ , K + ) Λ$
$0 0$
. Figure adapted from [
Figure 2. Energy of a $Λ$ hyperon in the single-particle states $s , p , d , f$ and g of several hypernuclei as a function of $A − 2 / 3$ deduced from emulsion, $( K − , π − )$ and $( π + , K + )$
reactions. The lines are drawn just to help the reader.
Figure 3.
Level scheme and
-ray transitions and
$Λ 16$
O measured at BNL. Figure adapted from [
Figure 4.
Weak decay rate
as a function of the total number of particles in units of the weak decay rate of the
in free space
$Γ Λ free$
. Dot, dashed and solid lines show, respectively, the theoretical predictions of the mesonic
$Γ M$
, non-mesonic
$Γ N M$
and total
$Γ T$
decay rates. Dot-dashed lines labeled
$Γ 1$
$Γ 2$
display the contributions of one-nucleon and two-nucleon induced decay modes to the non-mesonic decay rate (see Equations (
) and (
)). Experimental values of the total and non-mesonic decay rates are given by the squares and circle marks, respectively. Figure adapted from the original one in [
Figure 5. BHF approximation of the finite nucleus $Λ$ self-energy (a), split into the sum of a first order contribution (b) and a second order 2p1h correction (c).
Figure 6.
Gravitational mass as a function of the baryonic mass for neutrino-free (solid lines) and neutrino-trapped (dashed lines) matter. Panel (
) shows the results for matter containing nucleons and hyperons, whereas the results for pure nucleonic matter are shown in panel (
). Dotted horizontal and vertical lines show the window of metastability in the gravitational and baryonic masses. Figure adapted from [
Figure 7. Panel (a): r-mode instability region for a pure nucleonic and a hyperonic star with $1.27 M ⊙$. The frequency of the mode is taken as $ω = 10 4$ s$− 1$. Panel (b): Bulk viscosity as a
function of the density for $T = 10 9$ K and $ω = 10 4$ s$− 1$. Contributions from the direct and modified nucleonic Urca processes as well as from the weak non-leptonic process $n + n ↔ p + Σ −$ are
Table 1.
Energy of the
single-particle bound states for several hypernuclei from
$Λ 5$
He to
$Λ 209$
Pb. Results are shown for the chiral
$Y N$
interactions NLO13 [
] and NLO19 [
] of the Jülich–Bonn–Munich group for different values of the cutoff of the interaction. Available experimental data [
] for the closest measured hypernuclei are included.
The weak signal for
$Λ 40$
Ca [
] is not included in the recent compilation in [
NLO13 NLO19 Exp.
Cutoff (MeV) 500 550 600 650 700 500 550 600 650 700
$Λ 5$He $Λ 5$He
$s 1 / 2$ $− 0.73$ $− 0.15$ $− 0.63$ $− 2.36$ $− 4.90$ $− 2.16$ $− 1.36$ $− 1.77$ $− 3.42$ $− 5.63$ $− 3.12 ( 2 )$
$Λ 13$C $Λ 13$C
$s 1 / 2$ $− 4.44$ $− 2.24$ $− 3.72$ $− 8.91$ $− 13.40$ $− 8.91$ $− 6.42$ $− 7.22$ $− 10.81$ $− 14.98$ $− 11.69 ( 12 )$
$p 3 / 2$ − − − − $− 1.22$ − − − $− 0.12$ $− 1.76$ $− 0.8 ( 3 )$ (p)
$p 1 / 2$ − − − − $− 0.97$ − − − − $− 1.40$
$Λ 17$O $Λ 16$O
$s 1 / 2$ $− 6.07$ $− 3.46$ $− 5.35$ $− 10.51$ $− 16.37$ $− 11.46$ $− 8.61$ $− 9.55$ $− 13.60$ $− 18.18$ $− 13.0 ( 2 )$
$p 3 / 2$ − − − $− 1.22$ $− 4.04$ $− 1.26$ $− 0.14$ $− 0.53$ $− 2.40$ $− 4.89$ $− 2.5 ( 2 )$ (p)
$p 1 / 2$ − − − $− 0.66$ $− 3.31$ $− 0.51$ − − $− 1.69$ $− 4.10$
$Λ 41$Ca $Λ 40$Ca
$s 1 / 2$ $− 12.37$ $− 8.78$ $− 11.24$ $− 17.56$ $− 24.36$ $− 19.51$ $− 15.86$ $− 16.80$ $− 21.30$ $− 26.47$ $− 18.7 ( 1.1 ) †$
$p 3 / 2$ $− 4.95$ $− 2.54$ $− 3.98$ $− 8.82$ $− 13.43$ $− 9.91$ $− 6.93$ $− 7.48$ $− 11.04$ $− 15.06$ $− 11.0 ( 5 )$ (p)
$p 1 / 2$ $− 4.37$ $− 2.08$ $− 3.50$ $− 7.73$ $− 12.87$ $− 9.13$ $− 6.23$ $− 6.82$ $− 10.42$ $− 14.47$
$d 5 / 2$ − − − $− 0.40$ $− 3.59$ $− 1.47$ − − $− 1.99$ $− 4.67$ $− 1.0 ( 5 )$ (d)
$d 3 / 2$ − − − $− 0.50$ $− 4.02$ $− 0.56$ − − $− 1.20$ $− 3.84$
$Λ 91$Zr $Λ 89$Y
$s 1 / 2$ $− 19.36$ $− 14.66$ $− 17.83$ $− 25.10$ $− 32.50$ $− 27.72$ $− 22.57$ $− 23.19$ $− 28.94$ $− 34.61$ $− 23.6 ( 5 )$
$p 3 / 2$ $− 14.24$ $− 10.59$ $− 13.27$ $− 19.27$ $− 25.45$ $− 20.59$ $− 16.24$ $− 16.94$ $− 22.05$ $− 26.96$ $17.7 ( 6 )$ (p)
$p 1 / 2$ $− 13.95$ $− 10.39$ $− 13.05$ $− 19.07$ $− 25.31$ $− 20.45$ $− 15.96$ $− 16.67$ $− 21.86$ $− 26.82$
$d 5 / 2$ $− 6.21$ $− 3.33$ $− 5.24$ $− 10.30$ $− 15.27$ $− 11.92$ $− 8.10$ $− 8.44$ $− 12.68$ $− 16.78$ $− 10.9 ( 6 )$ (d)
$d 3 / 2$ $− 5.80$ $− 2.98$ $− 4.88$ $− 9.70$ $− 14.97$ $− 11.65$ $− 7.61$ $− 7.98$ $− 12.27$ $− 16.40$
$f 7 / 2$ − − − $− 1.68$ $− 5.63$ $− 4.04$ $− 0.98$ $− 0.89$ $− 3.97$ $− 7.04$ $− 3.7 ( 6 )$ (f)
$f 5 / 2$ − − − $− 1.28$ $− 5.23$ $− 3.59$ $− 0.33$ $− 0.28$ $3.39$ $− 6.54$
$Λ 209$Pb $Λ 208$Pb
$s 1 / 2$ $− 25.75$ $− 21.41$ $− 25.09$ $− 32.28$ $− 39.51$ $− 36.28$ $− 29.50$ $− 29.60$ $− 35.84$ $− 41.58$ $− 26.9 ( 8 )$
$p 3 / 2$ $− 21.88$ $− 15.77$ $− 18.33$ $− 25.13$ $− 31.83$ $− 33.72$ $− 26.73$ $− 25.27$ $− 30.26$ $− 34.71$ $− 22.5 ( 6 )$ (p)
$p 1 / 2$ $− 21.55$ $− 15.53$ $− 18.14$ $− 25.00$ $− 31.74$ $− 33.58$ $− 26.57$ $− 25.13$ $− 30.17$ $− 34.64$
$d 5 / 2$ $− 14.47$ $− 8.79$ $− 9.96$ $− 14.78$ $− 19.98$ $− 25.49$ $− 19.28$ $− 16.84$ $− 20.08$ $− 23.15$ $− 17.4 ( 7 )$ (d)
$d 3 / 2$ $− 14.35$ $− 8.71$ $− 9.83$ $− 14.62$ $− 19.83$ $− 25.29$ $− 18.98$ $− 16.57$ $− 19.85$ $− 22.97$
$f 7 / 2$ $− 4.46$ − − $− 5.91$ $− 12.57$ $− 16.23$ $− 10.15$ $− 7.91$ $− 11.90$ $− 15.80$ $− 12.3 ( 6 )$ (f)
$f 5 / 2$ $− 4.42$ − − $− 5.60$ $− 12.24$ $− 15.96$ $− 9.70$ $− 7.47$ $− 11.47$ $− 15.38$
$g 9 / 2$ $− 1.87$ − − $− 3.23$ $− 9.21$ $− 13.72$ $− 7.55$ $− 5.18$ $− 8.92$ $− 12.32$ $− 7.2 ( 6 )$ (g)
$g 7 / 2$ $− 1.38$ − − $− 2.91$ $− 8.94$ $− 13.38$ $− 7.03$ $− 4.69$ $− 8.53$ $− 12.00$
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:/
Share and Cite
MDPI and ACS Style
Vidaña, I. Hyperons in Finite and Infinite Nuclear Systems. Universe 2021, 7, 376. https://doi.org/10.3390/universe7100376
AMA Style
Vidaña I. Hyperons in Finite and Infinite Nuclear Systems. Universe. 2021; 7(10):376. https://doi.org/10.3390/universe7100376
Chicago/Turabian Style
Vidaña, Isaac. 2021. "Hyperons in Finite and Infinite Nuclear Systems" Universe 7, no. 10: 376. https://doi.org/10.3390/universe7100376
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
Article Metrics | {"url":"https://www.mdpi.com/2218-1997/7/10/376","timestamp":"2024-11-13T21:58:46Z","content_type":"text/html","content_length":"793175","record_id":"<urn:uuid:aac97233-9a54-4c59-8250-fe3e027a6708>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00742.warc.gz"} |
MTA 07005
Journal Home Page
Cumulative Index
List of all Volumes
Complete Contents
of this Volume
Next Article
Minimax Theory and its Applications 07 (2022), No. 1, 119--130
Copyright Heldermann Verlag 2022
Approximate Solutions to Nonsmooth Multiobjective Programming Problems
Mohammad Golestani
Dept. of Mathematics, Fasa University, Fasa, Iran
We consider a multiobjective mathematical programming problem with inequality and equality constraints, where all functions are locally Lipschitz. An approximate strong Karush-Kuhn-Tucker (ASKKT for
short) condition is defined and we show that every local efficient solution is an ASKKT point without any additional condition. Then a nonsmooth version of cone-continuity regularity is defined for
this kind of problem. It is revealed that every ASKKT point under the cone-continuity regularity is a strong Karush-Kuhn-Tucker (SKKT for short) point. Correspondingly, the ASKKTs and the
cone-continuity property are defined and the relations between them are investigated.
Keywords: Mathematical programming, optimality conditions, nonlinear programming, nonsmooth analysis and approximate conditions.
MSC: 90C46, 90C30, 90C29, 49J52.
[ Fulltext-pdf (118 KB)] for subscribers only. | {"url":"https://www.heldermann.de/MTA/MTA07/MTA071/mta07005.htm","timestamp":"2024-11-07T16:47:03Z","content_type":"text/html","content_length":"3201","record_id":"<urn:uuid:5585cd9d-eaad-4f5b-9d0b-2d08d5a58708>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00454.warc.gz"} |
Grade 11 – Functions Mathematics – Canada
Grade 11 – Functions Mathematics – Canada
Grade 11 – Functions Mathematics
# TOPIC TITLE
1 Study Plan Study plan – Grade 11 – Functions
Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision.
2 Surds/Radicals Introducing surds
Objective: To recognise and simplify numerical expressions involving surds
3 Surds/Radicals Some rules for the operations with surds
Objective: To learn rules for the division and multiplication of surds
4 Surds/Radicals Simplifying Surds
Objective: To simplify numerical expressions and solve equations involving surds
5 Surds/Radicals Creating entire surds
Objective: To write numbers as entire surds and compare numbers by writing as entire surds
6 Surds/Radicals Adding and subtracting like surds
Objective: To add and subtract surds and simplify expressions by collecting like surds
7 Surds/Radicals Expanding surds
Objective: To expand and simplify binomial expressions involving surds
8 Surds/Radicals Binomial expansions
Objective: To expand and simplify the squares of binomial sums and differences involving surds
9 Surds/Radicals Conjugate binomials with surds
Objective: To expand and simplify products of conjugate binomial expressions
10 Surds/Radicals Rationalising the denominator
Objective: To rationalise the denominator of a fraction where the denominator is a monomial surd
11 Surds/Radicals Rationalising binomial denominators
Objective: To rationalise the denominator of a fraction when the denominator is a binomial with surds
12 Algebra – Basic Simplifying easy algebraic fractions
Objective: To simplify simple algebraic fractions using cancellation of common factors
13 Algebra – Basic Simplifying algebraic fractions using the Index Laws
Objective: To use the index laws for division to simplify algebraic fractions
14 Algebra – Basic Algebraic fractions resulting in negative Indices
Objective: To simplify algebraic fractions using negative indices (as required) in the answer
15 Algebra – Basic Factorisation of algebraic fractions including binomials
Objective: To simplify algebraic fractions requiring the factorisation of binomial expressions
16 Algebra – Basic Cancelling binomial factors in algebraic fractions
Objective: To simplify algebraic fractions with binomials in both the numerator and denominator
17 Indices/Exponents Adding indices when multiplying terms with the same base
Objective: To add indices when multiplying powers that have the same base
18 Indices/Exponents Subtracting indices when dividing terms with the same base
Objective: To subtract indices when dividing powers of the same base
19 Indices/Exponents Multiplying indices when raising a power to a power
Objective: To multiply indices when raising a power to a power
20 Indices/Exponents Multiplying indices when raising to more than one term
Objective: To raise power products to a power
21 Indices/Exponents Terms raised to the power of zero
Objective: To evaluate expressions where quantities are raised to the power 0
22 Indices/Exponents Negative Indices
Objective: To evaluate or simplify expressions containing negative indices
23 Indices/Exponents Fractional Indices
Objective: To evaluate or simplify expressions containing fractional indices
24 Indices/Exponents Complex fractions as indices
Objective: To evaluate or simplify expressions containing complex fractional indices and radicals
25 Graphs part 1 The parabola: to describe properties of a parabola from its equation
Objective: To describe properties of a parabola from its equation and sketch the parabola
26 Graphs part 1 Quadratic Polynomials of the form y = ax^2 + bx + c
Objective: To describe and sketch parabolas of the form y = x^2 + bx + c
27 Graphs part 1 Graphing perfect squares: y=(a-x) squared
Objective: To describe and sketch parabolas of the form y = (x – a)^2
28 Graphs part 1 Graphing irrational roots
Objective: To determine the vertex (using -b/2a), and other derived properties, to sketch a parabola
29 Graphs part 1 Solving Simultaneous Equations graphically
Objective: To solve simultaneous equations graphically
30 Algebra – Quadratic equations Solving Quadratic Equations
Objective: To solve quadratic equations that need to be changed into the form ax^2 + bx + c = 0
31 Algebra – Quadratic equations Completing the square
Objective: To complete an incomplete square
32 Algebra – Quadratic equations Solving Quadratic Equations by Completing the Square
Objective: To solve quadratic equations by completing the square
33 Algebra – Quadratic equations The Quadratic Formula
Objective: To find the roots of a quadratic equation by using the quadratic formula
34 Algebra – Quadratic equations Problem solving with quadratic equations
Objective: To solve problems which require finding the roots of a quadratic equation
35 Algebra – Quadratic equations Solving Simultaneous Quadratic Equations Graphically
Objective: To determine points of intersection of quadratic and linear equations
36 Logarithms Powers of 2
Objective: To convert between logarithm statements and indice statements
37 Logarithms Equations of type log x to the base 3 = 4
Objective: To find the value of x in a statement of type log x to the base 3 = 4
38 Logarithms Equations of type log 32 to the base x = 5
Objective: To solve Logrithmic Equation where the variable is the base x = 5
39 Graphs part 2 Graphing complex polynomials: quadratics with no real roots
Objective: To graph quadratics that have no real roots, hence don’t cut the x-axis
40 Graphs part 2 General equation of a circle: determine and graph the equation
Objective: To determine and graph the equation of a circle with radius a and centre (h,k)
41 Graphs part 2 Graphing cubic curves
Objective: To graph cubic curves whose equation is of the form y = (x – a)^3 + b or y = (a – x)^3 + b
42 Graphs part 2 Absolute Value Equations
Objective: To graph equations involving absolute values
43 Graphs part 2 The Rectangular Hyperbola
Objective: To graph rectangular hyperbolae whose equations are of the form xy = a and y = a/x
44 Graphs part 2 The Exponential Function
Objective: To graph exponential curves whose exponents are either positive or negative
45 Graphs part 2 Logarithmic Functions
Objective: To graph and describe log curves whose equations are of the form y = log (ax + b)
46 Conic sections Introduction to Conic Sections and Their General Equation
Objective: To identify the conic from its equation by examining the coefficients of x^2 and y^2
47 Conic sections The Parabola
Objective: To examine the properties of parabolas of the forms x^2 = 4py and y^2 = 4px
48 Conic sections Circles
Objective: To graph circles of the form x^2 + y^2 = r^2 and to form the equation of the given circles
49 Conic sections The Ellipsis
Objective: To identify ellipses of the form x^2/a^2 + y^2/b^2 = 1 and to find the equation of ellipses
50 Conic sections The Hyperbola
Objective: To find the equation of a hyperbola and to derive properties (e.g. vertex) from its equation
51 Function Functions and Relations: domain and range
Objective: To identify and represent functions and relations
52 Function Function Notation
Objective: To write and evaluate functions using function notation
53 Function Selecting Appropriate Domain and Range
Objective: To determine appropriate domains for functions
54 Function Domain and Range from Graphical Representations
Objective: To determine the range of a function from its graphical representation
55 Function Evaluating and Graphing Piecewise Functions
Objective: To evaluate and graph piecewise functions
56 Function Combining Functions
Objective: To determine the resultant function after functions have been combined by plus, minus, times and divide
57 Function Simplifying Composite Functions
Objective: To simplify, evaluate and determine the domain of composite functions
58 Function Inverse Functions
Objective: To find the inverse of a function and determine whether this inverse is itself a function
59 Function Graphing Rational Functions Part 1
Objective: To determine asymptotes and graph rational functions using intercepts and asymptotes
60 Function Graphing Rational Functions Part 2
Objective: To determine asymptotes and graph rational functions
61 Function Parametric Equations
Objective: To interchange parametric and Cartesian equations and to identify graphs
62 Function Polynomial Addition: in Combining and Simplifying Functions
Objective: To evaluate, simplify and graph rational functions
63 Function Parametric Functions
Objective: To change Cartesian and parametric equations and to graph parametric functions
64 Polynomials Introduction to polynomials
Objective: To define polynomials by degree, leading term, leading coefficient, constant term and monic
65 Polynomials The Sum, Difference and Product of Two Polynomials
Objective: To add, subtract and multiply polynomials
66 Series and sequences part 1 General Sequences
Objective: To use the general form of the n’th term of a sequence to find the first 3 terms
67 Series and sequences part 1 Finding Tn Given Sn
Objective: To find the value of the n’th term in a sequence given the sum of the first n terms
68 Series and sequences part 1 The Arithmetic Progression
Objective: To find the common difference of a given arithmetic progression
69 Series and sequences part 1 Finding the position of a term in an A.P.
Objective: To find the position of a term in a sequence, given an arithmetic progression and a value term
70 Series and sequences part 1 Given two terms of A.P. find the sequence
Objective: To find the first term and the common difference in an A.P. given the values and positions of two terms
71 Series and sequences part 1 Arithmetic Means
Objective: To find the arithmetic mean of two values
72 Series and sequences part 1 The sum to n terms of an A.P.
Objective: To find the sum of n terms of an arithmetic progression given the first three terms
73 Series and sequences part 1 The Geometric Progression
Objective: To find the common ratio of a given geometric progression
74 Series and sequences part 1 Finding the position of a term in a G.P.
Objective: To find the place of a term in a given geometric progression
75 Series and sequences part 1 Given two terms of G.P. find the sequence
Objective: To find the first term given two terms of a geometric progression
76 Series and sequences part 2 Geometric Means
Objective: To find geometric means of a and b and insert geometric means between 2 endpoints
77 Series and sequences part 2 The sum to n terms of a G.P.
Objective: To find the sum of n terms of a sequence
78 Series and sequences part 2 Sigma notation
Objective: To evaluate progressions using sigma notation
79 Series and sequences part 2 Limiting Sum or Sum to Infinity
Objective: To find the limiting sum of a sequence
80 Series and sequences part 2 Recurring Decimals and the Infinite G.P.
Objective: To express recurring decimals as a G.P. and to express the limiting sum as a fraction
81 Series and sequences part 2 Compound Interest
Objective: To calculate the compound interest of an investment using A=P(1+r/100)^n
82 Series and sequences part 2 Superannuation
Objective: To calculate the end value of adding a regular amount to a fund with stable interest paid over time
83 Series and sequences part 2 Time Payments
Objective: To calculate the payments required to pay off a loan
84 Series and sequences part 2 Applications of arithmetic sequences
Objective: To learn about practical situations with arithmetic series
85 Probability The Binomial Theorem and Binomial Coefficients
Objective: To calculate binomial coefficients and expand binomial powers.
86 Probability Binomial probabilities using the Binomial Theorem
Objective: To calculate the binomial probability of a given number of successful trials
87 Trigonometry part 1 Trigonometric Ratios
Objective: To name the sides of a right-angled triangle and to determine the trig ratios of an angle
88 Trigonometry part 1 Using the Calculator
Objective: To determine trigonometric ratios using a calculator
89 Trigonometry part 1 Using the Trigonometric Ratios to find unknown length [Case 1 Sin]
Objective: To use the sine ratio to calculate the opposite side of a right-angled triangle
90 Trigonometry part 1 Using the Trigonometric Ratios to find unknown length [Case 2 Cosine]
Objective: To use the cosine ratio to calculate the adjacent side of a right-angle triangle
91 Trigonometry part 1 Using the Trigonometric Ratios to find unknown length [Case 3 Tangent Ratio]
Objective: To use the tangent ratio to calculate the opposite side of a right-angled triangle
92 Trigonometry part 1 Unknown in the Denominator [Case 4]
Objective: To use trigonometry to find sides of a right-angled triangle and the Unknown in denominator
93 Trigonometry part 1 Bearings: The Compass
Objective: To change from true bearings to compass bearings and vice versa
94 Trigonometry part 1 Angles of Elevation and Depression
Objective: To identify and distinguish between angles of depression and elevation
95 Trigonometry part 1 Trigonometric Ratios in Practical Situations
Objective: To solve problems involving bearings and angles of elevation and depression
96 Trigonometry part 1 Using the Calculator to Find an Angle Given a Trigonometric Ratio
Objective: To find angles in right-angled triangles given trigonometric ratios
97 Trigonometry part 1 Using the Trigonometric Ratios to Find an Angle in a Right-Angled Triangle
Objective: To use trigonometric ratios to determine angles in right-angled triangles and in problems
98 Trigonometry part 1 Trigonometric Ratios of 30, 45 and 60 Degrees: Exact Ratios
Objective: To determine the exact values of sin, cos and tan of 30, 45 and 60 degrees
99 Trigonometry part 1 The Cosine Rule to find an unknown side [Case 1 SAS]
Objective: To complete the cosine rule to find a subject side for given triangles
100 Trigonometry part 1 The Sine Rule to find an unknown side: Case 1
Objective: To complete the cosine rule to find a subject angle for given triangles
101 Trigonometry part 1 The Sine Rule: Finding a Side
Objective: To find an unknown side of a triangle using the sine rule
102 Trigonometry part 1 The Sine Rule: Finding an Angle
Objective: To find an unknown angle of a triangle using the sine rule
103 Trigonometry part 2 Reciprocal Ratios
Objective: To find the trigonometric ratios for a given right-angled triangle
104 Trigonometry part 2 Complementary Angle Results
Objective: To use complementary angle ratios to find an unknown angle given a trigonometric equality
105 Trigonometry part 2 Trigonometric Identities
Objective: To simplify expressions using trigonometric equalities
106 Trigonometry part 2 Angles of Any Magnitude
Objective: To assign angles to quadrants and to find trigonometric values for angles
107 Trigonometry part 2 Trigonometric ratios of 0°, 90°, 180°, 270° and 360°
Objective: To find trigonometric ratios of 0, 90, 180, 270 and 360 degrees
108 Trigonometry part 2 Graphing the Trigonometric Ratios I: Sine Curve
Objective: To recognise the sine curve and explore shifts of phase and amplitude
109 Trigonometry part 2 Graphing the Trigonometric Ratios II: Cosine Curve
Objective: To recognise the cosine curve and explore shifts of phase and amplitude
110 Trigonometry part 2 Graphing the Trigonometric Ratios III: Tangent Curve
Objective: To recognise the tangent curve and explore shifts of phase and amplitude
111 Trigonometry part 2 Graphing the Trigonometric Ratios IV: Reciprocal Ratios
Objective: To graph the primary trigonometric functions and their inverses
112 Trigonometry part 2 Using One Trig. Ratio to Find Another
Objective: To derive trig ratios complement from one given trig ratio + some other quadrant identifier.
113 Exam Exam – Grade 11 – Functions
Objective: Exam | {"url":"https://www.futureschool.com/canada-curriculum/mathematics-grade-11-functions/","timestamp":"2024-11-06T20:12:03Z","content_type":"text/html","content_length":"82618","record_id":"<urn:uuid:1d996b0d-fbcf-466b-8670-5af90c26e114>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00081.warc.gz"} |
Math Contest Repository
POTW #10 - March 3, 2024
MCR Problem of the Week, March 3, 2024 Edition
Let $S_n = 1 \cdot (n-1) + 2 \cdot (n-2) + 3 \cdot (n-3) + \cdots + (n-1) \cdot 1, \quad n \geq 4.$
Then find the value of $\left\lceil\sum_{n=4}^{\infty} \left( 2 \frac{S_n}{n!} - \frac{1}{(n-2)!} \right)\right\rceil$
This problem was borrowed from byjus.com.
Please login or sign up to submit and check if your answer is correct.
flag Report Content
You should report content if:
• It may be offensive.
• There is something wrong with it (statement or difficulty value)
• It isn't original.
Thanks for keeping the Math Contest Repository a clean and safe environment! | {"url":"https://mathcontestrepository.pythonanywhere.com/problem/potw10/","timestamp":"2024-11-04T04:57:10Z","content_type":"text/html","content_length":"9849","record_id":"<urn:uuid:36a5171d-18ac-4aba-a200-93da70286498>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00461.warc.gz"} |
Comments on Various Consequences: Recurrence, Averaging and PredictabilityA Rigorous ODE solver and Smale's 14th Problem...Interesting sounding paper, from the Abstract: Non...These neat little examples on parameter continuati...Yes interesting paper with many details .
They fin...Tom said: Focusing a moment on the "simple&qu...I just meant that I'm actually going to add a ...Argh I am muddled today .
The end should read N = ...Of course there is a nonsense in the above that I ...Joshua
My skills and experience with numerical me...Good paper Tom; thanks. I was going to approach i...Joshua :
I think I'll get to a forced Lorenz ...Is that a moment closure technique that Tom/Tomas ...Thanks, That answers my question. So it is really ...WHT, you must be spying on me; I've been noodl...I think this is very important work, essentially s...Found it; it went to the spam bucket.
Here's ...The comment I wanted to add was that this result c...Thanks again for your thoughtful comments Tom; I r...Very informed and excellent post !
I just found it...Dr Pielke has a post up about recent news coverage...
tag:blogger.com,1999:blog-5822805028291837738.post5211543931454957413..comments2024-08-04T05:45:32.347-04:00Joshua Stultshttp://www.blogger.com/profile/
03506970399027046387noreply@blogger.comBlogger20125tag:blogger.com,1999:blog-5822805028291837738.post-35098661854914870582012-08-10T06:42:15.207-04:002012-08-10T06:42:15.207-04:00<a href="http://
www2.math.uu.se/~warwick/main/rodes.html" rel="nofollow">A Rigorous ODE solver and Smale's 14th Problem</a>Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-76588001809116420252011-10-15T10:56:06.064-04:002011-10-15T10:56:06.064-04:00Interesting sounding paper, from
the Abstract: <i>Nonlinear systems driven by noise and periodic forces with more than one frequency exhibit the phenomenon of Ghost Stochastic Resonance (GSR) found in a wide and disparate variety of
fields ranging from biology to geophysics. The common novel feature is the emergence of a "ghost" frequency in the system's output which it is absent in the input. As reviewed here, the
uncovering of this phenomenon helped to understand a range of problems, from the perception of pitch in complex sounds or visual stimuli, to the explanation of climate cycles. Recent theoretical
efforts show that a simple mechanism with two ingredients are at work in all these observations. The first one is the linear interference between the periodic inputs and the second a nonlinear
detection of the largest constructive interferences, involving a noisy threshold. These notes are dedicated to review the main aspects of this phenomenon, as well as its different manifestations
described on a bewildering variety of systems ranging from neurons, semiconductor lasers, electronic circuits to models of glacial climate cycles. </i><br /><a href="http://arxiv.org/abs/1110.0136"
rel="nofollow">The Ghost of Stochastic Resonance: an Introductory Review</a>Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-88963493100889117822011-03-08T11:14:43.047-05:002011-03-08T11:14:43.047-05:00These neat little <a href="http:
//www.mathworks.com/help/techdoc/math/f1-713877.html#brfhdqp-1" rel="nofollow">examples on parameter continuation</a> were linked from the Maxima list in a completely unrelated context, but goes to
the IVP / BVP distinction. My question to folks at Dr Curry's site about why GCM's aren't using all the standard BVP and convergence acceleration techniques (dropping the time-derivative
term) if they are really solving a BVP has been met with limp-wristed hand-waving or silence. <br /><br />I guess they didn't know that Steven Schneider (certainly not on anyone's list of &#
39;skeptics') thought climate was probably an IVP problem too: <br />"A common view of the climate system and ecosystem structure and function is that of path independence (no memory of
previous conditions). However, the multiple stable equilibria for both THC and for atmosphere-biosphere interactions in West Africa suggest a more complex reality. In such systems, the equilibrium
state reached is dependent on the initial conditions of the system."<br /><a href="http://www.oecd.org/dataoecd/9/59/2482280.pdf" rel="nofollow">Abrupt Non-Linear Climate Change, Irreversibility
and Surprise</a>Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-76570570180500402672011-02-09T05:28:55.578-05:002011-02-09T05:28:55.578-05:00Yes interesting paper with many
details .<br />They find basically that :<br /><i>Increasing the<br />number of significant digits beyond the 420 digits used in this work increases the<br />interval on which the Lorenz solution is
computable.</i><br />what is an elaborated way to say what I wrote above in a fast 0 order approximation :<br /><i>T=dt/A</i><br /><br />There is a fine point of critics that I would make and that is
that T is an upper bound and not a constant of the system .<br />Depending on X0 , the numerical solution may become wrong (e.g an artefact) BEFORE T for a GIVEN hardware precision .<br />The real T
cannot be independent of X0 .<br />When I played with Lorenz solvers at low precision and with the same dt , I noticed that the time at which the trajectories intersect (e.g T when the numerical
solution becomes an artefact) depends on X0 .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-13298948146863492782011-02-08T13:17:30.109-05:002011-02-08T13:17:30.109-05:00
Tom said: <i>Focusing a moment on the "simple" interpretation .<br />A characteristics of a chaotic solution is , as I already wrote above , that :<br />f(X0+dX0,t) - f(X0,t) = g(X0).exp
(L.t)<br />From there follows trivially that for times less than 1/L the trajectories don't diverge too wildly and regardless of the "time step" which is necessarily much smaller than 1
/L , when the time step decreases you will get something that looks like "convergence" .</i><br />I think you'll appreciate the introductory error analysis in <a href="http://
lorenzsystem.net/paper/" rel="nofollow">the paper</a> that goes with the site from reference [8] linked above.<br /><br />Tom also said: <i>Normally what you should see if you plot differences
between trajectories with different time steps what is equivalent to take the differences between different initial conditions , e.g f(X0+dX0,t) - f(X0,t)(this should be proven if one wants to be
rigorous), is a horizontal 0 line untill approximately T and then a more or less sudden explosion .</i><br />Yes, that's exactly what I found; see the <a href="http://www.variousconsequences.com/
2010/01/lorenz-63-ensemble.html" rel="nofollow">plots in Figure 2 of this post</a>, differences between trajectories in an initial condition ensemble.Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-91214301450419377262011-02-08T11:00:09.218-05:002011-02-08T11:00:09.218-05:00I just meant that I'm
actually going to add a periodic forcing term rather than "force" the system by having a time-dependent parameter value; so it's a different system, but that's already coded up
because that's what I needed for verifying the numerical implementation with MMS.Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-35998934456763559452011-02-08T08:53:10.668-05:002011-02-08T08:53:10.668-05:00Argh I am muddled today .<br />
The end should read N = 1/A so as T =N.dt , T=dt/A .<br />Correct this time . And better for the dimensional consistency
.TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-1547216320046028682011-02-08T08:40:04.799-05:002011-02-08T08:40:04.799-05:00Of course there is a nonsense in the above
that I could not edit.<br />The time after which a numerical solution becomes an artefact can obviously NOT be greater than T because T is an upper bound .<br />It can only be smaller and what one
cannot know is by how much smaller it is because that depends on X0 (initial condition) .<br /><br />T can be easily computed .<br />If one supposes that the solution is normalised so that it is a
number between 0 and 1 , then if A is the accuracy of the hardware for [0,1] , and dt the step chosen , the maximum of steps you can compute is N = 1/(A.dt) so T = 1/A . Beyond this T surely , and
possibly earlier , your solution becomes an artefact
.TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-88838279124677584482011-02-08T07:44:26.520-05:002011-02-08T07:44:26.520-05:00Joshua<br /><br />My skills and experience
with numerical methods are far below yours and Dan Hugh's .<br />I am a physicist who does 95% of his physics with a pencil and a sheet of paper so when I look at your scripts , my eyes glaze
over .<br />Sure I learned some time ago things about Runge Kuttas and such but have never really do it in practice .<br />That's why your very target as in the link you gave above stays largely
mysterious to me .<br />My certainly imperfect interpretation is that you want to know things about the trajectory divergence/convergence and then I cannot decide if the questions you ask are simple
or on the contrary so complex that I did not understand them .<br /><br />Focusing a moment on the "simple" interpretation .<br />A characteristics of a chaotic solution is , as I already
wrote above , that :<br />f(X0+dX0,t) - f(X0,t) = g(X0).exp(L.t)<br />From there follows trivially that for times less than 1/L the trajectories don't diverge too wildly and regardless of the &
quot;time step" which is necessarily much smaller than 1/L , when the time step decreases you will get something that looks like "convergence" .<br />One of course doesn't know
exactly g(X0) but if it is small , it will contribute to keeping the trajectories near to each other for times below 1/L .<br /><br />On the other hand for times greater than 1/L (note that this is
just a convention , an order of magnitude) , the term exp(L.t) will overwhelm everything and the increase of the length of convergence by taking smaller and smaller dt will become marginal .<br />In
summary depending on the exact form of g(X0) and the value of L , there will be a time T beyond which no decrease of dt will improve the convergence significantly .<br /><br />Ultimately the smallest
possible dt is given by the construction properties of the computer .<br />The corollary of the above is that once one uses the smalles dt allowed by the hardware , everything computed beyond T is
just an artefact .<br /><br />For fun it is possible to mathematically prove that ANY numerical solution of a chaotic system is NECESSARILY an artefact LATEST beyond some finite T .<br />This doesn&#
39;t however mean that it becomes an artefact exactly at T . It may become an artefact even before T (for grossly large dt) or a long time after T . You simply can't know .<br /><br />A numerical
analysis of convergence will give hints about this T and eventually about the g(X0) .<br />Normally what you should see if you plot differences between trajectories with different time steps what is
equivalent to take the differences between different initial conditions , e.g f(X0+dX0,t) - f(X0,t)(this should be proven if one wants to be rigorous), is a horizontal 0 line untill approximately T
and then a more or less sudden explosion .<br /><br />Last but not least in order to be complete .<br />Of course it would be nice if things were as simple as that but obviously the divergence can&#
39;t go forever because the exponetial goes to infinity .<br />And we know that chaotic systems having an attractor (like Lorenz one) must stay in a finite volume of the phase space because the
attractor is always bounded .<br />From there follows that when f(X0+dX0,t) - f(X0,t) begins to reach the size of the attractor , the trajectories don't diverge exponentially anymore . They are
then so different and in so diferent places of the attractor that they have nothing in common anymore .<br />That's why the Lyapounov coefficient can only be used and makes sense for some finite
time interval [0,TL] and looses more and more its significance when one approaches TL
.TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-49184113141531792972011-02-07T15:18:31.041-05:002011-02-07T15:18:31.041-05:00Good paper Tom; thanks. I was going to
approach it from <a href="http://www.variousconsequences.com/2010/01/chaotic-time-convergence-and-mms.html" rel="nofollow">a slightly different angle</a>, but I'm going to incorporate a bit of
discussion of that one now too.Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-57458090401104499122011-02-07T09:24:57.798-05:002011-02-07T09:24:57.798-05:00Joshua :<br /><br /><i>I think I
'll get to a forced Lorenz system (just periodic forcing), before I get to stochastic resonance, but all these ideas are related. </i><br /><br />Well thought .<br />But before going there , read
that : http://arxiv.org/abs/chao-dyn/9405012<br /><br />Just to save your time :)
TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-68337239252888565602011-02-01T01:09:00.813-05:002011-02-01T01:09:00.813-05:00Is that a moment closure technique that Tom/
Tomas is using to show that the average diverges?<br /><br />I was looking at this paper a few weeks ago in the context of Monte Carlo sims, which struck a chord:<br />http://www.mas.ncl.ac.uk/~ncsg3
18297101284358849575noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-33256729123634260952011-01-31T09:41:52.529-05:002011-01-31T09:41:52.529-05:00Thanks, That answers my
question. So it is really dependent on the size of the forcing; if the forcing is not big enough, a stochastic resonance mechanism can kick in.@whuthttps://www.blogger.com/profile/
18297101284358849575noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-79225483884228511682011-01-31T08:05:32.184-05:002011-01-31T08:05:32.184-05:00WHT, you must be spying on me; I
've been noodling around a stochastic resonance post, but haven't really got a good "hook" for it yet. I don't think I'm understanding your question. Forcings that should be
too small to drive the oscillations are able to because the stochastic noise pushes the system "over the top" (and back again). I think I'll get to a forced Lorenz system (just periodic
forcing), before I get to stochastic resonance, but all these ideas are related.Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-54204447093850853352011-01-30T23:20:43.932-05:002011-01-30T23:20:43.932-05:00I think this is very important
work, essentially showing how robust the formulations are to uncertainties. It seems like the complement to this is the introduction of order out of noise and disturbances, which is the basic idea
behind stochastic resonance. If you could unify these two ideas, it would help clear things up in my mind. <br /><br />So in a particular situation, is the oscillation observed because it is always
there and just immune to disturbances, or does it emerge from the noise, resonantly amplified as a kind of principle component from the nonlinear positive feedback of the system? Or could these two
ideas be the same principle?@whuthttps://www.blogger.com/profile/
18297101284358849575noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-74853542171497176642011-01-26T12:46:07.834-05:002011-01-26T12:46:07.834-05:00Found it; it went to the spam
bucket.<br /><br />Here's a <a href="http://amath.colorado.edu/faculty/juanga/Papers/PhysRevLett_96_254103.pdf" rel="nofollow">clickable link for the paper</a>.Joshua Stultshttps://
comment I wanted to add was that this result can't be easily transported to the climate .<br />Indeed climate is spatio-temporal chaos and the result obtained here is only valid for chaotic
solutions in temporal chaos .<br /><br />This difficulty is due to the fact that you correctly identified - temporal chaos deals with functions while climate science deals with functionals .<br />I
know of only one person who could deal with functionals correctly and even did it so correctly that he got a Nobel for that - Feynman :)<br /><br />Once you said that you said almost everything .<br
/>A differential of a functional is not uniquely defined so you loose all the impressive and necessary weaponry of differential calculus once you start having only functionals .<br /><br />As a side
note I go ROFL everytime when I see expressions like dTg/dF (which is supposed to define the climate sensitivity) and where Tg is an average global temperature and F some forcing .<br />Tg being a
functional that makes dTg be .... what ?<br />A nonsense .<br />But nonsense won't of course stop climate "scientists" to come to far reaching conclusions .<br /><br />So it is a kind
of damned if you do , damned if you don't .<br />If you don't use functionals , you face a full blown spatio temporal chaos with its untractable infinite dimensionality (of the phase space) .
<br />If you use functionals , a miracle seems to happen because you obtain things that superficially look like functions of time only .<br />But it is only an illusion and you are strictly forbidden
to plug that thing in the good old temporal chaos theory .<br /><br />So what is left ?<br />I have already written about it here .<br />If a full spatio-temporal chaos theory is not feasible in the
next 100 years or so , then what is left is this : http://amath.colorado.edu/faculty/juanga/Papers/PhysRevLett_96_254103.pdf .<br />This is one of the ways to discretize space and thus transform the
world in a <b>FINITE</b> spatial network of coupled oscillators chaotic or not .<br /><br />And of course hope that this discretization will produce dynamics that are approximations of the real
system for a sufficiently long time .<br />As you are apparently expert in numerical treatements , you will intuitively see the difficulty at once - imagine that you want to study a convergence of an
algorithm depending on the grid size but you don't know the function that has to be represented by the algorithm .<br /><br />Actually it is even worse :)<br />On every <b>node</b> of the grid
you have an unknown and different function of time and you are supposed to combine them in some way to obtain a known result and this independently of the grid size .<br />And the amusing part is
that if you divide the grid step by 2 , not only you need X times more functions but it can't be excluded that the functions that worked with the previous size must be changed too
.TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-400802615615264152011-01-25T11:33:41.979-05:002011-01-25T11:33:41.979-05:00Thanks again for your thoughtful comments Tom;
I really appreciate your contributions.Joshua Stultshttps://www.blogger.com/profile/
03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-46676863519529237842011-01-25T09:30:30.618-05:002011-01-25T09:30:30.618-05:00Very informed and excellent post
!<br />I just found it following a kind mention by Dan Hughes and will perhaps comment more in detail later .<br />It resonates with a post I made here a year ago or so , pointing out that the
ergodicity was the single most important issue in climate matters .<br />It is largely the same thing as what Lorenz said in the quotes above .<br /><br />You wrote<br /><i>"Lorenz would seem to
agree, “most climatic elements, and certainly climatic means, are not predictable in the first sense at infinite range, since a non-periodic series cannot be made periodic through averaging [1].”
We’re not going to just take his word on it. We’ll see if we can demonstrate this with our toy model. </i><br /><br />So just for fun a much faster demonstration in 3 lines :)<br /><br />A solution
is chaotic if the trajectories diverge exponentially.<br />That is : f(X0+dX0,t) - f(X0,t) = g(X0).exp(L.t) where :<br />X0 are the initial conditions , f is the chaotic solution , g is some function
of IC only and L>0 (Lyapounov coefficient) .<br />Let's define the time average TA(X0,t) = 1/T Integral t to T + t (f(X0,x)dx).<br /><br />T is some arbitrary averaging period<br />Then<br />
TA(X0+dX0,t) - TA(X0,t) = 1/T Integral t to T + t [f(X0+dX0,x) - f(X0,x)]dx = 1/T Integral t to T + t [g(X0).exp(L.x)]dx = <br />Something(T,X0,L).exp(L.t)<br /><br />The trajectories of the average
diverge exponentially for any arbitrary averaging period T .<br />Therefore if a solution is chaotic , all its time averages are chaotic too .<br />QED
.TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-32601543502547594682011-01-24T16:24:01.107-05:002011-01-24T16:24:01.107-05:00Dr Pielke has a <a href="http://
pielkeclimatesci.wordpress.com/2011/01/24/confessions-in-the-news-on-the-predictability-of-the-climate-system/" rel="nofollow">post up</a> about recent news coverage on predictability. He was also
kind enough to link this post on one of <a href="http://pielkeclimatesci.wordpress.com/2011/01/23/recommended-reading-on-whether-climate-is-an-initial-value-problem/" rel="nofollow">his previous
entries</a>.Joshua Stultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.com | {"url":"http://www.variousconsequences.com/feeds/5211543931454957413/comments/default","timestamp":"2024-11-14T02:25:05Z","content_type":"application/atom+xml","content_length":"54042","record_id":"<urn:uuid:dc3669b2-8940-4f61-b543-84bbad90d8de>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00026.warc.gz"} |
Quadrupole and octupole collectivity in <sup>148</sup>Nd
The role of quadrupole and octupole collectivity in the shape-transitional nucleus ^148Nd has been studied by Coulomb excitation using beams of ^58Ni and ^92Mo, and a beam of ^148Nd (using a ^208Pb
target). The extracted E1, E2 and E3 matrix elements involving states up to 12^+ in the ground band and 13^- in the negative-parity band are presented, and compared to calculations that assume a
vibrational and rotational octupole nature for the negative-parity band. The positive-parity ground-band states are well described in terms of a prolate deformed shape with Q[20] ≈ 400 e fm[2] (β^rms
[2] ≈ +0.18). The present results suggest a vibrational octupole nature for the low-spin negative-parity states, with an intrinsic moment Q[30] ≈ 1500 e fm^3 (β^rms[3] ≈ 0.12). The E2 and E3 matrix
elements connecting these bands to the β-and γ-vibrational bands (and within these bands) are also presented, and compared to calculations incorporating the coupling between the rotational and
vibrational modes. These calculations describe reasonably well the E2 matrix elements involving the gamma band, but do not reproduce the measured E2 matrix elements for the beta band, implying a
complicated intrinsic structure for the beta band. The strong enhancement of the measured E3 matrix elements connecting the negative-parity band to the beta band could be indicative of a significant
component of the two-phonon octupole vibration in the wavefunction of the so-called beta band. | {"url":"https://pure.york.ac.uk/portal/en/publications/quadrupole-and-octupole-collectivity-in-sup148supnd","timestamp":"2024-11-06T08:06:27Z","content_type":"text/html","content_length":"50590","record_id":"<urn:uuid:e5b04c95-311e-4324-a980-7ec9b8c517f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00521.warc.gz"} |
Graph Formula: Concept and Solved Examples
Graph Formula
Graphs are mathematical constructs used to represent pairwise relationships between entities. A graph is composed of vertices (or nodes) and edges that connect these vertices. The formulas related to
graphs are employed to characterize their attributes and dynamics. In this article, students will learn in detail about graph formula, solved examples based on it etc,. in detail.
What is Graph Formula?
In mathematics and computer science, a graph formula is a mathematical expression or a set of rules that define the properties and structure of a graph. Graphs are utilized to represent relationships
between objects, and different formulas are applied to analyze and comprehend these relationships.
Definition of Graph
A graph is a mathematical framework designed to model pairwise relationships between objects. It comprises a set of vertices (or nodes) and a set of edges (or links) that connect pairs of these
vertices. Formally, a graph G is represented as an ordered pair (V,E), where V denotes the set of vertices and E denotes the set of edges. Each edge in E signifies a connection or relationship
between two vertices in V. Graphs are extensively used across various disciplines, including computer science, mathematics, and engineering, to depict and analyze networks and relational structures.
Graph Notations
Vertex Set: V (G) represents the collection of vertices in the graph G.
Edge Set: E (G) represents the collection of edges in the graph G.
Degree of a Vertex: This refers to the number of edges connected to a vertex. In the context of a directed graph, this is further categorized into in-degree (the number of edges coming into the
vertex) and out-degree (the number of edges going out from the vertex).
Basic Terminology
Path: A sequence of vertices where each consecutive pair is connected by an edge.
Cycle: A path that begins and ends at the same vertex, with no other vertices or edges repeated.
Adjacency Matrix: A square matrix A used to represent a finite graph. For a graph with n vertices, A is an n×n matrix where:
• A[i][j] = 1 if there is an edge from vertex i to vertex j.
• A[i][j]=0 otherwise.
Incidence Matrix: A matrix representing the relationship between vertices and edges. For a graph with n vertices and m edges, the incidence matrix B is an n×m matrix where:
• B[i][j]=1 if vertex i is incident to edge j.
• B[i][j]=0 otherwise.
Eulerian Path: A path that traverses every edge exactly once.
Eulerian Circuit: A circuit that traverses every edge exactly once and returns to the starting vertex.
Slope Intercept Formula for Graph
The slope-intercept formula for a straight line passing through two points, (a[1], b[1]) and (a[2], b[2]), is given by y=mx+b.
The relationship between the two points, used to plot the line on the graph, is described by the Graph Formula, also known as the slope-intercept form of the straight-line equation. The Graph Formula
simplifies the process of plotting graphs. It is derived using the coordinates of the two points on the line. The Graph Formula is written as y=mx+b, where mmm is the slope. The slope mmm is also
known as “rise over run,” indicating how many units the line moves up or down for each unit it moves horizontally. In the Graph Formula, bbb is the y-int/ercept, indicating the point where the line
crosses the y-axis.
To summarize, assuming the two points are (a[1], b[1]) and (a2, b[2]), the slope-intercept form of the straight line can be calculated using the following Graph Formula:
y = mx+b
• m is the slope, calculated as b2−b1/a2−a1.
• b is the y-intercept.
Types of Graph
A graph functions as a mathematical representation of networks, designed to visually convey mathematical relationships for better understanding. There are several types of graph formats, including:
Bar Graph
A bar graph, also referred to as a bar chart, visually represents data using rectangular bars or columns. Each bar corresponds to a specific category or data point, with the length or height of the
bar proportional to the value it signifies. Bar graphs are often used to compare different categories or to illustrate data changes over time. They offer a straightforward visual method for quickly
comprehending and analyzing numerical data and trends.
Pie Graph
A pie graph, also called a pie chart, is a circular visual representation used to depict the proportions or percentages of data within a whole. The circle represents the entirety of the data set,
divided into individual “slices” or sectors, each corresponding to a specific category or data point. The size of each slice is relative to the quantity it represents compared to the whole dataset.
Pie graphs are effective for displaying the distribution of categorical data and demonstrating how different segments contribute to the overall composition. They provide a visual means to quickly
understand the relative sizes of various components within a dataset. The angles of the slices are calculated based on the proportions they represent, facilitating easy comparison of the magnitudes
of different categories at a glance.
Line Graph
A line graph, also known as a line chart, is a graphical representation that presents data points as markers connected by lines. This graph is commonly utilized to illustrate the correlation between
two or more variables and to depict their fluctuations over a continuous or discrete period. Typically, in a line graph, the x-axis represents the independent variable, often time, while the y-axis
represents the dependent variable. Data points are plotted at specific positions on the graph, and lines are drawn to link these points. These lines offer a visual representation of the trend,
pattern, or variation in the data across the specified range. Line graphs are valuable tools for visualizing changes and trends in data over time or other ordered sequences.
Histogram Graph
A histogram graph visually represents data using rectangular bars of different heights, where each bar represents a particular range of quantitative values. Unlike a bar graph, which is used for
categorical variables, a histogram focuses on displaying the distribution of quantitative data. Each bar in a histogram corresponds to a specific range or interval of values along the horizontal
axis, known as bins or buckets. The height of each bar indicates the frequency or count of data points falling within that range. Typically, the bars in a histogram are adjacent and touch each other
to emphasize the continuous nature of the data.
Scatter Diagrams
A scatter diagram, commonly known as a scatter plot, is a visual representation used to depict the relationship between two variables. It utilizes Cartesian coordinates, with one variable plotted on
the horizontal axis (x-axis) and the other variable plotted on the vertical axis (y-axis). In a scatter plot, individual data points are represented by dots or markers on the graph, with each dot
corresponding to a specific combination of values for the two variables. By examining the pattern formed by these data points, one can assess the correlation or relationship between the variables.
Solved Examples on Graph Formula
Example 1: Determine the slope between the points (2, 4) and (6, 10).
Using the slope formula:
Slope (m)=(b2−b1)/(a2−a1)
Substitute the given coordinates:
Slope (m)=(10−4)/(6−2)
Slope (m)=1.5
Therefore, the slope between the points (2, 4) and (6, 10) is 1.5.
Example 2: Determine the slope between the points (-3, 5) and (2, -1).
Using the slope formula:
Slope (m)=(b2−b1)/(a2−a1)
Substitute the given coordinates:
Slope (m)=(-1−5)/(2−(-3)
Slope (m)=−1.2
Therefore, the slope between the points (-3, 5) and (2, -1) is -1.2.
Example 3: Find the slope between the points (-1, 3) and (-4, 7).
Using the slope formula:
Slope (m)=(b2−b1)/(a2−a1)
Substitute the given coordinates:
Slope (m)=(7−3)/(−4−(−1)
Slope (m)=−4/3
FAQs (Frequently Asked Questions)
1. What are the three techniques for graphing equations?
Three methods exist for graphing linear equations. Students have three options
(1) use the slope and y-intercept to find two points, (2) use the slope and y-intercept, or (3) use the x- and y-intercepts.
2. What is the slope?
The slope of a line on a graph represents its steepness or inclination. It quantifies how much the line rises or falls for every unit increase along the horizontal axis.
3. How to Calculate Slope?
To calculate the slope between two points (a1, b1) and (a2, b2) on a graph, the formula commonly used is:
Slope (m)=(b2−b1)/(a2−a1)
4. What role does 'm' play in the graph formula?
Within the equation y = mx + b, ‘m’ symbolizes the slope of the line, providing insight into its steepness or inclination.
5. What is the significance of 'b' in the graph formula?
In the graph formula y = mx + b, ‘b’ denotes the y-intercept, indicating the point at which the line intersects the y-axis. | {"url":"https://www.extramarks.com/studymaterials/formulas/graph-formula/","timestamp":"2024-11-08T15:36:21Z","content_type":"text/html","content_length":"638105","record_id":"<urn:uuid:8edef620-e768-4057-a124-11b6a19d3898>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00821.warc.gz"} |
Vestnik KRAUNC. Fiz.-Mat. nauki. 2022. vol. 39. no. 2. P. 32-41. ISSN 2079-6641
MSC 35J15
Research Article
On one boundary value problem for the fourth-order equation in partial derivatives
O. Sh. Kilichov¹, A. N. Ubaydullaev²
¹V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b University str., Tashkent, 100174, Uzbekistan
²Bukhara State University, M. Ikbol str. 11, Bukhara, 705018, Uzbekistan
E-mail: oybek2402@mail.ru
The initial-boundary problem for the heat conduction equation inside a bounded domain is considered. It is supposed that on the boundary of this domain the heat exchange takes place according to
Newton’s law. The control parameter is equal to the magnitude of output of hot air and is defined on a givenmpart of the boundary. Then we determined the dependence T(Θ) on the parameters of the
temperature process when Θ is close to critical value.
Key words: boundary value problem; Fourier method; the existence of a solution; the uniqueness of a solution.
DOI: 10.26117/2079-6641-2022-39-2-32-41
Original article submitted: 17.07.2022
Revision submitted: 10.08.2022
For citation. Kilichov O. Sh., Ubaydullaev A. N. On one boundary value problem for the fourth-order equation in partial derivatives. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 32-41. DOI: 10.26117
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Kilichov O. Sh., Ubaydullaev A. N., 2022
Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript
was approved by all authors.
Acknowledgments. The authors are deeply grateful to the referee for a number of comments that contributed to the improvement of the article.
1. Tikhonov A. N. About the boundary conditions containing derivatives of an order, exceeding an equation order, Mat. Sb., 1950. vol. 26(1), pp. 35–56 (In Russian).
2. Bitsadze A. V On the Neumann problem for harmonic functions, Dokl. Akad. Nauk SSSR, 1990. vol. 311 (1), pp. 11–13 (In Russian).
3. Bavrin I. I. Operators for harmonic functions and applications, Differential equations, 1985. vol. 21(1), pp. 9–15 (In Russian).
4. Karachik V. V, Turmetov B. H. About a problem for harmonic equation, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1990. vol. 4, pp. 17–21 (In Russian).
5. Karachik V. V. About solvability of a boundary value problem for Helmholtz’s equation with high order normal derivative on a boundary, Differ. Uravneniya, 1992. vol. 28 (5), pp. 907–909 (In
6. Karachik V. V. About a problem for Poisson’s equation with high order normal derivative on boundary, Differ. Uravneniya, 1996. vol. 32, no. 3, pp. 1501–1503 (In Russian).
7. Karachik V. V. Generalized Neumann problem for harmonic functions in space, Differ. Uravneniya, 1999. vol. 35, no. 7, pp. 1–6 (In Russian).
8. Sokolovskiy V. B. On a generalization of Neumann problem, Differ. Uravneniya, 1998. vol. 24, no. 4, pp. 714–716 (In Russian).
9. Amanov D. On a generalization of the first initial-boundary value problem for the heat conduction equation, Contemporary Analysis and Applied Mathematics, 2014. vol. 2, no. 1, pp. 88–97.
10. Amanov D., Ibragimov G., Kilicman A. On a Generalization of the Initial-Boundary Problem for the Vibrating String Equation, Symmetry, 2019. vol. 11, no. 1.
11. Amanov D., Yuldasheva A. Solvability and spectral properties of a self-adjoint problem for a fourth-order equation, Uzbekskii Matem. Zhurnal, 2007. vol. 4, pp. 3–8 (In Russian).
12. Amanov D., Murzambetova M. A boundary value problem for a fourth order equation with a lower term, Vestnik Udmurtsk. un-ta. Matem. Mekh. Komp’yut. Nauki, 2013. vol. 1, pp. 3–10 (In Russian).
13. Amanov D. On a nonlocal problem for the heat equation, Uzbekskii Matem. journal, 2016. vol. 2, pp. 21–25
14. Kilichov O. Sh. On a nonlocal boundary value problem for the equation fourth-order in partial derivatives, Vestnik KRAUNC. Phys.-Mat. Nauki, 2021. vol. 37, no. 4, pp. 16–23 (In Russian).
15. Kilichov O. Sh. Nonlocal boundary value problem for the heat conduction equation, Uzbek Mathematical Journal, 2021. vol. 2, pp. 110–116.
16. Kilichov O. Sh. A boundary value problem for a fourth-order equation, Bulletin of the Institute of Mathematics, 2021. vol. 4, no. 2, pp. 61–69 (In Russian).
17. Ashurov R. R., Mukhiddinova A. T. Initial-boundary value problems for hyperbolic equations with an elliptic operator of arbitrary order, Vestnik KRAUNC. Phys.-Mat. Nauki, 2020. vol. 30, no. 1,
pp. 8–19 (In Russian).
18. Yuldashev T. K. Nonlocal mixed-value problem for a Boussinesq-type integro-differential equation with degenerate kernel cubature formulas, Ukrainian Mathematical Journal, 2016. vol. 68, no. 8,
pp. 1278–1296.
19. Yuldashev T. K. Mixed problem for pseudo parabolic integro-differential equation with degenerate kernel, Differential equations, 2017. vol. 53, no. 1, pp. 99–108.
20. Amanov D., Kilichov O. Sh. Boundary value problem for a fourth-order mixed-type equation in a rectangular domain, Bulletin of the Institute of Mathematics, 2018. vol. 2, pp. 1–8 (In Russian).
21. Moiseev Y. I. On the solution by a spectral method of a single non-local boundary value problem, Differential Equations, 1999. vol. 8, no. 35, pp. 1094–1100 (In Russian).
22. Il’in V. A., Poznyak E. G. Osnovy matematicheskogo analiza [Fundamentals of mathematical analysis]. Nauka: Moscow, 1973. 448 pp. (In Russian)
23. lev V. I. Elementy funktsional’nogo analiza [Elements of Functional Analysis]. Nauka: Moscow, 1965. 520 pp. (In Russian)
Kilichov Oybek Sharafiddinovich – Doctoral student, Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan, ORCID 0000-0002-7673-943X.
Ubaydullaev Alisher Nematillayevich – Teacher of the Department of Mathematics of Bukhara State University, Bukhara, Uzbekistan, ORCID 0000-0002-4219-5155. | {"url":"https://krasec.ru/kilichov2022392eng/","timestamp":"2024-11-03T20:02:01Z","content_type":"text/html","content_length":"66970","record_id":"<urn:uuid:fdccec5b-8558-46d4-b236-71edf6f15c9d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00611.warc.gz"} |
Numpy - Check If an Element is NaN - Data Science Parichay
In this tutorial, we will look at how to check if an element in a Numpy array is a NaN (not a number) value or not with the help of some examples.
How to test if an element is NaN or not in a Numpy array?
You can use the numpy.isnan() function to check (element-wise) if values in a Numpy array are NaN or not. The following is the syntax –
# test for nan - pass scaler value or numpy array
If you apply the numpy.isnan() function to a scalar value, it returns a boolean value (True if the value is NaN otherwise False). If you apply it to an array, it returns a boolean array.
Let’s now look at some examples of using the above function to test for NaN.
Example 1 – Check if a value is NaN or not using numpy.isnan()
First, let’s pass scaler values to the numpy.isnan() function.
Let’s create two variables – one containing a NaN value and the other containing a non-Nan value respectively and then apply the numpy.isnan() function on each of these values.
import numpy as np
# create two variables
a = 21
b = np.nan
# check if nan
We get False as the output for the value 21 and True as the output for the NaN value.
📚 Data Science Programs By Skill Level
Introductory ⭐
Intermediate ⭐⭐⭐
Advanced ⭐⭐⭐⭐⭐
🔎 Find Data Science Programs 👨💻 111,889 already enrolled
Disclaimer: Data Science Parichay is reader supported. When you purchase a course through a link on this site, we may earn a small commission at no additional cost to you. Earned commissions help
support this website and its team of writers.
Example 2 – Element-wise check for NaN in a Numpy array using numpy.isnan()
If you apply the numpy.isnan() function on an array, it will return a boolean array containing with True for values that are NaN and False for the non-Nan values.
Let’s create a 1-D array and apply the numpy.isnan() function to it.
# create a numpy array
ar = np.array([1, 2, np.nan, 4, 5, np.nan, np.nan])
# element-wise check for nan value in ar
array([False, False, True, False, False, True, True])
We get a boolean array as an output. You can see that in the boolean array we get True for NaN values in the original array and False for the other values.
In this tutorial, we looked at how we can use the numpy.isnan() function to check if an element is NaN or not in a Numpy array. Keep in mind that if you pass a scaler value, it returns a boolean
value and if you pass a 1-D array it returns a boolean array.
You might also be interested in –
Subscribe to our newsletter for more informative guides and tutorials.
We do not spam and you can opt out any time. | {"url":"https://datascienceparichay.com/article/numpy-check-if-an-element-is-nan/","timestamp":"2024-11-13T19:05:16Z","content_type":"text/html","content_length":"258482","record_id":"<urn:uuid:785cc43e-e58b-492e-a8ba-5cdcfd5041e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00285.warc.gz"} |
Nodal sets of laplace eigenfunctions: Estimates of the hausdorff measure in dimensions two and three
Let Δ[M] be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u: Δ[M]u+λu = 0. In dimension n = 2 we refine the
Donnelly–Fefferman estimate by showing that H^1({u = 0}) ≤C^λ3/4−β for some β∈ (0, 1/4). The proof employs the Donnelly–Fefferman estimate and a combinatorial argument, which also gives a lower
(non-sharp) bound in dimension n = 3: H^2({u = 0}) ≥ cλ^α for some α∈ (0,1/2). The positive constants c, C depend on the manifold, α and β are universal.
Publication series
Name Operator Theory: Advances and Applications
Volume 261
ISSN (Print) 0255-0156
ISSN (Electronic) 2296-4878
All Science Journal Classification (ASJC) codes
• Harmonic functions
• Laplace eigenfunctions
• Nodal set
Dive into the research topics of 'Nodal sets of laplace eigenfunctions: Estimates of the hausdorff measure in dimensions two and three'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/nodal-sets-of-laplace-eigenfunctions-estimates-of-the-hausdorff-m","timestamp":"2024-11-04T12:10:52Z","content_type":"text/html","content_length":"49213","record_id":"<urn:uuid:a5965f8b-a918-4648-8a6b-bba1c7512009>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00827.warc.gz"} |
Python Booleans: Use Truth Values in Your Code – Real Python
Watch Now This tutorial has a related video course created by the Real Python team. Watch it together with the written tutorial to deepen your understanding: Python Booleans: Leveraging the Values of
The Python Boolean type is one of Python’s built-in data types. It’s used to represent the truth value of an expression. For example, the expression 1 <= 2 is True, while the expression 0 == 1 is
False. Understanding how Python Boolean values behave is important to programming well in Python.
In this tutorial, you’ll learn how to:
• Manipulate Boolean values with Boolean operators
• Convert Booleans to other types
• Convert other types to Python Booleans
• Use Python Booleans to write efficient and readable Python code
Free Bonus: 5 Thoughts On Python Mastery, a free course for Python developers that shows you the roadmap and the mindset you’ll need to take your Python skills to the next level.
The Python Boolean Type
The Python Boolean type has only two possible values:
1. True
2. False
No other value will have bool as its type. You can check the type of True and False with the built-in type():
>>> type(False)
<class 'bool'>
>>> type(True)
<class 'bool'>
The type() of both False and True is bool.
The type bool is built in, meaning it’s always available in Python and doesn’t need to be imported. However, the name itself isn’t a keyword in the language. While the following is considered bad
style, it’s possible to assign to the name bool:
>>> bool
<class 'bool'>
>>> bool = "this is not a type"
>>> bool
'this is not a type'
Although technically possible, to avoid confusion it’s highly recommended that you don’t assign a different value to bool.
Python Booleans as Keywords
Built-in names aren’t keywords. As far as the Python language is concerned, they’re regular variables. If you assign to them, then you’ll override the built-in value.
In contrast, the names True and False are not built-ins. They’re keywords. Unlike many other Python keywords, True and False are Python expressions. Since they’re expressions, they can be used
wherever other expressions, like 1 + 1, can be used.
It’s possible to assign a Boolean value to variables, but it’s not possible to assign a value to True:
>>> a_true_alias = True
>>> a_true_alias
>>> True = 5
File "<stdin>", line 1
SyntaxError: cannot assign to True
Because True is a keyword, you can’t assign a value to it. The same rule applies to False:
>>> False = 5
File "<stdin>", line 1
SyntaxError: cannot assign to False
You can’t assign to False because it’s a keyword in Python. In this way, True and False behave like other numeric constants. For example, you can pass 1.5 to functions or assign it to variables.
However, it’s impossible to assign a value to 1.5. The statement 1.5 = 5 is not valid Python. Both 1.5 = 5 and False = 5 are invalid Python code and will raise a SyntaxError when parsed.
Python Booleans as Numbers
Booleans are considered a numeric type in Python. This means they’re numbers for all intents and purposes. In other words, you can apply arithmetic operations to Booleans, and you can also compare
them to numbers:
>>> True == 1
>>> False == 0
>>> True + (False / True)
There aren’t many uses for the numerical nature of Boolean values, but there’s one technique you may find helpful. Because True is equal to 1 and False is equal to 0, adding Booleans together is a
quick way to count the number of True values. This can come in handy when you need to count the number of items that satisfy a condition.
For example, if you want to analyze a verse in a classic children’s poem to see what fraction of lines contain the word "the", then the fact that True is equal to 1 and False is equal to 0 can come
in quite handy:
>>> lines="""\
... He took his vorpal sword in hand;
... Long time the manxome foe he sought—
... So rested he by the Tumtum tree
... And stood awhile in thought.
... """.splitlines()
>>> sum("the" in line.lower() for line in lines) / len(lines)
Summing all values in a generator expression like this lets you know how many times True appears in the generator. The number of times True is in the generator is equal to the number of lines that
contain the word "the", in a case-insensitive way. Dividing this number by the total number of lines gives you the ratio of matching lines to total lines.
To see why this works, you can break the above code into smaller parts:
>>> lines = """\
... He took his vorpal sword in hand;
... Long time the manxome foe he sought—
... So rested he by the Tumtum tree
... And stood awhile in thought.
... """
>>> line_list = lines.splitlines()
>>> "the" in line_list[0]
>>> "the" in line_list[1]
>>> 0 + False + True # Equivalent to 0 + 0 + 1
>>> ["the" in line for line in line_list]
[False, True, True, False]
>>> False + True + True + False
>>> len(line_list)
>>> 2/4
The line_list variable holds a list of lines. The first line doesn’t have the word "the" in it, so "the" in line_list[0] is False. In the second line, "the" does appear, so "the" in line_list[1] is
True. Since Booleans are numbers, you can add them to numbers, and 0 + False + True gives 1.
Since ["the" in line for line in line_list] is a list of four Booleans, you can add them together. When you add False + True + True + False, you get 2. Now, if you divide that result by 4, the length
of the list, you get 0.5. The word "the" appears in half the lines in the selection. This is a useful way to take advantage of the fact that Booleans are numbers.
Boolean Operators
Boolean operators are those that take Boolean inputs and return Boolean results.
Note: Later, you’ll see that these operators can be given other inputs and don’t always return Boolean results. For now, all examples will use Boolean inputs and results. You’ll see how this
generalizes to other values in the section on truthiness.
Since Python Boolean values have only two possible options, True or False, it’s possible to specify the operators completely in terms of the results they assign to every possible input combination.
These specifications are called truth tables since they’re displayed in a table.
As you’ll see later, in some situations, knowing one input to an operator is enough to determine its value. In those cases, the other input is not evaluated. This is called short-circuit evaluation.
The importance of short-circuit evaluation depends on the specific case. In some cases, it might have little effect on your program. In other cases, such as when it would be computationally intensive
to evaluate expressions that don’t affect the result, it provides a significant performance benefit. In the most extreme cases, the correctness of your code can hinge on the short-circuit evaluation.
Operators With No Inputs
You can think of True and False as Boolean operators that take no inputs. One of these operators always returns True, and the other always returns False.
Thinking of the Python Boolean values as operators is sometimes useful. For example, this approach helps to remind you that they’re not variables. For the same reason you can’t assign to +, it’s
impossible to assign to True or False.
Only two Python Boolean values exist. A Boolean operator with no inputs always returns the same value. Because of this, True and False are the only two Boolean operators that don’t take inputs.
The not Boolean Operator
The only Boolean operator with one argument is not. It takes one argument and returns the opposite result: False for True and True for False. Here it is in a truth table:
A not A
True False
False True
This table illustrates that not returns the opposite truth value of the argument. Since not takes only one argument, it doesn’t short-circuit. It evaluates its argument before returning its result:
>>> not True
>>> not False
>>> def print_and_true():
... print("I got called")
... return True
>>> not print_and_true()
I got called
The last line shows that not evaluates its input before returning False.
You might be wondering why there are no other Boolean operators that take a single argument. In order to understand why, you can look at a table that shows all theoretically possible Boolean
operators that would take one argument:
A not A Identity Yes No
True False True True False
False True False True False
There are only four possible operators with one argument. Other than not, the remaining three operators all have somewhat whimsical names since they don’t actually exist:
• Identity: Since this operator simply returns its input, you could just delete it from your code with no effect.
• Yes: This is a short-circuit operator since it doesn’t depend on its argument. You could just replace it with True and get the same result.
• No: This is another short-circuit operator since it doesn’t depend on its argument. You could just replace it with False and get the same result.
None of the other possible operators with one argument would be useful.
The and Boolean Operator
The and operator takes two arguments. It evaluates to False unless both inputs are True. You could define the behavior of and with the following truth table:
A B A and B
True True True
False True False
True False False
False False False
This table is verbose. However, it illustrates the same behavior as the description above. If A is False, then the value of B doesn’t matter. Because of this, and short-circuits if the first input is
False. In other words, if the first input is False, then the second input isn’t evaluated.
The following code has a second input that has a side effect, printing, in order to provide a concrete example:
>>> def print_and_return(x):
... print(f"I am returning {x}")
... return x
>>> True and print_and_return(True)
I am returning True
>>> True and print_and_return(False)
I am returning False
>>> False and print_and_return(True)
>>> False and print_and_return(False)
In the last two cases, nothing is printed. The function isn’t called since calling it isn’t necessary to determine the value of the and operator. Being aware of short-circuits is important when
expressions have a side effect. In the last two examples, the short-circuit evaluation prevents the printing side effect from happening.
One example in which this behavior can be crucial is in code that might raise exceptions:
>>> def inverse_and_true(n):
... 1 // n
... return True
>>> inverse_and_true(5)
>>> inverse_and_true(0)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 2, in inverse_and_true
ZeroDivisionError: integer division or modulo by zero
>>> False and inverse_and_true(0)
The function inverse_and_true() is admittedly silly, and many linters would warn about the expression 1 // n being useless. It does serve the purpose of neatly failing when given 0 as a parameter
since division by 0 is invalid. However, the last line doesn’t raise an exception. Because of short-circuit evaluation, the function isn’t called, the division by 0 doesn’t happen, and no exception
is raised.
In contrast, True and inverse_and_true(0) would raise an exception. In that case, the value of the second input would be needed for the result of and. Once the second input was evaluated,
inverse_and_true(0) would be called, it would divide by 0, and an exception would be raised.
The or Boolean Operator
The value of the or operator is True unless both of its inputs are False. The or operator could also be defined by the following truth table:
A B A or B
True True True
False True True
True False True
False False False
This table is verbose, but it has the same meaning as the explanation above.
When used informally, the word or can have one of two meanings:
• The exclusive or is how or is used in the phrase “You can file for an extension or submit your homework on time.” In this case, you can’t both file for an extension and submit your homework on
• The inclusive or is sometimes indicated by using the conjunction and/or. For example, “If you do well on this task, then you can get a raise and/or a promotion” means that you might get both a
raise and a promotion.
When Python interprets the keyword or, it does so using the inclusive or. If both inputs are True, then the result of or is True.
Because it uses an inclusive or, the or operator in Python also uses short-circuit evaluation. If the first argument is True, then the result is True, and there is no need to evaluate the second
argument. The following examples demonstrate the short-circuit evaluation of or:
>>> def print_and_true():
... print("print_and_true called")
... return True
>>> True or print_and_true()
>>> False or print_and_true()
print_and_true called
The second input isn’t evaluated by or unless the first one is False. In practice, the short-circuit evaluation of or is used much less often than that of and. However, it’s important to keep this
behavior in mind when reading code.
Other Boolean Operators
The mathematical theory of Boolean logic determines that no other operators beyond not, and, and or are needed. All other operators on two inputs can be specified in terms of these three operators.
All operators on three or more inputs can be specified in terms of operators of two inputs.
In fact, even having both or and and is redundant. The and operator can be defined in terms of not and or, and the or operator can be defined in terms of not and and. However, and and or are so
useful that all programming languages have both.
There are sixteen possible two-input Boolean operators. Except for and and or, they are rarely needed in practice. Because of this, True, False, not, and, and or are the only built-in Python Boolean
Comparison Operators
Some of Python’s operators check whether a relationship holds between two objects. Since the relationship either holds or doesn’t hold, these operators, called comparison operators, always return
Boolean values.
Comparison operators are the most common source of Boolean values.
Equality and Inequality
The most common comparison operators are the equality operator (==) and the inequality operator (!=). It’s almost impossible to write any meaningful amount of Python code without using at least one
of those operators.
The equality operator (==) is one of the most used operators in Python code. You often need to compare either an unknown result with a known result or two unknown results against each other. Some
functions return values that need to be compared against a sentinel to see if some edge condition has been detected. Sometimes you need to compare the results from two functions against each other.
The equality operator is often used to compare numbers:
>>> 1 == 1
>>> 1 == 1.0
>>> 1 == 2
You may have used equality operators before. They’re some of the most common operators in Python. For all built-in Python objects, and for most third-party classes, they return a Boolean value: True
or False.
Note: The Python language doesn’t enforce that == and != return Booleans. Libraries like NumPy and pandas return other values.
Second only to the equality operator in popularity is the inequality operator (!=). It returns True if the arguments aren’t equal and False if they are. The examples are similarly wide-ranging. Many
unit tests check that the value isn’t equal to a specific invalid value. A web client might check that the error code isn’t 404 Not Found before trying an alternative.
Here are two examples of the Python inequality operator in use:
>>> 1 != 2
>>> 1 != (1 + 0.0)
Perhaps the most surprising thing about the Python inequality operator is the fact that it exists in the first place. After all, you could achieve the same result as 1 != 2 with not (1 == 2). Python
usually avoids extra syntax, and especially extra core operators, for things easily achievable by other means.
However, inequality is used so often that it was deemed worthwhile to have a dedicated operator for it. In old versions of Python, in the 1.x series, there were actually two different syntaxes.
As an April Fools’ joke, Python still supports an alternative syntax for inequality with the right __future__ import:
>>> from __future__ import barry_as_FLUFL
>>> 1 <> 2
This should never be used in any code meant for real use. It could come in handy for your next Python trivia night, however.
Order Comparisons
Another set of test operators are the order comparison operators. There are four order comparison operators that can be categorized by two qualities:
• Direction: Is it less than or greater than?
• Strictness: Is equality allowed or not?
Since the two choices are independent, you get 2 * 2 == 4 order comparison operators. All four are listed in this table:
Less than Greater than
Strict < >
Not strict <= >=
There are two options for direction and two options for strictness. This results in total of four order comparison operators.
The order comparison operators aren’t defined for all objects. Some objects don’t have a meaningful order. Even though lists and tuples are ordered lexicographically, dictionaries don’t have a
meaningful order:
>>> {1: 3} < {2: 4}
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: '<' not supported between instances of 'dict' and 'dict'
It’s not obvious how dictionaries should be ordered. As per the Zen of Python, in the face of ambiguity, Python refuses to guess.
While strings and integers are ordered separately, intertype comparisons aren’t supported:
>>> 1 <= "1"
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: '<=' not supported between instances of 'int' and 'str'
Again, since there’s no obvious way to define order, Python refuses to compare them. This is similar to the addition operator (+). Though you can add strings to strings and integers to integers,
adding strings to integers raises an exception.
When the order comparison operators are defined, in general they return a Boolean.
Note: Python doesn’t enforce that comparison operators return Booleans. While all built-in Python objects, and most third-party objects, return Booleans when compared, there are exceptions.
For example, comparison operators between NumPy arrays or pandas DataFrames return arrays and DataFrames. You’ll see more about the interaction of NumPy and Boolean values later in this tutorial.
Comparing numbers in Python is a common way of checking against boundary conditions. Note that < doesn’t allow equality, while <= does:
>>> 1 <= 1
>>> 1 < 1
>>> 2 > 3
>>> 2 >= 2
Programmers often use comparison operators without realizing that they return a Python Boolean value.
The is Operator
The is operator checks for object identity. In other words, x is y evaluates to True only when x and y evaluate to the same object. The is operator has an opposite, the is not operator.
A typical usage of is and is not is to compare lists for identity:
>>> x = []
>>> y = []
>>> x is x
>>> x is not x
>>> x is y
>>> x is not y
Even though x == y, they are not the same object. The is not operator always returns the opposite of is. There’s no difference between the expression x is not y and the expression not (x is y) except
for readability.
Keep in mind that the above examples show the is operator used only with lists. The behavior of the is operator on immutable objects like numbers and strings is more complicated.
The in Operator
The in operator checks for membership. An object can define what it considers members. Most sequences, such as lists, consider their elements to be members:
>>> small_even = [2, 4]
>>> 1 in small_even
>>> 2 in small_even
>>> 10 in small_even
Since 2 is an element of the list, 2 in small_even returns True. Since 1 and 10 aren’t in the list, the other expressions return False. In all cases, the in operator returns a Boolean value.
Since strings are sequences of characters, you might expect them to also check for membership. In other words, characters that are members of the string will return True for in, while those that
don’t will return False:
>>> "e" in "hello beautiful world"
>>> "x" in "hello beautiful world"
Since "e" is the second element of the string, the first example returns True. Since x doesn’t appear in the string, the second example returns False. However, along with individual characters,
substrings are also considered to be members of a string:
>>> "beautiful" in "hello beautiful world"
>>> "belle" in "hello beautiful world"
Since "beautiful" is a substring, the in operator returns True. Since "belle" is not a substring, the in operator returns False. This is despite the fact that every individual letter in "belle" is a
member of the string.
Like the operators is and ==, the in operator also has an opposite, not in. You can use not in to confirm that an element is not a member of an object.
Chaining Comparison Operators
Comparison operators can form chains. You can create comparison operator chains by separating expressions with comparison operators to form a larger expression:
The expression 1 < 2 < 3 is a comparison operator chain. It has expressions separated by comparison operators. The result is True because both parts of the chain are True. You can break up the chain
to see how it works:
Since 1 < 2 returns True and 2 < 3 returns True, and returns True. A comparison chain is equivalent to using and on all its links. In this case, since True and True returns True, the result of the
whole chain is True. This means that if any of the links are False, then the whole chain is False:
This comparison chain returns False since not all of its links are True. Because comparison chains are an implicit and operator, if even one link is False, then the whole chain is False. You can
break up the chain to see how it works:
>>> 1 < 3 and 3 < 2
In this case, the parts of the chain evaluate to the following Booleans:
• 1 < 3 is True
• 3 < 2 is False
This means that one of the results is True and one is False. Since True and False is equal to False, the value of the entire chain is False.
You can mix types and operations in a comparison chain as long as the types can be compared:
>>> 1 < 2 < 1
>>> 1 == 1.0 < 0.5
>>> 1 == 1.0 == True
>>> 1 < 3 > 2
>>> 1 < 2 < 3 < 4 < 5
The operators don’t have to be all the same. Not even the types have to be all the same. In the examples above, you have three numeric types:
1. int
2. float
3. bool
These are three different numeric types, but you can compare objects of different numeric types without issue.
Short-Circuit Chain Evaluation
If chains use an implicit and, then chains must also short-circuit. This is important because even in cases where an order comparison isn’t defined, it’s possible for a chain to return False:
>>> 2 < "2"
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: '<' not supported between instances of 'int' and 'str'
>>> 3 < 2 < "2"
Even though Python can’t order-compare integers and strings numbers, 3 < 2 < "2" evaluates to False because it doesn’t evaluate the second comparison. In this case, the short-circuit evaluation
prevents another side effect: raising an exception.
Short-circuit evaluation of comparison chains can prevent other exceptions:
Dividing 1 by 0 would have raised a ZeroDivisionError. However, because of the short-circuit evaluation, Python doesn’t evaluate the invalid division. This means that Python skips evaluating not only
the comparison but also the inputs to the comparison.
Another aspect that is important to understand about comparison chains is that when Python does evaluate an element in the chain, it evaluates it only once:
>>> def foo():
... print("I'm foo")
... return 1
>>> 0 < foo() < 2
I'm foo
>>> (0 < foo()) and (foo() < 2)
I'm foo
I'm foo
Because the middle elements are evaluated only once, it’s not always safe to refactor x < y < z to (x < y) and (y < z). Although the chain behaves like and in its short-circuit evaluation, it
evaluates all values, including the intermediate ones, only once.
Chains are especially useful for range checks, which confirm that a value falls within a given range. For example, in a daily invoice that includes the number hours worked, you might do the
>>> hours_worked = 5
>>> 1 <= hours_worked <= 25
If there are 0 hours worked, then there’s no reason to send the invoice. Accounting for Daylight Saving Time, the maximum number of hours in a day is 25. The above range check confirms that the
number of hours worked in a day falls within the allowable range.
Mixing Operators and Chaining
Until now, all our examples involved ==, !=, and the order comparisons. However, you can chain all of Python’s comparison operators. This can lead to surprising behavior:
>>> a = 0
>>> a is a < 1
>>> (a is a) < 1
>>> a is (a < 1)
Because a is a < 1 is a comparison chain, it evaluates to True. You can break the chain into its parts:
• The expression a is a is True, as it would be for any value evaluated against itself.
• The expression a < 1 is True since 0 is less than 1.
Since both parts are True, the chain evaluates to True.
However, people who are used to other operators in Python may assume that, like other expressions involving multiple operators such as 1 + 2 * 3, Python inserts parentheses into to the expression.
However, neither way of inserting parenthesis will evaluate to True.
You can see why both evaluate to False if you break up the expressions. If you break up the first expression, you get the following:
>>> a = 0
>>> a is a
>>> True == 1
>>> (a is a) < 1
You can see above that a is a returns True, as it would for any value. This means that (a is a) < 1 is the same as True < 1. Booleans are numeric types, and True is equal to 1. So True < 1 is the
same as 1 < 1. Since this is a strict inequality, and 1 == 1, it returns False.
The second expression works differently:
>>> a = 0
>>> a < 1
>>> 0 is True
<stdin>:1: SyntaxWarning: "is" with a literal. Did you mean "=="?
Since 0 is less than 1, a < 1 returns True. Since 0 != True, then it can’t be the case that 0 is True.
Note: Don’t take the above SyntaxWarning lightly. Using is on numbers can be confusing. However, specifically for cases in which you know the numbers are not equal, you can know that is will also
return False. While this example is correct, it’s not an example of good Python coding style.
The most important lesson to draw from this is that chaining comparisons with is usually isn’t a good idea. It confuses the reader and probably isn’t necessary.
Like is, the in operator and its opposite, not in, can often yield surprising results when chained:
>>> "b" in "aba" in "cabad" < "cabae"
To maximize the confusion, this example chains comparisons with different operators and uses in with strings to check for substrings. Again, this is not an example of well-written code! However, it’s
important to be able to read this example and understand why it returns True.
Finally, you can chain is not with not in:
>>> greeting = "hello"
>>> quality = "good"
>>> end_greeting = "farewell"
>>> greeting is not quality not in end_greeting
Note that the order of not in the two operators isn’t the same! The negative operators are is not and not in. This corresponds with the regular usage in English, but it’s easy to make a mistake when
modifying code.
Python Boolean Testing
The most popular use for a Python Boolean is in an if statement. This statement will execute if the value is True:
>>> 1 == 1
>>> if 1 == 1:
... print("yep")
>>> 1 == 2
>>> if 1 == 2:
... print("yep")
print() is called only when the expression evaluates to True. However, in Python you can give any value to if. The values that if considers True are called truthy, and the values that if considers
False are called falsy.
if decides which values are truthy and which are falsy by internally calling the built-in bool(). You’ve already encountered bool() as the Python Boolean type. When called, it converts objects to
None as a Boolean Value
The singleton object None is always falsy:
This is often useful in if statements that check for a sentinel value. However, it’s usually better to explicitly check for identity with is None. Sometimes None can be useful in combination with
short-circuit evaluation in order to have a default.
For example, you can use or to substitute None with an empty list:
>>> def add_num_and_len(num, things=None):
... return num + len(things or [])
>>> add_num_and_len(5, [1, 2, 3])
>>> add_num_and_len(6)
In this example, the list won’t be created if things is a non-empty list since or will short-circuit before it evaluates [].
Numbers as Boolean Values
For numbers, bool(x) is equivalent to x != 0. This means the only falsy integer is 0:
>>> bool(3), bool(-5), bool(0)
(True, True, False)
All nonzero integers are truthy. This is also true for floating-point numbers, including special floating-point numbers like infinity and Not a Number (NaN):
>>> import math
>>> [bool(x) for x in [0, 1.2, 0.5, math.inf, math.nan]]
[False, True, True, True, True]
Since infinity and NaN aren’t equal to 0, they’re truthy.
Equality and inequality comparisons on floating-point numbers are subtle operations. Since doing bool(x) is equivalent to x != 0, this can lead to surprising results for floating-point numbers:
>>> bool(0.1 + 0.2 + (-0.2) + (-0.1))
>>> 0.1 + 0.2 + (-0.2) + (-0.1)
Floating-point number computations can be inexact. Because of that, the results of bool() on floating-point numbers can be surprising.
Python has more numeric types in the standard library, and they follow the same rules. For non-built-in numeric types, bool(x) is also equivalent to x != 0. The fractions module is in the standard
library. Like other numeric types, the only falsy fraction is 0/1:
>>> import fractions
>>> bool(fractions.Fraction("1/2")), bool(fractions.Fraction("0/1"))
(True, False)
As with integers and floating-point numbers, fractions are false only when they’re equal to 0.
The decimal module is also in the standard library. Decimals are similarly falsy only when they’re equal to 0:
>>> import decimal, math
>>> with decimal.localcontext(decimal.Context(prec=3)) as ctx:
... bool(ctx.create_decimal(math.pi) - ctx.create_decimal(22)/7)
>>> with decimal.localcontext(decimal.Context(prec=4)) as ctx:
... bool(ctx.create_decimal(math.pi) - ctx.create_decimal(22)/7)
The number 22 / 7 is an approximation of Pi to two decimal places. This fact was discussed by Archimedes in the 3rd century BCE. When the difference between 22 / 7 and Pi is computed with this
precision, the result is falsy. When the difference is computed with higher precision, the difference isn’t equal to 0, and so is truthy.
Sequences as Boolean Values
In general, objects that have a len() will be falsy when the result of len() is 0. It doesn’t matter if they’re lists, tuples, sets, strings, or byte strings:
>>> bool([1]), bool([])
(True, False)
>>> bool((1,2)), bool(())
(True, False)
>>> bool({1,2,3}), bool(set())
(True, False)
>>> bool({1: 2}), bool({})
(True, False)
>>> bool("hello"), bool("")
(True, False)
>>> bool(b"xyz"), bool(b"")
(True, False)
All built-in Python objects that have a length follow this rule. Later, you’ll see some exceptions to this rule for non-built-in objects.
Other Types as Boolean Values
Unless types have a len() or specifically define whether they’re truthy or falsy, they’re always truthy. This is true for built-in as well as user-defined types. In particular, functions are always
>>> def func():
... pass
>>> bool(func)
Methods are always truthy, too. You might encounter this if a parenthesis is missing when you call a function or method:
>>> import datetime
>>> def before_noon():
... return datetime.datetime.now().hour < 12
>>> def greet():
... if before_noon:
... print("Good morning!")
... else:
... print("Good evening!")
>>> greet()
Good morning!
>>> datetime.datetime.now().hour
This can happen as a result of a forgotten parenthesis or misleading documentation that doesn’t mention that you need to call the function. If you expect a Python Boolean value but have a function
that returns a Boolean value, then it will always be truthy.
By default, user-defined types are always truthy:
>>> class Dummy:
... pass
>>> bool(Dummy())
Creating an empty class makes every object of that class truthy. All objects are truthy unless special methods are defined. If you want to make some instances of your class falsy, you can define
>>> class BoolLike:
... am_i_truthy = False
... def __bool__(self):
... return self.am_i_truthy
>>> x = BoolLike()
>>> bool(x)
>>> x.am_i_truthy = True
>>> bool(x)
You can also use .__bool__() to make an object neither truthy nor falsy:
>>> class ExcludedMiddle:
... def __bool__(self):
... raise ValueError("neither")
>>> x = ExcludedMiddle()
>>> bool(x)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 3, in __bool__
ValueError: neither
>>> if x:
... print("x is truthy")
... else:
... print("x is falsy")
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 3, in __bool__
ValueError: neither
The if statement also uses .__bool__(). It does so to evaluate whether the object is truthy or falsy, which determines which branch to execute.
If you define the __len__ method on a class, then its instances have a len(). In that case, the Boolean value of the instances will be falsy exactly when their length is 0:
>>> class DummyContainer:
... my_length = 0
... def __len__(self):
... return self.my_length
>>> x = DummyContainer()
>>> bool(x)
>>> x.my_length = 5
>>> bool(x)
In this example, len(x) would return 0 before the assignment and 5 afterward. The reverse, however, is not true. Defining .__bool__() doesn’t give instances a length:
>>> class AlwaysTrue:
... def __bool__(self):
... return True
>>> class AlwaysFalse:
... def __bool__(self):
... return False
>>> bool(AlwaysTrue()), bool(AlwaysFalse())
(True, False)
>>> len(AlwaysTrue())
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: object of type 'AlwaysTrue' has no len()
>>> len(AlwaysFalse())
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: object of type 'AlwaysFalse' has no len()
Defining .__bool__() doesn’t make instances of either class have a len(). When both .__bool__() and .__len__() are defined, .__bool__() takes precedence:
>>> class BooleanContainer:
... def __len__(self):
... return 100
... def __bool__(self):
... return False
>>> x=BooleanContainer()
>>> len(x)
>>> bool(x)
Even though x has a length of 100, it’s still falsy.
Example: NumPy Arrays
The above example may seem like something that only happens when you write a class intended to demonstrate edge cases in Python. However, it’s possible to get similar results using one of the most
popular libraries on PyPI: NumPy.
Arrays, like numbers, are falsy or truthy depending on how they compare to 0:
>>> from numpy import array
>>> x = array([0])
>>> len(x)
>>> bool(x)
Even though x has a length of 1, it’s still falsy because its value is 0.
When arrays have more than one element, some elements might be falsy and some might be truthy. In those cases, NumPy will raise an exception:
>>> from numpy import array
>>> import textwrap
>>> y=array([0, 1])
>>> try:
... bool(y)
... except ValueError as exc:
... print("\n".join(textwrap.wrap(str(exc))))
The truth value of an array with more than one element is ambiguous.
Use a.any() or a.all()
The exception is so wordy that in order to make it easy to read, the code uses text processing to wrap the lines.
An even more interesting edge case involves empty arrays. You might wonder if those are falsy like other sequences or truthy because they’re not equal to 0. As you saw above, those aren’t the only
two possible answers. The arrays could also refuse to have a Boolean value.
Interestingly, none of these options is entirely true:
>>> bool(array([]))
<stdin>:1: DeprecationWarning: The truth value of an empty array is ambiguous.
Returning False, but in future this will result in an error.
Use `array.size > 0` to check that an array is not empty.
While empty arrays are currently falsy, relying on this behavior is dangerous. In some future NumPy version, this will raise an exception.
Operators and Functions
There are a few more places in Python where Boolean testing takes place. One of those is in Boolean operators.
The operators and, or, and not accept any value that supports Boolean testing. In the case of not, it will always return a Boolean value:
>>> not 1
>>> not 0
The truth table for not is still correct, but now it takes the truthiness of the input.
In the case of and and or, in addition to short-circuit evaluation, they also return the value at which they stopped evaluating:
>>> 1 and 2
>>> 0 and 1
>>> 1 or 2
>>> 0 or 2
The truth tables are still correct, but they now define the truthiness of the results, which depends on the truthiness of the inputs. This can come handy when, for example, you want to give values
Assume you have a function called summarize() that, if the text is too long, takes the beginning and the end and adds an ellipsis (...) in the middle. This might be useful in some reports that can’t
fit the full text. However, some datasets have missing values represented by None.
Since summarize() assumes the input is a string, it will fail on None:
>>> def summarize(long_text):
... if len(long_text) <= 4:
... return long_text
... return long_text[:2] +"..." + long_text[-2:]
>>> summarize("hello world")
>>> summarize("hi")
>>> summarize("")
>>> summarize(None)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 2, in summarize
TypeError: object of type 'NoneType' has no len()
>>> for a in ["hello world", "hi", "", None]:
... print("-->", summarize(a or ""))
--> he...ld
--> hi
This example takes advantage of the falsiness of None and the fact that or not only short-circuits but also returns the last value to be evaluated. The code for printing the report adds or "" to the
argument to summarize(). The addition of or "" helps you to avoid errors with just a small code change.
The built-in functions all() and any() evaluate truthiness and also short-circuit, but they don’t return the last value to be evaluated. all() checks whether all of its arguments are truthy:
>>> all([1, 2, 3])
>>> all([0, 1, 2])
>>> all(x / (x - 1) for x in [0, 1])
In the last line, all() doesn’t evaluate x / (x - 1) for 1. Since 1 - 1 is 0, this would have raised a ZeroDivisionError.
any() checks whether any of its arguments are truthy:
>>> any([1, 0, 0])
>>> any([False, 0, 0.0])
>>> any(1 / x for x in [1, 0])
In the last line, any() doesn’t evaluate 1 / x for 0.
The Python Boolean is a commonly used data type with many useful applications. You can use Booleans with operators like not, and, or, in, is, ==, and != to compare values and check for membership,
identity, or equality. You can also use Boolean testing with an if statement to control the flow of your programs based on the truthiness of an expression.
In this tutorial, you’ve learned how to:
• Manipulate Boolean values with Boolean operators
• Convert Booleans to other types
• Convert other types to Python Booleans
• Use Booleans to write efficient and readable Python code
You now know how short-circuit evaluation works and recognize the connection between Booleans and the if statement. This knowledge will help you to both understand existing code and avoid common
pitfalls that can lead to errors in your own programs.
Watch Now This tutorial has a related video course created by the Real Python team. Watch it together with the written tutorial to deepen your understanding: Python Booleans: Leveraging the Values of
What Do You Think?
LinkedIn Twitter Facebook Email
What’s your #1 takeaway or favorite thing you learned? How are you going to put your newfound skills to use? Leave a comment below and let us know.
Commenting Tips: The most useful comments are those written with the goal of learning from or helping out other students. Get tips for asking good questions and get answers to common questions in our
support portal.
Looking for a real-time conversation? Visit the Real Python Community Chat or join the next “Office Hours” Live Q&A Session. Happy Pythoning! | {"url":"http://docs.pymars.org/index-989.html","timestamp":"2024-11-11T08:23:56Z","content_type":"text/html","content_length":"251110","record_id":"<urn:uuid:795d192a-cee1-4424-b255-f74f7c3e2d8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00839.warc.gz"} |
Math Practice Multiplication Worksheets With Coloring
Mathematics, specifically multiplication, creates the keystone of countless academic techniques and real-world applications. Yet, for many learners, grasping multiplication can present a challenge.
To resolve this obstacle, teachers and parents have welcomed a powerful device: Math Practice Multiplication Worksheets With Coloring.
Intro to Math Practice Multiplication Worksheets With Coloring
Math Practice Multiplication Worksheets With Coloring
Math Practice Multiplication Worksheets With Coloring -
FREE Multiplication Color By Number printables for early education classrooms Easily learn single digit multiplication with these color by multiplication worksheets A great fit for any classroom
These Multiplication Coloring Worksheets combine math and art allowing students to practice multiplication skills while creating a beautiful image with vibrant colors With each correct answer a
section of the image is revealed
Relevance of Multiplication Technique Comprehending multiplication is crucial, laying a solid foundation for innovative mathematical concepts. Math Practice Multiplication Worksheets With Coloring
supply structured and targeted method, fostering a much deeper understanding of this basic math procedure.
Evolution of Math Practice Multiplication Worksheets With Coloring
Third Grade Multiplication Practice
Third Grade Multiplication Practice
Free Downloadable Multiplication Color by Number Worksheet PDFs Ready to get your hands on some free color by code multiplication worksheets Check out this fun collection of printable multiplication
color by
In these spring themed multiplication coloring worksheets kids will solve each single digit multiplication problem and then use the color codes to color the picture This set includes 8 color by code
printables with one page each for
From typical pen-and-paper workouts to digitized interactive styles, Math Practice Multiplication Worksheets With Coloring have progressed, satisfying diverse learning styles and preferences.
Kinds Of Math Practice Multiplication Worksheets With Coloring
Basic Multiplication Sheets Simple workouts focusing on multiplication tables, assisting learners build a solid arithmetic base.
Word Issue Worksheets
Real-life scenarios incorporated right into problems, enhancing critical reasoning and application skills.
Timed Multiplication Drills Tests made to boost speed and precision, helping in rapid mental mathematics.
Benefits of Using Math Practice Multiplication Worksheets With Coloring
Printable Multiplication Worksheets 2 12 Printable Multiplication Flash Cards
Printable Multiplication Worksheets 2 12 Printable Multiplication Flash Cards
Keep reading to learn more about these free multiplication coloring pages and to find out how you can snag a few printable PDFs of your very own When it comes to free multiplication worksheets hidden
A huge selection of multiplication color by number math worksheets in holiday and seasonal themes perfect for building multiplication fact skills All coloring pages in printable PDF format
Improved Mathematical Abilities
Consistent technique sharpens multiplication effectiveness, boosting general math abilities.
Improved Problem-Solving Abilities
Word problems in worksheets develop analytical reasoning and method application.
Self-Paced Learning Advantages
Worksheets accommodate individual discovering speeds, fostering a comfortable and adaptable knowing atmosphere.
How to Create Engaging Math Practice Multiplication Worksheets With Coloring
Integrating Visuals and Colors Vibrant visuals and shades record focus, making worksheets aesthetically appealing and involving.
Including Real-Life Circumstances
Associating multiplication to daily situations includes relevance and usefulness to workouts.
Tailoring Worksheets to Various Skill Degrees Personalizing worksheets based upon varying proficiency levels makes sure inclusive learning. Interactive and Online Multiplication Resources Digital
Multiplication Tools and Gamings Technology-based resources use interactive understanding experiences, making multiplication interesting and enjoyable. Interactive Web Sites and Applications Online
platforms supply diverse and accessible multiplication technique, supplementing traditional worksheets. Personalizing Worksheets for Numerous Knowing Styles Visual Learners Aesthetic aids and
representations aid comprehension for students inclined toward aesthetic knowing. Auditory Learners Verbal multiplication issues or mnemonics cater to learners that comprehend concepts via auditory
ways. Kinesthetic Learners Hands-on tasks and manipulatives support kinesthetic students in comprehending multiplication. Tips for Effective Application in Understanding Consistency in Practice
Regular method enhances multiplication skills, promoting retention and fluency. Stabilizing Repetition and Selection A mix of repetitive workouts and diverse problem layouts maintains passion and
understanding. Giving Useful Comments Responses help in recognizing locations of improvement, motivating continued development. Difficulties in Multiplication Method and Solutions Inspiration and
Engagement Difficulties Boring drills can lead to disinterest; cutting-edge approaches can reignite motivation. Overcoming Fear of Mathematics Unfavorable perceptions around mathematics can impede
progression; developing a favorable learning setting is necessary. Impact of Math Practice Multiplication Worksheets With Coloring on Academic Performance Studies and Research Findings Research
suggests a favorable connection between constant worksheet use and enhanced math performance.
Math Practice Multiplication Worksheets With Coloring become functional devices, cultivating mathematical efficiency in learners while suiting diverse understanding designs. From basic drills to
interactive on-line sources, these worksheets not just improve multiplication abilities yet also promote vital reasoning and analytic abilities.
3 Digit By 2 Digit Multiplication Worksheets 99Worksheets
Color By Number Multiplication Best Coloring Pages For Kids
Check more of Math Practice Multiplication Worksheets With Coloring below
Free Multiplication Worksheets You Can Download Today Grades 3 5 Printable multiplication
3rd Grade Multiplication Coloring Worksheets Free Printable
Multiplication Worksheet For 3rd Grade 001 Multiplication worksheets Printable multiplication
Free Printable Math Coloring Worksheets For 4th Grade Math Worksheets Printable
Copy Of Multiplication Table Multiplication Table Multiplication Table Printable
Colour By Multiplication Fun math Activities Multiplication Math coloring
Color By Number Multiplication Worksheet Twinkl USA
https://www.twinkl.com › resource
These Multiplication Coloring Worksheets combine math and art allowing students to practice multiplication skills while creating a beautiful image with vibrant colors With each correct answer a
section of the image is revealed
Multiplication Coloring Worksheets Math Monks
https://mathmonks.com › worksheets › multiplication...
Test your multiplication skills with our fun coloring worksheets designed for 3rd 4th and 5th graders to make math practice engaging and creative
These Multiplication Coloring Worksheets combine math and art allowing students to practice multiplication skills while creating a beautiful image with vibrant colors With each correct answer a
section of the image is revealed
Test your multiplication skills with our fun coloring worksheets designed for 3rd 4th and 5th graders to make math practice engaging and creative
Free Printable Math Coloring Worksheets For 4th Grade Math Worksheets Printable
3rd Grade Multiplication Coloring Worksheets Free Printable
Copy Of Multiplication Table Multiplication Table Multiplication Table Printable
Colour By Multiplication Fun math Activities Multiplication Math coloring
15 Best Halloween Multiplication Coloring Printables Printablee
15 Best Halloween Multiplication Coloring Printables Printablee
10 Best Free Printable Multiplication Coloring Worksheets Printablee
Frequently Asked Questions (Frequently Asked Questions).
Are Math Practice Multiplication Worksheets With Coloring appropriate for any age groups?
Yes, worksheets can be customized to different age and skill degrees, making them versatile for different students.
Just how usually should pupils practice making use of Math Practice Multiplication Worksheets With Coloring?
Regular method is essential. Routine sessions, preferably a few times a week, can produce considerable renovation.
Can worksheets alone improve math abilities?
Worksheets are a beneficial device yet must be supplemented with diverse understanding methods for comprehensive ability growth.
Are there on-line systems offering free Math Practice Multiplication Worksheets With Coloring?
Yes, several academic sites use open door to a large range of Math Practice Multiplication Worksheets With Coloring.
How can parents support their kids's multiplication practice in the house?
Urging consistent technique, giving support, and producing a positive understanding environment are helpful steps. | {"url":"https://crown-darts.com/en/math-practice-multiplication-worksheets-with-coloring.html","timestamp":"2024-11-13T21:19:05Z","content_type":"text/html","content_length":"28201","record_id":"<urn:uuid:c33b7b28-0c1f-434b-b16c-07426a7c7d08>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00797.warc.gz"} |
What is the primary goal of supervised learning algorithms - ITEagers
Data Science - Question Details
What is the primary goal of supervised learning algorithms?
Similar Question From (Data Science)
What is the primary objective of time series forecasting in predictive modeling?
Similar Question From (Data Science)
In reinforcement learning, what does the term 'exploitation' refer to?
Similar Question From (Data Science)
What is the primary goal of reinforcement learning algorithms?
Similar Question From (Data Science)
Which algorithm is used in reinforcement learning to directly optimize the policy without value function estimation?
Similar Question From (Data Science)
What is the primary advantage of boosting in classification modeling?
Similar Question From (Data Science)
How does polynomial regression model nonlinear relationships?
Similar Question From (Data Science)
How does exponential smoothing model temporal dependencies in time series data?
Similar Question From (Data Science)
How does linear regression estimate the relationship between variables?
Similar Question From (Data Science)
Which regression model is suitable for predicting outcomes when the relationship between variables is linear?
Similar Question From (Data Science)
Which reinforcement learning algorithm uses a neural network to approximate the value function?
Read More Questions
Explore the power of data science with cutting-edge techniques. Unlock insights and drive decision-making with advanced analytics. Read More
Challenge Your Knowledge!
Engage in our interactive quizzes to put your learning to the test. Strengthen your understanding and reinforce your knowledge with our thought-provoking questions and activities.
Start Quiz
Data Science
Explore in-depth information about the subject to enhance your understanding.
Recent comments
Latest Comments section by users
Add a comment
Your Comment will appear after approval!
Computer Science
Solved Past Papers (SPSC)
Solved Past Papers (FPSC) | {"url":"https://iteagers.com/Computer%20Science/Data%20Science/21877_What-is-the-primary-goal-of-supervised-learning-algorithms","timestamp":"2024-11-11T17:14:59Z","content_type":"text/html","content_length":"104829","record_id":"<urn:uuid:14cf5a82-fef8-48b4-b2d7-0b338c88db12>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00824.warc.gz"} |
Did you know...
This selection is made for schools by a children's charity read more. SOS Children is the world's largest charity giving orphaned and abandoned children the chance of family life.
In physics and other sciences, energy (from the Greek ενεργός, energos, "active, working") is a scalar physical quantity that is a property of objects and systems which is conserved by nature. Energy
is often defined as the ability to do work.
Several different forms of energy, including kinetic, potential, thermal, gravitational, elastic, electromagnetic, chemical, nuclear, and mass have been defined to explain all known natural
Energy is converted from one form to another. This principle, the conservation of energy, was first postulated in the early 19th century, and applies to any isolated system. According to Noether's
theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.
Although the total energy of a system does not change with time, its value may depend on the frame of reference. For example, a seated passenger in a moving airplane has zero kinetic energy relative
to the airplane, but nonzero kinetic energy relative to the earth.
The concept of energy emerged out of the idea of vis viva, which Leibniz defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved. To
account for slowing due to friction, Leibniz claimed that heat consisted of the random motion of the constituent parts of matter — a view shared by Isaac Newton, although it would be more than a
century until this was generally accepted. In 1807, Thomas Young was the first to use the term "energy", instead of vis viva, in its modern sense. Gustave-Gaspard Coriolis described "kinetic energy"
in 1829 in its modern sense, and in 1853, William Rankine coined the term " potential energy." It was argued for some years whether energy was a substance (the caloric) or merely a physical quantity,
such as momentum.
He amalgamated all of these laws into the laws of thermodynamics, which aided in the rapid development of explanations of chemical processes using the concept of energy by Rudolf Clausius, Josiah
Willard Gibbs and Walther Nernst. It also led to a mathematical formulation of the concept of entropy by Clausius, and to the introduction of laws of radiant energy by Jožef Stefan.
During a 1961 lecture for undergraduate students at the California Institute of Technology, Richard Feynman, a celebrated physics teacher and Nobel Laureate, said this about the concept of energy:
“ There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law — it is exact so far we know. The law is called conservation of
energy; it states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a
mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a
strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same. ”
—The Feynman Lectures on Physics
Since 1918 it has been known that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time. That is,
energy is conserved because the laws of physics do not distinguish between different moments of time (see Noether's theorem).
Energy in various contexts since the beginning of the universe
The concept of energy and its transformations is useful in explaining and predicting most natural phenomena. The direction of transformations in energy (what kind of energy is transformed to what
other kind) is often described by entropy (equal energy spread among all available degrees of freedom) considerations, since in practice all energy transformations are permitted on a small scale, but
certain larger transformations are not permitted because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces.
The concept of energy is used often in all fields of science.
In chemistry, the energy differences between substances determine whether, and to what extent, they can be converted into other substances or react with other substances.
In biology, chemical bonds are broken and made during metabolic processes, and the associated changes in available energy are studied in the subfield of bioenergetics. Energy is often stored by
cells in the form of substances such as carbohydrate molecules (including sugars) and lipids, which release energy when reacted with oxygen.
In geology and meteorology, continental drift, mountain ranges, volcanos, and earthquakes are phenomena that can be explained in terms of energy transformations in the Earth's interior. While
meteorological phenomena like wind, rain, hail, snow, lightning, tornadoes and hurricanes, are all a result of energy transformations brought about by solar energy on the planet Earth.
In cosmology and astronomy the phenomena of stars, nova, supernova, quasars and gamma ray bursts are the universe's highest-output energy transformations of matter. All stellar phenomena
(including solar activity) are driven by various kinds of energy transformations. Energy in such transformations is either from gravitational collapse of matter (usually molecular hydrogen) into
various classes of astronomical objects (stars, black holes, etc.), or from nuclear fusion (of lighter elements, primarily hydrogen).
Energy transformations in the universe over time are characterized by various kinds of potential energy which has been available since the Big Bang, later being "released" (transformed to more active
types of energy such as kinetic or radiant energy), when a triggering mechanism is available.
Familiar examples of such processes include nuclear decay, in which energy is released which was originally "stored" in heavy isotopes (such as uranium and thorium), by nucleosynthesis, a process
which ultimately uses the gravitational potential energy released from the gravitational collapse of supernovae, to store energy in the creation of these heavy elements before they were incorporated
into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs. In a slower process, heat from nuclear decay of these atoms in the core of the Earth releases
heat, which in turn may lift mountains, via orogenesis. This slow lifting represents a kind of gravitational potential energy storage of the heat energy, which may be released to active kinetic
energy in landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store which has been produced ultimately from the same radioactive heat sources.
Thus, according to present understanding, familiar events such as landslides and earthquakes release energy which has been stored as potential energy in the Earth's gravitational field or elastic
strain (mechanical potential energy) in rocks; but prior to this, represents energy that has been stored in heavy atoms since the collapse of long-destroyed stars created these atoms.
In another similar chain of transformations beginning at the dawn of the universe, nuclear fusion of hydrogen in the Sun releases another store of potential energy which was created at the time of
the Big Bang. At that time, according to theory, space expanded and the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store
of potential energy which can be released by fusion. Such a fusion process is triggered by heat and pressure generated from gravitational collapse of hydrogen clouds when they produce stars, and some
of the fusion energy is then transformed into sunlight. Such sunlight from our Sun may again be stored as gravitational potential energy after it strikes the Earth, as (for example) water evaporates
from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbine/generators to produce electricity). Sunlight also drives all weather
phenomenon, including such events as those triggered in a hurricane, when large unstable areas of warm ocean, heated over months, give up some of their thermal energy suddently to power a few days of
violent air movement. Sunlight is also is captured by plants as chemical potential energy, when carbon dioxide and water are converted into a combustable combination of carbohydrates, lipids, and
oxygen. Release of this energy as heat and light may be triggered suddenly by a spark, in a forest fire; or it may be available more slowly for animal or human metabolism, when these molecules are
ingested, and catabolism is triggered by enzyme action. Through all of these tranformation chains, potential energy stored at the time of the Big Bang is later released by intermediate events,
sometimes being stored in a number of ways over time between releases, as more active energy. In all these events, one kind of energy is converted to other types of energy, including heat.
Regarding applications of the concept of energy
Energy is subject to a strict global conservation law; that is, whenever one measures (or calculates) the total energy of a system of particles whose interactions do not depend explicitly on time, it
is found that the total energy of the system always remains constant.
• The total energy of a system can be subdivided and classified in various ways. For example, it is sometimes convenient to distinguish potential energy (which is a function of coordinates only)
from kinetic energy (which is a function of coordinate time derivatives only). It may also be convenient to distinguish gravitational energy, electric energy, thermal energy, and other forms.
These classifications overlap; for instance thermal energy usually consists partly of kinetic and partly of potential energy.
• The transfer of energy can take various forms; familiar examples include work, heat flow, and advection, as discussed below.
• The word "energy" is also used outside of physics in many ways, which can lead to ambiguity and inconsistency. The vernacular terminology is not consistent with technical terminology. For
example, the important public-service announcement, "Please conserve energy" uses vernacular notions of "conservation" and "energy" which make sense in their own context but are utterly
incompatible with the technical notions of "conservation" and "energy" (such as are used in the law of conservation of energy).
In classical physics energy is considered a scalar quantity, the canonical conjugate to time. In special relativity energy is also a scalar (although not a Lorentz scalar but a time component of the
energy-momentum 4-vector). In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of space-time (= boosts).
Energy transfer
Because energy is strictly conserved and is also locally conserved (wherever it can be defined), it is important to remember that by definition of energy the transfer of energy between the "system"
and adjacent regions is work. A familiar example is mechanical work. In simple cases this is written as:
$\Delta{}E = W$ (1)
if there are no other energy-transfer processes involved. Here $\Delta{}E$ is the amount of energy transferred, and $W$ represents the work done on the system.
More generally, the energy transfer can be split into two categories:
$\Delta{}E = W + Q$ (2)
where $Q$ represents the heat flow into the system.
There are other ways in which an open system can gain or lose energy. If mass is counted as energy (as in many relativistic problems) then $E$ must contain a term for mass lost or gained. In chemical
systems, energy can be added to a system by means of adding substances with different chemical potentials, which potentials are then extracted (both of these process are illustrated by fueling an
auto, a system which gains in energy thereby, without addition of either work or heat). Winding a clock would be adding energy to a mechanical system. These terms may be added to the above equation,
or they can generally be subsumed into a quantity called "energy addition term $E$" which refers to any type of energy carried over the surface of a control volume or system volume. Examples may be
seen above, and many others can be imagined (for example, the kinetic energy of a stream of particles entering a system, or energy from a laser beam adds to system energy, without either being either
work-done or heat-added, in the classic senses).
$\Delta{}E = W + Q + E$ (3)
Where E in this general equation represents other additional advected energy terms not covered by work done on a system, or heat added to it.
Energy is also transferred from potential energy ($E_p$) to kinetic energy ($E_k$) and then back to potential energy constantly. This is referred to as conservation of energy. In this closed system,
energy can not be created or destroyed, so the initial energy and the final energy will be equal to each other. This can be demonstrated by the following:
$E_{pi} + E_{ki} = E_{pF} + E_{kF}'''$
The equation can then be simplified further since $E_p = mgh$ (mass times acceleration due to gravity times the height) and $E_k = \frac{1}{2} mv^2$ (half times mass times velocity squared). Then the
total amount of energy can be found by adding $E_p + E_k = E_{total}$.
Energy and the laws of motion
The Hamiltonian
The total energy of a system is sometimes called the Hamiltonian, after William Rowan Hamilton. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex
or abstract systems. These classical equations have remarkably direct analogs in nonrelativistic quantum mechanics.
The Lagrangian
Another energy-related concept is called the Lagrangian, after Joseph Louis Lagrange. This is even more fundamental than the Hamiltonian, and can be used to derive the equations of motion. In
non-relativistic physics, the Lagrangian is the kinetic energy minus potential energy.
Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (like systems with friction).
Energy and thermodynamics
Internal energy
Internal energy – the sum of all microscopic forms of energy of a system. It is related to the molecular structure and the degree of molecular activity and may be viewed as the sum of kinetic and
potential energies of the molecules; it comprises the following types of energy:
│ Type │ Composition of Internal Energy (U) │
│ Sensible energy │ the portion of the internal energy of a system associated with kinetic energies (molecular translation, rotation, and vibration; electron translation and spin; and nuclear spin) │
│ │ of the molecules. │
│ Latent energy │ the internal energy associated with the phase of a system. │
│ Chemical energy │ the internal energy associated with the different kinds of aggregration of atoms in matter. │
│ Nuclear energy │ the tremendous amount of energy associated with the strong bonds within the nucleus of the atom itself. │
│ Energy │ those types of energies not stored in the system (e.g. heat transfer, mass transfer, and work), but which are recognized at the system boundary as they cross it, which represent │
│ interactions │ gains or losses by a system during a process. │
│ Thermal energy │ the sum of sensible and latent forms of internal energy. │
The laws of thermodynamics
According to the second law of thermodynamics, work can be totally converted into heat, but not vice versa.This is a mathematical consequence of statistical mechanics. The first law of thermodynamics
simply asserts that energy is conserved, and that heat is included as a form of energy transfer. A commonly-used corollary of the first law is that for a "system" subject only to pressure forces and
heat transfer (e.g. a cylinder-full of gas), the differential change in energy of the system (with a gain in energy signified by a positive quantity) is given by:
$\mathrm{d}E = T\mathrm{d}S - P\mathrm{d}V\,$,
where the first term on the right is the heat transfer into the system, defined in terms of temperature T and entropy S (in which entropy increases and the change dS is positive when the system is
heated); and the last term on the right hand side is identified as "work" done on the system, where pressure is P and volume V (the negative sign results since compression of the system is needed to
do work on it, so that the volume change dV is negative when work is done on the system). Although this equation is the standard text-book example of energy conservation in classical thermodynamics,
it is highly specific, ignoring all chemical, electric, nuclear, and gravitational forces, effects such as advection of any form of energy other than heat, and because it contains a term that depends
on temperature. The most general statement of the first law — i.e. conservation of energy — is valid even in situations in which temperature is undefinable.
Energy is sometimes expressed as:
$\mathrm{d}E=\delta Q+\delta W\,$,
which is unsatisfactory because there cannot exist any thermodynamic state functions W or Q that are meaningful on the right hand side of this equation, except perhaps in trivial cases.
Equipartition of energy
The energy of a mechanical harmonic oscillator (a mass on a spring) is alternatively kinetic and potential. At two points in the oscillation cycle it is entirely kinetic, and alternatively at two
other points it is entirely potential. Over the whole cycle, or over many cycles net energy is thus equally split between kinetic and potential. This is called equipartition principle - total energy
of a system with many degrees of freedom is equally split among all these degrees of freedom.
This principle is vitally important to understanding the behaviour of a quantity closely related to energy, called entropy. Entropy is a measure of evenness of a distribution of energy between parts
of a system. This concept is also related to the second law of thermodynamics which basically states that when an isolated system is given more degrees of freedom (= is given new available energy
states which are the same as existing states), then energy spreads over all available degrees equally without distinction between "new" and "old" degrees.
Oscillators, phonons, and photons
In an ensemble (connected collection) of unsynchronized oscillators, the average energy is spread equally between kinetic and potential types.
In a solid, thermal energy (often referred to loosely as heat content) can be accurately described by an ensemble of thermal phonons that act as mechanical oscillators. In this model, thermal energy
is equally kinetic and potential.
In an ideal gas, the interaction potential between particles is essentially the delta function which stores no energy: thus, all of the thermal energy is kinetic.
Because an electric oscillator (LC circuit) is analogous to a mechanical oscillator, its energy must be, on average, equally kinetic and potential. It is entirely arbitrary whether the magnetic
energy is considered kinetic and the electric energy considered potential, or vice versa. That is, either the inductor is analogous to the mass while the capacitor is analogous to the spring, or vice
1. By extension of the previous line of thought, in free space the electromagnetic field can be considered an ensemble of oscillators, meaning that radiation energy can be considered equally
potential and kinetic. This model is useful, for example, when the electromagnetic Lagrangian is of primary interest and is interpreted in terms of potential and kinetic energy.
1. On the other hand, in the key equation $m^2 c^4 = E^2 - p^2 c^2$, the contribution $mc^2$ is called the rest energy, and all other contributions to the energy are called kinetic energy. For a
particle that has mass, this implies that the kinetic energy is $0.5 p^2/m$ at speeds much smaller than c, as can be proved by writing $E = mc^2$ √$(1 + p^2 m^{-2}c^{-2})$ and expanding the
square root to lowest order. By this line of reasoning, the energy of a photon is entirely kinetic, because the photon is massless and has no rest energy. This expression is useful, for example,
when the energy-versus-momentum relationship is of primary interest.
The two analyses are entirely consistent. The electric and magnetic degrees of freedom in item 1 are transverse to the direction of motion, while the speed in item 2 is along the direction of motion.
For non-relativistic particles these two notions of potential versus kinetic energy are numerically equal, so the ambiguity is harmless, but not so for relativistic particles.
Work and virtual work
Work is roughly force times distance. But more precisely, it is
$W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}$
This says that the work ($W$) is equal to the integral (along a certain path) of the force; for details see the mechanical work article.
Work and thus energy is frame dependent. For example, consider a ball being hit by a bat. In the centre-of-mass reference frame, the bat does no work on the ball. But, in the reference frame of the
person swinging the bat, considerable work is done on the ball.
Quantum mechanics
In quantum mechanics energy is defined in terms of the energy operator as a time derivative of the wave function. The Schrödinger equation equates energy operator to the full energy of a particle or
a system. It thus can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of the wave function of quantum
systems. The solution of this equation for bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta. In the solution of the
Schrödinger equation for any oscillator (vibrator) and for electromagnetic wave in vacuum, the resulting energy states are related to the frequency by the Planck equation $E = hu$ (where $h$ is the
Planck's constant and $u$ the frequency). In the case of electromagnetic wave these energy states are called quanta of light or photons.
When calculating kinetic energy (= work to accelerate a mass from zero speed to some finite speed) relativistically - using Lorentz transformations instead of Newtonian mechanics, Einstein discovered
unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest mass energy - energy which every mass must possess even when being at rest. The
amount of energy is directly proportional to the mass of body:
$E = m c^2$,
m is the mass,
c is the speed of light in vacuum,
E is the rest mass energy.
For example, consider electron- positron annihilation, in which the rest mass of individual particles is destroyed, but the inertia equivalent of the system of the two particles (its invariant mass)
remains (since all energy is associated with mass), and this inertia and invariant mass is carried off by photons which individually are massless, but as a system retain their mass. This is a
reversible process - the inverse process is called pair creation - in which the rest mass of particles is created from energy of two (or more) annihilating photons.
In general relativity, the stress-energy tensor serves as the source term for the gravitational field, in rough analogy to the way mass serves as the source term in the non-relativistic Newtonian
It is not uncommon to hear that energy is "equivalent" to mass. It would be more accurate to state that every energy has inertia and gravity equivalent, and because mass is a form of energy, then
mass too has inertia and gravity associated with it.
There is no absolute measure of energy, because energy is defined as the work that one system does (or can do) on another. Thus, only of the transition of a system from one state into another can be
defined and thus measured.
The methods for the measurement of energy often deploy methods for the measurement of still more fundamental concepts of science, namely mass, distance, radiation, temperature, time, electric charge
and electric current.
Conventionally the technique most often employed is calorimetry, a thermodynamic technique that relies on the measurement of temperature using a thermometer or of intensity of radiation using a
Throughout the history of science, energy has been expressed in several different units such as ergs and calories. At present, the accepted unit of measurement for energy is the SI unit of energy,
the joule.
Forms of energy
Classical mechanics distinguishes between potential energy, which is a function of the position of an object, and kinetic energy, which is a function of its movement. Both position and movement are
relative to a frame of reference, which must be specified: this is often (and originally) an arbitrary fixed point on the surface of the Earth, the terrestrial frame of reference. Some introductory
authors attempt to separate all forms of energy in either kinetic or potential: this is not incorrect, but neither is it clear that it is a real simplification, as Feynman points out:
These notions of potential and kinetic energy depend on a notion of length scale. For example, one can speak of macroscopic potential and kinetic energy, which do not include thermal potential
and kinetic energy. Also what is called chemical potential energy (below) is a macroscopic notion, and closer examination shows that it is really the sum of the potential and kinetic energy on
the atomic and subatomic scale. Similar remarks apply to nuclear "potential" energy and most other forms of energy. This dependence on length scale is non-problematic if the various length scales
are decoupled, as is often the case ... but confusion can arise when different length scales are coupled, for instance when friction converts macroscopic work into microscopic thermal energy.
Examples of the interconversion of energy
Mechanical energy is converted
into by
Mechanical energy Lever
Thermal energy Brakes
Electric energy Dynamo
Electromagnetic radiation Synchrotron
Chemical energy Matches
Nuclear energy Particle accelerator
Potential energy
Potential energy, symbols E[p], V or Φ, is defined as the work done against a given force (= work of given force with minus sign) in changing the position of an object with respect to a reference
position (often taken to be infinite separation). If F is the force and s is the displacement,
$E_{\rm p} = -\int \mathbf{F}\cdot{\rm d}\mathbf{s}$
with the dot representing the scalar product of the two vectors.
The name "potential" energy originally signified the idea that the energy could readily be transferred as work—at least in an idealized system (reversible process, see below). This is not completely
true for any real system, but is often a reasonable first approximation in classical mechanics.
The general equation above can be simplified in a number of common cases, notably when dealing with gravity or with elastic forces.
Gravitational potential energy
The gravitational force near the Earth's surface varies very little with the height, h, and is equal to the mass, m, multiplied by the gravitational acceleration, g = 9.81 m/s². In these cases, the
gravitational potential energy is given by
$E_{\rm p,g} = mgh$
A more general expression for the potential energy due to Newtonian gravitation between two bodies of masses m[1] and m[2], useful in astronomy, is
$E_{\rm p,g} = -G{{m_1m_2}\over{r}}$,
where r is the separation between the two bodies and G is the gravitational constant, 6.6742(10)×10^−11 m³kg^−1s^−2. In this case, the reference point is the infinite separation of the two bodies.
Elastic potential energy
Elastic potential energy is defined as a work needed to compress (or expand) a spring. The force, F, in a spring or any other system which obeys Hooke's law is proportional to the extension or
compression, x,
$F = -kx$
where k is the force constant of the particular spring (or system). In this case, the calculated work becomes
$E_{\rm p,e} = {1\over 2}kx^2$.
Hooke's law is a good approximation for behaviour of chemical bonds under normal conditions, i.e. when they are not being broken or formed.
Kinetic energy
Kinetic energy, symbols E[k], T or K, is the work required to accelerate an object to a given speed. Indeed, calculating this work one easily obtains the following:
$E_{\rm k} = \int \mathbf{F} \cdot d \mathbf{x} = \int \mathbf{v} \cdot d \mathbf{p}= {1\over 2}mv^2$
At speeds approaching the speed of light, c, this work must be calculated using Lorentz transformations, which results in the following:
$E_{\rm k} = m c^2\left(\frac{1}{\sqrt{1 - (v/c)^2}} - 1\right)$
This equation reduces to the one above it, at small (compared to c) speed. A mathematical by-product of this work (which is immediately seen in the last equation) is that even at rest a mass has the
amount of energy equal to:
$E_{\rm rest} = mc^2$
This energy is thus called rest mass energy.
Thermal energy
Examples of the interconversion of energy
Thermal energy is converted
into by
Mechanical energy Steam turbine
Thermal energy Heat exchanger
Electric energy Thermocouple
Electromagnetic radiation Hot objects
Chemical energy Blast furnace
Nuclear energy Supernova
The general definition of thermal energy, symbols q or Q, is also problematic. A practical definition for small transfers of heat is
$\Delta q = \int C_{\rm v}{\rm d}T$
where C[v] is the heat capacity of the system. This definition will fail if the system undergoes a phase transition—e.g. if ice is melting to water—as in these cases the system can absorb heat
without increasing its temperature. In more complex systems, it is preferable to use the concept of internal energy rather than that of thermal energy (see Chemical energy below).
Despite the theoretical problems, the above definition is useful in the experimental measurement of energy changes. In a wide variety of situations, it is possible to use the energy released by a
system to raise the temperature of another object, e.g. a bath of water. It is also possible to measure the amount of electric energy required to raise the temperature of the object by the same
amount. The calorie was originally defined as the amount of energy required to raise the temperature of one gram of water by 1 °C (approximately 4.1855 J, although the definition later changed), and
the British thermal unit was defined as the energy required to heat one gallon (UK) of water by 1 °F (later fixed as 1055.06 J).
Electric energy
Examples of the interconversion of energy
Electric energy is converted
into by
Mechanical energy Electric motor
Thermal energy Resistor
Electric energy Transformer
Electromagnetic radiation Light-emitting diode
Chemical energy Electrolysis
Nuclear energy Synchrotron
The electric potential energy of given configuration of charges is defined as the work which must be done against the Coulomb force to rearrange charges from infinite separation to this configuration
(or the work done by the Coulomb force separating the charges from this configuration to infinity). For two point-like charges Q[1] and Q[2] at a distance r this work, and hence electric potential
energy is equal to:
$E_{\rm p,e} = {1\over {4\pi\epsilon_0}}{{Q_1Q_2}\over{r}}$
where ε[0] is the electric constant of a vacuum, 10^7/4πc[0]² or 8.854188…×10^−12 F/m. If the charge is accumulated in a capacitor (of capacitance C), the reference configuration is usually selected
not to be infinite separation of charges, but vice versa - charges at an extremely close proximity to each other (so there is zero net charge on each plate of a capacitor). In this case the work and
thus the electric potential energy becomes
$E_{\rm p,e} = {{Q^2}\over{2C}}$
If an electric current passes through a resistor, electric energy is converted to heat; if the current passes through an electric appliance, some of the electric energy will be converted into other
forms of energy (although some will always be lost as heat). The amount of electric energy due to an electric current can be expressed in a number of different ways:
$E = UQ = UIt = Pt = U^2t/R$
where U is the electric potential difference (in volts), Q is the charge (in coulombs), I is the current (in amperes), t is the time for which the current flows (in seconds), P is the power (in
watts) and R is the electric resistance (in ohms). The last of these expressions is important in the practical measurement of energy, as potential difference, resistance and time can all be measured
with considerable accuracy.
Magnetic energy
There is no fundamental difference between magnetic energy and electric energy: the two phenomena are related by Maxwell's equations. The potential energy of a magnet of magnetic moment m in a
magnetic field B is defined as the work of magnetic force (actually of magnetic torque) on re-alignment of the vector of the magnetic dipole moment, and is equal:
$E_{\rm p,m} = -m\cdot B$
while the energy stored in a inductor (of inductance L) when current I is passing via it is
$E_{\rm p,m} = {1\over 2}LI^2$.
This second expression forms the basis for superconducting magnetic energy storage.
Electromagnetic fields
Examples of the interconversion of energy
Electromagnetic radiation is converted
into by
Mechanical energy Solar sail
Thermal energy Solar collector
Electric energy Solar cell
Electromagnetic radiation Non-linear optics
Chemical energy Photosynthesis
Nuclear energy Mössbauer spectroscopy
Calculating work needed to create an electric or magnetic field in unit volume (say, in a capacitor or an inductor) results in the electric and magnetic fields energy densities:
$u_e=\frac{\epsilon_0}{2} E^2$
$u_m=\frac{1}{2\mu_0} B^2$,
in SI units.
Electromagnetic radiation, such as microwaves, visible light or gamma rays, represents a flow of electromagnetic energy. Applying the above expressions to magnetic and electric components of
electromagnetic field both the volumetric density and the flow of energy in e/m field can be calculated. The resulting Poynting vector, which is expressed as
$\mathbf{S} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B},$
in SI units, gives the density of the flow of energy and its direction.
The energy of electromagnetic radiation is quantized (has discrete energy levels). The spacing between these levels is equal to
$E = hu$
where h is the Planck constant, 6.6260693(11)×10^−34 Js, and ν is the frequency of the radiation. This quantity of electromagnetic energy is usually called a photon. The photons which make up visible
light have energies of 270–520 yJ, equivalent to 160–310 kJ/mol, the strength of weaker chemical bonds.
Chemical energy
Chemical energy is the energy due to associations of atoms in molecules and various other kinds of aggregates of matter. It may be defined as a work done by electric forces during re-arrangement of
electric charges, electrons and protons, in the process of aggregation. If the chemical energy of a system decreases during a chemical reaction, it is transferred to the surroundings in some form of
energy (often heat); on the other hand if the chemical energy of a system increases as a result of a chemical reaction - it is by converting another form of energy from the surroundings. For example,
when two hydrogen atoms react to form a dihydrogen molecule, the chemical energy decreases by 724 zJ (the bond energy of the H–H bond);
when the electron is completely removed from a hydrogen atom, forming a hydrogen ion (in the gas phase), the chemical energy increases by 2.18 aJ (the ionization energy of hydrogen).
It is common to quote the changes in chemical energy for one mole of the substance in question: typical values for the change in molar chemical energy during a chemical reaction range from tens to
hundreds of kJ/mol.
The chemical energy as defined above is also referred to by chemists as the internal energy, U: technically, this is measured by keeping the volume of the system constant. However, most practical
chemistry is performed at constant pressure and, if the volume changes during the reaction (e.g. a gas is given off), a correction must be applied to take account of the work done by or on the
atmosphere to obtain the enthalpy, H:
ΔH = ΔU + pΔV
A second correction, for the change in entropy, S, must also be performed to determine whether a chemical reaction will take place or not, giving the Gibbs free energy, G:
ΔG = ΔH − TΔS
These corrections are sometimes negligible, but often not (especially in reactions involving gases).
Since the industrial revolution, the burning of coal, oil, natural gas or products derived from them has been a socially significant transformation of chemical energy into other forms of energy. the
energy "consumption" (one should really speak of "energy transformation") of a society or country is often quoted in reference to the average energy released by the combustion of these fossil fuels:
1 tonne of coal equivalent (TCE) = 29 GJ
1 tonne of oil equivalent (TOE) = 41.87 GJ
On the same basis, a tank-full of gasoline (45 litres, 12 gallons) is equivalent to about 1.6 GJ of chemical energy. Another chemically-based unit of measurement for energy is the "tonne of TNT",
taken as 4.184 GJ. Hence, burning a tonne of oil releases about ten times as much energy as the explosion of one tonne of TNT: fortunately, the energy is usually released in a slower, more controlled
Simple examples of chemical energy are batteries and food. When you eat the food is digested and turned into chemical energy which can be transformed to kinetic energy.
Nuclear energy
Examples of the interconversion of energy
Nuclear binding energy is converted
into by
Mechanical energy Alpha radiation
Thermal energy Sun
Electric energy Beta radiation
Electromagnetic radiation Gamma radiation
Chemical energy Radioactive decay
Nuclear energy Nuclear isomerism
Nuclear potential energy, along with electric potential energy, provides the energy released from nuclear fission and nuclear fusion processes. The result of both these processes are nuclei in which
strong nuclear forces bind nuclear particles more strongly and closely. Weak nuclear forces (different from strong forces) provide the potential energy for certain kinds of radioactive decay, such as
beta decay. The energy released in nuclear processes is so large that the relativistic change in mass (after the energy has been removed) can be as much as several parts per thousand.
Nuclear particles ( nucleons) like protons and neutrons are not destroyed (law of conservation of baryon number) in fission and fusion processes. A few lighter particles may be created or destroyed
(example: beta minus and beta plus decay, or electron capture decay), but these minor processes are not important to the immediate energy release in fission and fusion. Rather, fission and fusion
release energy when collections of baryons become more tightly bound, and it is the energy associated with a fraction of the mass of the nucleons (but not the whole particles) which appears as the
heat and electromagnetic radiation generated by nuclear reactions. This heat and radiation retains the "missing" mass, but the mass is missing only because it escapes in the form of heat and light,
which retain the mass and conduct it out of the system where it is not measured. The energy from the Sun, also called solar energy, is an example of this form of energy conversion. In the Sun, the
process of hydrogen fusion converts about 4 million metric tons of solar matter per second into light, which is radiated into space, but during this process, the number of total protons and neutrons
in the sun does not change. In this system, the light itself retains the inertial equivalent of this mass, and indeed the mass itself (as a system), which represents 4 million tons per second of
electromagnetic radiation, moving into space. Each of the helium nuclei which are formed in the process are less massive than the four protons from they were formed, but (to a good approximation), no
particles or atoms are destroyed in the process of turning the sun's nuclear potential energy into light.
Surface energy
If there is any kind of tension in a surface, such as a stretched sheet of rubber or material interfaces, it is possible to define surface energy. In particular, any meeting of dissimilar materials
that don't mix will result in some kind of surface tension, if there is freedom for the surfaces to move then, as seen in capillary surfaces for example, the minimum energy will as usual be sought.
A minimal surface, for example, represents the smallest possible energy that a surface can have if its energy is proportional to the area of the surface. For this reason, (open) soap films of small
size are minimal surfaces (small size reduces gravity effects, and openness prevents pressure from building up. Note that a bubble is a minimum energy surface but not a minimal surface by
Transformations of energy
One form of energy can often be readily transformed into another with the help of a device- for instance, a battery, from chemical energy to electric energy; a dam: gravitational potential energy to
kinetic energy of moving water (and the blades of a turbine) and ultimately to electric energy through an electric generator. Similarly, in the case of a chemical explosion, chemical potential energy
is transformed to kinetic energy and thermal energy in a very short time. Yet another example is that of a pendulum. At its highest points the kinetic energy is zero and the gravitational potential
energy is at maximum. At its lowest point the kinetic energy is at maximum and is equal to the decrease of potential energy. If one (unrealistically) assumes that there is no friction, the conversion
of energy between these processes is perfect, and the pendulum will continue swinging forever.
Energy can be converted into matter and vice versa. The mass-energy equivalence formula E = mc², derived independently by Albert Einstein and Henri Poincaré, quantifies the relationship between mass
and rest energy. Since $c^2$ is extremely large relative to ordinary human scales, the conversion of mass to other forms of energy can liberate tremendous amounts of energy, as can be seen in nuclear
reactors and nuclear weapons. Conversely, the mass equivalent of a unit of energy is minuscule, which is why a loss of energy from most systems is difficult to measure by weight, unless the energy
loss is very large. Examples of energy transformation into matter (particles) are found in high energy nuclear physics.
In nature, transformations of energy can be fundamentally classed into two kinds: those that are thermodynamically reversible, and those that are thermodynamically irreversible. A reversible process
in thermodynamics is one in which no energy is dissipated into empty quantum states available in a volume, from which it cannot be recovered into more concentrated forms (fewer quantum states),
without degradation of even more energy. A reversible process is one in which this sort of dissipation does not happen. For example, conversion of energy from one type of potential field to another,
is reversible, as in the pendulum system described above. In processes where heat is generated, however, quantum states of lower energy, present as possible exitations in fields between atoms, act as
a reservoir for part of the energy, from which it cannot be recovered, in order to be converted with 100% efficiency into other forms of energy. In this case, the energy must partly stay as heat, and
cannot be completely recovered as usable energy, except at the price of an increase in some other kind of heat-like increase in disorder in quantum states, in the universe (such as an expansion of
matter, or a randomization in a crystal).
As the universe evolves in time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or other kinds of increases in disorder). This has been referred to as the
inevitable thermodynamic heat death of the universe. In this heat death the energy of the universe does not change, but the fraction of energy which is available to do work, or be transformed to
other usable forms of energy, grows less and less.
Law of conservation of energy
Energy is subject to the law of conservation of energy. According to this law, energy can neither be created (produced) nor destroyed itself. It can only be transformed.
Most kinds of energy (with gravitational energy being a notable exception) are also subject to strict local conservation laws, as well. In this case, energy can only be exchanged between adjacent
regions of space, and all observers agree as to the volumetric density of energy in any given space. There is also a global law of conservation of energy, stating that the total energy of the
universe cannot change; this is a corollary of the local law, but not vice versa. Conservation of energy is the mathematical consequence of translational symmetry of time (that is, the
indistinguishability of time intervals taken at different time) - see Noether's theorem.
According to energy conservation law the total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system.
This law is a fundamental principle of physics. It follows from the translational symmetry of time, a property of most phenomena below the cosmic scale that makes them independent of their locations
on the time coordinate. Put differently, yesterday, today, and tomorrow are physically indistinguishable.
Thus because energy is quantity which is canonical conjugate to time, it is impossible to define the exact amount of energy during any definite time interval - making it impossible to apply the law
of conservation of energy. This must not be considered a "violation" of the law. We know the law still holds, because a succession of short time periods do not accumulate any violation of
conservation of energy.
In quantum mechanics energy is expressed using the Hamiltonian operator. On any time scales, the uncertainty in the energy is by
$\Delta E \Delta t \ge \frac {h} {2 \pi}$
which is similar in form to the Heisenberg uncertainty principle (but not really mathematically equivalent thereto, since H and t are not dynamically conjugate variables, neither in classical nor in
quantum mechanics).
In particle physics, this inequality permits a qualitative understanding of virtual particles which carry momentum, exchange by which and with real particles, is responsible for the creation of all
known fundamental forces (more accurately known as fundamental interactions). Virtual photons (which are simply lowest quantum mechanical energy state of photons) are also responsible for
electrostatic interaction between electric charges (which results in Coulomb law), for spontaneous radiative decay of exited atomic and nuclear states, for the Casimir force, for van der Waals bond
forces and some other observable phenomena.
Energy and life
Any living organism relies on an external source of energy—radiation from the Sun in the case of green plants; chemical energy in some form in the case of animals—to be able to grow and reproduce.
The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as a combination of oxygen and food molecules, the latter mostly carbohydrates and fats, of which glucose (C[6]H[12]O[6])
and stearin (C[57]H[110]O[6]) are convenient examples. The food molecules are oxidised to carbon dioxide and water in the mitochondria
C[6]H[12]O[6] + 6O[2] → 6CO[2] + 6H[2]O
C[57]H[110]O[6] + 81.5O[2] → 57CO[2] + 55H[2]O
and some of the energy is used to convert ADP into ATP
ADP + HPO[4]^2− → ATP + H[2]O
The rest of the chemical energy in the carbohydrate or fat is converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains when split and reacted
with water, is used for other metabolism (at each stage of a metabolic pathway, some chemical energy is converted into heat). Only a tiny fraction of the original chemical energy is used for work:
gain in kinetic energy of a sprinter during a 100 m race: 4 kJ
gain in gravitational potential energy of a 150 kg weight lifted through 2 metres: 3kJ
Daily food intake of a normal adult: 6–8 MJ
It would appear that living organisms are remarkably inefficient (in the physical sense) in their use of the energy they receive (chemical energy or radiation), and it is true that most real machines
manage higher efficiencies. However, in growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism tissue to be highly ordered with regard to the
molecules it is built from. The second law of thermodynamics states that energy (and matter) tends to become more evenly spread out across the universe: to concentrate energy (or matter) in one
specific place, it is necessary to spread out a greater amount of energy (as heat) across the remainder of the universe ("the surroundings"). Simpler organisms can achieve higher energy efficiencies
than more complex ones, but the complex organisms can occupy ecological niches that are not available to their simpler brethren. The conversion of a portion of the chemical energy to heat at each
step in a metabolic pathway is the physical reason behind the pyramid of biomass observed in ecology: to take just the first step in the food chain, of the estimated 124.7 Pg/a of carbon that is
fixed by photosynthesis, 64.3 Pg/a (52%) are used for the metabolism of green plants, i.e. reconverted into carbon dioxide and heat. | {"url":"https://ftp.worldpossible.org/endless/eos-rachel/RACHEL/RACHEL/modules/wikipedia_for_schools/wp/e/Energy.htm","timestamp":"2024-11-12T15:28:30Z","content_type":"text/html","content_length":"91298","record_id":"<urn:uuid:ccf4b14d-975e-45e0-913a-64e8b371ec94>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00078.warc.gz"} |
Blog Archives
Last month we discussed the importance of re-grounding ourselves in strategic fundamentals prior to calculating our deal valuation, and we also discussed the EBITDA Archives
multiple approach to those calculations. This month we’ll discuss the cash flow approach, as well as a few other considerations for calculating estimated deal value.
Stream of unlevered cash flows
A common valuation approach is to calculate the target company’s projected future cash flows- often over a period of the next 5 to 10 years. The cash flows are usually
first “unlevered’, meaning that interest expense is added back for the purpose of valuation. Now it’s time for a trip back to our college finance classes. Don’t worry, Sign up to receive a free
we’ll make it quick. 30 minute consultation and
our "PMO in a Box"
Recall that our basic formula for the present value of something is: toolkit!
Future value=present value *[ (1+rate)^number of periods] Sign Up
Present values are calculated using the company’s cost of capital, or the required hurdle rate (as set by leadership/board of directors) if higher. An example assuming
a 10 year horizon and 18% hurdle rate:
1. Calculate the estimated cash flows for the target over each of the next 10 years using reasonable (and documented!) assumptions.
2. Unlever those cash flows by adding back any interest expenses. These cash flows will become our “future values” for our calculation.
3. Calculate the present value of each years unlevered cash flow using the formula above where “rate” is 18% and “number of periods” equals 1 for the first year’s cash
flows, 2 for the second year, and so on.
4. Add up all the individual present values to get the actual total present value, or the amount we are theoretically willing to pay for the transaction.
5. Enjoy the beverage of your choice, the fun is just beginning.
Note that there are a few wrinkles with this approach. First, we are assuming that our ability to forecast out 10 years is solid. Second, we need a relatively near-term
positive cash flow stream to make the calculations useful. Third, we are essentially valuing anything beyond our horizon at zero.
It is in part to address this problem that investment bankers will often convert the net present value to an internal rate of return (IRR). The IRR is a fancy way of
saying what hurdle rate would we need to use in the NPV formula to give us a net present value of $0? Once calculated, this IRR rate is then applied to future projected
cash flows in the assumption that reinvestment would be available at that rate.
Real-World Example: Bad Sport
I was recently asked to assist a CFO in the preparation of a valuation model for an acquisition. The projected unlevered cash flows, along with forecasted financial
statements, had been provided by the investment banking team. The small target company had a single product, which they sold only domestically via their website, at a
fixed price of $25. For purposes of this discussion let’s just say that the product was a sporting goods item. Thus, you would only need one of these if you
participated in that sport, and even so the useful life should be between 5-10 years once purchased.
In looking at the sales figures, I noticed something didn’t look right. I quickly ran a US population analysis, and did some quick total addressable market
calculations. To sum up the results, every man, woman, and child in the United States would have needed to purchase 1.32 of these items every 5 years in order to reach
sales numbers in the financial projections.
Obviously, this didn’t seem realistic to me, so I checked in with the bankers. Were prices expected to rise? Were new products on the horizon? Markets being expanded?
Distribution increasing? Nope, nope, nope, and nope. They had simply applied the growth rate of the last 2 years to the next 5 years, without taking into account the
actual constraints of the market itself!
Bottom line, always check the financial models. Make sure all assumptions are stated, and run market analysis on any projections. Be prepared to explain what market
share capture is assumed, and why that is realistic.
This solves one problem but creates a few more. First, there can mathematically be more than one IRR- i.e. more than 1 hurdle rate that causes the net present value of
future cash flows to be zero- particularly if cash flows fluctuate between positive and negative values. Second, IRR valuations tend to be on the high side, because
they treat investment in the target the same as they would treat putting money in a bank account at that rate of compound interest. Since we know that all companies
experience a lifecycle of growth, stabilization, and decline, that is a fairly optimistic viewpoint- particularly in highly competitive industries or those subject to
higher than average risks.
Asset valuation
Note that all of the methods we have discussed thus far are highly dependent on the company’s results of operations. Normally this is appropriate; however, a lot of
time can be saved by taking a closer look at what assets the company may have.
I was recently asked by a small, privately-held company to evaluate a potential acquisition. We had some really good comps to use, which yielded an enterprise value of
$7 million for the target. NPV analysis suggested a value of about $6.2 million; however, the company had just achieved profitability, after accumulating over $100
million in losses during prior years. Assuming a 30% tax rate, that means that there could be upwards of $30 million of tax reduction value to the company that acquires
this target! This suggests a valuation far beyond what my client could afford, and one that would bear no resemblance to the comps or NPV of cash flows. Luckily, we
caught this prior to the CEO calling in the Board to review the acquisition, and he was able to save face!
Most acquirers will study the income and cash flow statements carefully, but remember to also look at the balance sheet, and to consider the actual market value of any
tax benefits, real estate, or other assets, getting the appropriate experts in to value these as needed.
Tax Considerations
This is a good time to mention tax considerations. In general, it is highly critical to have skilled tax personnel involved in the evaluation and calculation of the
value of any proposed deal. The example above is just one of many I could name. For instance, I once worked on an acquisition with a value in excess of $15 billion,
much of which was actually paid for via careful tax planning regarding the legal entity structure of the deal. On the flip side, I also worked on a deal where tax had
been left out of the discussion to date, and we discovered a potential loss of several million in value resulting from ignoring tax considerations in the deal
structure. As we discussed in our due diligence chapters, make sure tax is at the table, and that they have sufficient time and funding to avoid any unnecessary value
There is no “perfect” valuation method. Comparisons rely heavily on the judgement of others, and are often apples to oranges. NPV overlooks the value generated beyond
the consideration horizon. IRR tends to over-value the company’s long-term contribution to the bottom line. Both NPV and IRR rely heavily on the ability to accurately
estimate the future cash flows, which is no easy task in itself. In mature industries I personally prefer to use the median of at least 7 good comps paired with a
10-year NPV to provide a valuation range, with an understanding that am likely to be looking at lower valuations than my competitors that are using IRR. Overall,
remember that more than half of acquisitions have historically failed to deliver on their projected value, so our skepticism bias is certainly warranted.
0 Comments | {"url":"https://www.nearcoadvisors.com/nearco-ma-blog/archives/08-2019","timestamp":"2024-11-12T02:15:24Z","content_type":"text/html","content_length":"67015","record_id":"<urn:uuid:a7376ec9-8dc0-4ca4-b242-c4734bd35f78>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00208.warc.gz"} |
Square Yard to Kanal
Square Yard [yd2] Output
1 square yard in ankanam is equal to 0.125
1 square yard in aana is equal to 0.026296566837107
1 square yard in acre is equal to 0.00020661138759433
1 square yard in arpent is equal to 0.0002445605876639
1 square yard in are is equal to 0.0083612736
1 square yard in barn is equal to 8.3612736e+27
1 square yard in bigha [assam] is equal to 0.000625
1 square yard in bigha [west bengal] is equal to 0.000625
1 square yard in bigha [uttar pradesh] is equal to 0.00033333333333333
1 square yard in bigha [madhya pradesh] is equal to 0.00075
1 square yard in bigha [rajasthan] is equal to 0.00033057851239669
1 square yard in bigha [bihar] is equal to 0.00033063923585599
1 square yard in bigha [gujrat] is equal to 0.00051652892561983
1 square yard in bigha [himachal pradesh] is equal to 0.0010330578512397
1 square yard in bigha [nepal] is equal to 0.00012345679012346
1 square yard in biswa [uttar pradesh] is equal to 0.0066666666666667
1 square yard in bovate is equal to 0.000013935456
1 square yard in bunder is equal to 0.000083612736
1 square yard in caballeria is equal to 0.0000018580608
1 square yard in caballeria [cuba] is equal to 0.0000062304572280179
1 square yard in caballeria [spain] is equal to 0.0000020903184
1 square yard in carreau is equal to 0.000064816074418605
1 square yard in carucate is equal to 0.0000017204266666667
1 square yard in cawnie is equal to 0.0001548384
1 square yard in cent is equal to 0.020661138759433
1 square yard in centiare is equal to 0.83612736
1 square yard in circular foot is equal to 11.46
1 square yard in circular inch is equal to 1650.12
1 square yard in cong is equal to 0.00083612736
1 square yard in cover is equal to 0.00030990636026686
1 square yard in cuerda is equal to 0.00021275505343511
1 square yard in chatak is equal to 0.2
1 square yard in decimal is equal to 0.020661138759433
1 square yard in dekare is equal to 0.00083612791159358
1 square yard in dismil is equal to 0.020661138759433
1 square yard in dhur [tripura] is equal to 2.5
1 square yard in dhur [nepal] is equal to 0.049382716049383
1 square yard in dunam is equal to 0.00083612736
1 square yard in drone is equal to 0.000032552083333333
1 square yard in fanega is equal to 0.0001300353592535
1 square yard in farthingdale is equal to 0.00082621280632411
1 square yard in feddan is equal to 0.00020059358018867
1 square yard in ganda is equal to 0.010416666666667
1 square yard in gaj is equal to 1
1 square yard in gajam is equal to 1
1 square yard in guntha is equal to 0.0082644628099174
1 square yard in ghumaon is equal to 0.00020661157024793
1 square yard in ground is equal to 0.00375
1 square yard in hacienda is equal to 9.3317785714286e-9
1 square yard in hectare is equal to 0.000083612736
1 square yard in hide is equal to 0.0000017204266666667
1 square yard in hout is equal to 0.00058829869152616
1 square yard in hundred is equal to 1.7204266666667e-8
1 square yard in jerib is equal to 0.00041360294117647
1 square yard in jutro is equal to 0.00014528711728931
1 square yard in katha [bangladesh] is equal to 0.0125
1 square yard in kanal is equal to 0.0016528925619835
1 square yard in kani is equal to 0.00052083333333333
1 square yard in kara is equal to 0.041666666666667
1 square yard in kappland is equal to 0.0054202473745624
1 square yard in killa is equal to 0.00020661157024793
1 square yard in kranta is equal to 0.125
1 square yard in kuli is equal to 0.0625
1 square yard in kuncham is equal to 0.0020661157024793
1 square yard in lecha is equal to 0.0625
1 square yard in labor is equal to 0.0000011663997583457
1 square yard in legua is equal to 4.665599033383e-8
1 square yard in manzana [argentina] is equal to 0.000083612736
1 square yard in manzana [costa rica] is equal to 0.00011963544790641
1 square yard in marla is equal to 0.033057851239669
1 square yard in morgen [germany] is equal to 0.000334450944
1 square yard in morgen [south africa] is equal to 0.000097598617952609
1 square yard in mu is equal to 0.001254191033729
1 square yard in murabba is equal to 0.0000082644555037733
1 square yard in mutthi is equal to 0.066666666666667
1 square yard in ngarn is equal to 0.0020903184
1 square yard in nali is equal to 0.0041666666666667
1 square yard in oxgang is equal to 0.000013935456
1 square yard in paisa is equal to 0.10518934081346
1 square yard in perche is equal to 0.02445605876639
1 square yard in parappu is equal to 0.0033057822015093
1 square yard in pyong is equal to 0.25291208711434
1 square yard in rai is equal to 0.0005225796
1 square yard in rood is equal to 0.00082644628099174
1 square yard in ropani is equal to 0.0016435354273192
1 square yard in satak is equal to 0.020661138759433
1 square yard in section is equal to 3.228305785124e-7
1 square yard in sitio is equal to 4.645152e-8
1 square yard in square is equal to 0.09
1 square yard in square angstrom is equal to 83612736000000000000
1 square yard in square astronomical units is equal to 3.7361268155715e-23
1 square yard in square attometer is equal to 8.3612736e+35
1 square yard in square bicron is equal to 8.3612736e+23
1 square yard in square centimeter is equal to 8361.27
1 square yard in square chain is equal to 0.0020661072388484
1 square yard in square cubit is equal to 4
1 square yard in square decimeter is equal to 83.61
1 square yard in square dekameter is equal to 0.0083612736
1 square yard in square digit is equal to 2304
1 square yard in square exameter is equal to 8.3612736e-37
1 square yard in square fathom is equal to 0.25
1 square yard in square femtometer is equal to 8.3612736e+29
1 square yard in square fermi is equal to 8.3612736e+29
1 square yard in square feet is equal to 9
1 square yard in square furlong is equal to 0.000020661138759433
1 square yard in square gigameter is equal to 8.3612736e-19
1 square yard in square hectometer is equal to 0.000083612736
1 square yard in square inch is equal to 1296
1 square yard in square league is equal to 3.5869921157396e-8
1 square yard in square light year is equal to 9.3416320166653e-33
1 square yard in square kilometer is equal to 8.3612736e-7
1 square yard in square megameter is equal to 8.3612736e-13
1 square yard in square meter is equal to 0.83612736
1 square yard in square microinch is equal to 1295998856724700
1 square yard in square micrometer is equal to 836127360000
1 square yard in square micromicron is equal to 8.3612736e+23
1 square yard in square micron is equal to 836127360000
1 square yard in square mil is equal to 1296000000
1 square yard in square mile is equal to 3.228305785124e-7
1 square yard in square millimeter is equal to 836127.36
1 square yard in square nanometer is equal to 836127360000000000
1 square yard in square nautical league is equal to 2.7086192499848e-8
1 square yard in square nautical mile is equal to 2.4377551745274e-7
1 square yard in square paris foot is equal to 7.93
1 square yard in square parsec is equal to 8.7815509904538e-34
1 square yard in perch is equal to 0.033057851239669
1 square yard in square perche is equal to 0.016371526910969
1 square yard in square petameter is equal to 8.3612736e-31
1 square yard in square picometer is equal to 8.3612736e+23
1 square yard in square pole is equal to 0.033057851239669
1 square yard in square rod is equal to 0.033057723990282
1 square yard in square terameter is equal to 8.3612736e-25
1 square yard in square thou is equal to 1296000000
1 square yard in square yoctometer is equal to 8.3612736e+47
1 square yard in square yottameter is equal to 8.3612736e-49
1 square yard in stang is equal to 0.00030864797342193
1 square yard in stremma is equal to 0.00083612736
1 square yard in sarsai is equal to 0.29752066115702
1 square yard in tarea is equal to 0.0013297190839695
1 square yard in tatami is equal to 0.50585477645351
1 square yard in tonde land is equal to 0.00015158218999275
1 square yard in tsubo is equal to 0.25292738822675
1 square yard in township is equal to 8.9675081421151e-9
1 square yard in tunnland is equal to 0.00016937998541447
1 square yard in vaar is equal to 1
1 square yard in virgate is equal to 0.000006967728
1 square yard in veli is equal to 0.00010416666666667
1 square yard in pari is equal to 0.000082644628099174
1 square yard in sangam is equal to 0.00033057851239669
1 square yard in kottah [bangladesh] is equal to 0.0125
1 square yard in gunta is equal to 0.0082644628099174
1 square yard in point is equal to 0.020661318293118
1 square yard in lourak is equal to 0.00016528925619835
1 square yard in loukhai is equal to 0.00066115702479339
1 square yard in loushal is equal to 0.0013223140495868
1 square yard in tong is equal to 0.0026446280991736
1 square yard in kuzhi is equal to 0.0625
1 square yard in chadara is equal to 0.09
1 square yard in veesam is equal to 1
1 square yard in lacham is equal to 0.0033057822015093
1 square yard in katha [nepal] is equal to 0.0024691358024691
1 square yard in katha [assam] is equal to 0.003125
1 square yard in katha [bihar] is equal to 0.0066127847171198
1 square yard in dhur [bihar] is equal to 0.1322556943424
1 square yard in dhurki is equal to 2.65 | {"url":"https://hextobinary.com/unit/area/from/sqyd/to/kanal","timestamp":"2024-11-09T20:45:21Z","content_type":"text/html","content_length":"128277","record_id":"<urn:uuid:31d28225-4ebe-448c-a027-0f30cdfb7eb6>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00500.warc.gz"} |
The Learning-Theoretic Agenda: Status 2023 — AI Alignment Forum
TLDR: I give an overview of (i) the key problems that the learning-theoretic AI alignment research agenda is trying to solve, (ii) the insights we have so far about these problems and (iii) the
research directions I know for attacking these problems. I also describe "Physicalist Superimitation" (previously knows as "PreDCA"): a hypothesized alignment protocol based on infra-Bayesian
physicalism that is designed to reliably learn and act on the user's values.
I wish to thank Steven Byrnes, Abram Demski, Daniel Filan, Felix Harder, Alexander Gietelink Oldenziel, John S. Wentworth^[1] and my spouse Marcus Ogren for reviewing a draft of this article, finding
errors and making helpful comments and suggestions. Any remaining flaws are entirely my fault.
I already described the learning-theoretic agenda (henceforth LTA) in 2018. While the overall spirit stayed similar, naturally LTA has evolved since then, with new results and research directions,
and slightly different priorities. This calls for an updated exposition.
There are several reasons why I decided that writing this article is especially important at this point:
• Most of the written output of LTA focuses on the technical, mathy side. This leaves people confused about the motivation for those inquiries, and how they fit into the big picture. I find myself
having to explain it again and again in private conversations, and it seems more efficient to have a written reference.
• In particular, LTA is often conflated with infra-Bayesianism. While infra-Bayesianism is a notable part, it is not the only part, and without context it's not even clear why it is relevant to
• Soares has recently argued that alignment researchers don't "stack", meaning that it doesn't help much to add more researchers to the same programme. I think that for LTA, this is not at all the
case. On the contrary, there is a variety of shovel-ready problems that can be attacked in parallel. With this in mind, it's especially important to explain LTA in order to get more people on
• In particular, ALTER has announced a prize designed to encourage work on LTA. I expect this article to be a useful resource for researchers who decide to compete.
• It seemed important to explain Physicalist Superimitation better, and this is hard to do without the broader context.
I am fairly optimistic that LTA leads to a solution to alignment. On the other hand, it's much harder to say whether the solution will arrive in time. I think that the rate of progress relative to
person-hours invested was pretty good so far, and we didn't hit any obstacle that cast doubt on the entire endeavor. At the same time, in absolute terms we still have most of the work in front of us.
In the coming years, I am planning to put considerable effort into scaling up: getting more researchers on board, and hopefully accelerating progress by a significant factor.
The philosophy of LTA was covered in the original article, I mostly stand behind what I wrote there and don't want to repeat it. The goal of this section is merely to recap and add a couple of
The goal of LTA is creating a general mathematical theory of intelligent agents. There are a number of reasons why this is necessary for AI alignment:
• Having a mathematical theory enables constructing models in which we can prove that a particular AI design is aligned (or unaligned), or at least form rigorous and strongly supported conjectures
(similar to the role in cryptography). While such a proof doesn't imply absolute confidence, it does allow us to reduce the problem to becoming confident in the assumptions of the model. This is
a much better situation than dealing with a sequence of heuristic steps, each of which might be erroneous and/or hiding assumptions that are not stated clearly. Of course, similarly to how in
cybersecurity having a provably secure protocol doesn't protect you from e.g. someone using their birthday for a password, here too we will need to meticulously verify that the assumptions hold
in the actual implementation. This might require knowledge from domains outside of theoretical computer science, e.g. physics, cognitive science, evolutionary biology etc.
• Empirical data is insufficient in itself, since without an underlying theory it's very hard to be confident about how the results extrapolate to new domains, scales, architectures and algorithms.
On the other hand, the combination of theory and experiment can be extremely powerful for such extrapolation, even if the theory cannot produce quantitative predictions ab initio on its own.
• Even if we don't do any explicit calculations using the theory about the AI we are designing, merely knowing the theory leads to much better intuitions, and equips us with the right vocabulary to
reason about the problem. To give an analogy, it is much easier to reason about designing engines if you're familiar with concepts such as "heat", "work", "temperature", "entropy", even if you're
just reasoning informally rather than actually calculating.
After I wrote the original article about LTA, more was written about the feasibility and importance of mathematics for alignment, both by me and by others.
Why is LTA concerned specifically with agents? Aren't there AI systems which are not agents? The reason is: the sort of risks I want to address are risks that arise from AIs displaying agentic
behavior (i.e. building sophisticated models of the world and using these models to construct plans that pursue unaligned goals, with catastrophic consequences), and the sort of solution I envision
also relies on AIs displaying agentic behavior (i.e. building sophisticated models of the world and using these models to construct plans that pursue an aligned goal, s.t. the result is protecting
humanity from unaligned AI).
Finally, I want to address a common point of confusion. Sometimes I tell someone about subproblems in LTA and they respond by trying to understand why each individual subproblem (e.g.
nonrealizability, or cartesian privilege, or value ontology) is relevant to alignment. Often there are some specific ways in which the subproblem can be connected to alignment. But the more important
point is that any fundamental question about intelligent agency is relevant to alignment, just because without answering this question we cannot understand intelligent agency. For any aspect of
intelligent agency that we don't understand and any AI that we design, one of the two will hold: either the AI lacks this aspect, in which case it is probably insufficient to protect humanity, or the
AI has this aspect but we don't understand why or how, in which case it is probably unaligned (because alignment is a small target, and significant unaccounted for phenomena are likely to make you
Key Problems
The starting point of LTA is Marcus Hutter's AIXI: a simplistic model of ideal agency.
Intuitively, an "intelligent agent" is a system that builds sophisticated models of the world and uses them to construct and execute plans that lead to particular goals (see also Yudkwosky). Building
models implies starting from a state of ignorance and updating on observations, which can be formalized by Bayesian probability theory. Building sophisticated models requires using Occam's razor,
which can be formalized by the Solomonoff prior. Constructing and executing plans can be formalized as maximizing expected utility. Putting all these ingredients together gives us AIXI.
However, there are both important gaps both in our understanding of AIXI-like agents, and significant weaknesses in the AIXI framework itself. Solving these problems is the natural framing for the
entire foundational^[2] part of LTA.
Problem 1: Computational Resource Constraints
AIXI is uncomputable. Obviously, real-world agents are not only computable but operate under strong computational resource constraints. This doesn't mean AIXI is not a useful toy model for studying
the interaction of Occam's razor with Bayesian optimization or reinforcement learning. However, it is a major limitation.
One obvious attempted solution is to impose some computational resource constraints on the programs that appear in Solomonoff's prior, for example as in Schmiduber's speed prior or in some other way.
This typically leads to agents that are computable but still computationally intractable. Relatedly, a recent line of research connects the hardness of time-bounded Kolmogorov complexity with the
existence of one-way functions.
On the other hand, we know some priors for which asymptotic Bayes-optimality is possible in polynomial time, for example priors supported on Markov decision processes (MDPs) with a small (i.e.
polynomial in the security parameter) number of states. Some stronger feasible models are also known, e.g. MDPs with linear features or kernel-MDPs. However, all such priors have significant
• They require the MDP to be communicating, i.e. contain no irreversible transitions (a.k.a. "traps"). This is obviously not true in the real-world, where e.g. jumping off a cliff is often
irreversible. Indeed, without this assumption approximating the Bayes-optimal policy is NP-hard, even for a small number of deterministic hypotheses.
• They usually don't embody any sort of Occam's razor.
• They are not rich enough to capture sophisticated models of the real world.
Problem 2: Frequentist Guarantees
AIXI is Bayes-optimal (by definition) but is not necessarily optimal in any sense for any particular environment. In contrast, in statistical learning theory we usually demand a frequentist
guarantee, i.e. that a learning algorithm converges to optimality (in some sense) for any^[3] data source (subject to some assumptions which depend on the setting). Such guarantees are important for
several reasons:
• Learning is a key ability of intelligent agents, and played a fundamental role in AI progress in recent decades. A natural way to operationalize learning is: an agent learned a fact when its
behavior is optimal conditional on this fact. In other words, learning which of a set of hypotheses is true means converging to optimal behavior for the true hypothesis, i.e. precisely a
frequentist guarantee. This means that a theory of frequentist guarantees tell us both which facts agents can learn and how fast they learn them (or how much data they require to learn them).
These are important questions that a theory of intelligent agents has to answer. In particular, it is required to analyze the feasibility of alignment protocols: e.g. if we expect the AI to learn
human values, we need to make sure it has sufficient information to learn them within a practical time frame.
• A Bayes-optimal algorithm does well on average across some large ensemble of possible universes. But in reality, we only observe one universe (which is not even in the ensemble: see Subproblem
2.3 below). Without frequentist guarantees, it's not clear why would a Bayes-optimal algorithm do especially well in our specific universe, or why should algorithms that do well in our specific
universe be Bayes-optimal.
• Evolution selected humans by their performance in the ancestral environment, which involved e.g. gathering berries and running away from lions. But they turned out to perform reasonably well in
very different environments that require e.g. writing code or designing rockets. This hints at the existence of some underlying frequentist guarantee as a key property of intelligent agents.
Moreover, in addition to single-agent frequentist guarantees it is desirable to derive multi-agent frequentist guarantees. Interactions between agents are quite important in the real-world, and have
significance specifically for AI alignment as well (AI-human, AI-other AI, AI-acausal trade partner, to some extent human-human too). A multi-agent frequentist guarantee can take the form of
convergence to a particular game-theoretic solution concept, or an asymptotic lower bound on expected utilities corresponding to some notion of game value.
There are multiple difficulties in deriving frequentist guarantees for AIXI-like agents. The first two difficulties ("traps" and "password guessing games" below) are not problems with the agent, but
rather with the way conventional frequentist guarantees are defined (which are nonetheless non-trivial to solve). The third difficulty (nonrealizability) might be a problem with the way the agent is
Subproblem 2.1: Traps
The most common type of frequentist guarantee in reinforcement learning (RL) is regret bounds^[4]. However, AIXI doesn't satisfy any regret bound w.r.t. the underlying hypothesis class (i.e. the
class of all computable environments) because these hypotheses involve irreversible transitions (traps). For example, suppose that in environment 1 taking action A sends you to a sink state with
reward , whereas taking action B sends you to a sink state with reward , and in environment 2 the opposite is true. Obviously a hypothesis class which contains both environments doesn't admit a
regret bound.
This problem is especially acute in the multi-agent setting, because other agents have memory. Either it's impossible to erase memory in which case the environment is irreversible by design, or it's
possible to erase memory which breaks conventional learning theory in other ways.
Subproblem 2.2: Password guessing games
What if we arbitrarily rule out traps? Consider an agent with action set and observation set . Suppose that the reward is a function of the last observation only . We can specify an environment^[5]
by providing a communicating MDP with state set and action set augmented by representation mappings
Here, is required to be an onto mapping from to for any . Also, we require that for any and
However, this only enables a fairly weak regret bound. Naively, it seems reasonable to expect a regret bound of the form where is the Kolmogorov complexity of the MDP+representation, is the diameter^
[6] and is the time horizon^[7], and are constants s.t. . Indeed, can be regarded as the amount of information that needs to be learned and e.g. Russo and Van Roy showed that in a bandit setting
regret scales as the square root of the entropy of the prior (which also expresses the amount of information to be learned). However, this is impossible, as can be seen from the following example.
Fix and (the "password"), let the action set be and the state space be . We think of a state as "the agent entered the string of bits into the terminal". The initial state is (the empty string). The
transition kernel works as follows:
In state , taking action produces the state (i.e. the agent enters the digit ). In state , taking action produces the state if (the agent entered the correct password) or state (the agent entered an
incorrect password and has to start from the beginning). In state taking action stays in state and taking action produces state (restarts the game).
All states have reward except for the state which has reward .
Obviously the agent needs time to learn the correct password, in expectation w.r.t. the uniform distribution over passwords. At the same time, and . This is incompatible with having a regret bound of
the desired form.
Subproblem 2.3: Nonrealizability
As we discussed in Problem 1, a realistic agent cannot be Bayes-optimal for a prior over all computable environments. Typically, each hypothesis in the prior has to admit a computationally feasible
approximately optimal policy in order for it to be feasible to approximate Bayes-optimality (in particular this has to be the case if the prior is learnable), which usually implies the hypotheses are
computationally feasible to simulate^[8]. However, in this situation we can no longer assume the true environment is within the hypothesis class. Even for AIXI itself, it is not really fair to assume
the environment is computable since that would exclude environments that contain e.g. other AIXIs.
A setting in which the true environment is not in the hypothesis class is known in learning theory as "nonrealizable". For offline (classification/regression) and online learning, there is a rich
theory of nonrealizable learning^[9]. However, for RL (which, among classical learning theory frameworks, is the most relevant for agents), the nonrealizable setting is much less understood (although
there are some results, e.g. Zanette et al).
Therefore, even if we arbitrarily assume away traps, and are willing to accept a factor of in the regret bound, a satisfactory theory of frequentist guarantees would require good understanding of
nonrealizable RL.
This problem is also especially acute in the multi-agent case, because it's rarely the case that each agent is a realizable environment from the perspective of the other agent (roughly speaking, if
Alice would simulate Bob simulating Alice, Alice would enter an infinite loop). This is known as the "grain of truth" problem^[10]. One attempt to solve the problem is by Leike, Taylor and
Fallenstein, using agents that are equipped with "reflective oracles". However, there are many different reflective oracles, and the solution relies on all agents in the system to use the same one,
i.e. that the design of the agents has been essentially centrally coordinated to make them compatible.
Problem 3: Value Ontology
AIXI's utility function depends on its direct observations. This is also assumed in RL theory. While this might be a legitimate type of agent, it is insufficiently general. We can easily imagine
agents that place value on parameters they don't directly observe (e.g. the number of paperclips in the observable universe, or the number of people who suffer from malaria). Arguably, humans are
such agents.
The obvious workaround is to translate the utility function from its original domain to observations by taking a conditional expectation. That is, suppose the utility function is for some space , let
be the space of everything that is directly observed (e.g. action-observation time sequences) and suppose we have some prior over . Then, we can define by
However, there are problems with this:
• In order to have reasonable computational complexity and frequentist guarantees, we will need to make assumptions about the utility function. While such assumptions can be natural for , they make
much less sense for unless we can somehow justify them via the prior over . Without a theory that explicitly talks about and it is impossible to know whether such justification is possible.
• Even if satisfies the required assumptions, a frequentist guarantee about does not necessarily imply an analogous frequentist guarantee about . That's because the transformation above involves
taking expected value which implicitly mixes different "possible universes", while a frequentist guarantee is supposed to refer to a particular universe.
• Different agents have different observation spaces and there is no natural joint prior on all of them. This means it's impossible to talk about different agents having "aligned" preferences.
Therefore, it is desirable to have a theory in which it is possible to explicitly talk about utility functions with domains other than observations. (See also de Blanc.)
Problem 4: Cartesian Privilege
The formalization of Occam's razor in AIXI relates to the description complexity of hypotheses represented in terms of the agent's actions and observations (as functions ). However, this is not how
Occam's razor is used in science. When we talk about a "simple" or "elegant" scientific theory, we refer to the equations governing the fundamental degrees of freedom on that theory (e.g. particles,
or fields or strings), not to the equations that would be needed to describe the RGB values of points on a person's retina. Indeed, expressing physics via the latter equations would be horrifically
In other words, AIXI-like agents believe they occupy a privileged position in the universe, which has various pathological consequences. See "infra-Bayesian physicalism" for more discussion and
Demski and Garrabrant for related prior work.
Problem 5: Descriptive Agent Theory
The framing of AIXI is: given certain subjective choices (e.g. universal Turing machine and utility function), what is an ideal agent? However, in the real-world agents are not ideal. Moreover, we
want to be able to understand which systems are agents, and what kind of agents they are, without already assuming e.g. a specific utility function. In particular, such a theory would have
applications to:
• Value learning, by regarding humans as agents.
• Studying "accidental" formation of agents, e.g. via evolution or as mesa-optimizers (related: Wentworth's selection theorems programme).
Legg and Hutter defined an intelligence measure that for every policy produces a number determining its intelligence, s.t. AIXI has maximal intelligence. The measure is essentially the policy's
expected utility w.r.t. the Solomonoff prior. This allows us to talk how close any given system is to an ideal agent. However, it suffers from two major limitations (in addition to the other problems
of AIXI discussed before):
• It depends on the choice of universal Turning machine (UTM). While the same is true of everything in algorithmic information theory, usually we have theorems that limit this dependence. For
example, Kolmogorov complexity only changes by when we switch to a different UTM. On the other hand, the Legg-Hutter measure doesn't have any such property (except in the asymptotic in which it
approaches 0, which is pretty meaningless).
• It depends on the choice of utility function. Technically, they don't make an explicit choice but instead assume the reward is one of the observation channels. In practice, this is a choice (and
a very restrictive one).
Moreover, naive attempts to ascribe a utility function to a policy run into difficulties. Specifically, any policy can arguably be ascribed a utility function which rewards following precisely this
policy. With respect to such a utility function, the policy is Bayes-optimal for any prior. Armstrong and Mindermann have argued on the basis of this type of reasoning that ascribing a utility
function is essentially impossible in a purely behaviorist framework (but I argue the opposite, see Direction 17.5 below).
Nonproblem: Expected Utility
Finally, I want to briefly discuss one issue which I don't consider a key problem. Namely, the reliance of AIXI on expected utility maximization.
Expected utility is often justified by coherence theorems such as von Neumann–Morgenstern (VNM) and Savage. These theorems are indeed important: they show that a small number of natural assumptions
produce a fairly narrow mathematical object (expected utility). This should indeed increase our credence in expected utility as a useful model. Often, objections are put forward that argue with
individual assumptions. IMO these objections miss the point: a coherence theorem is a piece of evidence, not a completely watertight proof that expected utility is philosophically correct. And, if
expected utility is philosophically correct, such a watertight proof is still not something we should expect to exist (what would it even look like?)
The more important justification of expected utility is the large body of work it supports: control theory, game theory, reinforcement learning theory etc. The methodology I believe in is, start with
plausible assumptions and try to build a theory. If the theory that results is rich, has synergy with other bodies of knowledge, has explanatory power, is useful in practice, each of these is
evidence that it's on the right track. (And if we failed to build such a theory, we probably also learned something.)
Another objection that is often raised is "humans have no utility function". This is ostensibly supported by research in behavioral economics which shows that humans behave irrationally. I consider
this deeply unconvincing. For one thing, I suspect that a large part of that research doesn't replicate. But even putting that aside, the interesting thing about irrational behavior is that we
recognize it as irrational: i.e. learning about it makes you behave differently (unless it's so minor that it isn't worth the effort). This already indicates that this irrationality is better viewed
not as a fundamental fact about human preferences, but as some combination of:
• Computational or informational resource constraints
• Error terms that vanish in the relevant asymptotic regime
• Random noise or some other effect that's not part of the cognitive algorithm
All of this is not to say that expected utility cannot be questioned. Rather that a productive objection would be to either come up with a demonstrably superior alternative, or at least a good
argument why some extremely concrete problem cannot be solved without an alternative to expected utility. Lacking that, the best strategy is to press forward unless and until we encounter such a
Now, we actually do have examples where deviating from purist VNM in specific ways is useful, e.g.:
• Infra-Bayesianism, where the conventional notion of "expectation" is replaced by some non-linear concave functional.
• Taylor's quantilization, where instead of maximizing expected utility we sample a random choice out of some top fraction of choices.
• Nash bargaining, where we maximize the product of several expected utilities (see also Garrabrant's "geometric rationality" sequence).
However, these examples came about from studying various concrete problems, not from arguing with the VNM model per se. Moreover, all of these examples can still be mathematically recast in VNM form:
infra-Bayesianism can be regarded as a VNM zero-sum game (against "Murphy"), quantilization and Nash bargaining can be regarded as special cases of infra-Bayesianism (as will be discussed in an [S:
upcoming:S] article by Appel). Therefore, even here the theory built around the VNM model remains useful.
Research Directions
This section focuses entirely on the foundational part of the programme. I will mention some of the applied part in the next section (although different applications are also possible, e.g. studying
quantilization or IDA), but for the most part I believe the foundational part to be the top priority.
In the following, I roughly grouped the research directions by the problems they are trying to address. However, the real relationship between directions and problems is many-to-many: a single
direction can have implications on many problems. Moreover, there are many connections between the different directions. And, even for directions that are not especially connected at present, if they
are successful, we will be faced with the task of merging them into a unified theory.
Subproblem 1.1: Frugal Universal (Infra-)Prior
One part of solving Problem 1 (computational resource constraints) is finding a prior (more precisely a family of priors) with the following properties:
• Formalizes Occam's razor, i.e. assigns higher probability to hypotheses that are simpler to describe.
• Sufficiently rich to contain sophisticated models applicable to the real-world.
• The (approximately) optimal policy for each hypothesis is feasible to find. Possible operationalization of "feasible": polynomial time in history length and description length of hypothesis.
• If we assume away traps, there is a feasible learning algorithm with a good regret bound. Possible operationalization: polynomial time in history length, regret bound similar to what was
described in section "Problem 2.2" above. (This desideratum is probably strictly stronger than the previous bullet point.)
More generally, we might want an infra-prior (see Direction 3 below), or a metacognitive (infra-)prior (see Direction 6 below), or a physicalist infra-prior (see Direction 18 below) with analogous
Direction 1: Frugal Compositional Languages
The time complexity of finding the optimal policy for a generic (tabular) MDP scales with its number of states. The same is true of the sample complexity of learning an unknown generic MDP. However,
the number of states in a sophisticated model of the real world has to be enormous. For example, if I reason about the world as it's comprised of objects with possible states each, the over number of
states is already .
Therefore, any realistic RL algorithm has to exploit some structure in its environment. For example, the environment might be comprised of spatially separated parts, or different processes happening
on different spatial scales, or different processing happening on different temporal scales. We want to find an appropriate compositional language f | {"url":"https://www.alignmentforum.org/posts/ZwshvqiqCvXPsZEct/the-learning-theoretic-agenda-status-2023","timestamp":"2024-11-08T05:30:01Z","content_type":"text/html","content_length":"1048971","record_id":"<urn:uuid:c9d8ae53-7b67-4ec7-82e4-d0067c762537>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00296.warc.gz"} |
Test your knowledge
Question 1 of 8 Basic
Centrifugal pumps can be categorised into the following groups…
Question 2 of 8 Basic
In a pump curve, head, power consumption, efficiency and NPSH are shown as a function of…
Question 3 of 8 Basic
Are axial flow impellers used in closed pump designs?
Question 4 of 8 Basic
How much of the world’s total volume of water is fresh water?
Question 5 of 8 Basic
What makes mixed flow impellers different from axial flow impellers?
Question 6 of 8 Basic
What does QH curve show?
Question 7 of 8 Basic
What does ‘NPSH’ stand for in the NPSH-curve?
Question 8 of 8 Basic
What kind of pumps would normally NOT have a radial flow impeller?
Question 8 of 8 Basic
Happy with your answers?
Submit your current answers and get the final test result.
Test results
Congratulations. You have passed the test and completed the 36 - Basic Principles and Pump Types
Q: Centrifugal pumps can be categorised into the following groups…
A: Forward flow, backward flow, static flow
01: Forward flow, backward flow, static flow Your answer
02: Vertical flow, horizontal flow, diagonal flow
03: Radial flow, mixed flow, axial flow
Q: Centrifugal pumps can be categorised into the following groups…
A: Vertical flow, horizontal flow, diagonal flow
01: Forward flow, backward flow, static flow
02: Vertical flow, horizontal flow, diagonal flow Your answer
03: Radial flow, mixed flow, axial flow
Q: Centrifugal pumps can be categorised into the following groups…
A: Radial flow, mixed flow, axial flow
Q: In a pump curve, head, power consumption, efficiency and NPSH are shown as a function of…
A: Performance
01: Performance Your answer
Q: In a pump curve, head, power consumption, efficiency and NPSH are shown as a function of…
A: The type of pump
02: The type of pump Your answer
Q: In a pump curve, head, power consumption, efficiency and NPSH are shown as a function of…
A: Flow
Q: Are axial flow impellers used in closed pump designs?
A: Yes
Q: Are axial flow impellers used in closed pump designs?
A: Sometimes
02: Sometimes Your answer
Q: Are axial flow impellers used in closed pump designs?
A: No
Q: How much of the world’s total volume of water is fresh water?
A: 72%
Q: How much of the world’s total volume of water is fresh water?
A: About 50%
02: About 50% Your answer
Q: How much of the world’s total volume of water is fresh water?
A: 3.5%
Q: What makes mixed flow impellers different from axial flow impellers?
A: Mixed flow impellers deliver higher rotation speeds
01: Mixed flow impellers deliver higher rotation speeds Your answer
02: They subject the fluid to a degree of radial flow
03: They’re not nearly as good as a fully axial flow impeller
Q: What makes mixed flow impellers different from axial flow impellers?
A: They subject the fluid to a degree of radial flow
Q: What makes mixed flow impellers different from axial flow impellers?
A: They’re not nearly as good as a fully axial flow impeller
01: Mixed flow impellers deliver higher rotation speeds
02: They subject the fluid to a degree of radial flow
03: They’re not nearly as good as a fully axial flow impeller Your answer
Q: What does QH curve show?
A: Flow at a given head
01: Flow at a given head Your answer
Q: What does QH curve show?
A: Head at a given flow
Q: What does QH curve show?
A: Overall efficiency
03: Overall efficiency Your answer
Q: What does ‘NPSH’ stand for in the NPSH-curve?
A: Net potential surface heat
01: Net potential surface heat Your answer
02: Net peripheral suction height
03: Net positive suction head
Q: What does ‘NPSH’ stand for in the NPSH-curve?
A: Net peripheral suction height
01: Net potential surface heat
02: Net peripheral suction height Your answer
03: Net positive suction head
Q: What does ‘NPSH’ stand for in the NPSH-curve?
A: Net positive suction head
Q: What kind of pumps would normally NOT have a radial flow impeller?
A: Circulating pumps
01: Circulating pumps Your answer
03: Sewage treatment pumps
Q: What kind of pumps would normally NOT have a radial flow impeller?
A: Immersible pumps
02: Immersible pumps Your answer
03: Sewage treatment pumps
Q: What kind of pumps would normally NOT have a radial flow impeller?
A: Sewage treatment pumps | {"url":"https://www.grundfos.com/my/learn/ecademy/all-courses/basic-principles-and-pump-types/test-your-knowledge","timestamp":"2024-11-05T15:20:45Z","content_type":"text/html","content_length":"767322","record_id":"<urn:uuid:b577aa8c-5583-40a6-aecc-d7bfe6168d60>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00576.warc.gz"} |
How To Work Out Forming And Solving Equations - Tessshebaylo
How To Work Out Forming And Solving Equations
Forming and solving equations gcse maths steps examples worksheets practice questions answers cazoomy worksheet quadratic mr mathematics you algebra ssdd problems mathscast
Forming And Solving Equations Gcse Maths Steps Examples
Forming Solving Equations Worksheets Practice Questions And Answers Cazoomy
Solving Equations Gcse Maths Steps Examples Worksheet
Forming And Solving Equations Gcse Maths Steps Examples
Forming And Solving Equations Gcse Maths Steps Examples
Forming And Solving Equations Gcse Maths Steps Examples
Forming And Solving Quadratic Equations Mr Mathematics You
Forming Solving Equations Worksheets Practice Questions And Answers Cazoomy
Algebra Equations Forming And Solving Ssdd Problems
Forming And Solving Equations Gcse Maths Steps Examples
Forming Solving Equations Mathscast You
Forming And Solving Equations You
Forming And Solving Equations Homework Ks3 Maths
Forming And Solving Equations Ks3 Walkthrough Worksheet
Forming And Solving Equations From Shapes Foundation Higher Gcse Jaggersmaths You
Forming Equations Textbook Exercise Corbettmaths
Forming Equations Corbettmaths
Form And Solve Quadratic Equations Mr Mathematics Com
Forming Solving Equations Mathscast You
Forming Solving Equations Worksheets Practice Questions And Answers Cazoomy
Educating Mrmattock Forming And Solving Linear Equations A Change Of Focus
Forming And Solving Equations From Descriptions
Q17 Answers Paper 1 June 19 Aqa Gcse Maths Higher Elevise
Solving equations gcse maths forming worksheets steps and quadratic ssdd problems mathscast
Trending Posts
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.tessshebaylo.com/how-to-work-out-forming-and-solving-equations/","timestamp":"2024-11-12T17:02:33Z","content_type":"text/html","content_length":"59042","record_id":"<urn:uuid:191d3c44-e317-4bb7-b5a6-f53b84d390bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00843.warc.gz"} |
TGD diary{{post.title}}Zero energy ontology and holography make possible memories by coding the quantum jump as a conscious event to the final state of the quantum jump
We have memories about the conscious experiences of the past. How are these memories formed? Zero energy ontology (ZEO) (see
) and
) suggests a rather concrete model for the representations of the memories in terms of the geometry of the space-time surface.
Consider first a brief summary of ZEO.
1. The basic notions of ZEO are causal diamond (CD), zero energy state, and state function reduction (SFR). There are two kinds of SFRs: "small" SFRs (SSFRs) and "big" SFRs (BSFRs).
2. A sequence of SSFRs is the TGD counterpart for a sequence of repeated measurements of the same observables: in wave mechanics they leave the state unaffected (Zeno effect). Already in quantum
optics, one must loosen this assumption and one speaks of weak measurements. In the TGD framework, SSFRs give rise to a flow of consciousness, self.
3. BSFR is the counterpart of the ordinary SFR. In the BSFR the arrow of the geometric time changes and BSFR means the death of self and to a reincarnation with an opposite arrow of geometric time.
Death and birth as reincarnation with an opposite arrow of time are universal notions in the TGD Universe.
Consider now this view in more detail.
1. Causal diamond CD=cd× CP[2] (see this) is the intersection of future and past directed light-cones of M^4. In the simplest picture, cero energy states are pairs of 3-D many-fermion states at the
opposite light-like boundaries of the CD.
2. Zero energy states are superpositions of space-time surfaces connecting the boundaries of CD. These space-time surfaces obey holography, which is almost deterministic. Holography = holomorphy
principle allows their explicit construction as minimal surfaces and they are analogous to Bohr orbits when one interprets 3-surface as a generalization of a point-like particle. Already 2-D
minimal surfaces fail to be completely deterministic (a given frame can span several minimal surfaces). This non-determinism forces ZEO: in absence of it one could have ordinary ontology with 3-D
objects as basic geometric entities.
The failure of complete determinism makes 4-dimensional Bohr orbits dynamical objects by giving them additional discrete degrees of freedom. They are absolutely essential for the understanding of
memory and one can speak of a 4-dimensional brain.
3. The 3-D many-fermion states and the restriction of the wave function in WCW to a wave function to the space-of 3-surfaces as the ends of Bohr orbits at the passive boundary of CD are unaffected
by the sequence of SSFRs. This is the counterpart for the Zeno effect. This requires that a given SSFR must correspond to a measurement of observables commuting with the observables which define
the state basis at the passive boundary.
The states at the opposite, active, boundary of CD are however affected in SSFRs and this gives rise to self and flow of consciousness. Also the size of CD increases in a statistical sense. The
sequence of SSFRs gives rise to subjective time correlating with the increase of geometric time identifiable as the temporal distance between the tips of the CD. The arrow of time depends on
which boundary of CD is passive and the time increases in the direction of the active boundary.
4. Ordinary SFRs correspond in TGD to BSFRs. Both BSFRs and SSFRs are possible in arbitrarily long scales since the h[eff] hierarchy makes possible quantum coherence in arbitrary long scales.
The new element is that the arrow of geometric time changes in BSFR since the roles of the active and passive boundaries of CD change. BSFR occurs when the set of observables measured at the
active boundary no longer commutes with the set of observables associated with the passive boundary.
The density matrix of the 3-D system characterizing the interaction of the 3-surface at the active boundary with its complement is a fundamental observable and if it ceases to commute with the
observables at the active boundary, BSFR must take place.
Consider now what memory and memory recall could mean in this framework.
1. The view has been that active memory recall requires what might be regarded as communications with the geometric past. This requires sending a signal to the geometric past propagating in the
non-standard time direction and absorbed by a system representing the memory (part of the brain or of its magnetic/field body). In the ZEO this is possible since BSFRs change the arrow of the
geometric time.
2. The signal must be received by a system of geometric past representing the memory. This requires that 4-D space-time surfaces are not completely deterministic: Bohr orbits as 4-D minimal surfaces
must have analogs of frames spanning the 2-D soap film, at which determinism fails. The seats of memories correspond to the seats of non-determinism as singularities of the space-time surface as
a minimal surface.
3. How are the memories coded geometrically? This can be understood by asking what happens in SSFR. What happens is that from a set of 3-D final states at the active boundary some state is selected.
This means a localization in the "world of classical worlds" (WCW) as the space of Bohr orbits. The zero energy state is localized to the outcome of quantum measurement. In ZEO the outcome
therefore also represents the quantum transition to the final state! This is not possible in the standard ontology.
The findings of Minev et al (see this and this) that in quantum optics quantum jumps correspond too smooth classical time evolutions leading from the initial state to the final state provide a
direct support for this picture.
ZEO therefore gives a geometric representation of a subjective experience associated with the SSFR. One obtains conscious information of this representation either by passive or active memory
recall by waking up the locus of non-determinism assignable to the original conscious event. The slight failure of determinism for BSFRS is necessary for this. The sequence of SSFRs is coded to a
sequence of geometric representations of memories about conscious events.
This is how the Universe gradually develops representations of its earlier quantum jumps to its own state. Since the algebraic complexity of the Universe can only increase in a statistical sense
the quantum hopping of the Universe in the quantum Platonic defined by the spinor fields of WCW implies evolution.
See the article
TGD as it is towards end of 2024: part II
or the
with the same title.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this. | {"url":"https://matpitka.blogspot.com/2024/07/zero-energy-ontology-and-holography.html","timestamp":"2024-11-06T04:39:18Z","content_type":"application/xhtml+xml","content_length":"134068","record_id":"<urn:uuid:6accf4e5-e741-4a80-9ea1-afb5659f0c3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00676.warc.gz"} |
How to Add Labels to the X-Axis And Y-Axis In Matplotlib?
To add labels to the x-axis and y-axis in Matplotlib, you can use the xlabel() and ylabel() functions, which allow you to set the labels for the respective axes.
For the x-axis label, you can use the syntax plt.xlabel('label_text'), where label_text represents the desired label for the x-axis. Similarly, for the y-axis label, the syntax is plt.ylabel
('label_text'), where label_text represents the desired label for the y-axis.
Here is an example:
1 import matplotlib.pyplot as plt
3 # Example data
4 x = [1, 2, 3, 4, 5]
5 y = [2, 4, 6, 8, 10]
7 # Plotting the data
8 plt.plot(x, y)
10 # Adding labels to x-axis and y-axis
11 plt.xlabel('X-axis label')
12 plt.ylabel('Y-axis label')
14 # Displaying the plot
15 plt.show()
In the above example, the plot consists of some sample data points. The xlabel() function is used to add the label 'X-axis label' to the x-axis, and the ylabel() function is used to add the label
'Y-axis label' to the y-axis. The plt.show() function is then called to display the final plot with the labels.
By customizing the respective input strings in xlabel() and ylabel(), you can provide meaningful labels to describe the data being plotted on the x-axis and y-axis accordingly.
How to label data points in a scatter plot in Matplotlib?
To label data points in a scatter plot in Matplotlib, you can use the plt.text() function. Here is an example:
1 import matplotlib.pyplot as plt
3 # Create example data
4 x = [1, 2, 3, 4, 5]
5 y = [2, 4, 6, 8, 10]
6 labels = ['A', 'B', 'C', 'D', 'E']
8 # Create scatter plot
9 plt.scatter(x, y)
11 # Label each data point
12 for i, label in enumerate(labels):
13 plt.text(x[i], y[i], label)
15 # Show the plot
16 plt.show()
In this example, we have a scatter plot with five data points (x, y) and their corresponding labels. The plt.text() function is used in a for loop to iterate over each data point and add the label to
the plot at the respective coordinates.
You can customize the label's appearance by modifying the parameters of plt.text(), such as the font size, color, alignment, etc.
How to change the font size of axis labels in Matplotlib?
To change the font size of axis labels in Matplotlib, you can use the fontsize parameter of the set_xlabel() and set_ylabel() methods of the Axes object. Here's an example:
1 import matplotlib.pyplot as plt
3 # Create some data
4 x = [1, 2, 3, 4, 5]
5 y = [1, 4, 9, 16, 25]
7 # Create a figure and an axes object
8 fig, ax = plt.subplots()
10 # Plot the data
11 ax.plot(x, y)
13 # Set the font size of x-axis label
14 ax.set_xlabel('X-axis label', fontsize=16)
16 # Set the font size of y-axis label
17 ax.set_ylabel('Y-axis label', fontsize=16)
19 # Show the plot
20 plt.show()
In this example, the fontsize parameter is set to 16, but you can change it to any other font size that suits your needs.
How to create a scatter plot in Matplotlib?
To create a scatter plot in Matplotlib, you can follow these steps:
1. Import the necessary libraries:
1 import matplotlib.pyplot as plt
1. Prepare the data:
1 x = [1, 2, 3, 4, 5] # x-coordinates of the data points
2 y = [2, 4, 6, 8, 10] # y-coordinates of the data points
1. Create the scatter plot:
1. Customize the plot labels and title (if desired):
1 plt.xlabel('X-axis')
2 plt.ylabel('Y-axis')
3 plt.title('Scatter Plot')
1. Show the plot:
Putting it all together, here's an example of a scatter plot:
1 import matplotlib.pyplot as plt
3 x = [1, 2, 3, 4, 5]
4 y = [2, 4, 6, 8, 10]
6 plt.scatter(x, y)
7 plt.xlabel('X-axis')
8 plt.ylabel('Y-axis')
9 plt.title('Scatter Plot')
10 plt.show()
This will create a scatter plot with the given data points and display it on the screen.
What is the difference between plt.plot() and plt.scatter() in Matplotlib?
The main difference between plt.plot() and plt.scatter() in Matplotlib is the way they represent the data.
plt.plot() is used to create a line plot or a line connecting the data points. It is typically used to plot continuous data, such as time series or mathematical functions. When using plt.plot(), the
x-axis values are assumed to be evenly spaced.
On the other hand, plt.scatter() is used to create a scatter plot or a collection of individual data points. It is typically used to plot discrete or unstructured data. When using plt.scatter(), the
x-axis values do not need to be evenly spaced.
Another difference is that plt.plot() can accept multiple arguments such as color, line style, and marker style to customize the appearance of the line, while plt.scatter() accepts various arguments
to customize the appearance of the individual data points, such as color, size, and marker style.
In summary, plt.plot() creates a line plot connecting the data points, while plt.scatter() creates a scatter plot with individual data points. | {"url":"https://topminisite.com/blog/how-to-add-labels-to-the-x-axis-and-y-axis-in","timestamp":"2024-11-13T09:07:51Z","content_type":"text/html","content_length":"387150","record_id":"<urn:uuid:0a992cd0-af1f-4a07-a0f8-e2b005b78ccd>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00445.warc.gz"} |
SQL Workshop - Selecting columns without a non-aggregate column
Written by Nikos Vaggalis
Thursday, 19 December 2013
Thinking in terms of sets and set operations can be difficult at first but after a while you discover that you can do things without needed to drop down to procedural approaches.
This scenario requires us to be members of a hospital's Dietary Department and with the end of the year approaching we are assigned the task of estimating the amount of money needed for next year’s
resource shopping, to keep the patients fed for the coming twelve months.
So we need to find the sum of the mean amount spent on the resources/raw material (vegetables, fruit, meat etc) grouped by Account Category (i.e. the account that serves for fruit) and Account Id
(actual account number) used for their shopping, and use that as the basis for our new season’s budget estimate.
When a request for, say, fruit comes in, we translate that request into the amount of money consumed using a formula based on the fruit’s dynamically updated Mean Value, the Quantity of the request
and a Ratio.
When the unit of measurement is 'PIECES' then our formula is :
Amount = Mean Value x Quantity x Ratio
while when the unit of measurement is a ‘KGR’ then the formula becomes:
Amount = Mean Value x Quantity / Ratio
The nature and meaning of the two formulae isn't as important as the fact that they vary according to the value of the Unit field.
Year Material_id Unit Ratio Mean_ value
1/1/2013 TX002 PIECES 1.000 $36.00
1/1/2013 TX003 KGR 2.000 $22.67
Year Material_id Account_category Account_id
1/1/2013 TX002 01 220201511
1/1/2013 TX003 01 220201511
Year Req_id Material_id Quantity
1/1/2013 1 TX002 10
1/1/2013 1 TX003 60
It is this variation of the formulas that will create the most trouble, as we’ll soon find out.
Let’s get a preview of our data together with an attempt to implement the formulas:
r.material_id, m.unit,
WHEN m.unit = 'PIECES'
THEN m.mean_value * r.quantity * m.ratio
//formula 1
ELSE IF m.unit = 'KGR'
THEN m.mean_value * r.quantity / m.ratio
// formula 2
END as amount
from requests r, materials m
where r.year = '1/1/13' and
r.req_id = 1 and
r.material_id = m.material_id
The desired result set would be :
Result Set 1
Material_id Unit Amount
TX002 PIECES $360
TX003 KGR $680.01
If we would also require their Account Category and Account Id (where the money for their purchasing is deducted from), the above query would become :
m.mean_value * r.quantity * m.ratio as amount,
from requests r, materials m, accounts a
where r.year = '1/1/13' and
r.material_id = m.material_id and
r.material_id = a.material_id and
r.year = m.year and
r.year = a.year and
m.unit = 'PIECES'
m.mean_value * r.quantity / m.ratio as
from requests r, materials m, accounts a
where r.year = '1/1/13' and
r.material_id = m.material_id and
r.material_id = a.material_id and
r.year = m.year and
r.year = a.year and
m.unit = 'KGR'
Producing a result set like :
│ Result Set 2 │
│Material_id│ Unit │Amount │Account_Category │Account_Id│
│... │PIECES│$360 │01 │220201511 │
│… │KGR │$680.01│01 │220201511 │
│ │PIECES│$580 │01 │220201511 │
│ │KGR │$220 │01 │220201511 │
│ │PIECES│$139 │01 │220201511 │
│ │KGR │$200 │01 │220201511 │
│… │… │... │ │ │
Of course our result set would include more units of measurement than PIECES and KGR and in a moment we will generalize in order to include additional values.
Using procedural logic we could now write a little program in Perl or any programming language which would treat the result set as a collection of values, iterate through it, and add to a variable
called $AmountPieces when encountering a unit of PIECES and $AmountKGR when encountering a unit of KGR, doing our calculations row by row.
But let’s choose another path, that of pure SQL and set based logic.
In any case, since we are exclusively interested in the monetary aggregates, we can forgo the attributes of Material_id and Unit in the final answer as long as we get the formulas right.
Let's say that for reasons of simplicity, only PIECES are calculated differently and the rest of the units are all calculated using the same formula, something that simplifies our query from
something like:
WHEN m.unit = 'PIECES'
THEN m.mean_value * r.quantity * m.ratio
ELSE IF m.unit = 'KGR'
THEN m.mean_value * r.quantity / m.ratio
ELSE IF m.unit = 'LT'
ELSE IF m.unit=….
THEN ….
END as amount
WHEN m.unit = 'PIECES'
THEN r.mean_value * r.quantity * m.ratio
ELSE r.mean_value * r.quantity / m.ratio
END as amount
Our first attempt to get to the sums would be:
a.account_category, a.account_id,
WHEN m.unit = 'PIECES'
THEN sum(m.mean_value * r.quantity * m.ratio)
ELSE sum(m.mean_value * r.quantity / m.ratio)
END as amount
from requests r, materials m, accounts a
where r.year = '1/1/13' and
r.material_id = m.material_id and
r.material_id = a.material_id and
r.year = m.year and
r.year = a.year
GROUP BY account_category, account_id
but instead of the results we are expecting we are confronted with the following error message instead :
E_US0B63 line 1, The columns in the SELECT clause must be contained in the
GROUP BY clause.
Dead end?
Last Updated ( Thursday, 06 September 2018 ) | {"url":"https://www.i-programmer.info/programming/139-database/6730-sql-workshop-selecting-columns-without-including-a-non-aggregate-column-in-the-group-by-clause.html","timestamp":"2024-11-04T04:42:16Z","content_type":"text/html","content_length":"47735","record_id":"<urn:uuid:4d4421b0-23d4-484e-af58-cda4f3fee9b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00233.warc.gz"} |
The Kappa Fleiss coefficient and a test to examine its significance
This coefficient determines the concordance of measurements conducted by a few judges (Fleiss, 19711)) and is an extension of Cohen's Kappa coefficient, which allows testing the concordance of only
two judges. With that said, it should be noted that each of
Z test for significance of Fleiss' Kappa coefficient (2)) is used to test the hypothesis that the ratings of several judges are consistent and is based on the coefficient
The p-value, designated on the basis of the test statistic, is compared with the significance level
The determination of Fleiss's Kappa coefficient is conceptually similar to the Mantel-Haenszel method. The determined Kappa is a general measure that summarizes the concordance of all judge ratings
and can be determined as the Kappa formed from individual layers, which are specific judge ratings (Fleiss, 20033)). Therefore, as a summary of each layer, the judges' concordance (Kappa coefficient)
can be determined summarizing each possible rating separately.
The settings window with the test of the Fleiss's Kappa significance can be opened in Statistics menu →NonParametric tests→Fleiss Kappa.
20 volunteers take part in a game to determine their personality type. Each volunteer has a rating given by 7 different observers (usually people from their close circle or family). Each observer has
been introduced to the basic traits describing temperament in each personality type: choleric, phlegmatic, melancholic, sanguine. We examine observers' concordance in assigning personality types. An
excerpt of the data is shown in the table below.}
We observe an unimpressive Kappa coefficient = 0.24, but statistically significant (p<0.0001), indicating non-random agreement between judges' ratings. The significant concordance applies to each
grade, as evidenced by the concordance summary report for each stratum (for each grade) and the graph showing the individual Kappa coefficients and Kappa summarizing the total.
It may be interesting to note that the highest concordance is for the evaluation of phlegmatics (Kappa=0.48).
With a small number of people observed, it is also useful to make a graph showing how observers rated each person.
In this case, only person no 14 received an unambiguous personality type rating – sanguine. Person no. 13 and 16 were assessed as phlegmatic by 6 observers (out of 7 possible). In the case of the
remaining persons, there was slightly less agreement in the ratings. The most difficult to define personality type seems to be characteristic of the last person, who received the most diverse set of
Fleiss J.L. (1971), Measuring nominal scale agreement among many raters. Psychological Bulletin, 76 (5): 378–382
Fleiss J.L., Levin B., Paik M.C. (2003), Statistical methods for rates and proportions. 3rd ed. (New York: John Wiley) 598-626 | {"url":"https://manuals.pqstat.pl/en:statpqpl:zgodnpl:nparpl:kappaflpl","timestamp":"2024-11-07T00:19:10Z","content_type":"text/html","content_length":"59227","record_id":"<urn:uuid:a16d295d-0a04-4f17-bebc-815ea69869be>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00479.warc.gz"} |
How to graph a derivative on HP Prime?
09-28-2014, 07:35 PM
(This post was last modified: 09-28-2014 07:36 PM by mayradio0508.)
Post: #1
mayradio0508 Posts: 1
Junior Member Joined: Sep 2014
How to graph a derivative on HP Prime?
I'm currently taking a calculus course and I'm supposed to be able to graph a derivative of a function using a graphing calculator. The textbook says to input nDer(f(x),x) but I can't seem to figure
it out. I've tried various things and sometimes it comes out as a line at y=0 but most of the things I've tried don't graph anything at all. Can someone explain to me how to graph a derivative of a
function, such as f(x)= lnx?
09-28-2014, 08:05 PM
Post: #2
mkspence Posts: 12
Junior Member Joined: Aug 2014
RE: How to graph a derivative on HP Prime?
10-05-2014, 04:24 PM
Post: #3
Eddie W. Shore Posts: 1,615
Senior Member Joined: Dec 2013
RE: How to graph a derivative on HP Prime?
(09-28-2014 07:35 PM)mayradio0508 Wrote: I'm currently taking a calculus course and I'm supposed to be able to graph a derivative of a function using a graphing calculator. The textbook says to
input nDer(f(x),x) but I can't seem to figure it out. I've tried various things and sometimes it comes out as a line at y=0 but most of the things I've tried don't graph anything at all. Can
someone explain to me how to graph a derivative of a function, such as f(x)= lnx?
Two ways, you'll need the derivative template [Template Key, 1st row, 4th column]
Let d represent the "curvy derivative symbol".
Textbook entry:
F1(X) = LN(X)
F2(X) = (d F1(X))/(dX=X)
or directly:
F1(X) = (d LN(x))/(dX = X)
Algebraic Entry:
F1(X) = d( LN(X), X=X)
10-06-2014, 04:57 PM
(This post was last modified: 10-06-2014 05:26 PM by Chris Pem10.)
Post: #4
Chris Pem10 Posts: 22
Junior Member Joined: Dec 2013
RE: How to graph a derivative on HP Prime?
This seems like the most logical way to ask the calc to plot a function and its derivative... the syntax seems completely logical and "correct", but it just doesn't work.
But yeah; putting dX=X in the denominator works:
Of course this works as expected in the CAS environment... but that's old news here.
User(s) browsing this thread: 1 Guest(s) | {"url":"https://www.hpmuseum.org/forum/thread-2199.html","timestamp":"2024-11-14T22:01:22Z","content_type":"application/xhtml+xml","content_length":"25850","record_id":"<urn:uuid:360eabd7-b9ed-495b-97e8-c76eedee0e1f>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00529.warc.gz"} |
Analog Passband Modulation
In most media for communication, only a fixed range of frequencies is available for transmitting messages. One way to communicate a message whose frequency spectrum does not fall within that fixed
frequency range, or one that is otherwise unsuitable for the channel, is to alter a carrier signal according to the information in your message signal. This alteration is called modulation. The
transmitter sends the modulated symbols. The receiver then recovers the original message symbols through a process called demodulation.
Modulation Methods
Analog passband modulation modulates analog transmission signals into sinusoidal waveforms. Communications Toolbox™ software provides features to apply a variety of analog passband modulation
methods. The process by which a carrier signal is altered according to information in a message signal depends on the modulation method applied. The general form of the carrier signal, s(t), is
s(t) = A(t)cos[2πf[0]t+ϕ(t)]
The information-carrying component is the amplitude (A), frequency (f[0]), or phase (ϕ) individually, or in combination. To satisfy the Nyquist criterion when simulating analog modulation systems,
the sample rate of the system must be greater than twice the sum of the carrier frequency and the signal bandwidth. For more information, see Baseband vs. Passband Simulation.
You can design your analog modulation system using these passband methods.
Filter Design Decisions
Unless otherwise indicated by filtering configuration controls, the features for passband modulation and demodulation do not perform pulse shaping or filtering. After demodulating a signal, you might
want to filter out the carrier signal. You can select a particular filter, such as butter, cheby1, cheby2, and ellip, on the mask of the demodulator block. Different filtering methods have different
properties, and you might need to test your application with several filters before deciding which is most suitable.
Analog passband DSB AM modulates using double-sideband amplitude modulation. The output is a passband representation of the modulated signal. Both the input and output signals are real scalar
For an input u(t) varying as a function of time t, then the output is
• k represents the input signal offset and is commonly set to the maximum absolute value of the negative part of the input signal u(t).
• f[c] represents the carrier frequency.
• θ represents the initial phase.
Typically, an appropriate carrier frequency is much higher than the highest frequency of the input signal. By the Nyquist sampling theorem, 1 / T[s] > f[c], where T[s] represents the sample time of
the input signal.
DSB-SC AM
Analog passband DSB-SC AM modulates using double-sideband suppressed-carrier amplitude modulation. The output is a passband representation of the modulated signal. Both the input and output signals
are real scalar signals.
For an input u(t) varying as a function of time t, then the output is
• f[c] represents the carrier frequency.
• θ represents the initial phase.
Typically, an appropriate carrier frequency is much higher than the highest frequency of the input signal. By the Nyquist sampling theorem, 1 / T[s] > f[c], where T[s] represents the sample time of
the input signal.
Analog passband SSB AM modulates using single-sideband amplitude modulation. The output is a passband representation of the modulated signal. Both the input and output signals are real scalar
SSB AM transmits either the lower or upper sideband signal, but not both.
If the input is u(t) varying as a function of time t, then the output is
(u(t)cos(f[c]t + θ) ± û(t)sin(f[c]t + θ)
• f[c] represents the carrier frequency.
• θ represents the initial phase.
• û(t) represents the Hilbert transform of the input u(t).
• For ±, the minus sign indicates the upper sideband and the plus sign indicates the lower sideband.
Analog passband FM modulates using frequency modulation. The output is a passband representation of the modulated signal. The output signal's frequency varies with the input signal's amplitude. Both
the input and output signals are real scalar signals.
If the input is u(t) varying as a function of time t, then the output is
$\mathrm{cos}\left(2\pi {f}_{c}t+2\pi {K}_{c}{\int }_{0}^{t}u\left(\tau \right)d\tau +\theta \right)$
• f[c] represents the carrier frequency.
• θ represents the initial phase.
• K[c] represents the frequency deviation.
Typically, an appropriate carrier frequency is much higher than the highest frequency of the input signal. By the Nyquist sampling theorem, 1 / T[s] > f[c], where T[s] represents the sample time of
the input signal.
Analog passband PM modulates using phase modulation. The output is a passband representation of the modulated signal. The output signal's phase varies with the input signal's amplitude. Both the
input and output signals are real scalar signals.
If the input is u(t) varying as a function of time t, then the output is
$\mathrm{cos}\left(2\pi {f}_{c}t+{K}_{c}u\left(t\right)+\theta \right)$
• f[c] represents the carrier frequency.
• θ represents the initial phase.
• K[c] represents the phase deviation.
Typically, an appropriate carrier frequency is much higher than the highest frequency of the input signal. By the Nyquist sampling theorem, 1 / T[s] > f[c], where T[s] represents the sample time of
the input signal.
Accessing Analog Passband Modulation Blocks
In Simulink^®, open the Analog Passband Modulation sublibrary by double-clicking its icon in the Modulation library. The Analog Passband Modulation sublibrary contains modulator-demodulator block
pairs for these modulation methods.
[1] Peebles, Peyton Z, Jr. Communication System Principles. Reading, MA: Addison-Wesley, 1976.
Related Topics | {"url":"https://de.mathworks.com/help/comm/ug/analog-passband-modulation.html","timestamp":"2024-11-05T06:22:56Z","content_type":"text/html","content_length":"87046","record_id":"<urn:uuid:4ebae363-4a77-4a64-8208-77d59dd8cc9d>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00078.warc.gz"} |
Java - How to Create a Binary Search Tree - Analytics Yogi
Java – How to Create a Binary Search Tree
This article represents the high level concept and code samples which could be used to create a binary search tree in Java. Please feel free to comment/suggest if I missed to mention one or more
important points. Also, sorry for the typos.
Following are the key points described later in this article:
• What is a binary search tree?
• What are different kind of traversals?
• Code Samples
What is a binary search tree?
A binary search tree is a binary tree in which every node contains a key that satisfies following criteria:
• The key in left child is less than the key in the parent node
• The key in the right child is more than the parent node
• The left and right child are again binary search trees.
Following diagram represents a binary search tree:
What are different kind of traversals?
Following are three different kind of traversals:
• Preorder traversal: In preorder traversal, the node is visted first and then, left and right sub-trees.
• Inorder traversal: In inorder traversal, the node is visited between left and right sub-tree.
• Postorder traversal: In postorder traversal, the node is visited after left and right subtrees.
Code Sample – How to Create a Binary Search Tree
If the numbers such as {20, 15, 200, 25, -5, 0, 100, 20, 12, 126, 1000, -150} are to be stored in a BinaryTree (represented by code below), following would get printed using different kind of
traversal mechanism:
//Preorder traversal
20, 15, -5, -150, 0, 12, 200, 25, 20, 100, 126, 1000
// Inorder traversal
-150, -5, 0, 12, 15, 20, 20, 25, 100, 126, 200, 1000
//Postorder traversal
-150, 12, 0, -5, 15, 20, 126, 100, 25, 1000, 200, 20
Following is the code for creating binary tree that uses following BinaryTree class and traversals:
BinaryTree tree = new BinaryTree( 20 );
int[] nums = {15, 200, 25, -5, 0, 100, 20, 12, 126, 1000, -150};
for(int i : nums ) {
tree.addNode( i );
Following is the code for BinaryTree class:
public class BinaryTree {
private int data;
private BinaryTree left;
private BinaryTree right;
public BinaryTree(int num) {
this.data = num;
this.left = null;
this.right = null;
// As a convention, if the key to be inserted is less than the key of root node, then key is inserted in
// left sub-tree; If key is greater, it is inserted in right sub-tree. If it is equal, as a convention, it
// is inserted in right sub-tree
public void addNode(int num) {
if (num < this.data) {
if (this.left != null) {
} else {
this.left = new BinaryTree(num);
} else {
if (this.right != null) {
} else {
this.right = new BinaryTree(num);
// Visit the node first, then left and right sub-trees
public void traversePreOrder() {
System.out.println( this.data );
if( this.left != null ) {
if( this.right != null ) {
// Visit left sub-tree, then node and then, right sub-tree
public void traverseInOrder() {
if( this.left != null ) {
System.out.println( this.data );
if( this.right != null ) {
// Visit left sub-tree, then right sub-tree and then the node
public void traversePostOrder() {
if( this.left != null ) {
if( this.right != null ) {
System.out.println( this.data );
Latest posts by Ajitesh Kumar
(see all)
I found it very helpful. However the differences are not too understandable for me
Very Nice Explaination. Thankyiu very much,
in your case E respresent Member or Oraganization which include on e or more peers?
Such a informative post. Keep it up
Thank you....for your support. you given a good solution for me.
Ajitesh Kumar
I have been recently working in the area of Data analytics including Data Science and Machine Learning / Deep Learning. I am also passionate about different technologies including programming
languages such as Java/JEE, Javascript, Python, R, Julia, etc, and technologies such as Blockchain, mobile computing, cloud-native technologies, application security, cloud computing platforms, big
data, etc. I would love to connect with you on Linkedin. Check out my latest book titled as First Principles Thinking: Building winning products using first principles thinking.
Posted in Java, Web. Tagged with Java. | {"url":"https://vitalflux.com/java-create-binary-search-tree/","timestamp":"2024-11-13T14:27:58Z","content_type":"text/html","content_length":"102465","record_id":"<urn:uuid:b5fac35b-6599-4dc8-8ffa-9a9bdc8a6dc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00763.warc.gz"} |
Carburetor Size Calculator
How to calculate Carburetor Size
To calculate the correct carburetor size for your engine, you need to know the engine RPM, the engine displacement, and the volumetric efficiency. This calculator uses the formula:
Carburetor Size (CFM) = (RPM * Displacement * Volumetric Efficiency) / 3456
For example, to find the recommended carburetor size for an engine with a displacement of 350 cubic inches, a max speed of 6,500 RPM, and 85% volumetric efficiency:
$$\text{Carburetor Size (CFM)} = \frac{350 \times 6500 \times 0.85}{3456}$$
$$\text{Carburetor Size (CFM)} = \frac{1,928,750}{3456}$$
$$\text{Carburetor Size (CFM)} \approx 558.17$$ | {"url":"https://yconvert.com/automotive-calculators/carburetor-size-calculator.php","timestamp":"2024-11-12T16:10:16Z","content_type":"text/html","content_length":"30742","record_id":"<urn:uuid:a5c543da-5c00-42ee-a29d-7837b20362b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00308.warc.gz"} |
Identity Function
The identity function is a function which returns the same value, which was used as its argument. It is also called an identity relation or identity map or identity transformation. If f is a
function, then identity relation for argument x is represented as f(x) = x, for all values of x. In terms of relations and functions, this function f: P → P defined by b = f (a) = a for each a ϵ P,
where P is the set of real numbers. Both the domain and range of function here is P and the graph plotted will show a straight line passing through the origin.
Identity Function Definition
Let R be the set of real numbers. Thus, the real-valued function f : R → R by y = f(a) = a for all a ∈ R, is called the identity function. Here the domain and range (codomain) of function f are R.
Hence, each element of set R has an image on itself. The graph is a straight line and it passes through the origin. The application of this function can be seen in the identity matrix.
Mathematically it can be expressed as;
Where a is the element of set R.
For example, f(2) = 2 is an identity function.
In set theory, when a function is described as a particular kind of binary relation, the identity function is given by the identity relation or diagonal of A, where A is a set.
Also, read:
Identity Function Graph
If we plot a graph for identity function, then it will appear to be a straight line. Let us plot a graph for function say f(x) = x, by putting different values of x.
x -2 -1 0 1 2
f(x) = y -2 -1 0 1 2
Now as you can see from the above table, the values are the same for both x-axis and y-axis. Hence, let us plot a graph based on these values.
So, from the above graph, it is clear that the identity function gives a straight line in the xy-plane.
Let us solve some examples based on this concept.
Identity Function Example
Q.1: Prove f(2x) = 2x is an identity function.
Solution: Given, f(2x) = 2x
Let us put the values of x in the given function.
If x = 1, then;
f(2(1)) = 2(1) ⇒ f(2) = 2
If x = 2, then;
f(2(2)) = 2(2) ⇒ f(4) = 4
If x = 3, then;
f(2(3)) = 2(3) ⇒ f(6) = 6
If x = 0, then;
f(2(0)) = 2(0) ⇒ f(0) = 0
Let us try with some negative values of x.
If x =-1, then;
f(2(-1)) = 2(-1) ⇒ f(-2) = -2
If x = -2, then;
f(2(-2)) = 2(-2) ⇒ f(-4) = -4
If x = -3, then;
f(2(-3)) = 2(-3) ⇒ f(-6) = -6
Let us draw a table for all values of x.
x -3 -2 -1 0 1 2 3
y=f(x) -6 -4 -2 0 2 4 6
Let us draw the graph for these values.
You can see from the above graph. The function f(2x) = 2x plots a straight line, hence it is an identity function.
Properties of Identity Function
• It is a linear operator in case of application of vector spaces.
• For positive integers, it is a multiplicative function.
• For m-dimensional vector space, it is expressed as identity matrix I[m].
• In topological space, this function is always continuous. | {"url":"https://mathlake.com/Identity-Function","timestamp":"2024-11-13T07:55:11Z","content_type":"text/html","content_length":"12111","record_id":"<urn:uuid:aef2daad-3309-4d2f-864f-9ff5a68d5035>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00218.warc.gz"} |
Custom research paper proofreading service for masters
UCL Home · Mathematics · Courses & Modules · Undergraduates · Modules · Second Year Honours · MATH7501 Probability and Statistics. Temperatures as we have been observing recently. Committee Home ·
Membership · Terms of Reference · News and Announcements · Future Conferences · Past Conferences · Short Courses · Links · Home; ». A sample space S is the set of all possi- ble outcomes of an
experiment. Sometimes you want to represent a lot of complicated information from a large data set in a way that is. Download "STAT 473. MATH 2411 – Calculus II. Introductory course in probability
and statistics that examines collecting, organizing and summarizing data in order to draw valid conclusions. John Aldrich, University of Southampton, Southampton, UK.
We offer the most extensive. 7 essay checker grammar morning. Another major door that depends being studied means Twitters concept to include copywriters and how they spread. The first part of this
course covers the. UCSB Academic Calendar 2016–2017. Win Probability & Box Scores. 2143122: Probability and Statistics for Information and Communication Engineering. 1.1.1 Kolmogorov definition of
probability. It is jointly organized by the Bernoulli Society. STAT 221: Introductory Probability I (Pre-req. A succinct reference in probability theory and statistics.
The Congress is be the. Sci., 1,347–350. Probability, Statistics and Truth has 27 ratings and 2 reviews. Bayes stated the defining relationship expressing the probability you test. Unit 1:
Statistics: Mean, Median & Mode.
MAT 121 - Introduction to Probability and Statistics.
On structure good probability stat what topic about 5 the could Grey how Christian to who books will read of of will those Shop the by essay system an UKs. Graphs Index. Class contact. That the
outcome will be A. Period: 5 (Spring 2011) Lectures: Wednesdays 13:30--15:15. Kyle Walker Tottenham Hotspur · FC. Win probability. SCI.-W/ACCESS Edition: 9TH 16 ISBN: 9781305779372, $80.50 (Used)
$84.00 (New). A first course in probability intended to serve as a background for statistics and other applications.
Department of Statistics Faculty. Real Betis Probability Stat. An introduction to probability, with the aim of developing probabilistic intuition as well as techniques needed to analyze simple random
samples. “Estimating the dimension of a model', Ann.
Alberto Ohashi, Evelina Shamarova & Nikolai N. Shamarov pages: 1-31. The course requires a basic knowledge. ECE 3610 - Engineering Probability & Statistics. Written and video lessons.
While there have been dramatic advances in the range and scale of forensic techniques used to help solve legal cases, the way that the. I. Locator Information: Instructor.
Euro Basketball Stats. Discuss statistical research, data analysis, statistics homework questions, R, SAS. I guess one also expects power-law (log-Gaussian?) Lecturer, Zhiqiang Tan Office: 459 Hill
Center Email: ztan at. C & Data Structure OR Do Microprocessor Library myr Comp. From Exercise 4, we cannot expect \(Y_n\) itself to have a. The facts and figures that are collected and examined for
information on a given subject are statistics. Andrew Gelman, Department of Statistics and Department of Political.
Free delivery worldwide on over 17 million titles. Each session will feature one or two talks featuring external and/or internal speakers. SKU: 11254225-c By Devore Department: Mathematics ISBN:
1-305-25180-6 Edition: 9. Here are some examples, with links to further. Point spread. A secondary school revision resource for GCSE Maths about foundation and higher level data handling, collection
and representation, averages and probability. Explore Katie Gordon's board "Probability & Statistics" on Pinterest, the world's catalog of ideasSee more about Local colleges. Bhole baba chilam love
stat • Buy Web Stat & Value Calculator Script. Thu Uyr Probability Stat. Basic concepts of.
Phenomenom is the proportion of times the outcome would occur in a very long series of repetitions. (Definitions for Middle School Teachers). In probability theory, random experiment means a
repeatable process.
Probability course and homework discussion. H1-Pr, H1 (foreground*) prior class probability (stat.fit.goodness). Put out a little time and money to get the report you could not. Definition 2.1. The
expected value or ______. Throughout 2011 and 2012, the Board of Studies NSW developed new K–10 syllabuses for English, Mathematics, Science (incorporating Science and. Theory of Probability, Oxford
Univ. In probability, we're given a model, and asked what kind of data we're likely to see. Conditional probability: The probability of an event occurring given that a second event has occurred. Stat
518 will have more advanced assignments and examinations focusing on theoretical methods. Our solutions are written by Chegg experts so you can be assured of the highest quality! These decisions or.
3, Annals of Statistics, journal, 6.653 Q1, 113, 72, 269, 2486, 792, 256, 2.91. Pre-game CARM-Elo. Statistical analysis often uses probability distributions, and the two topics are often studied
together. Smytherobertson hesitated then said probability stat icily sir. CHINESE JOURNAL OF APPLIED PROBABILITY AND STATISTICS, 2016,32(6): 593-603>')" href="#">, The. Decisions or predictions are
often based on data—numbers in context. Course overview. This book is distributed on the Web as part of the Chance Project, which is de- voted to providing materials for beginning courses in
probability and statistics. | {"url":"http://www.keetoncustomgolf.com/probability-and-stat/","timestamp":"2024-11-11T02:59:06Z","content_type":"application/xhtml+xml","content_length":"14271","record_id":"<urn:uuid:145a0479-a08c-41a2-bc71-31f7c22a903a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00324.warc.gz"} |
History of Mathematics
History of Mathematics is a component of Encyclopedia of Mathematical Sciences in the global Encyclopedia of Life Support Systems (EOLSS), which is an integrated compendium of twenty one
The Theme on History of Mathematics discusses: Mathematics in Egypt and Mesopotamia; History of Trigonometryto 1550; Mathematics in Japan; The Mathematization of The Physical Sciences-Differential
Equations of Nature; A Short History of Dynamical Systems Theory:1885-2007; Measure Theories and Ergodicity Problems; The Number Concept and Number Systems; Operations Research and Mathematical
Programming: From War to Academia - A Joint Venture; Elementary Mathematics From An Advanced Standpoint; The History and Concept of Mathematical Proof; Geometry in The 20th Century; Bourbaki: An
Epiphenomenon in The History of Mathematics
This volume is aimed at the following five major target audiences: University and College Students Educators, Professional Practitioners, Research Personnel and Policy Analysts, Managers, and
Decision Makers, NGOs and GOs.
Editor(s) Biography
Vagn Lundsgaard Hansen is born September 27, 1940 in Vejle, Denmark. He is professor of mathematics at the Technical University of Denmark since 1980, and Scientific Director of the Learning Lab at
this university since August 2005. He earned a Master’s Degree in mathematics and physics from the University of Aarhus, Denmark, 1966, and a Ph.D. in mathematics from the University of Warwick,
England, 1972. He has held positions as assistant professor, University of Aarhus, 1966-69; research fellow, University of Warwick, 1969-72; associate professor, University of Copenhagen, Denmark,
1972-80. He was visiting professor fall 1986, University of Maryland, College Park, US. He is the author of numerous research papers in topology, geometry, and global analysis, and author of several
books including the general books “Geometry in Nature” (1993), “Shadows of the Circle” (1998) and “Matematikkens Uendelige Univers” (2002). He was chairman of the World Mathematical Year 2000
committee appointed by the European Mathematical Society 1995-2000. He is chairman of the committee for Raising Public Awareness of Mathematics appointed by the European Mathematical Society
2000-2006. He was Invited Speaker at the International Congress of Mathematicians, Beijing 2002, and Invited Regular Lecturer at the 10th International Congress on Mathematical Education,Copenhagen
2004. He is President of the Danish Academy of Natural Sciences since 1984. He was elected Member of European Academy of Sciences (Brussels) 2004. He was Member of the Danish Natural Science Research
Council 1992-98, four of the years as vice-chairman. He is referee for the Norwegian Research Council and INTAS.He was member of the Jury for the European Union Contest for Young Scientists in
Budapest (2003), Dublin (2004) and Moscow (2005).
Jeremy Gray, Centre for Mathematical Sciences, Open University, UK | {"url":"http://www.eolss.net/ebooklib/bookinfo/history-mathematics.aspx","timestamp":"2024-11-06T06:02:34Z","content_type":"text/html","content_length":"39488","record_id":"<urn:uuid:b09aef54-c624-49f3-8d72-c05f650703fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00861.warc.gz"} |
Poisson Tests
Two distance-based tests of Poissonity are applied in poisson.tests, "M" and "E". The default is to do all tests and return results in a data frame. Valid choices for test are "M", "E", or "all" with
default "all".
If "all" tests, all tests are performed by a single parametric bootstrap computing all test statistics on each sample.
The "M" choice is two tests, one based on a Cramer-von Mises distance and the other an Anderson-Darling distance. The "E" choice is the energy goodness-of-fit test.
R must be a positive integer for a test. If R is missing or 0, a warning is printed but test statistics are computed (without testing).
The mean distance test of Poissonity (M-test) is based on the result that the sequence of expected values E|X-j|, j=0,1,2,... characterizes the distribution of the random variable X. As an
application of this characterization one can get an estimator \(\hat F(j)\) of the CDF. The test statistic (see poisson.m) is a Cramer-von Mises type of distance, with M-estimates replacing the usual
EDF estimates of the CDF: $$M_n = n\sum_{j=0}^\infty (\hat F(j) - F(j\;; \hat \lambda))^2 f(j\;; \hat \lambda).$$
In poisson.tests, an Anderson-Darling type of weight is also applied when test="M" or test="all".
The tests are implemented by parametric bootstrap with R replicates.
An energy goodness-of-fit test (E) is based on the test statistic $$Q_n = n (\frac{2}{n} \sum_{i=1}^n E|x_i - X| - E|X-X'| - \frac{1}{n^2} \sum_{i,j=1}^n |x_i - x_j|, $$ where X and X' are iid with
the hypothesized null distribution. For a test of H: X ~ Poisson(\(\lambda\)), we can express E|X-X'| in terms of Bessel functions, and E|x_i - X| in terms of the CDF of Poisson(\(\lambda\)).
If test=="all" or not specified, all tests are run with a single parametric bootstrap. poisson.mtest implements only the Poisson M-test with Cramer-von Mises type distance. poisson.etest implements
only the Poisson energy test. | {"url":"https://www.rdocumentation.org/packages/energy/versions/1.7-12/topics/Poisson%20Tests","timestamp":"2024-11-12T12:07:06Z","content_type":"text/html","content_length":"72025","record_id":"<urn:uuid:61181eac-e8c4-47a5-9a35-d0e1e6e82915>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00750.warc.gz"} |
In general, the probability of an event occuring is: P\left(\text{Event}\right)=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
This can be applied to geometric contexts in one, two, or three dimensions.
For lengths: P\left(\text{Event}\right)=\dfrac{\text{Length of favorable segment}}{\text{Total length}}
For areas: P\left(\text{Event}\right)=\dfrac{\text{Area of favorable region}}{\text{Total area}}
For volumes: P\left(\text{Event}\right)=\dfrac{\text{Volume of favorable space}}{\text{Total volume}} | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1098/topics/Topic-21296/subtopics/Subtopic-275273/?ref=blog.mathspace.co","timestamp":"2024-11-14T16:36:01Z","content_type":"text/html","content_length":"306378","record_id":"<urn:uuid:9572d0f7-9541-4c30-9b59-5ccf460a0a3c>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00689.warc.gz"} |
Tl BASIC One-Liners
Michael A. Covington
The Tl BASIC DEF statement can become a powerful tool in your programmer's bag of tricks. Here's how to use it.
If you've been programming in BASIC for any time at all, you've surely come across, and used, some of the built-in functions that the language provides, such as INT, SIN, COS, TAN, ATN, and LOG. But
did you know that you can use the DEF statement to create functions of your own? Defining your own functions lets you type a complicated formula only once, and it allows you to build complex
functions out of simple ones in a most efficient way.
Suppose, for instance, that your LOG function gives you natural (base e) logarithms, and you want base 10 logarithms. (If you're not sure which you've got, type PRINT LOG(1O) - if the answer is 1,
you're in base 10, and if it's about 2.3026, you're in base e.) You can convert base e logarithms to base 10 by dividing them by 2.302585093, so one of the options open to you is obviously to write
LOG(X)/2.302585093 (or whatever) every time you need a base 10 log. But there's an easier way.
Creating Functions
To create your own function - let's call it LOG10, though some computers may insist that you name it something like FNL - just include, early in your program, a statement like this:
10 DEF LOG10 (X) = LOG(X)/2.302585093
From then on, you'll be able to use the new function LOG10 to get base 10 logarithms. Try it out with a program something like this:
1O DEF LOG1O(X)=LOG(X)/2.3O2585O93
2O FOR 1=1 TO 10 STEP O.1
3O PRINT I,LOG1O(I)
40 NEXT I
and compare the results against a table of logarithms.
The DEF statement is different from most BASIC statements in that it can't refer to variables. (The X in it - it could be any variable name - is used only as a placeholder for the number within the
parentheses; it is completely separate from any variable named X that you may use elsewhere in the program.) You can refer only to numbers or other functions. Some computers require that the name of
the function be three letters and that the first two be FN - FNA, FNB, FNL, and so forth -although the TI-99, and many other microcomputers, allow you to name functions with the same type of names
you use for variables.
Sample One Liners
So that's how it's done. Now let's look at some practical examples.
1. Base 10 logarithms. That's what we've just discussed. For reference, here is the statement:
DEF LOG10(X) = LOG(X)/2.302585093
(assuming your machine's LOG function gives you base e logs).
2. Base 2 logarithms. On a machine on which the LOG function gives base e logarithms, you can get base 2 logarithms by using:
DEF LOG2 (X) = LOG(X)/0.6931471806
If your machine's LOG function gives base 10 logarithms, you'll need to use DEF LOG2(X) = LOG(X)/0.3010299957 instead.
3. Degrees to radians. If X is the measure of an angle in degrees, then RAD(X) will be the same angle measured in radians, if you define the following function:
DEF RAD(X) = X/57.29577951/
4. Radians to degrees. The opposite function, converting X in radians to DEG(X) in degrees, is:
DEF DEG(X) = X*57.29577951
5. Arcsine (in radians). The following definition will give you the arcsine function (which is not usually provided in implementations of BASIC, although the arctangent is).
DEF ASN(X) = 2*ATN(X/(1 + SQR(1-X^2)))
If you look through a table of trigonometric identities, you may find an apparently equivalent, but simpler, formula that would lead to the statement DEF ASN(X) = ATN(X/SQR(1-X^2)). But note that
this version won't do ASN(l) correctly (it will try to divide by zero). Hence the first version is preferable.
6. Arccosine (in radians). If you have the arcsine function, you can get the arccosine, as follows:
DEF ACS(X) = 1.570796327-ASN(X)
Remember that the DEF statement for ASN must precede the DEF statement for ACS (you can't refer to a function until you've defined it).
7. Rounding to a particular number of decimal places. Where n stands for the number of decimal places you want, use the definition:
DEF ROU(X) = INT(((l0^N)*X) + 0.5)/ l0^N)
Note that you must substitute a number for n; in most implementations, n cannot be a variable. Hence, for example, if you want rounding to three decimal places, your statement will read DEF ROU(X) =
INT(((10^3)*X) + 0.5)/(l0^3). The number of decimal places can be negative, of course; if you want to round to the nearest 10, ask for -1 decimal place, and if you want to round to the nearest 1000,
ask for -3 decimal places.
8. Rounding to a particular number of significant digits. Often, you'll find that the most convenient type of rounding involves coming up with a particular number of significant digits rather than a
particular number of decimal places. You can accomplish this with the definition
DEF RSFl(X) = (N-l)-INT(LOGlO(X))
DEF RSF(X) = INT(((10*RSFl(X))*X) + 0.5)/(l0^ RSF1(X))
Here the definition is so complex that it is best done in two stages: first we define RSF1, which is a function used internally in RSF, and then we define RSF, which is the function we actually use.
n stands for the number of significant digits you want; as before, you must substitute a number for it when typing the definition into the computer.
A word of warning: RSF (with its subsidiary calls to RSF1, which in turn calls LOG10) can take quite a bit of time to execute (about half a second of realtime on the TI-99).
9. Sexagesimal output: minutes. Our practice of expressing time in hours, minutes, and seconds, and angles in degrees, minutes, and seconds, is a remnant of an ancient Babylonian base-60
(sexagesimal) number system. Often, in a computer program dealing with time or with angles, it is desirable to express the output in terms of units, minutes, and seconds. The units are obtained by
taking 1NT(X); thus the units part of 2.5 hours = INT(2.5) = 2 hours. Here is a function that gives the minutes part:
DEF MNT(X) = INT(60*(X-INT(X)))
That is, we take the non-integer part of the value, multiply it by 60, and take the INT of that.
10. Sexagesimal output: seconds. The seconds part of the value, in turn, is given by:
DEF SCD(X) = 60*(60*(X-INT(X))-MNT(X))
That is, we subtract the integer part and the minutes; what's left gets multiplied by 60 twice.
The sexagesimal output functions can be tested by means of a program such as the following:
10 DEF MNT(X)=INT(60*(X-INT(X)))
2O DEF SCD(X)=60*(6O*(X-INT(X))-MNT(X) )
3O FOR H=O TO 2 STEP O.O1
4O PRINT
5O PRINT H,"HOURS"
6O PRINT INT(H),MNT(H),SCD(H)
7O NEXT H
From this we learn, for example, that 0.01 of an hour is 36 seconds, and that 0.5 of an hour is 30 minutes. (If your computer uses binary, rather than BCD or Radix-100, internal representations of
numbers, you may get odd errors due to rounding or lack of it. The solution would be to round the number of hours to some reasonably small number of decimal places before invoking the conversions,
and perhaps to insert some rounding in the definitions of MNT and SCD themselves.)
Incidentally, for sexagesimal input, you don't need any special functions, only a bit of multiplication. For instance, the statements
1O PRINT "TYPE HOURS, MINUTES, SECONDS"
20 INPUT H,M,S
3O H=H+M/6O+S/36OO
will give you (as H) the number of hours expressed as a decimal.
11. Modulo 12 arithmetic. In dealing with hours, you'll often want to reduce numbers to modulo 12. For instance, if it's 11 a.m., then you can calculate the time four hours later by adding 11+4
(which gives you 15) and then taking the result modulo 12. The function definition is:
DEF MOD12(X) = 12*(X/12-INT(X/12))
(unless, of course, your computer has a built-in MOD function, which is even simpler to use). This particular function is likely to be bothered by rounding and truncation errors. On the TI-99, I get
accurate results for numbers under 1000 or so, but larger numbers give slightly erroneous answers; a binary machine might be plagued by worse problems.
12. Modulo 60 arithmetic. The same function, giving modulo 60 answers (for dealing with minutes and seconds), is:
DEF MOD60(X) =60*(X/60 - INT(X/60))
(as if you couldn't have guessed). The following program starts with a time expressed as H hours M minutes, and adds Ml minutes:
1O DEF MOD12(X)=12*(X/12-INT(X/12))
2O DEF MOD6O(X)=6O*(X/60 INT)X/6O))
30 INPUT H, M
4O INPUT Ml
5O M=MOD6O(M+M1)
60 H=H+INT(M1/6O)
70 PRINT H, M
Line 50 adds the right number to the minutes part, and line 60 adds to the hours part if necessary. | {"url":"https://www.atarimagazines.com/compute/issue36/108_TI_BASIC_One-Liners.php","timestamp":"2024-11-04T01:53:13Z","content_type":"text/html","content_length":"24533","record_id":"<urn:uuid:f6afe4df-0fb0-4330-a87c-db514d1d7e5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00681.warc.gz"} |
70.7 decimeters per square second to decameters per square second
71 Decimeters per square second = 0.71 Decameters per square second
Acceleration Converter - Decimeters per square second to decameters per square second - 70.7 decameters per square second to decimeters per square second
This conversion of 70.7 decimeters per square second to decameters per square second has been calculated by multiplying 70.7 decimeters per square second by 0.01 and the result is 0.707 decameters
per square second. | {"url":"https://unitconverter.io/decimeters-per-square-second/decameters-per-square-second/70.7","timestamp":"2024-11-11T23:36:53Z","content_type":"text/html","content_length":"27422","record_id":"<urn:uuid:3e566973-09fe-4e39-9244-40cb0e2d06db>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00789.warc.gz"} |
Math Terms Word Search Printable - Word Search Maker
Math Terms Word Search Printable - Browse and print Math word searches below You can also browse Math Crossword Puzzles or make your own Math word search crossword fill in the blank word scramble
matching bingo handwriting exercise open response worksheet or flashcards
Math Word Search These free maths word searches are for different grade students Download the free printables to engage the students in classroom with some fun activity while getting to know about
the geometry algebra or
Math Terms Word Search Printable
Math Terms Word Search Printable
Math word searches offer a dynamic and engaging way for K-8 students to enhance their understanding of math vocabulary terms. These puzzles serve as an effective tool to reinforce mathematical
concepts while simultaneously building essential vocabulary skills.
Math Word Search Puzzles for first through sixth grade including specific puzzles for geometry algebra and many other topic areas Free printable PDFs with color answer keys
Printable Math Word Search Cool2bKids
Algebra Word Search A fun free printable word search puzzle worksheet featuring algebra for math students or anyone looking for to review their knowledge of the world The 26 vocabulary words covered
in this puzzle are absolute additive binomial coefficient constant equation exponent expression factor function inequality inverse
Math Terms Word Search
If you need a fun learning activity for the kids to do on those indoor play days check out these free math word search printable game sheets
Math Terms Word Search WordMint
Math Word Search Free Printable Letter Words Unleashed
Math Word Word Searches My Word Search
Solve these free math word search puzzles on your computer or download and print them
Printable Math Word Search
A free math word search printable with 18 math terms to find including decimal algebra multiplication fraction and addition
Our collection of engaging free math word search worksheets combines the fun of word searches with the educational value of math concepts.
Math Word Searches Printable And Free Rudolphacademy
Word search contains 35 words Print save as a PDF or Word Doc Add your own answers images and more Choose from 500 000 puzzles
Math Terms Word Search WordMint
Free Math Word Search Printable
Math Terms Word Search Printable
A free math word search printable with 18 math terms to find including decimal algebra multiplication fraction and addition
Math Word Search These free maths word searches are for different grade students Download the free printables to engage the students in classroom with some fun activity while getting to know about
the geometry algebra or
Free Math Word Search Printable Printable Word Searches
7TH GRADE MATH VOCABULARY WORDS Word Search WordMint
Free Math Word Search Puzzles Printable Printable Math Word Search Cool2bkids Printable Word
Printable Math Word Search
Printable Math Word Search | {"url":"https://wordsearchmaker.net/math-terms-word-search-printable","timestamp":"2024-11-05T06:09:59Z","content_type":"text/html","content_length":"48599","record_id":"<urn:uuid:6e01d4f6-c41c-4722-8528-bf8938075eaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00708.warc.gz"} |
2,652 research outputs found
The quantum analogs of the derivatives with respect to coordinates q_k and momenta p_k are commutators with operators P_k and $Q_k. We consider quantum analogs of fractional Riemann-Liouville and
Liouville derivatives. To obtain the quantum analogs of fractional Riemann-Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives
for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier
transformation.Comment: 9 pages, LaTe
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments
with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with
fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also
derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The
model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.Comment: 25page
Using the quasistatic approximation, we show that in a subdiffusion--reaction system the reaction front $x_{f}$ evolves in time according to the formula $x_{f} \sim t^{\alpha/2}$, with $\alpha$ being
the subdiffusion parameter. The result is derived for the system where the subdiffusion coefficients of reactants differ from each other. It includes the case of one static reactant. As an
application of our results, we compare the time evolution of reaction front extracted from experimental data with the theoretical formula and we find that the transport process of organic acid
particles in the tooth enamel is subdiffusive.Comment: 18 pages, 3 figure
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite
polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between
them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All
this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].Comment: 15 page
Recently it was pointed out that the solutions found in literature for the space fractional Schr\"odinger equation in a piecewise manner are wrong, except the case with the delta potential. We
reanalyze this problem and show that an exact and a proper treatment of the relevant integral proves otherwise. We also discuss effective potential approach and present a free particle solution for
the space and time fractional Schr\"odinger equation in general coordinates in terms of Fox's H-functions
We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous model, where all characteristics and fields are defined
everywhere in the volume but they follow some generalized equations which are derived by using integrals of fractional order. The order of fractional integral can be equal to the fractal mass
dimension of the solid. Fractional integrals are considered as an approximation of integrals on fractals. We suggest the approach to compute the moments of inertia for fractal solids. The dynamics of
fractal solids are described by the usual Euler's equations. The possible experimental test of the continuous medium model for fractal solids is considered.Comment: 12 pages, LaTe
We propose a method to extract from experimental data the subdiffusion parameter $\alpha$ and subdiffusion coefficient $D_\alpha$ which are defined by means of the relation $=2D_\alpha/\Gamma(1+\
alpha) t^\alpha$ where denotes a mean square displacement of a random walker starting from $x=0$ at the initial time $t=0$. The method exploits a membrane system where a substance of interest is
transported in a solvent from one vessel to another across a thin membrane which plays here only an auxiliary role. Using such a system, we experimentally study a diffusion of glucose and sucrose in
a gel solvent. We find a fully analytic solution of the fractional subdiffusion equation with the initial and boundary conditions representing the system under study. Confronting the experimental
data with the derived formulas, we show a subdiffusive character of the sugar transport in gel solvent. We precisely determine the parameter $\alpha$, which is smaller than 1, and the subdiffusion
coefficient $D_\alpha$.Comment: 17 pages, 9 figures, revised, to appear in Phys. Rev.
We use the hyperbolic subdiffusion equation with fractional time derivatives (the generalized Cattaneo equation) to study the transport process of electrolytes in media where subdiffusion occurs. In
this model the flux is delayed in a non-zero time with respect to the concentration gradient. In particular, we obtain the formula of electrochemical subdiffusive impedance of a spatially limited
sample in the limit of large and of small pulsation of the electric field. The boundary condition at the external wall of the sample are taken in the general form as a linear combination of
subdiffusive flux and concentration of the transported particles. We also discuss the influence of the equation parameters (the subdiffusion parameter and the delay time) on the Nyquist impedance
plots.Comment: 10 pages, 5 figure
We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the
diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The
resulting distributions have the form of the L\'evy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the
system.Comment: 7 pages, 2 figure | {"url":"https://core.ac.uk/search/?q=author%3A(Oldham%20K.)","timestamp":"2024-11-08T21:36:23Z","content_type":"text/html","content_length":"153975","record_id":"<urn:uuid:8260d0ad-0116-4947-b9cc-a0051bc45eb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00500.warc.gz"} |
Extending SDP Integrality Gaps to Sherali-Adams with Applications to Quadratic Programming and MaxCutGain, with Siavosh Benabbas. To appear in Conference on Integer Programming and Combinatorial
Optimization (IPCO) 2010.
On Quadratic Threshold CSPs
With Per Austrin and Siavosh Benabbas. To appear in LATIN 2010.
On the Tightening of the Standard SDP for Vertex Cover with l1 Inequalities
With K. Georgiou and I. Tourlakis. 29th Foundations of Software Technology and Theoretical Computer Science (FSTTCS) 2009
Optimal Sherali-Adams Gaps from Pairwise Independence. 
With K. Georgiou and M. Tulsiani. APPROX 2009. pdf
Robust algorithms for Maximum Independent Set on Minor-free graphs based on the Sherali-Adams Hierarchy. 
With M. Moharrami. APPROX 2009 pdf
On the nonexistence of Dimension Reduction in $\ell_2^2$.
With M. Moharrami. CCCG 08. pdf
Nearly tight dimensionality reductions that preserve volumes
With A. Zouzias. RANDOM 2008. pdf
Vertex Cover Resists SDPs Tightened by Local Hypermetric Inequalities.
With K. Georgiou and I. Tourlakis. 13th Conference on Integer Programming and Combinatorial Optimization (IPCO 2008). pdf
Integrality gaps of 2-o(1) for vertex cover SDPs in the Lovasz-Schrijver hierarchy.
With K. Georgiou, T. Pitassi and I. Tourlakis.
FOCS 2007. pdf
Integrality gaps of semidefinite programs for Vertex Cover and relations to $\ell_1$ embeddability of Negative Type metrics. 
With H. Hatami and V. Markakis. APPROX 2007. ps pdf
A rigorous analysis for set-up time models - a metric perspective.
With E. Bachmat and T. Lam. Theoretical Computer Science. pdf
Monotone circuits for the majority function.
With S. Hoory and T. Pitassi. RANDOM 2006. pdf
How well can primal-dual and local-ratio algorithms perform?
With A. Borodin and D. Cashman. ICALP 2005. ps pdf
Toward a Model for Backtracking and Dynamic Programming.
With M. Alekhnovich, A. Borodin, J. Buresh-Oppenheim, R. Impagliazzo and T. Pitassi. CCC (Conference on Computational Complexity) 2005. ps pdf
On-line algorithms for market equilibria.
With S Angelopoulos, A. Das Sarma and T. Viglas. COCOON 2005. ps pdf
Approximating Range Searching in Higher Dimension.
With B. Chazelle and D. Liu.
CCCG 2004 - Canadian Conference on Computational Geometry.
Metric embeddings beyond one-dimensional distortion.
With R. Krauthgamer and N. Linial.
Discrete & Computational Geometry 31(3): 339-356 (2004). ps pdf
Simple Permutations Mix Well.
With S. Hoory, S. Myers and C. Rackoff. special issue of Theoretical Computer Science 348(2-3): 251-261 (2005) (preliminary version in ICALP 2004) ps pdf
Embedding Hamming-distance of Boxes into Euclidean Space (Manuscript).
Here are slides from a talk presented in DIMACS Workshop on Discrete Metric Spaces and their Algorithmic Applications, Princeton, Aug 2003.
On Cutting-plane Rank and the role of Expansion.
With J. Buresh-Oppenheim, N. Galesi, S. Hoory and T. Pitassi.
Theory of Computing (2): 65--90 (2006) (preliminary version in FOCS 2003). ps pdf
A Sublinear Algorithm for Weakly Approximating Edit Distance.
With T. Batu, F. Ergun, J. Kilian, S. Raskhodnikova, R. Rubinfeld and R. Sami
STOC 2003. ps pdf
Sublinear Geometric Algorithms.
With B. Chazelle and D. Liu.
SICOMP 35(3): 627-646 (2005) (preliminary version in STOC 2003). ps pdf
Sublinear approximation of Euclidean minimum spanning tree.
With A. Czumaj, F. Ergun, L. Fortnow, R. Rubinfeld, I. Newman, and C. Sohler.
SICOMP (1): 91-109 (2005) (preliminary version in SODA 2003). ps pdf
Dimensionality Reductions that Preserve Volumes and Distance to Affine Spaces, and their Algorithmic Applications. Discrete & Computational Geometry 38(1): 139-153 (2007). Preliminary version in
RANDOM 2002. ps pdf
On the Euclidicity of Metric Spaces. Ph.D. Thesis, Hebrew University (2002). Thesis advisor: Professor Nathan Linial.
Designing oligo libraries taking alternative splicing into account.
With A. Shoshan, V. Grebinskiy, A. Scolnicov, E. Fink, D. Lehavi and A. Wasserman.
Proc. SPIE 4266, May 2001.
Girth and Euclidean Distortion.
With N. Linial and A. Naor.
STOC 2002 ps pdf and
GAFA, Geometric and Functional Analysis. 12: pp 380-394 (2002). ps pdf
Least-distortion Euclidean embeddings of graphs: Products of cycles and expanders.
With N. Linial.
Journal of Combinatorial Theory ser. B, 79 pp 157-171 (2000). ps pdf
Trees and Euclidean Metrics.
With N. Linial and M. Saks.
STOC 98 version ps pdf and version in Israel Journal of Methematics, 106 pp 339-348 (1998). ps pdf | {"url":"http://www.cs.toronto.edu/~avner/pub.html","timestamp":"2024-11-02T14:58:06Z","content_type":"text/html","content_length":"8322","record_id":"<urn:uuid:4977420c-503c-4164-8c09-b38db4e1cef7>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00563.warc.gz"} |
TOC | Previous | Next | Index
26.1 Bracketing a Minimum (.NET, C#, CSharp, VB, Visual Basic, F#)
Minima of univariate functions must be bracketed before they can be isolated. A bracket is a triplet of points, x[lower] < x[interior] < x[upper], such that f(x[interior]) < f(x[lower]) and f(x
[interior]) < f(x[upper]). These conditions ensure that there is some local minimum in the interval (x[lower],x[upper],).
If you know in advance that a local minimum falls within a given interval, you can simply call the NMath minimization routines using that interval. Before beginning minimization, the routine will
search for an interior point that satisfies the bracketing condition.
Otherwise, construct a Bracket object. Beginning with a pair of points, Bracket searches in the downhill direction for a new pair of points that bracket a minimum of a function. For example, if
function is a OneVariableFunction:
Code Example – C# minimization
var bracket = new Bracket( function, 0, 1 );
Code Example – VB minimization
Dim Bracket As New Bracket( MyFunction, 0, 1 )
Once constructed, a Bracket object provides the following properties:
● Function gets the function whose minimum is bracketed.
● Lower gets a lower bound on a minimum of the function.
● Upper gets an upper bound on a minimum of the function.
● Interior gets a point between the lower and upper bound such that x[lower] < x[interior] < x[upper], f(x[interior]) < f(x[lower]), and f(x[interior]) < f(x[upper])
● FLower gets the function evaluated at the lower bound.
● FUpper gets the function evaluated at the upper bound.
● FInterior gets the function evaluated at the interior point. | {"url":"https://www.centerspace.net/doc/NMath/user/minimizing-univariate-functions-84707.htm","timestamp":"2024-11-03T13:02:19Z","content_type":"text/html","content_length":"16953","record_id":"<urn:uuid:2df18e38-1c75-40fd-86a6-f25097490b14>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00874.warc.gz"} |
How do you calculate a 5% decrease in Excel?
How do you calculate a 5% decrease in Excel?
To decrease a number by a percentage amount, multiply the original amount by 1- the percent of increase.
How do I calculate a percentage change in Excel?
The formula =(new_value-old_value)/old_value can help you quickly calculate the percentage change between two numbers. Please do as follows. 1. Select a blank cell for locating the calculated
percentage change, then enter formula =(A3-A2)/A2 into the Formula Bar, and then press the Enter key.
What is the formula for percentage decrease?
Calculate Percentage Decrease: First, work out the difference (decrease) between the two numbers you are comparing. Next, divide the decrease by the original number and multiply the answer by 100.
How do you find the percentage of a grand total in Excel?
When the Value Field Settings window appears, click on the “show values as” tab. Then select “% of total” from the drop down list. Click on the OK button. Now when you view your pivot table, you
should only see the Totals displayed as a percentage of the Grand Total.
How do you increase by a percentage in Excel?
Increase by Percentage
1. Enter a number in cell A1. Enter a decimal number (0.2) in cell B1 and apply a Percentage format.
2. To increase the number in cell A1 by 20%, multiply the number by 1.2 (1+0.2). The formula below does the trick.
3. To decrease a number by a percentage, simply change the plus sign to a minus sign.
How do you find percent decrease in area?
How to Calculate Percentage Decrease
1. Subtract starting value minus final value.
2. Divide that amount by the absolute value of the starting value.
3. Multiply by 100 to get percent decrease.
4. If the percentage is negative, it means there was an increase and not an decrease.
How do you calculate a percentage of a total?
How to calculate percentage
1. Determine the whole or total amount of what you want to find a percentage for.
2. Divide the number that you wish to determine the percentage for.
3. Multiply the value from step two by 100.
What is the formula for calculating percent decrease?
The formula for calculating percent decrease used in our percentage decrease calculator is: Percent decrease = 100 – new / old * 100. where new is the newer quantity or measure, and old is the older
quantity or measure.
What is the formula to calculate percent change?
To calculate the percent change, there are two options: either divide the new value by the original value and subtract 1 the formula = (A1-A2)-1, or subtract the new value from the original value and
divide the result by the original value with the formula = (A2-A1)/A1.
How do you calculate percent reduction in Excel?
Calculate a percentage of decrease Click any blank cell. Type =(2425-2500)/2500, and then press RETURN . The result is -0.03000. Select the cell that contains the result from step 2. On the Home tab,
click . The result is -3.00%, which is the percentage of decrease in earnings.
How do you increase or decrease in Excel?
Calculate Percentage Change. If want to calculate a percentage increase in Excel (i.e. increase a number by a specified percentage), this can be done by simply multiply the number by 1 + the
percentage increase. For example, if you want to increase the number 50 by 20%, this can be done in Excel by typing the following formula into any Excel cell: | {"url":"https://www.idcafe.net/how-do-you-calculate-a-5-decrease-in-excel/","timestamp":"2024-11-05T22:16:41Z","content_type":"text/html","content_length":"56543","record_id":"<urn:uuid:ff6e6e00-6130-4c4e-982f-c895ddaaf6bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00798.warc.gz"} |
Greatest common divider online calculator - Calc Works
Greatest common divider calculator
Enter the two numbers whose gcd is to be calculated and then click on "Calculate".
Greatest common divider - What does it mean?
The greatest common divider (gcd) is a term in mathematics. It is the largest natural number by which two integers can be divided without remainder. The gcd is at least 1 and at most the smaller of
the two numbers. If the gcd is 1, then one calls both numbers divider-alien. The gcd is mainly used in fractions, but also in number theory.
In fraction calculus it is used to "shorten" fractions.
This means that a common factor is removed from the numerator and denominator of a fraction, whereby the value of the fraction does not change. If you truncate a fraction with the greatest common
divider of the numerator and denominator, you get a fraction that you cannot truncate any further, called a fully or maximally truncated fraction. A fraction is usually truncated in order to simplify
further calculations with it.
How is the greatest common divider calculated?
For the calculation of the gcd there are 2 possibilities, on the one hand the prime factorization of both numbers and on the other hand over the so-called Euclidean algorithm. With the calculation by
means of a prime factorization one takes the prime factors, which occur in both decompositions, and multiplies them with one another around the gcd to receive.
However, this process is very time-consuming, especially for large numbers, which is why a more efficient method, the so-called Euclidean algorithm, is used for such numbers. Here a division with
remainder is carried out in successive steps, whereby the remainder becomes the new divider in the next step. The divider with the remainder 0 is the greatest common divider of both numbers.
Our online calculator uses the Euclidean algorithm to determine the gcd. Simply enter the two numbers whose gcd you want to determine and click on "Calculate".
Can't find your personal online calculator?
Contact us and tell us what kind of online calculator you need! | {"url":"https://www.calcworks.co/greatest-common-divider-calculator","timestamp":"2024-11-03T04:02:57Z","content_type":"text/html","content_length":"43240","record_id":"<urn:uuid:efbdaaeb-9a77-40e3-9e24-6339cfa3e73b>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00255.warc.gz"} |
Formula to return AM or PM
I am in need of a formula to look at a cell containing a time value (24H) and return whether the it is an AM or PM time.
Best Answer
• Is it formatted as 1400 and 0105? If so, I would think that you could do
=IF(Time@row < 1200, "AM", "PM")
• Is it formatted as 1400 and 0105? If so, I would think that you could do
=IF(Time@row < 1200, "AM", "PM")
• Thank you. I give it a try and see if it does what I need.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/100166/formula-to-return-am-or-pm","timestamp":"2024-11-10T22:26:20Z","content_type":"text/html","content_length":"397174","record_id":"<urn:uuid:16be225b-76f0-40b3-abb4-1ce0fd026153>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00885.warc.gz"} |
Lesson Narrative
In this lesson, students make connections between zeros of a function and solutions to associated equations. In particular, they recognize that the zeros they find from graphing technology do not
always provide exact solutions. Next, students match expressions in the form \((ax+b)(cx+d)\) to their associated functions in standard form. In the associated Algebra 1 lesson, students examine
quadratic expressions of the form \(ax^2 + bx + c\) for which \(a \neq 0\) and the factored form of the expressions. The work in this lesson supports students by giving them some additional
background in the relationships between the 2 forms. Students must construct viable arguments and critique the reasoning of others (MP3) when they describe why they matched a factored expression to
one in standard form. They also attend to precision (MP6) when they describe why their choice of option does not belong using mathematically accurate language.
Learning Goals
Teacher Facing
• Recognize equations in expanded form and standard form are equivalent
Student Facing
• Let’s explore zeros on a graph
Additional Resources
Google Slides For access, consult one of our IM Certified Partners.
PowerPoint Slides For access, consult one of our IM Certified Partners. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/4/7/10/preparation.html","timestamp":"2024-11-03T05:41:42Z","content_type":"text/html","content_length":"75662","record_id":"<urn:uuid:dc78401a-2922-4627-8814-d8648cd1ef01>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00151.warc.gz"} |
Coq devs & plugin devs
I have a proof where I can step through all the goals, but when I reach Qed, it hangs. I waited for 3 hours, and it was never completed. I checked this issue on both Coq versions 8.17 and 8.18.0.
Running Show Proof results in a relatively small output (~180 lines). This appears to be a serious bug. Should I file a report on the GitHub bug tracker? The environment to reproduce it is rather
heavy, with many dependencies. My second question is whether there is a possible workaround?
Does slowness correlate with size of proof object? Because my proof object (just before Qed) is very small and quite trivial.
No. Not until your proof object is hundreds or thousands of megabytes in size
Here is mine: https://gist.github.com/vzaliva/a30bfff4d277ace76c5fc94348b773ba
This is output of Show Proof. Just before Qed.
That is not the full proof object, you have .... Try Set Printing Depth 10000000000
You also want Set Printing All to see all the hidden terms
The correlation with proof object size is dwarfed by the time it takes to run conversion, universe checking, and, occasionally, guard checking. Of these, only guard checking has any sort of
meaningful dependency on proof term size, but it's not always monotonic.
(Also, the relevant measure of proof term size is not lines but words)
Ok, with depth it is 35K lines :)
Right. I just giving a quick assessment of the size from my emacs output buffer line count.
Yeah, the proof object you pasted initially is just from the first 9 calls to destruct
I was brute-forcing proof of two functions equivalence which had many match statements which indeed caused explosion of brahcnes
but most of them I was automatically proving by finding a contradiction.
I guess there is no easy way to decrease number of destructs
I will leave it overnight to see if it finishes the Qed and will try to see tomorrow morning what could be done to speed things up following the link you sent
It might be of more help if you post the proof. I had such issues before and I always found ways around them. The link Jason initially posted is currently the best resource of hints on this I am
aware of.
Usually what made Qed slow for me was doing a change in a hypothesis - using replace instead can result in astronomic speed up factors in Qed time. If a few rules are observed, Qed should not take
longer than the construction of the proof term.
This topic was moved to #Coq users > Coq hangs on Qed by Karl Palmskog.
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/Coq.20hangs.20on.20Qed.html","timestamp":"2024-11-05T16:13:49Z","content_type":"text/html","content_length":"13631","record_id":"<urn:uuid:55989355-dd2b-48b2-b4a9-154264bc6e0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00781.warc.gz"} |
Abstract: Over the last few decades, substantial progress has been made is developing a rigorous mathematical theory of ring polymers (like DNA minicircles and mitochondrial DNA), mostly be
researchers at the interface between knot theory and biology. Recently, chemists working on developing novel materials have made substantial progress in synthesizing so-called “topological polymers”
which are modeled on more complicated graphs, including lassos, θ-curves, and even $K_{3,3}$ and $K_4$. Predicting the material properties of these polymers in solution requires a mathematical theory
of random embeddings of graphs. Such a theory was developed by James, Guth, and Flory [1,2,3] in the 20th century to study elasticity, but only with simple Gaussian interactions between monomers.
This talk describes a generalization of that theory [4] which can handle arbitrary interaction potentials, including freely-jointed networks and steric interactions.Joint with Jason Cantarella,
Tetsuo Deguchi, and Erica Uehara. [1] James, H.M., Guth, E. 1943 Theory of the elastic properties of rubber. J. Chem. Phys. 11, 455–481. [2] James, H.M. 1947 Statistical properties of networks of
flexible chains. J. Chem. Phys. 15, 651–668. [3] Flory, P.J. 1976 Statistical thermodynamics of random networks. Proc. R. Soc. Lond. A 351, 351–380. [4] Cantarella, J., Deguchi, T., Shonkwiler, C.,
Uehara, E. 2022 Random graph embeddings with general edge potentials. arXiv: 2205.09049.
Keywords: topological polymers, random embeddings, graphs. | {"url":"https://seminargeotop-a.com/event-talks/114","timestamp":"2024-11-07T09:51:10Z","content_type":"text/html","content_length":"67022","record_id":"<urn:uuid:1ef8b128-331e-47d4-ab31-fc8f067c3daa>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00414.warc.gz"} |
First grid / CSS: Layout on the Grid
Once you have the basic terminology figured out, it's time to create your first grid. In this lesson, we'll create a simple grid of 12 columns and 12 rows.
Let's start making the grid. As in the case of the Flex module, we need a container. Let's create a container with the class grid-12:
<section class="grid-12"></section>
Next, we turn a section into a grid container, using the display property with the grid value.
It won't produce immediate visible results. So, let's add three child elements inside our container, give them a different design, and use these blocks as an example of a simple grid:
By not specifying the number of columns and rows, we ended up with a grid with three rows and one column. It is not usually the result we expect from grids. If that's the structure you want, you may
not need grids. Sometimes, it's easier not to let the browser process what it doesn't have to process.
Another possible value for the display property is inline-grid. Everything will be the same as with a grid inside this container, but the container will behave like an inline element. It'll take up
exactly as much space as necessary.
Other elements in the flow will be able to streamline it as needed and as space permits:
Creating a grid
It's time to create our first grid. Here we use the grid-template-columns and grid-template-rows properties. The first property is responsible for the size of the columns, and the second for the size
of the rows. They can take many different values. Here are just a few of them:
• Any CSS units you know. These can be px, em, rem, and so on
• The min-content value. At this value, the column width will equal the minimum possible width. It depends on the content within the columns.
• The max-contentvalue. The opposite of the previous value. The column width will equal the maximum possible, considering the content of the columns
• The minmax(min, max)value. Here we see an integer function taking two values: the minimum and maximum size. In other words, we set boundaries within which the browser chooses the width
• The auto value. The browser automatically adjusts all the columns so that the largest element in our grid fits snugly.
There are several other values, which we'll look at below.
Once you know the possible values for the strip, you may wonder where the specific number of columns or rows is specified. The point is that the grid-template-columns and grid-template-rows
properties can take multiple values, separated by a space. Each such value is the size of one strip. In other words, to create a grid with 12 columns and 12 rows, we can specify values in each
property 12 times.
Create an even grid with square cells. Let's make them 20 pixels each. Let's take the previous example and add new properties:
<section class="grid-12">
<div class="grid-element bg-gray"></div>
<div class="grid-element bg-red"></div>
<div class="grid-element bg-blue"></div>
.grid-12 {
display: grid;
grid-template-columns: 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px;
grid-template-rows: 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px;
Now we've removed the height values for the container. It is because we calculate the width and height values from the size of the strip. There is no need to set these values additionally if you
don't need a specific container size:
Note what happened to the elements inside the grid. Both grid-template-columns and grid-template-rows properties occupied the entire screen width available before we set them. Now each property
occupies one cell with a size of 20 pixels by 20 pixels. It is easy to check in a web inspector, such as Chrome DevTools.
You can open it and hover your mouse over any of these elements. You'll see the entire grid and how much space each item has here:
You don't have to use only one version of the values inside the grid-template-columns and grid-template-rows properties. You can set a unique value for each strip like this:
.grid-12 {
display: grid;
grid-template-columns: 20px 3em minmax(20px, 10em) auto 20px 8% 20px 20px 20px 10em 20px 20px;
grid-template-rows: 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px 20px;
Pay attention to the size of the strips here.
It is tedious to specify each column separately, especially if they're the same. You easily may make a mistake and get a grid you weren't expecting. To avoid repeating the same values, the
grid-template-columns and grid-template-rows have a specific repeat value. More precisely, it's a function that takes two values:
1. How many times to repeat the value?
2. What value needs to be repeated?
Let's rewrite our 12-column grid using the repeat function:
.grid-12 {
display: grid;
grid-template-rows: repeat(12, 20px);
grid-template-columns: repeat(12, 20px);
The result is the same, but there's much less code, and it is easier to read. Which is great :)
The repeat function is one of the possible values of the grid-template-columns and grid-template-rows properties. So it's possible to use it several times or combine it with other values. Let's
assume that the first six cells vertically and horizontally need 20 pixels each, and the rest need 30 pixels. Then the CSS can take the following form:
.grid-12 {
display: grid;
grid-template-rows: repeat(6, 20px) repeat(6, 30px);
grid-template-columns: repeat(6, 20px) repeat(6, 30px);
Now, the grid in the web inspector will look like this:
In addition to roughly specifying the number of columns we want to repeat, we can use several other values:
• The auto-fill value. The browser will repeat the columns as many times as it can fit. If the container is limited in width, the browser will place as many columns as possible without exceeding
the container size.
• The auto-fit value. It is almost the same as the previous one, apart from one thing. If there's free space left in the container after we place all columns and elements, the browser will
automatically compress all other strips to zero. It'll throw them out:
Be sure to look at what the grid looks like in the web inspector. You'll see auto-fill results in a 16x5 grid. Whereas, with the auto-fit value, the grid is 3x1. The browser just threw out unneeded
strips with no elements in them.
Fraction Units
No, this is not about politics. In creating CSS Grid Layout, the developers paid close attention to adaptability. Now you adapt any web page if you care about the users.
So how do you adapt the grid? You can make do with relative units of measurement. This way can lead to more problems, especially if the number of strips in the grid changes. Once the number of strips
has changed, you also need to change all the values for the size of the columns and rows. Not the most convenient way.
To solve this problem, the CSS Grid Layout standard introduced a new unit of measurement — fraction units. It allows you to specify how much free space should be occupied by the grid strip.
This unit works on the same principle as flex-grow, which we studied in the CSS: Flex course. Essentially, we're saying how many parts a cell should take relative to other parts.
Let's make a screen-wide grid with 12 columns and 12 rows. We should automatically determine the size of these strips based on the current viewport resolution. Now we have fr units, so the task is
super simple:
.grid-12 {
display: grid;
grid-template-rows: repeat(12, 1fr);
grid-template-columns: repeat(12, 1fr);
width: 100vw;
height: 100vh;
Do it yourself
Using the lesson materials, create a 24-column grid. Try different values for grid-template-columns and grid-template-rows
Are there any more questions? Ask them in the Discussion section.
The Hexlet support team or other students will answer you.
For full access to the course you need a professional subscription.
A professional subscription will give you full access to all Hexlet courses, projects and lifetime access to the theory of lessons learned. You can cancel your subscription at any time.
Get access | {"url":"https://hexlet.io/courses/css-grid/lessons/first-grid/theory_unit","timestamp":"2024-11-11T23:50:17Z","content_type":"text/html","content_length":"69213","record_id":"<urn:uuid:7b590667-f657-4099-9c69-5b38d877d539>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00599.warc.gz"} |
Presentation of the laboratory
LAGA is a laboratory associated with the CNRS (UMR 7539). It is attached to the Institut Galilée, part of the Sorbonne Paris Nord University, and to the University of Paris 8. LAGA is also associated
with several LaBeX: SMP, INFLAMEX and MME-DII.
It includes around 90 researchers and teacher-researchers (including around ten CNRS researchers), 7 ITA and BIATSS staff, more than 50 doctoral students and it receives more than thirty foreign
visitors and post-docs each year.
The main research themes currently being developed within the laboratory are as follows:
Arithmetic, algebraic geometry, number theory, category theory, algebraic topology, homotopy theory, representation theory, dynamical systems, ergodic theory, harmonic analysis, linear and nonlinear
partial differential equations, microlocal analysis, mathematical physics, spectral theory, numerical analysis, probability and statistics, stochastic analysis, coding and cryptography, image
The laboratory, headed by Grégory Ginot, is structured into eight research teams:
The scientific activity of the laboratory is notably concretized by permanent or episodic seminars, as well as by the existence of the Mathematics Library. Finally, the laboratory aims to offer solid
supervision to doctoral students from the Masters of the mathematics department of the University of Paris 13, other Masters of mathematics from the Paris region or the provinces, and possibly
similar training abroad.
Last (pre)publications | {"url":"https://www.math.univ-paris13.fr/en/laboratoire-analyse-geometrie-et-applications-english/","timestamp":"2024-11-02T21:10:09Z","content_type":"text/html","content_length":"59642","record_id":"<urn:uuid:085bbb19-e848-4eff-89f8-00751dac1916>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00776.warc.gz"} |
Convert inches to feet ( in to ft )
Last Updated: 2024-11-05 03:35:18 , Total Usage: 420193
The conversion from inches to feet is a common task in measurement, especially in regions where the imperial system is used. It's a straightforward process but essential for understanding and
comparing lengths and heights in everyday situations.
Historical Context and Significance
Inches and feet are both units in the imperial system of measurement, which is primarily used in the United States. Historically, an inch was based on the width of a human thumb, while a foot was
based on the length of a human foot. Today, these units are standardized, with 12 inches making up one foot.
Conversion Formula
The formula to convert inches to feet is:
$$ \text{Feet} = \frac{\text{Inches}}{12} $$
This formula is derived from the fact that one foot equals 12 inches.
Example Calculation
Let's convert 36 inches to feet.
Using the formula: $$ \text{Feet} = \frac{36}{12} = 3 $$
So, 36 inches is equal to 3 feet.
Why This Conversion Matters
This conversion is widely used in several contexts:
1. Construction and Carpentry: Accurate measurements are crucial, and often dimensions are given in inches but need to be understood or converted into feet.
2. Fashion and Textile Industry: For measuring body dimensions and fabric lengths.
3. General Everyday Use: For people living in countries using the imperial system, converting inches to feet is a part of daily life, such as measuring height.
Common Questions (FAQs)
1. Why are there 12 inches in a foot?
□ The number 12 was chosen historically due to its divisibility by several numbers (2, 3, 4, 6) which made calculations easier in times before calculators.
2. Are inches and feet used internationally?
□ The imperial system, including inches and feet, is primarily used in the United States. Most other countries use the metric system.
3. How can I easily convert inches to feet and vice versa?
□ To convert inches to feet, divide by 12. To convert feet to inches, multiply by 12.
In conclusion, converting inches to feet is a fundamental skill in everyday life and various professional fields in countries that use the imperial system. This conversion helps in achieving accurate
measurements and ease in understanding and communicating dimensions. | {"url":"https://calculator.fans/en/tool/in-to-ft-convertor.html","timestamp":"2024-11-06T18:35:11Z","content_type":"text/html","content_length":"12266","record_id":"<urn:uuid:d6ee88c1-5db1-425a-95c3-8d82580b2961>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00560.warc.gz"} |
How Much Water Evaporates From the Earth Every Second?
Most recent answer: 07/23/2016
hoe much water evaporates from earth in every second
- Anomitra Das (age 16)
Malda, West Bengal, India
Since we know that water is not accumulating in the atmosphere, because whenever it tries we get rain instead, the water evaporating from the Earth must be equal to the amount falling from the sky.
I found a collection of sources for the amount of rain that falls on the Earth every year:
Using the most recently published figure of annual worldwide rainfall of 5.36 x10^14 cubic meters of water per year, the next step is to do unit conversion to get the volume of water per second.
(5.36x10^14 m^3 / 1 [S:year:S]) * ( 1 [S:year:S]/ 365 [S:days:S]) * ( 1 [S:day :S]/ 24 [S:hours:S]) * ( 1 [S:hour :S]/ 60 [S:minutes:S])* (1 [S:minute:S]/60 seconds) = 17 million cubic meters per
or the equivalent of 78 Amazon Rivers flowing up into the sky
Sheldon S.
(published on 07/23/2016) | {"url":"https://van.physics.illinois.edu/ask/listing/43286","timestamp":"2024-11-07T00:57:46Z","content_type":"text/html","content_length":"28351","record_id":"<urn:uuid:b3bc2261-3f24-4776-a103-c11252173dfd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00550.warc.gz"} |
32-Bit HDR Images | Photoshop - Advanced
About this lesson
Combine a series of images into a single High Dynamic Range image using Lightroom and Photoshop
Exercise files
There are no related exercise files for this lesson, or we cannot provide them due to copyright issues.
Quick reference
Topic : 32Bit HDR
Combine a series of images into a single High Dynamic Range image using Lightroom and Photoshop
When to use
With a bit of preparation and a tripod you can take a series of photos at different light levels and combine all the best bits to create a single HDR image. This is great for shots that include high
contrasts of bright and dark parts (shadows, sunny exteriors).
Lightroom to Photoshop Combining Images
1. Open your images (taken at different exposures) in Lightroom.
2. With all images selected, right click and go to ‘Edit In/Merge to HDR Pro’ in Photoshop.
3. Photoshop will align the images and combine the exposure information.
4. In the Photoshop dialogue box change the mode to ‘32 Bit’.
5. Click OK.
6. After conversion click save.
Lightroom Editing
1. Switch to the new image in Lightroom.
2. Edit parameters like shadows, highlights, vibrance and clarity.
3. Use the selective adjustment tool for more editing.
4. Select the color with the tool and drag up or down for effect.
Login to download
• 00:04 Today I'm going to be showing you guys how Adobe Photoshop Lightroom 4.1 and
• 00:09 Adobe Photoshop CS6 can work hand in hand to produce stunning HDR images.
• 00:15 In the past, I would typically use one or the other: Lightroom with a plugin, or
• 00:19 directly merge the photos to Photoshop's HDR Pro.
• 00:23 So if I could only use one application in the past,
• 00:26 why the heck would I wanna use two?
• 00:28 Well there's a very good reason, 32 bit editing.
• 00:31 With Lightroom 4.1 we now have the ability to edit 32 bit
• 00:36 HDR images, which eliminates the need for an external plug-in, and
• 00:40 you're more likely to end up with a much more natural result.
• 00:44 Let's go ahead and take a look.
• 00:46 So here in Lightroom I have three images which I took at Oxford University.
• 00:50 As you can see each image has been taken at a different exposure to
• 00:54 capture the shadows, the midtones, and
• 00:56 the highlights, which a single image simply cannot do.
• 01:00 I'm only working with three images, but if you want an even higher dynamic range you
• 01:04 can easily use five brackets if your camera supports it.
• 01:07 So with all three images selected, I'm gonna right-click on any of them,
• 01:11 go to Edit In, and
• 01:13 then select Merge to HDR Pro in Photoshop, which has been there for a while.
• 01:17 At this point, Photoshop will spring into action and start the process of not only
• 01:21 combining the three exposures, but it will also attempt to line up the images.
• 01:26 Sometimes, especially if you're like me and
• 01:28 don't use a tripod, don't tell anyone, your photos may be slightly misaligned.
• 01:33 Lining them up will give you a much cleaner result.
• 01:37 And here's the HDR Pro window, which many of you may be familiar with.
• 01:40 But instead of going through all the sliders to create a 16 bit HDR image,
• 01:45 we're actually gonna switch it over to 32 bit here at the top.
• 01:49 This will simply merge all three of the images and
• 01:51 the data, allowing us to bring it back into Lightroom and
• 01:55 use all the wonderful adjustments that Camera Raw has to offer.
• 01:58 Which, again, should give you a much more natural result than HDR toning would.
• 02:03 Now don't worry too much about the White Point slider at the top.
• 02:06 It's simply there for preview purposes.
• 02:08 I'll click OK to finalize the merge, which could take a few minutes depending on
• 02:12 your computer and the amount of exposures that you're working with.
• 02:16 And once the merge is complete all you need to do is save.
• 02:19 A simple Cmd or Ctrl+S will pop your new 32 bit image in
• 02:23 the same location that the other images are stored.
• 02:27 Let's hop back over to Lightroom where our new image should be waiting for us.
• 02:31 And there it is.
• 02:32 From here you can use the same adjustments that you've been using previously, but
• 02:35 because we have a 32 bit image made up of three exposures there's a ton of
• 02:40 data to work with.
• 02:42 Take a look as I increase and decrease the exposure.
• 02:45 Obviously you wouldn't need to go this extreme, but
• 02:47 it gives you an idea as to what's actually available to you.
• 02:51 I'll leave the exposure increased ever so slightly.
• 02:54 I'm gonna dump the highlights to get rid of some of the unnecessary bright areas,
• 02:58 and then I'm gonna increase the shadows to allow us to see into some of
• 03:01 the more shaded areas of the photo.
• 03:03 And, of course, what's an edit without increasing the clarity?
• 03:07 This will increase the contrast of your midtones which looks great on
• 03:10 textures such as bricks and stone.
• 03:13 Finally, the overall color of this image is a little bit dull so
• 03:16 increasing the Vibrance a touch should do the trick.
• 03:19 I'm not gonna touch the Saturation slider as the stones have a lot of
• 03:22 yellow in them.
• 03:23 Increasing the saturation on images that contain a lot of yellow or
• 03:26 skin tones can result in some unnatural effects.
• 03:29 Sliding down the Develop Module,
• 03:31 the Selective Adjustment Tool is also available to us.
• 03:34 I love this tool.
• 03:36 You're able to selectively adjust the hue,
• 03:38 saturation, and luminous of any part of your image.
• 03:42 For example, if I wanted a slightly darker sky, under Luminance,
• 03:46 I can activate the Selective Adjustment Tool and then click and
• 03:49 drag on the color that I want affected, in this case the blue of the sky.
• 03:54 Dragging up will brighten it, while dragging it down will darken it.
• 03:58 The same goes for hue and saturation.
• 04:00 Let's say I wanted to slightly decrease the yellow tint in the stones.
• 04:04 Selecting Saturation will allow me to use the same tool to increase or
• 04:07 decrease the saturation of the stones and even the grass, if I wish.
• 04:14 Finally let's slide down to Lens Corrections to deal with some of
• 04:17 that chromatic aberration that I see in the trees.
• 04:19 Also new in Lightroom 4.1, I can use the Fringe Color Selector to sample any
• 04:24 blue or green fringes that may be present and then adjust the sliders if necessary.
• 04:29 And that should complete the edit.
• 04:31 Let's take a look at the final result in comparison to the original images.
• 04:35 We started with an overexposed image to capture the shaded areas of the scene,
• 04:39 an underexposed image to capture the lighter areas like the roof of
• 04:43 the building and a neutral image that captured everything in between.
• 04:47 And merging all three images into a 32-bit HDR file, and
• 04:51 performing some pretty basic edits, we're left with a beautiful photo that
• 04:55 captures a range that's more true to what the human eye sees.
• 04:58 So even though it may seem more convenient to use only one application,
• 05:03 utilizing all of your resources can leave you with a much more desirable result.
Lesson notes are only available for subscribers. | {"url":"https://www.goskills.com/Course/Photoshop-Advanced/Lesson/354/32-Bit-HDR-Images?autoplay=false","timestamp":"2024-11-02T18:04:08Z","content_type":"text/html","content_length":"88048","record_id":"<urn:uuid:1df788eb-95ba-4362-b127-ad0a79b2cc31>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00016.warc.gz"} |
Leetcode 27: Remove Element - Cse Nerd Leetcode Detailed Solutions
Leetcode 27: Remove Element
Category: Easy
Given an array nums and a value val, remove all instances of that value in-place and return the new length.
Do not allocate extra space for another array, you must do this by modifying the input array in-place with O(1) extra memory.
The order of elements can be changed. It doesn’t matter what you leave beyond the new length.
Example 1:
1 Input: nums = [3,2,2,3], val = 3
2 Output: 2, nums = [2,2]
3 Explanation: Your function should return length = 2,
4 with the first two elements of nums being 2.
5 It doesn't matter what you leave beyond the returned length.
6 For example if you return 2 with nums = [2,2,3,3] or nums = [2,2,0,0],
7 your answer will be accepted.
Example 2:
1 Input: nums = [0,1,2,2,3,0,4,2], val = 2
2 Output: 5, nums = [0,1,4,0,3]
3 Explanation: Your function should return length = 5,
4 with the first five elements of nums containing 0, 1, 3, 0, and 4.
5 Note that the order of those five elements can be arbitrary.
6 It doesn't matter what values are set beyond the returned length.
Solution Approach
In this problem of ‘Remove Element’, we have to remove the elements in place and then return the size of the array after the removal. The condition is that we do not have to use any extra space.
This problem can be solved by utilizing two pointers. The first will loop through the array and the second is where we insert the target value. (i.e value which is not in the ‘Val’ variable). The
second pointer will ignore the target values. It will move when non-target values are encountered. Whenever we find non-target values we will insert them in place of the target variable values in the
array. The second variable will be returned as the answer because it stores the number of non-target array elements.
As we are looping through the array only once, hence our time complexity will be O(n). No extra space is required, therefore the space complexity is O(1).
Coming to the code, of the same it’s pretty straightforward. We will initialize a variable (let’s call it index) with 0. Then we loop through the array with ‘i’ as an iterator. Whenever we find an
array element that is not our targeted ‘Val’ value, then we will change the array element at the index to that value. In the end, we will return the index as the answer.
Solution code
1 class Solution {
2 public int removeElement(int[] nums, int val) {
3 int index = 0;
4 for (int i = 0; i < nums.length; i++) {
5 if (nums[i] != val)
6 nums[index++] = nums[i];
7 }
8 return index;
9 }
10 }
For more Leetcode explained solutions visit Leetcode Solutions.
If you like capture the flag challenges visit here.
Check out my socials below in the footer. Feel free to ask any doubts in the comment section or contact me via the Contact page I will surely respond. Happy Leetcoding 🙂
Leave a Comment | {"url":"https://csenerd.com/leetcode-27-remove-element/","timestamp":"2024-11-05T05:47:09Z","content_type":"text/html","content_length":"81407","record_id":"<urn:uuid:708d5cfd-00e6-46e7-a873-eb2009bbfc24>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00050.warc.gz"} |
Ultraconservative online algorithms for multiclass problems
In this paper we study online classification algorithms for multiclass problems in the mistake bound model. The hypotheses we use maintain one prototype vector per class. Given an input instance, a
multiclass hypothesis computes a similarity-score between each prototype and the input instance and then sets the predicted label to be the index of the prototype achieving the highest similarity. To
design and analyze the learning algorithms in this paper we introduce the notion of ultraconservativeness. Ultraconservative algorithms are algorithms that update only the prototypes attaining
similarity-scores which are higher than the score of the correct label’s prototype. We start by describing a family of additive ultraconservative algorithms where each algorithm in the family updates
its prototypes by finding a feasible solution for a set of linear constraints that depend on the instantaneous similarity-scores. We then discuss a specific online algorithm that seeks a set of
prototypes which have a small norm. The resulting algorithm, which we term MIRA (for Margin Infused Relaxed Algorithm) is ultraconservative as well. We derive mistake bounds for all the algorithms
and provide further analysis of MIRA using a generalized notion of the margin for multiclass problems.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 2111
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Other 14th Annual Conference on Computational Learning Theory, COLT 2001 and 5th European Conference on Computational Learning Theory, EuroCOLT 2001
Country/Territory Netherlands
City Amsterdam
Period 7/16/01 → 7/19/01
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Ultraconservative online algorithms for multiclass problems'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/ultraconservative-online-algorithms-for-multiclass-problems-2","timestamp":"2024-11-05T03:02:10Z","content_type":"text/html","content_length":"55226","record_id":"<urn:uuid:665b7e77-aff5-47b9-8b2b-b5e72fcda93c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00041.warc.gz"} |
Explain Pascal’s Law
If the pressure is applied on any point of an enclosed liquid or gas it is transmitted in all directions. Pascal stated a law about this transmission of pressure.
Pascal’s law
External pressure applied to any portion of a liquid or a gas enclosed in a container is equally transmitted in all directions in the liquid or the gas without any trace of diminution and acts
perpendicularly on the surface of the container in contact with the liquid or gas.
Fig: Mathematical explanation of Pascal’s law
The mathematical explanation of Pascal’s law: the principle of multiplications of force
On any portion of a confined liquid if force is applied by a smaller piston, then forces of greater magnitudes are exerted on the pistons of greater cross-sectional area This is known as the
principle of multiplication of force.
Let, C[1] and C[2] be two cylinders (Figure) and A[1] and A[2] be their cross-sectional areas respectively. The two cylinders are connected by a pipe. There is an airtight piston in each cylinder.
The two cylinders are filled up with a liquid. Now a force F applied to the smaller piston generates a pressure F[1]/A[1]. According to Pascal’s law, this pressure is transmitted in all directions
through the liquid. Therefore the upward pressure exerted on the larger piston is F[1]/A[1]. Because of this pressure, the larger piston experiences an upward force equal to (F[1]/A[1]) x A[2]. If
the upward force of larger piston is F[2],
Then, F[2] = (F[1]/A[1]) x A[2]
So, (F[2] / F[1]) = (A[2 ]/ A[1])
So, greater is the cross-sectional area of the larger piston, the greater is the force exerted on it. If the cross-sectional area of the larger one is 100 times greater than that of the smaller one,
then a force of 1N applied to the smaller one will produce an upward force of 100N on the larger piston. | {"url":"https://qsstudy.com/explain-pascals-law/","timestamp":"2024-11-05T12:24:49Z","content_type":"text/html","content_length":"23612","record_id":"<urn:uuid:8e78f357-aba1-4bb0-bd97-513be02a5b97>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00606.warc.gz"} |
A nonlinear mixed-effects approach is developed for disease progression models that
A nonlinear mixed-effects approach is developed for disease progression models that incorporate variation in age in a Bayesian framework. in the entire study group and s ∈ [0 T] is the time spent in
Sp. This is an extension of the sensitivity of Kim and Wu (2014) where the sensitivity depends on the sojourn time and the time spent in the preclinical state. Note that sojourn time T is a random
variable in this model. Here in general the parameters α and γ are responsible for the maximum value and for the rate of the sensitivity respectively while the parameter β explains how the behavior
of the sensitivity changes with age. Namely the maximum sensitivity increases as the parameter α increases. When s/T is close to zero sensitivity increases rapidly if γ < 1 while sensitivity
increases gradually if γ > 1. Sensitivity is an increasing function of Prednisolone acetate (Omnipred) age when the parameter β is positive (e.g. see Figure 2). Figure 2 The sensitivity of JHLP and
HIP Let Dij be the probability of an individual correctly diagnosed at the jth scheduled exam given at ti j?1 and started the screening exam at age ti 0 (i.e. the ith age group) and Iij the
probability of an interval case in (ti j?1 ti j). These two probabilities for j = 1 2 … Ni are: is the survivor function of the sojourn time. The log-logistic distribution was used to model the
sojourn time (Wu et al. 2005 and of JHLP and HIP and the estimate of HIP. Table 1 Estimates of Fixed-effects and Mixed-effects using JHLP and HIP data Estimates of the variance-covariance matrix Σ of
ME-DM are shown in Table 2. Since only log (α) β log (γ) and μ are considered as random-effects the size of Σ is four by four. For both JHLP and HIP data there is greater variation in the parameters
log (α) and log (γ) than these in other parameters indicating that sensitivity is influenced by age at diagnosis. Forest plots of each individual-level Prednisolone acetate (Omnipred) estimate of
ME-DM are plotted in Figure 1. In case of the parameters β and μ the empirical means of the individual-level estimates are very close to that of the population-level estimate for both JHLP and HIP
data. On the other hand we can see a larger variation of the individual-level estimates of log (α) and log (γ). These imply that the parameters α and β have a large influence on Prednisolone acetate
(Omnipred) LASS2 antibody age so does sensitivity. The individual-level estimates of each age at diagnosis can be found in Supplementary Information Tables S1 and S2. Figure 1 The forest plots of
individual-level estimates of ME-DM Table 2 Estimates of variance-covariance matrices of ME-DM using JHLP and HIP data The developed sensitivity models depend on age at diagnosis the time spent in
the preclinical state and the sojourn time resulting in a function of age and the proportion of time spent in the preclinical state to the sojourn time. Note that the average age in Equation (1) is
globally set Prednisolone acetate (Omnipred) to 55 years for all age groups in both JHLP and HIP data. Figure 2 shows the posterior sensitivities estimated by FE-DM and ME-DM on JHLP and HIP data.
The population-level estimates of FE-DM are less than one (i.e. log (of JHLP and HIP are positive and negative respectively. In general both JHLP and HIP data show large differences in sensitivity
between FE-DM and ME-DM. The individual-level posterior sensitivities are shown in Supplementary Information Figures S3 and S4. In particular these predicted sensitivities show significant variations
among age groups which might be resulted from the large variations in parameters log (α) and log (γ) in Table 2. Figure 3 shows the posterior transition probability estimated by FE-DM and ME-DM. The
estimates of ME-DM for both JHLP and HIP are larger than these of FE-DM resulting that the modes of FE-DM are a little smaller than these of ME-DM (61 vs. 72 years and 51 vs. 73 years respectively
for JHLP and HIP). The individual-level variation of the transition probability can be seen in Supplementary Information Figure S5. The variation in age is larger in JHLP data than in HIP data.
Figure 3 The transition probability of JHLP and HIP The posterior sojourn Prednisolone acetate (Omnipred) time distributions are depicted in Figure 4. As expected by that the 95% CIs of the estimates
of HIP are not overlapped the sojourn time distributions of HIP are very different between FE-DM and ME-DM of HIP. The modes of sojourn time in HIP are 1.01 and Prednisolone acetate (Omnipred) 15.15
years for FE-DM and ME-DM respectively while these of JHLP are 21.21 and 38.38 years for FE-DM and ME-DM. | {"url":"http://researchensemble.com/2016/08/29/a-nonlinear-mixed-effects-approach-is-developed-for-disease-progression-models-that/","timestamp":"2024-11-14T18:52:15Z","content_type":"text/html","content_length":"58891","record_id":"<urn:uuid:646b5ad0-e1dd-4a4d-9d6d-d1ea9d837c69>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00197.warc.gz"} |
type, public :: tem_cylinder_type
This type provides information to create cylinder geometry
Type Visibility Attributes Name Initial
real(kind public :: vec(3) vector defining length and axis of cylinder
real(kind public :: radius
real(kind public :: origin(3)
logical, public :: only_surface To choose what to do with intersection of this object if only_surface = true than the only the surface of the object is intersected if
only_surface = false then the whole object is intersected default is set to false
Source Code
type tem_cylinder_type !> vector defining length and axis of cylinder
real(kind=rk) :: vec(3) real(kind=rk) :: radius !< radius of the cylinder
real(kind=rk) :: origin(3) !< origin of the cylinder
!> To choose what to do with intersection of this object
!! if only_surface = true than the only the surface of the object
!! is intersected
!! if only_surface = false then the whole object is intersected
!! default is set to false
logical :: only_surface end type tem_cylinder_type | {"url":"https://geb.inf.tu-dresden.de/doxy/treelm/type/tem_cylinder_type.html","timestamp":"2024-11-09T00:04:04Z","content_type":"text/html","content_length":"34764","record_id":"<urn:uuid:162e0423-86d1-4143-8a04-01967151eed8>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00653.warc.gz"} |
Calculus (3rd Edition) Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.3 Polar Coordinates - Exercises - Page 619 40
Since $r^2=\cos (2\theta)$, then we have $$r^2=\cos^2\theta-\sin^2\theta=\frac{x^2}{r^2}-\frac{y^2}{r^2}.$$ Hence, $r^4=x^2-y^2$, which can be written in the form $$(x^2+y^2)^2=x^2-y^2.$$
You can help us out by revising, improving and updating this answer.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"url":"https://www.gradesaver.com/textbooks/math/calculus/calculus-3rd-edition/chapter-12-parametric-equations-polar-coordinates-and-conic-sections-12-3-polar-coordinates-exercises-page-619/40","timestamp":"2024-11-03T14:04:11Z","content_type":"text/html","content_length":"68485","record_id":"<urn:uuid:c5f1bc7d-40aa-4ca4-ba1d-cf2a055058f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00821.warc.gz"} |
2.1: Pixels, Coordinates, and Colors
Last updated
Page ID
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\
evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\
newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}
\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}
{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}
[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}
{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\
#3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp
#2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\
wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\
newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}
{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\
bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
To create a two-dimensional image, each point in the image is assigned a color. A point in 2D can be identified by a pair of numerical coordinates. Colors can also be specified numerically. However,
the assignment of numbers to points or colors is somewhat arbitrary. So we need to spend some time studying coordinate systems, which associate numbers to points, and color models, which associate
numbers to colors.
Pixel Coordinates
A digital image is made up of rows and columns of pixels. A pixel in such an image can be specified by saying which column and which row contains it. In terms of coordinates, a pixel can be
identified by a pair of integers giving the column number and the row number. For example, the pixel with coordinates (3,5) would lie in column number 3 and row number 5. Conventionally, columns are
numbered from left to right, starting with zero. Most graphics systems, including the ones we will study in this chapter, number rows from top to bottom, starting from zero. Some, including OpenGL,
number the rows from bottom to top instead.
Note in particular that the pixel that is identified by a pair of coordinates (x,y) depends on the choice of coordinate system. You always need to know what coordinate system is in use before you
know what point you are talking about.
Row and column numbers identify a pixel, not a point. A pixel contains many points; mathematically, it contains an infinite number of points. The goal of computer graphics is not really to color
pixels—it is to create and manipulate images. In some ideal sense, an image should be defined by specifying a color for each point, not just for each pixel. Pixels are an approximation. If we imagine
that there is a true, ideal image that we want to display, then any image that we display by coloring pixels is an approximation. This has many implications.
Suppose, for example, that we want to draw a line segment. A mathematical line has no thickness and would be invisible. So we really want to draw a thick line segment, with some specified width.
Let’s say that the line should be one pixel wide. The problem is that, unless the line is horizontal or vertical, we can’t actually draw the line by coloring pixels. A diagonal geometric line will
cover some pixels only partially. It is not possible to make part of a pixel black and part of it white. When you try to draw a line with black and white pixels only, the result is a jagged staircase
effect. This effect is an example of something called “aliasing.” Aliasing can also be seen in the outlines of characters drawn on the screen and in diagonal or curved boundaries between any two
regions of different color. (The term aliasing likely comes from the fact that ideal images are naturally described in real-number coordinates. When you try to represent the image using pixels, many
real-number coordinates will map to the same integer pixel coordinates; they can all be considered as different names or “aliases” for the same pixel.)
Antialiasing is a term for techniques that are designed to mitigate the effects of aliasing. The idea is that when a pixel is only partially covered by a shape, the color of the pixel should be a
mixture of the color of the shape and the color of the background. When drawing a black line on a white background, the color of a partially covered pixel would be gray, with the shade of gray
depending on the fraction of the pixel that is covered by the line. (In practice, calculating this area exactly for each pixel would be too difficult, so some approximate method is used.) Here, for
example, is a geometric line, shown on the left, along with two approximations of that line made by coloring pixels. The lines are greately magnified so that you can see the individual pixels. The
line on the right is drawn using antialiasing, while the one in the middle is not:
Note that antialiasing does not give a perfect image, but it can reduce the “jaggies” that are caused by aliasing (at least when it is viewed on a normal scale).
There are other issues involved in mapping real-number coordinates to pixels. For example, which point in a pixel should correspond to integer-valued coordinates such as (3,5)? The center of the
pixel? One of the corners of the pixel? In general, we think of the numbers as referring to the top-left corner of the pixel. Another way of thinking about this is to say that integer coordinates
refer to the lines between pixels, rather than to the pixels themselves. But that still doesn’t determine exactly which pixels are affected when a geometric shape is drawn. For example, here are two
lines drawn using HTML canvas graphics, shown greatly magnified. The lines were specified to be colored black with a one-pixel line width:
The top line was drawn from the point (100,100) to the point (120,100). In canvas graphics, integer coordinates corresponding to the lines between pixels, but when a one-pixel line is drawn, it
extends one-half pixel on either side of the infinitely thin geometric line. So for the top line, the line as it is drawn lies half in one row of pixels and half in another row. The graphics system,
which uses antialiasing, rendered the line by coloring both rows of pixels gray. The bottom line was drawn from the point (100.5,100.5) to (120.5,120.5). In this case, the line lies exactly along one
line of pixels, which gets colored black. The gray pixels at the ends of the bottom line have to do with the fact that the line only extends halfway into the pixels at its endpoints. Other graphics
systems might render the same lines differently.
The interactive demo c2/pixel-magnifier.html lets you experiment with pixels and antialiasing. Interactive demos can be found on the web pages in the on-line version of this book. If you have
downloaded the web site, you can also find the demos in the folder named demos. (Note that in any of the interactive demos that accompany this book, you can click the question mark icon in the upper
left for more information about how to use it.)
All this is complicated further by the fact that pixels aren’t what they used to be. Pixels today are smaller! The resolution of a display device can be measured in terms of the number of pixels per
inch on the display, a quantity referred to as PPI (pixels per inch) or sometimes DPI (dots per inch). Early screens tended to have resolutions of somewhere close to 72 PPI. At that resolution, and
at a typical viewing distance, individual pixels are clearly visible. For a while, it seemed like most displays had about 100 pixels per inch, but high resolution displays today can have 200, 300 or
even 400 pixels per inch. At the highest resolutions, individual pixels can no longer be distinguished.
The fact that pixels come in such a range of sizes is a problem if we use coordinate systems based on pixels. An image created assuming that there are 100 pixels per inch will look tiny on a 400 PPI
display. A one-pixel-wide line looks good at 100 PPI, but at 400 PPI, a one-pixel-wide line is probably too thin.
In fact, in many graphics systems, “pixel” doesn’t really refer to the size of a physical pixel. Instead, it is just another unit of measure, which is set by the system to be something appropriate.
(On a desktop system, a pixel is usually about one one-hundredth of an inch. On a smart phone, which is usually viewed from a closer distance, the value might be closer to 1/160 inch. Furthermore,
the meaning of a pixel as a unit of measure can change when, for example, the user applies a magnification to a web page.)
Pixels cause problems that have not been completely solved. Fortunately, they are less of a problem for vector graphics, which is mostly what we will use in this book. For vector graphics, pixels
only become an issue during rasterization, the step in which a vector image is converted into pixels for display. The vector image itself can be created using any convenient coordinate system. It
represents an idealized, resolution-independent image. A rasterized image is an approximation of that ideal image, but how to do the approximation can be left to the display hardware.
Real-number Coordinate Systems
When doing 2D graphics, you are given a rectangle in which you want to draw some graphics primitives. Primitives are specified using some coordinate system on the rectangle. It should be possible to
select a coordinate system that is appropriate for the application. For example, if the rectangle represents a floor plan for a 15 foot by 12 foot room, then you might want to use a coordinate system
in which the unit of measure is one foot and the coordinates range from 0 to 15 in the horizontal direction and 0 to 12 in the vertical direction. The unit of measure in this case is feet rather than
pixels, and one foot can correspond to many pixels in the image. The coordinates for a pixel will, in general, be real numbers rather than integers. In fact, it’s better to forget about pixels and
just think about points in the image. A point will have a pair of coordinates given by real numbers.
To specify the coordinate system on a rectangle, you just have to specify the horizontal coordinates for the left and right edges of the rectangle and the vertical coordinates for the top and bottom.
Let’s call these values left, right, top, and bottom. Often, they are thought of as xmin, xmax, ymin, and ymax, but there is no reason to assume that, for example, top is less than bottom. We might
want a coordinate system in which the vertical coordinate increases from bottom to top instead of from top to bottom. In that case, top will correspond to the maximum y-value instead of the minimum
To allow programmers to specify the coordinates system that they would like to use, it would be good to have a subroutine such as
The graphics system would then be responsible for automatically transforming the coordinates from the specified coordinate system into pixel coordinates. Such a subroutine might not be available, so
it’s useful to see how the transformation is done by hand. Let’s consider the general case. Given coordinates for a point in one coordinate system, we want to find the coordinates for the same point
in a second coordinate system. (Remember that a coordinate system is just a way of assigning numbers to points. It’s the points that are real!) Suppose that the horizontal and vertical limits are
oldLeft, oldRight, oldTop, and oldBottom for the first coordinate system, and are newLeft, newRight, newTop, and newBottom for the second. Suppose that a point has coordinates (oldX,oldY) in the
first coordinate system. We want to find the coordinates (newX,newY) of the point in the second coordinate system
Formulas for newX and newY are then given by
newX = newLeft + ((oldX - oldLeft) / (oldRight - oldLeft)) * (newRight - newLeft))
newY = newTop + ((oldY - oldTop) / (oldBottom - oldTop)) * (newBotom - newTop)
The logic here is that oldX is located at a certain fraction of the distance from oldLeft to oldRight. That fraction is given by
((oldX - oldLeft) / (oldRight - oldLeft))
The formula for newX just says that newX should lie at the same fraction of the distance from newLeft to newRight. You can also check the formulas by testing that they work when oldX is equal to
oldLeft or to oldRight, and when oldY is equal to oldBottom or to oldTop.
As an example, suppose that we want to transform some real-number coordinate system with limits left, right, top, and bottom into pixel coordinates that range from 0 at left to 800 at the right and
from 0 at the top 600 at the bottom. In that case, newLeft and newTop are zero, and the formulas become simply
newX = ((oldX - left) / (right - left)) * 800
newY = ((oldY - top) / (bottom - top)) * 600
Of course, this gives newX and newY as real numbers, and they will have to be rounded or truncated to integer values if we need integer coordinates for pixels. The reverse transformation— going from
pixel coordinates to real number coordinates—is also useful. For example, if the image is displayed on a computer screen, and you want to react to mouse clicks on the image, you will probably get the
mouse coordinates in terms of integer pixel coordinates, but you will want to transform those pixel coordinates into your own chosen coordinate system.
In practice, though, you won’t usually have to do the transformations yourself, since most graphics APIs provide some higher level way to specify transforms. We will talk more about this in Section
Aspect Ratio
The aspect ratio of a rectangle is the ratio of its width to its height. For example an aspect ratio of 2:1 means that a rectangle is twice as wide as it is tall, and an aspect ratio of 4:3 means
that the width is 4/3 times the height. Although aspect ratios are often written in the form width:height, I will use the term to refer to the fraction width/height. A square has aspect ratio equal
to 1. A rectangle with aspect ratio 5/4 and height 600 has a width equal to \( 600 \times (5/4) \), or 750.
A coordinate system also has an aspect ratio. If the horizontal and vertical limits for the coordinate system are left, right, bottom, and top, as above, then the aspect ratio is the absolute value
(right - left) / (top - bottom)
If the coordinate system is used on a rectangle with the same aspect ratio, then when viewed in that rectangle, one unit in the horizontal direction will have the same apparent length as a unit in
the vertical direction. If the aspect ratios don’t match, then there will be some distortion. For example, the shape defined by the equation \( x^2 + y^2 = 9 \) should be a circle, but that will only
be true if the aspect ratio of the (x,y) coordinate system matches the aspect ratio of the drawing area.
It is not always a bad thing to use different units of length in the vertical and horizontal directions. However, suppose that you want to use coordinates with limits left, right, bottom, and top,
and that you do want to preserve the aspect ratio. In that case, depending on the shape of the display rectangle, you might have to adjust the values either of left and right or of bottom and top to
make the aspect ratios match:
We will look more deeply into geometric transforms later in the chapter, and at that time, we’ll see some program code for setting up coordinate systems.
Color Models
We are talking about the most basic foundations of computer graphics. One of those is coordi- nate systems. The other is color. Color is actually a surprisingly complex topic. We will look at some
parts of the topic that are most relevant to computer graphics applications.
The colors on a computer screen are produced as combinations of red, green, and blue light. Different colors are produced by varying the intensity of each type of light. A color can be specified by
three numbers giving the intensity of red, green, and blue in the color. Intensity can be specified as a number in the range zero, for minimum intensity, to one, for maximum intensity. This method of
specifying color is called the RGB color model, where RGB stands for Red/Green/Blue. For example, in the RGB color model, the number triple (1, 0.5, 0.5) represents the color obtained by setting red
to full intensity, while green and blue are set to half intensity. The red, green, and blue values for a color are called the color components of that color in the RGB color model.
Light is made up of waves with a variety of wavelengths. A pure color is one for which all the light has the same wavelength, but in general, a color can contain many wavelengths— mathematically, an
infinite number. How then can we represent all colors by combining just red, green, and blue light? In fact, we can’t quite do that.
You might have heard that combinations of the three basic, or “primary,” colors are sufficient to represent all colors, because the human eye has three kinds of color sensors that detect red, green,
and blue light. However, that is only an approximation. The eye does contain three kinds of color sensor. The sensors are called “cone cells.” However, cone cells do not respond exclusively to red,
green, and blue light. Each kind of cone cell responds, to a varying degree, to wavelengths of light in a wide range. A given mix of wavelengths will stimulate each type of cell to a certain degree,
and the intensity of stimulation determines the color that we see. A different mixture of wavelengths that stimulates each type of cone cell to the same extent will be perceived as the same color. So
a perceived color can, in fact, be specified by three numbers giving the intensity of stimulation of the three types of cone cell. However, it is not possible to produce all possible patterns of
stimulation by combining just three basic colors, no matter how those colors are chosen. This is just a fact about the way our eyes actually work; it might have been different. Three basic colors can
produce a reasonably large fraction of the set of perceivable colors, but there are colors that you can see in the world that you will never see on your computer screen. (This whole discussion only
applies to people who actually have three kinds of cone cell. Color blindness, where someone is missing one or more kinds of cone cell, is surprisingly common.)
The range of colors that can be produced by a device such as a computer screen is called the color gamut of that device. Different computer screens can have different color gamuts, and the same RGB
values can produce somewhat different colors on different screens. The color gamut of a color printer is noticeably different—and probably smaller—than the color gamut of a screen, which explain why
a printed image probably doesn’t look exactly the same as it did on the screen. (Printers, by the way, make colors differently from the way a screen does it. Whereas a screen combines light to make a
color, a printer combines inks or dyes. Because of this difference, colors meant for printers are often expressed using a different set of basic colors. A common color model for printer colors is
CMYK, using the colors cyan, magenta, yellow, and black.)
In any case, the most common color model for computer graphics is RGB. RGB colors are most often represented using 8 bits per color component, a total of 24 bits to represent a color. This
representation is sometimes called “24-bit color.” An 8-bit number can represent 28, or 256, different values, which we can take to be the positive integers from 0 to 255. A color is then specified
as a triple of integers (r,g,b) in that range.
This representation works well because 256 shades of red, green, and blue are about as many as the eye can distinguish. In applications where images are processed by computing with color components,
it is common to use additional bits per color component, to avoid visual effects that might occur due to rounding errors in the computations. Such applications might use a 16-bit integer or even a
32-bit floating point value for each color component. On the other hand, sometimes fewer bits are used. For example, one common color scheme uses 5 bits for the red and blue components and 6 bits for
the green component, for a total of 16 bits for a color. (Green gets an addition bit because the eye is more sensitive to green light than to red or blue.) This “16-bit color” saves memory compared
to 24-bit color and was more common when memory was more expensive.
There are many other color models besides RGB. RGB is sometimes criticized as being unintuitive. For example, it’s not obvious to most people that yellow is made of a combination of red and green.
The closely related color models HSV and HSL describe the same set of colors as RGB, but attempt to do it in a more intuitive way. (HSV is sometimes called HSB, with the “B” standing for
“brightness.” HSV and HSB are exactly the same model.)
The “H” in these models stands for “hue,” a basic spectral color. As H increases, the color changes from red to yellow to green to cyan to blue to magenta, and then back to red. The value of H is
often taken to range from 0 to 360, since the colors can be thought of as arranged around a circle with red at both 0 and 360 degrees.
The “S” in HSV and HSL stands for “saturation,” and is taken to range from 0 to 1. A saturation of 0 gives a shade of gray (the shade depending on the value of V or L). A saturation of 1 gives a
“pure color,” and decreasing the saturation is like adding more gray to the color. “V” stands for “value,” and “L” stands for “lightness.” They determine how bright or dark the color is. The main
difference is that in the HSV model, the pure spectral colors occur for V=1, while in HSL, they occur for L=0.5.
Let’s look at some colors in the HSV color model. The illustration below shows colors with a full range of H-values, for S and V equal to 1 and to 0.5. Note that for S=V=1, you get bright, pure
colors. S=0.5 gives paler, less saturated colors. V=0.5 gives darker colors.
It’s probably easier to understand color models by looking at some actual colors and how they are represented. The interactive demo c2/rgb-hsv.html lets you experiment with the RGB and HSV color
Often, a fourth component is added to color models. The fourth component is called alpha, and color models that use it are referred to by names such as RGBA and HSLA. Alpha is not a color as such. It
is usually used to represent transparency. A color with maximal alpha value is fully opaque; that is, it is not at all transparent. A color with alpha equal to zero is completely transparent and
therefore invisible. Intermediate values give translucent, or partly transparent, colors. Transparency determines what happens when you draw with one color (the foreground color) on top of another
color (the background color). If the foreground color is fully opaque, it simply replaces the background color. If the foreground color is partly transparent, then it is blended with the background
color. Assuming that the alpha component ranges from 0 to 1, the color that you get can be computed as
new color = (alpha)*(foreground color) + (1 - alpha)*(background color)
This computation is done separately for the red, blue, and green color components. This is called alpha blending. The effect is like viewing the background through colored glass; the color of the
glass adds a tint to the background color. This type of blending is not the only possible use of the alpha component, but it is the most common.
An RGBA color model with 8 bits per component uses a total of 32 bits to represent a color. This is a convenient number because integer values are often represented using 32-bit values. A 32-bit
integer value can be interpreted as a 32-bit RGBA color. How the color components are arranged within a 32-bit integer is somewhat arbitrary. The most common layout is to store the alpha component in
the eight high-order bits, followed by red, green, and blue. (This should probably be called ARGB color.) However, other layouts are also in use. | {"url":"https://eng.libretexts.org/Bookshelves/Computer_Science/Applied_Programming/Introduction_to_Computer_Graphics_(Eck)/02%3A_Two-Dimensional_Graphics/2.01%3A_Pixels_Coordinates_and_Colors","timestamp":"2024-11-07T12:08:04Z","content_type":"text/html","content_length":"151916","record_id":"<urn:uuid:7fda0b86-03ed-49d3-be35-567ad9168043>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00552.warc.gz"} |
Trudy Mat. Inst. Steklova, 2012, Volume 276
General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Search papers
Search references
Latest issue
Current issues
Archive issues
What is RSS
| General information | Contents | Forward links |
Number theory, algebra, and analysis
Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday
Volume Editor: A. N. Parshin^
Editor in Chief: A. G. Sergeev^
Abstract: This volume is dedicated to the memory of the outstanding scientist Anatolii Alekseevich Karatsuba. The volume contains papers dealing with a wide range of topics in analytic number theory
and related fields of algebra and analysis.
The volume will be of interest to specialists, postgraduates, and senior students of mathematical specialties.
ISBN: 5-7846-0121-0 (978-5-7846-0121-6)
Full text: Contents
Citation: Number theory, algebra, and analysis, Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 276, ed. A. N.
Parshin, A. G. Sergeev, MAIK Nauka/Interperiodica, Moscow, 2012, 288 pp.
Citation in format AMSBIB:
\book Number theory, algebra, and analysis
\bookinfo Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2012
\vol 276
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\ed A.~N.~Parshin, A.~G.~Sergeev
\totalpages 288
Linking options:
Additional information
Number theory, algebra, and analysis
Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday
This volume is dedicated to the memory of the outstanding scientist Anatolii Alekseevich Karatsuba. The volume contains papers dealing with a wide range of topics in analytic number theory and
related fields of algebra and analysis.
The volume will be of interest to specialists, postgraduates, and senior students of mathematical specialties. | {"url":"https://www.mathnet.ru/php/archive.phtml?wshow=issue&bshow=main&jrnid=tm&series=0&bookID=1375&year=2012&volume=276&issue=&option_lang=eng","timestamp":"2024-11-11T04:44:22Z","content_type":"text/html","content_length":"22616","record_id":"<urn:uuid:dee617cc-60a5-4b6b-b61f-3a4767e0ba40>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00190.warc.gz"} |
AP Calculus BC Exam Help | Online Calculus Tutoring
AP Calculus BC Exam Help
InteractiveMathTutor.com provides tutoring for students preparing for the AP Calculus BC Exam. The AP Calculus BC Exam requires knowledge of the same topics as AP Calculus AB exam, but goes beyond
this curriculum requiring students to have an even more thorough understanding of Calculus I material to score high enough on the exam to receive college credits.
We provide comprehensive AP Calculus BC Exam tutoring for students including the following AP Calculus BC Exam topics:
• Absolute and Conditional Convergence
• Alternating Series
• Antiderivatives by Partial Fractions
• Application of Derivatives
• Area of Functions
• Asymptotic Behavior of Functions
• Convergence Tests
• Curve Sketching
• Divergence Tests
• Euler’s Method
• First Derivative Test
• Fundamental theorem of calculus
• Geometric Series
• Harmonic Series
• Implicit Differentiation
• Integrals
• Limits of Functions
• Maclaurin Series
• Mean Value theorem
• Methods of Anti differentiation
• Numerical Approximations
• Parametric Functions
• Ratio Test
• Relationship between Differentiability and Continuity
• Riemann Sums
• Root Test
• Second Derivatives
• Simpson’s Rule for Approximations
• Taylor Series
• Trapezoid Rule for Approximation
• Vector Functions
• Volume of Solids | {"url":"http://www.interactivemathtutor.com/mathematics/calculus/ap-calculus-bc-exam/","timestamp":"2024-11-05T22:43:32Z","content_type":"application/xhtml+xml","content_length":"33406","record_id":"<urn:uuid:36fb5012-87c7-410d-a6b1-ebb12bd84ef3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00657.warc.gz"} |
Subtyping variance of dependent product types
The manual gives this subtyping rule for product types here:
If E[Γ] ⊢ T =_{βδιζη} U and E[Γ :: (x : T)] ⊢ T' ≤_{βδιζη} U' then E[Γ] ⊢ ∀ x : T, T' ≤_{βδιζη} ∀ x : U, U'.
What surprises me here is the T =_{βδιζη} U where I would have expected T ≥_{βδιζη} U. I think that in programming languages with static typing and subtyping, the function type A → B is usually
contravariant in A, but this definition says that in Coq it is invariant.
I tested this code:
Parameter X : Set.
Check X : Type.
Parameter Y : nat -> Set.
Check Y : nat -> Type.
Parameter Z : Type -> nat.
Check Z : Set -> nat.
The result is very surprising to me: the first to Check commands pass, as expected, and the third also passes, but instead of
Z : Set -> nat
Coq says
(fun x : Set => Z x) : Set -> nat
so it seems like Z was automatically η-expanded to yield the right type.
What’s going on behind the scenes here? What’s the reason product types are invariant in the “index” type? How does Coq decide when to η-expand a term to convert it to a certain type? Where can I
read more about this?
What’s the reason product types are invariant in the “index” type?
We generally call it the domain not the index.
It makes model construction easier, for instance in the set model.
How does Coq decide when to η-expand a term to convert it to a certain type?
The coercion system when trying to coerce f : A-> B to A' -> B' tries to produce fun x : A' => coe_B_to_B' (f (coe_A'_to_A x)).
in your example the sub coercions are the identity thanks to cumulativity
see also Activating contravariant subtyping of dependent function types by herbelin · Pull Request #13270 · coq/coq · GitHub
1 Like
From https://inria.hal.science/hal-04077552v2/document
1This restriction comes from the fact that it is difficult to model contravariant cumulativity in set-theoretic models Timany and Sozeau 2018. Whether cumulativity could be contravariant on the
left-hand side of an arrow or not is still the subject of ongoing theoretical investigations.
1 Like | {"url":"https://coq.discourse.group/t/subtyping-variance-of-dependent-product-types/2457","timestamp":"2024-11-06T05:43:33Z","content_type":"text/html","content_length":"29191","record_id":"<urn:uuid:b2e3492b-5891-41ed-a099-30878a4ca187>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00735.warc.gz"} |
Lesson 10—Matched-Pairs t tests
Lesson 10—Matched-Pairs t tests
First published on April 20, 2019
Learning Objectives
Perform a matched-pair hypothesis t test
Here is the dataset that we are using in this demo from #17.16 in our textbook:
Goal: To conduct a two-sided t test of no difference for a matched-pair t test
Here are the null and alternative hypotheses in this example:
\(H_0: \mu_1 = \mu_2 \)
\(H_a: \mu_1 \neq \mu_2 \)
\( \mu_1 \) = population mean angle in squatting
\( \mu_2 \)= population mean angle in sitting
Here are the steps:
Step 1: Import the dataset
Step 2: Create a column of difference
w = sitting_squatting$Sitting - sitting_squatting$Squatting
DO NOT use name your column of difference as diff because diff is a built-in function in R.
In this demonstration, we name our column of difference w.
Step 3: Draw a stemplot to check data
Draw a stemplot on the column of difference to check if data are roughly symmetric and without too many extreme outliers:
Step 3: Run the matched-pair t test via the R function t.test
t.test(sitting_squatting$Sitting, sitting_squatting$Squatting,
mu=0, paired=TRUE,
conf.level = 0.95)
Whenever R runs a hypothesis test, R automatically calculates the corresponding confidence interval —the range of values which the population mean is estimated to lie within.
Given a set of data, the corresponding hypothesis test result and the confidence interval are closely related. Therefore if we want the significance level \(\alpha \) to be 0.05, then we set the
argument conf.level = 0.95 because conf.level = 1 – \(\alpha \) .
By default, R automatically sets \(\mu = 0 \) and conf.level = 0.95 even if you don’t explicitly type these arguments. So you can skip typing these arguments into the t.test function if you are
testing a two-sided alternative hypothesis with \(\alpha =0.05\).
« Lesson 9—Import data via R scripts | COURSE | Lesson 11—Two-Sample t tests » | {"url":"https://www.corsbook.com/lesson/lesson-9-matched-pairs-t-tests/","timestamp":"2024-11-05T12:10:49Z","content_type":"text/html","content_length":"24328","record_id":"<urn:uuid:119c887e-a623-480f-999b-335d29eedec9>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00025.warc.gz"} |
Charged Pion Contribution to the Anomalous Magnetic Moment of the Muon
The model dependence inherent in hadronic calculations is one of the dominant sources of uncertainty in the theoretical prediction of the anomalous magnetic moment of the muon. In this thesis, we
focus on the charged pion contribution and turn a critical eye on the models employed in the few previous calculations of $a_\mu^{\pi^+\pi^-}$. Chiral perturbation theory provides a check on these
models at low energies, and we therefore calculate the charged pion contribution to light-by-light (LBL) scattering to $\mathcal{O}(p^6)$. We show that the dominant corrections to the leading order
(LO) result come from two low energy constants which show up in the form factors for the $\gamma\pi\pi$ and $\gamma\gamma\pi\pi$ vertices. Comparison with the existing models reveal a potentially
significant omission - none include the pion polarizability corrections associated with the $\gamma\gamma\pi\pi$ vertex. We next consider alternative models where the pion polarizability is produced
through exchange of the $a_1$ axial vector meson. These have poor UV behavior, however, making them unsuited for the $a_\mu^{\pi^+\pi^-}$ calculation. We turn to a simpler form factor modeling
approach, generating two distinct models which reproduce the pion polarizability corrections at low energies, have the correct QCD scaling at high energies, and generate finite contributions to $a_\
mu^{\pi^+\pi^-}$. With these two models, we calculate the charged pion contribution to the anomalous magnetic moment of the muon, finding values larger than those previously reported: $a_\mu^\mathrm
{I} = -1.779(4)\times10^{-10}\,,\,a_\mu^\mathrm{II} = -4.892(3)\times10^{-10}$.
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: magnetic moment; LBL; charged pion; chiral perturbation theory; g-2
Degree Grantor: California Institute of Technology
Division: Physics, Mathematics and Astronomy
Major Option: Physics
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Wise, Mark B.
Group: Caltech Theory
Thesis Committee: • Wise, Mark B. (chair)
• Carroll, Sean M.
• Cheung, Clifford W.
• Porter, Frank C.
Defense Date: 16 May 2013
Non-Caltech Author Email: kevin.t.engel (AT) gmail.com
Record Number: CaltechTHESIS:05212013-160540441
Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05212013-160540441
DOI: 10.7907/M463-8Z93
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 7732
Collection: CaltechTHESIS
Deposited By: Kevin Engel
Deposited On: 29 May 2013 23:47
Last Modified: 26 Oct 2021 17:57
Thesis Files
PDF - Final Version
See Usage Policy.
Repository Staff Only: item control page | {"url":"https://thesis.library.caltech.edu/7732/","timestamp":"2024-11-06T14:54:10Z","content_type":"application/xhtml+xml","content_length":"28518","record_id":"<urn:uuid:14e935cd-b71b-4cf2-aa9d-248ed7f50782>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00876.warc.gz"} |
Making a selection in a ruby script and applying command
Making a selection in a ruby script and applying command
I'm new to making ruby scripts and am attempting to automate a sequence of actions I use constantly. The basic idea is, I select an edge and my script will draw a triangle of my specified size
and select all the edges that are connected to my initial selection and perform a followme on the triangle. Everything works up until the followme. My triangle is created and my selection set is
accurate. If I stop the script there, I can manually click Followme and it works on the path. But I can't seem to get the script to recognize those additional selected lines. It will only do a
followme on the very first line I picked before running the script. It doesn't collect the new edges.
I used this language to attempt the followme:
Convert the selection to an array of edges
connected_edges_array = selected_edges.to_a
Perform follow me on triangle face using selected edges as path
What am I missing?
You're not helping us help you.
You have only posted a few lines of code, so we can't see where you're going wrong.
Also a screenshot of the starting setup would be helpful...
Also please use the 'Code' formatting for any Ruby code your are posting...
As you seem to be able to do it manually the path and face seem OK - although we haven't seen them !
### something sets a selection of edges up ??
connected_edges_array = selected_edges.to_a
### the face is somehow selected ??
How do you ensure that all of those selected edges which are to be used are connected ?
How do you select the face ?
More code please...
Here is all the code that works and results in a closed loop of the edges I want. Not sure how to get the results to make a followme. I am testing this on a simple rectangle to start.
Get the active model and selection
mod = Sketchup.active_model
sel = mod.selection
Define z and perpendicular measurements
z_measurement = 2 * perpendicular_measurement
perpendicular_measurement = 3.0
Get selected edges
selected_edges = sel.grep(Sketchup::Edge)
Exit if there are no selected edges
if selected_edges.empty?
UI.messagebox('Please select an edge to draw the triangle connected to.')
Select the first selected edge
selected_edge = selected_edges.first
Get the start and end vertices of the selected edge
start_point = selected_edge.start.position
end_point = selected_edge.end.position
Calculate the midpoint of the selected edge
midpoint = Geom::Point3d.linear_combination(0.5, start_point, 0.5, end_point)
Calculate the direction vector perpendicular to the edge
direction_vector = end_point - start_point
perpendicular_vector = Geom::Vector3d.new(-direction_vector.y, direction_vector.x, 0).normalize
perpendicular_vector.length = perpendicular_measurement
Calculate the third vertex of the triangle
third_vertex = midpoint.offset(perpendicular_vector)
Create a point at the desired height above the midpoint
point_above_midpoint = midpoint.offset([0, 0, z_measurement])
Define the vertices of the triangle
triangle_vertices = [midpoint, point_above_midpoint, third_vertex]
Create the triangle face
triangle_face = mod.entities.add_face(triangle_vertices)
Create an edge between midpoint and third_vertex
midpoint_third_vertex_edge = mod.entities.add_line(midpoint, third_vertex)
Get the active model and selection
mod = Sketchup.active_model
sel = mod.selection
Get the selected edge
selected_edge = sel.grep(Sketchup::Edge).first
Exit if no edge is selected
unless selected_edge
UI.messagebox('Please select an edge to determine its Z height.')
Get the Z coordinate of the selected edge
selected_edge_start_z = selected_edge.start.position.z
selected_edge_end_z = selected_edge.end.position.z
Define a tolerance for comparing Z coordinates
tolerance = 1e-6
Select all edges with starting and ending Z coordinates matching the selected edge
mod.entities.each do |entity|
if entity.is_a?(Sketchup::Edge)
edge = entity
next if edge == selected_edge
# Get the starting and ending Z coordinates of the edge
edge_start_z = edge.start.position.z
edge_end_z = edge.end.position.z
# Check if starting and ending Z coordinates match the selected edge
if (selected_edge_start_z - edge_start_z).abs < tolerance &&
(selected_edge_end_z - edge_end_z).abs < tolerance
Remove the edge between midpoint and third_vertex to the selection
If I add those last two lines I mentioned for followme, I get this:
Please format your code properly using the 'Code' option above
It's almost impossible to read.
I think you are over complicating it.
mod = Sketchup.active_model
sel = mod.selection
selected_edges = sel.grep(Sketchup::Edge)
### add the triangle code - where the face is located on the edges is academic, you can draw it on and selected edge.
### Add the face from the desired points. You can get a reference to it after it's added.
The selected_edges are not guaranteed to be all connected and in order [if they are connected there are somewhat complicated methods to order them].
A simpler way to get an array of those edges is to get the 'base_face' rather than the what you are selecting the edges around, and use that -
mod = Sketchup.active_model
sel = mod.selection
selected_base_face= sel.grep(Sketchup::Face)[0]
### make triangle...
### automatically an order array
@TIG ```
Sorry, this should read better.
Will the base face method work if there is no face, just lines that could be a face?
<# Get the active model and selection
mod = Sketchup.active_model
sel = mod.selection
<# Define z and perpendicular measurements
z_measurement = 2 * perpendicular_measurement
perpendicular_measurement = 3.0
<# Get selected edges
selected_edges = sel.grep(Sketchup::Edge)
<# Exit if there are no selected edges
if selected_edges.empty?
UI.messagebox('Please select an edge to draw the triangle connected to.')
<# Select the first selected edge
selected_edge = selected_edges.first
<# Get the start and end vertices of the selected edge
start_point = selected_edge.start.position
end_point = selected_edge.end.position
<# Calculate the midpoint of the selected edge
midpoint = Geom::Point3d.linear_combination(0.5, start_point, 0.5, end_point)
<# Calculate the direction vector perpendicular to the edge
direction_vector = end_point - start_point
perpendicular_vector = Geom::Vector3d.new(-direction_vector.y, direction_vector.x, 0).normalize
perpendicular_vector.length = perpendicular_measurement
<# Calculate the third vertex of the triangle
third_vertex = midpoint.offset(perpendicular_vector)
<# Create a point at the desired height above the midpoint
point_above_midpoint = midpoint.offset([0, 0, z_measurement])
<# Define the vertices of the triangle
triangle_vertices = [midpoint, point_above_midpoint, third_vertex]
<# Create the triangle face
triangle_face = mod.entities.add_face(triangle_vertices)
<# Create an edge between midpoint and third_vertex
midpoint_third_vertex_edge = mod.entities.add_line(midpoint, third_vertex)
<# Get the active model and selection
mod = Sketchup.active_model
sel = mod.selection
<# Get the selected edge
selected_edge = sel.grep(Sketchup::Edge).first
<# Exit if no edge is selected
unless selected_edge
UI.messagebox('Please select an edge to determine its Z height.')
<# Get the Z coordinate of the selected edge
selected_edge_start_z = selected_edge.start.position.z
selected_edge_end_z = selected_edge.end.position.z
<# Define a tolerance for comparing Z coordinates
tolerance = 1e-6
<# Select all edges with starting and ending Z coordinates matching the selected edge
mod.entities.each do |entity|
if entity.is_a?(Sketchup::Edge)
edge = entity
next if edge == selected_edge
# Get the starting and ending Z coordinates of the edge
edge_start_z = edge.start.position.z
edge_end_z = edge.end.position.z
# Check if starting and ending Z coordinates match the selected edge
if (selected_edge_start_z - edge_start_z).abs < tolerance &&
(selected_edge_end_z - edge_end_z).abs < tolerance
<# Remove the edge between midpoint and third_vertex to the selection
<# Convert the selection to an array of edges
connected_edges_array = selected_edges.to_a
<# Perform follow me operation on triangle face using selected edges as path
@TIG said in Making a selection in a ruby script and applying command:
selected_base_face= sel.grep(Sketchup::Face)[0]
Awesome, the base face works great when a face with edges is selected!
No, it doesn't read a lot better.
You must add the 'code' tag and put your Ruby code inside it.
Then the start and end tags [i.e. three back-quotes each: note I put spaces in front here to prevent the forum thinking it was more code...
### ruby code text goes here
] they must both be on their own lines.
Which looks like this in the post.
### ruby code text goes here
Simplify your process, and make it more complex only when you've got some good results.
Trying to do it all in one go is just too difficult.
How do you eat an elephant - take small bites ! | {"url":"https://community.sketchucation.com/topic/163586/making-a-selection-in-a-ruby-script-and-applying-command","timestamp":"2024-11-04T11:51:36Z","content_type":"text/html","content_length":"123709","record_id":"<urn:uuid:87049930-97a4-4148-961e-0b45806025d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00684.warc.gz"} |
How to Think about Risk
Acknowledge the role of risk in every decision you make.
A coin is flipped into the air.
There’s 50% chance it will land on heads; 50% chance it will land on tails.
The distribution of outcomes looks like this:
So you pick a side as it falls back down.
Get it right and you win.
This is luck.
Get it wrong and you lose.
That is risk.
No situation in life is as simple as a coin toss but the principle is the same.
On the opposite side of luck is risk
For every decision we make, we draw a line between the outcomes we want to happen and those we want to avoid.
The outcomes we desire, we call winning, success and achievement.
All the other outcomes—the ones we don't want—we call losing, failure and mistakes.
In various life situations, the split between luck and risk can look like this:
and it can look like this:
The point is, luck and risk always lie on opposite sides of each other. You cannot have one without the other.
This means risk is present in every decision you make, no matter how 'safe' that decision actually feels.
• Pursue a 'safe' career.
• Invest in a 'safe' asset.
• Settle for a 'safe' relationship.
There are varying degrees of risk involved in all of those.
Nothing is 100% guaranteed.
Think in terms of probabilities
No decision is ever entirely safe or risky, right or wrong, good or bad.
A more nuanced approach is to explore the varying degrees of probability each outcome has of occurring.
Asking 'how safe is it?' is more useful than asking 'is it safe?'; it implies an acknowledgement of the hidden side of risk.
Getting absolute numbers is not needed—even approximations offer invaluable insights.
For example, what is the probability that I'll hit the bullseye?
• More than 80%?
• About 50%?
• Less than 20%?
Depending on what the answer is, my willingness to bet on the outcome will be drastically different.
Risk is relative, and that's okay
That doesn't mean there is an objectively correct answer though.
The probability that feels safe/right/good for me to bet on might feel risky/wrong/bad to you.
Our tolerance for risk is highly personal.
If I were brought up during a bust period, my attitude to risk would be completely different than were I were brought up during a boom period.
Similarly, your attitude to risk would be different with $100 in the bank versus $1,000,000.
The point is, don't ever let someone tell you what is too risky—you have to come to your own conclusions on that.
What’s important is that your decision sits right with you.
• Every single decision comes with a certain degree of risk.
• Ball-park the probability of both experiencing luck or risk.
• Ask yourself if you—and you alone—are comfortable with making a decision based on that split.
Then let fate unfold and see on which side you will fall.
Good luck.
Get my next post in your Inbox. | {"url":"https://blog.bendetalle.com/p/how-to-think-about-risk","timestamp":"2024-11-06T23:16:55Z","content_type":"text/html","content_length":"147227","record_id":"<urn:uuid:90c9ad23-cb24-42d7-81f4-5e07bf754eee>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00024.warc.gz"} |
How Do You Calculate Final Marks?
How Do You Calculate Final Marks?
How Do You Calculate Final Examination Marks? Grade calculator
After an assignment or examination has been taken by a student and the assignment has been reviewed, each student has got a score. Most likely, the score does not represent the final result for the
student. Often, the score needs to be calculated into marks. Sometimes the score is corrected before it is turned into a mark.
Usually, as it is known among many schools, colleges and universities, teachers are the ones that are responsible for giving grades to their students at the end of an examination, but it will be a
great feeling for students to be able to calculate their final grades using marks they obtained at the end of every semester or examination. By maintaining a record of the scores from your
assignments and assessments, you can double-check your teacher’s grading and make sure the correct grade registers.
Calculating your grade allows you to determine the scores you need to reach or maintain your desired grade. For example, if you find that your current grade is a C, you can figure out what you need
to earn on your final exam to boost your final grade to a B. Alternatively, if you currently have an A or B, you can determine the minimum score you need to keep that grade.
How To Calculate Final Marks
The year-mark contribution towards the final examination mark is calculated as follows: 50% of the mark obtained for assignment 01 Plus 50% of the mark obtained for assignment 02.
If you only submit assignment 01, your year mark will be 50% of the mark obtained for this assignment. This will then be your year mark out of a possible 100%. If, for example, you obtain 80% for
assignment 01 and 0% for assignment 02, your year mark will be 40%.
According to university policy, you require a sub-minimum of 40% in the examination before your year mark is taken into consideration. In other words, if you do not obtain at least 40% in the
examination, you will automatically fail, and your final mark will be the mark you obtained in the examination.
A final mark of 50% is required to pass this module. This final mark is calculated as follows: (10% x of the year mark) + (90% x mark obtained in the examination)
READ: How Much Is Grade R Course At Unisa
Final Mark Calculation Options
Three main options can be used to calculate the final marks of an examination, which include: formula, table or none.
Marks Calculated Using Formular:
The formula option allows you to create a mark calculation based on two parameters:
• Points: the number of points earned by the student
• Total: the total amount of points that a student can earn for the assignment
There are multiple functions and operators available which can be used to set the formula:
• Math operators: + , – , * , / , ^
• Logic functions: IF, AND, OR
• Comparison operators: <, >, <=, >=, !=, =
• Numeric functions: MIN, MAX, ROUND, ROUNDUP, ROUNDDOWN
• It’s possible to insert parentheses to determine the order of (parts of) the formula
The default formula used by Ans is 1 + 9 * points/total. This translates the score into a mark on a scale of 1 to 10.
Marks Calculated Using The Table Option:
Within the table, there are two different possibilities to insert a table in the answer. The first option is to manually insert or import a table, the second option is to let answers create a table
based on the cut-off score.
Manually Create A Table
The first option is to manually create a table by either adding each row by hand or by importing a table from a .csv file. On the screen, you will see the following buttons and fields:
• Rounding: choose the desired rounding of the mark. This can be either two decimals, one decimal, halves or whole numbers. A table always rounds down. Every row in the table can be seen as a
threshold. A student needs to score the number of points or a row to receive the corresponding mark. In the example below, a student that has got 2.5 points, will receive a mark of 2 as the
number of points in the next row has not been scored. Final Examination Marks
• Import: it’s possible to import a table that you created outside Ans. A table can be imported via a .csv file, with at least the columns ‘Points’ and ‘Marks’. Additionally, you can add a letter
grade as a third column. A template can be found when you click on Import.
• Export: in case you have a table created that you want to reuse for other purposes or other assignments, you can export it. This button is not clickable when there is not a single row.
• Add row: to add a row to your column manually, you can click the button and insert the number of points with the corresponding mark. Optionally you can add a letter grade, such as sufficient, or
insufficient. To edit or delete a row, you can press the more_vert– icon.
• Determine cut-off score: this is the second way to create a table. This option is described below.
Automatically Generated Table
When grading with a table, you have the option to determine the grading of the results using the cut-off score. By determining the minimum grade, pass grade, maximum grade, maximum points and cut-off
percentage, you can create a grading table that supports a cut-off score:
Click on Determine cut-off score to start. If you want to edit an existing table, you can also click on this button. Ans will show the previously entered information which you can adjust.
In the pop-up that appears, you can enter the following information:
• Minimum mark: The minimum mark is the mark a student gets when scoring the minimum amount of points.
• Pass mark: The pass mark is the mark which counts as the threshold of passing the assignment. All students with a rounded mark lower than the pass mark have not passed the assignment. This
information is used to determine the pass rate for the assignment.
• Maximum mark: The maximum mark is the mark a student gets when scoring the maximum amount of points. Final Examination Marks
• Minimum points: This is the number of points a student needs to score to get the minimum mark.
• Maximum points: This is the number of points a student needs to score to get the maximum mark. Ans shows the maximum amount of points that can be scored for the assignment. Ans also provides
information that can be used to apply the caesura method of Cohen-Schotanus*. Based on this method you adjust the maximum score for an assignment after taking the assignment. The new maximum
score is then equalled by the score of either the 95th or 90th percentile.
• Cut-off (%): In this field, you fill in the score (in percentage) that equals the pass mark. As an example, we take an assignment with a maximum amount of points of 100 points, a pass mark of 5.5
and a cut-off of 70%. In this scenario a student needs to score at least 70% of 100 points (= 70 points) to get a 5.5. This would create a non-linear caesura.
• Mark rounding: The mark rounding needs to be determined in this menu, as Ans (re)calculates the table once you click on Save. You have the option to set the mark rounding to whole numbers, halves
and one decimal.
READ: What Is The Pass Mark For Unisa Exams 2025
No Mark Calculation
The third and easiest option to explain is the option None in the dropdown menu. When using no mark calculation, the score will not be translated into a mark. The mark is not visible in other parts
of the platform where the mark is visible when using the formula or the table method. Examples are the Results menu and the publication of an assignment.
How Do You Calculate Final Marks 2025
Example One
• The current grade is 70% (or C-).
• The final exam weight is 50%.
• The required grade is 80% (or B-).
The final exam grade is equal to the required grade, minus 100% minus the final exam weight (w) times the current grade (g), divided by the final exam weight (w):
Final exam grade =
• = ( required grade – (100% – w)×current grade ) / w
• = ( 80% – (100% – 50%)×70% ) / 50% = 90%
So the final exam grade should be 90% (or A-).
Final Examination Marks
Example Two
• Assignment 1: weight1=50%, grade1=16 out of 20.
• Assignment 2: weight2=30%, max grade=30.
• Assignment 3: weight3=20%, max grade=40.
• Find the average grade in assignments 2 and 3 needed to get a class grade of 85%.
Calculation Method:
Current grade = Assignment 1 grade = grade1 / max grade1 = 16/20 = 0.8 = 80%
• Required grade = 85%
• Final exam weight = w = weight2+weight3 = 30%+20% = 50%
Final exam grade =
• = ( required grade – (100% – w)×current grade ) / w
• = ( 85% – 50%×80% ) / 50% = 90%
This means that you have to get an average grade of 90% in assignments 2 and 3 to get a class grade of 85%.
• Assignment 2 grade = 90% × max grade = 90% × 30 = 27
• Assignment 3 grade = 90% × max grade = 90% × 40 = 36
How To Calculate Midterm And Final Examination Grades
The example below presents one way that faculties have found helpful to calculate grades. Please consult your department to ensure that your grading rubric aligns with any relevant department or
school policies. Below are the steps to be used to calculate Midterm Final Grades:
1. To calculate a weighted midterm grade, add the weighted percentages for each grade and divide by the % of grades completed by the midpoint of the semester. If some students, but not all students,
have been graded for an assignment at the midpoint, you should exclude that grade. In the example below, not all students have made their Presentations by the midpoint of the semester, so that
grade will not be included in midterm grade calculations:
• Midterm Grade = (.20*Homework) +(.25* Exam 1) + (.15*Paper 1) (.20+.25+.15)
• Midterm Grade = (.20*75) +(.25* 85) + (.15*90)/60
• Midterm Grade = 15 +21.25 + 13.5 /60
• Midterm Grade = 49.75 = 82.91% /60
1. To calculate a final grade – add the weighted percentages for each grade:
• Final Grade = (.20*Homework) +(.25* Exam 1) + (.15*Paper 1) +(.25*Pres 1) + (.15*Paper 2)
• Final Grade = (.20*75) +(.25* 85) + (.15*90) +(.25*60) + (.15*85)
• Final Grade = 15 +21.25 + 13.5 +15 + 12.75
• Final Grade = 77.5
READ: What Is The Pass Mark At Unisa?
Calculating final marks often depends on the grading system of your school or institution. A typical method is to use a weighted average, though grading systems can vary and may include different
methods or scales, such as letter grades or GPAs. Visit The Official website of The Department of Higher Education and Training. | {"url":"https://onlineunisa.co.za/how-do-you-calculate-final-marks/","timestamp":"2024-11-06T11:44:57Z","content_type":"text/html","content_length":"154696","record_id":"<urn:uuid:4a6fa2f1-7f57-42f7-abfa-5892e0859430>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00641.warc.gz"} |
What is MPa in lbs?
What is MPa in lbs?
Megapascal to Pound-force/square Inch Conversion Table
Megapascal [MPa] Pound-force/square Inch
1 MPa 145.03773773 pound-force/square inch
2 MPa 290.07547546 pound-force/square inch
3 MPa 435.11321319 pound-force/square inch
5 MPa 725.18868865 pound-force/square inch
Is PSI equal to LB in2?
lbf/in2↔psi 1 lbf/in2 = 1 psi.
How do you convert lbs to PSI?
Divide the number of pounds per square foot by 144. The quotient is the pounds per square inch. For example, 2,160 pounds per square foot converts to 15 pounds per square inch (2160 psf ÷ 144 = 15
Is MPa a megapascal?
The megapascal is a x1000000 multiple of the pascal unit which is the SI unit for pressure. 1 megapascal equals 1,000,000 pascals. Primarily used for higher range pressure measurement due to its
larger value (e.g. 1 MPa = 10 bar), the MPa is mainly used to describe the pressure ranges and ratings of hydraulic systems.
How do you convert kN to MPa?
When measuring area in millimeters, use the following conversion factor: 1 kN/mm2 = 1,000 MPa.
How do you convert MPa to KPSI?
A pressure value measured in megapascals can be converted to the corresponding value in kilopounds per square inch using the following math.
1. 1 ksi = 1,000 psi = 6,894,760 Pascals (Pa)
2. 1 MPa = 1,000,000 Pascals (Pa)
3. ksi value x 6,894,760 Pa = MPa value x 1,000,000 Pa.
4. ksi value = MPa value x 0.145038.
Is lbs same as PSI?
It is the pressure resulting from a force of one pound-force applied to an area of one square inch. In SI units, 1 psi is approximately equal to 6895 Pa….
Pound per square inch
Unit system Imperial units, US customary units
Unit of Pressure, Stress
Symbol psi or lbf/in2
What does lbs mean in pounds?
The libra (Latin for “scales / balance”) is an ancient Roman unit of mass that was equivalent to approximately 328.9 grams. It was divided into 12 unciae (singular: uncia), or ounces. The libra is
the origin of the abbreviation for pound, “lb”.
What is a lb in weight?
pound, unit of avoirdupois weight, equal to 16 ounces, 7,000 grains, or 0.45359237 kg, and of troy and apothecaries’ weight, equal to 12 ounces, 5,760 grains, or 0.3732417216 kg. The Roman ancestor
of the modern pound, the libra, is the source of the abbreviation lb.
How many PA are in a GPa?
1000000000 Pa
Gigapascal to Pascal Conversion Table
Gigapascal [GPa] Pascal [Pa]
1 GPa 1000000000 Pa
2 GPa 2000000000 Pa
3 GPa 3000000000 Pa
5 GPa 5000000000 Pa
Is MPa the same as N mm2?
MPa) – A basic unit of pressure or tension measurement in the International System of Weights and Measures. 1 MPa = 145 psi, 1 MPa = 1 N/mm2.
How is MPa calculated?
In fact, by definition, 1 Pascal is equal to 1 Newton/meter2, which means that 1 megaPascal (MPa) equals 1,000 kiloNewtons (kN)/m2. If you know the pressure exerted on a barrier of known area in MPa,
multiply by the area in square meters, and then multiply by 1,000 to get the total force exerted on the barrier in kN.
How many megapascals (MPa) are in a pound/square inch (lb/in²)?
You are currently converting Pressure units from Pound/Square Inch to Megapascal. 1 Pound/Square Inch (lb/in²) = 0.00689 Megapascal (MPa) Visit Megapascal to Pound/Square Inch Conversion
How to convert lbf/in2 to MPA?
How to convert lbf/in² to MPa? The formula to convert lbf/in² to MPa is 1 Pound-Force per Square Inch = 0.00689475729310432 Megapascal. lbf/in² is 144.927536231884 times Smaller than MPa. Enter the
value of lbf/in² and hit Convert to get value in MPa. Check our lbf/in² to MPa converter.
What is the unit of pressure in pounds per square inch?
lb/in^2 or mPa. The SI derived unit for pressure is the pascal. 1 pascal is equal to 0.00014503773800722 lb/in^2, or 1000 mPa. Note that rounding errors may occur, so always check the results. Use
this page to learn how to convert between pounds/square inch and millipascals.
What is MPA unit of pressure?
0.00689 Megapascal (MPa) Pound/Square Inch : Pounds or pound force per square inch (symbol: psi, lb/in², pfsi or lbf/in²) is a unit of measure for pressure based on avoirdupois units. It is widely
used in British and American. | {"url":"https://stylesubstancesoul.com/uncategorized/what-is-mpa-in-lbs/","timestamp":"2024-11-08T14:03:44Z","content_type":"text/html","content_length":"50562","record_id":"<urn:uuid:040cd1c9-b880-4dd1-915d-a5f096095b7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00818.warc.gz"} |
Fractions as Parts of a Set
Quadratic Characteristics
Writing Fractions as Decimals
Parts of a Circle / Circumference
Parts of a Controlled Experiment
Parts of a Chemical Equation
Parts of a Sentence (Thunderstorms)
Parts of a Circle Pre-Quiz
Introduction to Percent Decimals and Fractions
Explore Fractions as Parts of a Set Worksheets by Grades
Explore Other Subject Worksheets for class 8
Explore printable Fractions as Parts of a Set worksheets for 8th Class
Fractions as Parts of a Set worksheets for Class 8 are an essential resource for teachers looking to enhance their students' understanding of fractions in a practical and engaging manner. These
worksheets are specifically designed to cater to the mathematical needs of Class 8 students, focusing on the concept of fractions, fraction models, and their application in real-world scenarios. By
incorporating these worksheets into their lesson plans, teachers can provide their students with a variety of problems that challenge their critical thinking and problem-solving skills. The use of
visual fraction models in these worksheets helps students to better grasp the concept of fractions as parts of a whole, making it easier for them to apply this knowledge in more complex mathematical
problems. Fractions as Parts of a Set worksheets for Class 8 are an indispensable tool for any teacher aiming to make the learning of fractions an enjoyable and fruitful experience for their
Quizizz is an innovative platform that offers a wide range of educational resources, including Fractions as Parts of a Set worksheets for Class 8, to help teachers create interactive and engaging
learning experiences for their students. By incorporating Quizizz into their teaching strategies, educators can access a vast library of ready-made quizzes, worksheets, and other learning materials
that are specifically tailored to the needs of Class 8 students. The platform also allows teachers to create their own quizzes and worksheets, enabling them to customize the content to suit their
students' unique learning needs. With its user-friendly interface and extensive collection of resources, Quizizz is an invaluable tool for teachers seeking to enhance their students' understanding of
math, fractions, and fraction models in a fun and interactive way. | {"url":"https://quizizz.com/en/fractions-and-parts-of-a-set-worksheets-class-8?page=1","timestamp":"2024-11-08T14:34:50Z","content_type":"text/html","content_length":"158119","record_id":"<urn:uuid:411b8d5b-19b3-4de2-a0e6-c3055a25458a>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00527.warc.gz"} |
Journal Article
Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds
No external resources are shared
There are currently no full texts shared for your IP range.
There is no public supplementary material available
Kolasinski, S., & Menne, U. (2017). Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds. Nonlinear Differential Equations and Applications NoDEA, 24: 17. doi: 10.1007/
Cite as: https://hdl.handle.net/11858/00-001M-0000-0024-A97D-2
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power
decay rates of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. Secondly, counter-examples to
pointwise power decay rates of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in
most cases, for example if the varifold is not two-dimensional. Thirdly, these counter-examples demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second
fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature | {"url":"https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_2085977","timestamp":"2024-11-08T08:06:15Z","content_type":"application/xhtml+xml","content_length":"41786","record_id":"<urn:uuid:a412a7a5-0fc5-46a8-847b-eddb789d83fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00124.warc.gz"} |
Loving Menemen
This question has been prepared by our sponsor JotForm.
Cem goes to buy some bread every morning. He really likes hot bread. His family wants him to bring the loaves of bread to the home as soon as possible to eat the loaves of bread with good menemen.
So, let's help him to go home as quickly as possible.
The neighbourhood Cem lives in is represented as an \(\textbf{N x M}\) grid. The grid consists of only the characters #, ., A, B. # characters represent walls that Cem cannot pass through. .
characters represent open roads that Cem can move on. A is Cem's current position, he just bought the hot bread and waiting in the bakery. Finally, B is Cem's home. Can you find the minimum distance
that Cem needs to travel, in order to reach home as quickly as possible?
The cells on the border will be walls #. There will be only one A and only one B in each input. Cem may not reach his home every time, you should print -1 in such cases.
The initial line only includes \(\mathbf{N}\) and \(\mathbf{M}\).
In the next \(\mathbf{N}\) lines, there exists a grid with \(\mathbf{N}\) rows and \(\mathbf{M}\) columns.
• \(1 \leq \mathbf{N}, \mathbf{M} \leq 10^{3}\)
In a single line, print the minimum distance between Cem and his home. If B is not accessible from A print -1 instead. | {"url":"https://arsiv.cclub.metu.edu.tr/problem/21maze/","timestamp":"2024-11-10T11:33:52Z","content_type":"text/html","content_length":"11550","record_id":"<urn:uuid:e342d473-1256-48de-805d-422abc0520d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00725.warc.gz"} |
8 Astounding Facts About David Hilbert
Source: Aip.org
When it comes to the field of mathematics, there are few names that carry the same level of prestige and influence as David Hilbert. Born on January 23, 1862, in what is now Poland, Hilbert made
significant contributions to a wide range of mathematical disciplines, leaving an indelible mark on the field.
In this article, we will delve into eight astounding facts about David Hilbert, shedding light on his life and achievements that have earned him a place among the greatest mathematicians of all time.
From his groundbreaking work in geometry to his famous list of 23 unsolved problems, Hilbert’s impact on mathematics continues to reverberate to this day.
Key Takeaways:
• David Hilbert was a math superstar who made big waves in the world of numbers. His ideas still influence math today, and his famous problems are still unsolved!
• Hilbert’s love for math and his brilliant teaching skills inspired many famous mathematicians. His legacy lives on, shaping the world of numbers and quantum mechanics.
The Father of Modern Mathematics
David Hilbert, born on January 23, 1862, in Germany, was one of the most influential mathematicians of the 20th century. His groundbreaking work laid the foundation for many areas of mathematics and
his impact is still felt today.
A Master of Mathematical Logic
Hilbert was a pioneer in the field of mathematical logic. He developed a formal system known as “Hilbert’s Program” with the goal of establishing a complete and consistent foundation for all of
mathematics. This ambitious endeavor had a profound impact on the development of mathematical logic and set the stage for future advancements in the field.
Contributions to Geometry
Hilbert’s work in geometry was revolutionary. He introduced the concept of “Hilbert space,” a mathematical structure that has become a fundamental tool in the field of quantum mechanics. His
contributions to geometry also include the development of the axiomatic method, which provides a rigorous foundation for the study of geometry.
Hilbert’s Famous 23 Problems
In 1900, Hilbert presented a list of 23 unsolved problems in mathematics that he believed would shape the future of the field. Dubbed as “Hilbert’s Problems,” these challenges spurred significant
advancements in various branches of mathematics and many of them remain unsolved to this day.
Lecturer Extraordinaire
Hilbert’s skills as a lecturer were legendary. His lectures were known for their clarity and elegance, and he had the ability to captivate audiences with his deep insights into complex mathematical
concepts. His teaching influenced generations of mathematicians and his lectures are still studied by students today.
A Mentor to Prominent Mathematicians
Throughout his career, Hilbert mentored several prominent mathematicians who went on to make significant contributions to the field. Some of his notable students include Hermann Weyl, Ernst Zermelo,
and Carl Gustav Hempel. Hilbert’s guidance and support paved the way for their success.
Hilbert’s Impact on Quantum Mechanics
Hilbert’s work on Hilbert spaces and the mathematical foundations of quantum mechanics had a profound impact on the development of this field. His contributions provided the mathematical framework
necessary for the formulation and understanding of quantum theory, revolutionizing our understanding of the microscopic world.
A Legacy That Continues to Inspire
David Hilbert’s legacy extends far beyond his lifetime. His groundbreaking ideas and profound contributions to mathematics and logic continue to inspire and shape the field. His commitment to the
pursuit of knowledge and his devotion to the mathematical community have left an indelible mark on the world of mathematics.
In conclusion, David Hilbert was an extraordinary mathematician who made significant contributions to the field. Through his groundbreaking work in formalism, he transformed the way mathematics is
approached and understood. Hilbert’s famous list of 23 unsolved problems served as a roadmap for mathematicians around the world, inspiring generations of researchers to push the boundaries of
mathematical knowledge.Not only was Hilbert renowned for his brilliance in mathematics, but he was also a dedicated teacher and mentor. He played a vital role in shaping the next generation of
mathematicians, sparking their curiosity and nurturing their talents.The impact of Hilbert’s work and ideas continues to be felt today. His legacy lives on in the countless mathematicians who have
been influenced by his teachings and discoveries. David Hilbert’s remarkable achievements have cemented his place as one of the most celebrated mathematicians in history.
Q: Who was David Hilbert?
A: David Hilbert was a German mathematician who lived from 1862 to 1943. He is best known for his contributions to the field of mathematics, particularly in the areas of algebra, geometry, and
mathematical logic.
Q: What is formalism in mathematics?
A: Formalism is a mathematical philosophy that focuses on the manipulation of symbols and the logical rules governing them. Hilbert played a crucial role in the development of formalism, which aimed
to establish a strong foundation for mathematics.
Q: What are Hilbert’s 23 unsolved problems?
A: Hilbert’s 23 unsolved problems, also known as Hilbert’s Problems, were a set of challenges put forth by David Hilbert in 1900. These problems covered various areas of mathematics and served as
significant catalysts for mathematical research in the 20th century.
Q: How did Hilbert contribute to algebra and geometry?
A: Hilbert made groundbreaking contributions to both algebra and geometry. In algebra, he formulated the concept of an infinite-dimensional vector space, paving the way for the development of
abstract algebra. In geometry, Hilbert’s axiomatization provided a rigorous foundation for the subject.
Q: What was Hilbert’s impact on mathematical education?
A: Hilbert was a dedicated teacher and mentor who significantly influenced mathematical education. He played a crucial role in establishing the Göttingen School of Mathematics, which became a
renowned center for mathematical research and education.
Q: How is Hilbert’s work relevant today?
A: Hilbert’s ideas and contributions continue to have a profound impact on modern mathematics. Many of the concepts and techniques he developed are still widely used today, shaping the way
mathematicians approach and solve problems.
David Hilbert's groundbreaking work laid the foundation for modern mathematics, but his influence extends far beyond his famous 23 problems. Hilbert's Nullstellensatz, a fundamental theorem in
algebraic geometry, is another testament to his genius. This powerful result connects the abstract world of polynomial equations with concrete geometric objects, opening up new avenues for
mathematical exploration. Dive deeper into the fascinating world of Hilbert's mathematics and uncover the secrets behind his enduring legacy.
Our commitment to delivering trustworthy and engaging content is at the heart of what we do. Each fact on our site is contributed by real users like you, bringing a wealth of diverse insights and
information. To ensure the highest standards of accuracy and reliability, our dedicated editors meticulously review each submission. This process guarantees that the facts we share are not only
fascinating but also credible. Trust in our commitment to quality and authenticity as you explore and learn with us. | {"url":"https://facts.net/history/people/8-astounding-facts-about-david-hilbert/","timestamp":"2024-11-05T18:24:04Z","content_type":"text/html","content_length":"231681","record_id":"<urn:uuid:0e22fb59-25dd-4831-9586-f4cce63afbc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00610.warc.gz"} |
Psychology Research Methods | 2K plays | Quizizz
18. Multiple Choice
1 minute
1 pt
17. Multiple Choice
1 minute
1 pt
Which of the following is the mode for the data set: 4, 4, 5, 6, 7, 7, 7, 8, 9?
16. Multiple Choice
30 seconds
1 pt
This refers to the method of gaining enough participants that represent various backgrounds of people, allowing researchers to generalize results to the larger population:
Full sampling
Double-blind placebo experiment
Random sampling
Naturalistic study
15. Multiple Choice
30 seconds
1 pt
In experimental studies which group receives nothing? In other words, they do not get the variable that is under study.
Experimental group
Control group
14. Multiple Choice
30 seconds
1 pt
An individual suffered a major traumatic brain injury after getting in a bad accident at work. If a psychologist wants to know the effects of the trauma, what method would be best?
Naturalistic observation
Case study
Experimental research
13. Multiple Choice
30 seconds
1 pt
If I want to gather how all students are feeling about remote learning so far, which method would make the most sense?
Naturalistic observation
Case study
12. Multiple Choice
30 seconds
1 pt
Cardinal rule of naturalistic observation:
Setting up dependent and independent variables
Research and find enough participants
Not disturbing participants in the study
Getting enough different age groups for study
11. Multiple Choice
30 seconds
1 pt
Independent variable is to dependent variable as _________ is to ____________.
hypothesis | conclusion
effect | cause
cause | effect.
conclusion | hypothesis
10. Multiple Choice
30 seconds
1 pt
The mode is:
The average of all the values
The middle number
The most frequently occurring number
The first number you see
9. Multiple Choice
30 seconds
1 pt
Identify the type of relationship between weight and price.
The price is never 0 on the graph
As weight increases, price increases
There is a negative linear correlation between weight and price
The price gets high at the end of the graph
8. Multiple Choice
30 seconds
1 pt
You psychology professor wants to describe the average score on your first psychology test. Which measure of central tendency is your professor most likely to use?
harmonic mean
7. Multiple Choice
10 seconds
1 pt
The advantage of this descriptive method of research is that it creates an immense amount of data to be gathered quickly and inexpensively.
Case Study
6. Multiple Choice
10 seconds
1 pt
This is a method of research where an investigator manipulates one or more factors (independent variables) to observe the effect on some behavior or mental process (the dependent variable).
Case Study
5. Multiple Choice
10 seconds
1 pt
This research method is a descriptive technique in which one individual or group is studied in depth in the hope of revealing universal principles
Case Study
4. Multiple Choice
30 seconds
1 pt
Measure of central tendency calculated by adding up all scores and dividing by the number of scores in the data set.
3. Multiple Choice
30 seconds
1 pt
A type of observation conducted in the participants' normal environment without interference from the researchers.
Controlled observation
Naturalistic observation
2. Multiple Choice
30 seconds
1 pt
In an experiment, the _________________ variable is tested by the researcher to observe its effect(s) on the _________________ variable.
A.independent; dependent
B.first; second
C.dependent; independent
D.research; results
1. Multiple Choice
30 seconds
1 pt
Method of intense study of just one individual:
Naturalistic Observation
Experimental Study
Case Study
Which of the following is the median for the data set: 4, 4, 5, 6, 7, 7, 7, 8, 9?
19. Multiple Choice
2 minutes
1 pt
Which of the following is the mean for the data set: 4, 4, 5, 6, 7, 7, 7, 8, 15?
20. Multiple Choice
30 seconds
1 pt
What kind of relationship?
Positive Correlation
Negative Correlation
No Relationship
21. Multiple Choice
30 seconds
1 pt
What kind of relationship?
Positive Correlation
Negative Correlation
No Relationship
22. Multiple Choice
30 seconds
1 pt
What kind of relationship?
Positive Correlation
Negative Correlation
No Relationship | {"url":"https://quizizz.com/admin/quiz/5f5a20d5e52d7a001ccf890e/psychology-research-methods","timestamp":"2024-11-08T19:04:37Z","content_type":"text/html","content_length":"458621","record_id":"<urn:uuid:65947aaf-c718-4bf7-b661-af99897d250a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00650.warc.gz"} |
Browse by Authors
Number of items: 10.
Braack, M. and Lang, J. and Taschenberger, N. (2012) Hierarchical a posteriori residual based error estimators for bilinear finite elements. Comput. Meth. Appl. Mech. Eng. . (Submitted)
Gottermeier, B. and Lang, J. (2010) Adaptive two-step peer methods for thermally coupled incompressible flow. ., Proceedings of V. European Conference on Computational Fluid Dynamics ECCOMAS CFD
2010. (Unpublished)
Ullman, S. and Lang, J. (2010) A POD-Galerkin reduced model with updated coefficients for smagorinsky LES. ., Proceedings of V. European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010.
Huang, W. and Kamenski, L. and Lang, J. (2010) A new anisotropic mesh adaption method based upon hierarchical a posteriori error estimates. J. Comput. Phys., 229 . pp. 2179-2198.
Gerisch, A. and Lang, J. and Podhaisky, H. and Weiner, R. (2009) High-order linearly implicit two-step peer-finite element method for time-dependent PDEs. Applied Numerical Mathematics, 59 (3-4).
624-638 . ISSN 0168-9274
Löbig, S. and Dörnbrack, A. and Fröhlich, J. and Hertel, C. and Kühnlein, C. and Lang, J. (2009) Towards Large-Eddy Simulation on moving grids. PAMM · Proc. Appl. Math. Mech., 9 . 445 -446.
Teleaga, , I. and Lang, J. (2008) Higher-order linearly implicit one-step methods for three-dimensional incompressible Navier-Strokes equations. Studia Babes-Bolyai Matematica, 53 . pp. 109-121.
Lang, J. and Verwer, J. (2007) On Global Error Estimation and Control for Initial Value Problems. SIAM J. Sci. Comp., 29 (4). pp. 1460-1475. ISSN 1064-8275 (print) 1095-7197 (online)
Debrabant, K. and Lang, J. (2007) On Global Error Estimation and Control for Parabolic Equations, Report No. 2512. Appl. Num. Math. . (Submitted)
Erdmann, B. and Kober, C. and Lang, J. and Deuflhard, P. and Zeilhofer, H.-F. and Sader, R. (2001) Efficient and reliable finite element methods for simulation of the human mandible. In: Proceedings
of Medicine Meets Mathematics. Kloster Banz/Staffelstein. Report ZIB 01-14, 2001. (Submitted) | {"url":"http://publications.imp.fu-berlin.de/view/people/Lang=3AJ=2E=3A=3A.html","timestamp":"2024-11-13T00:58:15Z","content_type":"application/xhtml+xml","content_length":"14048","record_id":"<urn:uuid:f2d098d1-c129-4c70-9145-d1d50b414c4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00040.warc.gz"} |
Exact is not Exact
Copyright (c) 2019
All Rights Reserved
amortization.com Ltd.
Burlington, Ontario
Amortization Pro for iPhone/iPad/iPod
The 365 day year Exact day monthly payment mortgage has already been covered. There are some lenders that say they have an exact day monthly payment mortgage but neglect to mention it is based upon
the 360 day Bankers year. Again a 12% rate is used so that all the examples are consistent on this site for comparison purposes. If you had a traditional mortgage at 12% and monthly compounding based
upon the 360 day year (Bankers year in the USA) the monthly interest factor is 1% and the monthly interest cost at the end of the month, for a loan with a balance of $100,000 at the begining of the
month, would be $1,000 (that is .01 x $100,000= $1,000). I will use a 31 day month as an example as it covers 7 out of 12 of the months in a year which represents more than 58% of your payments.
If the lender was using the 365 day method the interest cost at the end of the 31 day month would be $1,019.28
If the lender was using the 360 day method the interest cost at the end of the 31 day month would be $1,033.51
There are two ways to know for sure which of the three interest calculation methods the lender is using. An amortization schedule would show the difference on the very first payment. The lender could
provide you with the effective interest rate, EIR, minus all the APR smoke and mirrors. The latter is difficult to attain because the disclosure laws are confusing to most people and most people do
not understand compounding and the real value of the EIR.
(Screenshot 1)
Below are the monthly interest costs on a $100,000 balance owing for the 28, 29, 30 and 31 day month utilizing 365 and 360 day years. These are significant differences and you should know about them
as it is your money.
(Screenshot 2)
amortizationdotcom Mortgage Calculator for iPhone
Introduction to Canadian and American Mortgages
Seminar on prepaying principal (Part A)
Seminar on prepaying principal (Part B)
Global TV Interview regarding 40 Year Mortgages
Look for this logo on the Apple Store! | {"url":"http://amortization.com/exact_is_not_exact.htm","timestamp":"2024-11-03T09:05:35Z","content_type":"text/html","content_length":"11852","record_id":"<urn:uuid:e9bf5d23-6476-4567-8d1c-5960c0dbcee7>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00065.warc.gz"} |
Chapter 5 | Profit, Loss And Discount | Class-8 DAV Secondary Mathematics | NCERTBOOKSPDF.COM
Chapter 5 | Profit, Loss And Discount | Class-8 DAV Secondary Mathematics
Are you looking for DAV Maths Solutions for class 8 then you are in right place, we have discussed the solution of the Secondary Mathematics book which is followed in all DAV School. Solutions are
given below with proper Explanation please bookmark our website for further update !! All the Best !!
DAV SOLUTIONS CLASS 8 Secondary Mathematics || Profit, Loss And Discount
|| Unit 5 Worksheet 2
1. The marked price of a paint is ₹ 1,250 and the shopkeeper allows a discount of 8% on it. Find the discount and the selling price of the paint.
2. The marked price of a water cooler is ₹ 5,400. The shopkeeper offers an off season discount of 20% on it. Find its selling price.
3. An almirah of marked price ₹ 4,000 is sold for ₹ 3,700 after allowing certain discount. Find the rate of discount.
4. Find the rate of discount being given on a ceiling fan whose selling price is ₹ 1,175 after allowing a discount of ₹75 on its marked price.
5. Find the marked price of a washing machine which is sold at ₹8,400 after allowing a discount of 16%.
6. A dinner set was bought for ₹ 2,464 after getting a discount of 12% on its marked price. Find the marked price of the dinner set.
7. The marked price of a computer is ₹ 22,000. After allowing a 10% discount, a dealer still makes a profit of 20%. Find the cost price of a computer.
8. The marked price of a double bed is ₹9,575. A shopkeeper allows a discount of 12% on its marked price and still gains 10%. Find the cost price of the double bed.
9. How much a shopkeeper must mark his goods so that after allowing a discount of 25% on the marked price, he gains 20%, if the cost price of the goods is ₹20,000?
10. Priti allows 8% discount on the marked price of the suits and still makes a profit of 15%. If her gain over the scale of a suit is ₹ 156, find the marked price of the suit.
2 thoughts on “Chapter 5 | Profit, Loss And Discount | Class-8 DAV Secondary Mathematics”
1. 1 question left
2. jiyasinha | {"url":"https://ncertbookspdf.com/chapter-5-profit-loss-and-discount-class-8-dav-secondary-mathematics-2/","timestamp":"2024-11-08T09:27:16Z","content_type":"text/html","content_length":"91538","record_id":"<urn:uuid:3be7ffca-43ff-476e-bbc5-b250860375f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00566.warc.gz"} |
Laws of exponents
Slide 1
Laws of Exponents
Whenever we have variables which contain exponents and have equal bases, we can do certain mathematical operations to them. Those operations are called the “Laws of Exponents”
b = base x = exponent
Slide 2
Laws of Exponents
Slide 3
Other Properties of Exponents
Any single number or variable is always to the first power
Slide 4
Basic Examples
Slide 5
Basic Examples
Slide 6
More Examples
Slide 7
More Examples | {"url":"https://www.sliderbase.com/spitem-261-1.html","timestamp":"2024-11-12T23:35:19Z","content_type":"text/html","content_length":"10260","record_id":"<urn:uuid:e5d5f84b-74a1-44ca-939a-9b7ab1f485f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00466.warc.gz"} |
COVAR | Spread.NET 17 Formula Reference
In This Topic
This function returns the covariance, which is the average of the products of deviations for each data point pair in two sets of numbers.
The two arrays of data in the arguments of this function should meet these criteria:
• The data should contain numbers, names, arrays, or references that are numeric. If some cells do not contain numeric data, they are ignored.
• The data sets should be the same size, with the same number of data points.
• The data sets should not be empty, nor should the standard deviation of their values equal zero.
Use this covariance function to determine the relationship between two sets of data. For example, you can examine whether greater income accompanies greater levels of education in a population.
The covariance is calculated as follows, where n is the size of the arrays and mu is the mean.
Data Types
Accepts arrays of numeric data for both arguments. Returns numeric data.
COVAR({7,5,6},{7,4,4}) gives the result 1
COVAR({5,10,15,20,25},{4,8,16,32,64}) gives the result 144
Version Available
This function is available in product version 1.0 or later.
See Also
CORREL | VAR | Statistical Functions | {"url":"https://developer.mescius.com/spreadnet/docs/latest/online-formula/FunctionCOVAR.html","timestamp":"2024-11-13T03:06:31Z","content_type":"application/xhtml+xml","content_length":"15624","record_id":"<urn:uuid:1b6de9e7-a916-4c0d-b4fb-e23e97d15d93>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00187.warc.gz"} |
Transactions Online
Junichi NAKAYAMA, Yasuhiko TAMURA, Kiyoshi TSUTSUMI, "Shadow Theory of Diffraction Grating: A Numerical Example for TE Wave" in IEICE TRANSACTIONS on Electronics, vol. E92-C, no. 3, pp. 370-373,
March 2009, doi: 10.1587/transele.E92.C.370.
Abstract: By use of the shadow theory developed recently, this paper deals with the transverse electric (TE) wave diffraction by a perfectly conductive periodic array of rectangular grooves. A set of
equations for scattering factors and mode factors are derived and solved numerically. In terms of the scattering factors, diffraction amplitudes and diffraction efficiencies are calculated and shown
in figures. It is demonstrated that diffraction efficiencies become discontinuous at an incident wave number where the incident wave is switched from a propagating wave to an evanescent one, whereas
scattering factors and diffraction amplitudes are continuous even at such an incident wave number.
URL: https://global.ieice.org/en_transactions/electronics/10.1587/transele.E92.C.370/_p
author={Junichi NAKAYAMA, Yasuhiko TAMURA, Kiyoshi TSUTSUMI, },
journal={IEICE TRANSACTIONS on Electronics},
title={Shadow Theory of Diffraction Grating: A Numerical Example for TE Wave},
abstract={By use of the shadow theory developed recently, this paper deals with the transverse electric (TE) wave diffraction by a perfectly conductive periodic array of rectangular grooves. A set of
equations for scattering factors and mode factors are derived and solved numerically. In terms of the scattering factors, diffraction amplitudes and diffraction efficiencies are calculated and shown
in figures. It is demonstrated that diffraction efficiencies become discontinuous at an incident wave number where the incident wave is switched from a propagating wave to an evanescent one, whereas
scattering factors and diffraction amplitudes are continuous even at such an incident wave number.},
TY - JOUR
TI - Shadow Theory of Diffraction Grating: A Numerical Example for TE Wave
T2 - IEICE TRANSACTIONS on Electronics
SP - 370
EP - 373
AU - Junichi NAKAYAMA
AU - Yasuhiko TAMURA
AU - Kiyoshi TSUTSUMI
PY - 2009
DO - 10.1587/transele.E92.C.370
JO - IEICE TRANSACTIONS on Electronics
SN - 1745-1353
VL - E92-C
IS - 3
JA - IEICE TRANSACTIONS on Electronics
Y1 - March 2009
AB - By use of the shadow theory developed recently, this paper deals with the transverse electric (TE) wave diffraction by a perfectly conductive periodic array of rectangular grooves. A set of
equations for scattering factors and mode factors are derived and solved numerically. In terms of the scattering factors, diffraction amplitudes and diffraction efficiencies are calculated and shown
in figures. It is demonstrated that diffraction efficiencies become discontinuous at an incident wave number where the incident wave is switched from a propagating wave to an evanescent one, whereas
scattering factors and diffraction amplitudes are continuous even at such an incident wave number.
ER - | {"url":"https://global.ieice.org/en_transactions/electronics/10.1587/transele.E92.C.370/_p","timestamp":"2024-11-04T05:55:48Z","content_type":"text/html","content_length":"59899","record_id":"<urn:uuid:1f3dd1a4-8de7-432d-af80-3a9d06e50bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00629.warc.gz"} |
Curiosities: Partial Differential Equations
From the flow of air to the collapsing of a star to the spreading of a pollutant
A variety of systems in natural sciences are described through physically measurable quantities that depend on “independent variables.” For instance, we routinely measure the pressure and the
temperature of the air in the Earth’s atmosphere, and such measurements depend upon the time and the location of the device used. Several fundamental laws discovered by scientists through the last
three centuries give relations among the rates of change of such physical quantities. The resulting mathematical objects, called partial differential equations, are therefore ubiquitous in modern
science and engineering: they are efficiently used to model a variety of different phenomena, like the flow of air past the wings of an airplane, the collapsing of a star into a black hole, or the
spreading of a pollutant in the air.
The theoretical study of partial differential equations is a branch of pure mathematics that dates back to the dawn of modern sciences, originating in the works of Bernoulli, Fermat, Newton,
Lagrange, Euler, and several others. Central theoretical questions are the existence of solutions, how they behave, what we need to know to determine them, and whether they break down, for instance,
when they get in a range where the validity of the equations can be challenged. The latter phenomenon is usually called singularity formation. Such questions, especially the formation of
singularities and their descriptions, are the main subjects of my research. The two topics in which I have spent most of my recent efforts are the calculus of variations and the equations of
incompressible fluid dynamics.
In the calculus of variations, one seeks the solution of a minimum problem, for instance a shape that optimizes a certain feature. A prominent example is named after the Belgian nineteenth-century
physicist Joseph Plateau, who proposed to study area-minimizing surfaces, namely surfaces which minimize their area among those which span a fixed contour. It is long known that such surfaces might
have singularities, for instance, the formation of certain type of corners, but a complete description of the type and size of singularities is a long-standing open problem. I have shown with my
collaborators that surprisingly many singularities can occur at the junction between an area-minimizing surface and its contour, even when the latter is quite simple and smooth. However, our work
also gives the first proven theoretical limitation to the size of the singularities without any special geometric assumption on the contour. When the contour is a real analytic curve, a conjecture by
White asserts that in fact there can be only finitely many singularities. A recent preprint authored by IAS Member Zihui Zhao and me gives a first step in that direction.
The first system of partial differential equations ever written down in fluid dynamics is given by the Euler equations, found by Leonhard Euler more than 250 years ago. The incompressible Euler
equations are in fact a limiting case of another well-known system, the Navier-Stokes equations.
Whether regular solutions of the Euler and Navier-Stokes equations might form singularities in finite time is one of the biggest open problems in mathematics: for the Navier-Stokes equations, it is
one of the famous millennium prize problems. In the last decade, I have shown with László Székelyhidi, Jr., that there are very irregular solutions, many more than expected, and that they might
behave in a very surprising way. Our new approach borrows from the pioneering work of John Nash of the 1950s on the isometric embedding problem, a thus far completely unrelated topic in differential
geometry, another branch of mathematics. My ideas with László are at the base of recent important developments, such as the resolution by Phil Isett of a 1949 fundamental conjecture of Lars Onsager
(Nobel Prize winner in chemistry) in the theory of turbulent flows, and the unexpected discovery by Tristan Buckmaster and Vlad Vicol that irregular solutions of the Navier-Stokes system are not
uniquely determined by the equations. | {"url":"https://www.ias.edu/ideas/curiosities-partial-differential-equations","timestamp":"2024-11-13T05:59:44Z","content_type":"text/html","content_length":"71314","record_id":"<urn:uuid:ed28339f-53c6-4748-8ee7-8f97ec267422>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00669.warc.gz"} |
2.3.1: Single Bar Graphs
Last updated
Page ID
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\
evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\
newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}
\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}
{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}
[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}
{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\
#3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp
#2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\
wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\
newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}
{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\
bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
Bar Graphs
Judy has planted a garden in her backyard. She decided to keep track of how many vegetables were growing each month in her garden. The data table below shows the amounts of vegetables she grew in
July and August.
│ July │ August │
│30 carrots │60 carrots │
│10 tomatoes │20 tomatoes │
│25 zucchini │30 zucchini │
│15 squash │25 squash │
│10 potatoes │20 potatoes │
Judy wants to visually display her data in a way that she can show the community board to earn recognition for her growing garden. She wants the data to be easy to read and understand.
In this concept, you will learn how to create a bar graph from data and answer questions about the data represented.
Bar graphs are created from a set of data. It is called a bar graph because it is a visual display of data using bars. The number of items tells us how many bars the graph will have. The amount of
each item tells us how long each bar will be. Look at the following data. A bar graph can be used to represent the data, which tells how many hours students in the fifth, sixth, seventh, and eighth
grade classes volunteered in a month.
│Class │Number of Hours │
│5th │51 │
│6th │88 │
│7th │75 │
│8th │39 │
You can see that this information has been written in the form of a frequency table. It shows us how many hours each class has worked.
Now a bar graph can be created to display the information.
A bar graph contains two axes. One axis represents the items, which goes across the bottom on what is called the X-axis. The other represents the amounts of each item, which goes along the side on
what is called the Y-axis. The “items” in this case are each class. The amounts are the number of hours the classes worked. Axes always need to be labeled to show what each axis represents.
Richard Croft - https://commons.wikimedia.org/wiki/File:Vegetable_garden_-_geograph.org.uk_-_821006.jpg;https://pixabay.com/en/photos/vegetables/ - CC BY-NC
Next, a scale needs to be chosen for the amounts on the left side of the bar graph. Scales of 1, 2, 5, 10, 20, 50, 100, 1,000, or more a typically used because they are easy numbers to count by. To
choose the scale, look at the amounts you’ll be graphing, especially the largest amount. In the example, the greatest value is 88. If a scale of 100 is used, the scale marks on the left side of the
graph would be 0, 100, 200, and so on. It would be very difficult to read most of the amounts on this scale because it is too big. On the other hand, if a small scale is used, such as 5, the graph
would have to be very large to get all the way up to 90 (since the greatest value is 88).
It makes the most sense to use a scale that goes from 0 to 90 counting by 10’s. That way each value can easily represent the hours that each class worked.
Here is what the graph looks like with the scale filled in.
USGS - http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php;http://pixabay.com/en/server-computer-case-controller-40240/ - CC BY-NC
Now the bars can be drawn in to represent each number of hours that the students worked.
USGS - http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php;http://pixabay.com/en/server-computer-case-controller-40240/ - CC BY-NC
Guided Practice
What number of 7th graders have a favorite activity of watching television?
Richard Croft - https://commons.wikimedia.org/wiki/File:Vegetable_garden_-_geograph.org.uk_-_821006.jpg;https://pixabay.com/en/photos/vegetables/ - CC BY-NC
First, look for the column that refers to television. This is the activity that the question is referring to; therefore, is the data you are targeting.
Next, look at the vertical axis. This will show you where the data falls on the scale representing the number of students who enjoy that activity.
Then, answer the question with the exact number for the Y-axis that matches up with the bar in the graph representing television.
The answer is 9 seventh graders have "watching TV" as their favorite activity.
Example 1
Which state has the highest average price for gasoline?
Richard Croft - https://commons.wikimedia.org/wiki/File:Vegetable_garden_-_geograph.org.uk_-_821006.jpg;https://pixabay.com/en/photos/vegetables/ - CC BY-NC
First, look at the bar graph to determine the longest bar in the graph (which would represent the highest price for gas).
Next, look at the X-axis to see which state the longest bar belongs to. Also check the Y-axis to make sure that bar in the graph is matched to the highest amount on the Y-axis.
Then, answer the question with the data bar that you found to have the highest price of gasoline as represented in the graph.
The answer is Hawaii.
Example 2
Using the same bar graph above, which state has the lowest average price?
First, look at the bar graph to determine the shortest bar in the graph.
Next, look at the X-axis to see which state is represented by the shortest bar. Check the Y-axis as well to make sure the bar is matched to the lowest amount of all states on the graph.
Then, answer the question with the data bar that displays the lowest price of gasoline.
The answer is Missouri.
Example 3
Which state has the second highest average price?
First, look at the graph to find not the highest or lowest price, but the 2nd highest bar in the graph. The highest price is in Hawaii, so then look for the bar that is next highest to Hawaii.
Next, look at the X-axis to see which state is represented by the bar that is 2nd highest.
Then, answer the question with the data you find on X-axis that shows which state has 2nd highest gasoline prices.
The answer is California.
Follow Up
Richard Croft - https://commons.wikimedia.org/wiki/File:Vegetable_garden_-_geograph.org.uk_-_821006.jpg
Remember Judy and her growing garden that she wants to display to the community board? She kept track of the amounts of vegetables that grew in her garden each month and she now wants to visually
display that data. A bar graph is a great way to represent her data.
First, Judy collects and organizes her data. In July, Judy grew:
30 carrots
10 tomatoes
25 zucchini
15 squash
10 potatoes
Next, Judy can make the bar graph. Her amounts range from 10 to 30, so she can start her graph at 0 and use a scale that has increments of five. Here is the bar graph.
Richard Croft - https://commons.wikimedia.org/wiki/File:Vegetable_garden_-_geograph.org.uk_-_821006.jpg;https://pixabay.com/en/photos/vegetables/ - CC BY-NC
Judy can do the same thing with her data from August.
60 carrots
20 tomatoes
30 zucchini
25 squash
20 potatoes
Judy has a range of 20 to 60 in her August data, so she uses a different scale in her August bar graph. She uses increments of 10 on her Y-axis up to 70.
Richard Croft - https://commons.wikimedia.org/wiki/File:Vegetable_garden_-_geograph.org.uk_-_821006.jpg;https://pixabay.com/en/photos/vegetables/ - CC BY-NC
Then, Judy can display her data to the board and prove her garden is growing and producing vegetables that would earn her the recognition she is seeking. The board may ask her questions, or Judy can
make conclusions using the data in her graph such as:
In both months, Judy grew the most carrots of any vegetable in her garden. Judy could also exactly explain the amount of carrots that were grown in both months by reading the Y-axis next to the
carrots data bar.
Explore More
Use the bar graph to answer the following questions.
USGS - http://earthquake.usgs.gov/earthquakes/eqarchives/year/graphs.php - CC BY-NC
1. How many students were asked if they have summer jobs?
2. What is the range of the data?
3. What are the three jobs that students have?
4. How many students do not have a summer job?
5. How many students babysit?
6. How many students do yard work in the summer?
7. How many students work at an ice cream stand in the summer?
8. If ten more students got a job this summer, how many students would have summer jobs?
9. If each category had double the number of students in it, how many students would have summer jobs?
10. How many students would babysit?
11. How many students would work at an ice cream stand?
12. How many students wouldn’t have a summer job?
13. What scale was used for this graph?
14. What interval was used in the scale?
15. What is the difference between working at an ice cream stand and doing yard work?
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 2.11.
Term Definition
bar graph A bar graph is a plot made of bars whose heights (vertical bars) or lengths (horizontal bars) represent the frequencies of each category, with space between each bar.
Analyze To analyze is to look at data and draw conclusions based on patterns or numbers. | {"url":"https://k12.libretexts.org/Bookshelves/Mathematics/Statistics/02%3A_Visualizing_Data_-_Data_Representation/2.03%3A_Bar_Graphs/2.3.01%3A_Single_Bar_Graphs","timestamp":"2024-11-07T10:56:23Z","content_type":"text/html","content_length":"148185","record_id":"<urn:uuid:ec60fe17-c17a-45ee-9b52-2d8fd72eb1b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00199.warc.gz"} |
Agrawal, R., and Srikant, R. (1994). “Fast Algorithms for Mining Association Rules.” In Proceedings of the 20th VLDB Conference. Santiago, Chile: IBM Almaden Research Center. Accessed July 5, 2016.
Baglama, J., and Reichel, L. (2005). “Augmented implicitly restarted Lanczos bidiagonalization methods.” SIAM Journal on Scientific Computing, 27:19–42.
Bates, D. M., and Watts, D. G. (1988). Nonlinear Regression Analysis and Its Applications. New York: John Wiley & Sons.
Benford, F. (1938). “The law of anomalous numbers.” Proceedings of the American philosophical society, 551–572.
Benjamini, Y., and Hochberg, Y. (1995). “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.” Journal of the Royal Statistical Society, Series B 57:289–300.
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice-Hall.
Candes, E. J., Li, X., Ma, Y., and Wright, J. (2009). “Robust Principal Component Analysis?” Journal of the ACM, 58:1–37.
Cleveland, W. S. (1994). Visualizing Data, Summit, NJ: Hobart Press.
Conover, W. J. (1999). Practical Nonparametric Statistics. 3rd ed. New York: John Wiley & Sons.
Cureton, E. E. (1967). “The Normal Approximation to the Signed-Rank Sampling Distribution when Zero Differences are Present.” Journal of the American Statistical Association 62: 319, 1068–1069.
Hahsler, M. (2015). “A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules.” Accessed September 17, 2018. http://michael.hahsler.net/research/association_rules/
Hand, D. J., Mannila, H., and Smyth, P. (2001). Principles of Data Mining. Cambridge, MA: MIT Press.
Hastie, T. J., Tibshirani, R. J., and Friedman, J. H. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd ed. New York: Springer-Verlag.
Hawkins D. M., and Kass G. V. (1982). “Automatic Interaction Detection.” In Topics in Applied Multivariate Analysis, edited by D. M. Hawkins, 267–300. Cambridge: Cambridge University Press.
Huber, P. J., and Ronchetti, E. M. (2009). Robust Statistics. 2nd ed. New York: John Wiley & Sons.
Hyndman, R. J., Koehler, A. B., Ord, J. K., and Snyder, R. D. (2008). Forecasting with Exponential Smoothing: The State Space Approach. Berlin: Springer-Verlag.
Jolliffe, I. T. (2002). Principal Component Analysis. New York: Springer-Verlag.
Kass, G. V. (1980). “An Exploratory Technique for Investigating Large Quantities of Categorical Data.” Journal of the Royal Statistical Society, Series C 29:119–127
Lehman, E. L. (2006). Nonparametrics: Statistical Methods Based on Ranks. 2nd ed. New York: Springer.
Lin, Z., Chen, M., and Ma, Y. (2013). The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv preprint arXiv:1009.5055.
Mason, R. L., and Young, J. C. (2002). Multivariate Statistical Process Control with Industrial Applications. Philadelphia: SIAM.
McCullagh, P., and Nelder, J. A. (1989). Generalized Linear Models. 2nd ed. London: Chapman & Hall.
Nagelkerke, N. J. D. (1991). “A Note on a General Definition of the Coefficient of Determination.” Biometrika 78:691–692.
Nelder, J. A., and Wedderburn, R. W. M. (1972). “Generalized Linear Models.” Journal of the Royal Statistical Society, Series A 135:370–384.
Parker, R. J. (2015). Efficient Computational Methods for Large Spatial Data. Ph.D. diss., Department of Statistics, North Carolina State University. Accessed June 30, 2016. https://
Platt, J. (1998). Sequential minimal optimization: A fast algorithm for training support vector machines. Technical Report MST-TR-98-14, Microsoft Research. https://www.microsoft.com/en-us/research/
Qian, P. Z., Huaiquing, W., and Wu, C. F. (2012). “Gaussian process models for computer experiments with qualitative and quantitative factors.” Technometrics 50:383–396.
Ramsay, J. O., and Silverman, B. W. (2005). Functional Data Analysis. 2nd ed. New York: Springer.
Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models. New York: Marcel Dekker.
Sall, J. (2002). “Monte Carlo Calibration of Distributions of Partition Statistics.” SAS Institute Inc.,Cary, NC. Accessed July 29, 2015. https://www.jmp.com/content/dam/jmp/documents/en/white-papers
Santer, T., Williams, B., and Notz, W. (2003). The Design and Analysis of Computer Experiments. New York: Springer-Verlag.
SAS Institute Inc. (2020). SAS/ETS 15.2 User’s Guide. Cary, NC: SAS Institute Inc. https://go.documentation.sas.com/api/docsets/etsug/15.2/content/etsug.pdf.
Schäfer, J., and Strimmer, K. (2005). “A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics.” Statistical Applications in Genetics and Molecular
Biology 4, Article 32.
Schuirmann, D. J. (1987). “A Comparison of the Two One-sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability.” Journal of Pharmacokinetics and
Biopharmaceutics 15:657–680.
Shiskin, J., Young, A. H., and Musgrave, J. C. (1967). The X-11 Variant of the Census Method II Seasonal Adjustment Program. Technical Report 15, US Department of Commerce, Bureau of the Census.
Shmueli, G., Patel, N. R., and Bruce, P. C. (2010). Data Mining For Business Intelligence: Concepts, Techniques, and Applications in Microsoft Office Excel with XLMiner. 2nd ed. Hoboken, NJ: John
Wiley & Sons.
Shmueli, G., Bruce, P. C., Stephens M. L., and Patel, N. R. (2017). Data Mining For Business Intelligence: Concepts, Techniques, and Applications with JMP Pro. Hoboken, NJ: John Wiley & Sons.
Westfall, P. H., Tobias, R. D., and Wolfinger, R. D. (2011). Multiple Comparisons and Multiple Tests Using SAS. 2nd ed. Cary, NC: SAS Institute Inc.
White, K., Szarka, J., and Jensen, W. (2018). “A Recommended Set of Indices for Evaluating Process Health.” Presentation at the 2018 Fall Technical Conference, sponsored by the American Statistical
Association and the American Society for Quality. | {"url":"https://www.jmp.com/support/help/ja/16.2/jmp/references-6.shtml","timestamp":"2024-11-05T06:31:52Z","content_type":"application/xhtml+xml","content_length":"15016","record_id":"<urn:uuid:93d6bc22-f5ca-4815-9831-24c48f2c20c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00237.warc.gz"} |
Square Foot to Square Centimeters
Square Foot to Square Centimeters Converter
Enter Square Foot
Square Centimeters
⇅ Switch toSquare Centimeters to Square Foot Converter
How to use this Square Foot to Square Centimeters Converter 🤔
Follow these steps to convert given area from the units of Square Foot to the units of Square Centimeters.
1. Enter the input Square Foot value in the text field.
2. The calculator converts the given Square Foot into Square Centimeters in realtime ⌚ using the conversion formula, and displays under the Square Centimeters label. You do not need to click any
button. If the input changes, Square Centimeters value is re-calculated, just like that.
3. You may copy the resulting Square Centimeters value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Square Foot to Square Centimeters?
The formula to convert given area from Square Foot to Square Centimeters is:
Area[(Square Centimeters)] = Area[(Square Foot)] × 929.0304
Substitute the given value of area in square foot, i.e., Area[(Square Foot)] in the above formula and simplify the right-hand side value. The resulting value is the area in square centimeters, i.e.,
Area[(Square Centimeters)].
Calculation will be done after you enter a valid input.
Consider that a living room has an area of 300 square feet.
Convert this area from square feet to Square Centimeters.
The area in square foot is:
Area[(Square Foot)] = 300
The formula to convert area from square foot to square centimeters is:
Area[(Square Centimeters)] = Area[(Square Foot)] × 929.0304
Substitute given weight Area[(Square Foot)] = 300 in the above formula.
Area[(Square Centimeters)] = 300 × 929.0304
Area[(Square Centimeters)] = 278709.12
Final Answer:
Therefore, 300 ft^2^ is equal to 278709.12 cm^2^.
The area is 278709.12 cm^2^, in square centimeters.
Consider that a small office space measures 500 square feet.
Convert this area from square feet to Square Centimeters.
The area in square foot is:
Area[(Square Foot)] = 500
The formula to convert area from square foot to square centimeters is:
Area[(Square Centimeters)] = Area[(Square Foot)] × 929.0304
Substitute given weight Area[(Square Foot)] = 500 in the above formula.
Area[(Square Centimeters)] = 500 × 929.0304
Area[(Square Centimeters)] = 464515.2
Final Answer:
Therefore, 500 ft^2^ is equal to 464515.2 cm^2^.
The area is 464515.2 cm^2^, in square centimeters.
Square Foot to Square Centimeters Conversion Table
The following table gives some of the most used conversions from Square Foot to Square Centimeters.
Square Foot (ft^2^) Square Centimeters (cm^2^)
0 ft^2^ 0 cm^2^
1 ft^2^ 929.0304 cm^2^
10 ft^2^ 9290.304 cm^2^
45 ft^2^ 41806.368 cm^2^
90 ft^2^ 83612.736 cm^2^
180 ft^2^ 167225.472 cm^2^
360 ft^2^ 334450.944 cm^2^
1000 ft^2^ 929030.4 cm^2^
Square Foot
A square foot (ft^2) is a unit of area measurement equal to the area of a square with sides one foot in length. It is commonly used in the United States, Canada, and the United Kingdom to measure
smaller areas such as the size of rooms, apartments, or other building spaces. Square feet are essential in real estate, interior design, and construction, providing a standard measure for comparing
and planning spaces.
Square Centimeters
A square centimeter (cm^2) is a unit of area measurement equal to the area of a square with sides that are one centimeter in length. It is a smaller unit of area often used in science, medicine, and
small-scale engineering projects. Square centimeters are useful for measuring smaller surfaces, such as the area of a piece of fabric, the surface area of a small object, or in medical imaging to
describe the size of lesions or wounds.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Square Foot to Square Centimeters in Area?
The formula to convert Square Foot to Square Centimeters in Area is:
Square Foot * 929.0304
2. Is this tool free or paid?
This Area conversion tool, which converts Square Foot to Square Centimeters, is completely free to use.
3. How do I convert Area from Square Foot to Square Centimeters?
To convert Area from Square Foot to Square Centimeters, you can use the following formula:
Square Foot * 929.0304
For example, if you have a value in Square Foot, you substitute that value in place of Square Foot in the above formula, and solve the mathematical expression to get the equivalent value in Square | {"url":"https://convertonline.org/unit/?convert=sq_foot-sq_cm","timestamp":"2024-11-03T10:45:09Z","content_type":"text/html","content_length":"75846","record_id":"<urn:uuid:c62fd072-4e23-41a7-91a4-e56938806cd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00856.warc.gz"} |
Teaching Ordinal Numbers Year 2 - OrdinalNumbers.com
Teaching Ordinal Numbers Year 2
Teaching Ordinal Numbers Year 2 – By using ordinal numbers, it is possible to count any number of sets. They can also be utilized to generalize ordinal numbers.But before you are able to use them,
you need to know the reasons why they exist and how they work.
The ordinal number is among the fundamental concepts in mathematics. It is a number that identifies where an object is in a list of objects. A number that is normally between one and 20 is utilized
as the ordinal number. Ominal numbers are used for a variety of purposes but are most commonly used in order to indicate the order of items on an agenda.
Ordinal numbers can be represented with charts, words, numbers and other methods. They can also be used to indicate how a collection of pieces are arranged.
Ordinal numbers mostly are classified into two groups. The infinite ordinal numbers are represented by lowercase Greek letters, while finite ones are represented with Arabic numbers.
According to the axiom, every well-ordered set should have at least one ordinal. For example, the top grade would be given to the first person in the class. The contest winner was the one who
received the highest grade.
Combinational ordinal numbers
Multidigit numbers are called compound ordinal numbers. They are made by multiplying ordinal numbers by its final number. These numbers are commonly used for dating and ranking. They don’t have a
unique ending to the last digit, as do cardinal numbers.
Ordinal numerals are used to identify the sequence of elements in the collection. They are used to identify the items within a collection. There are two kinds of ordinal numbers: regular and
By prefixing a cardinal number by the suffix -u, regular ordinals are created. After that, the number is entered as a word and then an hyphen is added to it. There are other suffixes you can use.
“-nd” is for numbers that begin with 2 is an instance. “-th”, for numbers with endings between 4 and 9, is another.
Suppletive ordinals can be created by prefixing words with the suffix -u or -e. The suffix is employed to count and is usually wider than the regular one.
Limits to the significance of ordinal numbers
Ordinal numbers that do not exceed zero are the maximum for ordinal numbers. Limit ordinal numbers have the drawback that there is no maximum element for them. They are created by joining non-empty
sets with no the possibility of having maximum elements.
Limits on ordinal numbers can also be utilized in transfinite definitions of Recursion. Based on the von Neumann model, each infinite cardinal number is also an ordinal limit.
The ordinal numbers that have limits are equivalent to the sum of all ordinals that are lower than it. Limit ordinal numbers are easy to calculate with arithmetic. However, they can also be expressed
in natural numbers.
These numbers are arranged according to ordinal numbers. They provide an explanation of an object’s numerical location. They are often applied in the context of set theory and arithmetic. Although
they share the same form, they aren’t classified the same way as a natural number.
In the von Neumann model, a well-ordered collection is used. Assume that fyyfy is a subfunction g’ of a function described as a singular operation. If the subfunction fy’s is (i, ii), and g meets the
criteria that g is an ordinal limit.
The Church Kleene oral is a limit-ordering order that functions in a similar way. The Church-Kleene oral defines an appropriate limit as a well arranged collection of the smaller ordinals. It also
includes an ordinal that is nonzero.
Stories with examples of ordinal numbers
Ordinal numbers are frequently used to indicate the hierarchy of entities and objects. They are essential for organising, counting as well as ranking reasons. They are also useful for indicating the
order of things and to describe objects’ positions.
The letter “th” is commonly used to indicate the ordinal number. Sometimes, however the “nd” letter could be used in place of “th”. Titles of books typically include ordinal numbers.
Even though ordinal figures are usually used in list format they can be written down in the form of words. They can also be written by way of numbers or acronyms. According to research, these numbers
are more comprehensible than cardinal ones.
There are three types of ordinal numbers. These numbers can be learned more through games, practice, or other activities. You can increase your arithmetic skills by learning more about the basics of
them. Consider a coloring activity as an easy and enjoyable approach to improving. A simple marking sheet can be used to keep track of your results.
Gallery of Teaching Ordinal Numbers Year 2
Ordinal Numbers Worksheet Year 2 Printable Worksheets And Activities
Ordinal Numbers Activity For Grade 2
Ordinal Numbers Descriptions Worksheet By Teach Simple
Leave a Comment | {"url":"https://www.ordinalnumbers.com/teaching-ordinal-numbers-year-2/","timestamp":"2024-11-04T21:52:17Z","content_type":"text/html","content_length":"63800","record_id":"<urn:uuid:77cd58ad-b6ac-4f31-a627-be3e01c33d6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00402.warc.gz"} |
Transfinite combinators and admissible ordinals
I’m back with another topic concerning ordinals!
The transfinite combinator calculus
The transfinite combinator calculus is an extension of normal combinator calculus. We define several related concepts as follows:
• If an expression \(x\) is beta-reducible, then \(\operatorname{cof}(x)=\operatorname{cof}(\operatorname{reduce}(x))\) and \(x[\alpha]=\operatorname{reduce}(x)[\alpha]\).
• If \(y\) is a limit, then \(\operatorname{cof}(xy)=\operatorname{cof}(y)\) and \((xy)[\alpha]=x(y[\alpha])\).
• If \(x\) is a limit, \(y\) is not a limit, and \(xy\) is not beta-reducible, then \(\operatorname{cof}(xy)=\operatorname{cof}(x)\) and \((xy)[\alpha]=(x[\alpha])y\).
• \(x\) is a limit iff \(\operatorname{cof}(x)\) is defined.
The base of the transfinite combinator calculus are \(S\) and \(K\) which are both not limits. Depending on our application we may choose various infinite combinators to add.
Ordinals in transfinite combinator calculus
I’ve decided for ordinals to be first-class objects due to their importance. (I tried to make one where ordinals are represented by extensions of Church numerals and is rather inelegant and has
several problems.) When an ordinal needs to be applied to something else it behaves like its Church numeral. Formally, we can say that \((\alpha+1)xy=\alpha x(xy)\), and the cofinality and
fundamental sequences agree with the normal ones. (We assume a system of fundamental sequences exists for all ordinals.) A limit of ordinals agrees with the regular limit. We then introduce two
ordinal-based combinators:
• \(Ax\) for ordinals is the successor ordinal, and otherwise is the Church numeral successor.
• \(Exy\) if \(y\) is an ordinal \(<\operatorname{cof}(x)\) and \(x\) is a limit is \(x[y]\), otherwise it is undefined.
• \(Qxy\) checks for equality between ordinals if both are ordinals, otherwise applies the Church numeral comparison operator to its arguments.
• \(Px\) is the predecessor ordinal if \(x\) is a successor. If \(x\) is a limit ordinal, it returns \(x\). Otherwise, it applies the Church numeral predecessor operator to \(x\).
An OCF for admissible ordinals
We use the transfinite combinator calculus to construuct an OCF for admissible orrdinals. Like Bachmann’s, ours is somewhat cumbersome as it depends on fundamental sequences for all limit ordinals.
For every ordinal \(\alpha\), define a transfinite combinator calculus system \(\mathrm{TCC}_\alpha\) as follows:
• The combinator \(S\) is not a limit, and satisfies \(Sxyz=xz(yz)\).
• The combinator \(K\) is not a limit, and satisfies \(Kxy=x\).
• The combinators \(A\), \(E\), \(Q\), and \(P\) are not limits.
• The combinator \(\theta_\alpha\) is a limit. If either \(x\) or \(y\) is not an ordinal, its behavior is undefined. If \(y\ge\alpha\), it returns 0. Otherwise, it returns \(\theta_x(y)\).
Then let \[\Omega_\nu=\begin{cases}1 &\mbox{if } \nu=0\\ \aleph_\nu &\mbox{otherwise}\end{cases}\] And then \(\theta_\nu(0)=\omega_\nu\), and otherwise, \(\theta_\nu(\alpha)\) is the supremum of
ordinals in \(\Omega_{\nu+1}\) that can be calculated in \(\mathrm{TCC}_\alpha\).
Obviously \(\theta_\nu(\alpha)\) has cardinality \(\aleph_\nu\). It seems like it takes only admissible values. (Hence why I made ordinals first-class objects, otherwise there is no clear way to stop
it from taking the value \(\omega_{\omega_1^\mathrm{CK}}^\mathrm{CK}\).)
• \(\theta_0(n)=\omega_n^\mathrm{CK}\) for finite \(n\).
• \(\theta_0(\omega)=\omega_{\omega+1}^\mathrm{CK}\). (It skips over the non-admissible \(\omega_\omega^\mathrm{CK}\).)
• \(\theta_0(\Omega_1)\) is the first admissible after the fisrt fixed-point of \(\alpha\mapsto\omega_\alpha^\mathrm{CK}\).
DISCLAIMER: These values are incorrect.
Leave a Comment | {"url":"https://evin.one/2021/10/16/transfinite-combinators-and-admissible-ordinals/","timestamp":"2024-11-10T14:46:00Z","content_type":"text/html","content_length":"113578","record_id":"<urn:uuid:646e98fe-2a43-4756-abbe-9b406c4d9172>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00042.warc.gz"} |
How to Master the Discount Rate Formula for T-bills: A Step-by-Step Guide - Gospel10
How to Master the Discount Rate Formula for T-bills: A Step-by-Step Guide
The discount rate formula for T-bills calculates the rate at which future cash flows are discounted to determine their present value, specifically for Treasury bills.
For example, if you invest $100 in a one-year T-bill with a 5% discount rate, the present value would be $95.24, reflecting the discounted future cash flow of $100.
Understanding the discount rate formula is crucial for investors navigating the financial markets, as it allows them to assess the value of future income streams and make informed investment
decisions. Its historical roots can be traced back to the development of time value of money concepts, highlighting its long-standing significance in financial analysis.
The following sections will explore the key components of the discount rate formula, its applications, and practical considerations for investors.
Discount Rate Formula for T-bills
The discount rate formula for T-bills is a crucial aspect of fixed income investing. Its key aspects include:
• Present Value Calculation
• Future Cash Flows
• Discount Rate
• Time Value of Money
• Investment Analysis
• Risk Assessment
• Yield Curve
• T-bill Pricing
Understanding these aspects is essential for investors to accurately value T-bills, assess their risk-return profile, and make informed investment decisions. The discount rate formula helps determine
the present value of future cash flows, considering the time value of money and the prevailing interest rate environment. It also plays a vital role in constructing yield curves, which are crucial
for understanding interest rate dynamics and predicting future economic conditions.
Present Value Calculation
Present value calculation is a key aspect of the discount rate formula for T-bills, as it determines the present worth of future cash flows. It involves discounting future cash flows by the discount
rate to reflect the time value of money.
• Formula: Present Value = Future Cash Flow / (1 + Discount Rate)^n
• Time Value of Money: Considers that money today is worth more than the same amount in the future due to its earning potential.
• Discount Rate: The rate used to discount future cash flows, typically based on prevailing interest rates or the yield curve.
• Example: A $100 T-bill maturing in one year with a 5% discount rate has a present value of $95.24.
Understanding present value calculation is crucial for investors to assess the value of T-bills and make informed investment decisions. It allows them to compare T-bills with other investment options
and determine their potential return on investment. Additionally, present value calculation plays a vital role in portfolio management and financial planning, helping investors plan for future
financial goals and mitigate risks.
Future Cash Flows
Future cash flows are a critical component of the discount rate formula for T-bills, as they represent the anticipated payments that investors will receive in the future. Understanding these cash
flows is crucial for accurate valuation and investment decision-making.
• Principal Repayment: The face value of the T-bill, which is repaid to the investor at maturity.
• Interest Payments: T-bills do not pay periodic interest payments. Instead, the difference between the purchase price and the maturity value represents the implied interest earned.
• Maturity Date: The date on which the T-bill matures and the investor receives the principal repayment.
• Expected Inflation: Investors need to consider the impact of inflation on future cash flows, as it can erode the purchasing power of the principal and interest payments.
By carefully considering these aspects of future cash flows, investors can accurately assess the present value of T-bills and make informed investment decisions. It is important to note that the
discount rate used in the formula reflects the time value of money and the prevailing interest rate environment, which can influence the present value calculation.
Discount Rate
In the context of T-bills, the discount rate is the key determinant of their present value and a crucial component of the discount rate formula. It represents the rate at which future cash flows are
discounted to reflect their present worth, considering the time value of money and the prevailing interest rate environment.
The discount rate plays a pivotal role in the discount rate formula for T-bills. By incorporating the discount rate, the formula accurately calculates the present value of the future cash flows
associated with the T-bill, including the principal repayment and implied interest earned. This calculation is essential for investors to assess the attractiveness and potential return of T-bill
Real-life examples of the discount rate’s impact on T-bill pricing are evident in the financial markets. When interest rates rise, the discount rate used in the formula increases, leading to a
decrease in the present value of T-bills. Conversely, when interest rates fall, the discount rate decreases, resulting in an increase in the present value of T-bills. This inverse relationship
highlights the significance of the discount rate in determining T-bill valuations.
Understanding the connection between the discount rate and the discount rate formula for T-bills empowers investors with the knowledge to make informed investment decisions. By carefully considering
the prevailing interest rate environment and the impact of the discount rate on T-bill valuations, investors can optimize their investment strategies and mitigate potential risks. This understanding
is particularly valuable in fixed income investing, where T-bills play a significant role in portfolio diversification and risk management.
Time Value of Money
The “time value of money” concept is fundamentally intertwined with the “discount rate formula for T-bills.” It acknowledges that the value of money today differs from its value in the future due to
its earning potential over time. This principle serves as the cornerstone for determining the present value of future cash flows associated with T-bills.
• Present vs. Future Value: The time value of money underscores that a certain amount of money today is worth more than the same amount in the future because of its potential to generate returns or
interest over time.
• Impact of Inflation: The time value of money also considers the corrosive effects of inflation, which can erode the purchasing power of money over time. As a result, future cash flows need to be
adjusted to account for inflation’s impact.
• Discounting Future Cash Flows: The discount rate formula for T-bills incorporates the time value of money by discounting future cash flows back to the present. This process involves applying a
discount rate that reflects the prevailing interest rates and inflation expectations.
• Investment Decisions: Understanding the time value of money is crucial for investors when evaluating T-bills and making investment decisions. It helps them assess the present value of future cash
flows and compare different investment options based on their time-adjusted returns.
In essence, the time value of money within the discount rate formula for T-bills provides a framework for valuing future cash flows in the present, taking into account the impact of interest rates,
inflation, and the opportunity cost of capital. This concept empowers investors to make informed investment decisions and optimize their returns in the context of fixed income investments.
Investment Analysis
Investment analysis is a critical component of the discount rate formula for T-bills, as it provides the foundation for determining the appropriate discount rate to apply when calculating the present
value of future cash flows. Without a thorough understanding of investment analysis, investors may struggle to accurately assess the value of T-bills and make informed investment decisions.
One of the key challenges in investment analysis is determining the appropriate discount rate to use. The discount rate should reflect the time value of money, the riskiness of the investment, and
the investor’s opportunity cost of capital. By carefully considering these factors, investors can ensure that the discount rate they use is appropriate for their individual circumstances and
investment goals.
Real-life examples of investment analysis within the discount rate formula for T-bills include the use of the yield curve to determine the appropriate discount rate. The yield curve plots the
relationship between interest rates and maturities for different types of fixed income securities. By using the yield curve, investors can identify the current market discount rate for T-bills of
different maturities. This information can then be used to calculate the present value of future cash flows and make informed investment decisions.
Understanding the connection between investment analysis and the discount rate formula for T-bills is crucial for investors looking to optimize their fixed income investments. By carefully
considering the factors that influence the discount rate, investors can make informed decisions about the appropriate discount rate to use, ultimately leading to more accurate valuations of T-bills
and better investment outcomes.
Risk Assessment
Risk assessment plays a critical role in the discount rate formula for T-bills, as it helps investors evaluate the potential risks associated with investing in these securities and make informed
decisions. By carefully considering the various risk factors, investors can mitigate potential losses and optimize their investment strategies.
• Inflation Risk: The risk that inflation will erode the purchasing power of future cash flows, reducing the real return on investment. For example, if inflation is higher than expected, the
present value of future cash flows will be lower, potentially leading to losses.
• Interest Rate Risk: The risk that interest rates will change, affecting the value of T-bills. If interest rates rise, the present value of future cash flows from T-bills will decrease,
potentially leading to losses.
• Default Risk: The risk that the issuer of the T-bill will default on its obligation to repay the principal. While T-bills issued by the U.S. government are considered very low risk, there is
always a possibility of default, especially in the case of T-bills issued by other countries or entities.
• Liquidity Risk: The risk that T-bills cannot be easily sold or converted into cash when needed. T-bills are generally considered liquid investments, but there may be times when market conditions
make it difficult to sell T-bills quickly at a fair price.
These are just a few of the key risk factors that investors should consider when evaluating T-bills and using the discount rate formula to calculate their present value. By carefully assessing these
risks, investors can make more informed investment decisions and mitigate potential losses.
Yield Curve
Yield curve plays a critical role in the context of the discount rate formula for T-bills, as it serves as a valuable tool for investors to assess the relationship between interest rates and
maturities. By analyzing the yield curve, investors can gain insights into the current and expected future interest rate environment, which is essential for determining the appropriate discount rate
to use when calculating the present value of future cash flows from T-bills.
• Term Structure of Interest Rates: The yield curve represents the graphical representation of the relationship between interest rates and the maturities of fixed income securities, providing a
snapshot of the term structure of interest rates at a specific point in time.
• Market Expectations: The yield curve reflects the market’s expectations about future interest rates. By analyzing the slope and shape of the yield curve, investors can infer market sentiment and
make informed predictions about the direction of future interest rates.
• Discount Rate Determination: The yield curve is a key input in determining the appropriate discount rate to use in the discount rate formula for T-bills. Investors often use the yield curve to
identify the current market discount rate for different maturities, which can then be used to calculate the present value of future cash flows.
• Risk Assessment: The yield curve can also be used to assess the risk associated with investing in T-bills. For example, an upward sloping yield curve typically indicates that investors expect
interest rates to rise in the future, which can lead to potential losses for investors holding T-bills with longer maturities.
In summary, the yield curve is an indispensable tool for investors seeking to accurately value T-bills using the discount rate formula. By carefully analyzing the yield curve, investors can gain
insights into the term structure of interest rates, market expectations, and potential risks, enabling them to make informed investment decisions and optimize their returns.
T-bill Pricing
T-bill pricing is a cornerstone of the discount rate formula for T-bills. By delving into the intricacies of T-bill pricing, investors can gain a comprehensive understanding of how T-bills are valued
and the factors that influence their pricing.
• Maturity Date: The maturity date of a T-bill significantly impacts its price. T-bills with longer maturities typically have higher yields and, consequently, higher prices.
• Interest Rate Environment: T-bill pricing is heavily influenced by the prevailing interest rate environment. When interest rates rise, T-bill prices tend to fall, and vice versa.
• Inflation Expectations: Inflation expectations play a role in T-bill pricing, as investors anticipate the impact of inflation on the purchasing power of future cash flows.
• Market Liquidity: The liquidity of the T-bill market affects its pricing. Higher liquidity generally leads to tighter bid-ask spreads and more efficient pricing.
Understanding these factors is crucial for investors to accurately value T-bills using the discount rate formula. By incorporating these considerations into their analysis, investors can make
informed investment decisions and optimize their returns in the fixed income market.
Discount Rate Formula for T-bills FAQs
This section provides answers to frequently asked questions (FAQs) about the discount rate formula for T-bills, covering key concepts and practical considerations.
Question 1: What is the discount rate formula for T-bills?
Answer: The discount rate formula for T-bills calculates the present value of future cash flows received from a T-bill investment, considering the time value of money and prevailing interest rates.
Question 2: How is the discount rate determined?
Answer: The discount rate is typically based on the yield curve, which represents the relationship between interest rates and maturities for different fixed income securities, including T-bills.
Question 3: What factors influence T-bill pricing?
Answer: T-bill pricing is influenced by factors such as the maturity date, interest rate environment, inflation expectations, and market liquidity.
Question 4: Why is it important to consider the time value of money when calculating the present value of T-bills?
Answer: The time value of money recognizes that the value of money changes over time due to its earning potential. Ignoring this concept can lead to inaccurate valuations of future cash flows.
Question 5: How can investors use the discount rate formula to make investment decisions?
Answer: The discount rate formula helps investors assess the potential return on T-bill investments by calculating the present value of future cash flows. This information can be used to compare
T-bills with other investment options.
Question 6: What are the limitations of the discount rate formula?
Answer: While the discount rate formula provides a valuable tool for valuing T-bills, it has limitations, such as its reliance on accurate estimates of the discount rate and its assumption of
constant interest rates over the investment horizon.
In summary, understanding the discount rate formula for T-bills is essential for investors seeking to accurately value and make informed investment decisions in the fixed income market. It is
important to consider the various factors that influence T-bill pricing and the limitations of the formula to ensure its effective use.
The following section will delve deeper into practical applications of the discount rate formula for T-bills, providing investors with strategies and examples to enhance their investment
decision-making process.
TIPS for Utilizing the Discount Rate Formula for T-bills
This section provides practical tips and strategies for investors to effectively use the discount rate formula for T-bills in their investment decision-making process.
Tip 1: Understand the Time Value of Money: Recognize that the value of money changes over time due to its earning potential. This concept is fundamental to accurately valuing T-bills and should not
be overlooked.
Tip 2: Choose an Appropriate Discount Rate: The discount rate should reflect the current market conditions, interest rate environment, and the riskiness of the T-bill investment. Consider using the
yield curve as a reference point for determining the appropriate discount rate.
Tip 3: Consider Inflation Expectations: Inflation can erode the purchasing power of future cash flows. Take into account inflation expectations when selecting the discount rate to ensure an accurate
assessment of T-bill value.
Tip 4: Analyze the Yield Curve: The yield curve provides insights into the relationship between interest rates and maturities. By analyzing the yield curve, investors can make informed predictions
about future interest rates and adjust their discount rate accordingly.
Tip 5: Calculate the Present Value: Use the discount rate formula to calculate the present value of future cash flows received from a T-bill investment. This value represents the current worth of the
investment and can be used to compare different T-bills.
Tip 6: Compare T-bills with Other Investments: The present value calculated using the discount rate formula allows investors to compare T-bills with other fixed income investments, such as bonds or
certificates of deposit, to determine the most suitable option.
Tip 7: Monitor Interest Rate Changes: Interest rates can fluctuate over time, impacting the value of T-bills. Regularly monitor interest rate changes and adjust the discount rate accordingly to
maintain an accurate valuation.
Tip 8: Seek Professional Advice: If needed, consult with a financial advisor or investment professional to gain further insights into the discount rate formula for T-bills and its application in
investment decision-making.
By following these tips, investors can enhance their understanding and effective use of the discount rate formula for T-bills, leading to more informed investment decisions and potentially improved
The following section will conclude the article by highlighting the importance of accurately valuing T-bills and the role of the discount rate formula in this process.
This article delved into the discount rate formula for T-bills, exploring its significance and practical applications in fixed income investing. Key takeaways include:
• The discount rate formula is crucial for valuing T-bills, considering the time value of money and prevailing interest rates.
• Factors such as the maturity date, interest rate environment, inflation expectations, and market liquidity influence T-bill pricing.
• Investors can use the discount rate formula to compare T-bills with other investments and make informed investment decisions.
Accurately valuing T-bills is essential for investors seeking to optimize their returns in the fixed income market. By understanding the discount rate formula and its components, investors can
effectively assess the present worth of future cash flows and make well-informed investment choices. Remember, the discount rate formula is a valuable tool, but its effective use requires careful
consideration of the underlying factors that influence T-bill pricing.
Leave a Comment | {"url":"https://www.gospel10.com/how-to-master-the-discount-rate-formula-for-t-bills-a-step-by-step-guide/","timestamp":"2024-11-13T11:53:13Z","content_type":"text/html","content_length":"207616","record_id":"<urn:uuid:9143b957-0df3-41bb-b304-580fe9ea7f49>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00177.warc.gz"} |
How to Read A Shock Graph | Wild Racing
top of page
How to Read A Shock Graph
-Where to start?
If you break down a shock graph, it can be very simple. At the bottom of the graph is the velocity of the shaft in (in/sec). This is in refence to inches per second, and in dirt car racing, industry
norm is to dyno them at 10in/sec. Vertically along the graph is force in pounds(lbs). A positive number is compression and a negative number is rebound.
-Shock Rating
Lots of racers, for example, will state that a shock has a valving of "5-5". But those numbers are recorded at a 10in/sec, and unless the track is extremally rough, the shock never travels at that
speed. So why do we rate shocks with that metric? Well its an old concept that we've kept over time, and is now obsolete. Now we refer to velocity at different in/sec, like: 1"=25# or 3"=-200
referencing the shock graph. This allows for multiple shock brands to be rated the same, by their displacement, not a number system that is in reference with a shaft velocity that the car never sees.
-Graph Shape
Graph shape is dictated by: weight of oil, temp of oil, valving, piston design, gas pressure, and bleed. All this things can be adjusted to achieve the shape wanted. Most of the time a perfect shape
is impossible to achieve, but we shoot for a pound number at at certain shaft velocity and manipulate the rest of the graph to best fit the the situation.
bottom of page | {"url":"https://www.wildracingparts.com/items/how-to-read-a-shock-graph","timestamp":"2024-11-15T00:17:06Z","content_type":"text/html","content_length":"802109","record_id":"<urn:uuid:4d937def-711c-4b8f-80d1-a166d7e9c04c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00447.warc.gz"} |
Proving Triangles Similar
Word problems allow you to see the real world uses of math! This tutorial shows you how to take a word problem and use indirect measurement to turn it into a proportion. Then see how to use the mean
extremes property of proportions to cross multiply and solve for the answer. Take a look! | {"url":"https://virtualnerd.com/texas-digits/txh-geo/similarity/proving-triangles-similar/","timestamp":"2024-11-13T11:21:17Z","content_type":"text/html","content_length":"17400","record_id":"<urn:uuid:da9d23be-252f-4ca4-9b02-e60d4776e2f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00034.warc.gz"} |
Related queries: Natural language processing interview questions / Natural language processing interview questions /NLP interview questions
Which of the following is NOT a common approach in conversational AI?
Intent recognition
Entity extraction
Dialogue management
Sentiment amplification
What is the main goal of neural text detoxification in NLP?
To remove or mitigate toxic content in text
To generate non-toxic text
To translate toxic text into non-toxic language
To perform sentiment analysis on toxic text
What is the main goal of contrastive learning in NLP?
To learn representations that group similar examples closer together while pushing dissimilar ones apart
To generate contrasting text
To translate between contrasting languages
To perform sentiment analysis on contrasting opinions
Which of the following is NOT a common approach in neural text infilling?
Masked language modeling
Autoregressive generation
Bidirectional generation
Sentiment-based infilling
What is the purpose of the UNK library in NLP?
To provide pre-trained word embeddings
To offer tools for handling out-of-vocabulary words
To perform sentiment analysis
To generate text summaries
What is the purpose of the SimpleTransformers library in NLP?
To provide pre-trained word embeddings
To offer a simplified interface for using Transformer models
To perform sentiment analysis
To generate text summaries
What is the main advantage of using Switch Transformers in NLP?
They are always faster
They can scale to very large models efficiently
They require less memory
They work only for English
What is the primary goal of Named Entity Recognition (NER)?
To identify parts of speech
To extract proper nouns and classify them into predefined categories
To determine the sentiment of a text
To generate text summaries
Which Python library is commonly used for creating custom question generation models?
What is the main advantage of using T5 (Text-to-Text Transfer Transformer) in NLP?
It's always faster
It can be fine-tuned for any text-based task
It requires less memory
It works only for English
Score: 0/10
Which of the following is NOT a common approach in neural text simplification?
Lexical simplification
Syntactic simplification
Sentence splitting
Sentiment simplification
Which Python library is commonly used for topic modeling?
What is the main goal of zero-shot learning in NLP?
To perform tasks without any task-specific training examples
To generate text without any input
To translate between languages without parallel data
To perform sentiment analysis without labeled data
What is the purpose of the spaCy library in NLP?
To provide pre-trained word embeddings
To offer industrial-strength natural language processing
To perform sentiment analysis
To generate text summaries
Which Python library is specifically designed for processing multilingual text?
What is the main goal of text summarization in NLP?
To classify text into categories
To extract key information and create a shorter version of the text
To perform machine translation
To generate new text based on input
Which Python library is commonly used for creating custom text readability assessment models?
What is the main goal of multi-task learning in NLP?
To train a model to perform multiple NLP tasks simultaneously
To generate multiple texts at once
To translate between multiple languages
To perform sentiment analysis for multiple aspects
Which Python library is commonly used for creating custom word sense disambiguation models?
Which of the following is NOT a common approach in neural text normalization?
Seq2seq models
Transformer-based models
Rule-based normalization
Sentiment-based normalization
Score: 0/10
What is the purpose of the PyKaldi library in NLP?
To provide pre-trained word embeddings
To offer Python bindings for the Kaldi speech recognition toolkit
To perform sentiment analysis
To generate text summaries
Which of the following is NOT a common approach in abstractive text summarization?
Sequence-to-sequence models
Pointer-generator networks
Reinforcement learning
Sentiment-based extraction
What is the main difference between stemming and lemmatization?
Stemming is faster but less accurate
Lemmatization is rule-based while stemming is not
Stemming works only for English
Lemmatization doesn't reduce words to their root form
What is the main goal of unsupervised domain adaptation in NLP?
To adapt models to new domains without labeled target domain data
To generate text in new domains
To translate between different domains
To perform sentiment analysis across domains
Which Python library is commonly used for creating custom authorship attribution models?
What is the purpose of the PyLaia library in NLP?
To provide pre-trained word embeddings
To offer tools for handwritten text recognition
To perform sentiment analysis
To generate text summaries
Which Python library is commonly used for creating custom text complexity analysis models?
Which of the following is NOT a common text similarity measure?
Cosine similarity
Jaccard similarity
Levenshtein distance
Sentiment similarity
What is the purpose of the Pytext library in NLP?
To provide pre-trained word embeddings
To offer a deep learning NLP framework built on PyTorch
To perform sentiment analysis
To generate text summaries
What is the purpose of the TextKernel library in NLP?
To provide pre-trained word embeddings
To offer tools for information extraction from resumes and job postings
To perform sentiment analysis
To generate text summaries
Score: 0/10
Which of the following is NOT a common approach in neural text style transfer?
Disentangled representation learning
Adversarial training
Reinforcement learning
Sentiment amplification
Which Python library provides tools for working with regular expressions in NLP tasks?
What is the main purpose of word embeddings in NLP?
To remove stop words
To represent words as dense vectors in a continuous vector space
To perform sentiment analysis
To identify named entities
What is the purpose of the FlairNLP library in NLP?
To provide pre-trained word embeddings
To offer a powerful NLP library with state-of-the-art models
To perform sentiment analysis
To generate text summaries
What is the purpose of the Optuna library in NLP?
To provide pre-trained word embeddings
To offer hyperparameter optimization for machine learning
To perform sentiment analysis
To generate text summaries
What is the purpose of the PyEMD library in NLP?
To provide pre-trained word embeddings
To offer Earth Mover's Distance calculations for text similarity
To perform sentiment analysis
To generate text summaries
What is the purpose of the TensorFlow Datasets library in NLP?
To provide pre-trained word embeddings
To offer a collection of datasets ready to use with TensorFlow
To perform sentiment analysis
To generate text summaries
Which of the following is NOT a common approach in text-to-speech synthesis?
Concatenative synthesis
Statistical parametric synthesis
Neural network-based synthesis
Sentiment-based synthesis
Which Python library is commonly used for creating custom text-to-speech models?
What is the purpose of the PyNLPl library in NLP?
To provide pre-trained word embeddings
To offer a collection of Python NLP libraries and tools
To perform sentiment analysis
To generate text summaries
Score: 0/10 | {"url":"https://coolgenerativeai.com/natural-language-processing-with-python/","timestamp":"2024-11-10T06:13:16Z","content_type":"text/html","content_length":"193008","record_id":"<urn:uuid:112f0dd4-fac2-45ab-94d0-5c3e6e6c424a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00036.warc.gz"} |
Class 11 maths syllabus consists of some of the most exciting and essential topics that act as the basis for your future studies if appropriately understood. The chapters are formulated keeping in
mind their practical use. NCERT is one of the best books to prepare for class 11 maths. Sequence and Series in Chapter 9 of the NCERT book is one of the most crucial chapters in the syllabus. The
chapter involves arithmetic mean, geometric mean, arithmetic progression, geometric progression, their general terms, the sum of series, and a few class 11 Maths NCERT solutions. With the assistance
of definition, theory, formulas, and examples, you can learn the application of sequence and series. This article brings to you all the information that you need regarding Sequence and Series along
with some Class 11 Maths Chapter 9 NCERT Solutions. Read the full article to get a firm hold on the concepts in this chapter! | {"url":"https://msvgo.com/cbse/ncert-solutions-class-11-maths-sequences-and-series","timestamp":"2024-11-06T20:23:00Z","content_type":"text/html","content_length":"484973","record_id":"<urn:uuid:c658e15f-be66-4820-b0b2-776f333823ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00774.warc.gz"} |
Rounding Rules for Nutrients
If you use nutrients as components, you can define rounding rules for quantities or recommended daily allowances (RDA) of nutrients in specification management in the PLM Web UI.
The system uses rounding rules when it determines the declared values for the component list from the calculated values.
The system uses the base unit of measure of the nutrient when rounding quantities, and rounds recommended daily allowances as percentages.
If the system cannot determine any rounding rules or any valid rounding rules for a nutrient, it rounds the values as specified in the Business Add-In (BAdI) BAdI: Data for Nutrition Label/
Quantitative Component Label: Execute Rules; in the standard system, this is to two decimal places. You also use this BAdI to specify how the rounding rules are defined.
In the standard system, you define rounding rules by specifying the settings and the beginning of the intervals in which the rounding rules are applied:
• Intervals
You specify intervals that build on each other. The interval with the smallest value is the starting point from where rounding is applied. Subsequent intervals specify sections in which the
rounding rule might be defined differently.
The last interval with the highest number is used for all numbers above the interval limit. The interval limit is defined as exact as four decimal places.
You have defined three rounding rules. You have specified the lower limit as Limit1 and Limit2. The system evaluates any calculated value to be rounded based on the following limits:
□ If the value is smaller than Limit1, no rounding is applied.
□ If the value is equal to or bigger than Limit1 and smaller than Limit2, the rounding defined for Limit1 is applied.
□ If the value is equal to or bigger than Limit2, the rounding defined for Limit2 is applied.
End of the example.
• Settings for rounding rules
□ Round to the Nearest:
The value of this field defines a stepping in which results are calculated.
☆ Round to the nearest is 5.
If the calculated value is 4, it is rounded up to 5. If the value is 7, it is rounded down to 5.
☆ Round to the nearest is 0.2.
If the calculated value is 8.15, it is rounded up to 8.2. If the value is 7.91, it is rounded down to 8.0.
End of the example.
□ Declared Value and Declared Value Statement:
You can enter a fixed numerical value, a phrase, or a combination of both, that is used for values of this interval.
You have set up a rounding rule as follows:
If the calculated value is 112.2456, then the returning value is 100.00 and the value statement is More than.
End of the example.
□ Do Not Display Item:
If this field is selected, the result will be that the Display checkbox is not selected. | {"url":"https://help.sap.com/saphelp_globext607_10/helpdata/en/9d/22feec6a384d2bac7b8fca6a69dadd/content.htm?no_cache=true","timestamp":"2024-11-12T03:01:17Z","content_type":"application/xhtml+xml","content_length":"14953","record_id":"<urn:uuid:930b109d-f774-41e1-add4-782b66cc3359>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00784.warc.gz"} |
requency response
Frequency response of discrete-time filter System object
[h,w] = freqz(sysobj) returns the complex frequency response h of the filter System object™, sysobj. The vector w contains the frequencies (in radians/sample) at which the function evaluates the
frequency response. The frequency response is evaluated at 8192 points equally spaced around the upper half of the unit circle.
[h,w] = freqz(sysobj,n) returns the complex frequency response of the filter System object and the corresponding frequencies at n points equally spaced around the upper half of the unit circle.
freqz uses the transfer function associated with the filter to calculate the frequency response of the filter with the current coefficient values.
[h,w] = freqz(sysobj,Arithmetic=arithType) analyzes the filter System object, based on the arithmetic specified in arithType, using either of the previous syntaxes.
freqz(sysobj) plots the magnitude and unwrapped phase of the frequency response of the filter object.
For more input options, see freqz in Signal Processing Toolbox™.
Frequency Response of the Filter
This example plots the frequency response of the lowpass FIR filter using freqz.
b = designLowpassFIR(FilterOrder=80,CutoffFrequency=0.5,Window="custom",CustomWindow=kaiser(81,8));
firFilt = dsp.FIRFilter(Numerator=b);
Input Arguments
sysobj — Input filter
filter System object
Input filter, specified as one of the following filter System objects:
n — Number of points over which the frequency response is computed
8192 (default) | positive integer
Number of points over which the frequency response is computed. For an FIR filter where n is a power of two, the computation is done faster using FFTs.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
arithType — Arithmetic type
'double' (default) | 'single' | 'Fixed'
Arithmetic used in the filter analysis, specified as 'double', 'single', or 'Fixed'. When the arithmetic input is not specified and the filter System object is unlocked, the analysis tool assumes a
double-precision filter. When the arithmetic input is not specified and the System object is locked, the function performs the analysis based on the data type of the locked input.
The 'Fixed' value applies to filter System objects with fixed-point properties only.
When the 'Arithmetic' input argument is specified as 'Fixed' and the filter object has the data type of the coefficients set to 'Same word length as input', the arithmetic analysis depends on whether
the System object is unlocked or locked.
• unlocked –– The analysis object function cannot determine the coefficients data type. The function assumes that the coefficients data type is signed, has a 16-bit word length, and is auto scaled.
The function performs fixed-point analysis based on this assumption.
• locked –– When the input data type is 'double' or 'single', the analysis object function cannot determine the coefficients data type. The function assumes that the data type of the coefficients
is signed, has a 16-bit word length, and is auto scaled. The function performs fixed-point analysis based on this assumption.
To check if the System object is locked or unlocked, use the isLocked function.
When the arithmetic input is specified as 'Fixed' and the filter object has the data type of the coefficients set to a custom numeric type, the object function performs fixed-point analysis based on
the custom numeric data type.
Output Arguments
h — Frequency response
Complex n-element frequency response vector. If n is not specified, the function uses a default value of 8192. The frequency response is evaluated at n points equally spaced around the upper half of
the unit circle.
Data Types: double
Complex Number Support: Yes
w — frequencies
Frequency vector of length n, in radians/sample. w consists of n points equally spaced around the upper half of the unit circle (from 0 to π radians/sample). If n is not specified, the function uses
a default value of 8192.
Data Types: double
There are several ways of analyzing the frequency response of filters. freqz accounts for quantization effects in the filter coefficients, but does not account for quantization effects in filtering
arithmetic. To account for the quantization effects in filtering arithmetic, refer to function noisepsd.
freqz calculates the frequency response for a filter from the filter transfer function Hq(z). The complex-valued frequency response is calculated by evaluating Hq(e^j^ω) at discrete values of w
specified by the syntax you use. The integer input argument n determines the number of equally-spaced points around the upper half of the unit circle at which freqz evaluates the frequency response.
The frequency ranges from 0 to π radians per sample when you do not supply a sampling frequency as an input argument. When you supply the scalar sampling frequency fs as an input argument to freqz,
the frequency ranges from 0 to fs/2 Hz.
Version History
Introduced in R2011a
R2024b: dsp.BiquadFilter object warns
The dsp.BiquadFilter object issues a warning and will be removed in a future release. Use the dsp.SOSFilter object instead. For more information on how to replace your existing code, see the
Compatibility Considerations section in the dsp.BiquadFilter reference page.
R2024b: Support for dsp.DCBlocker object
Starting in R2024b, this function supports the dsp.DCBlocker object.
R2024a: freqz function no longer uses Filter Visualization Tool
When you call the freqz function with no output arguments, the function no longer uses Filter Visualization Tool to plot the frequency response of the filter. Starting in R2024a, the function uses
the MATLAB^® plot instead.
You do not need to make any changes to your code.
R2023b: Support for dsp.ParallelFilter and dsp.Delay Objects
Starting in R2023b, the freqz analysis function supports the dsp.ParallelFilter and the dsp.Delay objects.
R2023b: dsp.BiquadFilter object will be removed
The dsp.BiquadFilter object will be removed in a future release. Use the dsp.SOSFilter object instead. | {"url":"https://ch.mathworks.com/help/dsp/ref/dsp.allpassfilter.freqz.html","timestamp":"2024-11-11T03:09:52Z","content_type":"text/html","content_length":"105582","record_id":"<urn:uuid:68589264-ecd6-4f5a-a512-6ef7860fd025>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00361.warc.gz"} |
Data on how much solutions differ in effectiveness — EA Forum
Click the link above to see the full article and charts. Here is a summary I wrote for the latest edition to the 80,000 Hours newsletter, or see the Twitter version.
Is it really true that some ways of solving social problems achieve hundreds of times more, given the same amount of effort?
Back in 2013, Toby Ord^1 pointed out some striking data about global health. He found that the best interventions were:
□ 10,000x better at creating years of healthy life than the worst interventions.
□ 50x better than the median intervention.
He argued this could have radical implications for people who want to do good, namely that a focus on cost-effectiveness is vital.
For instance, it could suggest that by focusing on the best interventions, you might be able to have 50 times more impact than a typical person in the field.
This argument was one of the original inspirations for our work and effective altruism in general.
Now, ten years later, we decided to check how well the pattern in the data holds up and see whether it still applies – especially when extended beyond global health.
We gathered all the datasets we could find to test the hypothesis. We found data covering health in rich and poor countries, education, US social interventions, and climate policy.
If you want to get the full picture on the data and its implications, read the full article (with lots of charts!):
The bottom line is that the pattern Toby found holds up surprisingly well.
This huge variation suggests that once you’ve built some career capital and chosen some problem areas, it’s valuable to think hard about which solutions to any problem you’re working on are most
effective and to focus your efforts on those.
The difficult question, however, is to say how important this is. I think people interested in effective altruism have sometimes been too quick to conclude that it’s possible to have, say, 1,000
times the impact by using data to compare the best solutions.
First, I think a fairer point of comparison isn’t between best and worst but rather between the best measurable intervention and picking randomly. And if you pick randomly, you expect to get the
mean effectiveness (rather than the worst or the median).
Our data only shows the best interventions are about 10 times better than the mean, rather than 100 or 1,000 times better.
Second, these studies will typically overstate the differences between the best and average measurable interventions due to regression to the mean: if you think a solution seems unusually good,
that might be because it is actually good, or because you made an error in its favour.
The better something seems, the greater the chance of error. So typically the solutions that seem best are actually closer to the mean. This effect can be large.
Another important downside of a data-driven approach is that it excludes many non-measurable interventions. The history of philanthropy suggests the most effective solutions historically have
been things like R&D and advocacy, which can’t be measured ahead of time in randomised trials. This means that restricting yourself to measurable solutions could mean excluding the very best
And since our data shows the very best solutions are far more effective than average, it’s very bad for your impact to exclude them.
In practice, I’m most keen on the “hits-based approach” to choosing solutions. I think it’s possible to find rules of thumb that make a solution more likely to be among the very most effective,
such as “does this solution have the chance of solving a lot of the problem?”, “does it offer leverage?”, “does it work at all?”, and “is it neglected?”
Hypothetically, if we could restrict ourselves to solutions that are among the top half and then pick randomly from what remains, we can expect a cost-effectiveness that’s about twice the mean.
And I think it’s probably possible to do better than that. Read more in our article on choosing solutions.
So, suppose you use a hits-based approach to carefully pick solutions within an area. How much more impact can you have?
My overall take is something like 10 times more. I feel pretty uncertain, though, so my range is perhaps 3-100 times.
A 10-times increase in impact given the same amount of effort is a big deal. It’s probably underrated by the world at large, though it may be overrated by fans of effective altruism.
A final thought: I think you can increase your impact by significantly more than 10 times by carefully choosing which problem area to focus on in the first place. This is a big reason why we
emphasise problem selection in career choice so much at 80,000 Hours. Overall, we’d say to focus on exploring and building career capital first, then start to target some problem areas, and only
later focus on choosing solutions.
jackva 16
Hey Ben, thanks for the replies -- adding some more to get closer to the same page 🙂
Re your 1), my criticism here is more one of emphasis and of the top-line messaging, as you indeed mention these cases of advocacy and research.
I just think that these cases are rather fundamental and affecting the conclusions very significantly -- because we are almost never in the situation that all we can choose from are direct
interventions so the solution space (and with it, the likely variance) will almost always look quite different than what is discussed as primary evidence in the article (that does not mean we will
never choose direct interventions, to be sure, just that the variance of solutions will mostly be one that emerges from the conjunction of impact differentials).
Re your 2), I think this is mostly a misunderstanding -- my comment was also very quickly written, apologies.
I am not saying we should always choose the most leveraged thing ever, but rather that the solution space will essentially always be structured by conjunction of multipliers. There are reasons to not
only choose the most leveraged solution, as you point out, but I don’t think this is enough to argue that the most effective actions will not usually be conjunctive ones.
I agree that the data in the article is useful for specifying the shape of a particular impact differential, I am mostly arguing that it understates the variance of the solution space.
(I worry that we are mixing expected and realized value here, I am mostly talking about conjunctive strategies affecting how the variance of the solution space looks like on expected value, this does
not preclude the realized value sometimes being zero (and that risk aversion or other considerations can drive us to prefer less leveraged actions.)).
Re your 3) & 4) I agree -- my understanding was that these are the factors that lead you to only 10x and my comment was merely that I think direct intervention space variance is not that informative
with regards to solution selection in most decision contexts.
Aside: I agree with you that I don’t think that advocacy by itself is a 100x multiplier in expectation.
One small extra data point that might be useful: I made a rough estimate for smallpox eradication in the post, finding it fell in the top 0.1% of the distribution for global health, so it seemed
EdoArad 2
Yea, I agree with your analyses in the article, though I'd be interested in understanding the relative effects
I agree different comparisons are relevant in different situations.
A comparison with the median is also helpful, since it e.g. tells us the gain that the people currently doing the bottom 50% of interventions could get if they switched.
Though I think the comparison to the mean is very relevant (and hasn't had enough attention) since it's the effectiveness of what the average person donates to, supposing we don't know anything about
them. Or alternatively it's the effectiveness you end up with if you pick without using data.
I think you'd need to show why this mean-over-median approach is correct to apply to strategy selection but incorrect to apply to cause area selection. Couldn't you equally argue that regression
to the mean indicates we'll make errors in thinking some cause areas are 1000x more important or neglected than others?
Yes absolutely.
I think regression to the mean is a bigger issue for cause selection than solution selection. I've tried to take this into account when thinking about between-cause differences, but could have
underestimated it.
Basically, I think it's easier to pick the top 1% of causes than the top 1% of solutions, and there's probably also greater variance between causes.
(One way to get an intuition for this is that only <0.001% of world GDP goes into targeted xrisk reduction or ending factory farming, while ~10% of world GDP is spent on addressing social issues in
rich countries.)
More posts like this
Sorted by Click to highlight new comments since:
jackva 29
I am worried that the cited data does not really inform this question -- as we can always choose solutions that leverage "conjunctions of multipliers" (e.g. advocacy, changing trajectories) so that
real variance also in solutions should be *much* larger than 10x for anyone being a funder within a cause area.
To make this more concrete, when choosing what to fund in climate one would not choose between different policies or lifestyle actions (the evidence for climate presented here), but between fundable
opportunities that stack different impact multipliers on top of each other, e.g. advocacy instead of direct action, supporting policies with large expected long-term consequences (e.g. by
accelerating technological change, whereas -- AFAICT -- the data from Gillngham and Stock displayed here is their static case which they describe as focused on current technology and project cost,
contrasted to their dynamic case studies which seems much more likely to be something attractive to fund), etc.
So, it seems to me the evidence presented significantly underplays the real variance in solution effectiveness that a funder faces because it uses data on single-variable direct intervention variance
as a proxy for variance in effectiveness of interventions, despite the most effective interventions not usually being direct actions (certainly outside GHD, most EA funding does not buy equivalents
of malaria nets for ex-risk etc.) and not actions where impact differentials can be easily quantified with certainty (despite being very large in expectation).
This also seems to potentially lead to biased comparisons between solution variance and cause level variance given how strongly differences in cause level variance are driven by expected value
calculations (value of the future, etc.) that are far more extreme / speculative to what people comparing interventions on single interventions would have data on.
Benjamin_Todd 21
Hey, thanks for the comments. Here are some points that might help us get on the same page:
1) I agree this data is missing difficult-to-measure hits based interventions, like research and advocacy, which means it'll understate the degree of spread.
I discuss that along with other ways it could understate the differences here:
2) Aside: I'm not sure conjunction of multipliers is the best way to illustrate this point. Each time you add a multiplier it increases the chance it doesn't work at all. I doubt the optimal degree
of leverage in all circumstances is "the most possible", which is why Open Philanthropy supports interventions with a range of degree of multipliers (including those without), rather than putting
everything into the most multiplied thing possible (research into advocacy into research into malaria..). (Also if adding multipliers is the right way to think about it, this data still seems
relevant, since it tells you the variance of what you're multiplying in the first place.)
3) My comparison is between the ex ante returns of top solutions and the mean of the space.
Even if you can pick the top 1% of solutions with certainty, and the other 99% achieve nothing, then your selection is only ~100x the mean. And I'm skeptical we can pick the top 1% in most cause
areas, so that seems like an upper bound. E.g. in most cases (esp things like advocacy) I think there's more than a 1% chance of picking something net harmful, which would already take us out of the
top 1% in expectation.
4) There are also major ways the data overstates differences in spread, like regression to the mean.
The data shows the top are ~10x the mean. If you were optimistic about getting a big multiplier on those, that maybe could get you to 1,000x. But then when we take into account regression to the
mean, that can easily reduce spread another 10x, getting us back to something like 100x.
That seems plausible but pretty optimistic to me. My overall estimate for top vs. mean is ~10x, but with a range of 3-100x.
>This also seems to potentially lead to biased comparisons between solution variance and cause level variance given how strongly differences in cause level variance are driven by expected value
calculations (value of the future, etc.) that are far more extreme / speculative to what people comparing interventions on single interventions would have data on.
I agree estimates of cause spread should be regressed more than solution spread. I've tried to take this into account, but could have underestimated it.
In general I think regression to the mean is a very interesting avenue for developing a critique of core EA ideas.
I'd also add it would be great if there was more work to empirically analyse ex ante and ex post spread among hits based interventions with multiple outcomes. I could imagine it leading to a somewhat
different picture, though I think the general thrust will still hold, and I still thinking looking at spread among measurable interventions can help to inform intuitions about the hits based case.
One example of work in this area is this piece by OP, where they say they believe they found some 100x and a few 1000x multipliers on cash transfers to US citizens by e.g. supporting advocacy into
land use reform. But this involves an element of cause selection as well as solution selection, cash transfers seem likely below the mean, and this was based on BOTECs that will contain a lot of
model error and so should be further regressed. Overall I'd say this is consistent with within-cause differences of ~10x from top to mean, and doesn't support > 100x differences.
jackva 4
I agree that this would be great to exist, though it is likely very hard and the examples that will exist soon will not be the strongest ones (given how effects can become visible over longer
time-frames, e.g. how OP discusses green revolution and other interventions that took many years to have the large effects we can now observe).
EdoArad 4
Some of these DCP cost-effectiveness estimates are terribly low: few dollars per QALY, compared to GiveWell's evaluation of their top charities (on the order of $100/QALY).
Even more surprisingly, looking into DCP3, the top 4 interventions had negative cost-effectiveness values.
This seems to me to be mostly because these cost-effectiveness analyses are from a decision-maker standpoint. Say, a hospital that can choose between different medications (e.g. for malaria, $4/DALY)
or a governmental policy that can reduce overall health costs (e.g. reducing salt intake, reduction of $1.4k per DALY).
I think it's mostly because these estimates aren't properly adjusted for regression to the mean – there's a ton of sources of model error, and properly factoring these in will greatly reduce the top
interventions. There are also other factors like the top interventions quickly running out of capacity. I discuss this in the article. I put a lot more trust in GiveWell's figures as an estimate of
the real marginal cost-effectiveness. Though I agree there could be some interventions accessible to policy-makers that aren't accessible to GiveWell.
firinn 3
"First, I think a fairer point of comparison isn’t between best and worst but rather between the best measurable intervention and picking randomly. And if you pick randomly, you expect to get the
mean effectiveness (rather than the worst or the median)."
I'm not sure if this is fair if you're trying to communicate the amount of value that could be created by getting more people to switch strategies.
Let's say everyone picks their strategy randomly. Then they read some information that suggests that some strategies are far more effective than others. Those who are already executing top-10%
interventions conclude that they should stick with their current strategies, while some fraction of the other 90% are persuaded to switch. If everyone who switches strategies comes from that
bottom-90% group, then the average change in value will look closer to 100x rather than 10x - because if you exclude the positive outliers then the mean will look much lower, and in fact closer to
the median.
If you're trying to suggest that choosing the correct cause area is more important than choosing the correct strategy, because there's "only" a 10x value difference in choosing the correct strategy,
I think you'd need to show why this mean-over-median approach is correct to apply to strategy selection but incorrect to apply to cause area selection. Couldn't you equally argue that regression to
the mean indicates we'll make errors in thinking some cause areas are 1000x more important or neglected than others? | {"url":"https://forum.effectivealtruism.org/posts/seFH9jcH3saXHJqin/data-on-how-much-solutions-differ-in-effectiveness","timestamp":"2024-11-05T13:57:13Z","content_type":"text/html","content_length":"361615","record_id":"<urn:uuid:eec86a67-3921-4a85-9090-f697f3570f74>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00714.warc.gz"} |
1 myCobot Series 6-Axis Collaborative Robotic Arm
myCobot 280 algorithm
1 Structural Parameters
1.1 DH Parameters of Robotic Arm
joint theta d a alpha offset
1 q1 131.22 0 1.5708 0
2 q2 0 -110.4 0 -1.5708
3 q3 0 -96 0 0
4 q4 63.4 0 1.5708 -1.5708
5 q5 75.05 0 -1.5708 1.5708
6 q6 45.6 0 0 0
1.2 Kinematic Model
2 Coordinate System Introduction
2.1 Tool Coordinate System
The figure shows the robot model of Mecharm270. Base in the figure represents the base coordinate system of the robot, O' represents the end flange coordinate system, and point P represents the
position of the end of the manipulator relative to the base coordinate system (x=152, y=0 , z=224)
Extend a certain pose on the basis of the end flange, and regard the set tool point as the end of the machine:
T in the figure is the set tool coordinate system. The posture of this coordinate system is consistent with O’, and the relative displacement of the origin has occurred. Use the python function to
set the tool coordinate system:
• set_tool_reference([x, y, z, rx, ry, rz]) //Set tool coordinate system
• set_end_type(1) //Set the end coordinate system type as tool
• Assume that the tool coordinate system T is not rotated relative to O' (rx = ry = rz = 0)
• Assume that the origin of the tool coordinate system T is in the coordinate system O’ at (x = 0, y = 0, z = 100mm)
• The final tool coordinate system parameter is set_tool_reference(0, 0, 100, 0, 0, 0)
Since the tool coordinate system is set, the end of the robot extends from O' to T at this time, and the coordinates of the end of the machine read at this time become (152+100, 0, 224), and the
coordinate posture movement will revolve around the tool point T rotate.
2.2 World Coordinate System
Section 2 introduces that by setting the tool coordinate system, the end coordinate system of the manipulator can be extended to a certain pose. We can also extend a certain pose on the basis of the
base coordinate system of the manipulator by setting the world coordinate system. The set world coordinate system will replace the original Base coordinate system and become the new base coordinate
W in the figure is the set world coordinate system. The posture of this coordinate system is consistent with Base, and the relative displacement of the origin has occurred. Use the python function to
set the world coordinate system:
• set_world_reference([x, y, z, rx, ry, rz]) //Set the world coordinate system
• set_reference_frame(1) //Set the base coordinate system type to the world
• Assuming that the world coordinate system W has not rotated relative to Base(rx = ry = rz = 0)
• Suppose the origin of the world coordinate system W is in the coordinate system Base (x = 0, y = 0, z = -100mm)
• The final world coordinate system parameter is set_world_reference(0, 0, -100, 0, 0, 0)
Since the world coordinate system is set, the origin of the robot extends from Base to W at this time, and the O’ coordinate read at this time becomes (152, 0, 224+100). | {"url":"https://docs.elephantrobotics.com/docs/pro600-en/2-serialproduct/2.1-280/Kinematics&Coordinate.html","timestamp":"2024-11-02T21:59:55Z","content_type":"text/html","content_length":"40750","record_id":"<urn:uuid:1181af5c-8a40-45e2-89aa-892018bb02b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00778.warc.gz"} |
Discount Calculator
Please provide any 2 values below to calculate.
The term discount can be used to refer to many forms of reduction in the price of a good or service. The two most common types of discounts are discounts in which you get a percent off, or a fixed
amount off.
A percent off of a price typically refers to getting some percent, say 10%, off of the original price of the product or service. For example, if a good costs $45, with a 10% discount, the final price
would be calculated by subtracting 10% of $45, from $45, or equivalently, calculating 90% of $45:
10% of $45 = 0.10 × 45 = $4.50
$45 – $4.50 = $40.50
90% of $45 = 0.90 × 45 = $40.50
In this example, you are saving 10%, or $4.50.
A fixed amount off of a price refers to subtracting whatever the fixed amount is from the original price. For example, given that a service normally costs $95, and you have a discount coupon for $20
off, this would mean subtracting $20 from $95 to get the final price:
$95 - $20 = $75
In this example, you are saving the fixed amount of $20.
The above examples are two of the most common discount methods. There are numerous others that can be more confusing, such as stackable discounts where you can get 20% off the original price, then
15% more off of that discounted price. If you need to do these kinds of calculations, refer to the Percent Off Calculator. | {"url":"https://www.calculator.net/discount-calculator.html","timestamp":"2024-11-09T16:01:33Z","content_type":"text/html","content_length":"10079","record_id":"<urn:uuid:adcc964b-a8ce-4fdd-9f15-da86f34fe65b>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00458.warc.gz"} |
Review of Ng’s deeplearning.ai Course 1: Neural Networks and Deep Learning
From Damien Kühn CC
Credit:: Damien Kühn CC
(See my reviews on Course 2 and Course 3.)
As you all know, Prof. Ng has a new specialization on Deep Learning. I wrote about the course extensively yet informally, which include two “Quick Impressions” before and after I finished Course 1 to
3 of the specialization. I also wrote three posts just on Heroes on Deep Learning including Prof. Geoffrey Hinton, Prof. Yoshua Bengio and Prof. Pieter Abbeel and Dr. Yuanqing Lin . And Waikit
and I started a study group, Coursera deeplearning.ai (C. dl-ai), focused on just the specialization. This is my full review of Course 1 after finish watching all the videos. I will give a
description on what the course is about, and why you want to take it. There are already few very good reviews (from Arvind and Gautam). I will write based on my experience as the admin of AIDL, as
well as a deep learning learner.
The Most Frequently Asked Question in AIDL
If you don’t know, AIDL is one of most active Facebook group on the matter of A.I. and deep learning. So what is the most frequently asked question (FAQ) in our group then? Well, nothing fancy:
How do I start deep learning?
In fact, we got asked that question daily and I have personally answered that question for more than 500 times. Eventually I decided to create an FAQ – which basically points back to “My Top-5 List
” which gives a list of resources for beginners.
The Second Most Important Class
That brings us to the question what should be the most important class to take? Oh well, for 90% of the learners these days, I would first recommend Andrew Ng’s “Machine Learning“, which is both
good for beginners or more experienced practitioners (like me). Lucky for me, I took it around 2 years ago and got benefited from the class since then.
But what’s next? What would be a good second class? That’s always the question on my mind. Karpathy cs231n comes to mind, or may be Socher’s cs224[dn] is another choice. But they are too
specialized in the subfields. E.g. If you view them from the study of general deep learning, the material in both classes on model architecture are incomplete.
Or you can think of general class such as Hinton’s NNML. But the class confuses even PhD friends I know. Indeed, asking beginners to learn restricted Boltzmann machine is just too much. Same can
be said for Koller’s PGM. Hinton’s and Koller’s class, to be frank, are quite advanced. It’s better to take them if you already know the basics of ML.
That narrows us to several choices which you might already consider: first is fast.ai by Jeremy Howard, second is deep learning specialization from Udacity. But in my view, those class also seems
to miss something essential – e.g., fast.ai adopts a top-down approach. But that’s not how I learn. I alway love to approach a technical subject from ground up. e.g. If I want to study string
search, I would want to rewrite some classic algorithms such as KMP. And for deep learning, I always think you should start with a good implementation of back-propagation.
That’s why for a long time, Top-5 List picked cs231n and cs224d as the second and third class. They are the best I can think of after researching ~20 DL classes. Of course, deeplearning.ai
changes my belief that either cs231n and cs224d should be the best second class.
Learning Deep Learning by Program Verification
So what so special about deeplearning.ai? Just like Andrew’s Machine Learning class, deeplearning.ai follows an approach what I would call program verification. What that means is that instead of
guessing whether your algorithm is right just by staring at the code, deeplearning.ai gives you an opportunity to come up with an implementation your own provided that you match with its official
Why is it important then? First off, let me say that not everyone believes this is right approach. e.g. Back when I started, many well-intentioned senior scientists told me that such a matching
approach is not really good experimentally. Because supposed your experiment have randomness, you should simply run your experiment N times, and calculate the variance. Matching would remove this
experimental aspect of your work.
So I certainly understand the point of what the scientists said. But then, in practice, it was a huge pain in the neck to verify if you program is correct. That’s why in most of my work I adopt the
matching approach. You need to learn a lot about numerical properties of algorithm this way. But once you follow this approach, you will also get an ML tasks done efficiently.
But can you learn in another way? Nope, you got to have some practical experience in implementation. Many people would advocate learning by just reading paper, or just by running pre-prepared
programs. I always think that’s missing the point – you would lose a lot of understanding if you skip an implementation.
What do you Learn in Course 1?
For the most part, implementing feed-forward (FF) algorithm and back-propagation (BP) algorithm from scratch. Since for most of us, we are just using frameworks such as TF or Keras, such
implementation from scratch experience is invaluable. The nice thing about the class is that the mathematical formulation of BP is fined tuned such that it is suitable for implementing on Python
numpy, the course designated language.
Wow, Implementing Back Propagation from scratch? Wouldn’t it be very difficult?
Not really, in fact, many members finish the class in less than a week. So the key here: when many of us calling it a from-scratch implementation, in fact it is highly guided. All the tough matrix
differentiation is done for you. There are also strong hints on what numpy functions you should use. At least for me, homework is very simple. (Also see Footnote [1])
Do you need to take Ng’s “Machine Learning” before you take this class?
That’s preferable but not mandatory. Although without knowing the more classical view of ML, you won’t be able to understand some of the ideas in the class. e.g. the difference how bias and
variance are viewed. In general, all good-old machine learning (GOML) techniques are still used in practice. Learning it up doesn’t seem to have any downsides.
You may also notice that both “Machine Learning” and deeplearning.ai covers neural network. So will the material duplicated? Not really. deeplearning.ai would guide you through implementation of
multi-layer of deep neural networks, IMO which requires a more careful and consistent formulation than a simple network with one hidden layer. So doing both won’t hurt and in fact it’s likely that
you will have to implement a certain method multiple times in your life anyway.
Wouldn’t this class be too Simple for Me?
So another question you might ask. If the class is so simple, does it even make sense to take it? The answer is a resounding yes. I am quite experienced in deep learning (~4 years by now) and I
learn machine learning since college. I still found the course very useful, because it offers many useful insights which only industry expert knows. And of course, when a luminary such as Andrew
speaks, you do want to listen.
In my case, I also want to take the course so that I can write reviews about it and my colleagues in Voci can ask me questions. But with that in mind, I still learn several things new through
listening to Andrew.
That’s what I have so far. Follow us on Facebook AIDL, I will post reviews of the later courses in the future.
[1] So what is a true from-scratch implementation? Perhaps you write everything from C and even the matrix manipulation part?
If you like this message, subscribe the Grand Janitor Blog’s RSS feed. You can also find me (Arthur) at twitter, LinkedIn, Plus, Clarity.fm. Together with Waikit Lau, I maintain the Deep Learning
Facebook forum. Also check out my awesome employer: Voci.
Nov 29, 2017: revised the text once. Mostly rewriting the clunky parts.
Oct 16, 2017: fixed typoes and misc. changes.
Oct 14, 2017: first published
One reply on “Review of Ng’s deeplearning.ai Course 1: Neural Networks and Deep Learning”
I also would recommend to start with Andrew Ng’s “Machine Learning”. Matlab/Octace codes match much better with the material of the lectures than Python. Particularly the implemantation of vectorized | {"url":"http://thegrandjanitor.com/2017/10/14/review-of-ngs-deeplearning-ai-course-1-neural-networks-and-deep-learning/","timestamp":"2024-11-08T12:19:42Z","content_type":"text/html","content_length":"54007","record_id":"<urn:uuid:51c30e2e-37d4-4ea9-b960-1cfa96e55b07>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00688.warc.gz"} |
Is a square a rhombus ? Thanks | HIX Tutor
Is a square a rhombus ? Thanks
Answer 1
Since a square has four equal sides, it is a quadrilateral, and a rhombus is an equilateral quadrilateral.
However, unless a rhombus is both equilateral and equiangular, meaning that the measures of all four angles are equal, it is not a square.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/is-a-square-a-rhombus-thanks-44068a3ebb","timestamp":"2024-11-10T11:56:51Z","content_type":"text/html","content_length":"575689","record_id":"<urn:uuid:bc85b1f0-ee90-4599-96db-c855ceb29bdc>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00854.warc.gz"} |
No mixing in light cone perturbation theory
3400 views
In hep-ph/0609090, Triumvirate of Running Couplings in Small-x Evolution, Kovchegov et. al. calculated the running coupling correction to the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and
Kovner (JIMWLK) equation using light cone perturbation methods.
The diagrams he was calculated are of two types:
1. Bubbles which comes from (regular) gluon and goes to (regular) gluon.
2. Bubbles which comes from instanteneous gluon and goes to instanteneous gluon.
However, he didn't look in the more involved case when we have some mixing between interactions.
The question I would like to ask doesn't requires to read Kovchegov's paper and in terms of LCPT I can phrase it also in the following way - is there an arument why the matrix elements involving both
regular and instanteneous interactions (such as $\left\langle g\left|H\right|qq\right\rangle \left\langle qq\left|H\right|0\right\rangle $ or $\left\langle qq\left|H\right|qq\right\rangle \left\
langle qq\left|H\right|g\right\rangle \left\langle g\left|H\right|0\right\rangle $ for example, where $H$ denotes the interaction part of the QCD Hamiltonian) should not taken into account when
calculating the wave function or any higher order corrections?
This post imported from StackExchange Physics at 2014-08-11 14:52 (UCT), posted by SE-user Yair
I finaly got the answer - a careful calculation shows that these are identially vanishing.
This post imported from StackExchange Physics at 2014-08-11 14:52 (UCT), posted by SE-user Yair | {"url":"https://www.physicsoverflow.org/21990/no-mixing-in-light-cone-perturbation-theory","timestamp":"2024-11-13T05:25:54Z","content_type":"text/html","content_length":"102379","record_id":"<urn:uuid:cc6f09a9-04ca-4ec4-8444-a8f39e5a43fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00132.warc.gz"} |
Part Five: Token Testing
13. Head over to Kreludor .
14. Click on the little space ship at the bottom of the page to be taken up to the Token Testing Facility - or simply click here !
You will then have four options to participate in - creating rods, mixing acids, collecting objects for later tests, and performing the actual tests. You can pick which you'd rather do or switch
between all four to give yourself some variety.
Heat Test: Create control rods
To begin this task, click on the Converter Furnace icon.
Thanks to Nikitachi for finding this method!
You will be taken to the 'making control rods' room where YRB-X1 will greet you and explain you that your task is to manufacture a control rod by selecting the appropriate piece of metal. Everybody
gets different pieces of metal and the maximum weight of each rod is 10kg. To make sure you don't exceed this value, you will have to add the Mass together.
Your goal is to manufacture a rod by using 1, 2, 3 of these pieces of metal, but only certain combinations will work. Each rod that you will create will be made of 1 main compound. There are 10
different compounds, thus 10 different types of rod that you could make. Here is a chart that is explaining you how much KG of a compound you need to make that rod. When you are making a rod, focus
on 1 compound and only on 1 compound. You can ignore all other value, but make sure to get the total mass of 1 of your compound as it appears in the chart.
Compound Name Mass needed to turn it into a rod
Dariganium 5.0kg
Faeryllium 5.3kg
Kadoatite 3.3kg
Meridellium 0.7kg
Neopium 4.0kg
Slothite 1.1kg
Tikium 2.5kg
Tyrannium 3.6kg
Xweetite 1.9kg
Zafarium 1.8kg
In order to select the right pieces of metal, you need to calculate the mass of each compound. This can be done by multiplying the percent of compound by the mass of the scrap piece of metal.
Mass: 2.1kg
• 9.5% Dariganium
• 4.8% Slothite
• 85.7% Tyrannium
To help you understand this process, we will work from an example, to the left. Your value WILL be different than the one of this example.
We can now calculate the mass of each of the compound of that piece of metal:
2.1kg X 0.095 = 0.20kg
2.1kg X 0.048 = 0.10kg
2.1kg X 0.857 = 1.80kg
You can keep track of your work using a spreadsheet that would look like this:
At this point, you have 2 choices: you can either find the mass of all your compounds (that's 30 numbers to gather, yes) OR you can move on your second metal piece and so on until you meet
requirement from the rod chart!
When you find 1, 2 or 3 compounds that add to the rod chart values, select the metal pieces and melt them down. This process will take 180 seconds.
If you select the right metal pieces, you will get a message similar to this one:
Dissolving Test: Create acid solutions
No solution as of yet - please check back later!
Gravitics Test: Gather singularities
Gathering singularities is the easiest thing of them all, it's dumb luck! Click the BIG black box, you will either be redirected to a completely random Neopets page, get "timed out", or find a
"singularity" - which is your ultimate goal. You will know you've found the singularity when you see the image below and a message that says "Congratulations...You have successfully retrieved a
Testing Station Part 1: Heat
To begin testing all the dulplicate Space Faerie Tokens check the box indicated below and click to "Engage telepotation".
If you already have a station number you can type the number in instead of checking the box - if you want to go to a station with your friends or you just like a particular number or whatever.
NOTE: You *might* be thrown onto a station where some people have already started, and thus you can actually skip this step and jump directly to the Dissolving or Gravitics. This happens when people
leave, or are thown out of, a station where parts of the puzzle have been done. If you don't want this to happen then gather a group of friends and pick a random high number to go to - chances are
no-one will have started that yet.
This part is a team effort - each station needs 4 people, and once you have 4 you will begin. Once you have started click the lever and drag it so that the bar moves so that it's right before the
little flashing dot in the half circle of light (see the screenie). After you think you've got it right click to green padlock to confirm your position. If all 4 people get it correct, you will move
onto the next round.
There are three rounds, if your team completes them all then you move on to the second step - dissolving. However if at any point before you move on one member of your team makes a mistake you will
be reset back to round one.
Testing Station Part 2: Dissolving
Once you've successfully passed the three rounds of heat tests (or if you enter a testing station that is already past them) you will be able to start testing your Token with acids. Again, this is a
group round and everyone needs to get each round right to progress.
You will be asked for a specific amount of a specific acid - for example in the screenie above I was asked for 1% of Meridellic acid. To work out how much of your acid you need to add, you'll need to
do a bit of basic math. You're given a percentage amount and you need to change this into a concrete value in milliliters.
Using the formula above, take your total acid (in red) and multiply it by 1000 (to convert to milliliters) and then multiply it by your percentage needed (at the top in light blue). Divide your total
by 100 and you will get a number. Enter that number into the white box by your pet's picture and then select the appropriate acid according to this chart:
Finally remember to lock your choices in by clicking the green padlock! As with the Heat tests there are three rounds to the dissolving step, before you move on to the final test - gravity.
Testing Station Part 3: Gravitics
For the final part of the Token testing step you work again with three other people, testing the Token's affect on singularities. This step can get quite confusing! To start with you will see a
screen with a grid on it, and the Token at the center.
The first two parts to this are relatively straightforward. You need to find two values - called A and B respectively (for simplicity, you can call them whatever you like). Click on the black
singularity and drag it til the co-ordinates reach 0,0. Make sure you remember to lock your position! Once its been through its flashing green show the singularity will turn white and a number will
appear under it - this is A. For your next singularity drag it to co-ordinates 400,0. Lock it, let it have its show and another number will appear under it - B.
Sub these numbers into the following formulas to find the co-ordinates for your final singularity. Remember, X is the first co-ordinate and Y is the second.
Gravitics with Excel
You may find it quite difficult to use the formulas - typing out all these calculations in the 30 or so seconds alloted to you for the final singularity is somewhat annoying. It is possible, but also
very irritating if you go through all of that just to find you were a few seconds to late. However, if you've got a basic spreadsheeting program, like Excel, then there is a quicker method! Its takes
a bit of work beforehand though, as you need to set up spreadsheet to deal with the formula. This isn't that hard, and it makes solving the singularity co-ordinates much, much faster.
Firstly, open up a new spreadsheet and set it up with the A=, B=, X= and Y= to look like the one in the screenie below.
Next, enter the following formulas into the appropriate cells of your spreadsheet (you can just copy and paste them in). Don't forget the equal sign, thats important for Excel to recognize it as a
Cell Formula
A3 =D1^2
A4 =F1^2
A8 =D6^2
D6 =(A3-A4+400^2)/800
F6 =SQRT(A3-A8)
With nothing in your A and B cells your spreadsheet will now look like this:
The #NUM! error comes because when the A and B values are 0 (or undefined and taken to be 0 by Excel) as you are trying to find the square root of a negative number - which is mathmatically
impossible. Once you enter your A and B values that will go away, you shouldn't be asked to use two values that give the error.
Your spreadsheet is now set up and you're ready to tackle some more Gravitics tests! Find A and B in the same way as above and enter them into your spreadsheet in the correct cells (see screenie
below) - and your correct co-ordinates will magically appear in the X and Y cells!
If anything comes out as a decimal (like 247.189) then just disgard the numbers after the decimal (so its 247 for the last example). Position your singularity as your calculations dictate and lock
your choice in. If everyone in your group has gotten the singularity correctly placed then you'll get a screen saying you've finished testing, but that sadly that was a dupe Token - and that you're
welcome to got back and help more people test more tokens! Presumably, when a group finds the correct token they'll get a message saying as much and the step will be over.
Got questions? Want to set up a group at the testing stations? Come and discuss the plot at our Neopets Forums!
Welcome to TDN,
The Snowager
Next sleep in 3h, 36m, 40s.
Next Possible Wake
Nov 3: 5 AM/PM NST
Nov 1: 10 OR 11 AM/PM NST
Nov 2: 3 OR 4 AM/PM NST
Nov 3: 8 OR 9 AM/PM NST
Neopia Today
The Runway Contest
Recently At Forums | {"url":"https://thedailyneopets.com/return-sloth/ros-part-five/","timestamp":"2024-11-04T02:23:19Z","content_type":"text/html","content_length":"37894","record_id":"<urn:uuid:683bad1b-4eac-424b-8e36-410a3e83d071>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00320.warc.gz"} |
Geometry Problem 706: Triangle, Cevian, Three Circumcenters,
In a triangle ABC, O is the circumcenter, and D is a point on AC. If E and F are the circumcenters of triangles ABD and BDC, respectively, prove that the points B, E, O, and F are concyclic. (See the
figure below.) | {"url":"https://www.gogeometry.com/problem/p706_triangle_cevian_three_circumcenter_concyclic_points_plane_geometry.htm","timestamp":"2024-11-10T02:56:31Z","content_type":"text/html","content_length":"7732","record_id":"<urn:uuid:f9f685f8-482f-4997-af89-0dbacb6757df>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00343.warc.gz"} |