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# Closed form expressions for harmonic sums I.
This question is a continuation of the topic in Closed form expressions for harmonic sums . By using the integral representation of harmonic numbers and by induction we have derived the following integral representation of an infinite harmonic sum. We have: $${\mathfrak S}^{(p)}_n(t):=\sum\limits_{m=0}^\infty H_m^p \cdot \frac{t^{m+1}}{(m+1)^n} = \int\limits_{[0,1]^p} \prod\limits_{\eta=0}^{p-1} \log\left(1-\xi_\eta\right) \cdot \frac{\left(\sum\limits_{l=0}^{p}(-1)^l \binom{p}{l} Li_{n-l}(t\prod\limits_{\eta=0}^{p-1} \xi_\eta )\right)}{\prod\limits_{\eta=0}^{p-1} \xi_\eta^2} \prod\limits_{\eta=0}^{p-1} d \xi_\eta$$ Here $Li_n()$ are poly-logarithms, $p$ and $n$ are strictly positive integers and $t\in (-1,1)$. Now the question is how can we use the result above to calculate those sums at unity,i.e. how do we calculate ${\mathfrak S}^{(n)}_p(1)$ ? Does the result always depend only on certain values of the zeta function (as it does in case $p=1$ as shown in the question quoted above) or instead does it also depend on something else?
• For $p+m$ is odd it is reducible to zeta values and there is a general formula. For the other case I have never seen one. – Zaid Alyafeai Apr 24 '17 at 17:08
• – Zaid Alyafeai Apr 24 '17 at 17:12
• @Zaid Alyafeai: Thank you for this document and for the comments. I am interested in this topic because those Euler sums appear to be related to solutions to certain recurrence relations with time dependent coefficients. Again, ${\mathfrak S}^{(1)}_n(1)$ is known -- you posted it yourself some time ago. I was thinking that the integral I gave above could help us to derive the result for $p=2$ or maybe $p=3$. I will be working on this and I will post results if I achieve any. – Przemo Apr 24 '17 at 17:20
• It seems difficult to work with $\mathrm{Li}_q(xy)$. Moreover in the link I provided they claim that Euler sums with large even weight should be considered as 'new' constants. Perhaps the best someone can do is consider ${\mathfrak S}^{(2)}_4(1)$. I guess you already know that ${\mathfrak S}^{(p)}_p(1)$ is trivial. – Zaid Alyafeai Apr 25 '17 at 6:45
• If you like working with Euler sums. You can take a look at my latest question math.stackexchange.com/questions/2169507/… – Zaid Alyafeai Apr 25 '17 at 6:50 |
# Is an anti-symmetric and asymmetric relation the same? Are irreflexive and anti reflexive the same?
I don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x).
But what if you have only one (not all) double arrow (e.g. one where if (x,y) then (y,x)). What is the name for this kind of relation? Anti symmetric?
Same goes for irreflexive and anti-reflexive. From my understanding, it is irreflexive if for no x then (x,x). But what if you have only one element that refers to itself?
I hope I made the question clear.
Thank you.
A binary relation $R$ on a set $X$ is symmetric when :
$\forall a,b \in X ( aRb \rightarrow bRa )$.
An example of symmetric relation : "... is married to ___".
A binary relation $R$ on a set $X$ is asymmetric when :
$\forall a,b \in X ( aRb \rightarrow \lnot(bRa) )$.
An example of asymmetric relation : "... is father of ___".
A binary relation $R$ on a set $X$ is antisymmetric if there is no pair of distinct elements of $X$ each of which is related by $R$ to the other; i.e. :
$\forall a,b \in X ((aRb \land bRa) \rightarrow a = b )$.
An example of antisymmetric relation : The usual order relation ≤ on the real numbers.
Every asymmetric relation is also an antisymmetric relation.
Your proposal for asymmetric is :
• if there is not any element where: if (x,y), then (y,x);
this "sounds" like :
$\lnot \exists x,y ( xRy \rightarrow yRx )$.
Moving "inward" $\lnot$, it is equivalent to :
$\forall x,y (\lnot ( xRy \rightarrow yRx ))$.
Now we need the equivalence between $\lnot ( p \rightarrow q )$ and $p \land \lnot q$ to get :
$\forall x,y (xRy \land \lnot (yRx) )$.
This is not the same as the formula in the definition.
According to the definition of asymmetric, if $a$ is father of $b$, then $b$ is not father of $a$, which is reasonable.
According to your "rewritten" condition, for every couple of individuals, we have that the first one is father of the second and the second is not father of the first one, which sound quite unreasonable.
Examples about reflexive and irreflexive (or anti-reflexive) :
• the relation "... is equal to _" between real numbers is reflexive;
• the relation "... is greater than ___" between real numbers is irreflexive.
Every asymmetric relation is also antisymmetric. But if antisymmetric relation contains pair of the form (a,a) then it cannot be asymmetric.
Antisymmetric means that the only way for both aRb and bRa to hold is if a = b. It can be reflexive, but it can't be symmetric for two distinct elements.
Asymmetric is the same except it also can't be reflexive. An asymmetric relation never has both aRb and bRa, even if a = b.
So an asymmetric relation is just one that is both antisymmetric and irreflexive.
Antisymmetricity is defined as:
∀a∀b(((a, b) ∈ R ∧ (b, a) ∈ R) → (a = b))
So if a = b then (a,b) = (a,a) = (b,b) = (b,a) ∈ R. So it implicitly says that if a ≠ b, then either (a, b) ∈ R or (b, a) ∈ R but not both, which indirectly is the part of the defition of asymmetricity
∀a∀b((a, b) ∈ R → (b, a) ∉ R)
However the definition above of asymmetricity indirectly adds that if a = b then
(a,b) = (a,a) = (b,b) = (b,a) ∉ R.
which is the reason for why asymmetric relation cannot be reflexive.
For example, {<1,1>, <1,2>, <2,3>} is not asymmetric because of <1,1>, but it is antisymmetric.
So in matrix representation of the asymmetric relation, diagonal is all 0s. And in digraph representation, there are no self-loops.
Let's take an analogy.
Relation : Hitting
$x$R$y$ $\equiv$ $x$ is related to $y$ $\equiv$ $x$ hits $y$.
Let's make two types of classification of people.
• First classification:
There are two types of people (y):
1. If you hit them, they will hit you back.
($x$ hits $y$ => $y$ hits $x$) $\equiv$ ($x$R$y$ => $y$R$x$).
1. If you hit them, they will ignore you.
($x$ hits $y$ but $y$ does not hits $x$) $\equiv$ ($x$R$y$ but $\lnot$$yRx) . • Second classification: Insane people, they hit themselves. (y hits y) \equiv (yRy) And Sane people, they do not hit themselves. (y does not hits y) \equiv (\lnot$$y$R$y$)
Now, we collect some people and make a set.
Symmetric relation:
If you make a set of all type (1) people and they are Insane people.
(type (1) $\wedge$ sane) $\equiv$ (($x$R$y$ => $y$R$x$)$\wedge$ ($y$R$y$))
So, Symmetric relation -
1. If (x,y) is in set, then (y,x) must be in the set.
2. (x,x) must be in the set.
Antisymmetric relation:
If you make a set of all type (2) people and they may be Sane or Insane people.
(type (2) $\wedge$ (sane $\lor$ Insane)) $\equiv$ (($x$R$y$ but $\lnot$$yRx)\wedge ((\lnot$$y$R$y$) $\lor$ ($y$R$y$)))
So, Antisymmetric relation -
1. If (x,y) is in set, then (y,x) must not be in the set.
2. (x,x) may or may not be in the set.
Asymmetric relation:
If you make a set of all type (2) people and they are Sane people.
(type (2) $\wedge$ sane) $\equiv$ (($x$R$y$ but $\lnot$$yRx)\wedge (\lnot$$y$R$y$))
So, Asymmetric relation -
1. If (x,y) is in set then, (y,x) must not be in the set.
2. (x,x) must not be in the set.
Point 2. tells the difference between Antisymmetric and Asymmetric set.
Reflexive relation:
If you make a set of all Insane people. They may be type (1) or type(2).
($y$R$y$) $\wedge$ (type(1) $\lor$ type(2))
So, Reflexive relation -
1. (x,x) must be in the set.
2. If (x,y) is in set then, (y,x) may or may not be in the set.
Irreflexive/Anti-reflexive relation:
If you make a set of all Sane people. They may be type (1) or type(2).
($\lnot$$y$R$y$) $\wedge$ (type(1) $\lor$ type(2))
1. (x,x) must not be in the set.
2. If (x,y) is in set then, (y,x) may or may not be in the set.
I will consider the matrix representation of relation Now check condition 1} if all diagonal elements are 0 condition 2} transpose of a matrix is not equal to itself matrix A.
If above conditions are satisfies then your relation is asymmetric relation. |
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# Solids and Surface Tension for JEE
## What is Surface Tension?
Last updated date: 19th Mar 2023
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Surface tension, which tends to minimise surface area, is the tension of a liquid's surface film brought on by the attraction of the surface layer's particles to the liquid's bulk. The forces of attraction between the particles in the provided liquid as well as the forces of attraction of the solid, liquid, or gas in contact with it affect surface tension. The effort or energy necessary to remove the surface layer of molecules in a unit area can be conceived of as being roughly equivalent to the energy responsible for the phenomenon of surface tension.
The forces of attraction between the particles in the given liquid, as well as the gas, solid, or liquid in contact with it, are the fundamental determinants of surface tension. Water has a surface tension of 0.07275 joule per square metre at 20 °C (68 °F).
## Surface Tension Formula
Surface tension can be defined mathematically as follows:
$T=\frac{F}{L}$
where F is the force applied per unit length, L is the length over which the force is applied, and T is the liquid's surface tension.
## Unit of Surface Tension
Newton per Metre, or N/m, is the metric measurement for surface tension (SI unit). Dyn/cm is the CGS unit for surface tension.
## Dimensional Formula of Surface Tension
The formula, as we all know, provides the surface tension.
Surface tension $=\frac{F}{L}$
Knowing that F = ma, we may substitute the value for it in the equation to obtain
Surface tension $=\frac{ma}{L}$
When we put the basic quantities into an equation, we get
Surface tension = $\left[ ML{{T}^{-2}}{{L}^{-1}} \right]$
Continuing to solve, we obtain
The dimensional formula of surface tension is $\left[ M{{T}^{-2}} \right]$.
## Surface Energy
The breakdown of intermolecular connections that takes place when a surface is formed is quantified as surface energy. Surface free energy and interfacial free energy are other names for it. The work performed by the force that forms the new surface per unit area can be used to define surface energy.
## Surface Tension in Solids
The tangential stress (force per unit length) in the surface layer is known as surface tension. This stress needs to be countered by either external forces or volume tensions within the body.
The relationship $y=F+A\left(\dfrac {dF}{dA} \right)$, where A is the surface area, links the surface free energy to the surface tension of a crystal face. Surface free energy and tension are equivalent for a liquid with a single component. Surface tension and surface energy are not identical for crystals.
## Contact Angle
When a gas-liquid interface comes into contact with a solid surface, such as the container's walls, surface tension arises; however, the interface typically curves up or down near the solid surface. A meniscus is a concave or convex surface with such a form.
It is possible to establish a connection between the liquid-solid surface tension and the liquid-gas surface tension using the contact angle, theta, as follows:
$\gamma{{a}_{ls}}=-\gamma{{a}_{\lg }}\cos \theta$
where $\gamma{{a}_{ls}}$ is the liquid-solid surface tension, $\gamma{{a}_{\lg }}$ is the liquid-gas surface tension, and $\theta$ is the contact angle.
Solid-Liquid-Gas Interface
One thing to remember about this equation is that the cosine component will be negative, which indicates that the liquid-solid surface tension will be positive, in situations where the meniscus is convex (that is, the contact angle is greater than 90 degrees).
The connection would produce a negative liquid-solid surface tension if, on the other hand, the meniscus is concave (i.e., dips down, causing the contact angle to be less than 90 degrees). In general, this means that the liquid is sticking to the container's walls and trying to increase the area in contact with the solid surface in order to reduce the total amount of potential energy.
## Methods of Measurement of Surface Tension
Using the following techniques, we may measure the surface tension:
1. Spinning Drop Method: Measurements are performed in a spinning horizontal tube that is filled with a dense fluid.
2. Stalagmometric Method: The system is built around a capillary glass tube with a wider centre.
3. Pendant Drop Method: The balance of forces, which includes the liquid's surface tension, determines the shape of the drop hanging from a needle.
4. Capillary Rise Method: The increase of a liquid above zero pressure level known as capillary rise is brought on by the net upward force generated by the attraction of water molecules to a solid surface.
5. Bubble Pressure Method: It enables monitoring the growth of surface tension at an emerging interface.
6. Sessile Drop Method: Using a superoleophobic surface created by chemical acid etching, anodizing, and fluorination treatment, the sessile drop method measures surface tension.
## Applications of Surface Tension
• Small insects, like the water strider, can only float on the water's surface due to their extremely low weight, which prevents them from penetrating it.
• On the water's surface, a tiny needle can be floated.
• Surface tension is crucial for the characterization of products used in food, medicine, and packaging.
• Different disorders in the human body can be identified by changes in surface tension in biological processes.
• The reason why water bubbles are round is because surface tension reduces the bubble into spherical shapes while still providing the tension necessary to form the bubble with the water.
• Surface tension is also the basis for how soaps and detergents function. They reduce the water's surface tension so that soaps and detergents can penetrate pores and holes with ease.
• When we use disinfectants in the field, they can float on the water and spread out on the cells to kill them because they are mostly low surface tension solutions.
• Surface tension phenomenon is used by R&D departments in all industrial plants to raise product quality.
## Conclusion
Fluid surfaces have a tendency to contract into the smallest surface area feasible due to surface tension. Surface tension is described as a phenomenon that happens when the surface of a liquid comes into contact with another phase (it can be a liquid as well). The least amount of surface area is often acquired by liquids. The liquid's surface behaves like an elastic sheet. The relationship between surface force F and the length L along which the force acts is known as surface tension.
Competitive Exams after 12th Science
## FAQs on Solids and Surface Tension for JEE
1. What Causes Surface Tension?
The liquid particles are drawn together by intermolecular forces like the Van der Waals force. The particles are drawn toward the liquid's interior along the surface. The relationship between surface force F and the length L along which the force acts is known as surface tension.
Surface tension can be defined mathematically as follows:
$T=\frac{F}{L}$
2. What Distinguishes Surface Energy from Surface Tension?
Surface tension and surface energy differ primarily in that surface tension measures the force per unit length of the surface whereas surface energy calculates the amount of effort required to stretch it per unit area. In contrast to surface energy, which is often used to describe the same forces between solid-state molecules, surface tension is typically used to describe the attractive interactions between liquid molecules. |
# A Tale of Three Cities: OmegaCAM discovers multiple sequences in the color-magnitude diagram of the Orion Nebula Cluster [SSA]
As part of the Accretion Discs in H$\alpha$ with OmegaCAM (ADHOC) survey, we imaged in r, i and H-alpha a region of 12×8 square degrees around the Orion Nebula Cluster. Thanks to the high-quality photometry obtained, we discovered three well-separated pre-main sequences in the color-magnitude diagram. The populations are all concentrated towards the cluster’s center. Although several explanations can be invoked to explain these sequences we are left with two competitive, but intriguing, scenarios: a population of unresolved binaries with an exotic mass ratio distribution or three populations with different ages. Independent high-resolution spectroscopy supports the presence of discrete episodes of star formation, each separated by about a million years. The stars from the two putative youngest populations rotate faster than the older ones, in agreement with the evolution of stellar rotation observed in pre-main sequence stars younger than 4 Myr in several star forming regions. Whatever the final explanation, our results prompt for a revised look at the formation mode and early evolution of stars in clusters.
G. Beccari, M. Petr-Gotzens, H. Boffin, et. al.
Mon, 29 May 17
-76/35
# HD 46487 is Now a Classical Be Star [SSA]
We present the first observations of hydrogen line emission detected around the B-type star HD 46487, a well-studied star in the CoRoT field of view. The emission is only evident in the H$\alpha$ line, for which the observed violet-red peak separation ($\Delta v_{p}$) is typical of a B$e$ star with a circumstellar disk. The absence of dust emission from the infrared spectral energy distribution excludes the possibility of a very young star. The star’s magnitude (V=5.079) and regular use in the literature for a variety of studies suggests that the line emission had a high probability of being found previously, had it been evident; since such was not the case, we believe that the B$e$ phenomenon for HD 46487 has only very recently “turned on.” We therefore recommend that this star be spectroscopically and photometrically monitored to track continued changes to its circumstellar morphology.
D. Whelan and R. Baker
Fri, 26 May 17
-32/63
Comments: 8 pages, 3 figures; accepted for publication in JAAVSO, Vol. 45
# Coronal Flux Ropes and their Interplanetary Counterparts [SSA]
We report on a study comparing coronal flux ropes inferred from eruption data with their interplanetary counterparts constructed from in situ data. The eruption data include the source-region magnetic field, post-eruption arcades, and coronal mass ejections (CMEs). Flux ropes were fit to the interplanetary CMEs (ICMEs) considered for the 2011 and 2012 Coordinated Data Analysis Workshops (CDAWs). We computed the total reconnected flux involved in each of the associated solar eruptions and found it to be closely related to flare properties, CME kinematics, and ICME properties. By fitting flux ropes to the white-light coronagraph data, we obtained the geometric properties of the flux ropes and added magnetic properties derived from the reonnected flux. We found that the CME magnetic field in the corona is significantly higher than the ambient magnetic field at a given heliocentric distance. The radial dependence of the flux-rope magnetic field strength is faster than that of the ambient magnetic field. The magnetic field strength of the coronal flux rope is also correlated with that in interplanetary flux ropes constructed from in situ data, and with the observed peak magnetic field strength in ICMEs. The physical reason for the observed correlation between the peak field strength in MCs is the higher magnetic field content in faster coronal flux ropes and ultimately the higher reconnected flux in the eruption region. The magnetic flux ropes constructed from the eruption data and coronagraph observations provide a realistic input that can be used by various models to predict the magnetic properties of ICMEs at Earth and other destination in the heliosphere.
N. Gopalswamya, S. Akiyamaa, S. Yashiroa, et. al.
Fri, 26 May 17
-23/63
Comments: 36 pages, 8 figures, 1 table, under review in J. Atmos. Solar-Terr. Phys. May 24, 2017
# On the discrepancy between asteroseismic and Gaia DR1 TGAS parallaxes [SSA]
Recently, a deviation of the Gaia TGAS parallaxes from the asteroseismic ones for giants was found. We show that for parallaxes $\varpi<1.5$ mas it can be explained by a selection effect in favour of bright and luminous giants in the Tycho-2 and TGAS catalogues. Another explanation of this deviation seems to be valid for $\varpi>1.5$ mas based on the best extinction estimates: the deviation may be caused not by a bias of parallax, but by an underestimation of the extinction (and, consequently, an overestimation of the calculated absolute magnitude) in the asteroseismic results. We demonstrate that the reliable estimates of the reddening and extinction (about 0.22 mag of the visual extinction for the Kepler field) better fit both the giants and main sequence stars to the PARSEC, MIST and YaPSI isochrones.
G. Gontcharov and A. Mosenkov
Fri, 26 May 17
-22/63
Comments: 5 pages, 4 figures, MNRAS Letters
# Increase in the amplitude of line-of-sight velocities of the small-scale motions in a solar filament before eruption [SSA]
We present a study on the evolution of the small scale velocity field in a solar filament as it approaches to the eruption. The observation was carried out by the Solar Dynamics Doppler Imager (SDDI) that was newly installed on the Solar Magnetic Activity Research Telescope (SMART) at Hida Observatory. The SDDI obtains a narrow-band full disk image of the sun at 73 channels from H$\alpha$ – 9.0 \AA\ to H$\alpha$ + 9.0 \AA, allowing us to study the line-of-sight (LOS) velocity of the filament before and during the eruption. The observed filament is a quiescent filament that erupted on 2016 November 5. We derived the LOS velocity at each pixel in the filament using the Becker’s cloud model, and made the histograms of the LOS velocity at each time. The standard deviation of the LOS velocity distribution can be regarded as a measure for the amplitude of the small scale motion in the filament. We found that the standard deviation on the previous day of the eruption was mostly constant around 2-3 km s$^{-1}$, and it slightly increased to 3-4 km s$^{-1}$ on the day of the eruption. It shows further increase with a rate of 1.1 m s$^{-2}$ about three hours before eruption and again with a rate of 2.8 m s$^{-2}$ about an hour before eruption. From this result we suggest the increase in the amplitude of the small scale motions in a filament can be regarded as a precursor of the eruption.
D. Seki, K. Otsuji, H. Isobe, et. al.
Fri, 26 May 17
-20/63
Comments: 8 pages, 5 figures, accepted for publication in ApJL on May 25, 2017
# Low mass variable stars in the globular cluster NGC 6397 [SSA]
We have conducted a photometric survey of the globular cluster NGC 6397 in a search for variable stars. We obtained ~11h of time-resolved photometric images with one ne European Southern Observatory-Very Large Telescope using the FOcal Reducer and low dispersion Spectrograph imager distributed over two consecutive nights. We analyzed 8391 light curves of stars brighter than magnitude 23 with the 465 nm-filter, and we identified 412 variable stars, reaching ~ 4.8 +- 0.2 per cent of variability with timescales between 0.004 and 2d, with amplitudes variation greater than +- 0.2 mag.
E. Martinazzi, S. Kepler and J. Costa
Fri, 26 May 17
-18/63 |
# What is the theoretical smallest size limit for a phased array antenna that can reach LEO?
Now that one provider is operating a phased array ground-to-satellite internet comm system, others will follow. A frequently asked question is "When can we use Starlink with a sat phone?". Does physics allow such a thing? I can believe the current dish can be shrunk, but what are the theoretical and practical limits?
The current Starlink constellation orbits at 540 to 590 km and Kuiper plans to use 590 to 630 km. However, Starlink proposed a later shell would orbit at 340 km, although idk if that is in their current planning. Nevertheless, 340 km could be used in your calculations of the ground antenna size limit.
• The question's a bit poorly formed. The size requirements depend on the wavelength, usable power, data rate requirements, and on the capabilities of the antennas at both ends of the link. Phones have additional problems in that any phased arrays integrated into them will be largely pointed away from overhead satellites while in use. Cell phone phased arrays are a real thing, but they're more intended for tracking towers that are more or less near the horizon. Sep 5, 2021 at 18:26
• Higher frequency means shorter wavelength and so a smaller phased array for a given performance. However, this answer shows that going up in frequency from a few GHz to a few dozen GHz has a down side- higher attenuation when it rains. 3 GHz and 30 GHz have wavelengths of 10 cm and 1 cm respectively, so individual raindrops are betting bigger compared to a wavelength and so can do some serious scattering. And if your phased array gets water droplets on its surface and doesn't shed them, they will screw up the radiation pattern.
– uhoh
Sep 5, 2021 at 22:58
• @uhoh Be careful with "smaller phased array for a given performance" - that is true, but only if you treat gain as your only performance metric. For a given antenna aperture on both ends, different frequencies, and all else being equal, the link budget will be identical. The free-space path loss is frequency-dependent and this effect exactly cancels out the gain increase at a given aperture. Imagine a dipole antenna - as you lower frequency, the antenna and its effective aperture get bigger. For a given aperture on both ends, frequency affects directivity through gain, but not received power. Sep 5, 2021 at 23:39
• @DodoDude700 wavelength appears in "free space path loss" only because of the way that link budget defines gain. Per square meter of spacecraft antenna and at a given distance, power received will be proportional to antenna diameter $d$ as $(d/ \lambda)^2$ Some other posts about link budgets: space.stackexchange.com/search?q=user%3A12102+%27link+budget%27
– uhoh
Sep 5, 2021 at 23:59
• @uhoh Okay, looking at this again, I'd like to correct my earlier comment here, because I realize I said "on both ends". For a given aperture on both ends, you are right, power received DOES depend on frequency, and higher is better. For a given directivity on both ends, lower is better. What I meant, and should have said, is that for a given aperture on one end (satellite reflector) and a given gain on the other (mobile terminal with unsteered antenna), frequency does not matter. As such, whether it matters depends on if terminal is limited by max size or by a need to be omnidirectional. Sep 6, 2021 at 4:31
It's all a matter of link budget.
Globalstar, for example, has satellites at 1400 kilometers, usable from a satellite phone with a low-gain omnidirectional antenna, as are Iridium's satellites at 780. Both provide narrowband to midband internet access. Inmarsat, Thuraya, and plenty of others have satellites in geostationary orbit at 35,786 kilometers that can provide narrowband-to-midband internet service to low-to-medium gain antennas (about 10 dBi, in the case of BGAN) by using large antennas on the satellite side.
Pictured: An Inmarsat I-4, above as deployed in GEO, below during assembly (humans for scale!)
These are, of course, L- and S-band services, which if your question is about Starlink, their satellites do not support (and are not licensed to use), but if it's a question of technical feasibility, it's very possible - there are plenty of companies already doing it.
However, working from this number we can come up with a basic link budget. The antenna reflector on the aforementioned Inmarsat I-4 has a diameter of 10 meters. For mobile-satellite service, you generally want the lowest frequency you can get away with in order to keep required user terminal gain low, but realistically, such a hypothetical "MSS Starlink" would not be able to get as much L-band spectrum as it would need - lot fewer total MHz to allocate out down there. However, we can work with it. At L-band (1.5 GHz), the gain of the giant I-4 like satellite-side antenna should be about 43 dBi. From the same calculator, at Ka-band, 20GHz (the lowest-frequency MSS-allocated spectrum I reckon they could actually use for this), it's 65 dBi.
The distance from the user to the satellite is very much up to the constellation design. A satellite that must serve users at its horizon needs to support users much further away than its orbital altitude. However, since this is a "what is theoretically possible" question we can assume a very large constellation, in which the distance to the user approaches orbital altitude. I'll say, as somewhat of a guess, that max distance to the user is 400 kilometers, for a very dense constellation at your 340km orbit.
Plugging all that into here, and assuming a user terminal antenna gain of 4 dBi (this is pretty typical for an unpointed mobile satellite service terminal antenna) and adjusting the transmitter power until we reach our link budget, we find that the satellite would need to put out around 150-200 watts. This is high, but not utterly inconceivable for satellite communications. If we instead assume 10 dBi on the user terminal (BGAN-like gain - would probably need a small, few-element phased array to get away with this for a non-GEO satellite, but could still be a small flat panel, maybe on the order of 10x10 cm with like, a 2x2 array or so) we can get the needed satellite-side TX power down to 50 watts, more reasonable.
At Ka-band, thanks to the higher gain of the large reflector on the satellites, the link budget actually works out relatively similar, though one thing that calculator won't account for unless you calculate separately and add it in is atmospheric losses which are larger at Ka-band than L-band. Note that on the satellite side, due to amplifier inefficiency, the effective power requirement to transmit at Ka-band is likely 2-3x what it would be to transmit at the same power output at L-band. Getting 200 watts for one transmit beam out of a satellite in the Ka-band would be rather challenging, I would think.
I would also like to note that simply taking a GEO satellite and putting it in LEO is going to require some design changes - among other things, GEO satellites are not typically designed to have their view of the sun eclipsed by earth every 100-ish minutes, and while they do have batteries for when the sun DOES go behind the earth for them, this is much more infrequent and not as meaningful to their power budget. Adapting a GEO satellite to LEO, you would probably need more solar panels and more, possibly more and better (higher cycle-life - you'll see a lot more cycles on them in LEO), batteries.
Finally, the user terminal having such greatly decreased gain is likely to require network changes to ensure two satellites, even far-apart ones, do not serve anything close to the same area. With a high-gain steerable user terminal you get some degree of choice in which satellite you "hear" - not so at 4 dBi. This is a significant, if not insurmountable, consideration.
TL;DR: With enough of a satellite, it is possible. However, you're talking about taking some of the largest, most specialized MSS satellites in GEO and launching thousands of them to reach a suitably dense LEO constellation. It would be an exceptionally expensive undertaking, even on the scale of the already-high costs of GEO satellites or LEO constellations. According to the best info on it I could find, Inmarsat's satellites cost between \$100M and \$400M each. Economies of scale of course apply when producing more of them, but I think at Starlink-like numbers this could very easily be a trillion dollar project (while I'm sure someone will be quick to raise dropping launch costs, I would like to point out that for such satellites, launch, even launch of such large satellites to such a distant orbit as GEO, accounts for only a relatively small fraction of the cost to build, launch, and operate). On a purely technical basis however, yes, it is possible.
EDIT: Also, as for shrinking the user terminal down this small on the current constellation, no chance. No amount of redesign work will get the aperture efficiency of an antenna above 100%. They could maybe make it a little better, or make the modulation a little more efficient, but they're not going to go this far just with improved user terminals. You can't beat the Shannon limit and you can't get aperture efficiency above 100%.
• That you for the very thorough answer. My simple-minded takeaway is the 10x10 cm array for a practical unit. Sep 5, 2021 at 22:58
• @SpaceInMyHead That's probably about roughly right but In the Ka-band case it could conceivably be smaller than that - a 4x4 array at Ka-band with a smaller 5x5cm footprint is probably possible per the link budget numbers I gave probably. Would need to run some simulations to see what atmospheric effects do to it at those frequencies though. Sep 5, 2021 at 23:23
• Why are you taking a GEO (Inmarsat-4) as a reference point for "sat phone" service and not an Iridium or a Globalstar? Sep 7, 2021 at 19:56
• @NgPh Because this is a question of providing broadband service to very small terminals. In their actual implementation, MSS services, whether GEO or LEO, generally have relatively similar, "mutually competitive" offerings. However, to provide broadband to low-gain terminals, you're, practically speaking, going to need BOTH the large antennas currently on GEO satellites, and the low orbits of LEO satellites. Iridium and Globalstar don't have (or need) anything close to the antennas seen on GEO MSS satellites. As such, a GEO MSS satellite in a lower orbit is a reasonable starting point I think. Sep 8, 2021 at 1:53
• We CANNOT delimit the OP's question by "broadband service to very small terminals" w/o first pinning down what "broadband" & "very small" mean. The 2nd gen. BigLEO have peak bitrates comparable to BGAN ( cf this). If we want to explore a solution involving a huge satellite antenna, there is this company. They are at advanced R&D phase (test satellite scheduled for 03/22), but it's worth mentioning them. Sep 8, 2021 at 10:05
The trick in this question is the term "theoretical", which the OP left open for you to show the "limit" (if any).
"Theoretical" could mean:
• At any cost.
Then you can try to increase the satellite antenna gain (in future Starlink generation(s)) to compensate for the loss of gain in the user terminal (as suggested in comments and one answer), to keep your link-budget closed, everything else being equal.
Practically, it means more (narrower) beams per satellite or more satellites. It could mean heavier satellites and hence more launches.
• At any service instability due to weather
You compensate the antenna gain loss by reducing the rain margin, which is high at the frequencies used by Starlink for the service link (Ku-band).
Practically, and especially for tropical regions, you mainly aim at providing service when the sky is clear. The engineering data for this is given by this ITU recommendation. You assume that the users have no other alternative, that is you are "better than nothing".
• At any throughput
You antenna gain loss is compensated by reducing the service bit rate, assuming that Starlink already operates close to the Shannon bound (the only theoretical limit I know of). Theoretically, if you reduce the peak bitrate by 10, you can recover 10dB in any loss of gain in your link-budget. Note that the previous "Big LEO" constellations offer bitrates in the order of 1/1000 of that of Starlink.
Practically, it means you are not selling the same type of service (i.e. Broadband Access), and hence your market size is smaller.
• Without due considerations to frequency regulations
Theoretically, you can try to transmit as much as you need ("to reach LEO").
Practically, if you don't have exclusive rights to a portion of frequencies, there are rules by the ITU to protect other users of these frequencies, especially those who have priority over yourself. Note that Starlink operates in the frequency already used by GEO satellites (the "incumbents", who have priority). The only way to protect GEO is by using "directional separation". That is to ensure that terminals of Non-GEO constellations do not transmit excessively in the direction of the Equator (where the GEO satellites operate). These rules impose directional antennas, which in turn impose large areas/dimensions, whether they are array-based or reflector-based, or a hybrid of both. For interferences reason, Starlink antennas cannot "look" at a large portion of the sky above, unless all satellites it sees (the GEO, non-GEO-non-Starlink, and the Starlink themselves but not the best in-sight one) operate in different frequencies.
The bottom line: a link-budget exercise is not a "simple" balance sheet. It is the engineer's art of making the right assessments of practical limits. |
Math Help - Polynomial Rigs _ Dummit and Foote Chapter 9
1. Polynomial Rigs _ Dummit and Foote Chapter 9
I am reading Dummit and Foote Chapter 9 on Polynomial Rings and am trying to get a good understanding of Propostion 2 (see Attachment - page 296) which states:
================================================== ==============
Let I be an ideal of the ring R and let (I) = I[x] denote the ideal of R[x] generated by I (set of polynomials with co-efficients in I). Then
R[x]/I[x] $\cong$ (R/I) [x]
In particular, if I is a prime ideal of R then (I) is a prime ideal of R[x]
================================================== ===============
I decided to generate a simple example using R= $\mathbb{Z}$ and I = 2 $\mathbb{Z}$
Then $\mathbb{Z}$ = { ..., -2, -1, 0, 1, 2, 3, ... } and 2 $\mathbb{Z}$ = { ..., -4, -2, 0, 2, 4, 6 .... }
also $\mathbb{Z}$/2 $\mathbb{Z}$ = { $\overline{0}, \overline{1}$ }
Then it appears to me that R[x] = $\mathbb{Z}$ [x] is the set of all polynomials with integer coefficients and I[x] is the set of polynomials with even integers as coefficients
Now how do I formally express R[x], I[x] and $\mathbb{Z}$/2 $\mathbb{Z}$ formally and algebraicly?? Is my text above OK?
It seem that $\mathbb{Z}$ [x] /2 $\mathbb{Z}$ [x] would have two elements - one which was all the polynomials with even co-efficients and one which contains all the polynomials with odd integer co-efficients - but again - how do I express this in formal algebraic symbolism
Further, returning to Propostion 2 above
R/I = $\mathbb{Z}$/2 $\mathbb{Z}$ = { $\overline{0}, \overline{1}$ } and so (R/I)[x] appears to have two elements - one the set of polynomials with coefficients in $\overline{0}$ and one with coefficients in [TEX]\overline{1}TEX] which seems to be correct.
Is my example and reasoning correct? Would appreciate an assurance from someone that all is OK.
How would my exampole ber expressed in more formal algebraic symbolism?
Note: I was also somewhat thrown by D&F's use of the symbolism (I) for I[x]. Previously (see attachment on Properties of Ideals, D&F ch 7 page 251) the symbol (I) was used to denote the smallest idea of R containing I which D&F point out is the set of all finite sums of elements of the form ra with r $\in$ R and a $\in$ I ie sums such as $r_1 a_1 + r_2 a_2 + ... + r_n a_n$. Isn't the use of (I) for I[x] somewhat inconsistent with the use of the symbolism just described.
Peter
2. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
First of all, yes your examples are correct.
$R[x] = \{a_nx^n+a_{n-1}x^{n-1}+...+a_0 | a_i \in R, n \in \mathbb{N} \}$
Same for I[x]
As for $\frac{\mathbb{Z}}{2\mathbb{Z}} = \{0 + 2\mathbb{Z}, 1 + 2\mathbb{Z} \}$
one which was all the polynomials with even co-efficients and one which contains all the polynomials with odd integer co-efficients - but again - how do I express this in formal algebraic symbolism
Since most of this is merely representation, i think if you are clear, you can write it anyway.
Like,
$(\frac{\mathbb{Z}}{2\mathbb{Z}})[X] = \{ \{a_0+a_1x+...+a_nx^n | a_i \in 0 + 2\mathbb{Z} \}, \{a_0+a_1x+...+a_nx^n | a_i \in 1 + 2\mathbb{Z} \} \}$
OR
$(\frac{\mathbb{Z}}{2\mathbb{Z}})[X] = \{ \{a_0+a_1x+...+a_nx^n | a_i = 2k \text{ for some } k \in \mathbb{N}\}, \{a_0+a_1x+...+a_nx^n | a_i = 2m + 1 \text{ for some } m \in \mathbb{N} \} \}$
3. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
let's look at Z[x]/(2Z)[x] and (Z/2Z)[x] separately.
the first consists of cosets of the form:
p(x) + (2Z)[x]
for example, we have the cosets 4x3+1 + (2Z)[x] and 2x2+4x+3 + (2Z)[x]. note that these are actually the SAME coset, since:
4x3+1 - (2x2+4x+3) = 4x3-2x2-4x-2, which is in (2Z)[x].
the second consists of polynomials in (Z/2Z)[x], which have coefficients in the ring (Z/2Z) = {[0],[1]} (and a common abuse of notation is just to call these 0 and 1). for example, x+1, x2+1, and x3+x+1 are all elements of
if we want to show that these two rings are isomorphic, we can (and probably SHOULD) display an isomorphism:
let's try this one:
we know we have a (ring) homomorphism k-->k (mod 2) of Z onto Z/(2Z). let's call this φ.
so now, let's define a map ψ:Z[x]-->(Z/2Z)[x] by:
ψ(a0+a1x+...+anxn) = φ(a0) + φ(a1)x +....φ(an)xn.
i leave it to you to verify ψ is actually a homomorphism (it's easy). essentially, we're just replacing aj with aj (mod 2), so for example, in our polynomials above:
ψ(4x3+1) = φ(4)x3 + φ(1) = 0x3 + 1 = 1,
ψ(2x2+4x+3) = φ(2)x2 + φ(4)x + φ(3) = 0x2 + 0x + 1 = 1.
if we can show ker(ψ) = (2Z)[x], we're done (by the first isomorphism theorem for rings)!
but if ψ(p(x)) = 0, we must have φ(aj) = 0, for every coefficient aj of p, thus p is in (2Z)[x], so ker(ψ) is in (2Z)[x].
on the other hand if p(x) is in (2Z)[x] = (ker(φ))[x], then ψ(p(x)) = 0 + 0x +....+0xn = 0, so (2Z)[x] is contained in ker(ψ).
(the same proof works, by the way, for R and the ideal I).
thus Z[x]/(2Z)[x] is isomorphic to (Z/2Z)[x].
**************************
(Z/2Z)[x] DOES NOT HAVE TWO ELEMENTS!!!!! it has infinitely many. all of them have "1's" as coefficients, but x and x2 are NOT the same polynomial in (Z/2z)[x]. x+1 and x are also distinct polynomials. in fact, for any degree n, we have 2n distinct polynomials of degree n. for example, we have, of degree 2:
x2
x2+1
x2+x
x2+x+1
**************************
a word about the notation (I). note that this is the smallest ideal of R[x] containing I. since (I) is an ideal of R[x], it must contain all monomials of the form axj, with a in I. thus it must contain all sums of these (since ideals are closed under addition), and thus I[x]. hence I[x] is contained in (I). on the other hand, since I[x] is an ideal of R[x] containing I (as the polynomials a*1, for a in I), we must have (I) in I[x]. hence (I) = I[x].
4. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
Originally Posted by Deveno
let's look at Z[x]/(2Z)[x] and (Z/2Z)[x] separately.
the first consists of cosets of the form:
p(x) + (2Z)[x]
for example, we have the cosets 4x3+1 + (2Z)[x] and 2x2+4x+3 + (2Z)[x]. note that these are actually the SAME coset, since:
4x3+1 - (2x2+4x+3) = 4x3-2x2-4x-2, which is in (2Z)[x].
the second consists of polynomials in (Z/2Z)[x], which have coefficients in the ring (Z/2Z) = {[0],[1]} (and a common abuse of notation is just to call these 0 and 1). for example, x+1, x2+1, and x3+x+1 are all elements of
if we want to show that these two rings are isomorphic, we can (and probably SHOULD) display an isomorphism:
let's try this one:
we know we have a (ring) homomorphism k-->k (mod 2) of Z onto Z/(2Z). let's call this φ.
so now, let's define a map ψ:Z[x]-->(Z/2Z)[x] by:
ψ(a0+a1x+...+anxn) = φ(a0) + φ(a1)x +....φ(an)xn.
i leave it to you to verify ψ is actually a homomorphism (it's easy). essentially, we're just replacing aj with aj (mod 2), so for example, in our polynomials above:
ψ(4x3+1) = φ(4)x3 + φ(1) = 0x3 + 1 = 1,
ψ(2x2+4x+3) = φ(2)x2 + φ(4)x + φ(3) = 0x2 + 0x + 1 = 1.
if we can show ker(ψ) = (2Z)[x], we're done (by the first isomorphism theorem for rings)!
but if ψ(p(x)) = 0, we must have φ(aj) = 0, for every coefficient aj of p, thus p is in (2Z)[x], so ker(ψ) is in (2Z)[x].
on the other hand if p(x) is in (2Z)[x] = (ker(φ))[x], then ψ(p(x)) = 0 + 0x +....+0xn = 0, so (2Z)[x] is contained in ker(ψ).
(the same proof works, by the way, for R and the ideal I).
thus Z[x]/(2Z)[x] is isomorphic to (Z/2Z)[x].
**************************
(Z/2Z)[x] DOES NOT HAVE TWO ELEMENTS!!!!! it has infinitely many. all of them have "1's" as coefficients, but x and x2 are NOT the same polynomial in (Z/2z)[x]. x+1 and x are also distinct polynomials. in fact, for any degree n, we have 2n distinct polynomials of degree n. for example, we have, of degree 2:
x2
x2+1
x2+x
x2+x+1
**************************
a word about the notation (I). note that this is the smallest ideal of R[x] containing I. since (I) is an ideal of R[x], it must contain all monomials of the form axj, with a in I. thus it must contain all sums of these (since ideals are closed under addition), and thus I[x]. hence I[x] is contained in (I). on the other hand, since I[x] is an ideal of R[x] containing I (as the polynomials a*1, for a in I), we must have (I) in I[x]. hence (I) = I[x].
Yes sir, probably should have said 2 cosets.
5. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
Originally Posted by jakncoke
Yes sir, probably should have said 2 cosets.
no, not even two cosets. infinitely many.
6. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
Originally Posted by Deveno
no, not even two cosets. infinitely many.
Yes sir, right again, i'm gonna go to bed.
7. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
A most helpful post - still working through the ideas in this post!
Thank you!!!
Peter
8. Re: Polynomial Rings _ Dummit and Foote Chapter 9
In the first part of your post above, you invite us to consider Z[x]/(2Z)[x].
This quotient ring of polynomials, you point out, consists of cosets of the form
p(x) + (2Z)[x]
and you give as an example the cosets of $4x^3 + 1 + (2Z)[x]$ and $2x^3 + 4x + 3 + (2Z)[x]$
and you point out that these are actually the same coset since $(4x^3 + 1) - (2x^3 + 4x + 3) \in (2Z)[x]$
Thus you seem to be using the following rule:
Two polynomials r(x) and s(x) belong to the same coset if $r(x) - S(x) \in (2Z)[x]$
Is this correct? Presumably you can generalise this from (2Z)[x] to the general ring R[x]
================================================== =============================
To try to get a better understanding of your post and the above in general I turned to Hungerford's Abstract Algebra Ch 5 Congruence in F[x] and Congruence-Class Arithmetic - see attachment for pages 119-121.
I tried to fit or reconcile your definitions and process above with Hungerford's definitions at the bottom on p119 and page 120 plus his two illustrative examples of congrience modulo $(x^2 + 1)$ in R[x] and congruence modulo $(x^2 = x + 1)$ in $Z_2[x]$, but was unable to formally do so.
Can you please show how your analysis fits with the approach of Hungerford. I know they deal with the same structures so they should be in harmony but I cannot formally reconcile the two approaches. Can you show how this works?
By the Hungerford deals with F[x] where F is a field. DOes it work if F is simply a ring?
Peter
9. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
in ANY ring (polynomial or not) R, if I is an ideal of R, we can form the factor ring R/I.
if R is a commutative ring, then we have to ways to "factor by I":
R[x]/I[x] (form the polynomial rings _[x] first, then factor the "big" polynomial ring by the "smaller" one).
(R/I)[x] (factor R by I first, THEN form a polynomial ring).
it turns out that this is "two paths to the same destination".
in general, for a factor ring R/I, we have a+I = b+I iff a-b in I (this is the abelian group version of aH = bH iff ab-1 in H). it turns out that for a ring, the additive structure "controls" how we form the cosets (ideals/kernels are the pre-images of the additive identity of the image).
the same rule holds for the factor ring F[x]/I, where I is an ideal of F[x], for a field F: f(x) + I = g(x) + I iff f(x) - g(x) is in I. however, F[x] where F is a field, has a property general polynomial rings R[x] do not: every ideal is principal (this is the corresponding idea to "cyclic group" for ideals: a principal ideal is generated by a single element, so every element in it is some multiple of the generator.
this is usually indicated in F[x] by <(p(x)> or (p(x)). so if f(x) - g(x) is in (p(x)), this means that f(x) - g(x) = k(x)p(x), for some polynomial k(x), or more pithily: p(x) DIVIDES f(x) - g(x).
to illustrate what goes "wrong" in a ring R[x], consider the ideal <2,x> in Z[x] (the ring of polynomials with integer coefficients). what do the elements of <2,x> look like?
they are the sum of a polynomial with all even coefficients and a polynomial with no constant term. a shorter description is: all polynomials with even constant term. so
2+x
3x + x2 <--0 constant terms (and thus a factor of x).
4 + 2x + x3 are all examples
now there is NO polynomial p(x) in Z[x] such that <2,x> = <p(x)>. to see this, note that such a polynomial would have to divide 2, and so must be 1,-1,2, or -2 (the only integer divisors of 2).
but 2 does NOT divide x (there is no polynomial k(x) in Z[x] with 2k(x) = x), and neither does -2. but <-1> = <1> = Z[x], and <2,x> is NOT equal to the entire ring (for example 1+x is NOT in <2,x> since it does not have even constant term).
a "plain old commutative ring" doesn't, in general, "have enough divisors" for the gcd of two polynomials to be a "maximal" generator. so the gcd "generates too much". when the ring is a field, this problem goes away, the "extra structure" we get with a field (namely, "more units") gives "more structure" to its corresponding polynomial field.
the example hungerford gives of R[x]/<x2+1> is an extremely important one.
here, our ideal is <x2+1>, consisting of all real polynomials that contain x2+1 as a factor. the elements of R[x]/<x2+1> are cosets of <x2+1>. i will write this ideal as I, to underscore the similarity with R/I for a general ring R (R[x] is a ring, after all).
as with ANY factor ring, we declare f(x) CONGRUENT to g(x) if they lie in the same coset of I = <x2+1> (this is by direct analogy with the factor ring Z/nZ = Z/(n) where we declare a = b (mod n) if a - b is in (n) = nZ, that is, if a and b differ by a multiple of n. the congruence class [a] = a (mod n) is actually the coset a + nZ = a + (n)).
as is the case with groups, a ring congruence creates a partition (or equivalence) on the ring. in this case, we seek to find "simple representatives" of the cosets (equivalence classes) f(x) + I.
it turns out that we can represent ANY coset as the coset of a LINEAR polynomial ax +b + I. rather than go about PROVING this, i will just show how it's done for a polynomial of degree 5.
say we have x5 + 3x + 1 in R[x]. then x5 + 3x + 1 = x3(x2 + 1) - x3 + 3x + 1 (take a minute, and work through the algebra on this).
thus x5 + 3x + 1 - (x2 + 3x + 1) = x3(x2 + 1), and the RHS is in I = <x2 + 1>, so
x5 + 3x + 1 = -x3 + 3x + 1 (mod x2+1), that is: x5 + 3x + 1 + I = -x3 + 3x + 1 + I.
so now we've reduced our "representative" down to a polynomial of degree 3. we can keep going:
-x3 + 3x + 1 = -x(x2 + 1) + x + 3x + 1 = -x(x2 + 1) + 4x + 1 so:
-x3 + 3x + 1 - (4x + 1) is in I, so -x3 + 3x + 1 = 4x + 1 (mod I).
if you are paying attention, you'll see i am just doing "polynomial long division" one step at a time. basically i am "dividing" x5 + 3x + 1 by x2 + 1 until i get a remainder of degree less than deg(x2 + 1) = 2.
so x5+ 3x+ 1 + I = 4x + 1 + I, 4x + 1 is the linear polynomial i promised i would produce.
now here is where it gets fun:
what do we get if we multiply:
(a+bx + I)(c+dx + I), after we "reduce mod x2 + 1"? (you 'll see later why i wrote the constant term first) let's do it:
(a+bx + I)(c+dx + I) = ac + (ad+bc)x + bdx2 + I. this is a quadratic, so we can reduce by one more degree. what we're going to do is add and subtract bd, so that we have a factor bd(x2 + 1).
ac + (ad+bc)x + bdx2 + I = ad - bc + (ad+bc)x + bd(x2 + 1) + I. but since bd(x2 + 1) is IN I, bd(x2 + 1) + I = I, so:
compare this with the product of complex numbers:
(a+bi)(c+di) = ac-bd + (ad+bc)i...see any similarities?
in fact, and this is MOST important: it turns out that the coset x+I is a ROOT of the polynomial x2+1 in R[x]/I:
(x+I)2 + (1+I) = (x2+I ) + (1+I) = x2+1 + I = I = 0+I (recall that "I" is the 0-element (additive identity) of the ring R[x]/I).
in other words, a+bx + I = a(1 + I) + b(x + I) acts "just like" the complex number a+bi with "1+I" playing the role of the real number 1, and "x+I" playing the role of the imaginary number i (which is a root of x2+1).
so one might suspect that R[x]/I is, in fact, a field isomorphic to the complex numbers, and it turns out this is true.
10. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
Thank you for that extensive post - I will now happily work through it!
Peter
11. Re: Polynomial Rigs _ Dummit and Foote Chapter 9
You write:
"now there is NO polynomial p(x) in Z[x] such that <2,x> = <p(x)>. to see this, note that such a polynomial would have to divide 2, and so must be 1,-1,2, or -2 (the only integer divisors of 2)."
I am struggling to see why in the above <2,x> = <p(x)> implies that such a polynomial p(x) in Z[x] would have to divide 2
Can you formally show why this is the case
Peter |
# Problem: When two or more capacitors are connected in series across a potential difference:a) the potential difference across the combination is the algebraic sum of the potential differences across the individual capacitors.b) the equivalent capacitance of the combination is less than the capacitance of any of the capacitors.c) each capacitor carries the same amount of charge.d) All of the above choices are correct.e) None of the above choices are correct.
###### FREE Expert Solution
Equivalent capacitance of series capacitors.
$\overline{)\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{e}\mathbf{q}}}{\mathbf{=}}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{1}}}{\mathbf{+}}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{2}}}{\mathbf{+}}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{3}}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{+}}\frac{\mathbf{1}}{{\mathbf{C}}_{\mathbf{n}}}}$
The equivalent capacitance of the combination is less than the capacitance of any of the capacitors.
###### Problem Details
When two or more capacitors are connected in series across a potential difference:
a) the potential difference across the combination is the algebraic sum of the potential differences across the individual capacitors.
b) the equivalent capacitance of the combination is less than the capacitance of any of the capacitors.
c) each capacitor carries the same amount of charge.
d) All of the above choices are correct.
e) None of the above choices are correct. |
# From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the tota
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closed
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest c m^2
by (71.6k points)
selected by
Total surface area=pir^2+2pirh+pirl
=pir^2+2pirh+pir sqrt(r^2+h^2)
=pir(r+2h+sqrt(r^2+h^2))
=22/7(0.7)(0.7+4.8+sqrt(2.4^2+0.7^2))
=2.2(5.5+2.5)
=17.6 cm^2 |
# How do you solve x+3y=11 and x+4y=14 using matrices?
Jul 12, 2016
x=2; y=3
#### Explanation:
you must solve by applying the basic propositions:
x=D_x/D; y=D_y/D
where D is the determinant of the matrix
$| \quad 1 \quad 3 |$
$| \quad 1 \quad 4 |$
Its determinant is obtained by:
$D = 1 \cdot 4 - 3 \cdot 1 = 1$
${D}_{x} \mathmr{and} {D}_{y}$ are respectively the determinants of the matrices
$| \quad 11 \quad 3 |$
$| \quad 14 \quad 4 |$
$| \quad 1 \quad 11 |$
$| \quad 1 \quad 14 |$
${D}_{x} = 11 \cdot 4 - 3 \cdot 14 = 44 - 42 = 2$
${D}_{y} = 1 \cdot 14 - 11 \cdot 1 = 3$
So
$x = {D}_{x} / D = \frac{2}{1} = 2$
$y = {D}_{y} / D = \frac{3}{1} = 3$
Jul 17, 2016
$x = 2 \text{ ; } y = 3$
Two methods given
#### Explanation:
$\textcolor{b l u e}{\text{Using Gauss Jordon elimination}}$
$\text{ "color(white)(.)xcolor(white)(.)ycolor(white)(.)"answer}$
" "[[1,3,"|",11],[1,4,"|",14]]
$\text{ } R o w 2 - R o w 1$
$\text{ } \downarrow$
" "[[1,3,"|",11],[0,1,"|",3]]
$\text{ } R o w 1 - 3 \left(R o w 2\right)$
$\text{ } \downarrow$
$\text{ "[[1,0,"|",2],[0,1,"|",3]]" } \leftarrow \left[\begin{matrix}x \\ y\end{matrix}\right]$
$\textcolor{w h i t e}{.}$
$\implies \left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right] \textcolor{w h i t e}{.} \left[\begin{matrix}x \\ y\end{matrix}\right] = \left[\begin{matrix}2 \\ 3\end{matrix}\right]$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Using Linear Algebra}}$
$A = \left[\begin{matrix}1 & 3 \\ 1 & 4\end{matrix}\right]$
$\textcolor{w h i t e}{.}$
$X = \left[\begin{matrix}x \\ y\end{matrix}\right]$
$B = \left[\begin{matrix}11 \\ 14\end{matrix}\right]$
$\implies A X = B$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Consider the Algebra method of determining } x}$
Given:$\text{ } 3 x = 4$
Divide both side by 3 $\to \frac{1}{3} \times 3 x = \frac{1}{3} \times 4$
We multiply the 3 by 3 inverse to turn the coefficient of $x$ into 1
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Using the Algebra method shown above}}$
Multiply both sides by $A \text{ inverse. written as } \textcolor{b l u e}{{A}^{-}}$
$\textcolor{b r o w n}{\implies \textcolor{b l u e}{{A}^{-}} A X \text{ "=" } \textcolor{b l u e}{{A}^{-}} B}$
$\implies X = {A}^{-} B$ ...................................Equation(1)
Checked in Maple:
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine } {A}^{-}}$
$\textcolor{b r o w n}{\text{Method 1}}$
Write the original matrix as:
" "[[1,3,"|",1,0],[1,4,"|",0,1]]
And conduct a Gauss Jordon elimination on the LHS
$\text{ } R o w 2 - R o w 1$
$\text{ } \downarrow$
" "[[1,3,"|",1,0],[0,1,"|",-1,1]]
$\text{ } R o w 1 - 3 \left(R o w 2\right)$
$\text{ } \downarrow$
" "[[1,0,"|",4,-3],[0,1,"|",-1,1]] => A^(-)=[[4,-3],[-1,1]]
,.....................................................................................
$\textcolor{b r o w n}{\text{Method 2 - Shortcut approach}}$
Multiply a modified version of $\left[\begin{matrix}1 & 3 \\ 1 & 4\end{matrix}\right]$ by its inverted determinant.
For $\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]$ the determinant is $a b - c d$
and the modified matrix is $\left[\begin{matrix}d & - b \\ - c & a\end{matrix}\right]$
$D = \left(1 \times 4\right) - \left(1 \times 3\right) = 1$
Thus ${A}^{-} = 1 \times \left[\begin{matrix}4 & - 3 \\ - 1 & 1\end{matrix}\right]$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Finish the calculation for Equation(1)}}$
$\implies X = {A}^{-} B$ ...................................Equation(1)
$\left[\begin{matrix}4 & - 3 \\ - 1 & 1\end{matrix}\right] \left[\begin{matrix}11 \\ 14\end{matrix}\right] = \left[\begin{matrix}2 \\ 3\end{matrix}\right]$ |
• anonymous
Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement. (a) lim x-> a [f(x) + g(x)] = infinity (b) lim x-> a [f(x)g(x)] = infinity if c > 0 (c) lim x-> a [f(x)g(x)] = negative infinity if c < 0 I need to prove it using the precise definition of a limit (i.e. no limit laws). Thanks so much!!!!!! I actually only need the proofs for a) and c)... if it helps, here's the link to the proof of a problem to (b): http://imageshack.us/photo/my-images/190/unledmkd.png/
Calculus1
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# Wierd sum of squares compilation to a single vector norm
1. Apr 17, 2012
### Sorento7
I've encountered a function like this:
S(x) = [M(x) - F(x)] ^2 + || G(x) || ^ 2
X being a 3*1 vector
M and F: vector----->scalar
G: vector------->vector and || G || meaning its norm
To change S(x) into a single square, authors have described it like this:
S(x) = || A + B || ^ 2 where A=(M(x) - F(x) , 0) and B=(0 , G(x))
I don't understand how two vectors could be added actually resulting a vector with the first eigenvalue being scalar and the second eigenvalue a 3*1 vector itself?
i.e. what is the "nature" of A + B? a vector?
2. Apr 17, 2012
### Office_Shredder
Staff Emeritus
Why are you describing the entries of the vector as eigenvalues?
All that they are doing is this. Suppose that
$$G(x)=(G_1(x),...,G_m(x))$$
where each Gi takes a vector to a scalar. Then
$$S(x) = (M(x)-F(x))^2+G_1(x)^2+...+G_m(x)^2$$
We are adding the squares of a bunch of numbers together. This is the same as taking the norm of a vector which has those numbers as its entries
$$S(x) = ||\left( M(x)-F(x),G_1(x),....,G_m(x) \right) ||^2$$
This vector whose norm we are taking is equal to A+B in your post
3. Apr 17, 2012
### Sorento7
Re: weird sum of squares compilation to a single vector norm
Thanks this sheds some light. Forgive my rudimentary mistakes since I am actually a doctor!
Well, the problem was more complicated than I described:
A(x) = (F(x) - M(x) + ∂M/∂x * x )2 taking 3*1 vector x to scalar
B(x) = || K.x ||2 k:3*3 matrix
I am confused how can (A + B) be represented as :
|| (F(x) - M(x) , 0) + (∂M/∂x , K).x ||2
Especially, what would be the dimension of the matrix (∂M/∂x , K) ? (discrepancy between the first 1 * 3 entry and second 3 * 3?) |
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Post-edited Network-guided feature selection in high-dimensional genomic data Azencott, Chloé (Auteur de la Conférence) | CIRM (Editeur )
Differences in disease predisposition or response to treatment can be explained in great part by genomic differences between individuals. This has given birth to precision medicine, where treatment is tailored to the genome of patients. This field depends on collecting considerable amounts of molecular data for large numbers of individuals, which is being enabled by thriving developments in genome sequencing and other high-throughput experimental technologies.
Unfortunately, we still lack effective methods to reliably detect, from this data, which of the genomic features determine a phenotype such as disease predisposition or response to treatment. One of the major issues is that the number of features that can be measured is large (easily reaching tens of millions) with respect to the number of samples for which they can be collected (more usually of the order of hundreds or thousands), posing both computational and statistical difficulties.
In my talk I will discuss how to use biological networks, which allow us to understand mutations in their genomic context, to address these issues. All the methods I will present share the common hypotheses that genomic regions that are involved in a given phenotype are more likely to be connected on a given biological network than not.
Differences in disease predisposition or response to treatment can be explained in great part by genomic differences between individuals. This has given birth to precision medicine, where treatment is tailored to the genome of patients. This field depends on collecting considerable amounts of molecular data for large numbers of individuals, which is being enabled by thriving developments in genome sequencing and other high-throughput ex...
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Post-edited Learning with differentiable perturbed optimizers Berthet, Quentin (Auteur de la Conférence) | CIRM (Editeur )
Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g. sorting, picking closest neighbors, finding shortest paths or optimal matchings). Although these discrete decisions are easily computed in a forward manner, they cannot be used to modify model parameters using first-order optimization techniques because they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform a block that outputs an optimal discrete decision into a differentiable operation. Our approach relies on stochastic perturbations of these parameters, and can be used readily within existing solvers without the need for ad hoc regularization or smoothing. These perturbed optimizers yield solutions that are differentiable and never locally constant. The amount of smoothness can be tuned via the chosen noise amplitude, whose impact we analyze. The derivatives of these perturbed solvers can be evaluated eciently. We also show how this framework can be connected to a family of losses developed in structured prediction, and describe how these can be used in unsupervised and supervised learning, with theoretical guarantees.
We demonstrate the performance of our approach on several machine learning tasks in experiments on synthetic and real data.
Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g. sorting, picking closest neighbors, finding shortest paths or optimal matchings). Although these discrete decisions are easily computed in a forward manner, they cannot be used to modify model parameters using first-order optimization techniques because they break the back-propagation of computational graphs. In order to expand the scope of learning ...
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Post-edited Signaling gradients in surface dynamics as basis for regeneration in flatworms Stevens, Angela (Auteur de la Conférence) | CIRM (Editeur )
We introduce and analyze a mathematical model for the regeneration of planarian flatworms. This system of differential equations incorporates dynamics of head and tail cells which express positional control genes that in turn translate into localized signals that guide stem cell differentiation. Orientation and positional information is encoded in the dynamics of a long range wnt-related signaling gradient.
We motivate our model in relation to experimental data and demonstrate how it correctly reproduces cut and graft experiments. In particular, our system improves on previous models by preserving polarity in regeneration, over orders of magnitude in body size during cutting experiments and growth phases. Our model relies on tristability in cell density dynamics, between head, trunk, and tail. In addition, key to polarity preservation in regeneration, our system includes sensitivity of cell differentiation to gradients of wnt-related signals measured relative to the tissue surface. This process is particularly relevant in a small tissue layer close to wounds during their healing, and modeled here in a robust fashion through dynamic boundary conditions.
We introduce and analyze a mathematical model for the regeneration of planarian flatworms. This system of differential equations incorporates dynamics of head and tail cells which express positional control genes that in turn translate into localized signals that guide stem cell differentiation. Orientation and positional information is encoded in the dynamics of a long range wnt-related signaling gradient.
We motivate our model in relation to ...
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Post-edited Estimating parameters in the initial phase of an epidemic Scalia Tomba, Gianpaolo (Auteur de la Conférence) | CIRM (Editeur )
In recent years, new pandemic threats have become more and more frequent (SARS, bird flu, swine flu, Ebola, MERS, nCoV...) and analyses of data from the early spread more and more common and rapid. Particular interest is usually focused on the estimation of $R_{0}$ and various methods, essentially based estimates of exponential growth rate and generation time distribution, have been proposed. Other parameters, such as fatality rate, are also of interest. In this talk, various sources of bias arising because observations are made in the early phase of spread will be discussed and also possible remedies proposed.
In recent years, new pandemic threats have become more and more frequent (SARS, bird flu, swine flu, Ebola, MERS, nCoV...) and analyses of data from the early spread more and more common and rapid. Particular interest is usually focused on the estimation of $R_{0}$ and various methods, essentially based estimates of exponential growth rate and generation time distribution, have been proposed. Other parameters, such as fatality rate, are also of ...
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Post-edited Evolutionary rescue over a phenotype fitness landscape: across the mutation rate spectrum Martin, Guillaume (Auteur de la Conférence) | CIRM (Editeur )
Evolutionary rescue (ER) is the process by which a population, initially destined to extinction due to environmental stress, avoids extinction via adaptive evolution. One of the widely observed pattern of ER (especially in the study of antibiotic resistance) is that it is more likely to occur in mild than in strong stress. This may be due either to purely demographic effects (extinction is faster in strong stress) or to evolutionary effects (adaptation is harder in strong stress). Disentangling the two and predicting the likelihood of ER has important medical or agronomic implications, but also has a strong potential for empirical testing of eco-evolutionary theory, as ER experiments are widespread (at least in microbial systems) and fairly rapid to perform.
Here, I will present results from three recent articles [1-3] where we considered the probability of ER, and the distribution of extinction times, in a classic phenotype-fitness landscape: Fisher’s geometric model (FGM). In our (classic) version of the FGM, fitness is a quadratic function of traits, with an optimum that depends on the environment. This model has received some empirical support with respects to its ability to reproduce or even predict patterns of context dependence in mutation effects on fitness (be it environmental or genetic context).
In our FGM-ER scenario, a population is initially adapted to the current optimum (either a clone or at mutation selection balance). The environment shifts abruptly and the optimum position, plus possibly peak height and width are modified. We follow the evolutionary and demographic response to this change, assuming a density-independent demography (which we approximate by continuous branching process CB process or Feller process).
In spite of its simplicity, the FGM displays fairly distinct behaviors depending on the relative strength of selection and mutation: this yields different approaches to deal with the FGM-ER scenario. I will thus present the different approaches we have used so far: from the strong selection, weak mutation regime to the weak mutation strong selection regime, and discuss possible extensions at the transition between these regimes.
Evolutionary rescue (ER) is the process by which a population, initially destined to extinction due to environmental stress, avoids extinction via adaptive evolution. One of the widely observed pattern of ER (especially in the study of antibiotic resistance) is that it is more likely to occur in mild than in strong stress. This may be due either to purely demographic effects (extinction is faster in strong stress) or to evolutionary effects ...
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Post-edited Unique ergodicity for foliations on compact Kähler surfaces Sibony, Nessim (Auteur de la Conférence) | CIRM (Editeur )
How to study the dynamics of a holomorphic polynomial vector field in $\mathbb{C}^{2}$? What is the replacement of invariant measure? I will survey some surprising rigidity results concerning the behavior of these dynamical system. It is helpful to consider the extension of this dynamical system to the projective plane.
Consider a foliation in the projective plane admitting a unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. With T.-C. Dinh, we showed that there is a unique positive $dd^{c}$-closed (1, 1)-current of mass 1 which is directed by the foliation. This is the current of integration on the invariant curve. A unique ergodicity theorem for the distribution of leaves follows: for any leaf $L$, appropriate averages on $L$ converge to the current of integration on the invariant curve (although generically the leaves are dense). The result uses our theory of densities for currents. It extends to Foliations on Kähler surfaces.
I will describe a recent result, with T.-C. Dinh and V.-A. Nguyen, dealing with foliations on compact Kähler surfaces. If the foliation, has only hyperbolic singularities and does not admit a transverse measure, in particular no invariant compact curve, then there exists a unique positive $dd^{c}$-closed (1, 1)-current of mass 1 which is directed by the foliation( it’s like uniqueness of invariant measure for discrete dynamical systems). This improves on previous results, with J.-E. Fornæss, for foliations (without invariant algebraic curves) on the projective plane. The proof uses a theory of densities for positive $dd^{c}$-closed currents (an intersection theory).
How to study the dynamics of a holomorphic polynomial vector field in $\mathbb{C}^{2}$? What is the replacement of invariant measure? I will survey some surprising rigidity results concerning the behavior of these dynamical system. It is helpful to consider the extension of this dynamical system to the projective plane.
Consider a foliation in the projective plane admitting a unique invariant algebraic curve. Assume that the foliation is ...
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Post-edited How to compute using quantum walks Kendon, Vivien (Auteur de la Conférence) | CIRM (Editeur )
Quantum walks are widely and successfully used to model diverse physical processes. This leads to computation of the models, to explore their properties. Quantum walks have also been shown to be universal for quantum computing. This is a more subtle result than is often appreciated, since it applies to computations run on qubit-based quantum computers in the single walker case, and physical quantum walkers in the multi-walker case (quantum cellular automata). Nonetheless, quantum walks are powerful tools for quantum computing when correctly applied. I will explain the relationship between quantum walks as models and quantum walks as computational tools, and give some examples of their application in both contexts.
Quantum walks are widely and successfully used to model diverse physical processes. This leads to computation of the models, to explore their properties. Quantum walks have also been shown to be universal for quantum computing. This is a more subtle result than is often appreciated, since it applies to computations run on qubit-based quantum computers in the single walker case, and physical quantum walkers in the multi-walker case (quantum ...
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Post-edited On determinants of random matrices Zeitouni, Ofer (Auteur de la Conférence) | CIRM (Editeur )
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Post-edited Emergence of wandering stable components Berger, Pierre (Auteur de la Conférence) | CIRM (Editeur )
In a joint work with Sebastien Biebler, we show the existence of a locally dense set of real polynomial automorphisms of $\mathbb{C}^{2}$ displaying a stable wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These wandering Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof follows from a real geometrical model which enables us to show the existence of an open and dense set of $C^{r}$ families of surface diffeomorphisms in the Newhouse domain, each of which displaying a historical, high emergent, wandering domain at a dense set of parameters, for every $2\leq r\leq \infty$ and $r=\omega$. Hence, this also complements the recent work of Kiriki and Soma, by proving the last Taken's problem in the $C^{\infty }$ and $C^{\omega }$-case.
In a joint work with Sebastien Biebler, we show the existence of a locally dense set of real polynomial automorphisms of $\mathbb{C}^{2}$ displaying a stable wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These wandering Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof follows from a real g...
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Post-edited Averages of Zagier L-functions Balkanova, Olga (Auteur de la Conférence) | CIRM (Editeur )
In 1976, Zagier established a functional equation for the generalized Dirichlet L-functions that are part of the Fourier-Whittaker expansion of halfintegral weight Eisenstein series. The special values of these L-functions at 1/2 and at 1 are of particular interest because of the connection with the Selberg trace formula, with moments of symmetric square L-functions and with the prime geodesic theorem. In this talk, we describe various properties of Zagier L-functions and consider several problems related to the asymptotic evaluation of averages of special L-values.
In 1976, Zagier established a functional equation for the generalized Dirichlet L-functions that are part of the Fourier-Whittaker expansion of halfintegral weight Eisenstein series. The special values of these L-functions at 1/2 and at 1 are of particular interest because of the connection with the Selberg trace formula, with moments of symmetric square L-functions and with the prime geodesic theorem. In this talk, we describe various ...
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Post-edited Failure of the Brauer-Manin obstruction for a simply connected fourfold, and an orbifold version of the Mordell theorem Kebekus, Stefan (Auteur de la Conférence) | CIRM (Editeur )
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov’s etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.
This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana’s "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.
Joint work with Pereira and Smeets.
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov’s etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.
This talk discusses a construction of simply connected fourfolds over global fields of positive characteristic for ...
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Post-edited Spectral decompositions and an extension of a theorem of Atzmon: a couple leading to spectral subspaces for Bishop operators Gallardo-Gutiérrez, Eva A. (Auteur de la Conférence) | CIRM (Editeur )
Bishop’s operator arose in the fifties as possible candidates for being counterexamples to the Invariant Subspace Problem. Several authors addressed the problem of finding invariant subspaces for some of these operators; but still the general problem is open. In this talk, we shall discuss about recent results on the existence of invariant subspaces which are indeed spectral subspaces for Bishop operators, by providing an extension of a Theorem of Atzmon (Joint work with M. Monsalve-Lopez).
Bishop’s operator arose in the fifties as possible candidates for being counterexamples to the Invariant Subspace Problem. Several authors addressed the problem of finding invariant subspaces for some of these operators; but still the general problem is open. In this talk, we shall discuss about recent results on the existence of invariant subspaces which are indeed spectral subspaces for Bishop operators, by providing an extension of a Theorem ...
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Post-edited Triality Elduque, Alberto (Auteur de la Conférence) | CIRM (Editeur )
Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the existence of two different types of geometric triality. One of them is related to the octonions, but the other one is better explained in terms of a class of nonunital composition algebras discovered by the physicist Okubo (1978) inside 3x3-matrices, and which has led to the definition of the so called symmetric composition algebras.
This talk will review the history, classification, and their connections with the phenomenon of triality, of the symmetric composition algebras.
Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the ...
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Post-edited On ellipsephic integers Dartyge, Cécile (Auteur de la Conférence) | CIRM (Editeur )
The term " ellipsephic " was proposed by Christian Mauduit to denote the integers with missing digits in a given basis. This talk is a survey on several results on the multiplicative properties of these integers.
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Post-edited $L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains Kozono, Hideo (Auteur de la Conférence) | CIRM (Editeur )
It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u$, there exist $h\in X^{r}_{har}\left ( \Omega \right )$, $w\in H^{1,r}\left ( \Omega \right )^{3}$ with div $w= 0$ and $p\in H^{1,r}\left ( \Omega \right )$ such that $u$ is uniquely decomposed as $u= h$ + rot $w$ + $\bigtriangledown p$.
On the other hand, if for the given $L^{r}$-vector field $u$ we choose its harmonic part $h$ from $V^{r}_{har}\left ( \Omega \right )$, then we have a similar decomposition to above, while the unique expression of $u$ holds only for $1< r< 3$. Furthermore, the choice of $p$ in $H^{1,r}\left ( \Omega \right )$ is determined in accordance with the threshold $r= 3/2$.
Our result is based on the joint work with Matthias Hieber, Anton Seyferd (TU Darmstadt), Senjo Shimizu (Kyoto Univ.) and Taku Yanagisawa (Nara Women Univ.).
It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Post-edited The local density approximation in density functional theory Seiringer, Robert (Auteur de la Conférence) | CIRM (Editeur ) We present a mathematically rigorous justification of the Local Density Approximation in density functional theory. We provide a quantitative estimate on the difference between the grand-canonical Levy-Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the Uniform Electron Gas energy of this density. The error involves gradient terms and justifies the use of the Local Density Approximation in situations where the density is very flat on sufficiently large regions in space. (Joint work with Mathieu Lewin and Elliott Lieb) We present a mathematically rigorous justification of the Local Density Approximation in density functional theory. We provide a quantitative estimate on the difference between the grand-canonical Levy-Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the Uniform Electron Gas energy of this density. The error involves gradient terms and justifies the use of the ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Post-edited Rationality of Fano 3-folds over non-closed fields Kuznetsov, Alexander (Auteur de la Conférence) | CIRM (Editeur ) In the talk I will discuss rationality criteria for Fano 3-folds of geometric Picard number 1 over a non-closed field$k$of characteristic 0. Among these there are 8 types of geometrically rational varieties. We prove that in one of these cases any variety of this type is k-rational, in four cases the criterion of rationality is the existence of a$k$-rational point, and in the last three cases the criterion is the existence of a$k$-rational point and a k rational curve of genus 0 and degree 1, 2, and 3 respectively. The last result is based on recent results of Benoist-Wittenberg. This is a joint work with Yuri Prokhorov. In the talk I will discuss rationality criteria for Fano 3-folds of geometric Picard number 1 over a non-closed field$k$of characteristic 0. Among these there are 8 types of geometrically rational varieties. We prove that in one of these cases any variety of this type is k-rational, in four cases the criterion of rationality is the existence of a$k$-rational point, and in the last three cases the criterion is the existence of a$k$-rational ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. Post-edited Chain conditions, unbounded colorings and the$C$-sequence spectrum Rinot, Assaf (Auteur de la Conférence) | CIRM (Editeur ) The productivity of the$\kappa $-chain condition, where$\kappa $is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of$kappa-cc$posets whose squares are not$\kappa-cc$were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which$\kappa = \aleph{_2}$, was resolved by Shelah in 1997. In the first part of this talk, we shall present analogous results regarding the infinite productivity of chain conditions stronger than$\kappa-cc$. In particular, for any successor cardinal$\kappa$, we produce a ZFC example of a poset with precaliber$\kappa$whose$\omega ^{th}$power is not$\kappa-cc$. To do so, we introduce and study the principle$U(\kappa , \mu , \theta , \chi )$asserting the existence of a coloring$c:\left [ \kappa \right ]^{2}\rightarrow \theta $satisfying a strong unboundedness condition. In the second part of this talk, we shall introduce and study a new cardinal invariant$\chi \left ( \kappa \right )$for a regular uncountable cardinal$\kappa$. For inaccessible$\kappa$,$\chi \left ( \kappa \right )$may be seen as a measure of how far away$\kappa$is from being weakly compact. We shall prove that if$\chi \left ( \kappa \right )> 1$, then$\chi \left ( \kappa \right )=max(Cspec(\kappa ))$, where: (1) Cspec$(\kappa)$:= {$\chi (\vec{C})\mid \vec{C}$is a sequence over$\kappa$}$\setminus \omega$, and (2)$\chi \left ( \vec{C} \right )$is the least cardinal$\chi \leq \kappa $such that there exist$\Delta\in\left [ \kappa \right ]^{\kappa }$and b :$\kappa \rightarrow \left [ \kappa \right ]^{\chi }$with$\Delta \cap \alpha \subseteq \cup _{\beta \in b(\alpha )}C_{\beta }$for every$\alpha < \kappa$. We shall also prove that if$\chi (\kappa )=1$, then$\kappa$is greatly Mahlo, prove the consistency (modulo the existence of a supercompact) of$\chi (\aleph_{\omega +1})=\aleph_{0}$, and carry a systematic study of the effect of square principles on the$C$-sequence spectrum. In the last part of this talk, we shall unveil an unexpected connection between the two principles discussed in the previous parts, proving that, for infinite regular cardinals$\theta< \kappa ,\theta \in Cspec(\kappa )$if there is a closed witness to$U_{(\kappa ,\kappa ,\theta ,\theta )}$. This is joint work with Chris Lambie-Hanson. The productivity of the$\kappa $-chain condition, where$\kappa $is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of$kappa-cc$posets whose squares are not$\kappa-cc$were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which$\kappa = \aleph{_2}\$, ...
Déposez votre fichier ici pour le déplacer vers cet enregistrement.
Post-edited Challenges in achieving scalable and robust linear solvers Grigori, Laura (Auteur de la Conférence) | CIRM (Editeur )
This talk focuses on challenges that we address when designing linear solvers that aim at achieving scalability on large scale computers, while also preserving numerical robustness. We will consider preconditioned Krylov subspace solvers. Getting scalability relies on reducing global synchronizations between processors, while also increasing the arithmetic intensity on one processor. Achieving robustness relies on ensuring that the condition number of the preconditioned matrix is bounded. We will discuss two different approaches for this. The first approach relies on enlarged Krylov subspace methods that aim at computing an enlarged subspace and obtain a faster convergence of the iterative method. The second approach relies on a multilevel Schwarz preconditioner, a multilevel extension of the GenEO preconditioner, that is basedon constructing robustly a hierarchy of coarse spaces. Numerical results on large scale computers, in particular for linear systems arising from solving linear elasticity problems, will discuss the efficiency of the proposed methods.
This talk focuses on challenges that we address when designing linear solvers that aim at achieving scalability on large scale computers, while also preserving numerical robustness. We will consider preconditioned Krylov subspace solvers. Getting scalability relies on reducing global synchronizations between processors, while also increasing the arithmetic intensity on one processor. Achieving robustness relies on ensuring that the condition ...
Déposez votre fichier ici pour le déplacer vers cet enregistrement.
Post-edited Twistor theory for LQG Eastwood, Michael (Auteur de la Conférence) | CIRM (Editeur )
Twistor Theory was proposed in the late 1960s by Roger Penrose as a potential geometric unification of general relativity and quantum mechanics. During the past 50 years, there have been many mathematical advances and achievements in twistor theory. In physics, however, there are aspirations yet to be realised. Twistor Theory and Loop Quantum Gravity (LQG) share a common background. Their aims are very much related. Is there more to it? This talk will sketch the geometry and symmetry behind twistor theory with the hope that links with LQG can be usefully strengthened. We believe there is something significant going on here: what could it be?
Twistor Theory was proposed in the late 1960s by Roger Penrose as a potential geometric unification of general relativity and quantum mechanics. During the past 50 years, there have been many mathematical advances and achievements in twistor theory. In physics, however, there are aspirations yet to be realised. Twistor Theory and Loop Quantum Gravity (LQG) share a common background. Their aims are very much related. Is there more to it? This ...
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03-01-2012 , 05:44 PM Posts: 245 | Thanked: 891 times | Joined on Feb 2012 #11
Originally Posted by javid
@itsnotabigtruck thankyou for taking the time to write this up
In your opinion is this safer than using some other application based flashing utility?
Do you know if NSU will allow me to use pre-downloaded files?
Whats the difference between nokia care and prod external in navifirm? (do you use + or regular navifirm?)
The reason i ask - i miyself have a N8-00 and my wife has the better phone - N9-00
She's currently where the internet is painfully slow - or non-existent.
So - right now - she is at PR1.0 - and now that 1.2 has come out - i need to update her twice - which means i need to risk the flashing twice...
Also - where do the files to be flashed need to be located? same folder as flasher?
And since i have your attention - if for some reason the internet is working well when i start an OTA instead- and the internet disappears mid download - will it hurt the phone or will i just need to start again when the internet returns?
Production External is what NSU uses to download firmware images; Nokia Care Suite is similar but has more files, including the eMMC content images.
You can indeed use NAVIFIRM to pre-download the firmware images - that's how I originally upgraded to PR1.1. Note that you need to download all the files listed under Production External for your phone's exact product code, then copy them into the right place. I don't remember off-hand exactly where the "right place" is, but it's some hidden directory somewhere.
The updater on the phone downloads everything before beginning the OS upgrade, so you don't need to worry about connection dropouts. However, it's a little buggy, and I had to reboot the phone in order to begin the download again after getting stuck.
Edit: Drop the files in C:\Users\<username>\AppData\Local\Nokia\NSU3\NOSSU 2\variants\Products\rm-696
Last edited by itsnotabigtruck; 03-01-2012 at 05:46 PM.
The Following User Says Thank You to itsnotabigtruck For This Useful Post:
03-01-2012 , 06:13 PM Posts: 16 | Thanked: 0 times | Joined on Dec 2011 @ houston, tx #12
Originally Posted by itsnotabigtruck
Production External is what NSU uses to download firmware images; Nokia Care Suite is similar but has more files, including the eMMC content images.
You can indeed use NAVIFIRM to pre-download the firmware images - that's how I originally upgraded to PR1.1. Note that you need to download all the files listed under Production External for your phone's exact product code, then copy them into the right place. I don't remember off-hand exactly where the "right place" is, but it's some hidden directory somewhere.
The updater on the phone downloads everything before beginning the OS upgrade, so you don't need to worry about connection dropouts. However, it's a little buggy, and I had to reboot the phone in order to begin the download again after getting stuck.
Edit: Drop the files in C:\Users\<username>\AppData\Local\Nokia\NSU3\NOSSU 2\variants\Products\rm-696
that is certainly obscure!
that is the location for Winflasher, correct?
03-01-2012 , 06:22 PM Posts: 203 | Thanked: 536 times | Joined on Oct 2009 @ Colombia #13
03-01-2012 , 07:22 PM Posts: 245 | Thanked: 891 times | Joined on Feb 2012 #14
Originally Posted by javid
that is certainly obscure!
that is the location for Winflasher, correct?
Nope - it's for Nokia Software Updater (NSU). Flasher loads firmware images from wherever you tell it to.
The Following User Says Thank You to itsnotabigtruck For This Useful Post:
03-02-2012 , 10:02 AM Posts: 22 | Thanked: 1 time | Joined on Feb 2012 #15
i changed my product code to get update 1.2 with nsu. At the end of installation it gave error about usb connection and nsu couldnt recover the phone as well. So i flashed same files with flasher with any probs. But it gave warning at start of phone about warranty.
i can go back my original 1.1 with error at the end of installation about
and phone starts but it gives a simcard problem and never recognizes it after some trying time.
Im at 1.2 again with warning at the start. What can i do?
P.s : i cant change my product code after first flash. How can i go local mode in this situation to change product code back or sth else.
if i go back 1.1 and connects it in suite mode. nss sees the product code but, because of cant get to local mode, it cant be changed.
waiting for help...
03-02-2012 , 10:16 AM Posts: 245 | Thanked: 891 times | Joined on Feb 2012 #16
Which variant are you trying to upgrade from, and which are you trying to flash?
Code:
flasher -f -F pr12-main.bin --flash-only=nolo,kernel,rootfs
flasher -f -F pr11-main.bin --flash-only=cmt
flasher -f -F pr12-emmc.bin
(pr11-main.bin is the old firmware, pr12-main.bin is the new firmware)
__________________
03-02-2012 , 10:28 AM Posts: 609 | Thanked: 194 times | Joined on May 2011 #17
what is the purpose of changing n9 product code? especially when we can flash our n9 with other country variant firmware with the same or higher legacy number(number after PR)
03-02-2012 , 10:37 AM Posts: 22 | Thanked: 1 time | Joined on Feb 2012 #18
original variant is 059M4z8. i changed it to 059J214 (both are turkey code)
And flashed it to 059J214 1.2 image.
flasher -i command says its still 1.1
The flasher codes that you've given me, i will try them now.
03-02-2012 , 10:49 AM Posts: 22 | Thanked: 1 time | Joined on Feb 2012 #19
-first with success.
-second failed with these;
bb5 rdc cert read failed (11 times)
cmt rfs failed
pipe 2 finishing in error state with status 1
do i need to do third one?
03-02-2012 , 12:14 PM Posts: 22 | Thanked: 1 time | Joined on Feb 2012 #20
from your codes. i understand that the only problem was cmt. So i flashed back to 1.1 and got cmt from 1.2 . so it did the trick. |
Qualification:
Ph.D, MPhil, MSc, BSc
[email protected]
Phone:
9916497200
Dr. M. Dharani currently serves as Assistant Professor(Sr. Gr.) at the department of Physics, Amrita School of Engineering, Banglore campus. Her areas of research include Quantum physics,Condensed matter physics and Biophysics. She has successfully defended her Ph.D. thesis titled “Investigation on threshold conditions, bound states, band structure and transmission by periodic delta potentials” at Amrita Vishwa Vidyapeetham, Coimbatore.
## Qualification
Year Degree Name of the university
2017 Ph.D Amrita Vishwa Vidyapeetham
2003 M.Phil Bharathiar University
2001 M.Sc Bharathiar University
1999 B.Sc Bharathiar University
## Publications
### Publication Type: Journal Article
Year of Conference Publication Type Title
2017
Journal Article
Ma Dharani and Shastry, C. S., “Investigation of bound states and transmission across orderly arranged pairs of attractive and repulsive delta potentials”, Physica B: Condensed Matter, vol. 516, pp. 27-31, 2017.[Abstract]
The pattern of bands generated by the transmission coefficient T for transmission across N ionic molecules in one dimension simulated by N alternating pairs of attractive and repulsive delta potential is studied by exploring its relation with the conditions for the occurence of threshold bound state. The number of peaks in the first band of T is found to be the difference between N and the number of negative energy bound states generated by this potential. Further we systematically study the nature of distribution of peaks in higher bands as a function of potential strength and distance parameters and compare it with the results obtained in our earlier works. © 2017 Elsevier B.V. More »»
2016
Journal Article
Ma Dharani and Shastry, C. S., “Threshold conditions, energy spectrum and bands generated by locally periodic Dirac comb potentials”, Physica B: Condensed Matter, vol. 481, pp. 104–117, 2016.[Abstract]
We derive expressions for polynomials governing the threshold conditions for different types of locally periodic Dirac comb potentials comprising of attractive and combination of attractive and repulsive delta potential terms confined symmetrically inside a one dimensional box of fixed length. The roots of these polynomials specify the conditions on the potential parameters in order to generate threshold energy bound states. The mathematical and numerical methods used by us were first formulated in our earlier works and it is also very briefly summarized in this paper. We report a number of mathematical results pertaining to the threshold conditions and these are useful in controlling the number of negative energy states as desired. We further demonstrate the correlation between the distribution of roots of these polynomials and negative energy eigenvalues. Using these results as basis, we investigate the energy bands in the positive energy spectrum for the above specified Dirac comb potentials and also for the corresponding repulsive case. In the case of attractive Dirac comb the base energy of the each band excluding the first band coincides with specific eigenvalue of the confining box whereas in the repulsive case it coincides with the band top. We deduce systematic correlation between band gaps, band spreads and box eigenvalues and explain the physical reason for the vanishing of band pattern at higher energies. In the case of Dirac comb comprising of orderly arranged attractive and repulsive delta potentials, specific box eigenvalues occur in the middle of each band excluding the first band. From our study we find that by controlling the number and strength parameters of delta terms in the Dirac comb and the size of confining box it is possible to generate desired types of band formations. We believe the results from our systematic analysis are useful and relevant in the study of various one dimensional systems of physical interest in areas like nanoscience.
More »»
2016
Journal Article
Ma Dharani and Shastry, C. S., “Band structures in transmission coefficients generated by Dirac comb potentials”, Physica B: Condensed Matter, vol. 500, pp. 66–76, 2016.[Abstract]
Using the threshold conditions and bound state energies investigated earlier by us as a critical input we systematically study the nature of band formation in the transmission coefficient generated by Dirac comb potentials having equispaced (i) attractive, (ii) repulsive and (iii) alternating attractive and repulsive delta terms having same strength and confined within a fixed range. We find that positions of the peaks of transmission coefficient generated by a combination of one attractive and one repulsive delta terms having same strength and separated by gap a is independent of the potential strength and coincide with the energy eigenvalues of 1D box of range a. We further study analytically and numerically the transmission across Dirac comb potentials containing two or three delta terms and these results are useful in the analysis of the transmission in the general case. In the case of Dirac comb potentials containing Na attractive delta terms we find that the nature of the first band and higher bands of the transmission coefficient are different, and if such a potential generates Nb number of bound states, the first band in the transmission coefficient generated by the potential has NT1=Na−Nb peaks. In the case of higher bands generated by delta comb potential having N delta terms each band has N−1 peaks. Further we systematically study the behavior of band gaps and band spread as a function of potential strength and number of terms in the Dirac comb. The results obtained by us provide a relation between bound state spectrum, number of delta terms in the Dirac comb and the band pattern which can be explored for potential applications.
More »»
2014
Journal Article
Ma Dharani, Sahu, Bb, and Shastry, C. Sc, “Threshold conditions and bound states for locally periodic delta potentials”, Central European Journal of Physics, vol. 12, no. 10, pp. 755-766, 2014.[Abstract]
We present a systematic study of the conditions for the generation of threshold energy eigen states and also the energy spectrum generated by two types of locally periodic delta potentials each having the same strength λV and separation distance parameter a: (a) sum of N attractive potentials and (b) sum of pairs of attractive and repulsive potentials. Using the dimensionless parameter g = λV a in case (a) the values of g = g n, n = 1, 2, ..., N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D 1(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D 1(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D 1(N, g). A similar study is presented for case (b). Using the properties of D 1(N, g) we establish that in case (a) the critical minimum value of g which guarantees the generation of the maximum possible number of bound states is g = 4. The corresponding result for case (b) is g = 2. A typical set of numerical results showing the pattern of variation of g n as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.
More »»
2013
Journal Article
Ma Dharani, Sahu, Bb, and Shastry, C. Sc, “Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions”, Central European Journal of Physics, vol. 11, pp. 995-1005, 2013.[Abstract]
This paper proves that for N attractive delta function potentials the number of bound states (Nb) satisfies 1 ≤ N b ≤ N in one dimension (1D), and is 0 ≤ N b ≤ N in three dimensions (3D). Algebraic equations are obtained to evaluate the bound states generated by N attractive delta potentials. In particular, in the case of N attractive delta function potentials having same separation a between adjacent wells and having the same strength λV, the parameter g=λVa governs the number of bound states. For a given N in the range 1-7, both in 1D and 3D cases the numerical values of gn, where n=1,2,..N are obtained. When g=gn, Nb ≤ n where Nb includes one threshold energy bound state. Furthermore, gn are the roots of the Nth order polynomial equations with integer coefficients. Based on our numerical calculations up to N=40, even when N becomes large, 0 ≤ g n ≤ 4 and (Formula presented.) and this result is expected to be generally valid. Thus, for g > 4 there will be no threshold or zero energy bound state, and if g≈ 2 for a given large N, the number of bound states will be approximately N/2. The empirical formula gn = 4/[1+exp((N 0 - n)/β)] gives a good description of the variation of gn as a function of n. This formula is useful in estimating the number of bound states for any N and g both in 1D and 3D cases. © 2013 Versita Warsaw and Springer-Verlag Wien.
More »»
### Publication Type: Conference Paper
Year of Conference Publication Type Title
2016
Conference Paper
Ma Dharani and Shastry, C. Sb, “Interesting features of transmission across locally periodic delta potentials”, in AIP Conference Proceedings, 2016, vol. 1731.[Abstract]
We study the theory of transmission of electrons through N delta potential barriers as well as wells. Some of the interesting features like the correlation between resonance peak positions and box states, number of peaks in transmission band and bound states are analyzed for locally periodic attractive, repulsive and pair of attractive and repulsive potentials. © 2016 Author(s). More »»
2012
Conference Paper
Ma Dharani, Shastry, C. S., Sahu, B., and Dr. Mahadevan S., “Selective suppression of Eigen states with an absorptive delta potential”, in Proceedings of the DAE-BRNS symposium on nuclear physics. V. 57, 2012.[Abstract]
Delta function potentials are used to explain short range elastic impurities and nature of the spectrum generated by the delta function potential embedded in a box and to construct model atomic systems interacting with the electromagnetic fields leading to multi photon absorption and ionization process. Similar study can be carried out in a nuclear system to analyze different properties of radioactive ions. The resonances generated when particle traverses across two delta potentials in one-dimension (ID) are studied. In this contribution, a novel feature is demonstrated that an absorptive delta potential suitably placed within a potential pocket can be used to selectively manipulate and suppress the resonances generated by the pocket
More »»
### Publication Type: Conference Proceedings
Year of Conference Publication Type Title
2015
Conference Proceedings
Ma Dharani, Shastry, C. S., Bhattacharyya, D., Chitra, R., and Sahoo, N. K., “Threshold conditions and bands in attractive Dirac comb”, AIP Conference Proceedings. AIP Publishing, 2015.[Abstract]
Taking into account the threshold conditions for the generation of bound states by an attractive delta comb in one dimension, we describe the band structure generated by the same when confined to a box demonstrating the correlation between the potential strength, band width and band gap.
More »»
2012
Conference Proceedings
K. Aditya, Dharani, Ma, Prema, P., Dr. Mahadevan S., and Shastry, C. S., “Calculation of Q values and Half lives of $\alpha$-decay for A= 152-181 using S-Matrix method”, Proceedings of the DAE-BRNS Symposium on Nuclear Physics, vol. 44. 2012.[Abstract]
The success in the description of half lives and Q-values of Super heavy elements (SHE) using the microscopic alpha daughter nucleus potential and S-Matrix method (SM) prompted us to extend the analysis to the nuclei in the rare earth regions with A=152-181. We have reported in 2010, the results of the half lives and Q-values for the nuclei A=152-181 using WKB method. In this paper we report the results obtained using the more accurate S-Matrix method. The microscopic alpha nucleus potential is generated in the double folding model (τρρ-approximation) using relativistic mean field (RMF) densities along with density dependent M3Ynucleon-nucleon interaction. This potential is then used in the SM method to calculate both Q-values and decay half lives
More »»
Faculty Research Interest: |
# American Institute of Mathematical Sciences
January 2009, 8(1): 275-293. doi: 10.3934/cpaa.2009.8.275
## Inelastic Collapse in a Corner
1 Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, Hong Kong 2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China
Received March 2008 Revised July 2008 Published October 2008
We consider the interaction of a rigid, frictionless, inelastic particle with a rigid boundary that has a corner. Typically, two possible final outcomes can occur: the particle escapes from the corner after experiencing a certain number of collisions with the corner, or the particle experiences an inelastic collapse in which an infinite number of collisions can occur in a finite time interval. For the former case, we determine the number of collisions that the particle will experience with the boundary before escaping the corner. For the latter case, we determine the conditions for which inelastic collapse can occur. For a corner composed of two straight walls, we derive simple analytic solutions and show that for a given coefficient of restitution, there is a critical corner angle above which inelastic collapse cannot occur. We show that as the corner angle tends to the critical corner angle from below, the process of inelastic collapse takes infinitely long. We also show a surprising phenomenon that if the corner has the form of a cusp, the particle can have an infinite number of collisions with the boundary in a finite time interval without losing all of its energy, and eventually escapes from the corner.
Citation: Ming Gao, Jonathan J. Wylie, Qiang Zhang. Inelastic Collapse in a Corner. Communications on Pure & Applied Analysis, 2009, 8 (1) : 275-293. doi: 10.3934/cpaa.2009.8.275
[1] Wolf-Jürgen Beyn, Thorsten Hüls. Continuation and collapse of homoclinic tangles. Journal of Computational Dynamics, 2014, 1 (1) : 71-109. doi: 10.3934/jcd.2014.1.71 [2] Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315 [3] N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco, G. B. Tanoglu. Analysis of a corner layer problem in anisotropic interfaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 237-255. doi: 10.3934/dcdsb.2006.6.237 [4] Christopher M. Kribs-Zaleta, Christopher Mitchell. Modeling colony collapse disorder in honeybees as a contagion. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1275-1294. doi: 10.3934/mbe.2014.11.1275 [5] Yvan Martel, Frank Merle. Inelastic interaction of nearly equal solitons for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 487-532. doi: 10.3934/dcds.2010.27.487 [6] Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445 [7] Zhongyi Huang, Peter A. Markowich, Christof Sparber. Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics. Kinetic & Related Models, 2010, 3 (1) : 181-194. doi: 10.3934/krm.2010.3.181 [8] José A. Carrillo, Stéphane Cordier, Giuseppe Toscani. Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 59-81. doi: 10.3934/dcds.2009.24.59 [9] Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145 [10] Shi Jin, Dongsheng Yin. Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction. Kinetic & Related Models, 2011, 4 (1) : 295-316. doi: 10.3934/krm.2011.4.295 [11] Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627 [12] Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264 [13] Bertrand Lods, Clément Mouhot, Giuseppe Toscani. Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models. Kinetic & Related Models, 2008, 1 (2) : 223-248. doi: 10.3934/krm.2008.1.223 [14] Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411 [15] Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic & Related Models, 2014, 7 (4) : 755-778. doi: 10.3934/krm.2014.7.755 [16] Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068
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Corn Grain Yield Estimator
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KurtHeckman.Corn Grain Yield Estimator
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The Corn Grain Yield calculator estimates the amount of corn produced in a field based on the acres (a), the results of a 1,000th acre sample including the number of ears in the (ears), the size estimate of your sample kernels (s), the average kernels per ear (KPE) in your sample.
INSTRUCTIONS: Enter the following information from your 1000th acre sample:
• (S) Choose a size indicator of the size of kernels [small, medium, large] (For more information, see Kernel Size description below.)
• (KPE) This is the average number of harvestable kernels per ear in your sample
• (ears) This is the number of ears in your 1000th acre sample.
• (a) This is the number of acres of corn. Note: the default is 1 acre which will give you your bushels of corn per acre. You can then enter the number of acres you have in corn to provide the total estimated yield for your land.
CORN YIELD: The calculator returns an estimate of the bushels of corn for a the number of acres specified. A more accurate Corn Yield Estimation is available in the CORN YIELD ESTIMATION CALCULATOR. It has a step by step process and includes moisture corrections for a more accurate estimation of corn yield.
General Information
Note: the rounded ends of the head are not considered since the combine will often not collect those kernels.
Kernel Size
This formula uses the default of average sized kernels of corn in a bushel. The ratios are as follows: Kernel Sizes
• small - small kernels require 120,000 kernels in a bushel
• medium - medium (average) sized kernels require 90,000 kernels in a bushel.
• large - large kernels only require 80,000 kernels for a bushel.
To compute the length of a thousandth (1/1000th) acre row, click HERE.
This equation, Corn Grain Yield Estimator, is listed in 2 Collections. |
## File: getURIAsynchronous.Rd
package info (click to toggle)
r-cran-rcurl 1.95-4.8-2
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124 \name{getURIAsynchronous} \alias{getURIAsynchronous} \alias{getURLAsynchronous} \title{Download multiple URIs concurrently, with inter-leaved downloads} \description{ This function allows the caller to specify multiple URIs to download at the same time. All the requests are submitted and then the replies are processed as data becomes available on each connection. In this way, the responses are processed in an inter-leaved fashion, with a chunk from one response from one request being processed and then followed by a chunk from a different request. Downloading documents asynchronously involves some trade-offs. The switching between different streams, detecting when input is available on any of them involves a little more processing and so increases the consumption of CPU cycles. On the other hand, there is a potentially large saving of time when one considers total time to download. See \url{http://www.omegahat.net/RCurl/concurrent.xml} for more details. This is a common trade-off that arises in concurrent/parallel/asynchronous computing. \code{\link{getURI}} calls this function if more than one URI is specified and \code{async} is \code{TRUE}, the default in this case. One can also download the (contents of the) multiple URIs serially, i.e. one after the other using \code{\link{getURI}} with a value of \code{FALSE} for \code{async}. } \usage{ getURIAsynchronous(url, ..., .opts = list(), write = NULL, curl = getCurlHandle(), multiHandle = getCurlMultiHandle(), perform = Inf, .encoding = integer(), binary = rep(NA, length(url))) } \arguments{ \item{url}{a character vector identifying the URIs to download.} \item{\dots}{named arguments to be passed to \code{\link{curlSetOpt}} when creating each of the different \code{curlHandle} objects.} \item{.opts}{a named list or \code{CURLOptions} object identifying the curl options for the handle. This is merged with the values of \dots to create the actual options for the curl handle in the request.} \item{write}{an object giving the functions or routines that are to be called when input is waiting on the different HTTP response streams. By default, a separate callback function is associated with each input stream. This is necessary for the results to be meaningful as if we use a single reader, it will be called for all streams in a haphazard order and the content interleaved. One can do interesting things however using a single object. } \item{curl}{the prototypical curlHandle that is duplicated and used in in } \item{multiHandle}{ this is a curl handle for performing asynchronous requests. } \item{perform}{a number which specifies the maximum number of calls to \code{\link{curlMultiPerform}} that are to be made in this function call. This is typically either 0 for no calls or \code{Inf} meaning process the requests until completion. One may find alternative values useful, such as 1 to ensure that the requests are dispatched. } \item{.encoding}{an integer or a string that explicitly identifies the encoding of the content that is returned by the HTTP server in its response to our query. The possible strings are \sQuote{UTF-8} or \sQuote{ISO-8859-1} and the integers should be specified symbolically as \code{CE_UTF8} and \code{CE_LATIN1}. Note that, by default, the package attempts to process the header of the HTTP response to determine the encoding. This argument is used when such information is erroneous and the caller knows the correct encoding. } \item{binary}{a logical vector identifying whether each URI has binary content or simple text.} } \details{ This uses \code{\link{curlMultiPerform}} and the multi/asynchronous interface for libcurl. } \value{ The return value depends on the run-time characteristics of the call. If the call merely specifies the URIs to be downloaded, the result is a named character vector. The names identify the URIs and the elements of the vector are the contents of the corresponding URI. If the requests are not performed or completed (i.e. \code{perform} is zero or too small a value to process all the chunks) a list with 2 elements is returned. These elements are: \item{multiHandle}{the curl multi-handle, of class \code{\link{MultiCURLHandle-class}}. This can be used in further calls to \code{\link{curlMultiPerform}}} \item{write}{the \code{write} argument (after it was potentially expanded to a list). This can then be used to fetch the results of the requests when the requests are completed in the future. } } \references{Curl homepage \url{http://curl.haxx.se}} \author{Duncan Temple Lang } \seealso{ \code{\link{getURL}} \code{\link{getCurlMultiHandle}} \code{\link{curlMultiPerform}} } \examples{ uris = c("http://www.omegahat.net/RCurl/index.html", "http://www.omegahat.net/RCurl/philosophy.xml") txt = getURIAsynchronous(uris) names(txt) nchar(txt) } \keyword{IO} \concept{Web} \concept{URI} \concept{Web services} |
PUMPA - THE SMART LEARNING APP
AI system creates personalised training plan based on your mistakes
Circle through a point
Given a point $$O$$, it is always possible to draw infinitely many circles through the given point.
In other words, an infinite number of circles can be drawn through a given point.
Illustration:
Circle through two points
Given any two points, it is always possible to draw infinitely many circles through the given points.
In other words, an infinite number of circles can be drawn, passing through a pair of points.
Illustration:
Circle through three points
Case 1: Collinear points
If the three points are collinear, then it is impossible to draw a circle using all three points.
Any set of points are said to be collinear if all the points lie on the same line.
Illustration:
Case 2: Non-collinear points
If the three points are non-collinear, then it is possible to draw only one circle using all the three points.
Any set of points are said to be non-collinear if all the points do not lie on the same line.
Illustration:
Theorem: There is one and only one circle passing through three non-collinear points. |
# Are the units of energy the same in higher dimensions?
In 3 spatial dimensions, $$[E] = [ML^2 T^{-2}]$$
Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?
• In the 4-dimensional case, would the fourth dimension be pretty much the same thing as the first three spatial dimensions? – Nat Oct 11 '18 at 23:02
• Note that in many problems and exercises on work and energy we limit the universe to lines / curves or planes / sufaces. This does not affect the interpretation or units of energy in any significant way. One could therefore empirically expect that it looks the same in higher dimensions as well. – Arthur Oct 12 '18 at 11:11
Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $$E=\frac{1}{2}m\vec{v}^2$$. I include the vector notation for the velocity here because it's useful to keep in mind that $$\vec{v}^2=\vec{v}\cdot\vec{v}$$.
If the particle is moving in two dimensions, then $$\vec{v}=v_x\hat{x}+v_y\hat{y}$$ and $$\vec{v}\cdot\vec{v}=v_x^2+v_y^2$$, which has units of velocity squared. In three dimensions, $$\vec{v} = v_x\hat{x}+v_y\hat{y}+v_z\hat{z}$$ and $$\vec{v}\cdot\vec{v} = v_x^2+v_y^2+v_z^2$$, which still has the units of velocity squared. In four dimensions, $$\vec{v} = v_x\hat{x}+v_y\hat{y}+v_z\hat{z}+v_w\hat{w}$$, and $$\vec{v}\cdot\vec{v}=v_x^2+v_y^2+v_z^2+v_w^2$$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.
We can also see this more generally by the formal definition of work, which is the definition of change in energy:
$$W=\int_C \vec{F}\cdot d\vec{s}$$
for an object being acted on by a force $$\vec{F}$$ moving along a path $$C$$ with arclength parameter $$d\vec{s}$$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.
Classical mechanics is already (effectively) multidimensional.
Consider the equation of motion for one particle in one dimension: \begin{align} m \ddot x(t) = F(x). \end{align} Now consider the equation of motion for one particle in two dimensions, \begin{align} m \ddot x(t) & = F_x(x,y)\\ m \ddot y(t) & = F_y(x,y), \end{align} and in three dimensions \begin{align} m \ddot x(t) & = F_x(x,y,z)\\ m \ddot y(t) & = F_y(x,y,z)\\ m \ddot z(t) & = F_z(x,y,z). \end{align} Now consider the equation of motion for two particles in one dimension, \begin{align} m_1 \ddot x_1(t) & = F_{1}(x_1,x_2)\\ m_2 \ddot x_2(t) & = F_{2}(x_1,x_2), \end{align} or in twodimensions, \begin{align} m_1 \ddot x_1(t) & = F_{x,1}(x_1,y_1,x_2,y_2)\\ m_1 \ddot y_1(t) & = F_{y,1}(x_1,y_1,x_2,y_2)\\ m_2 \ddot x_2(t) & = F_{x,2}(x_1,y_1,x_2,y_2)\\ m_2 \ddot y_2(t) & = F_{y,2}(x_1,y_1,x_2,y_2), \end{align} or three, \begin{align} m_1 \ddot x_1(t) & = F_{x,1}(x_1,y_1,z_1,x_2,y_2,z_2)\\ m_1 \ddot y_1(t) & = F_{y,1}(x_1,y_1,z_1,x_2,y_2,z_2)\\ m_1 \ddot z_1(t) & = F_{z,1}(x_1,y_1,z_1,x_2,y_2,z_2)\\ m_2 \ddot x_2(t) & = F_{x,2}(x_1,y_1,z_1,x_2,y_2,z_2)\\ m_2 \ddot y_2(t) & = F_{y,2}(x_1,y_1,z_1,x_2,y_2,z_2)\\ m_2 \ddot z_2(t) & = F_{z,2}(x_1,y_1,z_1,x_2,y_2,z_2). \end{align}
The pattern should be pretty obvious: adding spatial dimensions is identical to adding particles, and classical mechanics is already perfectly equipped for handling additional dimensions, so that e.g. if you added a fourth spatial dimension, the equations of motion \begin{align} m \ddot x(t) & = F_x(x,y,z,w)\\ m \ddot y(t) & = F_y(x,y,z,w)\\ m \ddot z(t) & = F_z(x,y,z,w)\\ m \ddot w(t) & = F_z(x,y,z,w) \end{align} would have a form which is exactly identical to that of two particles in two dimensions.
So, what does that mean for energy? Well following the core principle that
The same equations have the same solutions and the same properties,
the dynamics in four spatial dimensions will have a conserved energy which is exactly analogous to that of two particles in two dimensions, $$E = \frac12 m_1 \dot x_1^2 + \frac12 m_1 \dot y_1^2 + \frac12 m_2 \dot x_2^2 + \frac12 m_2 \dot y_2^2 + V(x_1,y_1,x_2,y_2)$$ i.e. a conserved energy of the form $$E = \frac12 m \dot x^2 + \frac12 m \dot y^2 + \frac12 m \dot z^2 + \frac12 m \dot w^2 + V(x,y,z,w)$$ where $$V(x,y,z,w)$$ is a potential energy. This should make it clear that the physical dimensionality of energy, $$[E] = [ML^2T^{-2}],$$ is not changed by the process - and its form from "normal" classical mechanics already fully incorporates effects spanning as many spatial dimensions as you want. |
# Is a $\sigma$ - compact Hausdorff space normal?
Are $\sigma$ -compact Hausdorff spaces normal?
-
Yes. Every $\sigma$-compact space is Lindelöf, every $\sigma$-compact Hausdorff space is $T_3$ (regular and $T_1$), and every $T_3$ Lindelöf space is $T_4$ (normal and $T_1$).
Correction: I was apparently not yet awake when I wrote that. There are countable Hausdorff spaces that are not regular, so the result is actually false in general. However, if you have any property that ensures regularity, you do get normality for free.
For an example of a countable Hausdorff space that isn’t regular, see this answer to an earlier, related question.
-
There may be some confusion based on definitions. Engelking's text, in particular, defines $\sigma$-compact spaces to be regular (T$_3$) spaces which are the countable union of compact subspaces. With this Brian's (original) answer works. (I cannot stress enough that his current answer works perfectly well.)
However if you remove regularity from the definition of $\sigma$-compactness, you can get counterexamples, as Brian mentions above. Another example is the following: Let $A = \{ \frac{1}{n} : n \in \mathbb{N} \}$, and give $\mathbb{R}$ the topology by declaring the open sets to be of the form $U \setminus B$ where $U \subseteq \mathbb{R}$ is open in the usual metric topology and $B \subseteq A$.
• As this topology is finer than the usual metric topology, it follows that this space is Hausdorff.
• It is $\sigma$-compact since you can cover $\mathbb{R}$ by countably many closed (bounded) intervals which each contain only finitely many elements of $A$.
• However this space is not regular because $A$ is closed in the new topology, but there is no pair of disjoint open sets around $0$ and $A$, respectively.
- |
# Thread in General Philosophy entitled "Universal Truths"
Discussion in 'SF Open Government' started by Rav, Jan 30, 2010.
1. ### Rav∞Valued Senior Member
Messages:
2,422
Hi guys
I am just wondering why this thread was closed. What motivates me to ask is the fact that I wanted to post in it. It has the potential to make for some interesting discussion, even if further discussion might have needed to initially focus on the fact that writing an essay is potentially unreasonable request to make of any contributor and/or the fact that there really isn't any such thing as a truth that no-one could possibly argue about. That's how I was planning to start anyway, before I moved on to something that I felt was a universal truth.
3. ### BenTheManDr. of Physics, Prof. of LoveValued Senior Member
Messages:
8,967
Maybe it would have worked in Free Thoughts. The OP, Illuminatingtherapy'', looks like a crackpot.
5. ### glaucontending tangentiallyRegistered Senior Member
Messages:
5,502
Rav,
I closed that thread for a number of reasons.
First, as you can see, it's ages old, with no activity since '06, with the exception of one post.
Second, and more importantly, as Ben noted, this discussion [sic] wasn't even remotely philosophical in nature, but rather little beyond babbling opining.
While you're correct that teh topic does have merit, even in Philosophy topics are required to be approached with some semblance of analysis.
If you do in fact, have something to post on this topic that could be characterized as analysis, or argument, and definitely not just blind supposition, feel free to PM me and I'll attempt to clean it up and re-open it. |
# Find $\frac{a^3}{a^6 + 1}$ given $a$ is a root of a quadratic equation
If $a$ is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$.
So, I figured we can use the quadratic formula, and formed the following equation: $$a=\frac{-(-3)+\sqrt{9-4}}{2(1)}\implies a=\frac{3+\sqrt5}2$$ But what I am thinking is, if I begin to find the required value, it will take me hours. And I believe, there must be some shortcut to this question. I had tried to solve this question with the manual process but it took me a lot of squares (one was $2012^2$!)
Thank you.
• $a^2-3a+1 = 0 \iff a+a^{-1} = 3$ and $\frac{a^3}{a^6+1} = \frac{1}{a^3+a^{-3}}$. – achille hui Feb 2 '14 at 9:13
• @achille Nice approach! – Gaurang Tandon Feb 2 '14 at 9:54
So $$a^2 + 1 = 3a$$ and this gives: $$\frac{a}{a^2 + 1} = \frac{1}{3},$$ and $$(a^2 + 1)^2 = 9a^2 \implies a^4 + 1 = 7a^2.$$ So $$\frac{a^2}{a^4 - a^2 + 1} = \frac{a^2}{6a^2} = \frac{1}{6}.$$ And finally $$\frac{a^3}{a^6 + 1} = \frac{a}{a^2 +1}\cdot \frac{a^2}{a^4 - a^2 + 1} = \frac{1}{3}\cdot \frac{1}{6} = \frac{1}{18}.$$
Since $a$ is a root of the quadratic, you know $a^2 -3a +1 = 0$, hence, $a^2 = 3a-1$.
This means any power of $a$ can be expressed as a linear combination of $a$ and $1$ in a natural fashion.
E.g. $a^3 = a(3a-1) = 3a^2 -a = 3(3a-1) -a = 8a -3$.
• Thank you. Help appreciated. – Gaurang Tandon Feb 2 '14 at 9:22
$a^2=3a-1$ then $$\frac{a^3}{a^6 + 1} \\=\frac{a^3}{(a^2+1)(a^4-a^2 + 1)} \\=\frac{a^3}{(3a)(9a^2-6a+1-3a+1+1)} \\=\frac{a^2}{(3)(9a^2-9a+3)} \\=\frac{a^2}{(9)(3a^2-3a+1)} \\=\frac{a^2}{(9)(9a-3-3a+1)} \\=\frac{a^2}{(9)(6a-2)} \\=\frac{3a-1}{(18)(3a-1)} \\=\frac{1}{18}$$
• Your method is flawed since $a^2+1\neq3a$. – user93957 Feb 2 '14 at 9:25
• @Aðøbe, no, this is true, it follows directly from the quadratic equation!! – Joachim Feb 2 '14 at 9:32
• @Aðøbe a is aroot – Semsem Feb 2 '14 at 9:34
• @Semsem Sorry, it's because in your first version you wrote $a^2=3a\color{red}+1$ so I naturally concluded from it that $a^2+1\neq 3a$. – user93957 Feb 2 '14 at 9:38
• @Semsem In your fourth equation, your numerator is $a$, shouldn't it be $a^2$ ? – Gaurang Tandon Feb 2 '14 at 10:00
As $a\ne0,$ we have $\displaystyle a^2-3a+1=0\iff a^2+1=3a\implies a+\frac1a=3$
and $\displaystyle a^3+\dfrac1{a^3}=\left(a+\frac1a\right)^3-3\left(a+\frac1a\right)^3$
$$\text{Again, }\frac{a^3}{a^6+1}=\frac1{a^3+\dfrac1{a^3}}$$
• Thanks ! I wish I knew these things before, lost 20 marks :( – Gaurang Tandon Feb 2 '14 at 12:59
• @GaurangTandon, I thought of adding a relatively simpler derivation. Try to solve similar problems – lab bhattacharjee Feb 2 '14 at 13:00 |
## Geometry: Common Core (15th Edition)
r=$\sqrt \frac{A}{\pi}$
We use the formula for the area of a circle to obtain: A=$\pi$$r^{2}$ $\frac{A}{\pi}$=$r^{2}$ $\sqrt \frac{A}{\pi}$=r |
## 85.3 Conventions and notation
The conventions from now on will be similar to the conventions in Properties of Spaces, Section 64.2. Thus from now on the standing assumption is that all schemes are contained in a big fppf site $\mathit{Sch}_{fppf}$. And all rings $A$ considered have the property that $\mathop{\mathrm{Spec}}(A)$ is (isomorphic) to an object of this big site. For topological rings $A$ we assume only that all discrete quotients have this property (but usually we assume more, compare with Remark 85.7.5).
Let $S$ be a scheme and let $X$ be a “space” over $S$, i.e., a sheaf on $(\mathit{Sch}/S)_{fppf}$. In this chapter we will write $X \times _ S X$ for the product of $X$ with itself in the category of sheaves on $(\mathit{Sch}/S)_{fppf}$ instead of $X \times X$. Moreover, if $X$ and $Y$ are “spaces” then we say "let $f : X \to Y$ be a morphism" to indicate that $f$ is a natural transformation of functors, i.e., a map of sheaves on $(\mathit{Sch}/S)_{fppf}$. Similarly, if $U$ is a scheme over $S$ and $X$ is a “space” over $S$, then we say "let $f : U \to X$ be a morphism" or "let $g : X \to U$ be a morphism" to indicate that $f$ or $g$ is a map of sheaves $h_ U \to X$ or $X \to h_ U$ where $h_ U$ is as in Categories, Example 4.3.4.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). |
## Closedness of the Tangent Spaces to the Orbits of Proper Actions
In this note we show that, for any proper action of a Banach-Lie group G on a Banach manifold M, the corresponding tangent maps g -> T-x(M) have closed range for each x is an element of M, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient M/G carries a natural manifold structure.
Published in:
Journal Of Lie Theory, 18, 517-521
Year:
2008
Keywords:
Laboratories: |
MA 339: Geometric Analysis
Pre-requisites :
1. A first course on manifolds (MA 338 should do).
2. Analysis (multivariable calculus, some measure theory, function spaces).
3. Functional analysis (The Hahn-Banach theorem, Riesz representation theorem, Open mapping theorem. Ideally, the spectral theory of compact self-adjoint operators too, but we will recall the statement if not the proof)
Basics of Riemannian geometry (Metrics, Levi-Civita connection, curvature, Geodesics, Normal coordinates, Riemannian Volume form), The Laplace equation on compact manifolds (Existence, Uniqueness, Sobolev spaces, Schauder estimates), Hodge theory, more general elliptic equations (Fredholmness etc), Uniformization theorem.
Suggested books :
1. Do Carmo, Riemannian Geometry .
2. Griffiths and Harris, Principles of Algebraic Geometry .
3. S. Donaldson, Lecture Notes for TCC Course “Geometric Analysis” .
4. J. Kazdan, Applications of Partial Differential Equations To Problems in Geometry .
5. L. Nicolaescu, Lectures on the Geometry of Manifolds .
6. T. Aubin, Some nonlinear problems in geometry .
7. C. Evans, Partial differential equations .
8. Gilbarg and Trudinger, Elliptic partial differential equations of the second order .
9. G. Szekelyhidi, Extremal Kahler metrics .
All Courses
Contact: +91 (80) 2293 2711, +91 (80) 2293 2265
E-mail: chairman.math[at]iisc[dot]ac[dot]in |
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# Automatic assessment of adverse drug reaction reports with interactive visual exploration
## Introduction
Drug safety has been well considered during the highly complex process of drug development, including phase I to phase III trials. However, pre-marketing trials lack sufficient power to reliably detect important adverse drug reactions (ADRs). Post-marketing surveillance for ADRs is indispensable since the possibility of rare, but serious ADRs or common delayed ADRs cannot be excluded even after approval to further ascertain safety for patient consumption. Post-marketing drug safety surveillance has primarily depended on spontaneous data from the ADR reporting system for decades, which relies on voluntary reporting by medical professionals, pharmaceutical companies, health authorities, or patients and their families1. The spontaneous reporting system (SRS) has been established in China since 1989 and supervision of ADRs now is a working priority for the drug authorities of China2.
In this paper, we advocate for assisting in reviewing and assessing ADR reports with automated assessment prediction and interactive visualization. Automatically assessing all reports could provide ADR experts with an overview of all reports and help ADR reporters to review and improve their reports. However, a perfect prediction model does not exist, and, therefore, automatic assessments cannot replace the judgment of humans. In fact, it is vital to have humans in the loop in drug safety practices. Interactive visualization is a human-centered technology that has demonstrated its usefulness and effectiveness in balancing the cognitive advantage of humans and the computational advantage of computers in data mining6 and decision making7. Therefore, we propose to use interactive visualization for users to explore ADR reports and examine reports of interest with their automatic assessments in detail. To facilitate filtering ADR reports with interactive visualization, a warning signal is devised such that characteristics of ADR reports could be conveniently measured quantitatively. Overall, our technique is a new approach for assessing and reviewing ADR reports in large quantities.
## Method
Unlike a large body of research focusing on “signal detection” from ADR reports, our work is intended to automatically assess the causality category of reports by learning from ADR experts and facilitate the exploration of ADR reports thereafter. In the remainder of this section, we will provide an overview of the method, and then, discuss each module of the method in detail, including the data, the automatic assessment model (ACA model), the warning signal computation, and the analysis with interactive visual exploration.
### Overview
The workflow of our method is shown in Fig. 1, which features an iterative human-in-the-loop approach. The input is report data from the SRS, which could be interactively assessed with the visualization tool (the orange box with dashed line) to provide labeled ADR reports. The labeled reports (typically, a small portion of all reports)—either assessed with or without the interactive visualization tool, e.g., previously manually assessed data—along with the input data are processed by our ACA model.
The ACA model predicts an assessment Y for each report in the SRS dataset. Our new warning signal $$\lambda$$ is calculated for each record using Y and the original ADR data, and $$\lambda$$ is used for filtering so that users could focus on particular groups of reports, e.g., reports with high absolute values of the warning signal as discussed in “Interactive visual exploration of the ACA model results”. The warning signal $$\lambda$$ is used for interactive exploration of ADR reports in our visualization tool. Interactive exploration could improve the assessment and allows users to examine specific report more closely. Moreover, ADR reporters could easily compare the difference between model assessment and theirs and possibly improve the quality of assessments after that. Therefore, the workflow of our method forms a loop that iteratively improves the quality of SRS reports.
### Data
The value of an assessment is categorized into five levels in an increasing order of certainty and coded as follows.
• 1 = unassessable/unclassifiable
• 2 = conditional/unclassified
• 3 = possible
• 4 = probable/likely
• 5 = certain
We use all variables in our interactive visualization tool and a subset of the variables are chosen for the automatic assessment model. Variables that are relevant for automatic assessment are first empirically selected by an ADR expert and then systematically refined by experimenting with the performance of the model. The resulting variables used for our automatic assessment model are listed and explained in Table 1.
### Automatic causality assessment model
The prediction of assessments of the data could be realized by logistic regression. Multinomial logistic regression is a typical method for such a task8. However, this requires the fitting of a large number of parameters, so the degree of freedom used in the model fitting process will put excessive demands on the data set. More problematic, multiple logistic regression does not take into account the ordinal properties of the response variables9.
In contrast, ordinal logistic regression is geared for data with ordinal response variables. Moreover, ordinal logistic regression has no specific requirements on variable types, considering that all dependent variables are classified variables, it is a simple and applicable method in this situation. Often, ordinal logistic regression is achieved through the proportional-odds cumulative logit model10. The model reads:
\begin{aligned} {\text {logit}}[P(Y_i\le j)]&=\ln \frac{P\left( Y_i\le j\mid X_i\right) }{1-P\left( Y_i\le j\mid X_i\right) }=\theta _j + \beta ^{T}X_i,\nonumber \\ j&=1,2,3,4,5\quad i=1,\ldots ,n, \end{aligned}
(1)
where $$Y_i$$ is the ordinal dependent variable, $$X_i$$ is the independent variable, and i is the index of the record within the dataset of n records. The probability function of the proportional-odds model is:
\begin{aligned} P(Y_i\le j)=\frac{\exp (\theta _j+\beta ^{T}X_i )}{1-\exp (\theta _j+\beta ^{T}X_i )}, j=1,2,3,4,5\quad i=1,\ldots ,n, \end{aligned}
(2)
where $$\theta _j$$ is the cut point of the jth logit function. This model is based on the assumption that the slope coefficients do not depend on the selection of cut-off points11, i.e., the parallel assumption. However, the parallel assumption can be easily violated.
As a natural extension of the proportional odds model, the partial proportional odds model is proposed10. This model allows some variables to violate the parallel assumption, in other words, allows some to vary among logit equations. The partial proportional odds model can be expressed as follows:
\begin{aligned} {\text {logit}}[P(Y_i\le j)]&=\ln \frac{P(Y_i\le j \mid X_{i})}{1-P(Y_i\le j \mid X_i)}=\theta _j+\beta ^{T}X_i+\tau _j^{T}T_i, \nonumber \\ j&=1,2,3,4,5\quad i=1,\ldots ,n. \end{aligned}
(3)
\begin{aligned} P(Y_i\le j)&=\frac{\exp (\theta _j+\beta ^{T}X_i+\tau _{j}^{T}T_i)}{1- \exp (\theta _j + \beta ^{T}X_i+ \tau _{j}^{T}T_i)}, \nonumber \\ j&=1,2,3,4,5\quad i=1,\ldots ,n, \end{aligned}
(4)
where $$T _ i$$ is a $$q\times 1$$ vector, $$q\le p$$, represents the values of q independent variables violating the parallel assumption, and $$\tau _ j$$ is the regression coefficients associated with those variables, also depend on the selection of j. If all variables follow the parallel assumption, it becomes the proportional odds model.
Therefore, the partial proportional odds model is used in our ACA model as it is more flexible and has a higher tolerance of independent variables compared to the proportional odds model. In our model, $$X _i$$ is the independent variable of p attributes, which is a $$p\times 1$$ vector, and $$Y_i$$ denotes the recorded causality category assessment of ordinal scales 1 through 5 of the ith report in the dataset that contains n records. Here, the number of p is determined to be 11 according to our variable selection as shown in Table 1.
We used the 2019 dataset that contains provincial-level ADR experts labeled records to train our model. The labeled data, i.e., the ground truth data, contains 20022 records in total, and we divide the records randomly into a training set (16,017 records) and a testing set (4005 records) to train the model. We implemented the ACA model in R aided by the ‘vgam’ package that readily implements the partial proportional odds model12. We compare the results of the ACA model to the ground truth data (expert labeled assessment), and those from the multinomial logistic regression and the decision tree methods. The accuracy, $$F_1$$ scores, and multiclass receiver operating characteristic (ROC) curves13 are calculated for each method. The multiclass ROC is used as the standard ROC is designed for binary classification, it is not suitable for our case of classification of multiple ordinal classes. The results can be found in “Results”.
### Warning signal computation
With our automatic assessment model, each report now has a predicted assessment trained with assessments by provincial-level ADR experts providing a reference for an ADR reviewer. However, exploring and analyzing the whole dataset even with automatic assessments is still challenging and laborious.
To this end, a warning signal that could quickly draw the attention of an ADR reviewer to reports of interest is helpful. Our goal is to encode the following information with the warning signal: I1, whether an agreement exists between assessments by the ACA model and a reporter, for example, both agree that a true adverse event occurred but assessments could vary on degrees; I2, the difference between the assessment of a reporter and the ACA model; I3, the severity of the adverse event. Moreover, the warning signal should use a medium value to encode the majority of reports, whereas high and low values indicate reports that are distinct as they are of potential interest.
We denote assessments from the ACA model and the spontaneous report as Y, and $$Y^0$$, respectively. The term s is designed to encode I1 and I3 using variables d and r:
\begin{aligned} d&= {\left\{ \begin{array}{ll} |Y - Y^0 |, (Y -3)(Y^0 - 3) > 0,\\ Y + Y^0 - 6, (Y -3)(Y^0 - 3) = 0,\\ -|Y - Y^0 |, (Y -3)(Y^0 - 3) < 0.\\ \end{array}\right. } \end{aligned}
(5)
\begin{aligned} r&= {\left\{ \begin{array}{ll} 1, {\text {normal ADE}},\\ 4, {\text {serious ADE}}, \end{array}\right. } \end{aligned}
(6)
where d describes the degree of disagreement of assessments (I1), and r describes the severity of the adverse event (I3). The variable d is designed in a way (Eq. 5) that a positive sign indicates that the assessment of ACA model agrees with that of the report, and a negative sign indicates that the two assessments disagree; whereas its absolute value measures the difference of the two assessments. To support intuitive user interactions in the visualization, s is designed to have a symmetric range by offsetting d by one ($$d+1$$) as in Eq. (8). The values of the term r are chosen to avoid the warning signal value overlapping for normal and serious ADE reports. As a result, a symmetric range is achieved for s with the possible values being a set $$S = \{-16, -12, -8, -4, -3, -2, -1, 0, 1, 2, 3, 4, 8, 12, 16\}$$. For a normal ADE, $$s \in [-3,3]$$, while $$s \in [-16,-4] \cap [4,16]$$ suggests a serious ADE.
At this point, I1 and I3 have been described with s, which has a symmetric range with severe cases spreading at two ends and normal cases at the center. To capture the difference information I2, we use another variable t that measures the difference between Y and $$Y^0$$. Therefore, our final warning signal $$\lambda$$ is a tuple of s ($$\lambda _s$$) and t ($$\lambda _t$$) for each report:
\begin{aligned} \lambda&= (\lambda _s, \lambda _t) = (s, t), \end{aligned}
(7)
\begin{aligned} {\text {where}}\, s&= r\cdot (d+1), \end{aligned}
(8)
\begin{aligned} t&= Y - Y^0. \end{aligned}
(9)
With $$\lambda$$, a reviewer could efficiently and flexibly filter the reports, and focus on the few reports that are of interest. The usefulness of the warning signal is demonstrated with our interactive visualization in “Interactive visual exploration of the ACA model results”.
### Interactive visualization
Parallel coordinates14,15 are a widely-used multidimensional visualization technique that arranges data dimensions (variables) as parallel vertical axes and visualizes data as polylines across these parallel axes. We choose parallel coordinates as our interactive visualization method for its two advantages: data dimensions are readily scalable by simply adding more vertical axes, and multidimensional data query can be achieved by interactively selecting ranges, namely, brushing, on specific axes. The arrangement of axes in parallel coordinates could be also modified for visualizing correlation information of intended orderings of dimensions.
Our dataset of 137,964 reports imposes a challenge as they are too large for interactive visualization and manipulation if rendered as a whole at every update, i.e., every user selection, window resizing, or axis reordering. To achieve interactivity, we adopt progressive rendering that draws a portion of the full dataset at every frame and generates the full visualization progressively.
The user interface of our interactive visualization method is shown in Fig. 2. The parallel coordinates plot is shown on the top of Fig. 2 with ordinal and Boolean variables as axes with the following default ordering: known or unknown ADE (New ADE), normal or severe ADE (Severe ADE), Gender, the assessment of the reporter (R assess), the assessment of the reporter’s institute (RI assess), the assessment of the municipal-level regulator (Muni assess), the assessment of the provincial-level regulator (Prov assess), suspected or co-occurring drug (Suspect), automatic assessment (Auto assess), warning signal $$\lambda _s$$ (Risk s), and warning signal $$\lambda _t$$ (Risk t). Moreover, we use “RI assess” as the assessment $$Y^0$$. These configurations are empirically determined by an ADR expert. To facilitate the exploration, a linked table view (bottom of Fig. 2) is used for showing the full information of each ADR report including the textual information of symptoms, the product name, and the batch number. Polylines in the parallel coordinates are colored based on our predicted assessment Y—with a red-yellow-blue color map indicating the causality category 1 (red) through 5 (blue). The table view is linked with the parallel coordinates so that changes on one view is reflected on the other, e.g., updates of brushing on parallel coordinates change the visibility of corresponding data items in the table view, and focusing on one row of the table highlights the corresponding polyline in the parallel coordinates. Moreover, the table supports convenient editing of elements enabling users to assess ADR reports interactively.
Our visualization tool is an interactive web page that can be conveniently accessed by users with modern web browsers. The tool is implemented in HTML and JavaScript aided by the Data-Driven Documents (D3.js)16, d3-parcoords17, and the SlickGrid18 libraries.
### Ethical approval
All data used in this study are anonymous. All methods were performed in accordance with the relevant guidelines and regulations. This study have been approved by the institutional review board ethics committee of the Peking University and obtained ethical exemption (Ethical approval number: IRB00001052-20012-exempt).
## Results
Results of our method are shown through the performance analysis of the ACA model and reports of interest found through interactive visual exploration. The performance of the ACA model is evaluated by comparing it to the ground truth and other competing models in “The ACA model evaluation”, and the analysis of reports is discussed in “Interactive visual exploration of the ACA model results”.
### The ACA model evaluation
We compute the accuracy, the $$F_1$$-score, and multiclass ROC curves as performance measures for the evaluation. The ACA model is compared to the ground truth, the multinomial logistic regression (MLR), and the decision tree (DT) on the testing set (4005 records, i.e., 20% of the labeled data). Note that the same set of variables as in Table 1 are used in the multinomial logistic regression and the decision tree. Performance measurements of these methods are summarized in Table 2.
The ACA model achieves an accuracy of $$85.99\%$$. In comparison, the multinomial logistic regression and decision tree achieve the accuracy as $$80.82\%$$ and $$85.39\%$$, respectively. The $$F_1$$-score that is widely used for performance measurement in classification problems is the harmonic mean of the precision and sensitivity. $$F_1$$-score can better measure the performance of models than the accuracy. In our case, $$F_1$$-score is calculated for each category. Note that category 1 was not used by provincial ADR experts leading to $$F_1 = 0$$ of category 1 for all methods, and are, therefore, omitted. As shown in Table 2, for category 2 (“conditional/unclassified”) through 5 (“certain”), the $$F_1$$ scores of our method are 0, 0.4, 0.8329, 0.8775, and 0.9083, respectively; multinomial logistic regression yields scores of 0, 0, 0.7716, 0.8482, and 0; scores of the decision tree are 0, 0, 0.8346, 0.8692, and 0.8559.
Multiclass ROC curves and the associated AUC are calculated for further evaluation. The averaged ROC curves can be seen in Fig. 3, where micro curves—ROC curves calculated using true positives, false positives, true negatives, and false negatives of each class—are solid, and macro curves—the average of ROC curves of each class—are dashed19. The ACA model, the multinomial logistic regression, and the decision tree are drawn as blue, red, and yellow curves, respectively. Note that a baseline has to be set for multinomial logistic regression. Different baseline settings affect the performance of the multinomial logistic regression: when the baseline is set to 4 (“probable/likely”), the macro AUC is 0.8603, when the baseline is set to 5 (“certain”), the macro AUC is about 0.8, and the macro AUC is about 0.65 (the minimum is 0.5) when the baseline is set to categories 1, 2, 3 (“unassessable/unclassifiable”, “conditional/unclassified”, “possible”). Detailed comparison of each causality category of our method and the multinomial logistic regression is shown in Fig. 4a, while b shows the comparison of results of ours and the decision tree. The AUC of the macro and micro curves, and the AUC of each category for all methods are listed in Table 2.
### Interactive visual exploration of the ACA model results
Evaluations of the ACA model in the previous section show the performance of the new model numerically. In this section, reports of interest that are found with interactive visual exploration are examined by ADR experts to demonstrate the usefulness of the new warning signal $$\lambda$$. ADR experts are interested in reports that have high absolute values of $$\lambda _s$$ and $$\lambda _t$$ in combination with conditions of other variables. Such reports of interest are not easily detected with traditional methods in the dataset, but can be quickly and flexibly identified through interactive visual exploration with our automatic assessment and warning signal. According to the design of $$\lambda _s$$, we are interested in negative $$\lambda _s$$ with high absolute values as they indicate that the ACA model provides opposite assessment predictions than reporters.
By selecting suspect drugs (Suspect = 1) with negative $$\lambda _s$$ with high absolute values ($$\lambda _s<-8$$), 62 reports were found when we examined the whole records of 2019 dataset (137,964 reports). Among them, nine reports have opposite assessments between provincial-level experts (Prov assess) and reporters (R assess). The “Prov assess” is greater than 3, i.e., categories of probably/likely, or certain. In contrast, “R assess”, “RI assess”, and “Muni assess” are all less than or equal to 3, i.e., categories of unassessable/unclassifiable, conditional/unclassified, or possible. These reports can be easily found through interactive brushing on parallel coordinates as shown in Fig. 5, where brushes of range selections on variables are shown as gray boxes on vertical axes.
Details of the reports are summarized in Table 3. Notably, assessments by our ACA model (Auto) are almost identical to the Prov assess with one exception that a 5 is given by our method with the report of drug “Levofloxacin Hydrochloride”. All of the ADRs included had been reported previously (National medical products administration of China (https://www.nmpa.gov.cn/) and U.S. Food and drug administration (https://www.fda.gov/) were used for ADR information acquisition), and most of them were listed in the ADR Section of the instruction of drug.
Reporters could also potentially benefit from using our method as the quality of their assessments may be improved even without assessments by higher-tier experts. Such a case is demonstrated in Fig. 6. We start by brushing on suspected drugs with no Prov assess (Prov assess = $$-1$$) in the 2019 dataset. Then, we select automatic assessment with the highest value (Auto assess = 5). As a result, ten records are found (see the table below the parallel coordinates in Fig. 5), and provincial-level ADR experts did not review those reports. Most of the event-drug pairs included in these ten reports were reported previously (with the aforementioned sources), but the assessment results between reporters, reporter’s institute, and the municipal-level regulator are inconsistent. Notably, assessments by reporter’s institute (RI assess) conflict with the municipal-level regulator: “RI assess” are “unassessable/unclassifiable” (RI assess = 1), whereas “Muni assess” vary from “possible” to “certain” (Muni assess $$\ge 3$$).
The analysis of reports with our warning signal and the interactive visualization tool is efficient. In fact, the whole process of each example took less than one minute, so that the reviewer could quickly identify the reports of interest.
## Discussion
In this study, we successfully developed an automatic causality assessment model based on ordinal logistic regression. We combined this model with the interactive visualization tool to establish an iterative workflow that will help reporters for self-review and improvement. Our method is not a new model for ADR causality assessment but is to assist reporters at primary-level medical institutions and experts at municipal- and provincial-level alike in facilitating and improving the process of reporting and reviewing.
Logistic regression has been wildly used in medical research for years, where the typical logistic regression model is the binary logistic regression model. When possible confounding factors need to be controlled for, or multiple factors need to be considered, particular multivariate logistic regression could be used. However, typically used multinomial logistic regression requires the fitting of many parameters, so the degree of freedom used in the model fitting process will put excessive demands on the data set. Moreover, the multinomial logistic regression does not consider the ordinal properties of the output variables9. In contrast, ordinal logistic regression is devised for an ordinal response variable8. In addition, the ordinal logistic regression has no specific requirements on variable types, considering that all dependent variables are classified variables. Therefore, the ordinal logistic regression is an effective and applicable method for problems with ordinal outcomes. Ordinal logistic regression has been previously used in many areas including medical researches20, for example, in risk prediction and life expectation21. An ordinal risk index is fitted using ordinal logistic regression to predict the risk of postoperative acute kidney infection22. A risk score is modeled using ordinal logistic regression to predict hand-and-foot syndrome dynamics in a longitudinal study23. However, to our knowledge, ordinal logistic regression has not been used for causality assessment of ADR reports.
Visualization converts data into interactive graphics to gain insights into the data through perception and interaction. Medicine is an important research topic in visualization, and an overview of drug safety-related visualization methods can be found elsewhere24. A web-based tool with a visual interface is available for validating clinical cases and assessing detection rules of ADEs of hospital record data25. The tool relies on structured query language for setting ADE rules and generates reports with lab test results and drug prescriptions. A visual analytic method26 is proposed for analyzing ADR data with an overview visualization of ADEs, interactive filtering of co-occurring drugs, and an advanced query interface supporting hypotheses generation. This method supports signal generation based on the odds ratio using the contingency table. However, these methods do not analyze datasets with causality assessments nor support the assessment of such categories. A combined automatic and interactive visualization ADR assessment approach is not available currently.
Multinomial logistic regression and decision tree were chosen for comparison. Multinomial logistic regression model had been used to estimate the probability of severe ADRs27. And the first causality assessment method for drug-induced liver injury was based on the decision tree28. As shown in “The ACA model evaluation”, our ACA model achieved similar accuracy with the decision tree method and is better than multinomial logistic regression. The results of $$F_1$$ scores for categories 2–5 (“conditional/unclassified”, “possible”, “probable/likely”, “certain”) of our method are superior to the multinomial logistic regression for all categories and $$F_1$$ scores of categories 2 (“conditional/unclassified”) and 5 (“certain”) for the multinomial logistic regression are 0; our method has comparable $$F_1$$ scores of categories 3–5 (“possible”, “probable/likely”, “certain”) to the decision tree, however, a 0 $$F_1$$ score is achieved by decision tree for category 2 (“conditional/unclassified”).
Multiclass ROC curves are commonly used as an evaluation method for multiclass regression models, it can reveal True Positive Rate (TPR) and False Positive Rate (FPR) for each paired classes19. Results of multiclass ROCs show that the ACA model has better AUC than multinomial logistic regression for all curves (micro, macro, categories 1–5). Furthermore, the ACA model has comparable or slightly better AUC than decision tree for the micro, macro, category 3–5 (“possible”, “probable/likely”, “certain”) curves, and better AUC for categories 1 (“unassessable/unclassifiable”) and 2 (“conditional/unclassified”).
There are still some limitations of our method. First, only reports from one province of China in one calendar year were used to train and test the ACA model. The generality of the ACA model requires further studies. We have performed a preliminary study of the ACA model in spontaneous reports collected by the Guangdong adverse drug reaction monitoring center in 2018 (anonymous data with 143,406 reports under the same ethical approval as of the 2019 data) and received a comparable accuracy of $$85.77\%$$. Second, unbalanced data, significantly, when many categories exist, could impact the performance of ordinal logistic regression. The predicted results are potentially biased due to considerable differences between sample sizes of categories29, which is the nature of ADR reports. Third, the variables used in the ACA model were selected based on the experience of ADR experts. An automatic selection method should be used to choose variables systematically in the future. Last, the current visualization method is effective but does not cover the many aspects of the ADR data, and, therefore, we plan to design a full-fledged integrated visual analysis method in the future.
## Conclusion
In this paper, an automatic assessment method with interactive visualization exploration is proposed to aid the causality assessment of ADR reports. The ACA model automatically predicts causality assessment for spontaneous reports by learning assessments of experienced ADR experts. A warning signal is derived based on the automatic assessments for filtering a large volume of reports. A parallel coordinates-based visualization method is used for interactive exploration of the ADR data extended by the automatic assessment and the warning signal. The ACA model achieves good accuracy, and potential reports of interest can be quickly and conveniently selected with our web-based interactive visualization tool. Our overall method could improve the efficiency of the assessment process and increase the coverage of ADR reports reviewing for ADR experts. Moreover, the method could also potentially improve the capability of analyzing ADR reports and is promising in reducing human labor and the chance of missing critical reports.
For future work, we would like to further improve the model for better prediction efficiency and automate the variable selection process. Another direction is to devise a comprehensive interactive visual analysis method to provide users an integrated environment for more efficient review and assessment of ADR reports and to gain more insights during the process.
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## Funding
This research was supported by the Data for Better Health Project of Peking University-Master Kong.
## Author information
Authors
### Contributions
Z.G. and Y.Y. equally contributed to the data analysis, model development and drafted the manuscript. R.M. and J.Y. contributed to the analysis and provided clinical support. L.Z. contributed to the visualization part, provided the overall supervision of the study, drafted the manuscript, and critically edited the manuscript. All authors reviewed the manuscript critically for scientific content, and all authors gave final approval of this version for publication.
### Corresponding author
Correspondence to Liang Zhou.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
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Gao, Z., Yang, Y., Meng, R. et al. Automatic assessment of adverse drug reaction reports with interactive visual exploration. Sci Rep 12, 6777 (2022). https://doi.org/10.1038/s41598-022-10887-5
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# instantaneous frequency
If my signal is
$$f(t)=\exp[i\phi(t)],$$ how do I show that
$$\int_{-\infty}^{\infty} {\xi}\left|S_f(u,\xi)\right|^2 d\xi = 2\pi\int_{-\infty}^{\infty} {\phi'(t)}\left|g(t-u)\right|^2 dt,$$
where $$S_f(u,\xi)=\int_{-\infty}^{\infty}f(t)g(t-u)\exp{(-i\xi t)} dt$$ is the short-time Fourier transform of $f(t)$?
• You might get a better response if you explain a little more where you're getting these equations from and what you hope to get out of it. As it is, the question is a little sparse. – Peter K. Sep 10 '13 at 16:19
• Is the greek chsi supposed to be frequency here? – Tom Kealy Sep 10 '13 at 17:01
• yes, you are right – meta_warrior Sep 11 '13 at 1:12
• I believe what the question is asking is: How do I show that the first moment in frequency of the short-time Fourier transform magnitude squared is equal to a smoothed version of the instantaneous frequency ($\phi'$) ? – Peter K. Sep 11 '13 at 8:02
• Given your signal, it's relatively easy to show that the spectrogram is just the absolute value of the window function squared. Then you just make the substitution that chsi = dpsi/dt and change the range of integration to be over the unit circle rather than R. – Tom Kealy Sep 12 '13 at 10:58 |
# 21 Geometric Series
A Geometric Series on the natural numbers is a series of the form $$\sum\limits_{i=1}^{\infty}a \cdot r ^ {i-1}$$ where $$a\neq 0,\ r\neq 0,1$$. Expanding the series gives
\begin{aligned} \sum\limits_{i=1}^{\infty} a \cdot r^{i-1} &= a \cdot r^0 + a \cdot r ^ 1 + a \cdot r ^ 2 + a \cdot r ^ 3 + \cdots \end{aligned}
Alternatively, a Geometric Series may be parameterized on the non-negative integers in the form $$\sum\limits_{i = 0}^{\infty} a \cdot r ^ i$$ where $$a\neq 0,\ r\neq 0,1$$. Expanding this series gives
$\sum\limits_{i=0}^{\infty} a \cdot r^{i} = a \cdot r^0 + a \cdot r ^ 1 + a \cdot r ^ 2 + a \cdot r ^ 3 + \cdots$
## 21.1 Partial Summation
### 21.1.1 Summation for the Series on the Natural Numbers
Let $$S_k$$ denote the sum of a series over $$k$$ terms (or the $$k^{th}$$ partial sum). For the Geometric Series\
\begin{aligned} S_k &= \sum\limits_{i=1}^{k} a \cdot r^{i-1} \\ &= a \cdot r^{1 - 1} + a \cdot r^{2 - 1} + a \cdot r^{3 - 1} + a \cdot r^{4 - 1} + \cdots + a \cdot r^{k-1} \\ &= a \cdot r^{0} + a \cdot r^{1} + a \cdot r^{2} + a \cdot r^{3} + \cdots + a \cdot r^{k-1} \\ &= a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1} \\ \end{aligned}
Notice that
\begin{aligned} r \cdot S_k &= r \cdot \left(a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1} \right) \\ &= a \cdot r + a \cdot r^2+ a \cdot r^3 + \cdots + a \cdot r^k \end{aligned}
Taking the difference of these equalities yields:
$\begin{array}{rrcl} & S_k - r \cdot S_k &=& \left(a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1}\right) - \\ &&& \left(a \cdot r + a \cdot r^2+ a \cdot r^3 + \cdots + a \cdot r^k\right) \\ &&=& a + a \cdot r - a \cdot r + a \cdot r^2 - a \cdot r^2 + \cdots + a \cdot r^{k - 1} - a \cdot r^{k - 1} - a \cdot r ^ k \\ &&=& a + a \cdot r ^ k \\ \Rightarrow & (1 - r) \cdot S_k &=& a - a \cdot r ^ k \\ \Rightarrow & S_k &=& \frac{a - a \cdot r ^ k}{1 - r} \end{array}$
### 21.1.2 Summation for the Series on the Non-negative Integers
Let $$S_k$$ be the summation of the first $$k$$ terms of the geometric series. Whereas the series is on the non-negative integers–that is, the first term is for $$i = 0$$, the $$k^{th}$$ terms is $$i = k - 1$$. This provides
\begin{aligned} S_k &= \sum\limits_{i=0}^{k - 1} a \cdot r^{i} \\ &= a \cdot r^{0} + a \cdot r^{1} + a \cdot r^{2} + a \cdot r^{3} + \cdots + a \cdot r^{k-1} \\ &= a + a \cdot r + a \cdot r^2 + a \cdot r^3 + \cdots + a \cdot r^{k-1} \end{aligned}
This is identical to the partial summation described in Section 21.1.1, and the derivation of the partial sum is identical.
## 21.2 Proofs of Convergence
$$\sum\limits_{i=1}^{\infty}ar^{i-1}$$ converges when $$|r|<1$$ and diverges when $$|r|>1$$.
Proof:
Recall that the $$k^{th}$$ partial sum of the Geometric Series is
$S_k = \frac{a-a \cdot r^k}{1-r}$
And let $$S$$ denote the sum of the infinite series, i.e. the sum as$$k\rightarrow\infty$$.
Note that since $$S_k$$ is identical whether the series is parameterized on the natural numbers or the non-negative integers, the result is the same for either case.
Case 1: $$|r|<1$$
\begin{aligned} S &= \lim\limits_{k\rightarrow\infty}S_k \\ &= \lim\limits_{k\rightarrow\infty} \frac{a-ar^k}{1-r} \\ &= \frac{a-\lim\limits_{k\rightarrow\infty}ar^k}{1-r} \\ &= \frac{a}{1-r} \end{aligned}
So $$\sum\limits_{k=1}^{\infty}ar^{k-1}$$ converges when $$|r|<1$$ and $$S=\frac{a}{1-r}$$
Case 2: $$|r|>1$$
\begin{aligned} S &= \lim\limits_{k\rightarrow\infty}S_k \\ &= \lim\limits_{k\rightarrow\infty} \frac{a-ar^k}{1-r} \\ &= \frac{a-\lim\limits_{k\rightarrow\infty}ar^k}{1-r} \\ &= \frac{a-\infty}{1-r} \end{aligned}
So $$\sum\limits_{k=1}^{\infty}ar^{k-1}$$ diverges whern $$|r|>1$$. |
Now showing items 1-2 of 2
• #### A Setup for Testing of Silicon Pixel Modules for the ITk Tracker in the ATLAS Experiment at CERN
(Master thesis, 2021-06-14)
New pixel sensors will be installed to the new Inner Tracker upgrade at the ATLAS detector. These sensor will have higher radiation tolerance than the ones currently on the Inner Detector. This thesis will concentrate on ...
• #### Prompt and non-prompt J/Psi cross-sections in proton-proton collisions at sqrt(s) = 5.02 TeV with ALICE
(Master thesis, 2019-06-25)
The study presented in this thesis is about the measurement of \jpsi\ production in proton-proton collisions at center-of-mass energy $\sqrt{s}$ = 5.02 TeV with the ALICE detector at the LHC, with a focus on the separation ... |
# Statistical Physics - blackbody radiation
1. Apr 2, 2014
### Matt atkinson
1. The problem statement, all variables and given/known data
A cavity contains black body radiation at temperature at T=500K. Consider a optical mode in the cavity with frequency w=2.5x10^(13) Hz. Calculate;
(a)the probability of finding 0 photons in the mode.
(b)the probability of finding 1 photon in the mode
(c)the mean number of photons in the mode.
2. Relevant equations
3. The attempt at a solution
Okay so I'm not sure where to start basically with part (a) and (b), but I made an attempt at part (c)
I used the equation;
$$\bar{n}=\frac{1}{e^{\frac{\hbar\omega}{k_b T}}-1}=2.51\times 10^{-17} Photons$$
would really love a nudge in the right way, I've just gone blank on probability it's been so long since I last did it.
Last edited: Apr 2, 2014
2. Apr 3, 2014
Bump! ;D
3. Apr 7, 2014
### az_lender
The probability distribution will be a Poisson dstribution, so
P(X=n) = λn e/n!
where λ is the mean of the distribution.
If your answer to (c) is correct, then (a) and (b) are easy,
just put in n=0 and n=1.
BTW if your formula for (c) is correct,
then your numerical answer for (c) is incorrect.
Last edited: Apr 7, 2014
4. Apr 8, 2014
### TSny
The equation is correct, but your numerical result is incorrect. I get roughly 0.1. You should recheck the calculation.
For information on calculating the probability that the mode contains n photons, see for example the following discussion
http://physics.ucsc.edu/~drip/5D/photons/photons.pdf |
On the Gauss maps of space curves in characteristic p, II
Compositio Mathematica, Volume 78 (1991) no. 3, pp. 261-269.
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volume = {78},
number = {3},
year = {1991},
zbl = {0756.14021},
mrnumber = {1106297},
language = {en},
url = {http://archive.numdam.org/item/CM_1991__78_3_261_0/}
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Kaji, Hajime. On the Gauss maps of space curves in characteristic p, II. Compositio Mathematica, Volume 78 (1991) no. 3, pp. 261-269. http://archive.numdam.org/item/CM_1991__78_3_261_0/
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Journal of Systems Science and Complexity
Short Title: J. Syst. Sci. Complex. Publisher: Springer, Berlin/Heidelberg; Chinese Academy of Sciences, Academy of Mathematics and Systems Science, Beijing ISSN: 1009-6124; 1559-7067/e Online: http://link.springer.com/journal/volumesAndIssues/11424http://www.sysmath.com/jweb_xtkxyfzx/EN/article/showTenYearOldVolumn.do Predecessor: Systems Science and Mathematical Sciences
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The strong law of large numbers for pairwise NQD random variables. Zbl 1227.60039
Wu, Qunying; Jiang, Yuanying
2011
Studying complex adaptive systems. Zbl 1105.93004
Holland, John H.
2006
Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over $$\mathbb Z_4$$. Zbl 1316.94111
Shi, Minjia; Wang, Yu
2014
Variable selection for covariate adjusted regression model. Zbl 1310.62086
Li, Xuejing; Du, Jiang; Li, Gaorong; Fan, Mingzhi
2014
On fuzzy $$h$$-ideals of hemirings. Zbl 1333.16046
Ma, Xueling; Zhan, Jianming
2007
Viability criteria for differential inclusions. Zbl 1269.93046
Gao, Yan
2011
Optimal target trajectory estimation and filtering using networked sensors. Zbl 1173.93377
Hu, Jiangping; Hu, Xiaoming
2008
Minimizing a linear fractional function subject to a system of sup-$$T$$ equations with a continuous Archimedean triangular norm. Zbl 1178.93084
Li, Pingke; Fang, Shu-Cherng
2009
Cooperation in two-stage games on undirected networks. Zbl 1369.91035
Gao, Hongwei; Petrosyan, Leon; Qiao, Han; Sedakov, Artem
2017
Good $$p$$-ary quasic-cyclic codes from cyclic codes over $$\mathbb F_p+ v\mathbb F_p$$. Zbl 1395.94351
Shi, Minjia; Yang, Shanlin; Zhu, Shixin
2012
Estimation in a semi-varying coefficient model for panel data with fixed effects. Zbl 1303.93168
Hu, Xuemei
2014
Maximum likelihood estimator of the parameter for a continuous one-parameter exponential family under the optimal ranked set sampling. Zbl 1402.93234
Chen, Wangxue; Tian, Yi; Xie, Minyu
2017
On limitations of the sampled-data feedback for nonparametric dynamical systems. Zbl 1033.93042
Xue, Feng; Guo, Lei
2002
Rational general solutions of higher order algebraic ODEs. Zbl 1285.34003
Huang, Yanli; Ngô, L. X. Châu; Winkler, Franz
2013
Uniform decay rate for the transmission wave equations with variable coefficients. Zbl 1226.93111
Chai, Shugen
2011
Bilevel programming model and solution method for mixed transportation network design problem. Zbl 1193.49041
Zhang, Haozhi; Gao, Ziyou
2009
Codimension-two bifurcations analysis and tracking control on a discrete epidemic model. Zbl 1338.93223
Yi, Na; Zhang, Qingling; Liu, Peng; Lin, Yanping
2011
General decay rate estimates for viscoelastic wave equation with variable coefficients. Zbl 1325.35211
Cao, Xiaomin; Yao, Pengfei
2014
Optimal $$p$$-ary codes from one-weight and two-weight codes over $$\mathbb{F}_p + v\mathbb{F}_p^*$$. Zbl 1336.94087
Shi, Minjia; Solé, Patrick
2015
Exponential stability of a reparable multi-state device. Zbl 1333.90033
Hu, Weiwei; Xu, Houbao; Yu, Jingyuan; Zhu, Guangtian
2007
Solvability and control design for synchronization of Boolean networks. Zbl 1294.93049
Xu, Xiangru; Hong, Yiguang
2013
Stability of switched polynomial systems. Zbl 1173.93030
Li, Zhiqiang; Qiao, Yupeng; Qi, Hongsheng; Cheng, Daizhan
2008
Robust consensus and soft control of multi-agent systems with noises. Zbl 1173.93309
Wang, Lin; Guo, Lei
2008
Accuracy analysis for Quasi-Carey element. Zbl 1173.65348
Shi, Dongyang; Hao, Xiaobin
2008
A comprehensive dea approach for the resource allocation problem based on scale economies classification. Zbl 1179.91115
Li, Xiaoya; Cui, Jinchuan
2008
On ($$\in,\in \vee q$$)-fuzzy filters of BL-algebras. Zbl 1184.03066
Ma, Xueling; Zhan, Jianming
2008
On LCD negacyclic codes over finite fields. Zbl 1443.94110
Pang, Binbin; Zhu, Shixin; Sun, Zhonghua
2018
On the single server retrial queue with priority subscribers and server breakdowns. Zbl 1256.90021
Wang, Jinting
2008
Asymptotic normality of some estimators in a fixed-design semiparametric regression model with linear time series errors. Zbl 1076.62042
You, Jinhong; Chen, Min; Chen, Gemai
2004
Crude oil price forecasting with TEI@I methodology. Zbl 1121.91340
Wang, Shouyang; Yu, Lean; Lai, K. K.
2005
Complete monotonicity of the probability of ruin and de Finetti’s dividend problem. Zbl 1259.91072
Dong, Hua; Yin, Chuancun
2012
Optimal decisions when balancing expected profit and conditional value-at-risk in newsvendor models. Zbl 1231.90070
Xu, Minghui; Li, Jianbin
2010
Global stability of endemic equilibrium point of basic virus infection model with application to HBV infection. Zbl 1402.92416
Zheng, Yu; Min, Lequan; Ji, Yu; Su, Yongmei; Kuang, Yang
2010
Extension of smoothing Newton algorithms to solve linear programming over symmetric cones. Zbl 1242.90100
Huang, Zhenghai; Liu, Xiaohong
2011
Stock loan valuation under a regime-switching model with mean-reverting and finite maturity. Zbl 1198.91212
Prager, David; Zhang, Qing
2010
Global stability in a competition model of plankton allelopathy with infinite delay. Zbl 1327.93041
Chen, Fengde; Xie, Xiangdong; Wang, Haina
2015
De Casteljau algorithm and degree elevation of toric surface patches. Zbl 1468.65015
Li, Jinggai; Ji, Ye; Zhu, Chungang
2021
Distributed nonsmooth convex optimization over Markovian switching random networks with two step-sizes. Zbl 1472.93007
Yi, Peng; Li, Li
2021
Finite-time and fixed-time synchronization of inertial neural networks with mixed delays. Zbl 1460.93088
Chaouki, Aouiti; El Abed, Assali
2021
A second-order sliding mode controller of quad-rotor UAV based on PID sliding mode surface with unbalanced load. Zbl 1460.93021
Kang, Bing; Miao, Yan; Liu, Fu; Duan, Jilu; Wang, Ke; Jiang, Shoukun
2021
New Lyapunov-Krasovskii functional for stability analysis of linear systems with time-varying delay. Zbl 1460.93073
Lin, Huichao; Zeng, Hongbing; Wang, Wei
2021
Distributed finite-time integral sliding-mode control for multi-agent systems with multiple disturbances based on nonlinear disturbance observers. Zbl 1460.93023
Yang, Yize; Liu, Fan; Yang, Hongyong; Li, Yuling; Liu, Yuanshan
2021
Brexit and its impact on the US stock market. Zbl 1460.91266
Qiao, Kenan; Liu, Zhengyang; Huang, Bai; Sun, Yuying; Wang, Shouyang
2021
A nonlinear optimal control approach for tracked mobile robots. Zbl 1472.93129
Rigatos, Gerasimos
2021
Distributed optimal control of nonlinear time-delay system subject to delayed measurements and communication disruptions. Zbl 1475.49054
Su, Baili; Duan, Yuxing
2021
Adaptive backstepping sliding mode control of uncertain semi-strict nonlinear systems and application to permanent magnet synchronous motor. Zbl 1460.93022
Wang, Fang; Wang, Jian-mei; Wang, Kun; Zong, Qun; Hua, Changchun
2021
Maximum principle for non-zero sum stochastic differential game with discrete and distributed delays. Zbl 1460.91029
Zhang, Qixia
2021
A dynamic multi-player bargaining game with veto players. Zbl 1460.91042
Liu, Jia; Wang, Xianjia
2021
Bootstrap inference on the variance component functions in the two-way random effects model with interaction. Zbl 1462.62456
Ye, Rendao; Ge, Wenting; Luo, Kun
2021
The dynamics characteristics of flexible spacecraft and its closed-loop stability with passive control. Zbl 1460.93075
Meng, Bin; Zhao, Yunbo
2021
Impulse controllability and impulse observability of stochastic singular systems. Zbl 1460.93050
Ge, Zhaoqiang
2021
Finite-time stochastic stability of random impulsive positive system. Zbl 1460.93104
You, Lijie; Mu, Xiaowu
2021
$$H_2/H_\infty$$ control for stochastic jump-diffusion systems with Markovian switching. Zbl 1460.93028
Wang, Meijiao; Meng, Qingxin; Shen, Yang
2021
Stabilization with arbitrary convergence rate for the Schrödinger equation subjected to an input time delay. Zbl 1460.93074
Li, Yanfang; Chen, Hao; Xie, Yaru
2021
Empirical likelihood based diagnostics for heteroscedasticity in semiparametric varying-coefficient partially linear models with missing responses. Zbl 1462.62473
Liu, Feng; Gao, Weiqing; He, Jing; Fu, Xinwei; Kang, Xinmei
2021
Interval type-2 fuzzy passive filtering for nonlinear singularly perturbed PDT-switched systems and its application. Zbl 07502185
Liu, Xinmiao; Xia, Jianwei; Wang, Jing; Shen, Hao
2021
Dynamic event-triggered control for nonlinear systems: a small-gain approach. Zbl 1455.93121
Zhao, Qingtao; Sun, Jian; Bai, Yongqiang
2020
Stochastic optimal control problem in advertising model with delay. Zbl 1455.93209
Chen, Li; Wu, Zhen
2020
A mean-field linear-quadratic stochastic Stackelberg differential game with one leader and two followers. Zbl 1455.91041
Wang, Guangchen; Zhang, Susu
2020
Nuclear norm subspace system identification and its application on a stochastic model of plague. Zbl 1437.92146
Yu, Miao; Liu, Jianchang; Zhao, Lichun
2020
On the topology and isotopic meshing of plane algebraic curves. Zbl 1439.14106
Jin, Kai; Cheng, Jinsan
2020
Robust stabilization of non-minimum phase switched nonlinear systems with uncertainty. Zbl 1447.93295
Khalil, Jouili
2020
Formation tracking for nonlinear multi-agent systems with input and output quantization via adaptive output feedback control. Zbl 1447.93109
Hu, Jinglin; Sun, Xiuxia; He, Lei
2020
New results on stability and stabilization of delayed Caputo fractional order systems with convex polytopic uncertainties. Zbl 1447.93272
Dinh, Cong Huong; Mai, Viet Thuan; Duong, Thi Hong
2020
The dynamics and control of the fractional forms of some rational chaotic maps. Zbl 1450.37092
Ouannas, Adel; Khennaoui, Amina-Aicha; Bendoukha, Samir; Wang, Zhen; Pham, Viet-Thanh
2020
Adaptive output feedback regulation for a class of uncertain feedforward time-delay nonlinear systems. Zbl 1447.93106
Belfeki, Mohsen
2020
General decay synchronization for recurrent neural networks with mixed time delays. Zbl 1447.93133
2020
Event-triggered consensus for discrete-time multi-agent systems with parameter uncertainties based on a predictive control scheme. Zbl 1447.93326
Liu, Xiaoyu; Xuan, Yongbo; Zhang, Zhongyu; Diao, Zhaoshi; Mu, Zhenxing; Li, Zhitao
2020
A symbolic-numeric approach for parametrizing ruled surfaces. Zbl 1446.14037
Pérez-Díaz, Sonia; Shen, Li-Yong
2020
New results on interval general Cohen-Grossberg BAM neural networks. Zbl 1455.93165
Aouiti, Chaouki; Dridi, Farah
2020
Predictive control with velocity observer for cushion robot based on PSO for path planning. Zbl 1455.93053
Sun, Ping; Shan, Rui
2020
Credit scoring based on the set-valued identification method. Zbl 1455.91263
Wang, Ximei; Hu, Min; Zhao, Yanlong; Djehiche, Boualem
2020
GMDH-based outlier detection model in classification problems. Zbl 1458.62133
Xie, Ling; Jia, Yanlin; Xiao, Jin; Gu, Xin; Huang, Jing
2020
Isogeometric least-squares collocation method with consistency and convergence analysis. Zbl 1458.65150
Lin, Hongwei; Xiong, Yunyang; Wang, Xiao; Hu, Qianqian; Ren, Jingwen
2020
Almost automorphic solutions for quaternion-valued Hopfield neural networks with mixed time-varying delays and leakage delays. Zbl 1437.93074
Li, Yongkun; Meng, Xiaofang
2020
Distributed second-order continuous-time optimization via adaptive algorithm with nonuniform gradient gains. Zbl 1460.93093
Mo, Lipo; Liu, Xiaodong; Cao, Xianbing; Yu, Yongguang
2020
A class of general transformation of characteristic functions in dynamic games. Zbl 1460.91041
Liu, Cui; Gao, Hongwei; Petrosian, Ovanes; Liu, Ying; Wang, Lei
2020
Average estimation of semiparametric models for high-dimensional longitudinal data. Zbl 1462.62280
Zhao, Zhihao; Zou, Guohua
2020
Periodic solution and ergodic stationary distribution of stochastic SIRI epidemic systems with nonlinear perturbations. Zbl 1418.92209
Zhang, Weiwei; Meng, Xinzhu; Dong, Yulin
2019
Consensus of linear multi-agent systems with persistent disturbances via distributed output feedback. Zbl 1414.93171
Mo, Lipo; Guo, Shaoyan
2019
On almost controllability of dynamical complex networks with noises. Zbl 1419.93007
Cai, Ning; He, Ming; Wu, Qiuxuan; Khan, M. Junaid
2019
Zhu, Wei; Wang, Dandan; Zhou, Qianghui
2019
Consensus of single integrator multi-agent systems with unbounded transmission delays. Zbl 1414.93172
Xu, Xiang; Liu, Lu; Feng, Gang
2019
Prescribed performance finite-time tracking control for uncertain nonlinear systems. Zbl 1414.93145
Jing, Yuanwei; Liu, Yang; Zhou, Shaowei
2019
The algebro-geometric method for solving algebraic differential equations – a survey. Zbl 1411.34007
Winkler, Franz
2019
The Ore-Sato theorem and shift exponents in the $$q$$-difference case. Zbl 1408.33030
Du, Hao; Li, Ziming
2019
Robust finite-time trajectory tracking control of wheeled mobile robots with parametric uncertainties and disturbances. Zbl 1423.93099
Guo, Yijun; Yu, Li; Xu, Jianming
2019
Recursive identification of quantized linear systems. Zbl 1419.93063
Xiao, Jianming; Song, Qijiang
2019
Finite-time synchronization for a class of dynamical complex networks with nonidentical nodes and uncertain disturbance. Zbl 1414.93163
Li, Qingbo; Guo, Jin; Sun, Changyin; Wu, Yuanyuan; Ding, Zhengtao
2019
Complete weight enumerator for a class of linear codes from defining sets and their applications. Zbl 1476.94038
Liu, Haibo; Liao, Qunying; Wang, Xiaofeng
2019
A proposal for the automatic computation of envelopes of families of plane curves. Zbl 1417.68177
Botana, Francisco; Recio, Tomás
2019
Novel criteria for exponential stability of linear non-autonomous functional differential equations. Zbl 1411.93144
Pham Huu Anh, Ngoc; Thai Bao, Tran; Cao Thanh, Tinh; Nguyen Dinh, Huy
2019
Robust finite-time stability and stabilization of a class of fractional-order switched nonlinear systems. Zbl 1428.93102
Mai, Viet Thuan; Dinh, Cong Huong
2019
The optimal control of fully-coupled forward-backward doubly stochastic systems driven by Itô-Lévy processes. Zbl 1419.93068
Wang, Wencan; Wu, Jinbiao; Liu, Zaiming
2019
Coordination control for a class of multi-agent systems under asynchronous switching. Zbl 1419.93060
Zhao, Xiaodan; Liu, Wenhui; Yang, Chunjie
2019
Tracking synchronization of networked Lagrangian systems via impulsive control and its applications. Zbl 1419.93058
She, Weiqiang; Ma, Mihua
2019
Heuristics for online scheduling on identical parallel machines with two GoS levels. Zbl 1420.90018
Cai, Shuang; Liu, Ke
2019
Test for heteroscedasticity in partially linear regression models. Zbl 1423.62065
Khaled, Waled; Lin, Jinguan; Han, Zhongcheng; Zhao, Yanyong; Hao, Hongxia
2019
Estimation in linear regression with Laplace measurement error using Tweedie-type formula. Zbl 1423.62067
Shi, Jianhong; Feng, Jing; Song, Weixing
2019
Guaranteed cost finite-time control of fractional-order nonlinear positive switched systems with $$D$$-perturbations via MDADT. Zbl 1414.93147
Liu, Leipo; Cao, Xiangyang; Fu, Zhumu; Song, Shuzhong; Xing, Hao
2019
Uniform pricing strategy vs. price differentiation strategy in the presence of cost saving and demand increasing. Zbl 1414.91138
Cai, Qiang; Luo, Chunlin; Tian, Xin; Wang, Shouyang
2019
Characteristic decomposition: from regular sets to normal sets. Zbl 1410.13018
Mou, Chenqi; Wang, Dongming
2019
Hilbert problem 15 and Ritt-Wu method. I. Zbl 1411.14061
Li, Banghe
2019
Ritt-Wu characteristic set method for Laurent partial differential polynomial systems. Zbl 1470.12003
Hu, Youren; Gao, Xiao-Shan
2019
Self-evident automated proving based on point geometry from the perspective of Wu’s method identity. Zbl 1468.68310
Zhang, Jingzhong; Peng, Xicheng; Chen, Mao
2019
Speeding up the GVW algorithm via a substituting method. Zbl 1417.68294
Li, Ting; Sun, Yao; Huang, Zhenyu; Wang, Dingkang; Lin, Dongdai
2019
Elimination theory in differential and difference algebra. Zbl 1448.12003
Li, Wei; Yuan, Chun-Ming
2019
Conditional congruence closure over uninterpreted and interpreted symbols. Zbl 1417.68077
Kapur, Deepak
2019
Recent results on constructing Boolean functions with (potentially) optimal algebraic immunity based on decompositions of finite fields. Zbl 1448.94313
Liu, Zhuojun; Wu, Baofeng
2019
Quantum algorithm design: techniques and applications. Zbl 1409.81033
Shao, Changpeng; Li, Yang; Li, Hongbo
2019
Robust finite-time guaranteed cost control for positive systems with multiple time delays. Zbl 1411.93060
Thuan, Mai Viet
2019
Control for a class of stochastic mechanical systems based on the discrete-time approximate observer. Zbl 1411.93167
Fu, Xinxin; Kang, Yu; Li, Pengfei; Yu, Peilong
2019
Exponential stability of the Euler-Bernoulli beam equation with external disturbance and output feedback time-delay. Zbl 1411.93153
Wu, Jilong; Shang, Yingfeng
2019
Closed-loop iterative learning control for discrete singular systems with fixed initial shift. Zbl 1411.93115
Gu, Panpan; Tian, Senping; Liu, Qian
2019
Designing uniform computer sequential experiments with mixture levels using Lee discrepancy. Zbl 1411.93073
Elsawah, A. M.
2019
Pinning synchronization of discrete-time complex networks with different time-varying delays. Zbl 1428.93104
Qunjiao, Zhang; Xiaoqun, Wu; Jie, Liu
2019
Robust chattering-free finite time attitude tracking control with input saturation. Zbl 1428.93101
Chen, Haitao; Song, Shenmin
2019
A class of second-order consensus protocol in multi-agent systems with multiple input delays. Zbl 1423.93357
Wu, Huai; Gao, Kai; An, Baoran
2019
Optimal switching integrity attacks on sensors in Industrial control systems. Zbl 1423.93131
Wu, Guangyu; Sun, Jian
2019
Relative degree structural hole centrality, $$\mathrm{C}_{\mathrm{RD-SH}}$$: a new centrality measure in complex networks. Zbl 1423.93049
2019
On LCD negacyclic codes over finite fields. Zbl 1443.94110
Pang, Binbin; Zhu, Shixin; Sun, Zhonghua
2018
Stabilization of discrete-time dynamical systems under event-triggered impulsive control with and without time-delays. Zbl 1390.93689
Liu, Bin; Hill, David John; Zhang, Changfan; Sun, Zhijie
2018
Distributed algorithm for robust resource allocation with polyhedral uncertain allocation parameters. Zbl 1390.93309
Zeng, Xianlin; Yi, Peng; Hong, Yiguang
2018
Optimal output feedback control and stabilization for NCSs with packet dropout and delay: TCP case. Zbl 1390.93639
Ma, Xiao; Qi, Qingyuan; Zhang, Huanshui
2018
Fuzzy static output feedback $$H_{\infty}$$ control for nonlinear systems subject to parameter uncertainties. Zbl 1401.93131
Zhao, Tao; Dian, Songyi
2018
Input-to-state stability of switched nonlinear delay systems based on a novel Lyapunov-Krasovskii functional method. Zbl 1401.93184
Zong, Guangdeng; Zhao, Haijuan
2018
Global disturbance rejection of switched nonlinear systems with switched internal model. Zbl 1390.93256
Long, Lijun
2018
A nonlinear small-gain theorem for large-scale infinite-dimensional systems. Zbl 1390.93104
Bao, Adiya; Liu, Tengfei; Jiang, Zhong-Ping; Zhang, Lina
2018
The convergence of least-squares progressive iterative approximation for singular least-squares fitting system. Zbl 07002011
Lin, Hongwei; Cao, Qi; Zhang, Xiaoting
2018
Existence and uniqueness results for quaternion-valued nonlinear impulsive differential systems. Zbl 1401.34026
Zhu, Jingwen; Sun, Jitao
2018
Dynamical criticality: overview and open questions. Zbl 1401.93007
Roli, Andrea; Villani, Marco; Filisetti, Alessandro; Serra, Roberto
2018
On construction of optimal two-level designs with multi block variables. Zbl 1397.62280
Zhao, Yuna; Zhao, Shengli; Liu, Minqian
2018
Adjusted empirical likelihood estimation of distribution function and quantile with nonignorable missing data. Zbl 1401.93192
Ding, Xianwen; Tang, Niansheng
2018
Observability of Boolean control networks with time-variant delays in states. Zbl 1401.93045
Jiang, Dapeng; Zhang, Kuize
2018
Saddle point criteria in nonsmooth semi-infinite minimax fractional programming problems. Zbl 1402.90208
Mishra, S. K.; Singh, Yadvendra; Verma, R. U.
2018
Tracking control of wheeled mobile robots with communication delay and data loss. Zbl 1401.93148
Zhang, Tian-Yong; Liu, Guo-Ping
2018
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Cited by 4,191 Authors
29 Chen, Lansun 25 Jiao, Jianjun 24 Shi, Dongyang 20 Meng, Xinzhu 19 Wang, Shouyang 17 Zhang, Weihai 14 Cai, Shaohong 14 Mo, Jiaqi 13 Shen, Liyong 13 Zhang, Bingyu 12 Alizadeh, Behrooz 12 Gao, Xiaoshan 12 Shi, Minjia 11 Hong, Yiguang 11 Hou, Ting 11 Wu, Qunying 11 Zhang, Jifeng 10 Cao, Jinde 10 Hu, Jiangping 10 Huang, Zhenyu 10 Lar’kin, Nikolaj Andreevich 10 Lin, Dongdai 10 Pérez-Díaz, Sonia 10 Shi, Jingtao 10 Tang, Yinghui 10 Wang, Dingkang 10 Wang, Leyi 10 Xie, Lihua 9 Chen, Wangxue 9 Lin, Jinguan 9 Zhao, Yong 8 Chen, Ge 8 Chen, Guanrong 8 Chen, Ping 8 Dong, XiWang 8 Gao, Qingwu 8 Guo, Lei 8 Jia, Yingmin 8 Li, Tao 8 Lin, Tao 8 Sun, Yao 8 Wei, Li 8 Zhao, Yanlong 8 Zhu, Quanxin 7 Agarwal, Ravi P. 7 Burkard, Rainer E. 7 Gao, Lixin 7 Gao, Yan 7 Liu, Guizhen 7 Solé, Patrick 7 Sun, Ji Tao 7 Teng, Zhi-dong 7 Wang, Ke 7 Wang, Xiangrong 7 Wang, Xuejun 7 Wang, Yuebao 7 Yang, Yuguang 7 Yu, Zhiyong 7 Zhan, Jianming 7 Zhang, Huanshui 7 Zhong, Yisheng 7 Zhu, Shixin 6 Cerpa, Eduardo 6 Chen, Mo 6 Chen, Zengqiang 6 He, Xiaoming 6 Hu, Xiaoming 6 Kai, Xiaoshan 6 Kapur, Deepak 6 Li, Wuquan 6 Li, Zhongfei 6 Liu, Yungang 6 Liu, Zhixin 6 Rosier, Lionel 6 Tang, Jingyong 6 Wang, Lei 6 Wen, Guanghui 6 Wu, Qinghe 6 Yin, Gang George 6 Yin, Li 6 Yu, Miaomiao 6 Yu, Wenwu 6 Yuan, Chunming 6 Zhang, Tongqian 6 Zhang, Xinyu 6 Zhao, Hui 6 Zhao, Yunbo 6 Zhong, Shou-Ming 6 Zhu, Detong 6 Zou, Guohua 5 Abdel-Hamid, Alaa H. 5 Adjerid, Slimane 5 Chen, Hanfu 5 Ding, Liwang 5 Djaidja, Sabir 5 Fan, Guoliang 5 Feng, Gang 5 Feng, Jun’e 5 Fiori, Simone 5 Ge, Zhaoqiang ...and 4,091 more Authors
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Cited in 392 Journals
338 Journal of Systems Science and Complexity 73 Mathematical Problems in Engineering 69 Applied Mathematics and Computation 69 Automatica 63 Journal of the Franklin Institute 52 Complexity 44 Journal of Computational and Applied Mathematics 41 International Journal of Control 39 Systems & Control Letters 34 Abstract and Applied Analysis 33 Journal of Mathematical Analysis and Applications 33 International Journal of Systems Science. Principles and Applications of Systems and Integration 30 Advances in Difference Equations 29 Computers & Mathematics with Applications 29 Communications in Statistics. Theory and Methods 29 Discrete Dynamics in Nature and Society 27 SIAM Journal on Control and Optimization 25 Chaos, Solitons and Fractals 25 Information Sciences 24 Nonlinear Dynamics 24 Asian Journal of Control 23 Acta Mathematicae Applicatae Sinica. English Series 23 European Journal of Operational Research 23 Journal of Inequalities and Applications 21 Journal of Applied Mathematics and Computing 18 Theoretical Computer Science 18 Applied Mathematical Modelling 17 International Journal of Robust and Nonlinear Control 17 Statistical Papers 17 Journal of Applied Mathematics 17 Journal of Industrial and Management Optimization 16 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 16 Journal of Statistical Computation and Simulation 15 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 14 Applied Mathematics and Optimization 14 Journal of Symbolic Computation 14 Journal of Scientific Computing 13 Journal of Optimization Theory and Applications 13 Statistics & Probability Letters 13 Applied Mathematics and Mechanics. (English Edition) 13 Asia-Pacific Journal of Operational Research 13 Applied Mathematics. Series B (English Edition) 12 Computer Aided Geometric Design 12 International Journal of Computer Mathematics 12 Computational and Applied Mathematics 12 Journal of the Korean Statistical Society 12 Science China. Mathematics 11 Mathematical Methods in the Applied Sciences 11 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 11 Discrete and Continuous Dynamical Systems. Series B 10 Physica A 10 Optimization 10 Acta Mathematica Sinica. English Series 9 Annals of Operations Research 9 Nonlinear Analysis. Real World Applications 9 Nonlinear Analysis. Hybrid Systems 8 Bulletin of Mathematical Biology 8 Insurance Mathematics & Economics 8 Designs, Codes and Cryptography 8 Numerical Algorithms 8 Automation and Remote Control 8 European Journal of Control 8 Wuhan University Journal of Natural Sciences (WUJNS) 8 Boundary Value Problems 8 Frontiers of Mathematics in China 8 International Journal of Biomathematics 7 Discrete Mathematics 7 Metrika 7 Applied Numerical Mathematics 7 Journal of Global Optimization 7 Computational Statistics and Data Analysis 7 Finite Fields and their Applications 7 Chaos 7 Journal of Dynamical and Control Systems 7 Science in China. Series F 7 Mathematics in Computer Science 7 Mathematical Control and Related Fields 7 Journal of Applied Analysis and Computation 7 AIMS Mathematics 6 Discrete Applied Mathematics 6 International Journal of Theoretical Physics 6 Fuzzy Sets and Systems 6 Journal of Differential Equations 6 Journal of Multivariate Analysis 6 Stochastic Analysis and Applications 6 Statistics 6 Computers & Operations Research 6 Science in China. Series A 6 Journal of Intelligent and Fuzzy Systems 6 Optimization Letters 6 Dynamic Games and Applications 5 Applicable Analysis 5 Journal of Computational Physics 5 Journal of Mathematical Physics 5 Journal of Statistical Planning and Inference 5 Kybernetika 5 Opsearch 5 Acta Applicandae Mathematicae 5 Applied Mathematics Letters 5 Mathematical Methods of Operations Research ...and 292 more Journals
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Cited in 54 Fields
855 Systems theory; control (93-XX) 334 Operations research, mathematical programming (90-XX) 308 Probability theory and stochastic processes (60-XX) 291 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 276 Ordinary differential equations (34-XX) 272 Partial differential equations (35-XX) 267 Computer science (68-XX) 261 Statistics (62-XX) 257 Numerical analysis (65-XX) 249 Biology and other natural sciences (92-XX) 173 Calculus of variations and optimal control; optimization (49-XX) 125 Information and communication theory, circuits (94-XX) 95 Combinatorics (05-XX) 83 Dynamical systems and ergodic theory (37-XX) 63 Operator theory (47-XX) 47 Number theory (11-XX) 43 Commutative algebra (13-XX) 28 Linear and multilinear algebra; matrix theory (15-XX) 28 Real functions (26-XX) 26 Fluid mechanics (76-XX) 24 Algebraic geometry (14-XX) 24 Difference and functional equations (39-XX) 24 Mechanics of deformable solids (74-XX) 23 Mathematical logic and foundations (03-XX) 22 Field theory and polynomials (12-XX) 21 Quantum theory (81-XX) 12 Integral equations (45-XX) 11 Mechanics of particles and systems (70-XX) 11 Statistical mechanics, structure of matter (82-XX) 10 Associative rings and algebras (16-XX) 9 Order, lattices, ordered algebraic structures (06-XX) 9 Special functions (33-XX) 9 Harmonic analysis on Euclidean spaces (42-XX) 9 Global analysis, analysis on manifolds (58-XX) 8 General and overarching topics; collections (00-XX) 8 Differential geometry (53-XX) 8 Geophysics (86-XX) 7 Optics, electromagnetic theory (78-XX) 6 Functions of a complex variable (30-XX) 6 Approximations and expansions (41-XX) 6 Functional analysis (46-XX) 4 Classical thermodynamics, heat transfer (80-XX) 3 Integral transforms, operational calculus (44-XX) 3 Geometry (51-XX) 2 History and biography (01-XX) 2 General algebraic systems (08-XX) 2 Group theory and generalizations (20-XX) 2 General topology (54-XX) 2 Algebraic topology (55-XX) 2 Relativity and gravitational theory (83-XX) 2 Mathematics education (97-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Sequences, series, summability (40-XX) 1 Astronomy and astrophysics (85-XX) |
00-01E
# 00-01E
By - Posted on 08 February 2012
UDC:
536.432.1; 536.44.
PACS:
05.70.Jk
Ab initio study of the vapour-liquid critical point of a symmetrical binary fluid mixture
O.V. Patsahan
M.P. Kozlovskii
R.S. Melnyk
A microscopic approach to the investigation of the behaviour of a symmetrical binary fluid mixture in the vicinity of the vapour-liquid critical point is proposed. It is shown that the problem can be reduced to the calculation of the partition function of a 3D Ising model in an external field. For a square-well symmetrical binary mixture we calculate the parameters of the critical point (critical temperature and critical density) as functions of the microscopic parameters: the parameter $r$ measuring the relative strength of interactions between the particles of dissimilar and similar species and the parameter $\lambda$ measuring the width of the potential well. The obtained results agree well with the ones of computer simulations.
Year:
2000
Pages:
24 |
Radiation resistance is that part of an antenna's feedpoint electrical resistance that is caused by the radiation of electromagnetic waves from the antenna.[1] In radio transmission, a radio transmitter is connected to an antenna. The transmitter generates a radio frequency alternating current which is applied to the antenna, and the antenna radiates the energy in the alternating current as radio waves. Because the antenna is absorbing the energy it is radiating from the transmitter, the antenna's input terminals present a resistance to the current from the transmitter. Unlike other resistances found in electrical circuits, radiation resistance is not due to the opposition (resistivity) of the material of the antenna conductors to electric current; it is a virtual resistance due to the antenna's loss of energy as radio waves.[2][1] The radiation resistance ${\displaystyle R_{\text{R}}}$ can be defined as the value of resistance that would dissipate the same amount of power as radiated as radio waves by the antenna with the antenna input current passing through it.[3][2][4] From Joule's law, it is equal to the total power ${\displaystyle P_{\text{R}}}$ radiated as radio waves by the antenna divided by the square of the rms current ${\displaystyle I_{\text{rms}}}$ into the antenna terminals:[4] ${\displaystyle R_{\text{R}}=P_{\text{R}}/I_{\text{rms}}^{2}}$.[5]
The radiation resistance is determined by the geometry of the antenna and the operating frequency.[6] The total feedpoint resistance at the antenna's terminals is equal to the radiation resistance plus the loss resistance due to ohmic losses in the antenna. In a receiving antenna the radiation resistance represents the source resistance of the antenna, and the portion of the received radio power consumed by the radiation resistance represents radio waves reradiated (scattered) by the antenna.[7][8]
## Cause
Electromagnetic waves are radiated by electric charges when they are accelerated.[1][9] In a transmitting antenna radio waves are generated by time varying electric currents, consisting of electrons accelerating as they flow back and forth in the metal antenna, driven by the electric field due to the oscillating voltage applied to the antenna by the radio transmitter.[10][6] An electromagnetic wave carries momentum away from the electron which emitted it. The cause of radiation resistance is the radiation reaction, the recoil force on the electron when it emits a radio wave photon, which reduces its momentum.[11][12][1] This is called the Abraham-Lorentz force. The recoil force is in a direction opposite to the electric field in the antenna accelerating the electron, reducing the average velocity of the electrons for a given driving voltage, so it acts as a resistance opposing the current.
## Radiation resistance and loss resistance
The radiation resistance is only part of the feedpoint resistance at the antenna terminals. An antenna has other energy losses which appear as additional resistance at the antenna terminals; ohmic resistance of the metal antenna elements, ground losses from currents induced in the ground, and dielectric losses in insulating materials. The total feedpoint resistance ${\displaystyle R_{\text{IN}}}$ is equal to the sum of the radiation resistance ${\displaystyle R_{\text{R}}}$ and loss resistance ${\displaystyle R_{\text{L}}}$
${\displaystyle R_{\text{IN}}=R_{\text{R}}+R_{\text{L}}}$
The power ${\displaystyle P_{\text{IN}}}$ fed to the antenna is split proportionally between these two resistances.[2][13]
${\displaystyle P_{\text{IN}}=I_{\text{IN}}^{2}(R_{\text{R}}+R_{\text{L}})}$
${\displaystyle P_{\text{IN}}=P_{\text{R}}+P_{\text{L}}}$
where
${\displaystyle P_{\text{R}}=I_{\text{IN}}^{2}R_{\text{R}}\quad }$ and ${\displaystyle \quad P_{\text{L}}=I_{\text{IN}}^{2}R_{\text{L}}}$
The power ${\displaystyle P_{\text{R}}}$ consumed by radiation resistance is converted to radio waves, the desired function of the antenna, while the power ${\displaystyle P_{\text{L}}}$ consumed by loss resistance is converted to heat, representing a waste of transmitter power.[2] So for minimum power loss it is desirable that the radiation resistance be much greater than the loss resistance. The ratio of the radiation resistance to the total feedpoint resistance is equal to the efficiency ${\displaystyle \eta }$ of the antenna.
${\displaystyle \eta ={P_{\text{R}} \over P_{\text{IN}}}={R_{\text{R}} \over R_{\text{R}}+R_{\text{L}}}}$
To transfer maximum power to the antenna, the transmitter and feedline must be impedance matched to the antenna. This means the feedline must present to the antenna a resistance equal to the input resistance ${\displaystyle R_{\text{IN}}}$ and a reactance (capacitance or inductance) equal to the opposite of the antenna's reactance. If these impedances are not matched, the antenna will reflect some of the power back toward the transmitter, so not all the power will be radiated. The antenna's radiation resistance is usually the main part of its input resistance, so it determines what impedance matching is necessary and what types of transmission line would be well matched to the antenna.
## Effect of the feedpoint
In a resonant antenna, the current and voltage form standing waves along the length of the antenna element, so the magnitude of the current in the antenna varies sinusoidally along its length. The feedpoint, the place where the feed line from the transmitter is attached, can be located at different points along the antenna element. Since radiation resistance depends on the input current, it varies with the feedpoint.[14] It is lowest for feedpoints located at a point of maximum current (an antinode), and highest for feedpoints located at a point of minimum current, a node, such as at the end of the element (theoretically, in an infinitesimally thin antenna element, radiation resistance is infinite at a node, but the finite thickness of actual antenna elements gives it a high but finite value, on the order of thousands of ohms).[15] The choice of feedpoint is sometimes used as a convenient way to impedance match an antenna to its feed line, by attaching the feedline to the antenna at a point at which its input resistance is equal to the characteristic impedance of the feed line.
In order to give a meaningful value for the antenna efficiency, the radiation resistance and loss resistance must be referred to the same point on the antenna, usually the input terminals.[16][17] Radiation resistance is usually calculated with respect to the maximum current ${\displaystyle I_{\text{0}}}$ in the antenna.[14] If the antenna is fed at a point of maximum current, as in the common center-fed half-wave dipole or base-fed quarter-wave monopole, that value ${\displaystyle R_{\text{R0}}}$ is the radiation resistance. However if the antenna is fed at another point, the equivalent radiation resistance at that point ${\displaystyle R_{\text{R1}}}$ can easily be calculated from the ratio of antenna currents[15][17]
${\displaystyle P_{\text{R}}=I_{\text{0}}^{2}R_{\text{R0}}=I_{\text{1}}^{2}R_{\text{R1}}}$
${\displaystyle R_{\text{R1}}={\Big (}{I_{\text{0}} \over I_{\text{1}}}{\Big )}^{2}R_{\text{R0}}}$
## Radiation resistance of common antennas
ohms
Source
Center-fed half-wave dipole 73.1[18] Kraus 1988:227
Balanis 2005:216
Short dipole of length ${\displaystyle \lambda /50 ${\displaystyle 20\pi ^{2}{\Big (}{L \over \lambda }{\Big )}^{2}}$ Kraus 1988:216
Balanis 2005:165,215
Base-fed quarter-wave monopole
over perfectly conducting ground
36.5 Balanis 2005:217
Stutzman & Thiele 2012:80
Short monopole of length ${\displaystyle L\ll \lambda /4}$
over perfectly conducting ground
${\displaystyle 40\pi ^{2}{\Big (}{L \over \lambda }{\Big )}^{2}}$ Stutzman & Thiele 2012:78-80
Resonant loop antenna, 1 ${\displaystyle \lambda }$ circumference ~100 Weston 2017:15
Schmitt 2002:236
Small loop of area ${\displaystyle A}$ with ${\displaystyle N}$ turns
(circumference ${\displaystyle \ll \lambda /3}$)
${\displaystyle 320\pi ^{4}{\Big (}{NA \over \lambda ^{2}}{\Big )}^{2}}$ Kraus 1988:251
Balanis 2005:238
Small loop of area ${\displaystyle A}$ with ${\displaystyle N}$ turns
on a ferrite core of effective relative permeability ${\displaystyle \mu _{\text{eff}}}$
${\displaystyle 320\pi ^{4}{\Big (}{\mu _{\text{eff}}NA \over \lambda ^{2}}{\Big )}^{2}}$ Kraus 1988:259
Milligan 2005:260
The above figures assume the antenna is made of thin conductors and that the dipole antennas are sufficiently far away from the ground or grounded structures.
The half-wave dipole's radiation resistance of 73 ohms is near enough to the characteristic impedance of common 50 and 75 ohm coaxial cable that it can usually be fed directly without need of an impedance matching network. This is one reason for the wide use of the half wave dipole as a driven element in antennas.[19]
### Relationship of monopoles and dipoles
The radiation resistance of a monopole antenna created by replacing one side of a dipole antenna by a perpendicular ground plane is one-half of the resistance of the original dipole antenna. This is because the monopole radiates only into half the space, the space above the plane, so the radiation pattern is identical to half of the dipole pattern and therefore with the same input current it radiates only half the power.[20] This is not obvious from the formulas in the table because the derived monopole antenna is only half the length of the original dipole antenna. This can be shown by calculating the radiation resistance of a short monopole of half the length of a dipole
${\displaystyle R_{\text{R}}=40\pi ^{2}{\Big (}{L/2 \over \lambda }{\Big )}^{2}=10\pi ^{2}{\Big (}{L \over \lambda }{\Big )}^{2}\qquad }$ (monopole of length L/2)
Comparing this to the formula for the short dipole shows the monopole has half the radiation resistance
${\displaystyle R_{\text{R}}=20\pi ^{2}{\Big (}{L \over \lambda }{\Big )}^{2}\qquad \qquad \qquad }$ (dipole of length L)
## Calculation
Calculating the radiation resistance of an antenna directly from the reaction force on the electrons is very complicated, and presents conceptual difficulties in accounting for the self-force of the electron.[1] Radiation resistance is instead calculated by computing the far-field radiation pattern of the antenna, the power flux (Poynting vector) at each angle, for a given antenna current.[21] This is integrated over a sphere enclosing the antenna to give the total power ${\displaystyle P_{\text{R}}}$ radiated by the antenna. Then the radiation resistance is calculated from the power by conservation of energy, as the resistance the antenna must present to the input current to absorb the radiated power from the transmitter, using Joule's law ${\displaystyle R_{\text{R}}=P_{\text{R}}/I_{\text{rms}}^{2}}$[5]
## Small antennas
Electrically short antennas, antennas with a length much less than a wavelength, make poor transmitting antennas, as they cannot be fed efficiently due to their low radiation resistance. As can be seen in the above table, for antennas shorter than their fundamental resonant length (${\displaystyle \lambda /2}$ for a dipole antenna, ${\displaystyle \lambda /4}$ for a monopole, circumference of ${\displaystyle \lambda }$ for a loop) the radiation resistance decreases with the square of their length.[22] As the length is decreased the loss resistance, which is in series with the radiation resistance, makes up a larger fraction of the feedpoint resistance, so it consumes a larger fraction of the transmitter power, causing the efficiency of the antenna to decrease.
For example, navies use radio waves of about 15 - 30 kHz in the very low frequency (VLF) band to communicate with submerged submarines. A 15 kHz radio wave has a wavelength of 20 km. The powerful naval shore VLF transmitters which transmit to submarines use large monopole mast antennas which are limited by construction costs to heights of about 300 metres (980 ft). Although these are tall antennas by ordinary standards, at 15 kHz this is still only about .015 wavelength high, so VLF antennas are electrically short. From the table a .015${\displaystyle \lambda }$ monopole antenna has a radiation resistance of about 0.09 ohm. It is extremely difficult to reduce the loss resistances of the antenna to this level. Since the ohmic resistance of the huge ground system and loading coil cannot be made lower than about 0.5 ohm, the efficiency of a simple vertical antenna is below 20%, so more than 80% of the transmitter power is lost in the ground resistance. To increase the radiation resistance, VLF transmitters use huge capacitively top-loaded antennas such as umbrella antennas and flattop antennas, in which an aerial network of horizontal wires is attached to the top of the vertical radiator to make a 'capacitor plate' to ground, to increase the current in the vertical radiator. However this can only increase the efficiency to 50 - 70% at most.
Small receiving antennas, such as the ferrite loopstick antennas used in AM radios, also have low radiation resistance and thus produce very low output. However at frequencies below about 30 MHz this is not such a problem, since the weak signal from the antenna can simply be amplified in the receiver.
At frequencies below 1 MHz the size of ordinary electrical circuits is so much smaller than the wavelength, that when considered as antennas they radiate an insignificant fraction of the power in them as radio waves. This explains why electrical circuits can be used with alternating current without losing energy as radio waves.
## Definition of variables
Symbol Unit Definition
${\displaystyle \lambda }$ meter Wavelength of radio waves
${\displaystyle \pi }$ none Constant = 3.14159
${\displaystyle \mu _{\text{eff}}}$ none Effective relative permeability of ferrite rod in antenna
${\displaystyle A}$ meter2 Cross sectional area of loop antenna
${\displaystyle f}$ hertz Frequency of radio waves
${\displaystyle I_{\text{IN}}}$ ampere RMS current into antenna terminals
${\displaystyle I_{\text{0}}}$ ampere Maximum RMS current in antenna element
${\displaystyle I_{\text{1}}}$ ampere RMS current at an arbitrary point in antenna element
${\displaystyle L}$ meter Length of antenna
${\displaystyle N}$ none Number of wire turns in loop antenna
${\displaystyle P_{\text{IN}}}$ watt Electric power delivered to antenna terminals
${\displaystyle P_{\text{R}}}$ watt Power radiated as radio waves by antenna
${\displaystyle P_{\text{L}}}$ watt Power consumed in loss resistances of antenna
${\displaystyle R_{\text{R}}}$ ohm Radiation resistance of antenna
${\displaystyle R_{\text{L}}}$ ohm Equivalent loss resistance of antenna at input terminals
${\displaystyle R_{\text{IN}}}$ ohm Input resistance of antenna
${\displaystyle R_{\text{R0}}}$ ohm Radiation resistance at point of maximum current in antenna
${\displaystyle R_{\text{R1}}}$ ohm Radiation resistance at arbitrary point in antenna
## Notes
1. Feynman, Leighton & Sands 1963, p. 32-1.
2. ^ a b c d Straw, R. Dean, Ed. (2000). The ARRL Antenna Book, 19th Ed. American Radio Relay League, Inc. p. 2.2. ISBN 0872598179.
3. ^ "Radiation Resistance". ATIS Telecom Glossary. Alliance for Telecommunications Industry Solutions. 2019. Retrieved 2020.
4. ^ a b Yarman, Binboga S. (2008). Design of Ultra Wideband Antenna Matching Networks. Springer Science and Business Media. p. 22. ISBN 9781402084171.
5. ^ a b Some derivations use the peak sinusoidal current ${\displaystyle I_{\text{p}}={\sqrt {2}}I_{\text{rms}}}$ instead of the rms current and the equivalent version of Joule's law: ${\displaystyle R_{\text{R}}=2P_{\text{R}}/I_{\text{p}}^{2}}$
6. ^ a b Balanis 2005, p. 10-11.
7. ^ a b c d Kraus 1988, p. 32.
8. ^ a b c d Balanis 2005, p. 83-85.
9. ^ Kraus 1988, p. 50.
10. ^ Serway, Raymond; Faughn, Jerry; Vuille, Chris (2008). College Physics, 8th Ed. Cengage Learning. p. 714. ISBN 978-0495386933.
11. ^ van Holten, Theo (2016). The Atomic World Spooky? It Ain't Necessarily So!. Springer. pp. 272-274. ISBN 9789462392342.
12. ^ McDonald, Kirk T. (29 May 2017). "The Radiation-Reaction Force and the Radiation Resistance of Small Antennas" (PDF). Joseph Henry Laboratory, Princeton University. Retrieved 2020. Cite journal requires |journal= (help)
13. ^ Ellingson, Steven W. (2016). Radio Systems Engineering. Cambridge University Press. ISBN 9781316785164.
14. ^ a b Kraus 1988, p. 227-228.
15. ^ a b Kraus 1988, p. 228.
16. ^ Rauch, Tom (2004). "Radiation resistance". W8JI personal website. Tom Rauch. Retrieved 2020.
17. ^ a b Balanis 2005, p. 179.
18. ^ Due to end effects a finite thickness dipole is not resonant at a length of one-half wavelength ${\displaystyle 0.5\lambda }$ but has inductive reactance. A typical thin dipole is actually resonant (has no reactance) at a slightly shorter length of around ${\displaystyle 0.475\lambda }$, at which its radiation resistance is about 67 ohms.Wallace, Richard; Andreasson, Krister (2005). Introduction to RF and Microwave Passive Components. Artech House. p. 77. ISBN 9781630810092.
19. ^ Huang, Yi; Boyle, Kevin (2008). Antennas: From Theory to Practice. John Wiley and Sons. p. 135. ISBN 9780470772928.
20. ^ Stutzman & Thiele 2012, p. 78-80.
21. ^ Balanis 2005, p. 154.
22. ^ Schmitt 2002, p. 232. |
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Math Central Quandaries & Queries
Question from Rahul: I want to know about limit proofs of composite functions. Like limit of log of a function equals log of limit of the function
Hi Rahul,
I think the result you want is
If $f$ is continuous at $b$ and $\lim_{x \rightarrow a} \; g(x) = b$ then
$\lim_{x \to a} f(g(x)) = f(b) = f\left(\lim_{x \to a} \; g(x) \right)$
Penny
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. |
# Defining Lookups
## What are Lookups?
Lookups define the corresponding ajax views used by the auto-completion fields and widgets. They take in the current request and return the JSON needed by the jQuery auto-complete plugin.
## Defining a Lookup
django-selectable uses a registration pattern similar to the Django admin. Lookups should be defined in a lookups.py in your application's module. Once defined you must register in with django-selectable. All lookups must extend from selectable.base.LookupBase which defines the API for every lookup.
from selectable.base import LookupBase
from selectable.registry import registry
class MyLookup(LookupBase):
def get_query(self, request, term):
data = ['Foo', 'Bar']
return filter(lambda x: x.startswith(term), data)
registry.register(MyLookup)
## Lookups Based on Models
Perhaps the most common use case is to define a lookup based on a given Django model. For this you can extend selectable.base.ModelLookup. To extend ModelLookup you should set two class attributes: model and search_field.
The syntax for search_field is the same as the Django field lookup syntax. You may optionally define a third class attribute filters which is a dictionary of filters to be applied to the model queryset. The keys should be a string defining a field lookup and the value should be the value for the field lookup. |
# Orthogonality In Software Design
Achieving orthogonality in the design of software is an investment into the future of the resulting software system because orthogonality increases its long-term fitness in terms of quality attributes such as flexibility, testability, and extensibility.
In this blog post, we’ll have a look at the concept of orthogonality in context of software engineering and how it can help us improve the quality attributes of the software we write. We’ll then explore two of the concepts that contribute to building orthogonal software systems.
# Orthogonality And Its Benefits
Orthogonality is a term borrowed from geometry, where two lines are orthogonal if they meet at right angles. Within the realms of software engineering, the term is used to refer to a kind of independence between the components in a software system. For example, say you have three modules and need to update one of them – in this case, the module is orthogonal to the others if a change made to it does not ripple through the other two. On the other hand, if you make a change to one module and you end up having to make changes in the other two modules as well, then the modules are not orthogonal.
Building orthogonal software modules can be tricky, but the effort is well worth it because orthogonality entails some important benefits for the quality attributes of the resulting software system:
• To build orthogonal modules, each module has to have a single, well-defined responsibility and needs to encapsulate the functionality required to fulfil this responsibility behind a neat API. This promotes reuse – modules can be easily combined in new and innovative ways if there is no responsibility (and, hence, functionality) overlap, and developers feel better about reusing existing modules if they know they don’t have to deal with a module’s internal complexity because it’s hidden behind an API.
• Reusing modules, in turn, facilitates adhering to the DRY principle (Don’t Repeat Yourself).
• In orthogonal modules, changes are localized. To achieve a certain goal, only one module has to be modified, and those modifications will not break things or otherwise necessitate change in other modules. Consequently, the system’s overall flexibility to adapt to new or changed requirements increases, and so does its extensibility, as developers can extend functionality with higher confidence.
• As an extension of the above, bugs are also isolated and can be fixed with less effort – and with more confidence that one fix will not magically introduce other bugs in completely unrelated parts of the system (a scenario that any developer will dread).
• It’s easier to write tests for and run tests on orthogonal modules, so the testability of the system as a whole increases.
Low coupling and high cohesion are both crucial to achieving orthogonality in a software system, so we’re going to cover the basics of both in the upcoming sections. Armed with that knowledge, we can then turn our attention to specific guidelines and principles on how to achieve both low coupling and high cohesion. This will be the topic of the next blog post.
# Low Coupling
The term low coupling can be found in pretty much any discussion on software design (often in combination with the term high cohesion). Coupling between classes or modules is a measure for how interconnected those classes or modules are. High coupling is considered bad because highly coupled classes or modules have to have a lot of knowledge about the internals and/or return values of the other, making it very difficult to change one without also having to change the other. To illustrate this, let’s have a look at the following piece of Java code (we’ll see an improved version of this a bit further down the line):
public class Customer1 {
private final long id;
private final String name;
private final List<Invoice> invoices;
public Customer1(long id, String name) {
this.id = id;
this.name = name;
this.invoices = new ArrayList<>();
initInvoices();
}
public void initInvoices() {
try {
var invoiceElements = line.split(",");
var date = invoiceElements[0];
var subject = invoiceElements[1];
var amount = Double.parseDouble(invoiceElements[2]);
});
} catch (FileNotFoundException e) {
// Exception handling
}
}
// Truncated
}
(This would become even uglier if you started to mix in a bit of error handling, like what to do if an invoice’s amount was not stored in a string that can be parsed into a double – but the code is sufficiently unelegant to drive home the point.)
Here, the Customer class uses an InvoiceReader class that returns CSV-formatted strings containing invoice data (a date, a subject, and an amount), which the Customer class uses to initialize some Invoice objects. To do so, Customer needs to know that the strings delivered by InvoiceReader are CSV-formatted, and which piece of information sits where in each line.
The way the invoice initialization works here is a nice example for very poor separation of concerns: The Customer class handles a task it shouldn’t be responsible for, and in doing so, becomes very tightly coupled to the InvoiceReader class as well as the values it returns. This is quite nasty because of the following reasons:
• The code is somewhat harder to read and understand because you wouldn’t expect the Customer class to encapsulate knowledge related to the storage format of invoices
• Changes made to InvoiceReader or the invoice format will ripple through the Customer class, necessitating the latter be modified, too
• Because the knowledge about the underlying invoice storage format and how to create Invoice objects is encapsulated in the Customer class, it cannot be reused separately
• An extension of the previous point, and maybe even more of an issue: The knowledge and its corresponding functionality cannot be tested separately
Therefore, building classes or modules to be loosely coupled benefits the following quality attributes:
• Flexibility, maintainability: Modifications made to fix bugs or implement a new feature are localized
• Testability: It’s a lot easier (and a lot more fun) to set up nice tests for decoupled classes or modules
• Reusability: Classes and modules can be easily combined in new and innovative ways
An additional benefit arises from the fact that designing classes and modules to be decoupled forces you to think about responsibility (essentially: “Which piece of functionality should reasonably go where?”) and proper separation of concerns, which further improves the system’s quality attributes. And, if you get the responsibilities right, you’ll also end up with a system in which classes or modules all have high cohesion That’s why the terms low coupling and high cohesion are so often used together – one influences the other. (Interestingly, low coupling and high cohesion don’t necessarily seem to be logical consequences of one another – you could build highly cohesive modules and still couple them together very tightly, and vice-versa, you could build loosely coupled modules that still have terribly cohesion. Therefore, both should be treated as individual goals to achieve.)
# High Cohesion
Cohesion is a measure of how closely pieces of code that contribute to the same goal relate to each other. For example, if a class or module contains code that accomplishes a lot of different things, then this class or module is said to have low cohesion. On the other hand, if all the code in a class or module is dedicated to the same functional goal, then that class or module has high cohesion. For example, the Customer class shown in the example above has low cohesion because it does something it should not be responsible for (parsing invoices from an underlying storage format, and encapsulating the knowledge for how to do so). Let’s have a look at an improved version of the previous example:
public class Customer2 {
private final long id;
private final String name;
private final List<Invoice> invoices;
public Customer2(long id, String name) {
this.id = id;
this.name = name;
}
// Truncated
}
The InvoiceReader class now directly returns a list of Invoice objects. Here’s how it looks:
public class InvoiceReader2 {
}
}
}
public List<Invoice> getInvoicesByCustomerID(long customerID) {
try {
return Files.lines(Paths.get(String.format("invoices/%d.csv", customerID)))
.map(invoiceString -> {
var invoiceData = invoiceString.split(",");
return new Invoice(invoiceData[0], invoiceData[1], Double.parseDouble(invoiceData[2]));
})
.collect(Collectors.toList());
} catch (IOException e) {
// Do some logging, or maybe rethrow exception to notify caller...
}
return Collections.emptyList();
}
}
This code is much easier to read and understand because it conforms to the expectation you’d probably have about where to find knowledge related to reading invoice data – in a class called InvoiceReader. The InvoiceReader class has high cohesion because its entire functionality is dedicated to only one thing, and so the effects of changes to that thing are limited to only that class – if we decided to, say, read invoice data from a database or modify the invoice format, the ensuing changes would be restricted to the InvoiceReader class, and the Customer class consuming it would be entirely indifferent to them. Hence, Customer and InvoiceReader are now loosely coupled. This increases the flexibility and maintainability of InvoiceReader, and also increases its functionality’s reusability. On top of that, testability is also improved – because the invoice reading functionality is now nicely encapsulated, it can be tested separately using module-level (or unit-level) tests, and such tests tend to be a lot easier to specify and implement than integration-level tests.
# Wrap-Up And Upcoming Topics
Building loosely coupled software modules with high cohesion contributes to achieving orthogonality and increases the quality attributes of the resulting software system. That’s great for developers since working on such a system is simply a lot more fun; it’s great for management since their developers will be able to spend more of their time building the things that deliver value to clients rather than fixing bugs; and it’s great for clients because they’ll get more valuable features and a more reliable system to interact with.
To achieve low coupling and high cohesion, it’s sometimes helpful to have a set of guidelines or principles that make flaws in existing classes or modules more obvious (or that help prevent them from occurring in the first place). It’s the job of the next blog post to provide you with just such guidelines – they will help you ask the right questions about some class or module you’ve just written or you were given to work with, and implementing the answers to those questions will help you reduce coupling and increase cohesion for the involved classes or modules. |
# How to get 3d view grid properties?
I am trying to store the grid scale, but keep getting an error for being in the text editor view.
I have tried
bpy.context.area.type = "VIEW_3D"
print(bpy.context.space_data.grid_scale)
but am getting an attribute error for "SpaceTextEditor" not having a "grid_scale". However, the space does change to the 3d view, so the first line is going through.
Any help is appreciated.
Your code works for me in 2.69, but there might be timing issues (grid_scale accessed but the view has not finished changing).
To get all 3D Views' grid_scale as a list, use:
grid_scales = [area.spaces[0].grid_scale for area in bpy.context.screen.areas if area.type == 'VIEW_3D'] |
# Exact functors on perverse coherent sheaves
Koppensteiner, C
September 2015
## Journal:
COMPOSITIO MATHEMATICA
## Last Updated:
2020-07-12T22:08:21.707+01:00
9
151
## DOI:
10.1112/S0010437X15007265
1688-1696
## abstract:
Inspired by symplectic geometry and a microlocal characterizations of
perverse (constructible) sheaves we consider an alternative definition of
perverse coherent sheaves. We show that a coherent sheaf is perverse if and
only if $R\Gamma_Z(\mathcal{F})$ is concentrated in degree 0 for special
subvarieties Z of X. These subvarieties Z are analogs of Lagrangians in the
symplectic case.
1037510 |
# Symbolic FindInstance
I'm wondering if there is any way of doing something like the following: FindInstance[x < y, {x}, Reals] and getting, as a possible return value, {x -> y - 1}.
Currently FindInstance will complain that y is a non-constant expression..
-
I am assuming you would not be satisfied with FindInstance[x - y < 0, {x, y}, Reals] (* {x->0,y->1} *) – chris Oct 26 '12 at 9:59
At this point, I think that I will just implement a function which uses CylindricalDecomposition. If I do successfully do that, I'll post it as an answer here. – jnhnum1 Oct 27 '12 at 16:33 |
# Denoting sets (permutations,tuples?),
I'll use and example to explain my question:
Example: Let there be two distinct types $$a$$ and $$b$$ and each type has equal number of sets.
• There are two sets of type $$a$$: $$\{1,2\}$$ and $$\{3,4\}$$.
• There are two sets of type $$b$$: $$\{2,4\}$$ and $$\{1,3,4\}$$.
I want to work with the following objects: $$\{1,2,2,4\}$$, $$\{1,1,2,3,4\}$$, $$\{2,3,4,4\}$$ and $$\{1,3,3,4,4\}$$.
One important property: $$\{1,2\}=\{1,1,2,2\}$$, $$\{1,2,2,4\}=\{1,2,2,2,4,4\}$$, etc.
So basically, each of these objects is constructed by conjoining exactly one set of type $$a$$ with exactly one set of type $$b$$ and the elements can have repetitions. Also, multiplying the elements of one set before conjoining it with another doesn't affect the object; e.g. $$\{1,2\}=\{1,1,2,2\}$$ is because RHS is just two copies of the LHS conjoined together, and $$\{1,2,2,4\}=\{1,2,2,2,4,4\}$$ because RHS is two copies of $$\{2,4\}$$ - a set of type $$b$$ conjoined with one copy of $$\{1,2\}$$ - a set of type $$b$$.
So my questions are:
1. Is there a name for these objects? They're not sets, not exactly tuples (but the cardinality of distinct objects is the same as that of tuples).
2. More importantly, how do you define a general rule to explain what they are?
3. Ultimately, I want to compare these objects. Specifically, I want to check whether one object is a subset (I know it's not a set but bear with me) of another. So I also want set relations to be meaningful, i.e. $$\{1,2,2,4\}\subset\{1,1,2,2,3,4\}$$ etc.
I hope it makes enough sense that somebody will help me. Thank you!
• The definition of equality is not very clear here. If repetition of elements doesn't matter at all, they're just sets and there's no reason to consider repetitions. But maybe that's not what you have in mind. Is there any example of two of these objects with the same underlying set that are not equal? – Jair Taylor May 21 at 22:31
• A copy of a set of a certain type doesn't matter but if an element comes from two different sets, then repetition matters. So $\{1,2\}=\{1,1,2,2\}$ because RHS is two copies of the LHS (a set of some type) but $\{1,2,2,4\}\neq\{1,2,4\}$ because RHS is not constructed from sets of any type. – Curious George May 21 at 22:39
• You're thinking of multisets, I think. – Asaf Karagila May 22 at 7:48
• Having repetitions etc. points to using multisets but there's also the point of taking exactly one set of each type to construct the object, so it's kind of like a tuple without order. – Curious George May 22 at 12:40 |
# Recover NLFSR description from given output
Given Fibonacci's Nonlinear Feedback Shift Register output, how can I recover the description (nonlinear feedback function and current seed)?
The clean output of consecutive and unmodified is large and the generator is available to ask for next values. The source code of PRNG is not available, the seed is unknown.
For the linear case there is already known Berlekamp-Massey (or Reed-Sloane) algorithm yielding the shortest polynomial. Is there a similar method for nonlinear case?
• I read it here, no the internals are unknown, it seems a bit complicated... But with all internals present recovering the seed is not a problem.
– Evil
Jan 22 '17 at 18:02
Normally we evaluate PRNGs under the model where the source code is available and fully known, and where the output is a deterministic function of the seed. This leaves multiple possibilities for how to build a NLFSR: we can have a fixed feedback function (hardcoded in the source code), and use the seed to specify the initial state; or we can use part of the seed to generate a feedback function and use the remainder of the seed to specify the initial state.
If the feedback function $f$ is known, it is trivial to predict the next outputs of the NLFSR from the past $n$ bits, where $n$ denotes the width of the state. Also, it is often possible to predict past output, but this depends on the feedback function and on $n$.
If $f$ is not known but derived from part of the seed, then the procedure depends on the details of how it is derived. There are $2^{2^n}$ possible feedback functions $f:\{0,1\}^n \to \{0,1\}$. If we use $2^n$ bits from the seed so we can sample uniformly at random from the set of all functions (i.e., all $2^{2^n}$ functions are possible and equally likely), then the PRNG is not very useful in practice. After seeing about $O(2^{n/2})$ bits of output, you can usually distinguish the output of the PRNG from true-randomness and predict future outputs, because (by the birthday paradox) it is likely that the internal state will repeat at some point in there and the generator will enter a cycle (of length at most $O(2^{n/2})$). Of course this design requires more than $2^n$ bits of true randomness as input, and can only produce $O(2^{n/2})$ bits of output before its output is recognizable, so this is a fairly crummy PRNG; you've basically wasted the randomness, and you might as well have just used the true-randomness directly without a PRNG.
Alternatively, perhaps you had in mind that part of the seed is used to select a feedback function $f$, via some algorithmic process -- but not all feedback functions are possible or equally likely. Then the ability to distinguish, predict future outputs, or recover the seed is dependent upon the details of that algorithmic process. However, we can say this much. The output of the PRNG can be distinguished from truly-random after $O(2^{n/2})$ outputs, for reasons sketched above (due to the birthday paradox).
The period of a NLFSR is determined by the function $F:\{0,1\}^n \to \{0,1\}^n$, defined by
$$F(x_1,\dots,x_n) = (x_2,\dots,x_n,f(x_1,\dots,x_n)).$$
Let me mention some typical cases.
First, if $f$ satisfies $f(0,x_2,\dots,x_n)=f(1,x_2,\dots,x_n) \oplus 1$ for all $x_2,\dots,x_n$, then $F$ will be bijective, the period of the NLFSR will typically be about $2^n$ (if $f$ has no other regularities), and given about $2^n$ bits of output you'll be able to predict all future outputs (since it cycles) or recover the original seed. Also, you'll be able to distinguish the output of the NLFSR from truly random with about $O(2^{n/2})$ bits of output (by the birthday paradox, a true random stream will have some $n$-bit value repeat twice within that range, but that'll never happen in the output from this type of NLFSR).
Second, if $f$ has no regularities or patterns and is basically random, then $F$ will fail to be bijective, and the period will typically be about $2^{n/2}$. So, after $O(2^{n/2})$ bits of output you'll be able to predict all future outputs (since it cycles), and you'll be able to distinguish the output of the NLFSR from truly random with about $O(2^{n/2})$ bits of output (a true-random stream won't cycle but this NLFSR will).
If the feedback function $f$ is chosen randomly from a small list of possibilities (e.g., the ones listed in the paper you cited), then it is easy to recognize output from such a PRNG: just try each possibility in turn, and use the method described for the case where $f$ is known. If there are $m$ possibilities for $f$, then $n+\lg m+O(1)$ bits of output suffice to infer which possibility is correct and verify that the output is consistent with such a NLFSR with high probability.
What this fails to account for is other attacks, which try to take advantage of structure in $f$. For instance, suppose $f$ is chosen to have small circuit size. Then it may be possible to recover $f$ with many fewer than $2^n$ bits of input: each consecutive $n+1$ bits of output gives us a known input-output pair $(x,y)$ for $f$ (namely, $x =$ the first $n$ bits, and $y =$ the $n+1$st bit), and given many such pairs, we can try to find a function $f$ that is consistent with all the pairs and has low circuit complexity. That might be enough to uniquely determine the function $f$; if so, it may or may not be possible to recover it explicitly within a reasonable amount of computation (e.g., with BDDs, Karnaugh maps, or other methods).
Since the question is rather vague, that makes it rather hard to assess the possibilities for the latter kind of method. So, I'll just end with a disclaimer that the analysis above might overestimate the number of bits of output needed to recognize/detect this kind of LFSR.
NLFSRs are intentionally designed/ used for cryptography to be highly "secure" and resistant to "weakness" aka finding their generating parameters except through brute force. this is essentially the process of "cryptoanalysis/ decryption of a PRNG" which is studied extensively in cryptography. but there is at least one case study for consideration. the KeeLoq code is based on a NLFSR and is used in a lot of automotive keylock systems, and therefore has been subject to heavy academic analysis/ experimental attacks. while some weaknesses have been found nevertheless it looks like most of the attacks are basically brute force, although there is some use of FPGAs to streamline the hardware attack. wikipedia notes Nicolas Courtois attacked KeeLoq using sliding and algebraic methods. see also [1] |
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# Calculating Profitability Index in Excel
Profitability index is an important measure in project finance to decide whether to invest in a project or not. It is calculated as the ratio of present value of a project cash flows and the initial investment.
$Profitability\ Index = \frac{PV\ of\ Future\ Cash\ Flows}{Initial\ Investment}$
If the profitability index is greater than 1, the project is accepted, and if it is less than 1, the project is rejected.
Let’s see how profitability index can be calculated in excel. Let us say that we are examining a project, which requires an initial investment of $10,000, and after the will give us cash flow of$3,000, $4,000,$2,000, 41,500, and \$1,800 in the next five years.
To calculate the profitability index:
Step 1: Assume a required rate of return, or cost of capital for the project. Let’s say the cost of capital is 10%.
Step 2: Calculate the present value of all future cash flows. You can use the PV() function in excel for this calculation.
Step 3: Take the total of PV of all future cash flows. In our example, the total is 9677.87.
Step 4: Calculate profitability Index as follows:
Profitability Index = 9677.87/10,000 = 0.97
Since the profitability index is less than one, this project should not be accepted. The cash flows and calculations are shown below: |
# Thermal Conductivity Paper Database
## Recommended Papers for: Takeshi Okutani
Total Papers Found: 3
#### Thermophysical properties of Zr–Cu–Al metallic glasses during crystallization
Zr-Cu-Al metallic glasses were formulated with differing compositions to determine their thermophysical properties. It was determined that by decreasing the zirconium content of the metallic glass, the thermal conductivity can be increased. The maximum value of thermal conductivity was found for the metallic glass with ...
Author(s):
#### Thermal conductivity measurement of molten indium antimonide in short-duration microgravity
In this study, a sensor measured the thermal conductivity of molten indium antimonide (InSb) using the transient plane source (TPS) method. The measurements were taken during a 10 m short-duration microgravity drop. Results showed that the thermal conductivities of molten InSb were higher in normal gravity ...
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#### Development of Sensor for Molten Metal, and the Thermal Conductivity Measurement of Molten Bismuth and Tin
The authors measured the thermal conductivity of a couple molten metals, namely bismuth and tin. This was accomplished by using a molybdenum sensor placed in between two aluminum nitride plates (to protect the molybdenum sensor from the high temperature of the molten metals. The thermal ...
Author(s): |
# an example of a representation
This is an example given in the representation theory book:
Let $\rho : S_3 → GL_2(\mathbb C)\qquad$ be specified on the generators $(1 2)$ and $(1 2 3)$
by $\rho _{(1 2)} =$ $\begin{bmatrix} -1 & -1 \\ 0 & 1 \\ \end{bmatrix}$,
$\rho_{(1 2 3)} =\begin{bmatrix} -1 & -1 \\ 1 & 0 \\ \end{bmatrix}$
Claim: $\rho _{(1 2)}$ and $\rho _{(1 23)}$ do not have a common eigenvector.
Indeed, direct computation reveals $\rho _{(1 2)}$ has eigenvalues $1$ and $−1$ with $V_{−1} = \mathbb C e_1$ and $V_1 =\mathbb C \begin{bmatrix} -1 \\ 2 \\ \end{bmatrix}$
First of all, how do they compute $V_{−1}$ and $V_{1}$? Also, what do $V_{−1}$ and $V_{1}$ specify?
Clearly $e_1$ is not an eigenvector of $\rho _{(1 2 3)}$ as $\rho _{(1 2 3)}\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} =\begin{bmatrix} -1 \\ 1 \\ \end{bmatrix}$ . Also, $\rho _{(1 2 3)}\begin{bmatrix} -1 \\ 2 \\ \end{bmatrix} = \begin{bmatrix} -1 \\ -1 \\ \end{bmatrix}$ , so (−1, 2) is not an eigenvector of $\rho _{(1 2 3)}$. Thus $\rho _{(1 2 )}$ and $\rho _{(1 2 3)}$ have no common eigenvector, which implies that $\rho$ is irreducible.
Secondly what is the reasoning made above while finding eigenvectors?
Also when I try to compute eigenvalues for $\rho _{(1 2 3)}$ , I get $\lambda^2+\lambda +1=0$ which does not have real roots
• Do you know how to compute eigenvectors? Because, they just computed them. Also, a matrix can have real valued eigenvectors even though the eigenvalues are complex. – Mathematician 42 Feb 24 '18 at 23:12
The characteristic polynomial of $A=\begin{pmatrix}-1&-1\\0&1\end{pmatrix}$ is $\det\begin{pmatrix}-1-X&-1\\0&1-X\end{pmatrix}=-(1+X)(1-X)=X^2-1$. Hence the eigenvalues are $\pm 1$. The space $V_1$ is $V_1=N(A-1\cdot Id)$ by definition, that is the solutions to the system $(A-Id)X=0$. Easy calculation shows that $V_1=\text{Span}[(-1,2)]$. Similarly, $V_{-1}=\text{Span}[(1,0)]$.
Let $B=\begin{pmatrix} -1&1\\1&0 \end{pmatrix}$. Simple calculations show that neither $(-1,2)$ or $(1,0)$ are eigenvectors of $B$. Thus $A$ and $B$ share no eigenvectors. No need to compute the eigenvectors of $B$ explicitly!
• Oh,thanks! I didn't think that by " the notation $\mathbb C$ " they were actually intending to mean spanning. – Leyla Alkan Feb 24 '18 at 23:51 |
# What is a good way to take fractional powers of a matrix in MATLAB?
I am working on a problem that involves taking fractional powers of particular matrices.
For the matrix A with 2 on the main diagonal and -1 on the sub and super diagonal (the finite difference matrix for the Dirichlet Laplacian in 1D), there was basically no difference between using MATLAB's sqrtm(A) function and diagonalizing the matrix ([V,D]=eig(A), A = V\D*V) and computing V\D^(1/2)*V even if the matrix was as large as 1000x1000. This made me think I could compute other fractional powers V\D^(1/n)*V that MATLAB does not have built in functions for.
However when I consider the matrix A for the Dirichlet Laplacian in 2D, there is a significant difference between MATLAB's sqrtm function and the diagonalization procedure I used above even if the matrix was as small as 100x100. In particular, MATLAB's sqrtm function is better and my results don't make any sense if I do it the other way.
I am wondering if anyone can explain why this happens and if there is a way to fix it so that I can compute fractional powers of a large matrix in MATLAB.
• You can directly use something like A^(2/3). Is there a reason this is undesirable? – Doug Lipinski Nov 19 '14 at 12:15
• It is not undesirable, I just didn't know it could be this easy! Someone has also informed me that expm(1/n*logm(A)) works too. Both of these are giving me matrices whose nth power differs from A by basically machine precision. Thanks again, if you submit that as an answer I will accept it. – RHP Nov 19 '14 at 17:42
MATLAB has the ability to compute fractional powers of matrices by using the normal ^ operator which calls the mpower function when either of the arguments is a matrix. You can read a discussion of when fractional powers of matrices can be computed here.
Your original issues may stem from the fact that you have mixed up the formula to compute matrices raised to a fractional power. The matrix should be diagonalized as $A = PDP^{-1}$. Then $A^\alpha = PD^\alpha P^{-1}$.
In MATLAB this can be done as follows (your code is a bit off):
[V,D]=eig(A);
Aalph = V*(D.^alph)/V;
Based on a quick test, it seems that this is exactly the method used by MATLAB inside mpower. The following code returns exactly 0 (no differences due to finite precision) on my machine.
A = rand(1000);
alph = 2/3;
[V,D]=eig(A);
Aalph = V*(D.^alph)/V;
B = A^(alph);
max(abs(Aalph(:)-B(:)))
• Thanks! This is what happens when you think you remember a formula and don't stop to check that it is correct. At least now I won't be forgetting V*D/V any time soon. Thanks again for clearing that up. – RHP Nov 19 '14 at 21:03 |
Differentiate a Chebyshev series with multidimensional coefficients in Python
PythonNumpyServer Side ProgrammingProgramming
To differentiate a Chebyshev series, use the polynomial.chebder() method in Python Numpy. The method returns the Chebyshev series of the derivative. Returns the Chebyshev series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl. The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*T_0 + 2*T_1 + 3*T_2 while [[1,2],[1,2]] represents 1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y) if axis=0 is x and axis=1 is y.
The 1st parameter is c, an array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. The 2nd parameter is m, Number of derivatives taken, must be non-negative. (Default: 1)
The 3rd parameter is scl, i.e. each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1). The 4th parameter is axis i.e. an axis over which the derivative is taken. (Default: 0).
Steps
At first, import the required libraries -
import numpy as np
from numpy.polynomial import chebyshev as C
Create a multidimensional array of Chebyshev series coefficients −
c = np.arange(4).reshape(2,2)
Display the coefficient array −
print("Our coefficient Array...\n",c)
Check the Dimensions −
print("\nDimensions of our Array...\n",c.ndim)
Get the Datatype −
print("\nDatatype of our Array object...\n",c.dtype)
Get the Shape −
print("\nShape of our Array object...\n",c.shape)
To differentiate a Chebyshev series, use the polynomial.chebder() method in Python −
print("\nResult...\n",C.chebder(c))
Example
import numpy as np
from numpy.polynomial import chebyshev as C
# Create a multidimensional array of Chebyshev series coefficients
c = np.arange(4).reshape(2,2)
# Display the coefficient array
print("Our coefficient Array...\n",c)
# Check the Dimensions
print("\nDimensions of our Array...\n",c.ndim)
# Get the Datatype
print("\nDatatype of our Array object...\n",c.dtype)
# Get the Shape
print("\nShape of our Array object...\n",c.shape)
# To differentiate a Chebyshev series, use the polynomial.chebder() method in Python Numpy.
# The method returns the Chebyshev series of the derivative.
print("\nResult...\n",C.chebder(c))
Output
Our coefficient Array...
[[0 1]
[2 3]]
Dimensions of our Array...
2
Datatype of our Array object...
int64
Shape of our Array object...
(2, 2)
Result...
[[2. 3.]]
Updated on 08-Mar-2022 07:34:31 |
# Tag Info
15
AIXI is a Bayesian, non-Markov, reinforcement learning and artificial general intelligence agent that is incomputable, given the involved incomputable Kolmogorov complexity. However, there are approximations of AIXI, such as AIXItl, described in Universal Artificial Intelligence: Universal Artificial Intelligence: Sequential Decisions based on Algorithmic ...
12
Exact Bayesian inference is (often) intractable (i.e. there is no closed-form solution, or numerical approximations are also computationally expensive) because it involves the computation of an integral over a range of real (or even floating-point) numbers, which can be intractable. More precisely, for example, if you want to find the parameters $\mathbf{\... 10 This question gets at a really interesting fact about AI research in general: AI is hard. In fact, almost every AI problem is computationally hard (typically NP-Hard, or #P-Hard). This means that most new areas of AI research starts out by characterizing some problem that is intractable, and proposing an algorithm that technically works, but is too slow to ... 5 The logical induction algorithm can make predictions about whether mathematical statements are true or false, which are eventually consistent; e.g. if A is true, its probability will eventually reach 1; if B implies C then C's probability will eventually reach or exceed B's; the probability of D will eventually be the inverse of not(D); the probabilities of ... 4 Hutter's "fastest and shortest algorithm for all well-defined problems" is the ultimate just-in-time compiler. It runs a given program and, in parallel, searches for proofs that some other program is equivalent but faster. The running program is restarted at exponentially-spaced intervals; if a faster program has been found, that is started instead. The ... 3 Levin's search algorithm is a general method of function inversion. Many AI tasks are of this sort, e.g. given a cost or reward function (object -> cost or object -> reward), its inverse (cost -> object or reward -> object) would find an object with the given cost/reward; we could ask this inverse function for an object with low cost or high ... 3 In general, partially-observable Markov decision processes (POMDPs) are also computationally intractable to solve exactly. However, there are several approximations methods. See, for example, Value-Function Approximations for Partially Observable Markov Decision Processes (2000) by Milos Hauskrecht. 3 If you are a freelancer, when a client asks to create a website we can easily measure how much the total cost is needed based on the requirements of the client. (the backend, UI/UX design, features, etc.). We can even measure the estimated time of completion. This is only the case when the full scope and design of the site is a well understood and ... 2 What is the time complexity? The time complexity of an algorithm is the number of basic operations, such as multiplications and summations, that the algorithm performs. The time complexity is usually expressed as a function of the input's size$n\$ (but this does not always have to be the case: for instance, you can express the time complexity as a function ...
1
Regarding your first point, it depends on what neural network you would like to use, the sensor temporal resolution, and the capabilities of the embedded system. You can figure out the number of operations required for a forward pass of your network, then when combined with the internal clock of the embedded system, you can calculate the approximate time it ...
1
I was struggling with the same question. This is what I came up with after thinking it through. With depth-first-search, you backtrack to a node that is a non-expanded child of your parent (or the parent of the parent when your parent has no more non-expanded children (and so on going up the tree)). So the space complexity is limited by your ancestors and ...
1
It's not completely clear from your question, but it looks like you want to prove that exact inference in a Bayesian Network is both NP-Hard and P-Hard. It appears that you have proven that it is NP-Hard, but are unsure how to show that it is also P-Hard. This is more of a TCS question than an AI question, but shouldn't be too difficult. You just need to ...
Only top voted, non community-wiki answers of a minimum length are eligible |
## Introduction
Complex systems store energy, process and, very often, efficiently exchange information to perform complex tasks. The universal mechanisms behind this behavior are unknown, although pioneering works have shown that the robustness of this type of systems to random failures or targeted attacks1 might emerge from the trade-off between the cost of exchanging information and the importance of guaranteeing communication dynamics for functioning2,3,4. Therefore, it is crucial for units in a complex network to route information through shortest paths, broadcasting, or according to some dynamics between these two extremes5,6, as it happens for instance in the Internet7. For several applications of interest, even the inverse problem, of identifying either the origin or the destination of the flow from the observation of pathways, is relevant8,9. This framework enables the description of a wide variety of systems, from cell signaling to individuals exchanging information in social/socio-technical systems such as human flows through different parts of a city by public or private transportation means. In the following we focus our attention on flow networks, systems characterized by the exchange of flows—e.g., number of streets between different parts of the city or human movements within a city, migration between different geographic areas, goods traded among countries, packets routed among servers, electricity in a power grid—through edges10,11,12,13. System’s units and their connections have a limited capacity and, in absence of sources and sinks, the sum of the overall incoming and outgoing flows is constant.
Two descriptors traditionally employed to characterize the structure, and hence indirectly the information flow, of unweighted, simple, sparse, and connected networks are the characteristic path length L and the clustering coefficient C2. L is defined as the average length over all shortest paths in the network, while C is the average local clustering coefficient over all nodes in the network and quantifies the network transitivity. Those networks having both a small characteristic path length L—typical of random graphs—and a large clustering coefficient C—typical of regular lattices—display the so-called “small-world” property2, which is found in real world networks and is related to how efficiently the information is exchanged in a system3,14.
One widely accepted measure of efficiency in information flow is the communication efficiency, that has been used to highlight the possible designing principles responsible for neural, man-made communication, and transportation systems3. This measure of efficiency was introduced in 20013, as a physically grounded and more general way to characterize networks displaying the small-world property. Instead of the two descriptors—L and C—for two apparently different kinds of properties of these networks, the communication efficiency evaluated at different scales is able to identify both structural features, indeed, 1/L and C have been shown to be approximations of the efficiency at the global and local scale, respectively. If the clustering coefficient finds a natural, physical generalization in the local communication efficiency, one main difference between the global communication efficiency and the characteristic path length remains: the first concerns the parallel, while the second the sequential information exchange in a system. This discrepancy is negligible if the distances in the network are not too diverse, while it becomes significant if they are highly heterogeneous, as, for instance, in the Internet3,14. A further advantage of the efficiency over the original characterization of small-world networks through L and C, is that it does not require the connectedness and sparseness of the network and, subject to an appropriate normalization, not even its unweightedness.
The topology of a complex network influences the information exchange among its units and is responsible for a rich repertoire of interaction patterns. For instance, the existence of a connection between two neurons allows them to exchange electrochemical signals and their communication dynamics is relevant for the functional organization of the brain. Similarly, human flows through different geographic areas shape the functional organization of a city and its neighborhood, or even email interactions among individuals in an organization determine how information reaches different teams. In these real systems we never see the everyone-is-connected-to-everyone structure, i.e., fully connected networks, because, even if it would be very efficient for the information exchange, it would also be extremely costly. The trade-off between the communication efficiency and the wiring cost characterizes complex systems and their robustness to perturbation in communication dynamics15,16.
Even more importantly, many empirical systems are characterized by connections with heterogeneous intensities and different correlations among weighted and purely topological network descriptors are ubiquitous17, from the human brain4,18,19,20 to transportation networks21. Therefore, it is essential to account for these underlying weighted architectures to gain real insights about the hidden construction principles and mechanisms used to transform, process, and exchange information14. An even broader scenario is possible: think for instance at infrastructure systems, where the units do not exchange information in parallel, where communication is subject to queues or priorities, where noise and failures may play an important role in the communication. In this case to assess the efficiency of the system one needs more information that may not be present (or be representable) in the topology or flows of the network.
However, even assuming that information is exchanged in parallel—which is assumed from henceforth so that when we use the terms efficiency or communication efficiency we mean the efficiency of parallel communication—for a wide class of weighted systems17 which are not embedded in space or for which metadata about the underlying geometry (nodes coordinates) are not available, the normalization of the weighted efficiency descriptor proposed by Latora and Marchiori3,14 may fail—due to a mathematical constraint which is not fulfilled—or may be difficult to compute—because of the nature of flows, encoded in edge weights. As a matter of fact, we observed that in many applications22,23,24,25,26,27 the weighted efficiency is not normalized by comparison with the most efficient version of the network at hand, as suggested by Latora and Marchiori3,14, but instead it is computed upon normalized weights. This latter descriptor, to the best of the authors’ knowledge, has not yet been studied in detail, so we will take care of it, underlining especially its lack of statistical robustness to fluctuations in the network topology or flows.
In this work we show that a mathematically rigorous, statistically robust, and physically grounded, normalized descriptor of the global efficiency of parallel information exchange can be computed without any knowledge on the system, but its weighted network representation. We demonstrate how to define a suitable "physical distance” between system’s units in terms of the flow they exchange across least resistance pathways. We also show that the quantification of the system efficiency might vary dramatically if flows are not adequately accounted for. In fact, discarding edge weights and considering only the topology of a network leads to an overestimation of its communication efficiency. In the opposite direction, incorporating the flows without normalizing the weighted efficiency descriptor leads to a measure that cannot be used to compare different systems. In between these two extremes lie several normalizing procedures for the weighted communication efficiency, which are discussed and compared in the remaining of this article. The normalizing procedures we propose in this work yields an efficiency descriptor that effectively summarizes both the topological and flow information encoded in the networks; in particular, on synthetic models we observe that the efficiency grows not only when the flows heterogeneity decreases, but also if the there is not a subset of privileged pathways monopolizing the whole information flow in the network.
## Results
### Flow exchange in complex topologies
Let us consider a complex network G = (V, E), whose weighted adjacency matrix $${\bf{W}}={({w}_{ij})}_{{i,j}\in V}$$ characterizes both its topology—indeed, wij = 0 if i, j are not adjacent, while wij > 0 if they are—and flows—by the magnitude of the weights wij.
The efficiency ϵij in the communication between two nodes i ≠ jV is assumed to be inversely proportional to their distance dij3. It follows that if i and j belong to different connected components, i.e., dij = , ϵij = 0. The global communication efficiency of the network G is the average over pairwise efficiencies
$$E(G)=\frac{1}{N(N-1)}\mathop{\sum}\limits_{i\ne {j}\in V}{d}_{ij}^{-1}.$$
(1)
The natural metric on unweighted networks is the shortest-path distance. In this case the topological distances satisfy $$0\le {d}_{ij}^{-1}\le 1$$, implying 0 ≤ E(G) ≤ 1, with equality holding when G is a clique and, since each pairwise communication occurs without mediators, information propagates the most efficiently. In case of weighted networks, distances should also account for weights and for what they stand for28. As a matter of fact, the algorithm proposed by Dijkstra in 195929 (and used mostly) involves the sum of the cost of connections to find the path of least resistance, which means that if the edge weights encode the intensity of interactions, their costs have to be derived before computing weighted distances. Furthermore, weighted distances are real valued so that, in general, E(G) [$$0,\infty$$) and depends on the scale of the weights. For this reason, a global indicator of efficiency should be rescaled in [0, 1] considering an idealized proxy of G, called Gideal, having maximum efficiency.
In3,14 the authors propose to build Gideal based on pairwise physical distances ij, which are supposed (i) “to be known even if in the graph there is no edge between i and j”, i.e. ij > 0 for all i ≠ j, (ii) should fulfill the constraint ij ≤ dij for all i, jV, and (iii) should be considered along with topological information in the computation of weighted shortest-path distances dij. Then, $$E({G}_{\text{ideal}})=\frac{1}{N(N-1)}\mathop{\sum}\nolimits_{i\ne {j}\in V}{\ell }_{ij}^{-1}\ge E(G)$$ and $$\frac{E(G)}{E({G}_{\text{ideal}})}$$—which is henceforth denoted by GCE(G)—is correctly normalized. For some spatial networks—e.g., transportation systems like the railway or infrastructures such as the power grid—the physical distances are well-defined by the underlying geometry, for others—among which power stations and water resources—it might be difficult to calculate physical distances because of the lack of direct information about spatial coordinates of units. For nonspatial systems—such as social and socio-technical systems—$${({\ell }_{ij})}_{i,j \in V}$$ can be found as ad hoc transformations of connection strengths (weights) into connection costs. For instance, in a biological network, where wij represents the velocity of chemical reaction along a direct connection between i and j, ij could be taken as its inverse3; or, ij could be the minimum between 1 and the inverse number of edges between i and j in network with multiple unweighted edges14. Unfortunately, this apparently straightforward procedure hides several issues, e.g., if there is no direct connection between two biochemical units in a connected network, their physical distance is infinite according to the previous definition, while their weighted shortest-path distance will be some positive real number, violating (ii). Furthermore, in case of real positive weight $${w}_{ij}\in {{\mathbb{R}}}_{+}$$ one cannot take $${\ell }_{ij}=\min \left\{1,\frac{1}{{w}_{ij}}\right\}$$, since this introduces a cut-off on weights smaller than 1. We indicate by $$E^{\mathrm{LM}} = \frac{E^{G}} {E(G_{\mathrm{ideal}})}$$ the weighted efficiency of $$G$$ when $$G_{\mathrm{ideal}}$$ is built according to3,14. Another common method for obtaining a normalized efficiency indicator22,23,24,25,26, assuming that the weight encode the interaction intensities, consists in firstly, rescaling the weights into [0, 1], then transforming them to costs (usually taking their reciprocals), applying Dijkstra’s algorithm for evaluating the pairwise distances and finally computing the efficiency by (8), without any further comparison with a Gideal. See the “Methods” section for further details. For instance, let us mention the max-normalization of weights $${\tilde{w}}_{ij}=\frac{{w}_{ij}}{\mathop{\max }\limits_{i,j}\{{w}_{ij}\}}$$, which leads to $${E}^{\text{MN}}(G)=\frac{E(G)}{\mathop{\max} \limits_{i, j} \{{w}_{ij}\}}$$ and will be used for comparison in the rest of this study. Observe as of now, that this rescaling is particularly sensitive to outliers or extreme values of the link weights and that, differently from the original definition by Latora and Marchiori14, it compares every pairwise weighted efficiency to the maximum possible efficiency in the whole network. In conclusion, in a broad spectrum of scenarios of practical interest for applications, there is no general recipe to compute E(Gideal).
### Rethinking efficiency of information flow in weighted architectures
To overcome the above issues, we build Gideal from the weighted graph G so that physical distances (i) are not necessarily calculated from metadata or accessible spatial information and (ii) preserve a local feature.
We assume hereafter that edge weights are nonnegative real values and represent the strength of connections. Recall that a path is the sequence of vertices in a nonintersecting walk across the network; the length of the path is the number of edges in—or the sum of edge costs along—that path. Weighted shortest-path distances are then computed minimizing the sum of the reciprocals of weights30,31, which can be seen as costs, over all paths between node pairs (other weighted metrics may be used28 but are not discussed here). Let us denote by SP(i, j) a weighted and possibly directed shortest-path from i to j; its length, $${d}_{ij}=\mathop{\sum}\nolimits_{n,m\in {\mathrm{SP}}({i,j})}{w}_{nm}^{-1}$$, is the shortest-path distance between i and j, while $${\phi }_{ij}=\mathop{\sum}\nolimits_{n,m\in {\mathrm{SP}}({i,j})}{w}_{nm}$$ is the total flow along SP(i, j).
The matrix $${\mathbf{\Phi }}={({\phi }_{ij})}_{i,j\in V}$$ represents an artificial connectivity made of shortcuts, where total flows along shortest paths are delivered in one topological step, as shown in Fig. 1. Gideal is then obtained averaging between the true structure W and the artificial connectivity, i.e., $${{\bf{W}}}_{\text{ideal}}=\frac{{\mathbf{\Phi }}+{\bf{W}}}{2}$$. We finally define $${\ell }_{ij}={({w}_{ij}^{\text{ideal}})}^{-1}$$ and, hereafter, $${\mathrm{GCE}}(G)$$ indicates the global communication efficiency normalized w.r.t. our $$G_{\mathrm{ideal}}$$. Note that a stronger option than averaging between the real and the artificial connectivity would be to take Wideal = Φ, or to define ϕij as the average flow along SP(i, j).
When G is connected, Gideal is completely connected and ij is finite i ≠ j. If otherwise G is not connected, Gideal will be disconnected as well. If there is no path between i, j both ij = dij = and their pairwise efficiency contribute neither to E(G) nor to E(Gideal). Note that in this case we are computing the average communication efficiency, a global indicator, of disconnected subnetworks, which may not be meaningful. Finally, it is possible to prove (using the Cauchy–Schwarz inequality, see “Methods” section) that the constraint ij ≤ diji ≠ j is always satisfied, hence ij are well-defined physical distances that can be calculated for any weighted systems. Having defined the mathematical tools, we now analyze some synthetic networks with a tunable structure. This enables us to separate the effects of topology and flows on the global efficiency of the network.
### Global efficiency of synthetic networks
We start with the simplest combination of topology and weights: upon a full network (a clique) with N = 30 nodes, we generate two ensembles of weighted networks sampling edge weights from different families of probability distributions. The topological efficiency ET, i.e., (1) with shortest-path distances computed ignoring weights, is 1 for all networks, since they are fully connected. We therefore focus on the weighted descriptors ELM, EMN, and GCE. The trivial case, wij = w > 0 constant, leads to ELM = EMN = GCE = 1. We impose more realistic homogeneous flows sampling from a Poisson distribution $${\mathcal{P}}(\lambda )$$ with varying λ. Since zero belongs to the support of the distribution, we add one to each sample to keep the complete connectedness of the network. The heterogeneity in the weighted structure is instead modeled with wij following power-laws(α) with a lower bound xmin = 532. For each value of the parameters λ and α we take 30 random samples from the respective distribution and generate 30 synthetic weighted networks. Figure 2a) shows their GCEs summarized through boxplots, as a function of λ and α. The five statistics of the GCEs distribution shown in the boxplots are: the first (Q1) and third (Q3) quartiles, or quantiles of order 25 and 75% (resp. lower and upper box hinges)—the width of the box shows the interqaurtile range (IQR = Q3 − Q1)—the median (middle line in the box), the smallest observation greater than or equal to Q1 − 1.5 IQR (lower whisker) and the largest observation smaller than or equal to Q3 + 1.5 IQR. Outliers, observations falling outside the expanded IQR are shown as dots. All these synthetic networks are topologically equally efficient since they are fully connected, however, accounting for the weights can lead to dramatically different results. The extreme heterogeneity of edge weights, characteristics of power-law distributions with small scaling exponent, strongly reduces the average communication efficiency of the network. Furthermore, as the tails of the weight distributions become lighter, the weighted GCE tends to the topological one. As the parameters λ and α grow the heterogeneity of weights decreases, since the tailness of the distributions decreases. A measure of the tailness of a distribution is the kurtosis, the standardized central moment of order 4. Usually, one evaluates the kurtosis minus three, which is called the excess kurtosis, and represents the excess w.r.t. the kurtosis of any normal distribution, which is always equal to three. For the Poisson distribution, the excess kurtosis is λ−1; for the power-law the excess kurtosis is finite only for α > 5 and, for α > 5, it decreases as a function of α, tending to 6 as α → . More details in the Supplementary Note 1. This reduction in the weights distribution tailness can also be seen in Fig. 2b), where we show the probability mass (resp. density) functions for λ = 1, 12 and α = 1.5, 7.
We next study the interplay between weights heterogeneity and topology through bond percolation, i.e., the targeted attack and removal of the links in the network26,33. By removing edges in decreasing weight order, we trim the tail of the weights distributions, reducing their heterogeneity. In Fig. 2c) we plot the four efficiency quantifiers as functions of the fraction f of removed edges and averaged over 30 random realizations of each model. Shaded areas indicate the standard deviation from the mean. We denote by Gf the damaged network obtained from G removing f % of its heavier links. G0 is, topologically, a clique, so ET(G0) = 1. In ET(Gf) the denominator is always N(N − 1), hence Gf is compared with a clique, by definition and ET(Gf) decreases monotonically. On the other side EMN, ELM (Fig. S4), and GCE use the flows of Gf to build the corresponding $${G}_{f}^{\text{ideal}}$$ and are, consequently non-monotone functions of f. It might seem a limitation nevertheless, it allows us to compare a series of networks Gf with slightly different topologies, increasingly sparser, and flows that become increasingly homogeneous. In the “Methods” section we propose a modification of the GCE to overcome this possible limitation in percolation applications. EMN and GCE behave similarly, although EMN has larger fluctuations because at each step the edge with maximum weight is removed. As expected, there are clear differences in the percolation plots of Poissonian and power-law network flows, but in both cases removing the heaviest links produces an increase in the average communication efficiency. In both cases, when the flows become more homogeneous the GCE depends largely on the topology, Finally, when the network is disrupted—near the critical threshold fc indicated by the maximum of the second largest component size34,35 (insets of Fig. 2)—the GCE has a break-down point, since we are averaging the efficiencies of many, small, distinct (and maybe individually efficient) disconnected networks.
Before moving to synthetic networks with realistic topologies, let us spend few words on the comparison between the GCE and EMN. The latter is, apparently, more attractive than the GCE, because it is easier to compute. A more attentive look, however, reveals some issues: firstly, the sample maximum is the least robust order statistic, i.e., it is very sensitive to extreme values and outliers. If this is not a strong enough reason to avoid it, have a look at Fig. 3. The GCE converges to 1 as the weights become more homogeneous, while EMN remains approximately below 0.5; and these are very specific networks, they are fully connected. What is then the meaning of a descriptor normalized into [0, 1], when the maximum value is so difficult to reach?
Finally, we compare the two descriptors on synthetic networks generated from models of real-systems, in particular small-world networks (Watts–Strogatz (WS) model2) and scale-free networks (Barabási–Albert (BA) model36). Again we consider 30 realizations of each topology, each having N = 256 nodes, average degree 〈k〉 12 and around 5% of all possible edges. We indicated by $${\bf{A}}=({A}_{ij})$$ the adjacency matrix of each network. Upon these, the edge weights are assigned following the two following rules:
$${w}_{ij}={k}_{i}^{\beta }$$
(2)
$${w}_{ij}={e}_{ij}^{\beta }$$
(3)
where $${k}_{i}=\mathop{\sum }\nolimits_{j = 1}^{N}{A}_{ij}$$ is the degree of node i, eij is the (topological) edge-betweenness of the link {i,j}, and β is a free parameter allowing us to tune the flow structure. The betweenness37 of the edge {i,j} is the number of the shortest paths between any pairs of nodes s, t that go through {i,j}, here indicated by gsijt, over the total number of shortest paths between the nodes s and t: $${e}_{ij}=\mathop{\sum}\nolimits_{s\ne t}\frac{{g}_{sijt}}{{g}_{st}}$$. First observe that (2)38 generates asymmetric weights; for positive β hubs have strong out-going links, while for negative values of β the intensity of the connections decreases with the degree. The case β = 0 leads to unweighted systems. Results are summarized in Fig. 4. The poor robustness of EMN emerges in the plot: the distributions of EMN over the ensembles are skewed and have greater variance. Therefore, we focus on the GCE for the between models comparison. Topologically, BA networks are slightly more efficient than WS networks, but as soon as weights are introduced the panorama changes: the degree distribution of small-world networks is less heterogeneous w.r.t. the scale-free leading to less heterogeneous weights generated by (2) and higher efficiency, independently from the sign of β. On the other side, when links strengths are related to their topological betweenness, BA networks are generally more efficient than WS networks and when β > 0 the networks are more efficient, because those edges that are very in-between shortest paths are also very efficient. Notice that here, differently from the previous example where all possible links were present, communication paths are unlikely to be able to “reorganize” (i.e., choose a different sequence of edges) in response to weights changes.
### Global efficiency of real interconnected systems
We use our framework to study the efficiency of four real systems (see Table 1). From the FAO worldwide food trade network we selected the layers of cocoa, coffee, tea, and tobacco. From the migration dataset we selected internal migration flows inside three Asian regions: India, China, and Vietnam. From the worldwide air traffic network we extracted the traffic in and between Europe and Africa. Finally, we consider the structural connectivity of human brain, quantified through diffusion tensor imaging (DTI) and fiber tractography methods.
These real networks have different properties, among which edge density and weight distribution. Based on the results of our analysis of synthetic networks and on previous studies14,26, we expect the weighted efficiency of these real networks to be smaller than their topological efficiency. Thanks to our normalizing procedure, which can be applied unchanged to all networks, it is legitimate to compare the weighted efficiencies of diverse systems and we expect the trade networks to have the smallest efficiency. As a matter of fact, observing the boxplots in Fig. 5 (and Fig. S8 of the Supplementary Note 2) we can see that the distributions of the network flows in the trade networks are highly heterogeneous. The whiskers in the boxplots extend from the minimum to the maximum of the distribution to highlight the presence of extreme outliers or very heavy tails. Let us then look at the results of the analysis.
Figure 6 shows the curves corresponding to ET(Gf) and GCE(Gf). Independently from the system, ignoring the network flows leads to an overestimation of the average efficiency, especially when flows are highly heterogeneous. The network of internal migration, is the most efficient, but it also has the highest cost being a clique. The tea trade network is the most inefficient. Finally the brain and the airports network have similar GCEs until the first 25% of their edges are removed, with the brain remaining afterwards more efficient w.r.t the reduced flows. Observe that the total flow could be restored, while keeping a specific efficiency value, redistributing the removed flow on the remaining links. In general, removing those edges monopolizing shortest paths forces their reallocation inducing an increase of the global weighted efficiency.
## Conclusion
Exchanging information is one of the main functions of many real complex systems and quantifying how efficiently they perform this task is of great interest for different disciplines. Consequently, the concept of communication and transport efficiency is relevant for a broad range of applications, from public transportation to the human brain and the Internet. Defining the efficiency and telling which system is efficient from which is not, is still an open and debated question. It depends on the context—“Are we interested in the efficiency of a system in relation to an objective, such as a maximum cost, a performance level, or in an absolute efficiency measure?”—on the amount of information available on the system—“Do its units have to wait before passing a massage?”, “Is this information encoded (or even, can this information be encoded) in the network representing the system?”—etc. For instance, communication and transport processes in many real systems may involve waiting times, error rates and technical inefficiency of the networked system, and possibly, also other variables that cannot be encoded as purely topological network features. In this case, characterizing the efficiency of the system by link weights and weighted shortest-path distances is an oversimplification.
However, given a network, without any metadata, we can reasonably assume that nodes connected by a link are closer or more similar than disconnected nodes, and that connected nodes communicate more easily and efficiently than disconnected ones. Furthermore, we can assume that strong, heavy links bring the nodes nearer, reducing the cost of their interaction and communication. Hence, our approach on system’s efficiency is based on the assumptions (i) of parallel information exchange between units and (ii) that the network representation of the system is enough for assessing its communication efficiency. Under these assumptions, we studied systems represented by flow networks, where links encode volumes of people, electrochemical junctions, packets, etc. While there is a widely adopted descriptor for the global communication efficiency in case of unweighted networks, we have found that its generalization to the case of weighted networks might not be suitable in all relevant cases. In this work we have identified and explained the mathematical limitations of the current measures.
A direct consequence of our analysis is that an estimation of global efficiency can be trusted only under specific conditions: i.e., the analysis of efficiency in the case of real network flows cannot be performed or, alternatively, when it is performed it might lead to important underestimation or overestimation of results, which agrees with previous results26. Since flow networks are ubiquitous, here we have proposed the most general definition of the global communication efficiency for weighted directed networks, which does not assume any other (meta-)information on the system.
Using our physically grounded definition of flow network efficiency, our results indicate that one can achieve a desired level of efficiency by wisely redistributing weights, instead of altering the underlying topology. This result is relevant for practical applications, since it is not always guaranteed that one can rewire or dramatically change with other interventions the network connectivity. In fact, altering network structure is usually expensive in economic or energetic terms. Our framework works under mild assumptions about the underlying topology and about the ideal and most efficient network, with no metadata, nor additional spatial (e.g., geographic) information on the system, allowing for trustworthy applications to empirical problems. It also allows for a complementary view of bond percolation from a functional perspective, allowing us to gain new insights about critical phases of information exchange and network flows in addition to topological ones.
## Methods
### Mathematical details on the normalizing procedure
We provide the proof of dij ≥ iji ≠ jV, which is a sufficient condition for the GCE to be correctly normalized in [0, 1]. Recall that SP(i, j) denotes a weighted (directed) shortest-path from i to j and $${d}_{ij}=\mathop{\sum}\nolimits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}^{-1}$$. Observe also that, if the shortest-path between i, j coincides with their link (i, j) the number of vertices in the sequence is SP(i, j) = 2 and their shortest-path distance is $$d(i,j)=\frac{1}{{w}_{ij}}$$. The total flow between i and j through the shortest-path SP(i, j) is defined as $${\phi }_{ij}=\mathop{\sum}\nolimits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}$$.
Before proving our main statements we write an inequality, which will be extensively used in the following proofs. The Cauchy–Schwarz inequality for vectors u, v in an inner product space reads u, v2≤〈u, u〉 〈v, v〉. Taking $${\bf{u}}=\left(\frac{1}{\sqrt{{x}_{1}}},\ldots ,\frac{1}{\sqrt{{x}_{n}}}\right)$$ and $${\bf{v}}=\left(\sqrt{{x}_{1}},\cdots \ ,\sqrt{{x}_{n}}\right)$$ the inequality becomes
$${n}^{2}={\left(\mathop{\sum }\limits_{i = 1}^{n}\frac{\sqrt{{x}_{i}}}{\sqrt{{x}_{i}}}\right)}^{2} \le \, \left(\mathop{\sum }\limits_{i=1}^{n}\frac{1}{{x}_{i}}\right)\left(\mathop{\sum }\limits_{i=1}^{n}{x}_{i}\right)\\ {n}^{2}{\left(\mathop{\sum }\limits_{i = 1}^{n}{x}_{i}\right)}^{-1} \le \, \left(\mathop{\sum }\limits_{i=1}^{n}\frac{1}{{x}_{i}}\right).$$
(4)
(4) states that for nonnegative real numbers x1, …, xn the inverse of their sum is smaller or equal to the sum of their reciprocals.
Since we have assumed edges weights to be positive we can apply the inequality, which leads us to
$${\left(\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}\right)}^{-1}\le | {\mathrm{SP}}(i,j){| }^{2}{\left(\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}\right)}^{-1}\le \mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}^{-1}.$$
(5)
Observe that SP(i, j)≥2 if G is connected, therefore the first inequality is actually strict.
From (5) we can derive useful inequalities involving wij, ϕij, dij, and ij:
$${\phi }_{ij}^{-1}={\left(\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}\right)}^{-1}\le \mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}^{-1}={d}_{ij}$$
(6)
note that if wij ≠ 0, it also holds $${d}_{ij}\le \frac{1}{{w}_{ij}}$$.
It is also possible to prove that ϕij ≥ wij, i, jV. Indeed, if i, j are not adjacent then wij = 0 but, since G is connected, there is a path between them with ϕij > 0. If instead, they are adjacent, either ϕij = wij meaning that the weighted shortest-path coincides with the edge (i, j), or there is a shortest-path going through other vertices, such that $${d}_{ij}=\mathop{\sum}\nolimits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}^{-1}<\frac{1}{{w}_{ij}}$$ and the claim follows from (6).
Starting from the definition of physical distances ij, using simple inequalities and (5)
$${\ell }_{ij} = \, 2{\left({w}_{ij}+{\phi }_{ij}\right)}^{-1}\,\le\, 2{\left(\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}\right)}^{-1}\,\le\, | {\mathrm{SP}}(i,j){| }^{2}{\left(\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}\right)}^{-1}\\ \le \,\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}^{-1}={d}_{ij}.$$
(7)
Again, for a connected network G the strict inequality ij < dij holds.
Finally, ϕij = 0 if and only if i and j lie in disconnected components and, consequently, the ideal network is disconnected as the original one. In this case both $${d}_{ij}=\frac{1}{{\phi }_{ij}}=\infty$$ and the missing links among disconnected components will not produce an underestimation of the efficiencies of the subgraphs. Of course, if the network is very fragmented, the GCE, a global descriptor, will not be very informative. Below, we propose a variant of the GCE, which is most appropriate in this case and in percolation simulations in general.
### Comparisons with other weighted efficiency measures
In this work we introduced existing measures of topological and weighted efficiency, more specifically, ET the topological efficiency defined in3, ELM defined by Latora and Marchiori in ref. 14, EMN obtained evaluating the efficiency on the network with max-normalized weights.
Let us recall the definition of efficiency3
$$E(G)=\frac{1}{N(N-1)}\mathop{\sum}\limits_{i\ne j\in V}{d}_{ij}^{-1}.$$
(8)
where we have the sum of the reciprocal values of pairwise distances $$\mathop{\sum}\nolimits_{{\mathrm{i}}\ne {\mathrm{j}}\in V}{d}_{{\mathrm{ij}}}^{-1}$$ divided by the number of non-diagonal entries in the distances matrix, i.e., N(N − 1). In the topological case this last term plays the role of a normalizing factor, since the sum of inverse shortest-path distances in a clique is exactly equal to N(N − 1). We refer to the efficiency (8) evaluated without edge weights, or in other words with topological shortest-path distances $$({d}_{ij})$$, as the topological efficiency and indicate it by ET. Then ET naturally lies in [0, 1].
The main difficulty arising in the definition of a weighted efficiency descriptor, have to do with the diversity of information that can be encoded as edge weights in a network. Usually weights represent connection strengths and connection costs are obtained as a function (e.g., inverse) of weights. Given the connection costs, one can compute weighted shortest-path distances28,29,30,31, which vary in [0 + ] and therefore, the efficiency computed according to (1) E(G) [0 + ) and needs to be rescaled (or normalized) in order to be comparable among different systems.
EMN, which has been used, for instance, in refs. 22,24,25,26 is the simplest generalization of ET to the weighted case: rescaling the weights to [0, 1] implies that shortest-path distances dij ≥ 1, since weighted shortest paths are those paths minimizing the sum of edge costs, that is, inverse weights. Consequently, being dij ≥ 1, (8) results to be normalized. Different rescaling transformations of weights are possible, the most common is the max-normalization (from which the superscript MN) $${\tilde{w}}_{ij}=\frac{{w}_{ij}}{\mathop{\max }\limits_{i,j}{w}_{ij}}$$. The cost of edges is then $${\tilde{w}}_{ij}^{-1}$$. We show, now, that $${E}^{\text{MN}}(G)=\frac{E(G)}{{w}_{max}}$$, where E(G) is (8) calculated on weighted geodesic distances without the max-normalization of weights. Let wmax be the maximum weight over all edges of a weighted network G = (V, E) and let SP(i, j) be a weighted shortest path between i, jV. Observe that the max-normalization of weights does not affect the shortest path, but it does affect the shortest-path distance
$${\tilde{d}}_{ij}=\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}\frac{1}{{\tilde{w}}_{ij}}=\mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}\frac{{w}_{\max }}{{w}_{ij}}={w}_{\max }{d}_{ij}.$$
Finally,
$${E}^{\text{MN}}(G)=\frac{1}{N(N-1)}\mathop{\sum}\limits_{i\ne j\in V}{\tilde{d}}_{ij}^{-1}=\frac{1}{{w}_{\max }}E(G).$$
This fact, could be appealing for computational reasons, but it is definitely not from a statistical point of view: the sample maximum and minimum are the least robust statistics, they are maximally sensitive to outliers. For this reason EMN may have very wild fluctuations over topologically similar networks but with different maximum weights, which makes this indicator not well suited for comparisons between different systems. To make it clearer, using this descriptor you might not be able to tell if your systems show different global efficiency values because they are characterized by different topologies and interplay between topology and flows, or because their maximum weights are, or are not, outliers to their weights distributions, a very global and extreme feature of the network. Of course, not only the max-normalization and inverse are available, for instance, in ref. 23 weights are wavelet correlation coefficient between regions in the brain and the cost of the connection between regions i and j is defined as cij = 1 − wij.
ELM is the weighted generalization of E(G) proposed by Latora and Marchiori in ref. 14. The idea is to normalize E(G) considering an ideal case Gideal, where all possible edges are present in the idealized graph and the information propagated most efficiently. Then,
$$\frac{E(G)}{E({G}_{\text{ideal}})}=\frac{\frac{1}{N(N-1)}\mathop{\sum}\limits_{i\ne j\in V}{d}_{ij}^{-1}}{\frac{1}{N(N-1)}\mathop{\sum}\limits_{i\ne j\in V}{l}_{ij}^{-1}}\le 1.$$
(9)
Observing that a sufficient condition for (9) is 0 ≤ ij ≤ dij for all i, j, V, defining Gideal reduces to building the matrix $${({\ell }_{ij})}_{i,j}$$. They called ij physical distances, in contrast to shortest-path distances, highlighting that the latter are computed using “the information contained both in the binary adjacency matrix and in $${({\ell }_{ij})}_{i,j}$$”. Observe that the matrix $${({\ell }_{ij})}_{i,j}$$ is, in every respect, a matrix of connection costs. In ref. 14 (Sec. 3) the authors give some examples to ij from edge weights. For instance, if weights wij ≥ 1 one can define $${\ell }_{ij}=\min \{1,\frac{1}{{w}_{ij}}\}$$, which is the transformation adopted in this work to compute ELM.
We refer to the Supplementary Note 2 (Figs. S2 and S3) for the full plot corresponding to Fig. 2b) of this study. The GCE converges faster to 1 as the weight distributions become less heterogeneous (in terms of kurtosis). We claim that the maximum of the GCE is obtained not only for full networks with constant edge weight distribution, but it is sufficient to have a uniform edge betweenness, as shown in Fig. S3.
Finally, the panel c) of Fig. 2 without the cut on the range of y − values is reported in Fig. S4 of the Supplementary Note 2. The percolation simulation consists in removing edges from an undirected weighted full network G, in decreasing weight order. We indicate by f the fraction of edges removed from G and by Gf the resulting, damaged, network with G0 = G. We then evaluate the efficiency of Gf by means of the already described measures: ET, EMN, ELM, and GCE. We repeat the process 30 times, sampling the edge weights from a Poisson distribution with parameter λ = 2 and 30 times, sampling the edge weights from a power-law distribution with free parameter α = 2.5. We include also the plots for common percolation indicators, such as the total weight of—i.e., the sum of the weights in—the largest connected component (LCC) rescaled in [0, 1], the size of the second LCC—divided by N = V—and the number of clusters (components)—also divided by N = V; see Fig. S5. The second LCC size has proven better at pinpointing the critical threshold in percolating lattices35, as well as at distinguishing between percolation regimes34.
Our normalization procedure can also be used to build a slightly modified version of the GCE that plays the role of a weighted integrity descriptor for percolation analysis. Let $${G}_{0}^{\,\text{ideal}\,}$$ be the idealized network corresponding to G0 build as described in our study. Then
$${\text{GCE}}^{* }({G}_{f})=\frac{E({G}_{f})}{E({G}_{0}^{\,\text{ideal}\,})}$$
(10)
is normalized in [0, 1] and it is a monotone decreasing function w.r.t. f.
This variant of the GCE has been evaluated for real networks in the Supplementary Note 2, Fig. S9.
### On artificial flows
We choose to build the artificial flows integrating weights over paths, but this is not the only possibility, provided that constraint ij ≤ dij is satisfied. Now, using the same definitions and notation adopted in this study,
$${d}_{ij} = \, \mathop{\sum}\limits_{n,m\in {\mathrm{SP}}(i,j)}\frac{1}{{w}_{nm}}\\ \ge \, \frac{1}{\min \{{w}_{nm}:n,m\in {\mathrm{SP}}(i,j)\}}$$
(11)
$$\, \ge \frac{| {\mathrm{SP}}(i,j)| }{{\sum }_{n,m\in {\mathrm{SP}}(i,j)}{w}_{nm}}\qquad\qquad$$
(12)
$$\; \;\;\ge \frac{1}{\max \{{w}_{nm}:n,m\in {\mathrm{SP}}(i,j)\}}$$
(13)
$$\ge \frac{1}{{\phi }_{ij}}\qquad\qquad\qquad\qquad\quad$$
So both the minimum, the maximum, and the average weight over the path are valid choices, as well as, the sum (our choice) and the maximum over all edges (the already discussed max-normalization). Now, when using the sum of flows over paths, we can combine the two sources of information W and Φ through the arithmetic mean, while this strategy is not possible if we define $${\phi }_{ij}^{* }=\min \{{w}_{nm}:n,m\in {\mathrm{SP}}(i,j)\}$$ (resp. $$\max$$) because we cannot prove that $${\ell }_{ij}=\frac{2}{{\phi }_{ij}^{* }+{w}_{ij}}\le {d}_{ij}$$, so we should drop W and simply define Wideal = Φ*. We implemented these choices corresponding to the minimum and maximum and show the results on our synthetic networks ensembles in Fig. S6. The two variants, called here GCEmin, $${\text{GCE}}^{\max }$$, converge faster to 1, since both minimizations (11)–(13) are less strict than ours (the sum). Taking the minimum (11), in particular, may result in values of efficiency spanning a narrow range, near 1, with a consequent difficulty in distinguishing networks on the basis of efficiency. Furthermore, in the bottom panel of Fig. S6 we can see a decreasing-increasing behavior of GCEmin which is, from our point of view, not desirable. $${\text{GCE}}^{\max }$$ displays, in general, a larger variability. We opt for the sum, since it has a physical meaning in terms of total flow of a subgraph (a path SP(i, j)), it allows us to average the artificial flows matrix with the original flows given by W and, last but not least, it is easily worked with in mathematical terms (simplifies rigorous proofs).
### On the normalized weighted efficiency of Latora and Marchiori14
Let us take G as the subgraph consisting of vertices q, r, v of Fig. 1 (indicated now by the indices {1, 2, 3}) and suppose that the weights are the result of the aggregation of multiple binary connections. Its weighted adjacency matrix is
$${\bf{W}}=\left(\begin{array}{lll}\cdot &4&1\\ 4&\cdot &2\\ 1&2&\cdot \end{array}\right)$$
We can compute physical distances ij following the suggestions in ref. 14 and shortest-path distances dij minimizing the sum of costs (i.e., inverse weights)
$${\bf{L}}=\left(\begin{array}{lll}\cdot &\frac{1}{4}&1\\ \frac{1}{4}&\cdot &\frac{1}{2}\\ 1&\frac{1}{2}&\cdot \end{array}\right)\quad {\bf{D}}=\left(\begin{array}{lll}\cdot &\frac{1}{4}&\frac{3}{4}\\ \frac{1}{4}&\cdot &\frac{1}{2}\\ \frac{3}{4}&\frac{1}{2}&\cdot \end{array}\right).$$
The global communication efficiency defined in ref. 14 is given by $${E}^{{\rm{LM}}}=\frac{E(G)}{E\left({G}_{{\rm{ideal}}}\right)}$$, where $$E(G)=\frac{1}{N(N-1)}\mathop{\sum}\nolimits_{i\ne j}\frac{1}{{d}_{ij}}$$ and $$E({G}_{{\rm{ideal}}})=\frac{1}{N(N-1)}\mathop{\sum}\nolimits_{{\mathrm{i}}\ne {\mathrm{j}}}\frac{1}{{\ell }_{{\mathrm{ij}}}}$$. Observe that the condition (which is sufficient for $$\frac{E(G)}{E({G}_{{\rm{ideal}}})}\le 1$$)
$${d}_{{\mathrm{ij}}}\ge {\ell }_{{\mathrm{ij}}}\quad \forall {\mathrm{i}}\ne {\mathrm{j}}\in {V}$$
(14)
is not satisfied for i = 1, j = 3 and this causes $$\,\text{GCE}\,=\frac{E(G)}{E({G}_{{\rm{ideal}}})}=\frac{22}{9}{\left(\frac{7}{3}\right)}^{-1}> 1$$.
This counter-example on the statement of (14) is not a pathological case: (14) is violated whenever the weighted shortest-path between adjacent nodes i, j does not traverse the direct link eij, i.e., $${d}_{{\mathrm{ij}}}<\frac{1}{{w}_{{\mathrm{ij}}}}$$ and it may often happen in real networks with large heterogeneous weights.
Trying to reproduce the results in ref. 14, we considered the neural network of the C. elegans2,14, with data from http://www-personal.umich.edu/mejn/netdata/. Firstly, we aggregate multiple edges, obtaining a simple, directed, weighted network with N = 297 nodes, m = 2345 edges, and weights in the range [1, 70]. If we consider the network as undirected, we obtain m = 2148 edges and weights in the range [1, 72]. The data are not the same used in ref. 14, so we cannot reproduce their results exactly. Let us focus on the undirected network: Fig. S7 shows the distance matrix D evaluated using Dijkstra’s algorithm with the reciprocal of edge weights, and the matrix of physical distances L, with $${\ell }_{{\mathrm{ij}}}=\min \{1,\frac{1}{{w}_{{\mathrm{ij}}}}\}$$.
### Real interconnected systems, additional results
Here we first apply the variant of the GCE, i.e., GCE*(Gf) to the networks of migrations inside Vietnam and of the human brain; secondly, we report the detailed percolation results for the real network flows discussed in this study.
We refer to the Supplementary Note 2, Fig. S9, to show the behavior of GCE*(G) for two of the real networks from Table 1.
Finally, we show the percolation plots for the remaining datasets studied in this work, see Supplementary Figs. S11 and S12. |
# JavaScript: Make a Real typeof
The problems with JavaScript’s typeof operator are manifold. It is so restricted, that its only use is in testing for undefined. The main culprit cause these restrictions is the loose typing together with the automatic type coercion: "2" + 2 results in 22, "2" == 2 is true, t = 2;typeof t returns "number" and t = new Number(2);typeof t returns "object" to name just a few.
The worst case is probably:
// somewhere on top of the code
var a = 2;
// some thousand lines and/or several scripts later
var b = new Number(2);
if(a === b){
// CPR: Cardiopulmonary resuscitation
console.log("Go on with CPR!");
} else {
}
# Conversion of Binary Floating Point Numbers—Second Try
The conversion of a string to a binary encoded floating point number is not easy, as shown in my last post. The information given there is a bit too much on the theoretical side. I’ll try to change that here with some real code: String to Bigfloat and Bigfloat to String. Continue reading
# Factoring an Integer in JavaScript with a Wheel
Update: bugfix in code
Factoring an integer with a wheel is a known algorithm for which many examples exist for many programming languages. Google might be of some help to find them.
Most of them are either way too simple or way too complicated; I would even say that all of them fall into one of these two categories but I do not know all of them.
One example that could fit both categories at once is the program factor from the GNU-coreutils which uses a wheel to do its work. I did a JavaScript version of it if that language suits you more. The program itself is quite simple yet hard to understand.
For example: what are these numbers at the start and where do they come from? Continue reading
# Multiple Precision in JavaScript: Rational Arithmetic
Now that we have a (hopefully) working big integer implementation we can go on, either to arbitrary floating point numbers or multiple precision rational numbers. The latter is slightly simpler(1), so let’s take that one first. Continue reading
# Multiple Precision in JavaScript: Fast Division III
## Division by Way of Multiplication with the Reciprocal (Newton Division)
For divisions with participants in the range of millions of bits we can use this method. It is probably the simplest method of all four division algorithms but also the one with the largest overhead.
The algorithm itself is probably the most published one (mainly because of its simplicity, I guess) but the version for integers: not so much, I think. A short search at Google brought up: nothing. Especially nothing about the problems you’ll meet.
At first the actual algorithm for large integers with base-2k digits (226 in the case of my implementation) in pseudocode:
divisionNewton(a,b){
// checks and balances omitted
// length of numerator in bits
alen = bitlength(a);
// length of denominator in bits
blen = bitlength(b);
// length of remainder in bits
rlen = alen - blen;
// We need some extra precision to avoid
// needing an expensive extra round
// this value is most probably too high
// but doesn't add much to the runtime
extra = ilog2(blen) + ilog2(alen) + 1;
// at least three extra-bits are needed
if(extra < 3){
extra = 3;
}
// these extra bits get added to the
// Numerator by multiplying with 2^extra, that is,
// shifting it left
as = a * 2^extra;
alen = alen + extra;
rlen = rlen + extra;
// We do not need to do every step in full precision
// but we must start somewhere.
// By choosing the start precision like that we could
// do the first approximation purely with normal numbers
// but the effect is so abysmally small, it is wa below
// measure-noise on my machine
startprecision = int (MP_DIGIT_BIT/2);
// You can pre-compute the individual precisions needed
// in the iterations, like I described in the previous post
giantsteps = computeGiantsteps(startprecision,rlen,2);
steps = giantsteps.length;
// the first approximation
// for integers the first approximation can be done directly
// by calculating 2^k/b and adjust the precision
t1 = 2^(2 * startprecision);
bs = b / 2^(blen - startprecision);
bs = round(bs, ROUND_UP);
t1 = t1 / bs;
// These iterations look slightly different from most
// descriptions but they are the very same, I thought it
// can be done faster that way but I was wrong.
// you do not need that much temporary variables, it
// is for legibility only
for(var i = 0; i < steps ; i++){
gsi = giantsteps[i];
// Adjust numerator (2^k) to new precision
t3 = t1 * 2^(gsi - gs0 + 1);
// Adjust denominator to new precision
t4 = b / 2^(blen - gsi);
// Do the squaring of the Newton-Raphson algorithm
t1 = t1^2;
// Do the multiplication of the Newton-Raphson algorithm
t1 = t1 * t4;
// The division of N-R gets replaced by a simple shift
t4 = t1 / 2^(2 * gs0);
// Do the subtraction of the Newton-Raphson algorithm
t1 = t3 - t4;
gs0 = gsi;
}
// the reciprocal is in t1, do the final multiplication to get the quotient
// Adjust the numerator's precision to the precision of the denominator
// but leave the extra bits in place
as = as / 2^(blen);
// Do the actual multiplication N * 1/D to get the raw quotient
q = as * t1;
// Get the real quotient by dividing by the 2^k from the top plus the
// extra bits
q = q / 2^(rlen + extra);
// Compute the remainder, we need it
r = a - q * b;
// The algorithm as it is implemented can be off by one.
// do one round of Barrettt reduction as known from the
// Barrett division treated in my last post to correct
if( r < 0){
r = r + b;
q = q -1;
}
else if( r >= b){
r = r - b;
q = q + 1;
}
return [q, r];
}
The same for my JavaScript implementation of a bigint library:
Bigint.prototype.divisionNewton = function(bint){
var tlen, blen, rlen, extra;
var t1, t2, t3, t4, ts, q, r;
var giantsteps, steps, gs0, gsi, startprecision;
tlen = this.highBit() + 1;
blen = bint.highBit() + 1;
rlen = tlen - blen;
// probably too much and should be adjusted to fill a limb if possible, too.
extra = blen.highBit() + tlen.highBit() + 1;
// should also have three bits at least
if(extra < 3){
extra = 3;
}
ts = this.lShift(extra);
tlen += extra;
rlen += extra;
// The value of startprecision has been chosen to keep the first
// approximation in the Number range but is was found to be of about the
// same speed as with Bigints. YMMV, so please try it out yourself.
startprecision = 15;
// precompute individual precisions to keep the iteration loop legible.
giantsteps = computeGiantsteps(startprecision,rlen,2);
steps = giantsteps.length;
t1 = new Bigint(1);
t1.lShiftInplace(2 * giantsteps[0]);
t1 = t1.div(bint.rShiftRounded(blen - giantsteps[0]));
// the first entry of giantsteps is not necessarily equal to startprecision
gs0 = giantsteps[0];
for(var i = 0; i < steps ; i++){
gsi = giantsteps[i];
// Adjust numerator (2^k) to new precision
t3 = t1.lShift(gsi - gs0 + 1);
// Adjust denominator to new precision
t4 = bint.rShift(blen - gsi);
// Do the squaring of the Newton-Raphson algorithm
t1 = t1.sqr();
// Do the multiplication of the Newton-Raphson algorithm
t1 = t1.mul(t4);
// The division of N-R gets replaced by a simple shift
t4 = t1.rShift(2 * gs0);
// Do the subtraction of the Newton-Raphson algorithm
t1 = t3.sub(t4);
gs0 = gsi;
}
// the reciprocal is in t1, do the final multiplication to get the quotient
// Adjust the numerator's precision to the precision of the denominator
ts.rShiftInplace(blen);
// Do the actual multiplication N*1/D
q = ts.mul(t1);
// Divide by 2^k to get the quotient
q.rShiftInplace(rlen + extra);
// compute the remainder
r = this.sub(q.mul(bint));
// The N_R algorithm as implemented can be off by one, correct it
if( r.sign == MP_NEG){
q.decr();
}
else if( r.cmp(bint) == MP_GT){
r = r.sub(bint);
q.incr();
}
return [q, r];
};
If you have implemented your bigint with a decimal base (e.g.: 107) just replace all 2k with 10k and use base 10 integer logarithms where I used base 2 integer logarithms (but both do the same: they count digits).
Next post: a needful little helper to speed up unbalanced multiplication which speeds up all of these fast divsion algorithms.
# Multiple Precision in JavaScript: Fast Division II
## Barrett Division
The Barrett division uses Barrett’s correction algorithm (Barrett reduction) meant for polynomial division, which are in this case the big-integers. It does work for fractions of the form $N \leq 2D$ only so it needs some precomputations for other fractions.
It basically works by computing a low-precision reciprocal of the denominator and using slices of it with slices of the numerator to build the result.
Bigint.prototype.barrettDivision = function(bint) {
var m = this.highBit() + 1;
var n = bint.highBit() + 1;
var mu, largemu, start, q, r, mask, digit, c;
if (m < n) {
return [new Bigint(0), this.copy()];
} else if (m <= 2 * n) {
mu = bint.inverse(m - n);
return this.barretDivisionCorrection(bint, mu);
} else {
// do school-division with big-digits of a size chosen such that
// the condition N<=2*D holds.
// Overall mu, gets splitted later
largemu = bint.inverse(n);
// choose the startpoint as an integer multiple of n
start = Math.floor(m / n) * n;
q = new Bigint(0);
// first part of the new numerator
r = this.rShift(start);
while (start > 0) {
start -= n;
// Snip a large digit from the LSB side of the original numerator
// make room for it in the new numerator
r.lShiftInplace(n);
// put the digit there
// get the right amount of mu (still under the condition N<=2*D)
mu = largemu.rShiftRounded(2 * n - (r.highBit() + 1));
// correct the result
c = r.barretDivisionCorrection(bint, mu);
// make room for the quotient-part
q.lShiftInplace(n);
// put it there
// the remainder is the new numerator
r = c[1];
}
return [q, r];
}
};
The reciprocal gets calculated with some rounds of Newton-Raphson root-finding with the precision of the denominator. This already gives a hint that this algorthm works best for fractions of the form $N \leq kD$ with $k \ggg 2$ and a sufficiently large denominator.
Some tests resulted in a breakeven to the Burnikel-Ziegler algorithm described earlier at $N = 10 D$ with a bit length of about 50k for the denominator but not only might your mileage vary but it is also different with different relations and sizes.
var BARRETT_NEWTON_CUTOFF = 100
Bigint.prototype.inverse = function(n) {
var m = this.highBit() + 1;
var giantsteps;
var steps, gs, gs0, i;
var r, s, t, u, w, a, b;
// truncated division
if (n <= BARRETT_NEWTON_CUTOFF) {
var ret = new Bigint(1);
ret.lShiftInplace(2 * n);
return ret.div(this.rShiftRounded(m - n));
}
// some rounds of Newton-Raphson
giantsteps = computeGiantsteps(MP_DIGIT_BIT >> 1, n, 2);
steps = giantsteps.length;
r = new Bigint(1);
r.lShiftInplace(2 * giantsteps[0]);
r = r.div(this.rShiftRounded(m - giantsteps[0]));
gs0 = giantsteps[0];
for (i = 0; i < steps; i++) {
gs = giantsteps[i];
a = r.lShift(giantsteps[i] - gs0 + 1);
b = r.sqr().mul(this.rShift(m - giantsteps[i])).rShift(2 * gs0);
r = a.sub(b);
gs0 = giantsteps[i];
}
return r;
};
The function computeGiantsteps(start, end, stepsize) is a small function to calulate the precision needed for the respective iteration rounds.
// Computes iteration steps for e.g. Newton-Raphson
// "stepsize" is the length of the steps and a multiplicator.
// For example stepsize=2 for quadratic convergences (Newton), stepsize=3
// for cubic ones (Housholder), etc.
// Yep, just like the similarily named Python function
function computeGiantsteps(start, end, stepsize) {
var ret = [ end ],
i = 1;
if (arguments.length != 3) {
return MP_VAL;
}
while (true) {
if (ret[ ret.length - 1 ] <= start * stepsize) {
break;
}
ret[ i++ ] = Math.floor(ret[ret.length - 1] / stepsize) + 2;
}
return ret.reverse();
}
The actual Barrett reduction is astonishingly small. It is just:
Bigint.prototype.barretDivisionCorrection = function(b, mu) {
var m = this.highBit() + 1;
var n = b.highBit() + 1;
var digit = this.rShift(n - 1);
var q = digit.mul(mu).rShift(m - n + 1);
var r = this.sub(b.mul(q));
while (r.sign < 0 || r.cmp(b) != MP_LT) {
if (r.sign < 0) {
q.decr();
} else {
r = r.sub(b);
q.incr();
}
}
return [q, r];
};
This works quite well if the reciprocal is not too much off but there are exceptions. Assuming that I did not make an error in the implementation teh following numbers made with the function Bigint.random() from my Bigint implementation
var C = new Bigint(0);
var D = new Bigint(0);
var N = 100;
C.random(100 * 26 * N, 123);
D.random(40 * 26 * N , 124);
it will “hang” in the loop trying to correct a ~5,300 limbs large result with a 4,000 limbs “small” denominator by way of stepwise subtraction.
Funnily, it works with
var N = 100;
C.random(99 * 26 * N, 123);
D.random(40 * 26 * N , 124);
and
var N = 100;
C.random(101 * 26 * N, 123);
D.random(40 * 26 * N , 124);
I was not able to find another example of this behaviour but if you know the reason: please let me know!
# Multiple Precision in JavaScript: Fast Division I
## The Burnikel-Ziegler Algorithm
A method to divide two numbers in an asymptotically faster way then simple schoolbook-division but also usable below the large limits of Barret reduction and multiplication with the reciprocal generated by some rounds of Newton-Raphson root-finding have been found by Christoph Burnikel and Joachim Ziegler and described in their paper “Recursive Division”[1].
The cutoff-point in my JavaScript Bigint library for Little is about 500 limbs for the numerator and 300 limbs for the denominator, about 4,000 and 2350 decimal digits respectively.
The algorithm itself is a more or less simple divide&conquer algorithm, taken quite literally here. It is described well and clearly in the paper linked to above. The implementation on the other side is not so simple, it took me some time.
The most complicated part was the main function that slices the numerator in pieces of the length of the denominator. The paper has a complicated method to squeeze the very last out of the algorithm which I was not able to implement successfully in JavaScript, something was always wrong; correct at the end but still very slow. So I had the weird idea to take the first sentence in this paragraph literally: cut the numerator in slices of the (bit)length of the denominator. This method leaves some rest most of the times which has to be handled specially. The rest worked—astonishingly!—quite smoothly and was fast at the end. Not as fast as the paper promised (mostly due to JavaScript), but still quite fast.
Without further ado, here is the first function, slicing the input which consists of two positive integers a and b with a>=b:
function divrem(a, b) {
// (max) size of one block
var n = b.highBit() + 1;
var tlen = a.highBit() + 1;
// # of blocks
var nblocks = Math.ceil((a.highBit() + 1) / n);
// # of n-sized blocks
var mblocks = Math.floor((a.highBit() + 1) / n);
var firstblock;
var r, q, qr, t;
var count;
count = 0;
if (firstblock.cmp(b) != MP_LT) {
r = new Bigint(0);
} else {
r = firstblock;
mblocks--;
nblocks--;
}
q = new Bigint(0);
while (nblocks--) {
t = qr[0];
r = qr[1];
mblocks--;
}
return [q, r];
}
The numerator gets sliced MSB first, so the very first block is the one which needs special treatment: if it is equal of greater the the denominator we preset the remainder to zero or to the first block otherwise. This gets done such that the input to div2n1n is always 2n large. It is not necessary to use two indices, I have done it for legibility. You might also use a logical or (addition base 2) instead of normal addition but I couldn’t find it faster (YMMV etc. p.p.).
The simplicity of the starting function does come with a small cost: the other two functions are slightly more complicated.
The first function that gets called is div2n1n:
function div2n1n(a, b, n) {
var mask, q1, q2, r, qr, a3, b1, b2;
var half;
if (n <= BURN_ZIEG_CUTOFF) {
return a.divrem(b);
}
var isodd = n & 1;
if (isodd) {
a.lShiftInplace(1);
b = b.lShift(1);
//b.lShiftInplace(1);
n++;
}
half = n >>> 1;
b1 = b.rShift(half);
r = a.rShift(n);
qr = div3n2n(r, a3, b1, b2, half);
q1 = qr[0];
r = qr[1];
qr = div3n2n(r, a3, b1, b2, half);
q2 = qr[0];
r = qr[1];
if (isodd) {
r.rShiftInplace(1);
}
return [q1.lShift(half).or(q2), r];
}
The variable BURN_ZIEG_CUTOFF is set to 3,000 bits in my case. It depends a bit on the size of the denominator but not very much in contrast to the original code where it was taken as the variable m which had to fit an integral times into the denominator which means that the denominator had to be expanded if that was not the case together with the numerator causing a lot of error-prone code writing.
One optimization of the algorithm has been implemented here: the original method does not handle the case of odd n and throws it to the normal division function, too. That is slower than making it even (proof omitted for brevity) for the recursion to consume. The original has the computation of the variables a12, a3, b1, and b2 put into the div3n2n but b1 and b2 is the same in both instances for example.
That makes the function div3n2n quite short:
function div3n2n(a12, a3, b1, b2, n) {
var t = a12.rShift(n), qr, q, r;
if (t.cmp(b1) == MP_EQ) {
q = new Bigint(1);
q = q.lShift(n);
q.decr();
} else {
qr = div2n1n(a12, b1, n);
q = qr[0];
r = qr[1];
}
r = r.lShift(n).or(a3).sub(q.mul(b2));
while (r.sign == MP_NEG) {
q.decr();
}
return [q, r];
}
The differences are not very large (and some were made larger for legibility without harm to the speed)
Compare to the according functions that follow the paper nearly to the letter:
function divide2n1n(a,b){
var n = b.used;
var odd = n&1;
if(n < BURNIKEL_ZIEGLER_THRESHOLD){
return a.divrem(b);
}
if(odd){
a.lShiftInplace(1);
b = b.lShift(1);//b.lShiftInplace(1);
n++;
}
n = n/2;
var c1 = divide3n2n(a.slice(n,a.used), b);
var a4 = a.slice(0,n);
if(odd){
c2[1].rShiftInplace(1);
}
}
var ret = [];
while (n--) {
}
return ret;
}
function divide3n2n( a, b){
var n = b.used >>> 1;
var a1 = a.slice(2 * n, a.used);
//var a2 = a.slice(n,2 * n);
var a3 = a.slice(0, n);
var b1 = b.slice(n, b.used);
var b2 = b.slice(0, n);
var q, r1,d,r,c;
var a12 = a.slice(n, a.used);
if(a1.cmp(b1) == MP_LT){
c = divide2n1n(a12, b1);
q = c[0];
r = c[1];
d = q.mul(b2);
} else {
q = new Bigint(0);
q.used = q.dp.length;
d = b2.dlShift(n);
d = d.sub(b2);
}
while (r.cmp(d) == MP_LT) {
q.decr();
}
r = r.sub(d);
q.clamp();
return [q, r];
}
(I spare you the mess that I made of the slicing function 😉 )
As a side-note: it does not matter which algorithm you use for the final division; it can be schoolbook division or Barret division or even Newton division as long as it is not burnikel-ziegler division itself for obvious reasons. So it might be a good idea to do some branching at that point to chose the right algorithm depending on the size of the input. Or not, to be tested.
In the next post I will talk about Barret reduction and it’s use for an even faster algorithm. The post after that, if nothing comes in the way, is about multiplication with the reciprocal (with the Newton-Raphson root-finding algorithm). All of the algorithms are already implemented in the file bignum.js in the lib subdirectory in Little.
[1] I have a lot of left-over punctuation marks. Anyone interested? |
# 30. Describe the one-year rule. Explain why the following scenario does not satisfy this rule and therefore
###### Question:
30. Describe the one-year rule. Explain why the following scenario does not satisfy this rule and therefore must be in writing: Mary is employed by a local college, whereby Mary and the college agree to one academic year of teaching beginning August 31, 2020, and ending May 20, 2021. They verbally agree to these terms on April 3, 2020. •
31. Describe the rule on collateral promises. Explain why the following promise by Hank must be in writing to be enforceable: Hank and his mom Cathy enter a bank together. Cathy wants a $10,000 loan from the bank. She is willing to sign the written document regarding the loan’s terms, but the bank officer is hesitant. Hank states “do not worry, if my mom cannot pay the loan, I will pay it.” The written agreement between the bank and Cathy is signed; Cathy defaults on her loan 3 months later, and the bank goes after Hank for payment. 32. Describe why, under the Uniform Commercial Code (UCC), the following scenario must be in writing to be enforceable: James and Teesha agree that James will delivery$1,000 worth of toys to Teesha’s toy shop by October 1, 2020, and Teesha agrees to pay \$1,000 by November 1, 2020.
### Help help help help help help help pls help
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### A small 350 gram ball on the end of a thin, light rod is rotated horizontal circle of radius 1.2 m. Calculate a. the moment of inertia
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Kattis
# Smart Phone
You have a very smart phone. Here is the user experience when entering a word:
1. You decide on the word you want to write.
2. You start typing it.
3. At some point, three suggestions appear. The list of suggestions never changes after it has first been presented, even if you keep typing.
You always only get three suggestions. This would mean that the suggestions would not always fit on the screen. However, when this happens they are read aloud by a text to speech plugin that cost $9.99. Yes, it reads all the suggestions for every character you enter until you take one of the suggestions. Since it’s a very smart phone, you can only do the following: • Replace what you have written so far with one of the suggestions. • Hit the backspace button. • Enter a letter. Each of these options takes one keypress. An example: you want to enter the text CAKEEATER. When the suggestions appear, you had gotten to CAK. Let’s say you decide to take the suggestion CAKEMONSTER, remove MONSTER and enter EATER. This would mean that you used 1+7+5=13 keypresses to get to the word CAKEEATER. The only time you would want to do this is when you want to annoy the cookie monster. Also, a lot of keypresses could have been avoided here if we had made different decisions. Write a program that minimize the amount of keypresses given the current input, the desired word and the three suggestions from your very smart phone. ## Input The first line of the input consists of a single integer,$T$, the number of test cases. For each test case there will be five lines: • The word you want to type. • What has been written so far. • Suggestion 1 • Suggestion 2 • Suggestion 3. •$1 \leq T \leq 50\$
• All line lengths will be greater than zero, and less than or equal to 25.
• All lines consist of only uppercase characters from the English alphabet (AZ).
## Output
For each test case, output the minimum number of keypresses needed to get to the desired word.
Sample Input 1 Sample Output 1
2
CAKEEATER
CAK
CAKEMONSTER
CARNIVAL
CAKEEATUR
IDIOPEN
HODOR
KEG
IPHONE
FLUXCAPACITATOR
5
11 |
# Humorous Script Oppositions in Classical Instrumental Music *
## James K. Palmer
KEYWORDS: humor, topics, formal functions, opposition, Classical, valence, wit, Haydn, Mozart, symphony, string quartet
ABSTRACT: Most of us can recall chuckling, or even laughing out loud, at a humorous musical passage and perhaps recalling how much that experience increased our enjoyment of the music. This article focuses on musical humor in passages from instrumental works by Joseph Haydn, Michael Haydn, and Mozart. In the most general sense, musical humor arises when composers play with established conventions of musical discourse by writing something incongruous according to the stylistic context.
I begin by briefly discussing the role of contrast in establishing musical humor in both historical and modern writings. I then introduce a strategy by which Classical composers created musical humor. I call this strategy “script opposition,” following linguistic theories of verbal humor. In my analytical discussion, I explain how “valence shifts” between implications of “high” and “low” create script oppositions, and demonstrate how these valence shifts are produced primarily by musical topics, but are bolstered by formal functions and cues in other musical parameters. My analytical and theoretical approach to musical humor draws on recent studies of musical topics, form, and communication in the Classical style, as well as concepts from recent linguistic theories of verbal humor.
Volume 23, Number 1, March 2017
### [1] Introduction
[1.1] Musical humor is one of the most novel and distinctive features of what we now call the Classical style. Many contemporaneous music theorists and critics were cognizant of the ways Classical composers played with stylistic conventions by incorporating sudden, incongruous shifts in affect.(1) According to Felix Diergarten, the “juxtaposition of varying styles and gestures . . . is one of the most characteristic attributes of the Italian-influenced instrumental music of the 18th century.”(2) The “unexpected twists” caused by these juxtapositions merited special attention from late-eighteenth- and early-nineteenth-century writers,(3) many of whom agreed that such departures from convention “should serve a humorous purpose.”(4) For example, Johann Adam Hiller referred to the “odd mixture of styles, of the serious and the comic, of the lofty and the vulgar” (1970, 3:107; quoted in Wheelock 1992, 43); Heinrich Christoph Koch explained that, to depict an absent-minded person, Haydn “connects sections which properly do not belong together”;(5) Friedrich Rochlitz (1806) specifically noted Mozart’s “funny obligatory treatment of the low basses against the high notes of the wind or string instruments”;(6) and Christian Friedrich Michaelis (1807) wrote,
The departure from convention, the unusual combination of the strange and remote . . . at first presents the appearance of inconsistency. But since this . . . immediately ceases to be inconsistent in the greater context, this music conveys the impression of comedy and can make one laugh (Diergarten 2008, [13], quoting Michaelis 1807, 729).
Each of these authors—like their early Enlightenment contemporaries in Germany—seems to view contrast as a crucial component of musical wit and humor (Diergarten 2008, [13]). Furthermore, the idea that humor involves sudden, incongruous contrasts remains integral to modern scholarship on the subject.(7)
[1.2] Most concertgoers today can appreciate the overt humor of Mozart’s parodic Ein musikalischer Spaß, or the second movement of Haydn’s Symphony no. 93 with its evocation of flatulence, but modern listeners are less likely to apprehend the more pervasive and subtle types of contrast to which eighteenth-century listeners were more closely attuned than most concertgoers today. But what were eighteenth-century listeners listening for? What sorts of norms and codes were their favorite composers using? How did these composers manipulate conventions in ways that might create humor? Vasili Byros explains,
What requires recuperation are not simply norms and genres as things in themselves, but also their customary usage, their interactions on numerous syntactic and semantic axes, how these norms are addressed to various audiences and subjected to compositional play, and how deviations from norms become a source of metaphoric forms of communication such as wit and humor.(8)
Melanie Lowe also recognizes the existence and communicative function of such “codes”:
To be sure, more than one person may arrive at the same, or at least very similar, set of meanings for a particular musical composition, an actuality that suggests the presence within a musical work of certain features we can recognize as a style or code (Lowe 2007, 20).
If we can recover Classical composers’ conventional usages and playful deployments of syntactic norms and semantic codes, we may be able to apprehend some of the metaphoric forms of communication that occurred between these composers and their listeners. Such an endeavour seeks to elevate listeners, performers, and analysts alike to a level of what Robert Hatten calls “stylistic competency” (2004a), in which we can construct intersubjective meanings and, ideally, become part of an “interpretive community” (Fish 1980, 171; quoted in Lowe 2007, 20). Members of this community may, according to Lowe, “choose to hear meanings that are sensitive to the musical, aesthetic, and historical circumstances of a musical work’s composition, and perhaps even aligned with the presumed intentions of its composer” (Lowe 2007, 20). This communal intersubjectivity does not preclude our ability to construct individual interpretations and meanings. Rather, it ensures that such readings are informed by a set of consistent “musical, aesthetic, and historical circumstances.”
[1.3] In this study, I seek to recuperate some of the semantic and syntactic usages and deployments mentioned above, in order to uncover specifically how Classical composers created humor in their instrumental music. I begin by clarifying what I mean by “humor” and describing how composers used “script opposition” and “valence shifts” to create humor. I then discuss specific passages by Joseph Haydn, Michael Haydn, and W. A. Mozart, focusing on incongruous deployments of musical topics (semantics) and formal functions (syntax) in particular.
### [2] Humor, Incongruity, “High” and “Low”
[2.1] I understand “humor” to encompass effects of varying intensity and sophistication, which might elicit responses ranging from quiet amusement to boisterous laughter among listeners familiar with the idioms of the Classical instrumental style. “Humor” in general is a problematic concept. Even in linguistics, where syntactic and semantic meanings are arguably more accessible and explicit than in music, a definition of “humor” has been notoriously difficult to pin down.(9) In musicological literature, “humor” encompasses a broad range of effects from the amusing to the comic, and from the witty to the absurd.(10) I adopt an inclusive definition of humor that encompasses effects that seem to have been designed to elicit a wide range of emotional responses, from relatively mild amusement to uproarious laughter. These various reactions may lead to the impression that something is entertaining, jovial, jocular, comical, satirical, ironic, parodic, farcical, clever, crafty, or witty. Each of these impressions falls along what I call the humor spectrum.
[2.2] Several eighteenth-century authors tried to draw distinctions among these terms, between “humor” and “wit” in particular.(11) Some modern scholars have also attempted to distinguish between humor and wit. Gretchen Wheelock notes the historical “ambivalence” of the term “wit,” while, in music scholarship in general, various usages of “wit” span from barely a shade beyond the strictly conventional to the incredibly abnormal.(12) According to Poundie Burstein, “Wit is a kind of cleverness that uncovers and expresses paradoxical relationships between unlike things. Though witty statements can be quite funny, they are not necessarily so” (1999, 69). Where wit is concerned, I am interested in those examples that seem to be intended to amuse, rather than merely to impress.
Example 1. Mozart’s Ein musikalischer Spaß, K. 522, iii, mm. 72–78
(click to enlarge and listen)
Example 2. Haydn’s Symphony no. 60, i, mm. 67–86
(click to enlarge and listen)
[2.3] For instance, Examples 1 and 2 present two very different passages that both exist somewhere along the humor spectrum. Compare the grandiloquent whole-tone scale, the brazen pizzicato-qua-whoopee cushion, and the willfully awkward trill-turned-tremolo in the violin cadenza from Mozart’s Ein musikalischer Spaß, K. 522 (Example 1)(13) to Haydn’s compositional play in Symphony no. 60, “Il distratto,” where the subdominant is prolonged to the point of excess in order to delay the cadential dominant (Example 2).(14) The first excerpt is a blatant, almost slapstick, example of rampant virtuosity, while the second is a more controlled and measured deployment of proportional play.(15) Put another way, the former is as gag-like as the latter is sophisticated, but both are humorous. I contend that the mechanisms and strategies underlying humorous passages like these are more or less the same across the humor spectrum for instrumental music in the Classical instrumental style.
[2.4] As a modern listener, my sense of humor is different from those of eighteenth-century listeners and composers. In an attempt to bridge this gap between historical and modern listening, I have selected examples discussed by eighteenth-century critics and/or modern scholars, and similar examples I have encountered. By emphasizing “classic” examples of musical humor, I focus on how these examples are funny, without worrying about whether or not they are funny.
[2.5] Burstein offers some general principles on what makes music funny. He writes, for example, “I find untenable the notion that humor results merely from the reconciliation of incongruities.”(16) While he accepts the view that incongruity is a necessary condition for humor to exist in any medium (a view widely shared among humor theorists in linguistics), Burstein rejects the notion that incongruity could be a sufficient condition for humor in music. To clarify this point, Burstein offers a “humor equation”: he writes that humor arises through “the linking and contrasting of things that are somehow serious, sensible, logical, or ‘lofty’ with things that are trivial, silly, illogical, or base.”(17) In other words, humor cannot occur without both “high” and “low” elements, a notion that echoes forward from the eighteenth- and nineteenth-century ideas about musical humor mentioned above.
[2.6] According to philosopher and psychologist John Morreall, the act of linking of “high” and “low” elements is achieved by a “cognitive shift”: “a rapid change in our perceptions” that occurs when one “set of background assumptions” moves to an “opposed” one (2009, 49–51). Morreall differs from other authors when he explains that we can only experience such cognitive shifts as humorous when we enter a non-serious “play mode” where we are “disengaged from conceptual and practical concerns” (2009, 50). Morreall’s “play mode” is required for humor to exist, since “There is nothing automatic about enjoying cognitive shifts” (52). Furthermore, “cognitive shifts are potentially disturbing” in cases where we are puzzled, “momentarily disoriented,” and might even “see our lives as in danger” (52). But according to Morreall, “We have several ways of taking a playful attitude toward problems rather than reacting with cognitive or practical concern. The most obvious is by fictionalizing them” (53). Musical experiences, however, do not seem to require this “fictionalizing” action because there are no real problems to begin with. Listening to music, it seems, automatically places us in a kind of play mode within which we are free to enjoy cognitive shifts at a composer’s whim: the way we construe “highs” and “lows” could be considered a kind of fictionalizing impulse as we connect sonically real, musical things to fictional ideas in what might be considered an idiomatically musical play mode.
[2.7] Notions of incongruity and contrasts between “high” and “low” have been at the forefront of many discussions of wit and humor since the eighteenth century. Modern linguistic theories of verbal humor tend to take incongruity as a point of entry, and it is particularly relevant to a theoretical model for sources of humor in Classical instrumental music, especially in light of the central role that incongruity plays in the remarks of eighteenth-century commentators on musical humor and related insights from recent scholarship.(18) In Maria Goeth’s words, “All psychological models of incongruity are based on the premise that humor derives from some clash between expectation/cognitive schemata and a new stimulus in a particular set of circumstances.”(19) Some specific ways in which linguistic theories of verbal humor articulate these incongruous clashes can help to provide greater clarity and explanatory power when applied analogically to music.
[2.8] A seminal work by Victor Raskin and subsequent writings by his student Salvatore Attardo are foremost among linguistic theories of humor, and provide some basic tenets that effectively cross the conceptual gap between linguistics and music (Raskin 1985 and Attardo 1994, 1998, and 2001).(20) Most importantly, Raskin posits that humor arises from an incongruous “opposition” between overlapping “scripts” (1985, 104). He defines a script as
[A] large chunk of semantic information surrounding the word or evoked by it. The script is a cognitive structure internalized by the native speaker and it represents the native speaker’s knowledge of a small part of the world. (Raskin 1985, 81)
In linguistic theories of verbal humor, an instance of Raskin’s and Attardo’s “script opposition”(21) —for example, good/bad, life/death, obscene/decent, rich/penniless, or high/low stature(22) —is required for a joke to be comprehended and enjoys such widespread appeal that a script opposition “has become synonymous with an instance of humor.”(23) In my analytical discussions below, I use the term “script opposition” (henceforth simply “opposition”) rather than “incongruity” because the former more vividly captures the psychological agency of musical passages in the way they activate response.
[2.9] To better understand opposition, consider the following verbal joke:
There are two fish in a tank. One fish turns to the other and says, “Do you have any idea how to drive this thing?”
We begin with “two fish in a tank.” For most of us, this likely paints a picture of two smallish imported exotic fish, perhaps one red and one blue, that reside in a tank (read: smallish aquarium), probably in someone’s living room. If the joke recipient is familiar with the most common semantic meanings of this opening phrase (i.e. they imagine two fish in an aquarium), then when the joke teller continues, “One fish turns to the other and says, ‘Do you have any idea how to drive this thing?’” the listener will be surprised at the jarring improbability of the new “fish in a military vehicle” scenario because of the opposition between “high and mighty military tank” and “lowly fish tank.” Although the semantic specificity that comes from this joke text surpasses any that could be produced in music, we can simplify this opposition to one that occurs frequently in musical jokes: high/low stature. Indeed, over three decades ago, Leonard Ratner specifically remarked on “what was probably the most profound stylistic opposition—the high versus the low” (1980, 364). Although opposition is not the sole requirement for musical humor, this powerful concept plays a crucial role in the present study as the primary strategy by which composers can articulate the incongruous clash of contrasting musical parameters in various contexts.(24)
Figure 1. “Fish tank” joke text script oppositions
(click to enlarge)
[2.10] Raskin explains that, in the most general sense, (script) opposition can occur between a “normal, expected state of affairs” and an “abnormal, unexpected state of affairs,”(25) but notes that greater semantic specificity can create more specific oppositions like obscene/non-obscene, money/no-money, etc. (Attardo and Raskin 1991). In the three-node opposition network shown in Figure 1, for example, contrasting scripts suggested by the “fish tank” joke are noted in the two upper nodes, while some salient oppositions they create are noted in the lower node. Likewise, in a musical joke “seemingly pure silliness will almost inevitably be contrasted with an unnamed, general standard” (Burstein 1999, 70). But as we will see, Classical composers can deploy musical topics to take advantage of more specific oppositions—most notably high/low stature—that fall under the umbrella of the more abstract scripts, good/bad and (even more abstract) normal/abnormal.(26)
[2.11] In the present study, incongruous oppositions delineate the sorts of things that can create humor in Classical instrumental music when they expose a shift between “high” and “low” elements. In general, the elements of “high” and “low” are what psychologists refer to as “valence.” However, a broader understanding of the term as it applies to music allows us to engage not only extramusical associations with “high” and “low” social status, but also implications of positive and negative emotional valences, and “high” and “low” arousal levels of the sort discussed in perception and cognition studies by David Huron (2006).
[2.12] As both Burstein and Morreall note, “The greater the contrast between the two states in the cognitive shift, the greater the possible amusement” (Morreall 2009, 51; see also Burstein 1999, 68). However, the degree of contrast between “highs” and “lows” does not tell the whole story. For example, Huron notes that the concept of “contrastive valence”—an influential factor in surprise, increased pleasure, and unexpectedly positive outcomes like humor—explains that (for reasons not yet understood) an unexpected good outcome is actually experienced as more positive than an expected good outcome (Huron 2006, 21–25). A recent study demonstrates that listeners rate emotional intensity higher for passages that “expressed a contrasting emotion” from the previous passage, and lower for passages that “expressed the same emotion” (Schellenberg et al. 2012, 574). This suggests that a shift in emotional contrast toward either a positive or negative emotion emphasizes the intensity of the newly expressed emotion. This is particularly pertinent because it indicates that the sudden expression of a positive emotion on the heels of a negative emotion—like that implied by a sudden pivot from a minor-mode serioso to a major-mode musette—may produce a stronger affective punch of the sort that often creates humor. Furthermore, high and low arousal (or energy) levels—as in a Sturm und Drang or contredanse passage, respectively—are associated with degrees of tension (see Huron 2006, 305–30) and may be simply evoked through dynamics, rhythm, texture, melody, harmony, etc. Thus, shifts between negative and positive emotional valence and/or high and low arousal level can bolster more salient shifts from high to low social status.
### [3] Creating Opposition using Topics and Formal Functions
[3.1] Opposition in the domains of musical semantics (topics) and syntax (formal functions) gives rise to valence shifts and humorous effects in the Classical instrumental music studied here. While the different musical contexts of the examples below necessitate varied interpretations and analytical treatments, the humorous effects of these examples are nonetheless produced by opposition.(27)
[3.2] In its most basic sense, opposition creates the impression of a sudden and immediate pull in an unexpected direction. Oppositions engage expectations within the music’s discursive flow when one or more strange contrasts occur between successive musical ideas. That is, oppositions occur when contrasting musical materials are juxtaposed to highlight semantic and/or syntactic contrasts in a manner that suggests stylistic impropriety. As Burstein explains, “If the contrast between high and low elements is strong enough, and these elements are linked convincingly through harmonic, motivic, or other structural means, then humor will result.”(28)
[3.3] In the Classical style, initiating, medial, and concluding formal functions are important parts of the musical syntax, while topics are the most accessible conveyors of musical semantics.(29) In the instrumental music of the first Viennese school, these two conventional aspects play central roles in the construction of musical humor.(30) No matter how we choose to describe a passage—witty or humorous, sophisticated or gag-like—these musical devices responsible for our recognition of a funny event or passage remain essentially the same.
Figure 2. Typical opposition
(click to enlarge)
[3.4] Since Classical audiences were familiar with the meanings or connotations of the musical topics of their time, topics were particularly effective couriers of affect and usually the primary agents of humorous opposition.(31) Most oppositions foreground a local semantic clash between topics with opposed valences, while contrasts in other parameters can heighten the humorous effect. Incongruities between initiating, medial, and concluding formal functions, for example, tend to support topics, with textural, metric, dynamic, timbral, melodic, and harmonic contrasts often in a peripheral role. Due to these characteristics, oppositions seem to jump suddenly out of an otherwise conventional musical context, presenting an abrupt, even visceral, punch line and most often creating “gag-like” or “slapstick” humorous effects. The general impressions of oppositions are captured well by Wheelock: “[W]hen expectations are grounded as habits and tendencies in patterned motions and gestures, discontinuities of motion and disturbances in metric and phrase patterns can have a visceral impact” (1992, 89). A simple diagram of a typical opposition is presented in Figure 2.
[3.5] My analytical perspective is particularly attuned to topical and form-functional deployments that appear incongruous according to their surrounding context. In other words, I am interested in intentional lapses of musical continuity resulting from topical and/or form-functional attributes and implications that fly in the face of Classical stylistic conventions. This focus necessitates, among other things, a careful application of topic theories (see Ratner 1980, Agawu 1991, Hatten 2004a and 2004b, and Monelle 2006) and form (see Caplin 1998 and Hepokoski 2006) to forge a productive synthesis between these two large and complex areas of music-theoretic inquiry. But before addressing how topics and formal functions can create humor in combination, we must look briefly at their individual potentials.
[3.6] As William Caplin defines them, formal functions are initiating, medial, or concluding syntactical phenomena that are informed most prominently by harmony, but also by other musical attributes like melody and rhythm (Caplin 1998). In a performance context, formal functions are expected to engage with one another in specific, stylistically appropriate ways: they follow a logical order that listeners grow accustomed to and expect a composition to follow (Caplin 1998, Agawu 1999). When a composer disrupts the syntactical flow of a passage by excessively prolonging or dislocating an expected formal function, the unconventional progression will create an incongruity.
[3.7] Hepokoski and Darcy are interested in such disruptions, and explain how “displacement[s] of typical function[s]” (or “modular dislocations”) are characteristic of Haydn and create impressions of “ideas in ‘wrong places’ . . . and surprises of different kinds.”(32) While playing with syntactic expectations can create incongruities in different places, this sort of play is found most frequently at beginnings and endings. Agawu also recognizes the affective potency of playing with dislocation, but broadens its application to topics intertwined with formal functions: “dislocating a beginning topos from its harmonic attachment . . . is only one instance of the sort of play that lies at the heart of music of the Classical period” (Agawu 1999, 156).
[3.8] While formal functions can create incongruities in the musical flow, topics provide the crucial piece of the interpretive puzzle when it comes to apprehending humor specifically: topics allow us to interpret bizarre syntactical happenings as humorous (or otherwise) because their sociocultural associations provide the requisite semantic information. I adopt a broad, essentially Ratnerian, understanding of topics, effectively summarized by Lowe: “Topics are musical commonplaces. Some are conventionalized musical types—full worked-out pieces—like the minuet, funeral march, and operatic aria; others are characteristic styles—rhythms, textures, melodies, accompanimental patterns, etc.—that imitate, evoke, or are otherwise associated with the music, sounds, and objects an ordinary listener was likely to encounter” (Lowe 2007, 13). In Michael Klein’s still broader terms, a topic is any “semiotic code that associates a conventional label with a constellation of musical signs” (Klein 2005, 56, quoted in Lowe 2007, 13).
[3.9] Wye J. Allanbrook observes, “Topoi articulate each other’s difference . . . by juxtaposition and opposition” and notes that “contrasting topoi” are examples of “the kaleidoscopic expressive mutations that are a crucial element of [the Classical] style” (2002, 214–15). In a similar vein, Wheelock explains, “Constituting a ready lexicon of the familiar . . . topics . . . provided opportunities for the subversion of traditional categories and hierarchies when used in incongruous combinations and contexts” (1992, 203). Composers that took advantage of this “ready lexicon”—that is, the rich sociocultural encoding of topics—could effectively juxtapose them to create salient or “marked” (see Hatten 2004b) musical moments loaded with semantic information. For example, Byros explains, “Musical topics were the extramusical contexts and situation-defining frames that ‘semanticized’ a particular figure for the attention, particularly if the figure is accompanied by a shift in topical discourse.”(33) That is, even though topics do not follow a stylistically defined syntax, a single topic’s most common semantic associations with social statuses or emotions tend to remain relatively consistent (see Hatten 2004b, 81; Allanbrook 1992, 170; and Caplin 2005, 114).
[3.10] Wheelock recognizes that topics were “associated with [social] musical functions and genres of court, urban, and country life” and notes, “These and other traditional associations provided musical frames of affective and social reference in instrumental music” (Wheelock 1992, 202–3; see also Ratner 1980, 3–30). The sociocultural associations of musical topics are particularly poignant since “eighteenth-century society was built on the inescapable premise of engrained inequality” (Munck 2000, 195–96; quoted in Lowe 2007, 134). Thus, according to Lowe, “In conceiving of our theoretical listeners primarily in terms of social status, we reflect the principal category for self-definition in late-eighteenth-century Europe.”(34) In other words, even though topics progress in no set order, they can still have a profound effect because of their culturally construed, extramusical associations. Byros agrees, explaining, “In music, such [extramusical and social] contexts are provided by topoi, which clarify the overall mood or sentiment of a situation” (Byros 2013, 240; Byros’ emphasis). Agawu elaborates, “a change of topic can reinforce, highlight, or even constitute the primary oppositions within a narrative trajectory” (Agawu 1999, 70). These associations, born as they are out of an “oppositional network of meanings” (Hatten 2004b, 81; see also Almén 2008, 72–73), allow uncommonly juxtaposed topics to provide the contextual clarification that listeners need in order to apprehend the particular incongruities that attend humor, as opposed to other kinds of non-normative musical discourse (which might produce different psychological and/or physiological effects).(35)
Figure 3. Emotional valence and social status conveyed by musical topics
(click to enlarge)
[3.11] Topics are most effectively poised to create a humorous effect when they emphasize implications of “high” and “low” social status, “negative” and “positive” emotional valence, or “high” and “low” arousal, especially in combination. To be sure, not all topics convey strong associations with social status, emotional valence, or arousal level, and those that convey such associations in one composition may be less clear in another.(36) A selection of topics with relatively consistent affiliations appears in Figure 3,(37) where the x and y axes represent the continua for social status and emotion respectively, and the resulting quadrants are labeled according to the affects typically produced by the topics and common adjectives located within them.
[3.12] The relatively few topics included in Figure 3 were chosen because they are common, often appear in the musical examples below, and—unlike many musical topics—often convey the relatively specific extramusical implications in the quadrants shown. For example, a musette topic typically conveys positive emotional valence (it moves at a moderately quick tempo, usually in a major mode) and low stature (it is associated with pastoral, peasant life away from the courts, especially because of its bagpipe-like, harmonically static drone bass), while Sturm und Drang typically conveys both negative emotional valence (it usually moves rather violently through many chromatically-inflected keys, while being rooted in a minor key) and high stature (it has denser textures and more intricate counterpoint). If a musette suddenly appears on the heels of a Sturm und Drang passage, the rapid shift from a negative, high topic to a positive, low one will be a particularly effective source of affective potential, heightened due to the “contrastive valence” noted earlier (see [2.12]), and could create humor.
[3.13] When we want to assess musical humor, it is crucial to consider not only the valences (high/low and/or negative/positive) implied by the two scripts, but also the order of their presentation. Humor is generally more effective when the emotional and status implications of the opposed topics shift suddenly from negative to positive and/or high to low, rather than vice versa. Sudden shifts from high to low topics may be more effective because of the human error suggested by the topically implied “fall from grace”: it is much easier to laugh at the country bumpkin at court than at the general who walks into a peasant’s circle. It is difficult to create a humorous opposition from “low” to “high” because, unless the composer performs the shift ironically, this will generally give an impression of improvement or an increase in social status: exactly the opposite of that conducive to humor.(38)
[3.14] Though two topics may appear to create an opposition on their own, both topics and formal functions influence the effect of humorous passages and are almost always intertwined. Agawu remarks, “Since beginnings, middles, and endings are defined in terms of conventional harmonic, melodic, and rhythmic activity, the role of topoi needs to be understood in reference to these more primary dimensions” (Agawu 1999, 156). That is, like formal functions, topics are parametric amalgams: they are the conceptual top of a bottom-up construction of individual musical dimensions that interrelate in innumerable ways. Oppositions between successive formal functions and/or between successive topics are therefore the result of contrasts between one or more constituent parameters: meters, melodic gestures, harmonies, textures, contrapuntal devices, dynamics, articulations, registers, timbres, etc. Since the “primary dimensions” (harmony, melody, and rhythm) of topics and formal functions always occur together, composers can enhance the affective potential when these dimensions are used to create oppositions between two formal functions and two topics at the same time. This is more easily achieved with some topics than with others. According to Agawu, “It seems clear that in Classical instrumental music . . . certain topoi occur characteristically at beginnings of pieces, while others are used in closing situations” (Agawu 1999, 156). Caplin provides some specific examples of topics that are commonly associated with formal functions of beginning, middle, or ending in Classical instrumental music, but recognizes that not all topics have these functional affiliations.(39)
[3.15] My analyses below discuss how each humorous passage features oppositions of high vs. low stature and/or negative vs. positive emotion. Each of the examples—whether by Joseph Haydn, Michael Haydn, or Mozart—can be funny even from a “first hearing only” perspective: an inescapable commonplace for the majority of Classical listeners. In addition, I believe it is probable that a modern audience can find all of these examples humorous, and perhaps even laugh at them, as I have. I make this assertion not as proof that they are funny, but because I contend, as does Lowe, that there is no reason to “remain uncomfortable admitting personal and subjective listening experience not as merely an inevitable part, but in fact a desirable part, of musical scholarship” (Lowe 2007, 19). My aim, then, is to expand an “interpretive community” that finds new and valuable listening experiences, while perceiving humorous events and passages in this repertoire (see Fish 1980, 171).
### [4] Haydn’s Symphony no. 60, “Il distratto,” third movement
[4.1] My first example occurs in Haydn’s Symphony no. 60, originally written as a theater symphony to accompany Jean François Regnard’s play Le distrait (Il distratto).(40) Diergarten unearths Koch’s negative criticism toward Haydn’s use of “wit” (Witz) in this particular symphony.(41) Koch’s polemic against this composition centers on Haydn’s “clash of styles,” which Koch claims are employed “without reason” (Diergarten 2008, [6]). Referring implicitly to Koch and Sulzer (among others), Elaine Sisman notes, “Haydn and Mozart both created downward trajectories from grand style to popular style, over the course of an exposition, in ways that aroused the ire of German critics who disliked such mixtures of serious and comic” (2014, 98; see also Burnham 2005, 63). While such “downward trajectories” are evident in these composers’ symphonic works (Lowe 1998), in the case of “Il distratto” it seems clear that Koch’s quarrel is not with such large-scale descending trajectories from high to low style, but rather with the “witty,” abrupt, local oppositions between the disparate styles common to the symphonic genre.(42) Although critics like Koch and Sulzer worried that such techniques would undermine the “premise of aesthetic illusion” in music, Michaelis recognized the comic potential of such “departure from convention” and lauded Haydn’s “unusual combination of thoughts” in the symphonies and string quartets in particular (Diergarten 2008, [13], quoting Bonds 1991, 55; see also Zenck 2008).
[4.2] In addition to this symphony’s ability to pique Koch’s interest in particular, it provides a suitable starting point for a discussion of humorous script oppositions, since it is Haydn’s “most thoroughly programmatic composition” (Webster 2004, 248). Of course, this symphony is not an anomaly in terms of its extramusical implications. As Lowe notes, “much of Haydn’s instrumental music incorporates extra-musical associations and [Webster] explains that Haydn ‘never intended to compose “absolute music” in the nineteenth-century sense’” (Lowe 2007, 10, quoting Webster 2004, 355). But in this theater symphony, we may expect Haydn to provide overt associations with the drama for which it was written. We will see that “Haydn’s extramusical associations [which] were an essential aspect of his art throughout his life” (Webster 2004, 249) are made particularly accessible in this theater symphony, when Haydn alludes to Regnard’s absent-minded protagonist (Leander).(43)
Example 3a. Haydn’s Symphony no. 60, iii, mm. 39–48
(click to enlarge and listen)
[4.3] In the third movement of Symphony no. 60, Haydn suggests Leander’s absent-mindedness with a profusion of topical contrasts that create humorous oppositions. The minuet movement would seem an ideal place for Haydn to employ opposing topics since it is “arguably the most socially readable movement type in the Classical symphony” (Lowe 2007, 6). In particular, this minuet’s trio section exhibits some of the strangest topical juxtapositions in the symphony, each of which involves a sudden shift to bizarre new material (Example 3a).
[4.4] The first opposition occurs when a determined serioso (mm. 39–42) shifts suddenly to a capricious musette (m. 43–48). The dogged serioso’s unison texture, minor-mode emphasis (through the opening minor third and the final semitonal step from the submediant to the dominant), chromaticism, long note durations, forte dynamic, and relatively rapid implied harmonic rhythm illustrate the topic’s associations of negative valence and high stature. Then, out of the blue, the meandering musette’s scalar motion, faster surface rhythm, piano dynamic, lighter articulation (staccato), and static (drone) harmony convey its opposing associations with positive emotional valence and lowly peasant life.(44) Though Classical composers often used the musette topic to inject an element of quirkiness, this one is unmistakably odd; it sounds particularly ditzy when compared to the confident, authoritative serioso that precedes it, and in this sense it presents an emphatic antipode to what Charles Rosen calls Haydn’s “heroic pastoral style.”(45) The musette’s aimless melody and harmonic stasis are particularly glaring sources of amusement.(46) Sisman also finds the musette “unusually passive” because “[it] is the consequent of a particularly forceful unison opening, and [because of] its harmonic stasis, uniform scoring, and treble-dominated texture.”(47) Here, Sisman implicitly acknowledges that the serioso’s ridiculous plunge into the musette creates a sudden valence shift from a “high” serioso to a “low” musette. And, though both topics are set in the minor mode, a shift from a characteristically negative valence (serioso) to a characteristically positive valence (musette) is also apparent.(48) Both of these valence shifts (high-to-low status and negative-to-positive emotion) occur in the directions that most frequently produce humor in Classical instrumental music.
[4.5] Although the topics take precedence here, form-functional implications magnify the valence shift between the two topics by embodying the characteristics of each. The resolute serioso’s initiation and half-cadential arrival project a form-functional progression that knows what it is supposed to do, but appears to have overconfidently arrived at a half cadence too early. Immediately following this, the would-be functional ending—pre-empted by the serioso’s early dominant arrival—meanders absent-mindedly above the harmonic stasis of the minor(!) dominant, resulting in the musette’s ambivalent, and ultimately failed, attempt at closure. In other words, since the opening four measures reached the goal dominant (or so we should assume at the arrival of the octave Gs), we expect a consequent phrase to provide an authentic cadence to the tonic C minor. Instead, the harmonic motion stops and stagnates on the inactive minor dominant, and we hear six measures that do nothing form-functionally useful: it is as if they assumed that the first four measures did everything required. This lack of a balancing cadence is almost inevitable because Haydn has chosen to end the phrase with a musette, a topic that almost invariably employs a pedal or drone and cannot bring the necessary cadential closure. When Sulzer discusses the musette in general, he notes, “Very frequently the piece is set over the sustained bass note; therefore the composer must know how to vary the harmony over the same bass.”(49) Here, the lackadaisical prolongation of the minor dominant creates the impression that the music (or Leander) was unable to conjure an appropriate melody with other harmonic implications. Perhaps due to the harmony’s functional impotence, the musette’s melody simply climbs upward, before wobbling its way back down toward the dominant, having apparently not found anything better to do. The decidedly awkward musette provides a surprising and unnecessary post-cadential function—technically, “after-the-end” rather than “ending” (Caplin 1998, 256)—and its six measures exist merely to balance the duration of the opening phrase (which is not even accomplished, since the musette is two measures longer than the serioso).
Figure 4. Haydn’s Symphony no. 60, iii script oppositions
(click to enlarge)
[4.6] Figure 4 highlights some of the oppositions Haydn creates between the confident serioso and the absent-minded musette in this trio’s opening 10 measures. This opposition network shows the salient topics of the opposing passages in the upper two circles and (in this case) anthropomorphizes them with characteristics that might be imparted to the dramatic protagonist, Leander. When Haydn juxtaposes these opposing topics and characteristics with one another, they create common script oppositions, shown in the lower circle.(50) These oppositions can be interpreted fruitfully in different ways, but Haydn invites us to hear the absent-minded Leander in the passages above. Indeed, Sisman singles out “juxtapositions of lyrical and martial/fanfare material” like those above as one kind of “inappropriate treatment” by which Haydn “makes known his theatrical intentions” (Sisman 1990, 320), and the sudden shift from a high and negative to a low and positive topic surely evokes an impression of the absent-minded protagonist. Such a shift from “Leander the confident” to “Leander the absent-minded” is further emphasized by the sudden change from high arousal (serioso) to low arousal (musette). The topical shift might also represent a fall in status or simply a human error without recourse to Regnard’s dramatic narrative. Either of these interpretations might be satisfactory, but as we will see, Haydn’s joke does not end here. Instead, having galloped comically right out of the trio’s starting gate, Haydn introduces further topical and form-functional oddities as the trio progresses. In the trio’s contrasting middle, the topical juxtapositions become increasingly oppositional and the possible connection between Regnard’s protagonist and Haydn’s compositional quirks becomes even more apparent.
Example 3b. Haydn’s Symphony no. 60, iii, mm. 49–71
(click to enlarge and listen)
[4.7] Following the trio’s small ternary exposition, a minuet topic beginning in m. 49 abuts an implied march topic that begins with the anacrusis to m. 59 (Example 3b). Although it would be incorrect to label the interrupting passage (mm. 59–62) as a typical march because of the triple meter, this passage’s dotted rhythms, unison voicing, and forte dynamic are all typical of the march topic.(51) The contrasting middle begins as a typical minuet, but its rather pedestrian melody and uninteresting supporting harmony border on dull (especially when considered alongside other Haydn minuets). And when the rustic-sounding oboes repeat the last four measures in a functionally superfluous—and stylistically uncharacteristic—manner, they bring with them yet another impression of absent-mindedness. That is, the oboes here are like “Carpani’s caricature of a second-violin persona as an unimaginative character, capable of imitation, but not invention” (Klorman 2013, 113). Here, Haydn once again appears to be portraying Leander’s unfortunate affinity for losing track of what is supposed to happen next: in this case, the orchestra has completed a rather perfunctory sentence, but cannot seem to remember how to proceed. The oboes, acting as an “avatar” for Leander (and his absent-mindedness) attempt to continue with an uncreative and pathetic-sounding echo of the preceding phrase that further stands out because it has no precedent anywhere in the movement.(52) Fortunately for the orchestra and Leander (and humorously for us), the insistent and self-important march (skewed by the triple meter) cuts off the floundering oboes (m. 58), pulling everyone in a more serious direction, while making the unnecessary and puzzling oboe echo sound even more bizarre in hindsight.
[4.8] But Haydn does not simply follow the minuet with the triple time march, he actually interrupts the minuet using a typical dotted anacrustic figure emblematic of the march topic. The early arrival of the march topic creates a conspicuous and startling overlap between topics in the trio’s contrasting middle. This “topical overlap” is far more uncommon in Classical instrumental music than simple juxtapositions, and here it creates an emphatic opposition between the quasi-serious march and the absent-minded minuet.(53) The overlap is further emphasized because Haydn writes a forte dynamic under the unison march entry while the oboes’ gesture remains at piano, making it a far more salient and unusual elision than the preceding one in m. 56. The dynamic discrepancy—emphasized by the particularly distinctive dotted anacrusis—results in this rare example where, in a topical juxtaposition, an entering group of instruments actually completely drowns out an already sounding group.(54)
[4.9] The topical shift appears to occur here in reverse. That is, the “low” (because of the absent-minded oboes) minuet is interrupted by a “high” march: this shift seems backward when compared to typical examples of humorous opposition where high topics shift suddenly to low ones.(55) However, this “low-to-high” shift only occurs if we understand these topics in their typical forms: minuet in a major mode in triple meter; march in either mode in duple meter. While the minuet fits its surroundings, the march clearly does not belong here. Though the march seems to push forward confidently and unequivocally with all voices in unison and with typical grace-note rhythms, it is in ${\text{}}_{4}^{3}$ meter!(56) The march—emphatic and proud as it may be—is clearly out of its element: this is minuet territory. If Haydn had wanted to more clearly evoke a march topic, he could have written this brief passage (mm. 59–62) in a more strongly implied duple meter (as he does in some of his other minuets).(57) But had he done so, he might have distracted his listeners from the focus of the trio: the absent-minded musettes and the oboe echo. As if to clarify this focus (and the march’s very brief and ultimately inconsequential intrusion), Haydn over-writes the march’s fourth measure (m. 62) by eliding it with the first measure of the returning serioso and forcing the march from an implied four-measure phrase group (typical of the march topic) to an apparent three-measure version. The result is a march that should be all about duple groupings skewed unnaturally into triple divisions at both the metric and phrase levels.
Example 4. Haydn’s String Quartet op. 33 no. 2, iv, mm. 148–153
(click to enlarge)
Figure 5. Haydn’s Symphony no. 60, iii script oppositions
(click to enlarge)
[4.10] It is important to properly consider the musical context of this march, since marches—like all topics—admit to some degree of variation in their emotional and social status associations. For example, even a serioso—though “serious” by definition and almost invariably negative—can “flip” its implied valence in the hands of a Classical master to become a low and positive parody of itself. According to Wheelock, this is exactly what occurs in Example 4, where Haydn composes “a somber intrusion, clearly at variance with the prevailing climate of the movement” (Wheelock 1992, 12; see also Almén 2008, 84). This well-known adagio serioso arrives in such a ridiculous and surprising manner that it becomes what Wheelock calls a “mock-serioso [whose] imposture is confirmed by the presto return of the theme” (Wheelock 1992, 12). Like this “mock-serioso” from the finale of String Quartet op. 33 no. 2, we can interpret the ${\text{}}_{4}^{3}$ march caricature as a “mock-march” because of its bizarre metrical setting. This reinterpretation of the march flips its associated social status from high to low and clarifies the valence shift between the two topics: now a high(er) minuet moves to a low(er) mock-march in ${\text{}}_{4}^{3}$ (due to its inappropriate usage). The mock-march appears initially to be interrupting the absent-minded minuet to get things back on track, but it also fails to do what it should. We are left with another humorous gag—albeit a clever one—whose most notable oppositions are summarized in Figure 5.
[4.11] With the arrival of the trio’s small ternary recapitulation, we end up again at the opening serioso (beginning in m. 62), rendered ridiculous once more by the returning musette. This concluding musette does, at least, arrive on the minuet’s tonic C, but appears to do so with a perfunctory and premature “resolution” (again with the inconclusive scale degree 5 in the upper voice) when the topics abut (m. 66). The implied half cadence (m. 65) leads to the meandering musette, altered this time from its opening incarnation as a stagnant and unnecessary minor-dominant prolongation to a concluding one with an unnecessary and strangely post-cadential tonic extension.(58) Like the first musette, this one is awkwardly and humorously forced into a phrase-ending position contrary to the phrase-beginning position in which we would expect to find a musette.
[4.12] By now, the latter portion of this ridiculous trio has run clumsily and humorously through a minuet, out-of-place oboe echo, ${\text{}}_{4}^{3}$ mock-march, overstated serioso recapitulation, and stagnant musette, all in the space of 23 measures (mm. 49–71).(59) In this trio, then, Haydn appears to have taken the galant style’s “complex equilibrium between the matching and opposition of material” (Sutcliffe 2014, 124) and employed solely the latter half of the equation. With this many quick and jarring topical juxtapositions, the individual oppositions, though remarkable, seem less critical than the overall impression of memory failure throughout. In other words, while it seems the trio only needed to run a 100-meter hurdle, it managed to trip and half-fall over every single gate along the way. Finally, the event is over and we return to a more conventional, Classical minuet section.(60)
[4.13] There is a possible connection in this trio to specific dramatic events in Regnard’s play. These events, rather than flattening the narrative capacity of the trio’s many humorous oppositions, can enliven the topical interplay Haydn composed in this section. Elsewhere in the symphony, Haydn uses a French dance and other more serious topics to represent Leander’s dramatic antithesis: the sensible, courtly, gallant Chevalier.(61) Here also, Haydn may be referencing the Chevalier and pitting his common sense and clarity of purpose against Leander’s absent-mindedness.(62) Following this interpretation, the serioso/musette and minuet/mock-march juxtapositions can be heard as a dialogue between the Chevalier and Leander. Haydn has certainly portrayed Leander’s absent-mindedness, but why not poke fun at the self-important Chevalier? As Burstein notes, “Characters who intimidate us or whom we regard as powerful or exalted (such as politicians or clergymen) are prime material for humor. With such people, the serious aspect of the humor equation is so accentuated that comic contrast can easily be established” (1999, 68). So it is with musical topics: the more exalted a topic, the more easily it may be parodied, as Haydn does here by chopping a leg out from under the Chevalier’s “march.”(63) In a nutshell, we might interpret the two examples above as involving Leander and Chevalier in the following situations: in the first section (Example 3a), the Chevalier makes a clear statement and Leander seems to lose track of what should occur, or simply does not understand; and in the second section (Example 3b), Leander begins on the right track, but seems to lose his way before the Chevalier (who does not belong here) interrupts by marching right through Leander’s nonsense to set him straight and get him back on track.
[4.14] This back-and-forth is certainly entertaining, and it may well be what Haydn intended, but I find it less humorous than a reading that places the forgetful Leander alone with his thoughts. In such a reading of the trio, a confident Leander begins with a single thought and great clarity of purpose (mm. 39–42): “I know what I have to do!” before forgetting how to proceed (mm. 43–48). Seeming to have lost his way entirely (on a minor dominant), Leander makes another attempt in the form of a minuet (mm. 49–55): “I can do it this time!” but he cannot even hang onto this simple idea without getting distracted (mm. 56–58): “Hey! Is that a bird?” He then rallies again defiantly, cutting off his distracted, absent-minded self (mm. 59–61) before he realizes that he is again on the wrong track (the mock-march in ${\text{}}_{4}^{3}$ doesn’t work) and stumbles back to his first idea (mm. 62–65) when the mock-march unison melody elides awkwardly with the beginning of the returning serioso.
### [5] Haydn’s Symphony no. 60, “Il distratto,” sixth movement
Example 5. Haydn’s Symphony no. 60, vi, mm. 1–36
(click to enlarge and listen)
[5.1] In the finale of the same symphony, Haydn gives us one of the most outrageous ruptures of convention to be found in the Classical instrumental repertoire. Here, he emphatically conveys Leander’s absent-mindedness when he abandons the declamatory opening prestissimo for a two-measure grand pause, after which he instructs the violins to tune their low string from F up to G.(64) Since the violins have not used their G strings, it has not yet caused them any embarrassment, but Haydn’s performance direction makes it seem that they have just noticed their error (“Wait! Everyone stop!”), so they regroup to tune their strings. After the surprising interruption, the piece begins again (“Okay, we’re ready now”) without the four-measure introduction and continues relatively unhindered (Example 5).(65) In the words of a Preßburger Zeitung contributor, “The musicians begin the piece with great pomposity and remember shortly that they have forgotten to tune their instruments.”(66)
[5.2] This tuning interruption may be Haydn’s cleverest and most successful attempt to represent the orchestra as the stage character Leander. The Preßburger Zeitung notes that in the first performance, “[T]he finale had to be repeated in view of the insistent applause of the listeners.”(67) And another reviewer remarks on the effect of the opening on a Salzburg audience: “The absentmindedness of the orchestra is surprising, amusing, a hearty good effect. One must laugh out loud at the idea.”(68) Indeed, the degree of stylistic incongruity in this example is rivalled only by some passages in Mozart’s Ein musikalischer Spaß, K. 522.(69) Haydn’s joke is very accessible and seems straightforward, but many different facets of humor are in operation.
[5.3] The tuning passage grants a funny, surprising, and jarring escape from the musical discourse proper and a sudden shift to what would normally occur before the performance. In her discussion of this opening, Wheelock explains, “Haydn breaks the musical frame with the shocking intrusion of ‘realism’” (1992, 155).(70) In a way, Haydn’s notation for tuning is a kind of “extra-musical” reference, though not in the usual sense; we might call this a “work-extraneous” reference instead.(71) That is, Haydn alludes to a phenomenon unique to the realm of musical performance (preparatory tuning), but this musical occurrence is never a part of the composition proper, at least not in the relevant historical context. When we listen to the opening, it is easy to perceive this humorous opposition, but the scripts suggested by the two opposing passages are unusual, since one passage belongs to normative Classical discourse while the other forms no part of such a discourse. Since most examples of opposition employ topics as the primary perpetrators of valence shifts, they illustrate a less radical and more common type of Classical gambit because both topics necessarily belong to normative Classical discourse when taken individually, but become abnormal due to the incongruous nature of their immediate juxtaposition. Although the work opens with a declamatory fanfare topic, the tuning interruption cannot properly be called a topic since there is no convention or precedent associated with this compositional maneuver. But Haydn’s tuning passage is nevertheless part of a contemporary listener’s “ready lexicon” (Wheelock 1992, 203): it is easily recognizable as part of the typical preparation for a musical performance. Thus, we might simply refer to this more general frame of reference as tuning and recognize its function as analogous to a typical musical topic.
Figure 6. Haydn’s Symphony no. 60, vi script oppositions
(click to enlarge)
[5.4] Even without a stylistically established topic, the opposition is clear. There is a conspicuous valence shift between two case-specific scripts that we could call the composition and not the composition.(72) The first script—the composition—is conveyed by the opening fanfare and Gjerdingen’s Prinner schema.(73) Although the Prinner schema is most commonly a riposte rather than an opening gambit (Gjerdingen 2007, 45), Haydn’s artful pairing of the common galant schema alongside a declamatory fanfare topic clearly conveys the high social status associated with artful composition. Fanfares are particularly apt beginnings, useful for “generating expectancy” (Lowe 2007, 34), and they strongly imply the succession of further artistic (or at least appropriate) compositional endeavours. Instead of the further artful composition we expect, the second script—not the composition—conveys the low social status associated with the unprecedented, ludicrous notion that the orchestra has forgotten to tune, and immediately grabs the audience’s attention. As Burstein notes, “Audiences tend to recognize only the lower side of the humor equation by focusing on the ridiculous” (1999, 68). This will inevitably be the case here—especially since the tuning follows such a raucous, “noise killer” opening (Somfai 1981, 390; see also Wheelock 1992, 225n6). As always, it is the preceding “high” that throws sudden, conspicuous relief onto the ridiculous “low” that follows and produces the humorous opposition. Although both scripts suggest positive emotional valence, the magnitude of the high-to-low stature shift easily produces the oppositions shown in Figure 6.
[5.5] Perhaps the most likely “target” (or butt) of this joke is the absent-minded Leander.(74) In this passage, Leander’s persona is like the many operatic characters Edward T. Cone calls “Petrouchkas, brought to life by the composer but thenceforth driven by their own wills and desires” (Cone 1974, 23). Edward Klorman notes that these “Petrouchkas” are “able to act freely within the confines of the opera’s [in this case, the drama’s] fictional frame” (Klorman 2013, 121). In this reading, the protagonist seems to start off on his merry way (a confident introduction and first phrase of the theme) before coming to a complete stop . . . he’s missing something . . . oh yeah! And the major blunder forces him to go back and start all over again (minus the introduction). We do not have to look far into the drama of Regnard’s Il distratto for suitable analogues to this event. This serious error in the finale sounds as though Leander has botched a rather important event, but, according to the Preßburger Zeitung, the tuning interruption “is meant to show . . . the absentminded bridegroom tying a knot in his handkerchief in order not to forget his wedding.”(75) Whether or not Haydn is depicting a deliberate reminder (as the Preßburger Zeitung suggests) or a forgotten event (as I find more likely), the interruption to the narrative and the resulting opposition create a palpable and highly effective joke.
[5.6] In addition, the clever manner in which Haydn allows the listener to change perspective on the music is of particular interest in this opening. That is, we can change our perception of the “target” of humor from Leander, as represented by the instruments, to either the performers or musical instruments themselves. Here, Haydn seems to cross a boundary between the extramusical and the musical when he represents Leander’s forgetfulness with perhaps the most embarrassing mistake on the part of the instrumentalists: they forgot to tune! This interpretation is only viable because the opposition the composition/not the composition engages “the music” in a more direct and absolute manner than most examples of opposition. Goeth offers the following analogy: “If, in a theatre play, a prologue appeared unexpectedly after the beginning of the last act, an effect analogous to Haydn’s tuning joke might be produced” (2013, 242). But she notes that this analogy is weaker than it may seem, since “the course of a recitation is not to the same extent standardized as the course of a concert” (Goeth 2013, 243). Thus, it is more crucial for humor in the musical medium that the tuning passage’s “mock dilettantism serves as a meta-commentary on musical incompetence” and that this meta-commentary arises due to the “ritualization” of contemporary performance practice (Goeth 2013, 241). This particular opposition allows Haydn to draw the listeners into both the music and the drama in a unique and humorous way. For me, the resulting impression is akin to a screen actor turning suddenly to the camera mid-scene and saying something like, “What’s my line?” except that the violins are the ones who have forgotten the compositional script.
### [6] Michael Haydn’s Symphony in D major, Perger 13, third movement
[6.1] Michael Haydn’s “Entr’actes” to Voltaire’s Zaïre fulfill the same role as Joseph Haydn’s theater symphony, and were later published as the five-movement “Symphony in D Major,” which could be performed without recourse to Voltaire’s drama. In their respective theater symphonies, both composers suggested characters from the drama with implied topical affiliations, and created oppositions that led to humorous outbursts recorded by contemporary listeners.(76)
[6.2] In the Adagio third movement, Michael Haydn composed a drastic contrast between two topics, creating a jarring opposition. This movement’s typical variation form is particularly well suited to direct diachronic juxtaposition of musical topics when one variation moves to the next: few Classical forms so strongly imply topical change with the regularity found in variation forms. For example, rondo forms have several relatively brief formal sections, but they do not typically present such a variety of topics.(77) While the “varied repetition” often found in rondo refrains is not usually achieved via topical change, variation forms often create the expectation that a new topic may occur with the beginning of a new variation.(78) This is not always the case, but the way is paved for the composer to alter the topical underpinning of the theme from variation to variation.
Example 6. Michael Haydn’s Symphony in D major, Perger 13, iii, mm. 78–83
(audio begins 10 seconds before notated example)
(click to enlarge and listen)
[6.3] Michael Haydn takes advantage of the variation form to play with our expectations for topical change. He begins with a theme that conveys aspects of learned style (florid counterpoint and steadily moving bass line) and fanfare (dotted rhythms, declamatory repetitions, and melodic arpeggiation), which create an introductory character that continues far into the movement. The overall result is a theme that sounds somewhat stilted, formal, and archaic (not galant), which, for the purposes of this discussion, I will refer to as the “conservative” style. Though we may think of “high” topics as those more ingeniously and creatively employed by Classical masters, this conservative style epitomizes (almost in caricature) the “high” social stature of the Viennese courts. Through the first three variations, increasing virtuosity and contrapuntal richness repeatedly bolster the conservative style’s “high” status until the fourth variation brings an abrupt and unwelcome topical shift (Example 6). When the fourth variation begins at the pickup to m. 81, the composer interjects an especially raucous and unruly Turkish topic. This sudden “out-of-character” variation, with its Turkish cymbals crashing in at a subito forte, intrudes on the conservative style (“business-as-usual”) and its piano dynamic. The rupture was so pronounced in performance that Leopold Mozart remarked on it specifically in a letter to his son in October of 1777: “Following a variation which was piano, there was one on the Turkish music, which was so sudden and unexpected that all the women looked terrified and the audience burst out laughing” (Anderson 1966, 1: 297).
[6.4] This effect on the audience is understandable, since in the highly nuanced topical landscape of the symphony genre, “the Turkish topic might mean comedy or war” (Sisman 2014, 115). The Turkish topic is particularly unexpected here because, rather than reproducing the high level of topical diversity present in many variation forms, Michael Haydn reined in the topics up to this point by retaining the conservative style throughout each of the first three variations, varying it relatively slightly until the marked arrival of the Turkish topic. By employing the relatively unchanging conservative style up to m. 80, Michael Haydn treads on our normal expectations for topical change, such that we are less likely to expect such a change—and certainly not such a drastic one—with the arrival of the next variation.(79) SURPRISE! is the result of the sudden new topic, and it is exaggerated considerably by the magnitude of the shift.
[6.5] The music’s subsequent return to the original conservative style for the beginning of the fifth variation leaves the Turkish intrusion unresolved in retrospect and heightens the ironic play between typical and atypical topical conventions in variation form.(80) Instead of hearing topical contrast between successive variations, as we would in most variation sets, there has been virtually no contrast between variations (only textural and timbral changes) until the Turkish fourth variation. This first topical contrast is further heightened because the Turkish fourth variation signals its arrival with the only anacrustic gesture in the movement so far (the subito forte pickup to m. 81)—to say nothing of the loud cymbals that enter on the following downbeat. Thus, while the audience politely waits for the next downbeat to bring about a new—and, based on the intraopus context thus far, largely similar and somewhat uninteresting—variation, this anacrusis creates an overlap between the two topics. The topical overlap brought in by the “early” Turkish ambush creates a particularly forceful opposition that threatens the stately, proper conservative style.(81)
Figure 7. Michael Haydn’s Symphony in D, iii script oppositions
(click to enlarge)
[6.6] The “sudden and unexpected” entrance of the distinctly demotic Turkish band is particularly pronounced and incongruous because its affiliations with low status, negative valence, and high arousal so clearly oppose the high (if somewhat stilted) status, positive valence, and low arousal of the preceding conservative music (see Figure 7). In fact, Classical composers often employed the Turkish topic specifically to create a humorous effect because of its uniquely multivalent, evocative nature:(82) the topic is uniquely poised to deliver an affective punch to the musical discourse because of its clear affiliations with both the exotic “other” and social conflict (see Head 2005).
[6.7] The Turkish topic’s special significance arose largely because the Turks had besieged Vienna in 1683 (fewer than 100 years before Leopold’s comment). According to Raymond Monelle, this topic carried “an aspect of fashionable orientalism in a Vienna released from the threat of invasion” (Monelle 2006, 6; see also Head 2000). A better-known example of this phenomenon is provided by Mozart (though not in an instrumental work), who wrote to his father about the interpolation of the Turkish topic in Die Entführung aus dem Serail, explaining, “Osmin’s rage is rendered comical by the use of the Turkish music” (Anderson 1966, 769).(83) The medium of music (especially Classical music) was particularly well suited as an outlet for this humor, since it was one in which the high accomplishments of Viennese culture could be made to stand out against the low, boisterous, and easily parodied Turkish idiom. Matthew Head notes that, in examples like these, “Sometimes the exotic and the humorous are indistinguishable.”(84)
[6.8] The audience laughter reported by Leopold likely arose out of initial feelings of relief followed by those of superiority. The relief came from the fact that the Turks no longer represented a threat, and superiority by virtue of the enemy’s defeat from the perspective of the eighteenth-century Viennese. A relief theorist like Freud would likely point to the act of laughing at the humor of a sudden, grotesque appearance of the Turkish topic as a way for the cultivated Viennese to slough off, and make light of, the buried tension once associated with the Turkish threat.(85) A superiority theorist like Hobbes would likely propose that the audience’s laughter expresses feelings of superiority over a vanquished adversary.(86) In either case, Leopold Mozart’s account of the listeners’ responses strongly suggests that the relief and superiority they experienced on the heels of sudden terror resulted in laughter at this juncture. Although laughter is not necessarily an indicator of humor (nor is laughter required to prove that something is humorous; see Huron 2006 and Morreall 2009), it is likely an indicator in this example since—as we have seen—the high/low status opposition required for humor is created by the musical parameters described above.(87)
### [7] Mozart’s Serenade in D major, K. 250, first movement
[7.1] Mozart’s Serenade no. 7 in D major, K. 250 is the largest symphonic serenade he wrote, and it is notable in that many movements contain humorous events.(88) This “festive” serenade was the first of two serenades written for the Haffners(89) whom Mozart knew and had befriended, and was commissioned by Siegmund Haffner, the mayor of Salzburg, for the wedding of his daughter Marie Elisabeth Haffner to Franz Xaver Späth.(90) Since the Mozarts knew the Haffners well, and given the Haffners’ roles as Salzburg benefactors, a wedding in the family would have been an excellent forum for festive music suitable for a convivial atmosphere. Such music, especially a work as long as K. 250, could hardly be entirely serious and still achieve the desired effect. Instead, a light-hearted, entertaining, and humorous musical demeanour would more effectively compliment the social situation. This is exactly the demeanour we find throughout K. 250.
[7.2] The first movement follows sonata form with typically Haydnesque Sturm und Drang features in the developmental core and is formally unremarkable. Mozart endows the movement with a conspicuously pedestrian tone throughout. Stanley Sadie pointedly remarks, “It is hard to understand how a movement of such nobility can be built of material of no real distinction or individuality” (Sadie 2006, 403). Indeed, the movement’s material sounds somewhat vacuous in general: it is filled with motives that are exceptionally dull or inane to the point that they become strange, allowing us to perceive an exaggerated degree of playfulness on the composer’s part, even though, in isolation, none of these motives is necessarily humorous.(91) Had a Classical master like Mozart not composed this, we might easily have associated some level of incompetence with the (at times) seemingly haphazard presentation of motives in this movement. Instead, the extent of Mozart’s playfulness helps to predispose listeners toward a humorous reaction to later events.
Example 7. Mozart’s Serenade in D major K. 250, i, mm. 267–83
(click to enlarge and listen)
[7.3] At the end of this movement, we find a humorous opposition that relies on form-functional implications and a bizarre use of the coup d’archet gesture. Mozart emphasizes the humorous opposition with a tutti rest immediately before the movement’s conclusion (Example 7).(92) This notated rest is longer than the two rests that occur before it in mm. 273 and 275. The earlier rests occur in the middle of a cadential harmonic progression and thus could not have satisfactorily concluded the piece. The final rest, however, could mark a satisfactory conclusion to this first movement. The arrival of any further music after the apparent conclusion suggested by the rest therefore presents a conspicuous and humorous incongruity.
[7.4] With this “fake” ending, Mozart’s ploy is somewhat similar to Haydn’s well-known joke at the end of String Quartet op. 33 no. 2.(93) Both composers engage deliberately with the expected end of the movement. But whereas Haydn’s primary joke ends the piece earlier than expected (after the first two-measure idea of the rondo refrain), Mozart composes his joke by writing a protracted conclusion, adding extra material and ending it later than expected.(94) But there are more factors in play than simply a surprise ending. Indeed, unlike the humorous oppositions we have seen so far, each of which has focused on extramusical associations borne out of topical contrast, Mozart’s joke features a topical dislocation that relies heavily on formal function.
Audio Example 8. Mozart Serenade in D major, K. 250, i, mm. 1–4
[7.5] The form-functional incongruity that arises is predicated upon the expectation that, in the very unlikely event that the composer chooses to prolong the ending after the rest, the prolongational material will be post-cadential in nature and based on ideas from the preceding measures. Neither of these expectations proves to be true. Instead, the final material is based on the thematic introduction and is therefore emphatically not post-cadential. Although the introductory material can be reinterpreted as post-cadential, it is imbued with a “beginning” function because its original (and only other) iteration was as an introductory idea (Audio Example 8).(95) This allows us to understand the functionally-driven opposition between the “ending” function we expected and the returning “beginning” idea foisted into its position. The opposition is further emphasized because the returning “beginning” idea does not relate motivically to any of the preceding post-cadential material. In addition, the effect is amplified by the mere existence of further musical material after what appeared to be the end.(96)
[7.6] Although the form-functional opposition is responsible for much of the surprise in the opening movement of K. 250, the primary source for humor is the specific material Mozart uses to recall “beginning” function: this bizarre conclusion is not simply an incongruous reuse of just any introductory idea. Mozart’s ploy is particularly effective because this specific introductory idea is a clear example of le premier coup d’archet, a topic and common musical gesture inextricably bound up with beginning rhetoric in French orchestral music of the period. Mozart found many French customs laughable, and not the least ridiculous to him was their insistence on including le premier coup d’archet in symphonic works. In a letter to his father in 1778, Mozart wrote:
I still hope, however, that even asses will find something in [Mozart’s symphony K. 297] to admire—and, moreover, I have been careful not to neglect le premier coup d’archet—and that is quite sufficient. What a fuss the oxen here make of this trick! The devil take me if I can see any difference! They all begin together, just as they do in other places. It is really too much of a joke (Anderson 1966, 2: 553).
Although Mozart wrote this letter around two years after the “Haffner” Serenade was written, he was undoubtedly aware of, and likely already held disdain for, this French habit. We may assume, therefore, that composing an extra such rhetorical flourish at the end of the work might have been particularly humorous to Mozart.
Figure 8. Mozart Serenade in D major, K. 250, i script oppositions
(click to enlarge)
[7.7] Apart from Mozart’s personal reasons for finding this humorous, a broader audience may also find his deployment of this technique humorous in K. 250 because of the opposition that occurs between the coup d’archet and the fanfare that precedes it. But the opposition Mozart sets up is not based on opposition between topics per se; rather, it involves the way a conventional beginning, used as such in this movement, supersedes what listeners initially take to have been the ending. At first, it also seems that the humor here is not predicated upon an opposition between high and low status, since both fanfare and coup d’archet are typically “high” topics. However, because Mozart deliberately and surreptitiously subverts the typical “beginning” usage of the coup d’archet and forces it into and “ending” role, he flips the usual “high” status of this topic to an opposed, “low,” parodic version of itself by using it in an inappropriate way (as Haydn did with the serioso in the finale of op. 33 no. 2).(97) While the fanfare acts “appropriately,”(98) the misbehaving coup d’archet (or mock-coup d’archet) acts “inappropriately,” creating an opposition between “appropriate” (“high”) and “inappropriate” (“low”) usages of these topics.(99) Mozart uses the tutti rest and form-functional opposition to amplify this valence shift. In the end, he makes the laughable French coup d’archet more laughable by using the topic he sees as pretentiously and thoughtlessly overdone in its normal context in a concluding role where it never belongs (see Figure 8).
[8] Conclusion
[8.1] My analytical discussions have focused on the strategy of “opposition,” which Classical composers employed to create humor in their instrumental music. The different humorous articulations above have highlighted how contrasts between semantic (topical) and syntactic (form-functional) elements create such oppositions. These contrasts pinpoint specific valence shifts from high to low stature that often occur in conjunction with shifts in emotional valence (positive or negative) and/or arousal levels (high or low). In humorous articulations involving what I labelled as “mock” topics (following Wheelock 1992), as well as the other more straightforward situations, musical context plays an important role in guiding our interpretations, allowing a fuller interpretation of the elements of contrast at work and enabling us to apprehend the humor therein. Although a composer’s personal style necessarily influences the deployment of these strategies, each of the humorous examples above can be explained according to (script) opposition.
[8.2] This game we play when trying to make sense of an incongruity between scripts leads us to an enjoyable and at times laugh-worthy musical encounter. Listening attentively and engaging in this playful game with the composer and/or performer(s) can help to recover some of the historical and cultural distance between us and Classical-era listeners. And if we are sensitive to the crafty endeavours of Classical composers and aware of the humorous potential of much of their instrumental repertoire, this kind of “listening for fun” can help us to recuperate ways of engaging with and enjoying this music that were native to original audiences and intended by the music’s creators.
James K. Palmer
University of British Columbia
School of Music
6361 Memorial Rd.
Vancouver, BC V6T 1Z2
[email protected]
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Recordings
Recordings
Haydn, Joseph, Symphony no. 60. David Blum, Esterházy Orchestra. Vanguard Classics OVC 5000 CD (1992).
Haydn, Joseph, Symphony no. 60. David Blum, Esterházy Orchestra. Vanguard Classics OVC 5000 CD (1992).
Haydn, Michael, Symphony in D, Perger 13. Johannes Goritzki, German Chamber Academy. CPO CD (1997).
Haydn, Michael, Symphony in D, Perger 13. Johannes Goritzki, German Chamber Academy. CPO CD (1997).
Huss, Manfred (conductor). Haydn: Three Theatrical Symphonies. Vienna Haydn Sinfonietta. BIS BIS-SACD-1815, CD (2010).
Huss, Manfred (conductor). Haydn: Three Theatrical Symphonies. Vienna Haydn Sinfonietta. BIS BIS-SACD-1815, CD (2010).
Mozart, Ein musikalischer Spaß, K. 522. Lincoln Center Chamber Society. Arabesque Recordings Z6617 CD (1991).
Mozart, Ein musikalischer Spaß, K. 522. Lincoln Center Chamber Society. Arabesque Recordings Z6617 CD (1991).
Mozart, Serenade in D major, K. 250. Charles Mackerras, Pražský komorní orchestr. Telarc International CD (1988).
Mozart, Serenade in D major, K. 250. Charles Mackerras, Pražský komorní orchestr. Telarc International CD (1988).
### Footnotes
* The author wishes to extend a special thanks to William Benjamin, Alan Dodson, and Poundie Burstein, as well as to the editors and anonymous reviewers of Music Theory Online, for their invaluable comments and insightful suggestions during the preparation of this article.
The author wishes to extend a special thanks to William Benjamin, Alan Dodson, and Poundie Burstein, as well as to the editors and anonymous reviewers of Music Theory Online, for their invaluable comments and insightful suggestions during the preparation of this article.
1. See, for example, Sulzer [1792–94] 1967, “Witz”; Hiller [1766–70] 1970, 3: 107, Koch [1782–93] 1969, 2: 40ff; English translation from Baker and Christensen (1995, 155); Weber 1800 col. 143; Rochlitz 1806, col. 247; and Michaelis 1807, 725ff. See also Byros 2013, Mirka 2009, Diergarten 2008, and Zenck 2008 for succinct surveys of these and other writers’ writings on the “broad aesthetic category” (Mirka 2009, 295) of humor. The first distinct treatment of “humor” appears in a discussion of the quality designated as “Comisch” in Sulzer’s Allgemeine Theorie der schönen Kunste (1967, 1: 485–86), but “humor” is addressed by many of Sulzer’s contemporaries. For example, Johann Reichardt (quoted in Sulzer (1967, 1: 485), remarking on a concert by Antonio Lolli, wrote that Lolli “was the first to show us that instrumental music by and for itself is capable of highest comical expression.” This positive view of musical humor gained currency in the writings of Weber (1800), Rochlitz (1806), and Michaelis (1807), who wrote about the comic or humorous in Leipzig’s Allgemeine musikalische Zeitung (see Mirka 2009, 295).
2. Diergarten (2008, [12]). Diergarten notes that these juxtapositions were “especially true of Viennese instrumental music starting in the 1770s” (2008, [12]).
3. Byros (2013, 240) explains that Johann Nikolaus Forkel (1801), Johann Philipp Kirnberger (1779), Johann Mattheson (1739), Johann Adolph Scheibe (1745), and Koch “described a variety of techniques for playing with convention . . . which served to create unexpected twists” and highlights Koch’s fascination with the “unexpected” (das Unerwartet).
4. Zenck (2008, 55–56). She explains that Sulzer, Koch, Weber, Rochlitz and Michaelis all agreed on this.
5. Diergarten (2008, [4]), quoting Koch (1969, 2: 1); English translation from Baker and Christensen (1995, 155). Although Koch does not refer to Haydn by name, but only to an instrumental depiction of an absent-minded person (den Zerstreuten), Diergarten (2008, [5]) clearly associates his remarks with Haydn’s Symphony no. 60, “Il distratto.”
6.Mozart oft so überaus glücklich war, wie in der komischen obligaten Behandlung der tiefen Basse gegen die hohen Tone der Blas, oder Saiteninstrumente”). See Rochlitz 1806, col. 247; my translation.
7. See especially Wheelock 1992, Burstein 1999, Huron 2006, Mirka 2009 and Byros 2013. See also Sisman 1990, 312, who notes Haydn’s pervasive use of a “clash of styles” in Symphony no. 60, “Il distratto.”
8. Byros 2013, 219. Modes of communication in music are the focus of Byros’s article, which explores compositional play and discusses humorous utterances with an aim to recuperate a historical listening practice. Byros is interested in the impact of numerous syntactic and semantic axes, and formal functions and topics are among these, but he focuses his analytical efforts squarely on galant schemata and the ways in which they can “become a source of metaphoric forms of communication such as wit and humor” (2013, 219).
9. According the linguist Thomas Veatch, “Humor is an inherently mysterious and interesting phenomenon which pervades human life” (2004, 161). The issue is further clouded by the existence of three often conflicting historical/methodological branches of humor—incongruity, superiority, and relief theories (introduced below)—and, more recently, the diverging strands of incongruity theory in linguistics. See Raskin 1985 and 2008; Attardo and Raskin 1991; Attardo 1994, 1998, and 2001; Davies 2004; Brock 2004; Veatch 2004; Norrick 2004; Veale 2004; and Morreall 2009 for discussions of the elusive nature of humor.
10. “Wit,” in particular, is a ubiquitous topic in critical, musicological, and analytical literature on Haydn. See, for example, Tovey 1949 and 1981; Wheelock 1992; Burstein 1999; Mirka 2009; Hepokoski 2006; and Mastic 2015. Many other authors, such as Meyer (1956), Webster (2004), and Agawu (1991 and 1999) use “wit” in different ways. Still other scholars use “wit” in discussions of works by Mozart and Beethoven. Levy (1992), for example, discusses humor in two of Beethoven’s op. 33 Bagatelles (see also Laul 2000), while other authors such as Lister (1994), Lowry (1974), and Godt (1986), for example, discuss humor in Mozart’s Ein musikalischer Spaß and elsewhere. Since not all wit is humorous—some is merely clever without necessarily being funny or amusing—but some humor is witty, the same kinds of musical incongruity that create wit also often create humor. Thus, the examples of wit found in this study are only those that fall under the umbrella of humor.
11. See Wheelock 1992, 21 and Diergarten 2008, [12]. See also Sulzer’s entries on “Witz” and “Laune” (1967) and Michaelis, who observed “das Launige (witty or jocular) in the character of particular rhythms or in the ‘unexpected entrance of certain voices or instruments’” (1807, col. 726, quoted in Zenck 2008, 56).
12. Wheelock 1992, 21–32. Like Wheelock, I seek “to locate the strategies of musical jests” (1992, x), but my theoretical approach to musical humor aims to uncover an economical set of principles with relatively broad explanatory power. However, the role of musical topics and syntactic elements in producing contrast in examples of Classical wit and humor is a point of contact in our work. More recently, Mastic (2015) addressed eighteenth-century notions of “wit” in the deformations found in Haydn’s sonata form recapitulations. Mastic (2015) approaches eighteenth-century “wit” from a contemporary listener’s point of view. He focuses, however, on a broader, less humorous, kind of wit than I do. Mirka focuses on metric witticisms in Haydn’s string quartets, approaching humor as a “broad aesthetic category” and considers metric manifestations of comic “absent-mindedness”—an idea she draws from Friedrich August Weber’s writings (Mirka 2009, 295).
13. Ein musikalischer Spaß is replete with gags like these. For discussions of various passages in K. 522, see Lister 1994, Godt 1986, Lowry 1974, and Kalisch 1992.
14. Sisman (1990, 311ff.) discusses this symphony at length.
15. The concept of “play with proportion” cannot be properly addressed within the scope of this article, but it factors heavily into Goeth’s (2013, 247) discussion of humor in music where she invokes notions of “misproportion,” “oversize,” and “excessive repetition.”
16. Burstein 1999, 69. Most modern humor theorists would agree with this statement. In addition, Burstein may have been the first music scholar to suggest that incongruity is not enough to describe musical humor, but he does not specify which authors have claimed that the reconciliation of incongruities is sufficient for humor in music.
17. Burstein 1999, 69. Burstein’s claim finds a near-analog in Lowe’s work when she notes “the expressive incompatibility of the tempestuous and the trivial” (2007, 142).
18. See Raskin 1985 and 2008, Attardo and Raskin 1991, and Attardo 1994, 1997, and 2001. Implication-realization approaches to music theory largely draw on this conception of humor. See, for example, Narmour 1992 and Meyer 1956. Along with incongruity, two other schools of thought in humor theory, superiority and relief, can be relevant to musical humor in certain special contexts (see, for instance, Example 6 below). For overviews of these three branches of humor theory, see Palmer 2015, 22–29 and Morreall 2009, 1–26. For excellent summaries of humor research in other disciplines (linguistics, literature, sociology, and psychology), see Raskin 2008 and Paulson 2011, 11–17.
19. Goeth 2013, 238. Goeth’s view is similar to Kant’s original formulation of incongruity as a sign of thwarted expectation that comes from the introduction of surprising oddities into otherwise typical discourse (see Kant [1790] 1951). In his view, humor arises from a “sense of falling short of . . . expectations” (Dalmonte 1995, 173). See also Kierkegaard 2004, Bergson 1911, and Morreall 1983, 1987, 1989a, and 2009. Kierkegaard followed many of Kant’s ideas of incongruity closely, whereas Bergson and (much more recently) Morreall incorporated the concept of “incongruity” into their specific approaches to humor. For Bergson, incongruity is most manifest as the dissonance between the mechanical and living. This idea—his most influential—is borne out in the phrase most often translated as “something mechanical encrusted on the living” (“quelque chose de mécanique plaqué sur du vivant”) and was adopted by Levy (1992) and Dalmonte (1995).
20. These writings also discuss many aspects of humor that focus too specifically on verbal humor to be directly applicable here. See Attardo 1994 for a summary of humor research in the field of linguistics.
21. “Script opposition,” originally called “script overlap” by Raskin (1985), has been widely adopted by Raskin and others in subsequent writings. See Raskin 2008, Attardo and Raskin 1991, Attardo 1994, Triezenberg 2004, Brock 2004, Norrick 2004, and Davies 2004.
22. Raskin (1985, 113–14 and 127; quoted in Attardo 1994, 204). The “high/low stature” opposition is particularly common and effective in music when it arises between Classical topics.
23. Triezenberg 2004, 412. Raskin and Attardo have since revised their earlier theories to incorporate more of linguistics in general (see Attardo and Raskin 1991), but such further theoretical specializations of script opposition have become too linguistically robust to be of any practical music-theoretic application (see especially Attardo 1994). Thus, I employ “script opposition” only, since it lies in a Goldilocks zone of explanatory power: the concept is neither too psychologically abstract, nor too field specific.
24. Rochlitz may have provided the earliest mention of opposition with respect to musical humor when he wrote, “[P]urpose and means should ‘oppose each other abruptly’” (Zenck 2008, 56, quoting Rochlitz 1806, col. 7).
25. Raskin 1985, 111. The normal/abnormal opposition is one of three general types of opposition. The other two types are “actual situation”/non-actual, non-existing situation” and “possible, plausible situation”/“fully or partially impossible or much less plausible situation” (111).
26. Attardo 1994, 204. As I explain below (see esp. paragraphs 5.5–5.11), such script oppositions are usually most strongly implied by the musical topics involved in such passages. The analytical discussions below explain how such script oppositions are created. See esp. Figures 4–8 for brief lists of some possible script oppositions.
27. In Palmer (2015), I contend that two strategies for the deployment of formal functions and topics are especially prominent in the Classical instrumental style: “opposition” and “excess.” Both of these notions factor heavily in many scholarly musical discussions of humor (see Burstein 1999, Goeth 2013, and also Sisman 1990, Dalmonte 1995, and Huron 2004).
28. Burstein 1999, 72. Burstein demonstrates the subtle extent to which Haydn employs harmonic connections and motivic parallelisms to produce humor in Symphonies nos. 58, 78, 83, 90, and 93.
29. Musical syntax also encompasses prolongational structure (see Burstein 1999 and Lerdahl 1983) and schemata (see Gjerdingen 2007, Byros 2013), but this study focuses primarily on connections between musical topics and local functional incongruities between beginnings, middles, and ends. The semantic content of music is far less specific and robust than the semantic content of language. See Agawu 1991, 33 and also Swain 1996.
30. The role of formal functions in creating musical humor or wit is mentioned by Byros (2013, 219), Goeth (2013, 237–38 and 247), Levy (1992, 239), Huron (2004), Sisman (1990, 320), Mirka (2009, 309), Hepokoski and Darcy (2006, 66–67), and Almén (2008, 169); See also Edwards (1998) and Burstein (2015). The role of musical topics in musical humor is explicitly discussed in Byros (2013, 219–20 and 240–43) and mentioned in Paulson (2011, 95–96 and 137–40), Agawu (1999, 156), Diergarten (2008, [13]), and Sisman (1990, 320).
31. Topics are not always involved: other more general or specific scripts—other semantic information that does not properly fall within the realm of Ratnerian topics—can perform a similar context-clarifying role. The extent to, and manner in, which this is possible is clarified alongside Example 5.
32. Hepokoski 2006, 66. Goeth notes that, as in verbal humor, humor in music arises “through the dislocation of phraseme-like patterns, and . . . through parody” (2013, 235) and explains, “the term phraseme is used for conventionalized expressions also referred to as idioms, collocations, fixed or set expressions” (237; Goeth’s emphasis). As such, formal functions could be considered a kind of phraseme in music, but the analogy is better applied to “conventionalized expressions” like topics and schemata.
33. Byros 2013, 241; see also Mirka 2009, 302–3. For example, Mirka writes about a passage in the first movement of Haydn’s String Quartet op. 50 no. 2: “[I]n the most immediate context of the preceding canons this [Ländler] topic brings a dramatic contrast between high and low style” (2009, 273).
34. Lowe 2007, 134. In her thorough discussion of Symphony no. 93, Lowe notes its “unusually frequent and salient juxtapositions of stylistic registers, points of contact between high, middle, and low musical expressions that would have been clearly audible and immediately apparent to Haydn’s late-eighteenth-century listener” (136).
35. Huron notes, for example, that a “pivotal factor” for creating specifically musical humor is “the underlying extramusical context and social situation” (Byros 2013, 240n100; Byros’ emphasis and parentheses, quoting Huron 2006, 287). He also explains, “In the case of musical humor, the context appears to be overtly one of playfulness and parody” (Huron 2006, 288). According to Huron, “playfulness” may be signaled by “the interjection of ‘low art’ gestures into ostensible ‘high art’ contexts” (288). The notion that contextually inappropriate compositional use of “high” or “low” elements can create humor resonates in the work of Byros (2013) and Diergarten (2008). Huron (2006, 288) also implies that a common source of parody is “the use of absurd sounds in an ostensibly normal musical context.” The flatulent expulsion from the bassoons in the second movement of Haydn’s Symphony no. 93 is an excellent example.
36. Furthermore, not all topical oppositions create humor. A prime example is the valence shift between the minuet and trio in the third movement of Haydn’s Symphony no. 67, which creates both negative/positive valence and high/low stature oppositions, but is not likely to be perceived as humorous (see Palmer 2015, 72–75). Lowe (1998) and Burnham (2005) discuss how motions toward lower, more popular styles (valences) occur often at different levels of structure in Haydn’s symphonies. Burnham (2005, 63), in particular, notes that symphonic finales commonly possess a “downhill trajectory” from higher to lower styles, while Lowe (2002) discusses the pivotal, and often ironic, role that Haydn’s minuets play in the symphony cycle.
37. Musical topics are italicized (as they are throughout this article), while genres and adjectives appear in normal script.
38. A discussion of how “low” elements move to “high” elements in an ironic manner is introduced alongside Examples 3a and b.
39. See Caplin 2005. For example, some common topical/functional affiliations are French overture with beginning, Sturm und Drang with middle, and lament with ending (Caplin 2005, 115). However, the topics generally referred to as “genre” or “style topics”—for example, aria, sarabande, or ombra—frequently have no clear functional affiliation (115).
40. According to Sisman, nearly every reference to Haydn’s Il distratto being composed specifically for a play suggests that such a deliberate compositional pairing was rare (Sisman 1990, 301). For example, she mentions briefly a record that a “new” (unknown) Mozart symphony was performed between acts of Die eingebildeten Philosophe on July 26, 1785, but this symphony—like most performed in the theater—was not written specifically to accompany the drama (Sisman 1990, 300–1). Although Symphony no. 60’s nickname often appears as the Italian Il distratto, its original incarnation as incidental music was written by Haydn to accompany Regnard’s French play, Le distrait. The accepted German equivalent was Der Zerstreute.
41. Diergarten 2008, [4] and [8]; from Koch 1787. Although Koch does not refer to Haydn or to the symphony by name, Diergarten demonstrates how Koch’s description of the instrumental depiction of an absent-minded person (den Zerstreuten) clearly levels his criticism at Haydn’s composition. Diergarten’s admirable sleuthing further reinforces this connection, noting that Koch’s paraphrase of Horace’s “at times even Homer nods off” strongly references an “important, recognized, and renowned composer”: Haydn is the clear referent in this case (2008, [5]).
42. Sisman observes of the Classical symphony: “The symphony . . . could move easily among social levels and locations, and could appeal by the very variety of topics at those levels” (2014, 98).
43. Comic “absent-mindedness” is a particular focus of Haydn’s Symphony no. 60. Mirka tackles Haydn’s use of “absent-mindedness,” drawing on Friedrich August Weber’s writings (2009, 295).
44. Even the musette’s minor mode is undercut slightly by the first violin’s opening major tetrachord.
45. Rosen 1972, 162–63; see also Webster 2004, 242. Sisman notes the “single most pervasive quality of a unison passage: its aura of authoritative control” (Sisman 2014, 108, quoting Levy 1982, 507).
46. Similarly incongruous musette passages where instruments seem to get “carried away” can be found in the fourth movement of Mozart’s String Quintet K. 516 (mm. 56–58), the fourth movement of String Quartet K. 590, and in the first movement of Haydn’s String Quartet op. 20 no. 3 (mm. 24, 107, 145, 151, and 240). Haydn is also fond of writing string quartet cello parts that sound absent-minded (see especially the second movement of op. 77 no. 2 and first movement of op. 33 no. 4).
47. Sisman 1990, 315. In the same discussion, Sisman also explains that, unlike in this trio, “Haydn [usually] played with such conventions [of phrase rhythm] in minuets . . . by an excess of activity, especially short motives in hemiola (e.g., String Quartet op. 33, no. 5).”
48. In a sense, the minor mode heightens this opposition because, as Sutcliffe explains, “Use of the minor mode not only typically brings certain kinds of material to the fore, it also tends to mute topical variety” (Sutcliffe 2014, 131). In other words, since the serioso and musette are forcibly juxtaposed in a minor context where such juxtapositions are usually “muted,” the contrast and resulting opposition are more pronounced.
49. Haringer 2014, 205, quoting Sulzer [1792–94] 1967, 2: 421. Sulzer also notes, “The dance under this name is designed for naïve rustic entertainments but it can portray noble shepherd characters as well as lowly peasants” (Haringer 2014, 205, quoting Sulzer [1792–94] 1967, 2: 421). Here, Haydn’s harmonic stasis clearly portrays the “lowly peasants.”
50. The three-node opposition networks I employ somewhat resemble Lawrence Zbikowski’s conceptual integration networks (2002, 63–95). While both types of networks highlight contrasting elements in upper nodes that are then “blended” into the lower node, the interactions between the nodes communicate different things. For example, while Zbikowski’s networks can map the domains of text and music onto one another to create a blended metaphorical space where the two are intertwined, my upper nodes begin with a “blended” space, laden with metaphor (topics and human characteristics). Furthermore, any blending of these upper nodes into the lower node cannot properly be called “blending” since, psychologically speaking, “script oppositions” are not a blend, but rather a persistent clash.
51. See Ratner 1980, 20. I have labelled this atypical topic “march” (in quotations) in Example 3b. Monelle (2000, 54–55) provides examples in Tchaikovsky’s fifth and sixth symphonies where march and waltz topics are mixed, but Haydn’s is the only Classical example I have encountered where a march implication occurs in a thoroughly triple-time movement. Monelle (2006, 115–16) discusses an example from Jean-Baptiste Lully in which the opposite occurs: a triple-time intrusion (a measure in ${\text{}}_{2}^{3}$) occurs in a march that is otherwise in duple meter.
52. Klorman (2013, 127–28) provides a concise discussion of analytical uses for Monahan’s (2013) “avatars” in Mozart’s music.
53. See Agawu 1991, 34 for a discussion of “topical overlap.”
54. This depends, of course, on the performers’ choices regarding this effect. I believe the best strategy for performing humor in this passage is to almost completely drown out the oboes as they finish. Most performances seem to allow the oboes to be heard relatively clearly to the end of their phrase, while a select few (my favorites) almost completely obscure their conclusion with the interrupting unison entry. Listen, for example, to the recording by the Haydn Sinfonietta Wien under Manfred Huss (2010) for a particularly effective (if slightly slow) example.
55. One might interpret this march in a positive light right from the outset, but because of its interrupting or intruding role, my initial impression of this march is negatively valenced before it is playfully contradicted. Both the minuet and march are “high” topics, so no obvious valence shift between social statuses occurs here. The particular social implications of this opposition are discussed below.
56. Haringer discusses how marches “can even be in triple meter, if only they preserve their pride and warlike nature” (2014, 200).
57. For example, the minuet second movement of Haydn’s String Quartet op. 77 no. 2 often strongly implies duple meter.
58. For a different reading of the cadence structure here, see Burstein 2014, 16.
59. Remarkably, Koch (1787, 41) provides a very similar “recipe” for depicting absent-mindedness in instrumental music, where he asks: “‘How, for example, does the composer represent an absent-minded person in an instrumental piece?’ and answers: ‘He connects sections which properly do not belong together; he makes a triple rhythm [Rhythmus] where we expect a duple; without reason he alternates the minor mode with the major, and so on’” (quoted in Mirka 2009, 300, translated by Baker and Christensen 1995, 155).
60. The minuet section has some oddities in it as well, however. The dotted march makes its first appearance in m. 24 and the absent-minded minuet is suggested from mm. 25 to 29. But neither of these topics is nearly as emphatically or bizarrely employed as they are in the trio.
61. In this trio, we might even hear the Chevalier as one of Eero Tarasti’s “negactants”: an agent-antagonist that acts in opposition to the central work-persona (Tarasti 1994). Klorman 2013, 147 discusses the role of “negactants” in Mozart’s string quartets.
62. This character-driven interpretation finds a precedent in Haydn’s own statements. That is, the “explicit connection [made by Forkel] of the theater style with moral sentiments finds an echo in Haydn’s statement to his biographer Griesinger that he often tried to portray ‘moral characters’ in his symphonies” (Sisman 1990, 310).
63. Sisman observes of the second movement, “The parody of a French dance introduced in the development section probably satirizes the Chevalier” (Sisman 1990, 314).
64. Die Violinen stimmen um von f auf g (mm. 23–30).
65. There is a later hindrance, which I do not discuss here, but which Wheelock (1992, 160) and Webster (2004, 245) have noted and discussed: further along in the movement Haydn includes a notably out-of-the-ordinary quotation of “The Night Watchman’s Song.” While Haydn’s “Night Watchman” quotation is surely added for comic effect (as Webster notes), it has—in my opinion—more the effect of general amusement or entertainment that I associate with Burnham’s (2005, 61) “clown stepping onstage” than with humorous opposition, per se. That is, it provides contrast at a point where we expect contrast, but does so in a slightly odd, somewhat sensible, and surely programmatic way.
66. Preßburger Zeitung, 23 July, 1774, quoted in Wheelock 1992, 155, quoting Angermüller 1978, 88. Winkler (2000, 103–16) also discusses the gag-like quality of Haydn’s tuning passage and relates it to the “surreal gags” that can be found in modern music.
67. Preßburger Zeitung, 23 July, 1774, quoted in Wheelock 1992, 155, quoting Angermüller 1978, 88.
68. Theaterwochenblatte für Salzburg no. 21 (27 January 1776), quoted in Wheelock 1992, 155–58. See Angermüller 1978, 90–92 for the complete review.
69. Comparable oddities in K. 522 occur in the violin cadenza at the end of the third movement, the horns’ parallel-thirds passage in the second movement (m. 17–20), and the polytonal cadence that concludes the serenade.
70. The notion of a musical “frame” and the idea of an out-of-frame reference in Haydn are discussed by Webster (2004).
71. The idea that Haydn is referring to something outside of the composition proper strongly suggests a connection to the extraneous comic references of Laurence Sterne’s Tristram Shandy. See Bonds 1991 and Irving 1984 for specific insights on connections between Haydn and Sterne.
72. Attardo (1996, 88) calls higher-level scripts like these “complex scripts.”
73. Gjerdingen 2007, 45. This Prinner is echoed in an entertainingly skewed way when Haydn quotes the “Night Watchman” song later in the movement.
74. “Target” is the term used in linguistic theories of verbal humor to refer to the “butt” of the joke. See Attardo and Raskin 1991, 301–2 and Davies 2004, 377.
75. Sisman 1990, 318; my emphasis. According to Sisman, “This detail from La Bruyère’s character sketch, absent in Regnard’s original play, is restored in the German text” (1990, 318). See Sisman (1990, 320) for a point-form summary of the musical events she finds representative of Haydn’s musical interaction with characters or events in the play.
76. In Michael Haydn’s case, this contemporary listener was Leopold Mozart (see Anderson 1966, 1: 297) and below.
77. Rondo couplets are often differentiated topically from refrains, but the repeated refrains themselves tend to present less variety of topics. Topical contrast manifests most often as a character change occurring alongside a tonal/modal shift (i.e. to minore in major-mode rondos or maggiore in minor-mode ones). See Caplin 1998, 231–35.
78. Varied refrain repetitions most often contrast in phrase structure or thematic design, where later refrains are often “abridged or incomplete” (Caplin 1998, 235). Topical change is not a requirement, but is certainly a likely possibility. Consider, for example, the nearly obligatory minore variation found in most Classical variation forms, which not only changes the tonality of the variation, but tends to alter other characteristics that influence the variation’s topic. See Caplin 1998, 218.
79. The reverse is also possible: so much of the same topic in the opening three variations might suggest more potential for topical change. Regardless, there is no way to expect the drastic contrast Michael Haydn presents here.
80. Michael Haydn may also have been thinking of more specific programmatic elements. When writing this theater symphony, he likely decided to include references to the story of Voltaire’s Zaïre, for which this symphony was written as accompaniment. As his brother Joseph did in Il distratto, Michael Haydn seems to have turned to musical topics to represent key characters in the play and, to some extent, the actions or events that involved them. One might, for example, suppose an affiliation between the conservative style and the non-combative character of the female prisoner Zara (Zaïre). Likewise, the Turkish music could easily represent the Turkish sultan Osman. Michael Haydn returns to the Turkish topic later in the movement in order to get the most out of his comic device.
81. The effect here is similar to the topical overlap between the minuet and march in the third movement of Haydn’s Symphony no. 60 (see Example 3b).
82. See Monelle 2006, 117–23 for a concise discussion of “Turkish music.” See also Hepokoski 2006, 397 and 401.
83. This letter was written in 1781 and refers to the character from Mozart’s opera Die Entführung aus dem Serail, which was completed the following summer. Webster (2004, 245) notes that Turkish elements were more commonly employed in operatic works than in instrumental ones. Mozart also treats the topic with increasing levels of impropriety in the seventh movement of his Serenade in B, K. 361. Other examples of the Turkish topic occur in the third movement of Mozart’s Sonata in A K. 331, Beethoven’s The Ruins of Athens, and the finale of his Ninth Symphony. See Monelle 2006, 6.
84. Head 2005, 83. A similar reference to the exotic occurs with the sudden “intrusions” of a Hungarian-inflected minore in the final of Haydn’s String Quartet op. 33 no. 3; see Palmer 2015, 206–10.
85. The relief branch of humor theory was primarily articulated by Herbert Spencer (1883) and later by Sigmund Freud (1960), and holds that the sense of relief or release that comes with the removal of a threat can lead to laughter—one possible physiological response tied to humor (Huron 2006, 287). In other words, “Release/relief theories perceive humor and laughter as a release of the tensions and inhibitions generated by societal constraints” (Carrell 2008, 313). Although some modern humor theories include elements of relief theories, “‘pure’ relief theorists, explaining all of humor and laughter as release of tension or ‘safety valve’, cannot be found anymore in humor scholarship” (Kuipers 2008, 362).
86. Superiority theory is the most ancient of the primary branches of humor theory. It can be traced back to Plato and Aristotle (see Bremmer 1997 and Smuts 2006), but its strongest proponent was Thomas Hobbes (1840). Superiority theories hold that the audience of a joke will find humor in the feeling of superiority over the joke’s “target”—the guy who walked into the bar, for example. But, while superiority may be necessary for our reaction to a shift between high and low stature, it is not fundamental to our perception of humor.
87. Many Classical composers engaged in what could be called “racist” musical jokes. Racial humor appears to have been a particularly common manifestation of the ubiquitous Classical valence shift from high to low status, especially when focused on the Turks, Balkans, or Hungarians. This fact should not be surprising to modern audiences, who know that many comedy acts once centered on the deprecation of other cultures (and some still do). Webster discusses the “deliberate invocation of ‘ethnic’ moods [which] is found primarily in two contexts: ‘Turkish’ elements in operas, and Balkan/Hungarian style in instrumental music” (Webster 2004, 245). Mirka also notes, “the mix of Gypsy and Hungarian national elements . . . behind the metrical dissonance in Haydn’s theme” from the finale of String Quartet op. 55 no. 2 (Mirka 2009, 275). Elsewhere, Allanbrook explains how “absurd horn calls . . . mutate rapidly into a Turkish-style minor” in Haydn’s Piano Sonata in E Major, Hob. XVI:52 (Allanbrook 2002, 210).
88. See Palmer 2015 for a discussion of humorous aspects of the third movement of K. 250.
89. Tovey [1935] 1981, 177. The second “Haffner” Serenade in D Major, K. 385, was written for the coronation of Siegmund Haffner the younger (brother to Marie Elisabeth and friend of Mozart), and in a later incarnation, became Mozart’s famous “Haffner” Symphony no. 35 in D Major, K. 385.
90. The Loreto convent first performed the work on the evening of 21 July, 1776 (the eve of the wedding) in the summerhouse in the garden of the Paris-Lodrongasse, near the Mozarts’ house (Eisen and Keefe 2006, 205; Sadie 2006, 403). The eight-movement serenade totals over an hour of music and on the occasion was flanked by Mozart’s March in D Major, K. 249.
91. The exception to this humorlessness is “one of Mozart’s most Rossinian themes” (Tovey [1935] 1981, 178), which begins first in m. 80 and sounds excessive because of its motivic inanity. The (post-cadential) form-functional redundancy is conspicuous and definitely makes me laugh. The passage is full of dull, scalar motion and, while it is unnecessarily repetitive, its overtly playful character suggests more a kind of “clowning around” than the conspicuous and more jarring form-functional redundancies often found in Haydn and Beethoven’s excessive passages. This is a prime example of what we might call “Mozartian excess.”
92. Haydn is best known for the grand-pause gambit, but he and Mozart both use it for humorous (or at least bizarre) ends in many works. Mozart’s most conspicuous uses occur in the finale of String Quartet no. 16, K. 428 (he also quotes Haydn’s “How do you do?” from the first movement of op. 33 no. 5 in this “Haydn Quartet”) and the finale of String Quintet no. 5, K. 593. The most emblematic cases written by Haydn occur in the second movement of his Symphony no. 101 “The Clock” and in many string quartets, including the second movement of String Quartet op. 55 no. 2, the first movement of op. 33 no. 3, the first movement of op. 33 no. 5, the first movement of op. 5 no. 3. See also Mirka 2009, 304.
93. This play with “ending” is also present in Beethoven’s Bagatelle op. 33 no. 2. See Levy 1995 and Palmer 2015.
94. The “joke” immediately before Haydn’s notated conclusion, however, is nearly identical to Mozart’s: a rest that appeared to be the end was not the real ending.
95. This material also answers the question “What does the Jupiter Symphony sound like upside down?” since the Mozart’s Symphony no. 41 has an incredibly similar opening coup d’archet motion to the tonic that ascends.
96. Of course, transplanting formal functions in this way is often not humorous. For example, Beethoven’s conclusion of the opening movement of his Symphony no. 8 is undoubtedly clever without being humorous: the final idea is motivically similar to the other closing material, and all of Beethoven’s orchestral rests suspend the dominant (and thus are unable to suggest a possible conclusion for the work). The first movement of Haydn’s String Quartet op. 33 no. 5 also reuses an opening idea at the conclusion without an explicitly humorous goal. This relatively common technique is frequently described as “clever” or “witty,” but is not necessarily humorous (see Levy 1995, 155 and Hepokoski 2006, 66–67).
97. According to Goeth (2013, 251), this kind of “context-quotation clash is one of the most common tools for parody construction in music.”
98. It is true that fanfares often occur with beginning function, but they are also frequently (and therefore appropriately) found at endings. According to Caplin (2005, 117), “[the fanfare’s] arpeggiated melodic configurations create a degree of harmonic stasis appropriate for a formal beginning. But the potential for fanfare motives both to ascend and to descend also makes them useful in other formal contexts.” Caplin (2005, 117) also notes that “fanfare gestures seem frequently to lead to a cadence, especially in Mozart” and that the fanfare’s “more general characteristics” (as compared to the coup d’archet) give it more functional flexibility.
99. The opposition here between “appropriate” and “inappropriate” is analogous to the common high-level script opposition between “normal, expected state of affairs” and “abnormal, unexpected state of affairs” (Raskin 1985, 111). There is no clear shift between negative and positive emotional implications in this example, nor is one needed because of the strong high-to-low shift.
See, for example, Sulzer [1792–94] 1967, “Witz”; Hiller [1766–70] 1970, 3: 107, Koch [1782–93] 1969, 2: 40ff; English translation from Baker and Christensen (1995, 155); Weber 1800 col. 143; Rochlitz 1806, col. 247; and Michaelis 1807, 725ff. See also Byros 2013, Mirka 2009, Diergarten 2008, and Zenck 2008 for succinct surveys of these and other writers’ writings on the “broad aesthetic category” (Mirka 2009, 295) of humor. The first distinct treatment of “humor” appears in a discussion of the quality designated as “Comisch” in Sulzer’s Allgemeine Theorie der schönen Kunste (1967, 1: 485–86), but “humor” is addressed by many of Sulzer’s contemporaries. For example, Johann Reichardt (quoted in Sulzer (1967, 1: 485), remarking on a concert by Antonio Lolli, wrote that Lolli “was the first to show us that instrumental music by and for itself is capable of highest comical expression.” This positive view of musical humor gained currency in the writings of Weber (1800), Rochlitz (1806), and Michaelis (1807), who wrote about the comic or humorous in Leipzig’s Allgemeine musikalische Zeitung (see Mirka 2009, 295).
Diergarten (2008, [12]). Diergarten notes that these juxtapositions were “especially true of Viennese instrumental music starting in the 1770s” (2008, [12]).
Byros (2013, 240) explains that Johann Nikolaus Forkel (1801), Johann Philipp Kirnberger (1779), Johann Mattheson (1739), Johann Adolph Scheibe (1745), and Koch “described a variety of techniques for playing with convention . . . which served to create unexpected twists” and highlights Koch’s fascination with the “unexpected” (das Unerwartet).
Zenck (2008, 55–56). She explains that Sulzer, Koch, Weber, Rochlitz and Michaelis all agreed on this.
Diergarten (2008, [4]), quoting Koch (1969, 2: 1); English translation from Baker and Christensen (1995, 155). Although Koch does not refer to Haydn by name, but only to an instrumental depiction of an absent-minded person (den Zerstreuten), Diergarten (2008, [5]) clearly associates his remarks with Haydn’s Symphony no. 60, “Il distratto.”
Mozart oft so überaus glücklich war, wie in der komischen obligaten Behandlung der tiefen Basse gegen die hohen Tone der Blas, oder Saiteninstrumente”). See Rochlitz 1806, col. 247; my translation.
See especially Wheelock 1992, Burstein 1999, Huron 2006, Mirka 2009 and Byros 2013. See also Sisman 1990, 312, who notes Haydn’s pervasive use of a “clash of styles” in Symphony no. 60, “Il distratto.”
Byros 2013, 219. Modes of communication in music are the focus of Byros’s article, which explores compositional play and discusses humorous utterances with an aim to recuperate a historical listening practice. Byros is interested in the impact of numerous syntactic and semantic axes, and formal functions and topics are among these, but he focuses his analytical efforts squarely on galant schemata and the ways in which they can “become a source of metaphoric forms of communication such as wit and humor” (2013, 219).
According the linguist Thomas Veatch, “Humor is an inherently mysterious and interesting phenomenon which pervades human life” (2004, 161). The issue is further clouded by the existence of three often conflicting historical/methodological branches of humor—incongruity, superiority, and relief theories (introduced below)—and, more recently, the diverging strands of incongruity theory in linguistics. See Raskin 1985 and 2008; Attardo and Raskin 1991; Attardo 1994, 1998, and 2001; Davies 2004; Brock 2004; Veatch 2004; Norrick 2004; Veale 2004; and Morreall 2009 for discussions of the elusive nature of humor.
“Wit,” in particular, is a ubiquitous topic in critical, musicological, and analytical literature on Haydn. See, for example, Tovey 1949 and 1981; Wheelock 1992; Burstein 1999; Mirka 2009; Hepokoski 2006; and Mastic 2015. Many other authors, such as Meyer (1956), Webster (2004), and Agawu (1991 and 1999) use “wit” in different ways. Still other scholars use “wit” in discussions of works by Mozart and Beethoven. Levy (1992), for example, discusses humor in two of Beethoven’s op. 33 Bagatelles (see also Laul 2000), while other authors such as Lister (1994), Lowry (1974), and Godt (1986), for example, discuss humor in Mozart’s Ein musikalischer Spaß and elsewhere. Since not all wit is humorous—some is merely clever without necessarily being funny or amusing—but some humor is witty, the same kinds of musical incongruity that create wit also often create humor. Thus, the examples of wit found in this study are only those that fall under the umbrella of humor.
See Wheelock 1992, 21 and Diergarten 2008, [12]. See also Sulzer’s entries on “Witz” and “Laune” (1967) and Michaelis, who observed “das Launige (witty or jocular) in the character of particular rhythms or in the ‘unexpected entrance of certain voices or instruments’” (1807, col. 726, quoted in Zenck 2008, 56).
Wheelock 1992, 21–32. Like Wheelock, I seek “to locate the strategies of musical jests” (1992, x), but my theoretical approach to musical humor aims to uncover an economical set of principles with relatively broad explanatory power. However, the role of musical topics and syntactic elements in producing contrast in examples of Classical wit and humor is a point of contact in our work. More recently, Mastic (2015) addressed eighteenth-century notions of “wit” in the deformations found in Haydn’s sonata form recapitulations. Mastic (2015) approaches eighteenth-century “wit” from a contemporary listener’s point of view. He focuses, however, on a broader, less humorous, kind of wit than I do. Mirka focuses on metric witticisms in Haydn’s string quartets, approaching humor as a “broad aesthetic category” and considers metric manifestations of comic “absent-mindedness”—an idea she draws from Friedrich August Weber’s writings (Mirka 2009, 295).
Ein musikalischer Spaß is replete with gags like these. For discussions of various passages in K. 522, see Lister 1994, Godt 1986, Lowry 1974, and Kalisch 1992.
Sisman (1990, 311ff.) discusses this symphony at length.
The concept of “play with proportion” cannot be properly addressed within the scope of this article, but it factors heavily into Goeth’s (2013, 247) discussion of humor in music where she invokes notions of “misproportion,” “oversize,” and “excessive repetition.”
Burstein 1999, 69. Most modern humor theorists would agree with this statement. In addition, Burstein may have been the first music scholar to suggest that incongruity is not enough to describe musical humor, but he does not specify which authors have claimed that the reconciliation of incongruities is sufficient for humor in music.
Burstein 1999, 69. Burstein’s claim finds a near-analog in Lowe’s work when she notes “the expressive incompatibility of the tempestuous and the trivial” (2007, 142).
See Raskin 1985 and 2008, Attardo and Raskin 1991, and Attardo 1994, 1997, and 2001. Implication-realization approaches to music theory largely draw on this conception of humor. See, for example, Narmour 1992 and Meyer 1956. Along with incongruity, two other schools of thought in humor theory, superiority and relief, can be relevant to musical humor in certain special contexts (see, for instance, Example 6 below). For overviews of these three branches of humor theory, see Palmer 2015, 22–29 and Morreall 2009, 1–26. For excellent summaries of humor research in other disciplines (linguistics, literature, sociology, and psychology), see Raskin 2008 and Paulson 2011, 11–17.
Goeth 2013, 238. Goeth’s view is similar to Kant’s original formulation of incongruity as a sign of thwarted expectation that comes from the introduction of surprising oddities into otherwise typical discourse (see Kant [1790] 1951). In his view, humor arises from a “sense of falling short of . . . expectations” (Dalmonte 1995, 173). See also Kierkegaard 2004, Bergson 1911, and Morreall 1983, 1987, 1989a, and 2009. Kierkegaard followed many of Kant’s ideas of incongruity closely, whereas Bergson and (much more recently) Morreall incorporated the concept of “incongruity” into their specific approaches to humor. For Bergson, incongruity is most manifest as the dissonance between the mechanical and living. This idea—his most influential—is borne out in the phrase most often translated as “something mechanical encrusted on the living” (“quelque chose de mécanique plaqué sur du vivant”) and was adopted by Levy (1992) and Dalmonte (1995).
These writings also discuss many aspects of humor that focus too specifically on verbal humor to be directly applicable here. See Attardo 1994 for a summary of humor research in the field of linguistics.
“Script opposition,” originally called “script overlap” by Raskin (1985), has been widely adopted by Raskin and others in subsequent writings. See Raskin 2008, Attardo and Raskin 1991, Attardo 1994, Triezenberg 2004, Brock 2004, Norrick 2004, and Davies 2004.
Raskin (1985, 113–14 and 127; quoted in Attardo 1994, 204). The “high/low stature” opposition is particularly common and effective in music when it arises between Classical topics.
Triezenberg 2004, 412. Raskin and Attardo have since revised their earlier theories to incorporate more of linguistics in general (see Attardo and Raskin 1991), but such further theoretical specializations of script opposition have become too linguistically robust to be of any practical music-theoretic application (see especially Attardo 1994). Thus, I employ “script opposition” only, since it lies in a Goldilocks zone of explanatory power: the concept is neither too psychologically abstract, nor too field specific.
Rochlitz may have provided the earliest mention of opposition with respect to musical humor when he wrote, “[P]urpose and means should ‘oppose each other abruptly’” (Zenck 2008, 56, quoting Rochlitz 1806, col. 7).
Raskin 1985, 111. The normal/abnormal opposition is one of three general types of opposition. The other two types are “actual situation”/non-actual, non-existing situation” and “possible, plausible situation”/“fully or partially impossible or much less plausible situation” (111).
Attardo 1994, 204. As I explain below (see esp. paragraphs 5.5–5.11), such script oppositions are usually most strongly implied by the musical topics involved in such passages. The analytical discussions below explain how such script oppositions are created. See esp. Figures 4–8 for brief lists of some possible script oppositions.
In Palmer (2015), I contend that two strategies for the deployment of formal functions and topics are especially prominent in the Classical instrumental style: “opposition” and “excess.” Both of these notions factor heavily in many scholarly musical discussions of humor (see Burstein 1999, Goeth 2013, and also Sisman 1990, Dalmonte 1995, and Huron 2004).
Burstein 1999, 72. Burstein demonstrates the subtle extent to which Haydn employs harmonic connections and motivic parallelisms to produce humor in Symphonies nos. 58, 78, 83, 90, and 93.
Musical syntax also encompasses prolongational structure (see Burstein 1999 and Lerdahl 1983) and schemata (see Gjerdingen 2007, Byros 2013), but this study focuses primarily on connections between musical topics and local functional incongruities between beginnings, middles, and ends. The semantic content of music is far less specific and robust than the semantic content of language. See Agawu 1991, 33 and also Swain 1996.
The role of formal functions in creating musical humor or wit is mentioned by Byros (2013, 219), Goeth (2013, 237–38 and 247), Levy (1992, 239), Huron (2004), Sisman (1990, 320), Mirka (2009, 309), Hepokoski and Darcy (2006, 66–67), and Almén (2008, 169); See also Edwards (1998) and Burstein (2015). The role of musical topics in musical humor is explicitly discussed in Byros (2013, 219–20 and 240–43) and mentioned in Paulson (2011, 95–96 and 137–40), Agawu (1999, 156), Diergarten (2008, [13]), and Sisman (1990, 320).
Topics are not always involved: other more general or specific scripts—other semantic information that does not properly fall within the realm of Ratnerian topics—can perform a similar context-clarifying role. The extent to, and manner in, which this is possible is clarified alongside Example 5.
Hepokoski 2006, 66. Goeth notes that, as in verbal humor, humor in music arises “through the dislocation of phraseme-like patterns, and . . . through parody” (2013, 235) and explains, “the term phraseme is used for conventionalized expressions also referred to as idioms, collocations, fixed or set expressions” (237; Goeth’s emphasis). As such, formal functions could be considered a kind of phraseme in music, but the analogy is better applied to “conventionalized expressions” like topics and schemata.
Byros 2013, 241; see also Mirka 2009, 302–3. For example, Mirka writes about a passage in the first movement of Haydn’s String Quartet op. 50 no. 2: “[I]n the most immediate context of the preceding canons this [Ländler] topic brings a dramatic contrast between high and low style” (2009, 273).
Lowe 2007, 134. In her thorough discussion of Symphony no. 93, Lowe notes its “unusually frequent and salient juxtapositions of stylistic registers, points of contact between high, middle, and low musical expressions that would have been clearly audible and immediately apparent to Haydn’s late-eighteenth-century listener” (136).
Huron notes, for example, that a “pivotal factor” for creating specifically musical humor is “the underlying extramusical context and social situation” (Byros 2013, 240n100; Byros’ emphasis and parentheses, quoting Huron 2006, 287). He also explains, “In the case of musical humor, the context appears to be overtly one of playfulness and parody” (Huron 2006, 288). According to Huron, “playfulness” may be signaled by “the interjection of ‘low art’ gestures into ostensible ‘high art’ contexts” (288). The notion that contextually inappropriate compositional use of “high” or “low” elements can create humor resonates in the work of Byros (2013) and Diergarten (2008). Huron (2006, 288) also implies that a common source of parody is “the use of absurd sounds in an ostensibly normal musical context.” The flatulent expulsion from the bassoons in the second movement of Haydn’s Symphony no. 93 is an excellent example.
Furthermore, not all topical oppositions create humor. A prime example is the valence shift between the minuet and trio in the third movement of Haydn’s Symphony no. 67, which creates both negative/positive valence and high/low stature oppositions, but is not likely to be perceived as humorous (see Palmer 2015, 72–75). Lowe (1998) and Burnham (2005) discuss how motions toward lower, more popular styles (valences) occur often at different levels of structure in Haydn’s symphonies. Burnham (2005, 63), in particular, notes that symphonic finales commonly possess a “downhill trajectory” from higher to lower styles, while Lowe (2002) discusses the pivotal, and often ironic, role that Haydn’s minuets play in the symphony cycle.
Musical topics are italicized (as they are throughout this article), while genres and adjectives appear in normal script.
A discussion of how “low” elements move to “high” elements in an ironic manner is introduced alongside Examples 3a and b.
See Caplin 2005. For example, some common topical/functional affiliations are French overture with beginning, Sturm und Drang with middle, and lament with ending (Caplin 2005, 115). However, the topics generally referred to as “genre” or “style topics”—for example, aria, sarabande, or ombra—frequently have no clear functional affiliation (115).
According to Sisman, nearly every reference to Haydn’s Il distratto being composed specifically for a play suggests that such a deliberate compositional pairing was rare (Sisman 1990, 301). For example, she mentions briefly a record that a “new” (unknown) Mozart symphony was performed between acts of Die eingebildeten Philosophe on July 26, 1785, but this symphony—like most performed in the theater—was not written specifically to accompany the drama (Sisman 1990, 300–1). Although Symphony no. 60’s nickname often appears as the Italian Il distratto, its original incarnation as incidental music was written by Haydn to accompany Regnard’s French play, Le distrait. The accepted German equivalent was Der Zerstreute.
Diergarten 2008, [4] and [8]; from Koch 1787. Although Koch does not refer to Haydn or to the symphony by name, Diergarten demonstrates how Koch’s description of the instrumental depiction of an absent-minded person (den Zerstreuten) clearly levels his criticism at Haydn’s composition. Diergarten’s admirable sleuthing further reinforces this connection, noting that Koch’s paraphrase of Horace’s “at times even Homer nods off” strongly references an “important, recognized, and renowned composer”: Haydn is the clear referent in this case (2008, [5]).
Sisman observes of the Classical symphony: “The symphony . . . could move easily among social levels and locations, and could appeal by the very variety of topics at those levels” (2014, 98).
Comic “absent-mindedness” is a particular focus of Haydn’s Symphony no. 60. Mirka tackles Haydn’s use of “absent-mindedness,” drawing on Friedrich August Weber’s writings (2009, 295).
Even the musette’s minor mode is undercut slightly by the first violin’s opening major tetrachord.
Rosen 1972, 162–63; see also Webster 2004, 242. Sisman notes the “single most pervasive quality of a unison passage: its aura of authoritative control” (Sisman 2014, 108, quoting Levy 1982, 507).
Similarly incongruous musette passages where instruments seem to get “carried away” can be found in the fourth movement of Mozart’s String Quintet K. 516 (mm. 56–58), the fourth movement of String Quartet K. 590, and in the first movement of Haydn’s String Quartet op. 20 no. 3 (mm. 24, 107, 145, 151, and 240). Haydn is also fond of writing string quartet cello parts that sound absent-minded (see especially the second movement of op. 77 no. 2 and first movement of op. 33 no. 4).
Sisman 1990, 315. In the same discussion, Sisman also explains that, unlike in this trio, “Haydn [usually] played with such conventions [of phrase rhythm] in minuets . . . by an excess of activity, especially short motives in hemiola (e.g., String Quartet op. 33, no. 5).”
In a sense, the minor mode heightens this opposition because, as Sutcliffe explains, “Use of the minor mode not only typically brings certain kinds of material to the fore, it also tends to mute topical variety” (Sutcliffe 2014, 131). In other words, since the serioso and musette are forcibly juxtaposed in a minor context where such juxtapositions are usually “muted,” the contrast and resulting opposition are more pronounced.
Haringer 2014, 205, quoting Sulzer [1792–94] 1967, 2: 421. Sulzer also notes, “The dance under this name is designed for naïve rustic entertainments but it can portray noble shepherd characters as well as lowly peasants” (Haringer 2014, 205, quoting Sulzer [1792–94] 1967, 2: 421). Here, Haydn’s harmonic stasis clearly portrays the “lowly peasants.”
The three-node opposition networks I employ somewhat resemble Lawrence Zbikowski’s conceptual integration networks (2002, 63–95). While both types of networks highlight contrasting elements in upper nodes that are then “blended” into the lower node, the interactions between the nodes communicate different things. For example, while Zbikowski’s networks can map the domains of text and music onto one another to create a blended metaphorical space where the two are intertwined, my upper nodes begin with a “blended” space, laden with metaphor (topics and human characteristics). Furthermore, any blending of these upper nodes into the lower node cannot properly be called “blending” since, psychologically speaking, “script oppositions” are not a blend, but rather a persistent clash.
See Ratner 1980, 20. I have labelled this atypical topic “march” (in quotations) in Example 3b. Monelle (2000, 54–55) provides examples in Tchaikovsky’s fifth and sixth symphonies where march and waltz topics are mixed, but Haydn’s is the only Classical example I have encountered where a march implication occurs in a thoroughly triple-time movement. Monelle (2006, 115–16) discusses an example from Jean-Baptiste Lully in which the opposite occurs: a triple-time intrusion (a measure in ${\text{}}_{2}^{3}$) occurs in a march that is otherwise in duple meter.
Klorman (2013, 127–28) provides a concise discussion of analytical uses for Monahan’s (2013) “avatars” in Mozart’s music.
See Agawu 1991, 34 for a discussion of “topical overlap.”
This depends, of course, on the performers’ choices regarding this effect. I believe the best strategy for performing humor in this passage is to almost completely drown out the oboes as they finish. Most performances seem to allow the oboes to be heard relatively clearly to the end of their phrase, while a select few (my favorites) almost completely obscure their conclusion with the interrupting unison entry. Listen, for example, to the recording by the Haydn Sinfonietta Wien under Manfred Huss (2010) for a particularly effective (if slightly slow) example.
One might interpret this march in a positive light right from the outset, but because of its interrupting or intruding role, my initial impression of this march is negatively valenced before it is playfully contradicted. Both the minuet and march are “high” topics, so no obvious valence shift between social statuses occurs here. The particular social implications of this opposition are discussed below.
Haringer discusses how marches “can even be in triple meter, if only they preserve their pride and warlike nature” (2014, 200).
For example, the minuet second movement of Haydn’s String Quartet op. 77 no. 2 often strongly implies duple meter.
For a different reading of the cadence structure here, see Burstein 2014, 16.
Remarkably, Koch (1787, 41) provides a very similar “recipe” for depicting absent-mindedness in instrumental music, where he asks: “‘How, for example, does the composer represent an absent-minded person in an instrumental piece?’ and answers: ‘He connects sections which properly do not belong together; he makes a triple rhythm [Rhythmus] where we expect a duple; without reason he alternates the minor mode with the major, and so on’” (quoted in Mirka 2009, 300, translated by Baker and Christensen 1995, 155).
The minuet section has some oddities in it as well, however. The dotted march makes its first appearance in m. 24 and the absent-minded minuet is suggested from mm. 25 to 29. But neither of these topics is nearly as emphatically or bizarrely employed as they are in the trio.
In this trio, we might even hear the Chevalier as one of Eero Tarasti’s “negactants”: an agent-antagonist that acts in opposition to the central work-persona (Tarasti 1994). Klorman 2013, 147 discusses the role of “negactants” in Mozart’s string quartets.
This character-driven interpretation finds a precedent in Haydn’s own statements. That is, the “explicit connection [made by Forkel] of the theater style with moral sentiments finds an echo in Haydn’s statement to his biographer Griesinger that he often tried to portray ‘moral characters’ in his symphonies” (Sisman 1990, 310).
Sisman observes of the second movement, “The parody of a French dance introduced in the development section probably satirizes the Chevalier” (Sisman 1990, 314).
Die Violinen stimmen um von f auf g (mm. 23–30).
There is a later hindrance, which I do not discuss here, but which Wheelock (1992, 160) and Webster (2004, 245) have noted and discussed: further along in the movement Haydn includes a notably out-of-the-ordinary quotation of “The Night Watchman’s Song.” While Haydn’s “Night Watchman” quotation is surely added for comic effect (as Webster notes), it has—in my opinion—more the effect of general amusement or entertainment that I associate with Burnham’s (2005, 61) “clown stepping onstage” than with humorous opposition, per se. That is, it provides contrast at a point where we expect contrast, but does so in a slightly odd, somewhat sensible, and surely programmatic way.
Preßburger Zeitung, 23 July, 1774, quoted in Wheelock 1992, 155, quoting Angermüller 1978, 88. Winkler (2000, 103–16) also discusses the gag-like quality of Haydn’s tuning passage and relates it to the “surreal gags” that can be found in modern music.
Preßburger Zeitung, 23 July, 1774, quoted in Wheelock 1992, 155, quoting Angermüller 1978, 88.
Theaterwochenblatte für Salzburg no. 21 (27 January 1776), quoted in Wheelock 1992, 155–58. See Angermüller 1978, 90–92 for the complete review.
Comparable oddities in K. 522 occur in the violin cadenza at the end of the third movement, the horns’ parallel-thirds passage in the second movement (m. 17–20), and the polytonal cadence that concludes the serenade.
The notion of a musical “frame” and the idea of an out-of-frame reference in Haydn are discussed by Webster (2004).
The idea that Haydn is referring to something outside of the composition proper strongly suggests a connection to the extraneous comic references of Laurence Sterne’s Tristram Shandy. See Bonds 1991 and Irving 1984 for specific insights on connections between Haydn and Sterne.
Attardo (1996, 88) calls higher-level scripts like these “complex scripts.”
Gjerdingen 2007, 45. This Prinner is echoed in an entertainingly skewed way when Haydn quotes the “Night Watchman” song later in the movement.
“Target” is the term used in linguistic theories of verbal humor to refer to the “butt” of the joke. See Attardo and Raskin 1991, 301–2 and Davies 2004, 377.
Sisman 1990, 318; my emphasis. According to Sisman, “This detail from La Bruyère’s character sketch, absent in Regnard’s original play, is restored in the German text” (1990, 318). See Sisman (1990, 320) for a point-form summary of the musical events she finds representative of Haydn’s musical interaction with characters or events in the play.
In Michael Haydn’s case, this contemporary listener was Leopold Mozart (see Anderson 1966, 1: 297) and below.
Rondo couplets are often differentiated topically from refrains, but the repeated refrains themselves tend to present less variety of topics. Topical contrast manifests most often as a character change occurring alongside a tonal/modal shift (i.e. to minore in major-mode rondos or maggiore in minor-mode ones). See Caplin 1998, 231–35.
Varied refrain repetitions most often contrast in phrase structure or thematic design, where later refrains are often “abridged or incomplete” (Caplin 1998, 235). Topical change is not a requirement, but is certainly a likely possibility. Consider, for example, the nearly obligatory minore variation found in most Classical variation forms, which not only changes the tonality of the variation, but tends to alter other characteristics that influence the variation’s topic. See Caplin 1998, 218.
The reverse is also possible: so much of the same topic in the opening three variations might suggest more potential for topical change. Regardless, there is no way to expect the drastic contrast Michael Haydn presents here.
Michael Haydn may also have been thinking of more specific programmatic elements. When writing this theater symphony, he likely decided to include references to the story of Voltaire’s Zaïre, for which this symphony was written as accompaniment. As his brother Joseph did in Il distratto, Michael Haydn seems to have turned to musical topics to represent key characters in the play and, to some extent, the actions or events that involved them. One might, for example, suppose an affiliation between the conservative style and the non-combative character of the female prisoner Zara (Zaïre). Likewise, the Turkish music could easily represent the Turkish sultan Osman. Michael Haydn returns to the Turkish topic later in the movement in order to get the most out of his comic device.
The effect here is similar to the topical overlap between the minuet and march in the third movement of Haydn’s Symphony no. 60 (see Example 3b).
See Monelle 2006, 117–23 for a concise discussion of “Turkish music.” See also Hepokoski 2006, 397 and 401.
This letter was written in 1781 and refers to the character from Mozart’s opera Die Entführung aus dem Serail, which was completed the following summer. Webster (2004, 245) notes that Turkish elements were more commonly employed in operatic works than in instrumental ones. Mozart also treats the topic with increasing levels of impropriety in the seventh movement of his Serenade in B, K. 361. Other examples of the Turkish topic occur in the third movement of Mozart’s Sonata in A K. 331, Beethoven’s The Ruins of Athens, and the finale of his Ninth Symphony. See Monelle 2006, 6.
Head 2005, 83. A similar reference to the exotic occurs with the sudden “intrusions” of a Hungarian-inflected minore in the final of Haydn’s String Quartet op. 33 no. 3; see Palmer 2015, 206–10.
The relief branch of humor theory was primarily articulated by Herbert Spencer (1883) and later by Sigmund Freud (1960), and holds that the sense of relief or release that comes with the removal of a threat can lead to laughter—one possible physiological response tied to humor (Huron 2006, 287). In other words, “Release/relief theories perceive humor and laughter as a release of the tensions and inhibitions generated by societal constraints” (Carrell 2008, 313). Although some modern humor theories include elements of relief theories, “‘pure’ relief theorists, explaining all of humor and laughter as release of tension or ‘safety valve’, cannot be found anymore in humor scholarship” (Kuipers 2008, 362).
Superiority theory is the most ancient of the primary branches of humor theory. It can be traced back to Plato and Aristotle (see Bremmer 1997 and Smuts 2006), but its strongest proponent was Thomas Hobbes (1840). Superiority theories hold that the audience of a joke will find humor in the feeling of superiority over the joke’s “target”—the guy who walked into the bar, for example. But, while superiority may be necessary for our reaction to a shift between high and low stature, it is not fundamental to our perception of humor.
Many Classical composers engaged in what could be called “racist” musical jokes. Racial humor appears to have been a particularly common manifestation of the ubiquitous Classical valence shift from high to low status, especially when focused on the Turks, Balkans, or Hungarians. This fact should not be surprising to modern audiences, who know that many comedy acts once centered on the deprecation of other cultures (and some still do). Webster discusses the “deliberate invocation of ‘ethnic’ moods [which] is found primarily in two contexts: ‘Turkish’ elements in operas, and Balkan/Hungarian style in instrumental music” (Webster 2004, 245). Mirka also notes, “the mix of Gypsy and Hungarian national elements . . . behind the metrical dissonance in Haydn’s theme” from the finale of String Quartet op. 55 no. 2 (Mirka 2009, 275). Elsewhere, Allanbrook explains how “absurd horn calls . . . mutate rapidly into a Turkish-style minor” in Haydn’s Piano Sonata in E Major, Hob. XVI:52 (Allanbrook 2002, 210).
See Palmer 2015 for a discussion of humorous aspects of the third movement of K. 250.
Tovey [1935] 1981, 177. The second “Haffner” Serenade in D Major, K. 385, was written for the coronation of Siegmund Haffner the younger (brother to Marie Elisabeth and friend of Mozart), and in a later incarnation, became Mozart’s famous “Haffner” Symphony no. 35 in D Major, K. 385.
The Loreto convent first performed the work on the evening of 21 July, 1776 (the eve of the wedding) in the summerhouse in the garden of the Paris-Lodrongasse, near the Mozarts’ house (Eisen and Keefe 2006, 205; Sadie 2006, 403). The eight-movement serenade totals over an hour of music and on the occasion was flanked by Mozart’s March in D Major, K. 249.
The exception to this humorlessness is “one of Mozart’s most Rossinian themes” (Tovey [1935] 1981, 178), which begins first in m. 80 and sounds excessive because of its motivic inanity. The (post-cadential) form-functional redundancy is conspicuous and definitely makes me laugh. The passage is full of dull, scalar motion and, while it is unnecessarily repetitive, its overtly playful character suggests more a kind of “clowning around” than the conspicuous and more jarring form-functional redundancies often found in Haydn and Beethoven’s excessive passages. This is a prime example of what we might call “Mozartian excess.”
Haydn is best known for the grand-pause gambit, but he and Mozart both use it for humorous (or at least bizarre) ends in many works. Mozart’s most conspicuous uses occur in the finale of String Quartet no. 16, K. 428 (he also quotes Haydn’s “How do you do?” from the first movement of op. 33 no. 5 in this “Haydn Quartet”) and the finale of String Quintet no. 5, K. 593. The most emblematic cases written by Haydn occur in the second movement of his Symphony no. 101 “The Clock” and in many string quartets, including the second movement of String Quartet op. 55 no. 2, the first movement of op. 33 no. 3, the first movement of op. 33 no. 5, the first movement of op. 5 no. 3. See also Mirka 2009, 304.
This play with “ending” is also present in Beethoven’s Bagatelle op. 33 no. 2. See Levy 1995 and Palmer 2015.
The “joke” immediately before Haydn’s notated conclusion, however, is nearly identical to Mozart’s: a rest that appeared to be the end was not the real ending.
This material also answers the question “What does the Jupiter Symphony sound like upside down?” since the Mozart’s Symphony no. 41 has an incredibly similar opening coup d’archet motion to the tonic that ascends.
Of course, transplanting formal functions in this way is often not humorous. For example, Beethoven’s conclusion of the opening movement of his Symphony no. 8 is undoubtedly clever without being humorous: the final idea is motivically similar to the other closing material, and all of Beethoven’s orchestral rests suspend the dominant (and thus are unable to suggest a possible conclusion for the work). The first movement of Haydn’s String Quartet op. 33 no. 5 also reuses an opening idea at the conclusion without an explicitly humorous goal. This relatively common technique is frequently described as “clever” or “witty,” but is not necessarily humorous (see Levy 1995, 155 and Hepokoski 2006, 66–67).
According to Goeth (2013, 251), this kind of “context-quotation clash is one of the most common tools for parody construction in music.”
It is true that fanfares often occur with beginning function, but they are also frequently (and therefore appropriately) found at endings. According to Caplin (2005, 117), “[the fanfare’s] arpeggiated melodic configurations create a degree of harmonic stasis appropriate for a formal beginning. But the potential for fanfare motives both to ascend and to descend also makes them useful in other formal contexts.” Caplin (2005, 117) also notes that “fanfare gestures seem frequently to lead to a cadence, especially in Mozart” and that the fanfare’s “more general characteristics” (as compared to the coup d’archet) give it more functional flexibility.
The opposition here between “appropriate” and “inappropriate” is analogous to the common high-level script opposition between “normal, expected state of affairs” and “abnormal, unexpected state of affairs” (Raskin 1985, 111). There is no clear shift between negative and positive emotional implications in this example, nor is one needed because of the strong high-to-low shift.
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# [Smart Math] Ratio Proportion Problem 29
Here’s and example of a SMART MATH problem for RATIO PROPORTION.
### Problem
7 men can dig a canal in 14 weeks. After 2 weeks of work, they are joined by another 7 men. In how many weeks more is the work finished?
1. 5
2. 6
3. 8
4. 9
5. 11
### The Usual Method
If 7 men can dig a canal in 14 weeks, than 7 men can dig $\frac{1}{14}$th of the canal in 1 week. Since, 7 men work for 2 weeks, $\frac{2}{14}$ or $\frac{1}{7}$th of the canal is dug. Hence, amount of work remaining = $\frac{6}{7}$.
Additional 7 men join, so now it becomes 14 men. If 7 men dig a canal in 14weeks, 14 men will take 7 weeks to dig the canal. Hence, $\frac{1}{7}$th of the canal will be dug by 14 men in 1 week.
Since it takes 1 week to do $\frac{1}{7}$ of the work, to do $\frac{6}{7}$ of the work, it will take 6 weeks.
(Ans: 2)
Estimated Time to arrive at the answer = 45 seconds. |
# ISEE Middle Level Math : How to find a square on a coordinate plane
## Example Questions
### Example Question #1 : How To Do Coordinate Geometry
Which of the following is a vertex of the square?
Explanation:
The coordinates of a point are determined by the distance from the origin. The first point in the ordered pair is the number of units to the left or right of the origin. Negative numbers indicate the number of units to the left while positive numbers indicate the number of units to the right. The second number indicates the number of units above or below the origin. Positive numbers indicate the number of units above while negative numbrs indicate the number of units below the origin. The vertices of the square are: |
• 研究论文 •
不同参考温度取值对三温模型反演植被蒸腾精度的影响
1. 北京大学深圳研究生院环境与能源学院, 广东深圳 518055
• 收稿日期:2021-06-08 接受日期:2021-07-05 出版日期:2022-04-20 发布日期:2021-08-02
• 通讯作者: 鄢春华,邱国玉
• 作者简介:([email protected])
*([email protected]);
• 基金资助:
国家自然科学基金(42001022);中国博士后科学基金(2019M660330)
Effects of different reference temperature values on the accuracy of vegetation transpiration estimation by three-temperature model
XIONG Bo-Wen, LI Tong, HUANG Ying, YAN Chun-Hua*(), QIU Guo-Yu*()
1. School of Environment and Energy, Peking University Shenzhen Graduate School, Shenzhen, Guangdong 518055, China
• Received:2021-06-08 Accepted:2021-07-05 Online:2022-04-20 Published:2021-08-02
• Contact: YAN Chun-Hua,QIU Guo-Yu
• Supported by:
National Natural Science Foundation of China(42001022);China Postdoctoral Science Foundation(2019M660330)
Abstract:
Aims Parameterization of reference temperature has always been the key and difficult part in calculating evapotranspiration and its evaporation and transpiration components by using three-temperature model. In this paper, the best value of reference temperature was determined by quantifying and comparing the influence of different reference temperature values on the accuracy of transpiration estimation by three-temperature model.
Methods Based on the Bowen ratio and thermal infrared observation data of a typical urban lawn, sensitivity analysis and error analysis were carried out on the input variables involved in the sub-model of the three-temperature model to determine the most critical variables for the accuracy of transpiration estimation. Then the influence of input variables parameterization on the calculation of transpiration was quantified and compared to determine the best value of reference temperature.
Important findings When using the three-temperature model, the best estimation is to select the maximum temperature of the whole piece of paper as the reference leaf temperature (R2 = 0.91, root mean square error (RMSE) = 0.078 mm·h-1). When the maximum value of the vegetation canopy temperature was used as the reference temperature, it is directly assumed that the transpiration at the maximum temperature of the vegetation is zero (there is a certain transpiration rate in fact). Therefore, it is easy to underestimate the actual transpiration, resulting in that the estimation accuracy of the three-temperature model was slightly lower than the accuracy of using the maximum value of the reference leaf temperature, but the estimation effect is still good (R2 = 0.87, RMSE = 0.080 mm·h-1). Therefore, considering the limitations of the reference leaf settings, if the reference leaf temperature cannot be measured in practical applications, the maximum temperature of the vegetation canopy as the reference temperature can be used to achieve good estimate results. |
What's the deal with biocLite's "Update all/some/none? [a/s/n]"?
1
3
Entering edit mode
kjo ▴ 60
@kjo-11078
Last seen 5.6 years ago
Whenever I install a package with biocLite(...), at the end of the installation I get a prompt like this one:
Old packages: 'RCurl', 'boot', 'class', 'cluster', 'codetools', 'foreign',
'KernSmooth', 'lattice', 'MASS', 'Matrix', 'mgcv', 'nlme', 'nnet',
'rpart', 'spatial', 'survival'
Update all/some/none? [a/s/n]:
My experience is that answering a(ll) does not prevent my being asked the same question (for the same "Old packages") the next time I install a package with biocLite(...).
In fact, if I run biocLite('some.package'), answer a to the Update all/some/none? [a/s/n]: prompt, and then immediately repeat biocLite('some.package'), I get the exact same Update all/some/none? [a/s/n]: prompt at the end, for the same "Old packages" that I supposedly updated a few minutes before.
What's going on? How can I get the "Old packages" to be updated in such a way that biocLite will recognize?
NB: Just be clear, I don't want to simply suppress the "Update all/some/none? [a/s/n]" prompt. I just want to deal with it once and for all.
bioclite • 8.2k views
3
Entering edit mode
@martin-morgan-1513
Last seen 1 day ago
United States
You have packages installed in libraries that you do not have write access to, or you have two versions of the package installed and R finds the old version but updates the new one. So arrange your libraries so that there is only one version of each package, and non-base packages (base packages do not change within an R release) are installed in a directory where you have write access.
0
Entering edit mode
Just to be clear, the situation described in my question arises both in systems for which I don't have superuser access, and those for which I do.
In the former case, there is no way I can update packages without ending up with multiple versions of the package.
In the latter case, it seems that you're suggesting either that
a. I should always run biocLite(...) as root (so that it updates the system packages); OR
b. after installing updates (as non-root) I selectively remove old system packages (as root).
If this is what you had in mind, then I must say that I'm amazed that the use of biocLite would entail such inconvenient and error-prone maneuvers.
I don't understand why biocLite can't consider only those packages visible to the user (through the user's search path) to decide which need to be updated.
Also, I'm puzzled to learn that "non-base packages ... do not change within an R release". I thought it would be exactly the opposite: only non-base packages would be liable to change within an R release.
1
Entering edit mode
I meant that base packages do not change, not that non-base packages do not change.
Yes, biocLite should just say 'these packages are old but can't be updated'; we'll fix that. Mostly the behavior here is inherited from R's update.packages(), but that's not a very good excuse.
I suggest that you install base R (and nothing else) as root, then all other packages as regular user via biocLite().
0
Entering edit mode
Got it. Thanks!
But what about the case where I'm working on a system for which I don't have root access, such as the shared computer cluster at work? The problem is not simply the annoyance of being prompted every time. The more serious problem is that it is difficult to distinguish between those packages that really need updating from those that only look that way due to update.packages's shortcomings. One is left with no choice but to answer "a" every time, thereby greatly increasing the time it takes to install or update a single package.
The matter is serious. Yesterday R was segfault-ing on me, and the only way that I could fix the problem was to re-install my entire personal library from scratch. There must have been a version mismatch somewhere in there.
Worse yet, after that incident I learned from a coworker that she did a similarly radical overhaul of her personal library because she was getting "screwy results". In this case, "screwy results" is shorthand for "a paper retraction waiting to happen." A few segfaults are nothing in comparison.
Therefore, I'm hellbent on keeping my R library up-to-date, and looking for ways to make the process more convenient and less error-prone. I'm sorry to discover that update.packages is not much of a help here.
0
Entering edit mode
I don't know what you mean by packages that 'really need updating' versus those that only look that way. There really are packages that are not current, and it's possible to load them (e.g., by specifying lib.loc= in library()) in preference to newer packages occurring earlier in .libPaths(). As I said, I'll work to make biocLite() only prompt about packages that can be updated, while still being informative about packages that are out of date and should be updated (by someone with appropriate permissions).
FWIW, and I know that you were likely taking a pragmatic approach, but there are more effective ways of troubleshooting a segfault than reinstalling all packages. BiocInstaller::biocValid() might point to packages that are too old / too new for the version of Bioconductor in use. Running R under a debugger and figuring out where a segfault occurs can be challenging but not super-impossible; some general approaches are outlined here. And oddly, I don't think that keeping packages current is the way to avoid retraction -- a static set of packages means a static analysis, providing a calm environment to carefully assess each step being performed.
1
Entering edit mode
Let me clarify that, even though I've been writing code for a long time, I am quite new to R. I've found your answer and comments extremely instructive. Thank you, I really appreciate them.
Regarding the "really need updating": if I understand the situation correctly, when one runs biocLite, the test for whether a package is out-of-date (i.e. "needs updating") is performed only on system packages. This means that packages for which an up-to-date version exists in the user's personal library can still be reported as out-of-date when one runs biocLite.
More specifically, if one chooses to update one such package X (e.g. by answering "a" or "s" to the Update all/some/none? [a/s/n]: prompt), and one does not have permission to modify the system packages, the updated X package will be placed in one's personal library. This means that this newly updated X package will not be detected next time that biocLite checks for packages that need updating; instead, it will, as before, check only the system's X package, and therefore, as before, it will report that X needs updating. But this is a spurious report, because I already have an updated X package in my personal library. The package X in this scenario is an example of what I mean by a package that does not "really need updating."
If, on the other hand, I don't have an updated version of the X package in my personal library, and the system-wide version of the X package is out-of-date, then in this case I'd say that the X package "really needs updating".
(Sorry if I've beaten a dead horse here.)
Thanks for the pointers on debugging a segfault. Indeed, as you guessed, I did take a pragmatic route. (I was under severe time pressure.) As it turns out, I did not erase my personal library; I just renamed the directory. Therefore I may try to follow your suggestions when I have a chance.
Your point on "a static set of packages" is well taken. In practice, however, I often add new packages to an existing library. If there is some subtle incompatibility between a newly added package Z and a pre-existing environment, then what is one supposed to do? The problem becomes more difficult to diagnose (and therefore deal with) if this incompatibility does not become manifest until well after the Z package was installed. When this happens, all one knows is that something is broken in the library. Finding what that is can take a long time. The pragmatic course of action, as you alluded to, is a full re-installation.
This all points to the need for R's analog of Python's virtualenv. Maybe this can be achieved simply by having multiple personal library directories, and selecting among them by, e.g., setting the R_LIBS_USER library appropriately; e.g.
R_LIBS_USER=/path/to/my/production/personal/R_library R |
# How do you find the roots, real and imaginary, of y=2x^2 + 5x - 12 using the quadratic formula?
Feb 25, 2016
Look at $\left({b}^{2} - \left(4 \times a \times c\right)\right)$ part of the formula.
#### Explanation:
In a quadratic equation the formula to find the roots is
$x = \frac{- b \pm \sqrt{{b}^{2} - \left(4 \times a \times c\right)}}{2 a}$
If $- \sqrt{{b}^{2} - \left(4 \times a \times c\right)}$ is positive, the roots of the given function are real.
If $\left({b}^{2} - \left(4 \times a \times c\right)\right)$ is negative, the roots of the given function are imaginary.
In our case -
(5^2-(4 xx 2 xx (-12)
(25-(-96)
$\left(25 + 96\right) = 121 > 0$
Since $\left({b}^{2} - \left(4 \times a \times c\right)\right) = 121 > 0$ positive, the roots of the given function are real. |
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Driving home from school one day, you spot a ball rolling out into the street (see the figure ). You brake for 1.30 s, slowing your 980-{\rm kg} car from 16.0 \rm m/s to 9.50 \rm m/s.
What was the magnitude of the average force exerted on your car during braking?
F =
\rm N
Part B
What was the direction of the average force exerted on your car during braking?
To the direction of motion.
Opposite to the direction of motion. |
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# Math of Poker
You aren't a true card shark unless you know your probabilities.
# A preview of "Math of Poker"Join Brilliant Premium
What is the probability of drawing a Heart or a King from a shuffled poker deck?
4 cards are drawn from a shuffled poker deck. How many possible combinations of these cards (without regard to order) are there?
You are playing a game of Five-Card Draw poker. You are dealt the following hand:
During the draw phase, you discard the $$2\clubsuit$$ and $$\text{Q}\spadesuit$$ cards and request two new cards.
What is the probability that you will improve your hand to a straight?
Note: You have no knowledge of the cards the other players have.
× |
## anonymous 5 years ago What is the distance from the point (6,0) to the line y=3x+2?
1. amistre64
which distance you want? shortest most likly
2. amistre64
the shortest distance is a perpendicular line....
3. amistre64
so to find it, we need the line that is perp to this and the point they intersect
4. anonymous
Possible answers are 4, $2\sqrt{10} , \sqrt{26} , and\ 3\sqrt{6}$ so I guess which ever equals one of those
5. amistre64
the perp has a product of slopes that equals -1 3x = -1 x = -1/3 0 = -1/3(6) +b b = 2 y = (-1/3)x +2
6. amistre64
these lines meet at: 3x+2 = (-1/3)x +2 3x + 1/3x = 0 10/3x = 0 x = 0
7. amistre64
they appear to meet at the origin...
8. amistre64
they meet at (0,2) if anything lol
9. amistre64
id go with 4
10. anonymous
Per distance formula is $d= \frac{ \left| ax +by +c \right| }{ \sqrt{a^2 +b^2}}$
11. amistre64
im prolly wrong since i guessed that lol
12. amistre64
y = 3x+2 ..... 0 = -6/3 +b 0 = -2 +b b = 2 y = (-1/3)x +2.... that seems to work out
13. anonymous
y= 3x+2 get into general form 3x - y + 2 =0 so this means that a=3 , b=-1 and c=2 in our formulas above, and (x,y) are the coordinates of the point ( 6,0) so x=6 , y=0
14. anonymous
d= (9 +2 ) / 2 = 4.5 units
15. anonymous
wait thats wrong
16. amistre64
3x+2 = (-1/3)x +2 3x = -1/3x when x=0 right?
17. amistre64
(0,2) (6,0) 36 + 4 = 40 = sqrt(40)
18. amistre64
2sqrt(10)
19. amistre64
got it ;)
20. anonymous
d = $\frac { \left| 3(6) + 0 + 2 \right|}{\sqrt{10} }$
21. amistre64
22. anonymous
20/ sqrt (10)
23. anonymous
wtf
24. amistre64
20sqrt(10) --------- 10
25. anonymous
yes I know
26. amistre64
2 sqrt(10)
27. amistre64
need a calculator? :)
28. anonymous
Very puzzling question lol
29. amistre64
just had to get the sleep of my ears to see what i was doin
30. anonymous
Well thank you |
# Hitzer, Helmstetter and Ablamowicz: Square Roots of −1 in Real Clifford Algebras
Eckhard Hitzer, Jacques Helmstetter and Rafal Ablamowicz, Square Roots of −1 in Real Clifford Algebras, submitted to E. Hitzer, S. Sangwine (eds.), “Quaternion and Clifford Fourier transforms and wavelets”, Trends in Mathematics, Birkhauser, Basel, 2013.
Abstract It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of −1. Adv. Appl. Clifford Algebras 16(1) (2006), 63–68.] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$. Further research on general algebras $Cl(p,q)$ has explicitly derived the geometric roots of -1 for $p+q \leq 4$ [E. Hitzer, R. Ab lamowicz, Geometric Roots of −1 in Clifford Algebras Cℓ(p, q) with p+q ≤ 4. Adv. Appl. Clifford Algebras, Online First, 13 July 2010. DOI: 10.1007/s00006-010-0240-x.]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of -1 found in the different types of Clifford algebras, depending on the type of associated ring ($\mathbb{R}$, $\mathbb{H}$, $\mathbb{R}^2$, $\mathbb{H}^2$, or $\mathbb{C}$). At the end of the paper explicit computer generated tables of representative square roots of -1 are given for all Clifford algebras with $n=5,7$, and $s=3 \, (mod 4)$ with the associated ring $\mathbb{C}$. This includes, e.g., $Cl(0,5)$ important in Clifford analysis, and $Cl(4,1)$ which in applications is at the foundation of conformal geometric algebra. All these roots of -1 are immediately useful in the construction of new types of geometric Clifford Fourier transformations. |
# Singular
### 3.3.1 Examples of ring declarations
The exact syntax of a ring declaration is given in the next two subsections; this subsection lists some examples first. Note that the chosen ordering implies that a unit-elements of the ring will be among the elements with leading monomial 1. For more information, see Monomial orderings.
Every floating point number in a ring consists of two parts, which may be chosen by the user. The leading part represents the number and the rest is for numerical stability. Two numbers with a difference only in the rest will be regarded equal.
• the ring with degree reverse lexicographical ordering. The exact ring declaration may be omitted in the first example since this is the default ring:
ring r1; ring r2 = 32003,(x,y,z),dp; ring r3=(ZZ/32003)[x,y,z]; ring r4 = (ZZ/32003),(x,y,z),dp;
• similar examples with indexed variables. The ring variables of r1 are going to be x(1)..x(10); in r2 they will be x(1)(1), x(1)(2), ..., x(1)(8), x(2)(1), ..., x(5)(8):
ring r1 = 32003,(x(1..10)),dp; ring r2 = 32003,(x(1..5)(1..8)),dp; ring r3 = (ZZ/32003)[x(1..5)(1..8)]; ring r4 = (ZZ/32003),(x(1..5)(1..8)),dp;
• the ring with lexicographical ordering:
ring r1 = 0,(a,b,c,d),lp; ring r2 = QQ,(a,b,c,d),lp;
• the ring with local degree reverse lexicographical ordering. The non-prime 10 is converted to the next lower prime in the second example:
ring r1 = 7,(x,y,z),ds; ring r2 = 10,(x,y,z),ds; ring r3 = (ZZ/7),(x,y,z),ds;
• the ring with lexicographical ordering for and degree reverse lexicographical ordering for :
ring r1 = 7,(x(1..6)),(lp(3),dp); ring r2 = (ZZ/7),(x(1..6)),(lp(3),dp);
• the localization of at the maximal ideal
:
ring r1 = 0,(x,y,z,a,b,c),(ds(3), dp(3)); ring r2 = QQ,(x,y,z,a,b,c),(ds(3), dp(3));
• the ring with weighted reverse lexicographical ordering. The variables , , and have the weights 2, 1, and 3, respectively, and vectors are first ordered by components (in descending order) and then by monomials: ring r1 = 0,(x,y,z),(c,wp(2,1,3)); ring r2 = QQ,(x,y,z),(c,wp(2,1,3));
For ascending component order, the component ordering C has to be used.
• the ring , where denotes the transcendental extension of by , and with degree lexicographical ordering:
ring r = (7,a,b,c),(x,y,z),Dp;
• the ring , where denotes the algebraic extension of degree 2 of by In other words, is the finite field with 49 elements. In the first case, denotes an algebraic element over with minimal polynomial ,in the second case,
refers to some generator of the cyclic group of units of :
ring r = (7,a),(x,y,z),dp; minpoly = a^2+a+3; ring r = (7^2,a),(x,y,z),dp;
• the ring , where denotes the field of real numbers represented by simple precision floating point numbers. This is a special case:
ring r = real,(x,y,z),dp;
• the ring , where denotes the field of real numbers represented by floating point numbers of 50 valid decimal digits and the same number of digits for the rest:
ring r = (real,50),(x,y,z),dp;
• the ring , where denotes the field of real numbers represented by floating point numbers of 10 valid decimal digits and with 50 digits for the rest:
ring r = (real,10,50),(x,y,z),dp;
• the ring , where denotes the field of real numbers represented by floating point numbers of 30 valid decimal digits and the same number for the rest. denotes the imaginary unit.
ring r = (complex,30,j),(x,y,z),dp;
• the ring , where denotes the field of real numbers represented by floating point numbers of 6 valid decimal digits and the same number for the rest. is the default for the imaginary unit.
ring r = complex,(x,y,z),dp;
• the quotient ring modulo the square of the maximal ideal :
ring R = 7,(x,y,z), dp; qring r = std(maxideal(2));
• the ring :
ring R = integer,(x,y,z), dp;
• the ring :
ring R = (integer, 6, 3),(x,y,z), dp;
• the ring :
ring R = (integer, 100),(x,y,z), dp; |
# Probability Theory/Independence
note to self: in the case of indep., the rules of the kind ${\displaystyle P(\sum \prod )=\sum \prod P}$ should be derived.
Definition (independence of events):
Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space and let ${\displaystyle A,B\in {\mathcal {F}}}$. ${\displaystyle A}$ and ${\displaystyle B}$ are said to be independent iff ${\displaystyle P(A\cap B)=P(A)P(B)}$.
Remark (independence and conditional probability):
Using the definition of conditional probability, if e.g. ${\displaystyle P(A)\neq 0}$ we may rephrase the independence of ${\displaystyle A}$ and ${\displaystyle B}$ as ${\displaystyle P(B)=P(B|A)}$. |
## blackstreet23 one year ago A cylindrical tank lying on its side is filled with liquid weighing 50lb/ft^3. Find the work required to pump all the liquid to a level 1 ft above the top of the tank.The diameter of the tank is 4 feet.The depth of the tank is 10 feet. NOTE: Please remember the tank is lying on its SIDE, it's NOT upright. A hint my professor gave is that the answer should be a number multiplied by pi. I'm not really sure where my slice goes and how to set up my integral because it's not upright.
1. blackstreet23
2. blackstreet23
I found a picture of it in that website
3. phi
|dw:1434933735712:dw|
4. phi
The work to lift x lbs up y ft is x*y ft-lbs Each thin "plate" is lifted from its "y" to y=3 and the distance is 3-y the weight of the place is density * volume or in this case 50 lbs/ft^3 * 2*x*10*dy ft^3 and it is lifted 3-y feet thus $\int_{-2}^2 50\cdot 2 \cdot x \cdot 10 \cdot (3-y) \ dy$ because we are integrating over y, replace x with the equivalent $x= \sqrt{4-y^2}$ we have $1000 \int_{-2}^2 \sqrt{4-y^2}\ (3-y) \ dy$
5. phi
to integrate that, write it as two separate integrals the first one requires some kind of trig substitution the second is more straightforward.
6. blackstreet23
I do not understand how you found the width :s
7. phi
a cylinder has a circle as its base and (in this case) a height of 10 tip the cylinder on its side, and look at its base (a circle) I put the circle with radius 2 (i.e. diameter/2) at the origin the equation of a circle at the origin is x^2 + y^2 = r^2 follow?
8. phi
if we solve for x, we get $x = \sqrt{4-y^2}$ if we take the positive value for the square root, then for a given y , we have found the point on the circle on the right side of the origin, in other words, the distance from the y-axis to the circle on its right side. double that value to get the entire width of the "plate"
9. blackstreet23
Thanks a lot !! Sorry I didn't answer before I have been busy. |
# zbMATH — the first resource for mathematics
On the supremum of an infinitely divisible process. (English) Zbl 0633.60025
A distribution function F is said to be long-tailed if we have $\lim_{x\to \infty}(1-F(x-y))/(1-F(x))=1$ for all $$y\in {\mathbb{R}}$$. Let $$X=\{X_ t\}_{t\geq 0}$$ be an infinitely divisible process and define $$Y_ t=\sup_{0\leq s\leq t}X_ s$$ for all $$t\geq 0.$$
The principal result of this paper, which generalizes a result of S. M. Berman [ibid. 23, 281-290 (1986; Zbl 0612.60063)], asserts that the distribution of each $$X_ t$$ is long-tailed if and only if the distribution of each $$Y_ t$$ is long-tailed, in which case, for each t, the probabilities $$P(X_ t\geq u)$$ and $$P(Y_ t\geq u)$$ are asymptotic as $$u\to \infty$$.
Reviewer: B.K.Horkelheimer
##### MSC:
60E07 Infinitely divisible distributions; stable distributions 60F10 Large deviations 60J99 Markov processes
##### Keywords:
long-tailed; infinitely divisible process
Full Text:
##### References:
[1] Berman, S.M., The supremum of a process with stationary independent and symmetric increments, (), 281-290 · Zbl 0612.60063 [2] Berman, S.M., Limit theorems for sojourns of stochastic processes, (), 40-71 [3] Embrechts, P.; Goldie, C.M.; Veraverbeke, N., Subexponentiality and infinite divisibility, Z. wahrsch. verw. geb., 49, 335-347, (1979) · Zbl 0397.60024 [4] Feller, W., () [5] Sgibnev, M.S., Infinitely divisible distributions belonging to the class $$L$$(γ), Sibirsk mat. zh., (1987), to appear in [6] Wilekens, E., Higher order theory for subexponential distributions (in Dutch), (), K.U. Leuven [7] Willekens, E., Subexponentiality on the real line, Technical report, (1986), K.U. Leuven
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. |
Course:CPSC522/Rao Blackwellized Particle Filtering
Rao Blackwellized Particle Filtering for Grid Mapping
Monte Carlo methods are a common approach for large dimensional problems such as grid mapping, Rao-Blackwellized Particle Filters aim to do the needed sampling as efficiently as possible.
Principal Author: Jocelyn Minns
Collaborators:
Abstract
Rao-Blackwellized Particle Filters (RBPF) incorporates the Rao–Blackwell theorem to improve the sampling done in a particle filter by marginalizing out some variables. Particle filters alone struggle in high dimensional spaces since a large number of particles must be used, but with improved RBPFs the amount of particles needed decreases greatly.
The two papers discussed are:
1. Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks[1]
2. Improved Techniques for Grid Mapping With Rao-Blackwellized Particle Filters [2]
Builds on
Particle fillers are a Monte Carlo methods that are used to update nonlinear system that can be modelled as a Hidden Markov network.
Background
Particle Filters
Main article: Particle fillers
Particle filtering is used to estimate the updated state of a system given noisy sensor data and noisy state transition information. Unlike other filters, such as a Kalman Filter, particle filtering works with nonlinear systems and non-Gaussian noise distributions.
The system must first be modelled as a hidden Markov model where the observed value are the measurements ${\displaystyle y_{t}}$ and the hidden variables are the true values ${\displaystyle z_{t}}$. It is also needed to define the transition probability density ${\displaystyle p(z_{t}|z_{t-1})}$ with an initial probability density ${\displaystyle p(z_{0})}$ and the conditional distribution ${\displaystyle p(y_{t}|z_{t})}$.
Initially, a set of N particles are sampled from the initial probability density ${\displaystyle p(x_{0})}$. The size of the set of particles will depend on the complexity of the problem and the accuracy that is desired. For the initial set, all particles will be given equal weight. Meaning each particle is equally likely to be the true status of the system.
The algorithm for a particle filter can be described in three steps:
1. Sample a new set of particles given the weights of the previous particles. Particles with a heaver weight will be sampled more often.
2. Sample the updated state given the previous state and the current measurement data${\displaystyle p(z_{t}|z_{t-1},y_{k})}$
3. Update the weights of each particle ${\displaystyle w=p(y_{t}|z_{t})}$ and normalize the weights.
Given a large enough set of particles, this method has been shown to be effective in estimating state transitions for a variety of nonlinear problems.
Rao-Blackwell theorem
Main article: Rao-Blackwell theorem
The Rao-Blackwell theorem states that given any kind of estimator${\displaystyle \delta (X)}$ of a parameter ${\displaystyle \theta }$ then the conditional expectation of ${\displaystyle \delta (X)}$ given a sufficient statistic ${\displaystyle T(X)}$ is typically a better estimator of ${\displaystyle \theta }$.
While this may seem to be a simple statement, in practice the improvement of the estimator can be very large.
Papers
Paper 1: Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks
The aim of Rao-Blackwellised Particle Filtering is to find an estimator of the conditional distribution ${\displaystyle p(y_{t}|z_{t})}$ such that less particles will be required to reach the same accuracy as a typical particle filter.
The objective remains the same as a particle filter: to estimate the distribution ${\displaystyle p(z_{t}|y_{1:t})}$.
The posterior gives the recession:
${\displaystyle p(z_{0:t}|y_{1:t})=p(z_{0:t-1}|y_{1:t-1}){\frac {p(y_{t}|z_{t})p(z_{t}|z_{t-1})}{p(y_{t}|y_{1:t-1})}}}$
However, this cannot be solved analytically hence why sampling is typically used. To decrease the state space into something that will be easier to sample from, they break the hidden variables ${\displaystyle z}$ into two groups: ${\displaystyle r_{t}}$ and ${\displaystyle x_{t}}$ such that ${\displaystyle p(z_{t}|z_{t-1})=p(x_{t}|r_{t-1:t},x_{t-1})p(r_{t}|r_{t-1})}$. The original problem assumes no structure to the hidden variables, but by breaking them into two groups as stated above, they can determine part of the posterior ${\displaystyle p(x_{0:t}|y_{1:t},r_{0:t})}$ analytically then marginalize out the variables ${\displaystyle x_{0:t}}$.
The posterior can be decompose as such:
${\displaystyle p(r_{0:t},x_{0:t}|y_{1:t})=p(x_{0:t}|y_{1:t},r_{0:t})p(r_{0:t}|y_{1:t})}$
The dimensions of ${\displaystyle p(r_{0:t}|y_{1:t})}$ are smaller than the original ${\displaystyle p(r_{0:t},x_{0:t}|y_{1:t})}$ so they can obtain better results.
Rao-Blackwellisation
For sampling, It is needed to define an estimator for ${\displaystyle p(r_{0:t},x_{0:t}|y_{1:t})}$.
First they define an estimate of any function f of the hidden variables as
${\displaystyle {\overline {I_{N}}}(f)={\frac {1}{N}}\sum (f(r_{0:t}^{(i)},x_{0:t}^{(i)}))}$
The paper goes on to use the Central Limit theorem to prove why this is an estimate, but that has been excluded from this page as it is unnecessary to understand the implementation of RBPF.
Since each particle may not have even weights, they replace the ${\displaystyle {\frac {1}{N}}}$ factor with the normalized weights of each particle giving:
${\displaystyle {\hat {I}}_{N}^{1}(f)=\sum ({\tilde {w}}_{0:t}^{(i)}f(r_{0:t}^{(i)},x_{0:t}^{(i)}))}$
However, since they have shown that they can marginalize out ${\displaystyle x_{0:t}}$, they can write a Rao-Blackwell estimator as
${\displaystyle {\hat {I}}_{N}^{2}(f)={\frac {\sum \mathbb {E} _{p(x_{0:t}|y_{1:t},r_{0:t}^{(i)})}(f(r_{0:t}^{(i)},x_{0:t}))w(r_{0:t}^{(i)})}{\sum w(r_{0:t}^{(i)})}}}$
Using this estimator can be proven to reduce the number of particles needed to reach a desired level of accuracy.
Algorithm
The algorithm for Rao-Blackwellised Particle Filtering can be described as three steps:
1. Sequential importance sampling step where they sample N particles from our given estimator, then compute and normalize their importance weights
2. Selection step where resample N particles based on the weights calculated above
3. Apply the transition step based on the probability distribution ${\displaystyle p(r_{0:t}|y_{1:t})}$
Application for Grid Mapping
A Factorial HMM with 3 hidden chains
This form of filtering has been shown to be useful in Simultaneous Localization and Mapping (SLAM) of a grid. Consider a robot that can move in a discrete, dimensional grid. Both the motors of the robot and the sensors are imperfect and have some noise to their readings. Say you want to colour the map grid as the robot moves around the room to indicate obstacle that it should avoid. Let ${\displaystyle L_{t}}$ be the location at time t, ${\displaystyle M_{t}(i)}$ be the colour of grid cell i and time t. The hidden Markov network is show in the figure for a 3 cell grid. ${\displaystyle Y}$ is the observation model.
Murphy describes using RBPF for this purpose further in his paper[3]. For this problem, if ${\displaystyle L_{0:t}}$ is known then that they can marginalize out ${\displaystyle M_{t}}$ analytically and only sample the reduced state space ${\displaystyle L_{t}}$.
This paper only tested the problem on a ${\displaystyle 10\times 10}$ grid map which in comparison to real world problems, is unrealistically small. This experiment does show to give equivalent results as other methods while requiring few particles.
Paper 2: Improved Techniques for Grid Mapping With Rao-Blackwellized Particle Filters
The original paper outlined the RBPF problem as a general solution for any problem that uses particle filters, even though they chose to use SLAM to evaluate the method. The second paper focuses only on the SLAM problem and improves the performance of the filter only for SLAM problems were the observation data is highly accurate as it would be if the robot was using a Laser Range Finder (LRF).
With this, they propose two approaches to improve RBPF:
1. A proposal distribution that considers the accuracy of the observed readings
2. An adaptive resampling technique that reduces the risk of particle depletion.
Improved Proposal Distribution
Likelihood of each sensor measurements. L is the location given from the LRF and is far more accurate than the motion model itself
In the original RBPF algorithm, the particles are updated using the odometry motion model as the proposal distribution ${\displaystyle p(x_{t}|x_{t-1},u_{t-1})}$ where ${\displaystyle u_{t-1}}$ is the odometry measurements at time t-1. Then the weights of each particle are determined by both the sensor measurements and the odometry measurements. However, the sensor measurements are far more accurate than the odometry measurements, meaning particle weights can differ significantly from each other. The figure shows the difference in the models. To overcome this problem, they incorporate the sensor readings into the proposal such as ${\displaystyle p(x_{t}|m_{t-1},x_{t-1},z_{t},u_{t-1})}$ where ${\displaystyle z_{t}}$ is the sensor readings at time t.
This model may then be hard to sample since ${\displaystyle z_{t}}$ is focused on such small area. To simplify the computation, they use the samples to estimate a Gaussian distribution with parameters
${\displaystyle \mu _{t}^{(i)}={\frac {1}{\eta ^{(i)}}}\sum x_{j}p(z_{t}|m_{t-1}^{(i)},x_{j})p(x_{j}|x_{j-1}^{(i)},u_{t-1})}$
${\displaystyle \Sigma _{t}^{(i)}={\frac {1}{\eta ^{(i)}}}\sum x_{j}p(z_{t}|m_{t-1}^{(i)},x_{j})p(x_{j}|x_{j-1}^{(i)},u_{t-1})(x_{j}-\mu _{t}^{(i)})(x_{j}-\mu _{t}^{(i)})^{T}}$
Where
${\displaystyle \eta ^{(i)}=p(z_{t}|m_{t-1}^{(i)},x_{j})p(x_{j}|x_{j-1}^{(i)},u_{t-1})}$
This does limit the type of noise that can be expected from the sensor though, which was one of the main advantages of using a particle filter, but since they are only focusing on LRF sensors, this is not an unreasonable assumption.
Since the samples are now highly focused on one area, they now must deal with the risk of particle depletion. This occurs when particles become overconfident in their positions and are incorrect due to poor observations or too small of an overlapping area between successive scans.
To mitigate this occurrence, they propose adding a resampling step based on a parameter ${\displaystyle N_{eff}}$ such that
${\displaystyle N_{eff}={\frac {1}{\sum ({\tilde {w}}^{(i)})^{2}}}}$
where ${\displaystyle {\tilde {w}}^{(i)}}$ are normalized weights. This means that as the weights grow and become more confident, ${\displaystyle N_{eff}}$ will drop. Once ${\displaystyle N_{eff}}$ drops below the threshold N/2 where N is the number of particles, then they will resample the particles. In the extensive testing done, this was shown to help decrease the risk of particle depletion.
Evaluation
To evaluate this improved RBFP, testing was done are three different locations: Intel Reseach Lab, Freibugrg Campus, and MIT Killian Court. These location varied in size, moving obstacles, and types of locations environments such as indoor vs outdoor maps. Each environment was tested multiple times with different sizes of particle sets to see the map accuracy that was able to be achieved.
Outdoor environments proved to be an issue since LRF have a limited range and open areas such as a parking lot caused the robot to get lost.
In all these experiments, the number of particles needed decrease by approximately one order of magnitude. As expected, smaller maps with fewer moving obstacles required the lease particles and larger maps required more. This was tested against the leading SLAM algorithm at the time known as FastSLAM [4].
Conclusion
The difference between the two papers shows the transition from theory to application. The first paper proposed the RBPF providing a large amount of mathematical background and performing 'toy' experiments to show that the theory did have promise. The second paper took the theory and looked at what could be done to improve it in the real world environments. Doing this came with its own limitations such that this approach would not transfer easily to another similar application if it does not fit model given. This means that someone may choose to create their own improved RBPF for their own system and disregard the approaches proposed by the second paper. For further studies, it would be interesting to see if the RBPF could be improved for a general case without adding application specific limitation.
Annotated Bibliography
1. Arnaud Doucet, Nando de Freitas, Kevin Murphy, and Stuart Russell. 2000. Rao-blackwellised particle filtering for dynamic Bayesian networks. In Proceedings of the Sixteenth conference on Uncertainty in artificial intelligence (UAI'00), Craig Boutilier and Moisés Goldszmidt (Eds.). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 176-183.
2. G. Grisetti, C. Stachniss and W. Burgard, "Improved Techniques for Grid Mapping With Rao-Blackwellized Particle Filters," in IEEE Transactions on Robotics, vol. 23, no. 1, pp. 34-46, Feb. 2007. doi: 10.1109/TRO.2006.889486
3. Kevin P. Murphy. 1999. Bayesian map learning in dynamic environments. In Proceedings of the 12th International Conference on Neural Information Processing Systems (NIPS'99), S. A. Solla, T. K. Leen, and K. Müller (Eds.). MIT Press, Cambridge, MA, USA, 1015-1021.
4. D. Hahnel, W. Burgard, D. Fox and S. Thrun, "An efficient fastSLAM algorithm for generating maps of large-scale cyclic environments from raw laser range measurements," Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453), 2003, pp. 206-211 vol.1.
Permission is granted to copy, distribute and/or modify this document according to the terms in Creative Commons License, Attribution-NonCommercial-ShareAlike 3.0. The full text of this license may be found here: CC by-nc-sa 3.0 |
# Computer Programs – Concepts
Let's start with the concepts of computer programming: computer programs, algorithms, programming languages, code compilation and execution.
## Video: Computer Programs, Compilers, Interpreters
Watch a video lesson about the concepts of programming, programs, compilers and interpreters here: https://youtu.be/U16C61p6m1k.
## What It Means "To Program"?
To program means to give commands to the computer, for example "to play a sound", "to print something on the screen" or "to multiply two numbers". When the commands are one after another, they are called a computer program. The text of computer programs is called a program code (or a source code, or even shorter – code).
Example of command for the computer:
Console.WriteLine("Welcome to coding");
Run the above code example: https://repl.it/@nakov/welcome-to-coding-csharp.
When executed, the above command prints the following text:
Welcome to coding
## Computer Programs
Computer programs represent a sequence of commands that are written in certain programming language, like C#, Java, JavaScript, Python, C++, PHP, C, Ruby, Swift, Go or another.
Example of computer program in C#:
using System;
class SquareArea
{
public static void Main()
{
var size = 5;
Console.WriteLine("Size = " + size);
Console.WriteLine("Area = " + size * size);
}
}
Run the above code example: https://repl.it/@nakov/square-area-csharp.
The above program defines a class SquareArea, holding a method Main(), which holds a sequence of 3 commands:
1. Declaring and assigning a variable: var size = 5;
2. Calculating and printing an expression: Console.WriteLine("Size = " + size);
3. Calculating and printing an expression: Console.WriteLine("Area = " + size * size);
The result (output) from the above program is as follows:
Size = 5
Area = 25
We shall explain in detail how to write programs in C#, why we need to define a class and why we need to define a method Main() a bit later. Now, assume that the C# language requires all the above code in order to execute a sequence of command.
In order to write commands, we should know the syntax and the semantics of the language which we are working with, in our case – C#. Therefore, we are going to get familiar with the syntax and the semantics of the language C#, and with programming generally, in the current book, by learning step by step code writing from the simpler to the more complex programming constructions.
## Algorithms
Computer programs usually execute some algorithm. Algorithms are a sequence of steps, necessary for the completion of a certain task and for gaining some expected result, something like a "recipe".
For example, if we fry eggs, we follow some recipe (an algorithm): we warm up the oil in a pan, break the eggs inside it, wait for them to fry and move them away from the stove.
Similarly, in programming the computer programs execute algorithms: a sequence of commands, necessary for the completion of a certain task. For example, to arrange a sequence of numbers in an ascending order, an algorithm is needed, e.g. find the smallest number and print it, then find the smallest number among the rest of the numbers and print it, and this is repeated until there are no more numbers left.
For convenience when creating programs, for writing programming code, for execution of programs and other operations related to programming, we need a development environment, for example Visual Studio.
## Programming Languages, Compilers, Interpreters and Development Environments
Let's review some concepts from computer programming: programming languages, compilers, interpreters and development environments (IDEs).
### Programming Languages
A programming language is an artificial language (syntax for expression), meant for giving commands that we want the computer to read, process and execute. Using programming languages, we write sequences of commands (programs), which define what the computer should do. Examples of programming languages are C#, Java, JavaScript, Python, C, C++, PHP, Swift, Go and many others. These languages differ in their philosophy, syntax, purpose, programming constructions and execution environment. The execution of computer programs can be done with a compiler or with an interpreter.
### Compilers
The compiler translates the code from programming language to machine code, as for each of the constructions (commands) in the code it chooses a proper, previously prepared fragment of machine code and in the meantime it checks the text of the program for errors. Together, the compiled fragments comprise the program into a machine code, as the microprocessor of the computer expects it. After the program has been compiled, it can be executed directly from the microprocessor in cooperation with the operating system. With compiler-based programming languages the compilation of the program is done obligatory before its execution, and syntax errors (wrong commands) are found during compile time. Languages like C++, C#, Java, Swift and Go work with a compiler. This an example how the compiler execution may look (the console-based dotnet compiler):
### Interpreters
Some programming languages do not use a compiler and are being interpreted directly by a specialized software called an "interpreter". The interpreter is "a program for executing programs", written in some programming language. It executes the commands in the program one after another, as it understands not only a single command and sequences of commands, but also other language constructions (evaluations, iterations, functions, etc.). Languages like Python, PHP and JavaScript work with an interpreter and are being executed without being compiled. Due to the absence of previous compilation, in interpreted languages the errors are being found during the execution time, after the program starts running, not previously. This is an example how an interpreter may look (the python interpreter in the console):
### Development Environments (IDE)
An environment for development (Integrated Development Environment – IDE) is a combination of traditional tools for development of software applications. In the development environment we write code, compile and execute the programs. Development environments integrate in them a text editor for writing code, a programming language, a compiler or an interpreter and a runtime environment for executing programs, a debugger for tracking the program and seeking out errors, tools for user interface design and other tools and add-ons.
Environments for development are convenient, because they integrate everything necessary for the development of the program, without the need to exit the environment. If we don't use an environment for development, we will have to write the code in a text editor, to compile it with a command on the console, to run it with another command on the console and to write more additional commands when needed, which is very time consuming. That is why most of the programmers use an IDE in their everyday work.
For programming with the C# language the most commonly used IDE is Visual Studio, which is developed and distributed freely by Microsoft and can be downloaded from: https://www.visualstudio.com/downloads/. Alternatives of Visual Studio are Rider (https://www.jetbrains.com/rider/) and MonoDevelop / Xamarin Studio (http://www.monodevelop.com) and SharpDevelop (http://www.icsharpcode.net/OpenSource/SD/). In the current book, we are going to use the development environment Visual Studio. This an example how a development IDE may look (the Visual Studio IDE for C#): |
# Optimization algorithm selection for 3 variable integer
I have a cost function: $f(x,y,z) \rightarrow \mathbb{R}$
• it is very expensive to evaluate
• $x,y,z \in \mathbb{Z}$
• 0 < x < 10
• 0 < y < 30
• 0 < z < 100
• I thought it was convex, not sure now based on @Brian's commentary.
• $f$ can only be evaluated at integers ( I pondered trying to evaluate at real's but it would be fairly insanely difficult to try & the result would be totally bogus).
• $f$ is smooth but only at the integer values. So the gradient could only be computed using re-evaluating $f$ at intervals - but again that's really expensive.
I'm looking for suggestions on algorithms, and libraries ( C++ preferred, C acceptable ).
I've spent a couple of days trying to brush up on the topics, but its pretty dense. I've been looking at COIN-OR, specifically OSI, but I can't seem to figure out how to formulate my problem into the API. I've also looked at EasyLocal++, but haven't really dug in yet.
• If $f$ can only be evaluated at points where $x$, $y$, and $z$ are integer, then it isn't a differentiable function and isn't smooth. It's also not convex (since the definition of a convex function involves a function defined on a convex set, and the set of integer points in the box isn't a convex set.) Branch and bound approaches that depend on solving a continuous relaxation of the problem aren't going to work, because you can't evaluate that continuous relaxation of the problem. – Brian Borchers May 12 '13 at 4:16
• An interpolated response surface will approximate the function and will get you derivative information for algorithms that require once (or twice) continuously differentiable functions. Is the interpolant meaningful if it doesn't make sense to evaluate the function at noninteger values? What about the result would make it bogus? Even derivative-free optimization methods generally assume that $f$ is at least Lipschitz continuous, if not differentiable, and assume that derivative information is unavailable, not that it doesn't exist at all (even in a generalized gradient sense). – Geoff Oxberry May 12 '13 at 22:43
• I was under the impression ( aparently wrongly ) that that $f$ was convex if its result was convex - not the input set. I have visualized the input set - its a fairly nice 3D grid with some corners missing. – Zac May 13 '13 at 6:16
• I can evaluate one example of the set $x,y,z$, but there are many such sets. Evaluating the full permutation of one set of $x,y,z$ takes ~2 minutes. I need to be able to evaluate hundreds of different sets of $x,y,z$ in a matter of seconds. I could go massively parallel, but I'd rather do it smarter. – Zac May 13 '13 at 6:20
You haven't said whether $f$ can be evaluated at points where $x$, $y$, and $z$ are real numbers within the specified ranges that aren't integers. If this makes no sense, than the function isn't actually convex. You also haven't said whether $f$ is smooth (e.g. is it twice continuously differentiable?)
Conventional gradient based optimization algorithms will typically require quite a few function evaluations (e.g. for line searches) as well as requiring the gradient of $f$ (which can be approximated by finite differences, but that implies even more function evaluations.) If the function is convex but not smooth, and you can't describe the non-smoothness in more detail, then these algorithms aren't appropriate anyway.
For a low dimensional problem with very expensive (or noisy) function evaluations, its often a good choice to use a "response surface method" to build a regression model of the function, minimize over the simple (e.g. quadratic)function that you've built using regression, and then (as many times as you can afford) narrow your search and build a new regression model.
Can you afford to do 27 function evaluations at $x=0, 5, 10$, $y=0, 15, 30$, and $z=0, 50, 100$? That should be enough to construct a first quadratic model of the response surface.
• I'd suggest rounding off to the nearest integers the minimum of the response surface or simply trying all of the nearby integer points. Actually running branch and bound to prove optimality of your solution is probably impractical. – Brian Borchers May 11 '13 at 19:38
• The idea of the 'response surface' is really interesting. I think i'm going to try evaluating $f$ a whole bunch of times and regressing a simple version of it that is cheap to evaluate. – Zac May 12 '13 at 4:02
• How expensive is it to evaluate $f$ at a given point? Does it take minutes? hours? days? How many total function evaluations can you afford to perform? – Brian Borchers May 12 '13 at 4:18
The convexity of $f$ (over the continuous relaxation of the domain) does not matter as far as choosing the outermost solver: you will need a branch-and-bound algorithm if you choose to solve your problem deterministically because you have binary decision variables. I presume $f$ is nonlinear? If that is the case, you will need an MINLP solver like Bonmin. The convexity of $f$ will help in solving lower bounding problems; if you relax all of the binary variables to continuous variables, you should be able to find a lower bound on your solution with a nonlinear programming solver like IPOPT. Feasible solutions will give you an upper bound. I'll see if I can dig around to find a good online reference about branch-and-bound algorithms in integer programming. You could also solve the nonlinear problems using derivative-free methods, to save on function evaluations.
If $f$ is linear, you'll probably want to use something like CBC, which is a branch-and-cut framework (similar in spirit to branch-and-bound). However, given that $f$ is expensive to evaluate, I doubt it is linear.
These methods don't lend themselves to problems with expensive function evaluations. Are you willing to give up some some accuracy in the solution for a faster algorithm, and if so, how much?
• I am willing to give up accuracy for a faster algorithm. Lots in fact. – Zac May 12 '13 at 4:14 |
# Find all values of m for which $y-e^{mx}$ is solution of equation $y'''-2y''-y'+2y=0$
$(a)\;1,2,3 \\ (b)\;1,-1,0 \\ (c)\;1,-1,2\\ (d)\;1,0,2$
## 1 Answer
$y'= me^{mx}$
$y''=m^2 e^{mx}$
$y'''=m^3e^{mx}$
$m^3-2 m^2-m+2=0$
$(m-1)(m^2-m-2)=0$
$(m-1)((m-2)(m+1))=0$
$m=1, -1,2$
Hence c is the correct answer
answered Feb 7, 2014 by
1 answer
1 answer
1 answer
1 answer
1 answer
1 answer
1 answer |
# Fourier transform of a differential operator
I have a differential operator defined by its Fourier transform:
$\left(\alpha k_x^2 + \beta k_y^2 + \gamma k_x k_y \right)^\delta \hspace{20pt} \alpha,\beta,\gamma,\delta \in \mathbb{R}$
I don't know how to do the inverse transform, but I know that it is impossible to compute in the most general case. However I don't need it, as I would only like to demonstrate that this operator, in real space (i.e. after antitrasforming) when applied to a constant term, yields zero. This is just my guess, but seeing how the operator is written, I suppose that after anti-transforming (if that would be possible) then it would be a function of $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$ and $\frac{\partial ^2}{\partial x \partial y}$, so being null when operating on constants, I suppose.
Is my guess correct? If so, how can I prove my guess?
Thank you very much!
Disclaimer: I am not a mathematician, I am a (wannabe) physicist, so some of my definitions could be inaccurate, I hope the question is clear, though.
-
In general, your operator is not correctly defined because the function written in your question may be very singular. After its multiplication on the Fourier transform of a test function, the result may happen not to be a Fourier transform, at least in the classical case. Therefore the result you ask for is possible only in special cases some of which are given in the book on fractional calculus by Samko, Kilbas, and Marichev. – Anatoly Kochubei Dec 9 '11 at 6:34
Making sense of this requires the theory of tempered distributions. Assuming everything makes sense, the Fourier transform of the constant function is the dirac distribution, so your guess is correct. – Antoine Levitt Dec 9 '11 at 10:26
Anatoly: thank you! What if I add some regularities conditions, such as $\delta=\frac{1}{2}$ and a constraint on $\alpha$,$\beta$,$\gamma$ so that what appears under the square root is always $\geq 0$? Would that help? Antoine: thank you for your answer! However, wouldn't the Dirac's distribution have to be convoluted with the operator, as a product in the real space becomes a convolution in reciprocal Fourier space? If I'm correct, then the Dirac's distribution would leave the operator unchanged. – zakk Dec 9 '11 at 10:51
This is not a multiplication, but an operator application. When you differentiate a function, you can either differentiate it in real space, or multiply by $i\omega$ its Fourier transform. Same applies here. – Antoine Levitt Dec 9 '11 at 11:08
Also, your operator looks, modulo rotation in the x-y plane, like a fractional laplacian, and there's quite a bit of litterature on that (also lookup sobolev space with non-integer exponent on wikipedia). – Antoine Levitt Dec 9 '11 at 11:55
Let me change slightly your notations and consider the quadratic form in $\mathbb R_{\xi,\eta}^2$ $$Q(\xi,\eta)=\alpha \xi ^2+2\gamma \xi \eta+\beta \eta^2,$$ where $\alpha, \beta$ are real parameters. This is a Fourier multiplier : $Fourier\bigl(Q(D_x,D_y)u\bigr)(\xi,\eta)=Q(\xi,\eta)\hat u(\xi,\eta).$ Let us for instance assume that $\gamma^2<\alpha\beta, \alpha >0$, i.e. $Q$ is positive-definite. $Q$ has positive characteristic roots and can be transformed by an orthogonal transformation into $\lambda \xi ^2+\mu \eta^2,\lambda, \mu >0.$ Changing scales leads to $\xi ^2+\eta^2$. This multiplier is the $-$Laplace operator in two dimensions, with the fundamental solution $$\frac{-1}{2\pi}\ln(\sqrt{x^2+y^2})=E(x,y).$$ So the inverse of the Fourier multiplier $D_x^2+D_y^2$ is the convolution by $E$. Some analogous things can be done when $\gamma^2\ge\alpha\beta$. |
# Does an object storing more internal energy emit more thermal radiation?
There are two objects of the same material, similar geometry but different dimensions and same temperature. The larger object is supposed to contain more quantity of heat because it has larger volume.
Now I know that the quantity of emitted thermal radiation is directly proportional to temperature, but is it also regarding to internal energy?
How do I know which object emits more thermal radiaion? (without measurement)
• For similar shaped objects the larger object will emit more radiation, but this is because it has a greater surface area, not because of it's volume or internal energy. Note that for objects with a very different shape it may be that the object with the smaller volume and/or heat capacity can have a larger surface area and so may emit more radiation – By Symmetry Aug 14 at 10:30
• Objects can't contain heat. Heat is energy that is transferred due to temperature differences. – Aaron Stevens Aug 14 at 10:37
• Per Aaron Stevens objects don't contain heat. They have internal energy. Suggest you edit your question and change the terms "quantity of heat" and "heat" to "internal energy" – Bob D Aug 14 at 11:22
• Aaron and Bob, there are contexts where objects do contain heat. It is standard in chemical engineering to establish a base temperature, and use that temperature to calculate the enthalpy, or heat content, of a give mass of a substance such as steam. Typically, zero enthalpy occurs for an ideal gas at zero Kelvin, but the basis can be defined at other conditions as needed because the concept of enthalpy is usually used to calculate enthalpy differences. – David White Aug 14 at 13:48
• @DavidWhite It may be that in chemical engineering objects do "contain" heat, but in thermodynamics there are no contexts where objects contain heat. Objects have internal energy. In thermodynamics heat is energy transfer due solely to temperature difference. – Bob D Aug 14 at 22:42
@Andreas Sundstrom is correct that the equation for radiation does not depend on the volume (mass) of the radiating object. But that is because the equation assumes the temperature of the emitting body is constant.
As the body radiates heat its internal energy is reduced thus reducing the temperature of the body and the rate of radiant heating. How quickly this occurs depends, of course, on the mass of the body and its specific heat and the variables in the radiant heat transfer equation. I should think the temperature of a sheet of paper would would drop quickly due to its low mass (volume).
Bottom line, to the extent that the temperature of the radiating body depends on its internal energy and the rate of radiant heat transfer depends on temperature, the rate of radiant heat transfer depends on the volume of the emitting object.
Hope this helps
If you just mean the blackbody radiation, then that the radiated power both depends on the object's surface area, $$A$$, and its temperature to the forth power, $$T^4$$, according to $$P_{\rm rad} = \epsilon\sigma A T^4$$, where $$\epsilon$$ is the object's emissivity (material dependent) and $$\sigma=5.67\times10^{-8}{\rm\,W\,m^{-2}\,K^{-4}}$$ is the Stefan-Boltzmann constant. Hence it's the surface area which determines the radiation.
Since two objects can have the same volume but vastly different surface areas (think of a sheet of paper which has a large surface area but a very small volume), the thermal radiation is not dependent on the volume. However if you have two objects of the same shape, e.g. two cubes or two spheres, then the surface area scales with the volume as $$A\propto V^{2/3}$$, which would then also affect the thermal radiation, but only for two object of the same shape.
To answer your final question: If you have two objects at the same temperature and material, then the one with the larger surface area will radiate more energy.
• You assumed a convex surface. – my2cts Aug 14 at 10:52
• @Andréas Thank you – hello there Aug 14 at 12:28
• @my2cts Yes indeed, I did. Thanks for pointing that out. – Andréas Sundström Aug 15 at 6:35 |
# Project: Millikan Oil Drop Experiment. Millikan experiment several identical mass il drops were given charges of Te The voltage was
Project: Millikan Oil Drop Experiment. Millikan experiment several identical mass il drops were given charges of Te The voltage was kept constant at 500 V The plate separation was manipulated and the resulting accelertion of the oil drops was measured. plate Acceleration (mls?) separation (mm) 183 Using cuTVe straightening techniques determine the mass of the oil drop in this experiment. You will need piece of 'graph paper to do this question manually: will show you how do this on Tl calculator and within a Google Sheet as well: Let $set up the straightening part as class. ## Answers #### Similar Solved Questions 1 answer ##### Can someone help me create and initialize a pointer to function of DisplayInfo for a template... can someone help me create and initialize a pointer to function of DisplayInfo for a template specific data type (double) here is DisplayInfo prototype template <typename ObjType> void DisplayInfo(const ObjType &object)... 5 answers ##### Genetic drift is most likely to occur when:Genetic diversity is highPopulations have low heterozygosityEnvironmental selection is strongPopulation size is low Genetic drift is most likely to occur when: Genetic diversity is high Populations have low heterozygosity Environmental selection is strong Population size is low... 5 answers ##### Consider the following equilibrium system: 2PbS(s) + 3Oz(g) 2PbO(s) + 2SO2(g) Predict the direction of the net reaction as a result of 'increasing the Pressure on the system at constant temperature:Reaction will shift to the leftIncreasing pressure has no effect on the reactionReaction will shift to the rightNo enough data are given Consider the following equilibrium system: 2PbS(s) + 3Oz(g) 2PbO(s) + 2SO2(g) Predict the direction of the net reaction as a result of 'increasing the Pressure on the system at constant temperature: Reaction will shift to the left Increasing pressure has no effect on the reaction Reaction will... 5 answers ##### Show that20 for I > 0. Vi+cUse Your result frO. question 6 L fiud[dx V1+I1 Show that 20 for I > 0. Vi+c Use Your result frO. question 6 L fiud[ dx V1+I1... 1 answer ##### Amino Acid Examples The following amino acid is aspartic acid. Which of the following best describes... Amino Acid Examples The following amino acid is aspartic acid. Which of the following best describes the properties its side chain has as part of a protein? A) Acidic он B) Basic C) Hydrophobic D) Hydrophilic H2N E) Special role ОН... 1 answer ##### Open-Closed Principle: (answer all the questions) a. Give the one sentence summary b Describe what it... Open-Closed Principle: (answer all the questions) a. Give the one sentence summary b Describe what it means c Why is it a good idea? d. What pattern uses it to add functionality, and how?... 5 answers ##### A steel ball is dropped from building roof and passes window, taking 0.2 to fall from the top t0 the bottom 0f the window_ distance of' 2m, It then falls t0 sidewalk and bounces back past the Window. moving from bottom to top in 0.2 Assunie that the upward flight is an exact reVerse of the fall. The time the ball spends below the bottom of the window is 4 s. How tall is the building? (g m$-)
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# peerreviewed_2014.bib
@article{Teixeira_GCB_2014,
author = {Anderson-Teixeira, Kristina J. and Davies, Stuart J. and Bennett,
Amy C. and Gonzalez-Akre, Erika B. and Muller-Landau, Helene C. and
Joseph Wright, S. and Abu Salim, Kamariah and Almeyda Zambrano, Ang{\~e}lica
M. and Alonso, Alfonso and Baltzer, Jennifer L. and Basset, Yves
and Bourg, Norman A. and Broadbent, Eben N. and Brockelman, Warren
Y. and Bunyavejchewin, Sarayudh and Burslem, David F. R. P. and Butt,
Nathalie and Cao, Min and Cardenas, Dairon and Chuyong, George B.
and Clay, Keith and Cordell, Susan and Dattaraja, Handanakere S.
and Deng, Xiaobao and Detto, Matteo and Du, Xiaojun and Duque, Alvaro
and Erikson, David L. and Ewango, Corneille E.N. and Fischer, Gunter
A. and Fletcher, Christine and Foster, Robin B. and Giardina, Christian
P. and Gilbert, Gregory S. and Gunatilleke, Nimal and Gunatilleke,
Savitri and Hao, Zhanqing and Hargrove, William W. and Hart, Terese
B. and Hau, Billy C.H. and He, Fangliang and Hoffman, Forrest M.
and Howe, Robert W. and Hubbell, Stephen P. and Inman-Narahari, Faith
M. and Jansen, Patrick A. and Jiang, Mingxi and Johnson, Daniel J.
and Kanzaki, Mamoru and Kassim, Abdul Rahman and Kenfack, David and
Kibet, Staline and Kinnaird, Margaret F. and Korte, Lisa and Kral,
Kamil and Kumar, Jitendra and Larson, Andrew J. and Li, Yide and
Li, Xiankun and Liu, Shirong and Lum, Shawn K.Y. and Lutz, James
A. and Ma, Keping and Maddalena, Damian M. and Makana, Jean-Remy
and Malhi, Yadvinder and Marthews, Toby and Mat Serudin, Rafizah
and McMahon, Sean M. and McShea, William J. and Memiaghe, Her{v\
e} R. and Mi, Xiangcheng and Mizuno, Takashi and Morecroft, Michael
and Myers, Jonathan A. and Novotny, Vojtech and de Oliveira, Alexandre
A. and Ong, Perry S. and Orwig, David A. and Ostertag, Rebecca and
den Ouden, Jan and Parker, Geoffrey G. and Phillips, Richard P. and
Sack, Lawren and Sainge, Moses N. and Sang, Weiguo and Sri-ngernyuang,
Kriangsak and Sukumar, Raman and Sun, I-Fang and Sungpalee, Witchaphart
and Suresh, Hebbalalu Sathyanarayana and Tan, Sylvester and Thomas,
Sean C. and Thomas, Duncan W. and Thompson, Jill and Turner, Benjamin
L. and Uriarte, Maria and Valencia, Renato and Vallejo, Marta I.
and Vicentini, Alberto and Vr{\v s}ka, Tom{\ a}{\v s} and Wang,
Xihua and Wang, Xugao and Weiblen, George and Wolf, Amy and Xu, Han
and Yap, Sandra and Zimmerman, Jess},
title = {{CTFS-ForestGEO}: a worldwide network monitoring forests in an era
of global change},
journal = {Global Change Biology},
year = {2014},
abstract = {Global change is impacting forests worldwide, threatening biodiversity
and ecosystem services including climate regulation. Understanding
how forests respond is critical to forest conservation and climate
protection. This review describes an international network of 59
long-term forest dynamics research sites (CTFS-ForestGEO) useful
for characterizing forest responses to global change. Within very
large plots (median size 25 ha), all stems â¥1 cm diameter are
identified to species, mapped, and regularly recensused according
to standardized protocols. CTFS-ForestGEO spans 25°Sâ61°N latitude,
is generally representative of the range of bioclimatic, edaphic,
and topographic conditions experienced by forests worldwide, and
is the only forest monitoring network that applies a standardized
protocol to each of the world's major forest biomes. Supplementary
standardized measurements at subsets of the sites provide additional
information on plants, animals, and ecosystem and environmental variables.
CTFS-ForestGEO sites are experiencing multifaceted anthropogenic
global change pressures including warming (average 0.61 °C), changes
in precipitation (up to ±30% change), atmospheric deposition of
nitrogen and sulfur compounds (up to 3.8Â g NÂ mâ2Â yrâ1 and
3.1Â g SÂ mâ2Â yrâ1), and forest fragmentation in the surrounding
landscape (up to 88% reduced tree cover within 5Â km). The broad
suite of measurements made at CTFS-ForestGEO sites makes it possible
to investigate the complex ways in which global change is impacting
forest dynamics. Ongoing research across the CTFS-ForestGEO network
is yielding insights into how and why the forests are changing, and
continued monitoring will provide vital contributions to understanding
worldwide forest diversity and dynamics in an era of global change.},
doi = {10.1111/gcb.12712},
file = {pubs/Teixeira_GCB_2014.pdf},
issn = {1365-2486},
keywords = {biodiversity, Center for Tropical Forest Science (CTFS), climate change,
demography, forest dynamics plot, Forest Global Earth Observatory
(ForestGEO), long-term monitoring, spatial analysis},
owner = {jkumar},
timestamp = {2014.09.26},
url = {http://dx.doi.org/10.1111/gcb.12712}
}
@article{Warren_NewPhytologist_2014,
author = {Warren, Jeffrey M. and Hanson, Paul J. and Iversen, Colleen M. and
Kumar, Jitendra and Walker, Anthony P. and Wullschleger, Stan D.},
title = {Root structural and functional dynamics in terrestrial biosphere
models: evaluation and recommendations},
journal = {New Phytologist},
author = {Warren, Jeffrey and Hanson, Paul and Iversen, Colleen and Kumar,
Jitendra and Walker, Anthony and Wullschleger, Stan},
title = {Root Structural and Functional Dynamics in Terrestrial Biosphere
Models – Evaluation and Recommendations},
journal = {New Phytologist (Accepted)},
year = {2014},
file = {pubs/Warren_NewPhytologist_2014.pdf},
abstract = {
There is wide breadth of root function within ecosystems that
should be considered when modeling the terrestrial biosphere. Root
structure and function are closely associated with control of plant
water and nutrient uptake from the soil, plant carbon (C) assimilation,
partitioning and release to the soils, and control of biogeochemical
cycles through interactions within the rhizosphere. Root function
is extremely dynamic and dependent on internal plant signals, root
traits and morphology, and the physical, chemical and biotic soil
environment. While plant roots have significant structural and functional
plasticity to changing environmental conditions, their dynamics are
noticeably absent from the land component of process-based Earth
system models used to simulate global biogeochemical cycling. Their
dynamic representation in large-scale models should improve model
veracity. Here, we describe current root inclusion in models across
scales, ranging from mechanistic processes of single roots to parameterized
root processes operating at the landscape scale. With this foundation
we discuss how existing and future root functional knowledge, new
data compilation efforts, and novel modeling platforms can be leveraged
to enhance root functionality in large-scale terrestrial biosphere
models by improving parameterization within models, and introducing
new components such as dynamic root distribution and root functional
traits linked to resource extraction.},
doi = {10.1111/nph.13034},
issn = {1469-8137},
keywords = {hydraulic redistribution, nitrogen uptake, root function, root model,
root plasticity, water uptake},
owner = {jkumar},
timestamp = {2014.09.26},
url = {http://dx.doi.org/10.1111/nph.13034}
}
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@article{Hoffman_JGRB_20140201,
author = {Forrest M. Hoffman and James T. Randerson and Vivek K. Arora and Qing Bao and Patricia Cadule and Duoying Ji and Chris D. Jones and Michio Kawamiya and Samar Khatiwala and Keith Lindsay and Atsushi Obata and Elena Shevliakova and Katharina D. Six and Jerry F. Tjiputra and Evgeny M. Volodin and Tongwen Wu},
title = {Causes and Implications of Persistent Atmospheric Carbon Dioxide Biases in {E}arth {S}ystem {M}odels},
journal = jgrb,
volume = 119,
number = 2,
pages = {141--162},
doi = {10.1002/2013JG002381},
day = 1,
month = feb,
year = 2014,
abstract = {The strength of feedbacks between a changing climate and future CO$_2$ concentrations are uncertain and difficult to predict using Earth System Models (ESMs). We analyzed emission-driven simulations---in which atmospheric CO$_2$ levels were computed prognostically---for historical (1850--2005) and future periods (RCP~8.5 for 2006--2100) produced by 15 ESMs for the Fifth Phase of the Coupled Model Intercomparison Project (CMIP5). Comparison of ESM prognostic atmospheric CO$_2$ over the historical period with observations indicated that ESMs, on average, had a small positive bias in predictions of contemporary atmospheric CO$_2$. Weak ocean carbon uptake in many ESMs contributed to this bias, based on comparisons with observations of ocean and atmospheric anthropogenic carbon inventories. We found a significant linear relationship between contemporary atmospheric CO$_2$ biases and future CO$_2$ levels for the multi-model ensemble. We used this relationship to create a contemporary CO$_2$ tuned model (CCTM) estimate of the atmospheric CO$_2$ trajectory for the 21$^\textnormal{st}$ century. The CCTM yielded CO$_2$ estimates of 600 $\pm$ 14 ppm at 2060 and 947 $\pm$ 35 ppm at 2100, which were 21 ppm and 32 ppm below the multi-model mean during these two time periods. Using this emergent constraint approach, the likely ranges of future atmospheric CO$_2$, CO$_2$-induced radiative forcing, and CO$_2$-induced temperature increases for the RCP~8.5 scenario were considerably narrowed compared to estimates from the full ESM ensemble. Our analysis provided evidence that much of the model-to-model variation in projected CO$_2$ during the 21$^\textnormal{st}$ century was tied to biases that existed during the observational era, and that model differences in the representation of concentration--carbon feedbacks and other slowly changing carbon cycle processes appear to be the primary driver of this variability. By improving models to more closely match the long-term time series of CO$_2$ from Mauna Loa, our analysis suggests uncertainties in future climate projections can be reduced.}
}
@article{Lindsay_JClim_20141215,
author = {Keith Lindsay and Gordon B. Bonan and Scott C. Doney and Forrest M. Hoffman and David M. Lawrence and Matthew C. Long and Natalie M. Mahowald and J. Keith Moore and James T. Randerson and Peter E. Thornton},
title = {Preindustrial-Control and Twentieth-Century Carbon Cycle Experiments with the {E}arth System Model {CESM1(BGC)}},
journal = jclim,
volume = 27,
number = 24,
pages = {8981--9005},
doi = {10.1175/JCLI-D-12-00565.1},
day = 15,
month = dec,
year = 201,
abstract = {Version 1 of the Community Earth System Model, in the configuration where its full carbon cycle is enabled, is introduced and documented. In this configuration, the terrestrial biogeochemical model, which includes carbon?nitrogen dynamics and is present in earlier model versions, is coupled to an ocean biogeochemical model and atmospheric CO$_2$ tracers. The authors provide a description of the model, detail how preindustrial-control and twentieth-century experiments were initialized and forced, and examine the behavior of the carbon cycle in those experiments. They examine how sea- and land-to-air CO$_2$ fluxes contribute to the increase of atmospheric CO$_2$ in the twentieth century, analyze how atmospheric CO$_2$ and its surface fluxes vary on interannual time scales, including how they respond to ENSO, and describe the seasonal cycle of atmospheric CO$_2$ and its surface fluxes. While the model broadly reproduces observed aspects of the carbon cycle, there are several notable biases, including having too large of an increase in atmospheric CO$_2$ over the twentieth century and too small of a seasonal cycle of atmospheric CO$_2$ in the Northern Hemisphere. The biases are related to a weak response of the carbon cycle to climatic variations on interannual and seasonal time scales and to twentieth-century anthropogenic forcings, including rising CO$_2$, land-use change, and atmospheric deposition of nitrogen.}
}
@article{Sun_PNAS_20141104,
author = {Ying Sun and Lianhong Gu and Robert E. Dickinson and Richard J. Norby and Stephen G. Pallardy and Forrest M. Hoffman},
title = {Impact of Mesophyll Diffusion on Estimated Global Land {CO$_2$} Fertilization},
journal = pnas,
volume = 111,
number = 44,
pages = {15774--15779},
doi = {10.1073/pnas.1418075111},
day = 4,
month = nov,
year = 2014,
abstract = {In C3 plants, CO$_2$ concentrations drop considerably along mesophyll diffusion pathways from substomatal cavities to chloroplasts where CO$_2$ assimilation occurs. Global carbon cycle models have not explicitly represented this internal drawdown and therefore overestimate CO$_2$ available for carboxylation and underestimate photosynthetic responsiveness to atmospheric CO$_2$. An explicit consideration of mesophyll diffusion increases the modeled cumulative CO$_2$ fertilization effect (CFE) for global gross primary production (GPP) from 915 to 1,057~PgC for the period of 1901--2010. This increase represents a 16\% correction, which is large enough to explain the persistent overestimation of growth rates of historical atmospheric CO$_2$ by Earth system models. Without this correction, the CFE for global GPP is underestimated by 0.05~PgC/y/ppm. This finding implies that the contemporary terrestrial biosphere is more CO$_2$ limited than previously thought.}
}
@article{Wang_Biogeosci_20140407,
author = {Y. P. Wang and B. C. Chen and W. R. Wieder and M. Leite and B. E. Medlyn and M. Rasmussen and M. J. Smith and F. B. Agusto and F. M. Hoffman and Y. Q. Luo},
title = {Oscillatory Behavior of Two Nonlinear Microbial Models of Soil Carbon Decomposition},
journal = biogeosci,
volume = 11,
number = 7,
pages = {1817--1831},
doi = {10.5194/bg-11-1817-2014},
day = 7,
month = apr,
year = 2014,
abstract = {A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to $\sqrt{\left(\epsilon^{-1} - 1\right) K_s / V_s}$ in the two-pool model, and to $\sqrt{\left(\epsilon^{-1} - 1\right) K_l / V_l}$ in the three-pool model, where $\epsilon$ is microbial growth efficiency, $K_s$ and $K_l$ are the half saturation constants of soil and litter carbon, respectively, and $V_s$ and $V_l$ are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period of between 5 and 15 years depending on other parameter values, and has smaller amplitude at soil temperatures between 0 and 15$^\circ$C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether $\epsilon$ varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input are notable features in these nonlinear models that are somewhat unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.}
} |
How do you solve log(x-15)=2-logx?
Sep 18, 2015
$x = 20$
Explanation:
Put everything that's a log on the same side
$\log \left(x - 15\right) + \log \left(x\right) = 2$
Remember that $\log \left(m\right) + \log \left(n\right) = \log \left(m n\right)$
$\log \left(x \left(x - 15\right)\right) = 2$
If ${\log}_{a} \left(b\right) = c$, then $b = {a}^{c}$
$x \left(x - 15\right) = {10}^{2}$
Expand and solve the quadratic equation
${x}^{2} - 15 x = 100 \rightarrow {x}^{2} - 15 x - 100 = 0$
$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \left(- 100\right)}}{2} = \frac{15 \pm \sqrt{225 + 400}}{2}$
$x = \frac{15 \pm \sqrt{625}}{2} = \frac{15 \pm 25}{2}$
${x}_{1} = \frac{15 + 25}{2} = \frac{40}{2} = 20$
${x}_{2} = \frac{15 - 25}{2} = - \frac{10}{2} = - 5$
Remember that since we were dealing with logarithms, we can't have null or negative arguments, so
$x - 15 > 0 \rightarrow x > 15$
$x > 0$
We conclude that any answers must follow $x > 15$, which only of the two answers do, thus, the answer is $x = 20$ |
# Sesquipower
2010 Mathematics Subject Classification: Primary: 68R15 [MSN][ZBL]
A sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.
Formally, let $A$ be an alphabet and $A^*$ be the free monoid of finite strings over $A$. Every non-empty word $w$ in $A^+$ is a sesquipower of order 1. If $u$ is a sequipower of order $n$ then any word $w = uvu$ is a sesquipower of order $n+1$.[1] The degree of a non-empty word $w$ is the largest integer $d$ such that $w$ is a sesquipower of order $d$'.[2]
The Zimin words $Z_n$ over the alphabet $\{x_1,\ldots,x_n\}$ are defined by $Z_1 = x_1$ and $Z_n = Z_{n-1} x_n Z_{n-1}$ for $n>1$. The infinite Zimin word $\mathbf{Z}$ may be defined as the word whose $i$-th position, for $i\ge 1$, contains $x_{j+1}$ where $2^j$ is the largest power of 2 dividing $i$. Each $Z_n$ is the initial segment of $\mathbf{Z}$ of length $2^n-1$.
A bi-ideal sequence is a sequence of words $f_i$ where $f_1 \in A^+$ and $$f_{i+1} = f_i g_i f_i$$ for some $g_i \in A^*$ and $i \ge 1$. The degree of a word $w$ is thus the length of the longest bi-ideal sequence ending in $w$.[2]
For a finite alphabet $A$ on $k$ letters, there is an integer $M$, depending on $k$ and $n$, such that any word of length $M$ has a factor which is a sesquipower of order at least $n$. We express this by saying that the sesquipowers are unavoidable patterns.[3][4]
Given an infinite bi-ideal sequence, we note that each $f_i$ is a prefix of $f_{i+1}$ and so the $f_i$ converge to an infinite sequence $$\mathbf{f} = f_1 g_1 f_2 g_2 \cdots \ .$$
We define an infinite word to be a sesquipower if is the limit of an infinite bi-ideal sequence.[5] An infinite word is a sesquipower if and only if it is a recurrent word,[5][6] that is, every factor occurs infinitely often.[7]
Fix a finite alphabet $A$ and assume a total order on the letters. For given integers $p$ and $n$, every sufficiently long word in $A^*$ has either a factor which is a $p$-power or a factor which is an $n$-sesquipower; in the latter case the factor has an $n$-factorisation into Lyndon words.[6]
## References
1. Lothaire (2011) p. 135
2. Lothaire (2011) p. 136
3. Lothaire (2011) p. 137
4. Berstel et al (2009) p.132
5. Lothiare (2011) p. 141
6. Berstel et al (2009) p.133
7. Lothaire (2011) p. 30
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V.; "Combinatorics on words. Christoffel words and repetitions in words", CRM Monograph Series 27, American Mathematical Society (2009) ISBN 978-0-8218-4480-9 Zbl 1161.68043
• Lothaire, M.; "Algebraic combinatorics on words", Encyclopedia of Mathematics and Its Applications 90 Cambridge University Press (2011) ISBN 978-0-521-18071-9 Zbl 1221.68183
• Pytheas Fogg, N.; "Substitutions in dynamics, arithmetics and combinatorics", Lecture Notes in Mathematics 1794 Springer-Verlag (2002) ISBN 3-540-44141-7 Zbl 1014.11015
• Sapir, Mark V., "Combinatorial algebra: syntax and semantics" Springer Verlag (2014) ISBN 978-3-319-08030-7
How to Cite This Entry:
Sesquipower. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sesquipower&oldid=39622 |
# Tag Info
15
Most probably, he was referring to the Brunswick Magnetic Anomaly: http://www.geosociety.org/news/pr/2014/14-22.htm This is an area of lower-than-normal magnetism present in North America in the Southwestern United States. Note that altough it COULD be a piece of Africa, it could also be caused by something else.
9
The boundary between the upper and lower mantle is today believed to be due to the phase change because of increasing pressure of: $$\text{Ringwoodite} \xrightarrow{\text{660 km, about 24.5 GPa}} \text{Mg-Perovskite} + \text{Mg-Wüstite}$$ The boundary was originally found in the form of a seismic discontinuity (the "660 km discontinuity"). This information ...
9
The principle does not apply to all materials on earth as observed by us but is primarily concerned with sediments and rocks that form from sediments. Since sedimentation is a process driven by gravity grains settle one on top of others and there is no possibility to change that without adding additional processes. So from this perspective the principle is ...
7
No, we occasionally get fragments of upper mantle overthrust onto the Earth's crust, but then the specific gravity (density) contrast is not so great, roughly 2.7 to 3.0 for crust, about 3.2 for mantle. But the core is another matter. The density contrast is huge - the liquid outer core is between 9.9 and 12.2, whilst the solid core is estimated to be 12.6 ...
4
The diagram is perhaps a little mis-leading. Think of the ground/rock between the electrodes as a big resistor. The surrounding rock can be modeled as a set of parallel resistors. Most of the electrons will take a direct straight line route between the electrodes, but some of them flow through this surrounding rock, so increasing the current and reducing the ...
4
Here is another example where the law of superposition breaks. Mind you this is a natural process: meteorite impacts. A meteorite impact will excavate material from the bottom, and deposit it above the crater rims. Here's how: This will result in something that looks like this: Source for both images is http://www.lpi.usra.edu/exploration/education/...
4
As Peter correctly responds, for sedimentary rocks the principle of superposition is pretty much watertight except where tectonic superposition takes place. The latter can either repeat the sedimentary sequence, overturn it, or overthrust earlier sequences. There are many possible geometries, which creationists tend to misrepresent. The principle of ...
4
The inner core rotates slightly faster than the rest of the planet. This passage from National Geographic explains: The liquid outer core separates the inner core from the rest of the Earth, and as a result, the inner core rotates a little differently than the rest of the planet. It rotates eastward, like the surface, but it’s a little faster, making an ...
4
Im am currently doing my masters in geophysics (last semester) and before that I did a bachelor in geoscience. I assume by layers you mean the crust, the mantle and the core. These all have different composition and also different densities. But the earth rotates as a whole, not the individual layers, all layers have the same angular velocity. That means ...
3
The terrain examples show some striking similarities to what is referred to in the United States as "Badlands" - a type of terrain formed by layered sedimentary strata of rock that is soft enough to be eroded by wind and water into recognizably similar features. For example, here is South Dakota Badlands National Park: Anecdotally, I have seen similar ...
2
If you were hoping for a smooth succession from Archean all the way down to Quaternary, you're going to be disappointed. Geology is messier than rocks being serenely laid down continuously through time like that. A drive from the Appalachians to the East Coast would get you everything but the Archaean rocks, although the only Proterozoic rocks would be found ...
1
The divisions in the geologic time scale have evolved over time. Its origins can be traced back to Nicolaus Steno in 1669 described two basic geologic principles. The first stated that sedimentary rocks are laid down in a horizontal manner. The second stated that younger rock units were deposited on top of older rock units. Work by James Hutton, Charles ...
Only top voted, non community-wiki answers of a minimum length are eligible |
Usually, paramagnetic. necessity of unpaired electrons in their orbitals. In addition, many of the strongly magnetic minerals discussed below are paramagnetic above a critical temperature (the Curie temperature or … Elemental iron is diamagnetic, this means that all the electrons of the frontier orbitals of iron are paired. It has enormous permanent magnetic moment. Since there are 2 unpaired electrons in this case, it is paramagnetic … In the presence of the external field the sample moves toward the strong field, attaching itself to … Susceptibility is inversely proportional to the absolute temperature. Most elements and some compounds are paramagnetic. the excess number of electrons in the same spin, determines the magnetic moment. Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. I'll tell you the Paramagnetic or Diamagnetic list below. Share this. number of electrons in Fe. ligands. The susceptibility and relative permeability are very large. Nonmagnetic (diamagnetic) metals include copper, gold, and silver. Indicate whether boron atoms are paramagnetic or diamagnetic. Iron, nickel, and cobalt are examples of ferromagnetic materials. Elemental iron and iron (III) are paramagnetic because of the Paramagnetic compounds contain one or more unpaired situation. In the presence of the external field the sample moves toward the strong field, attaching itself to … Paramagnetic materials include aluminium, oxygen, titanium, and iron oxide (FeO). Thus paramagnetic materials are permanent magnets by their intrinsic property, but the magnetic moment is too weak to detect it physically. Ferromagnetic substances are those in which each atom or molecule has a strong spontaneous net magnetic moment. Properties Diamagnetic Paramagnetic Ferromagnetic; Definition: It is a material in which there is no permanent magnetic moment. Therefore, it does not lead to the pairing of unpaired 3d electrons. If you want to quickly find the word you want to search, use Ctrl + F, then type the word you want to search. However, materials can react quite differently to … However if you were What does contingent mean in real estate? Diamagnetic Up to date, curated data provided by Mathematica 's ElementData function from Wolfram Research, Inc. Click here to buy a book, photographic periodic table poster, card deck, or 3D print based on the images you see here! Diamagnetic has no unpaired e-, while paramagnetic does. If the ligands attached to the Fe (II) metal are strong-field Indicate whether Fe 2 + ions are paramagnetic or diamagnetic. In addition, many of the strongly magnetic minerals discussed below are paramagnetic above a critical temperature (the Curie temperature or Néel temperature).In Table 2 are given susceptibilities for some iron-bearing minerals. Paramagnetism: Paramagnetic is basically a type of magnetism in which substances are getting attracted by an extrinsic magnetic field. ... Is Iron ( fe ) Paramagnetic or Diamagnetic ? Answer (a): The O atom has 2s 2 2p 4 as the electron configuration. Add up the amount bonding valence electrons it has. What is the difficulty of this problem? Some sulfides are paramagnetic, but some are strongly magnetic (see below). It has permanent magnetic moment. However if you were heat up iron you would be able to promote an electron the LUMO … Elemental iron and iron (III) are paramagnetic because of the necessity of unpaired electrons in their orbitals. (lowest unoccupied molecular orbital. paramagnetic or diamagnetic, respectively. So, most magnetic materials are metals. All Rights Reserved. same position most of the time. But in insulating solids like FeO and Fe2O3 the Fe2+ and Fe3+ ions will arrange themselves, in local bonding with oxygen, as antiferromagnets due to superexchange. paramagnetic or diamagnetic, respectively. Technically, they are repelled by the poles of a magnet, but this repulsion is Paramagnetic or Diamagnetic. (For example : ? electrons and are attracted to the poles of a magnet. If diamagnetic gas is introduced between pole pieces of magnet, it spreads at a right angle to the magnetic field. Question: Is Iron ( fe ) a Paramagnetic or Diamagnetic ? ligands in an octahedral configuration, a low-spin situation is created in the d Iron is strongly ferromagnetic: Iron shows strong magnetic properties. Why don't libraries smell like bookstores? If diamagnetic gas is introduced between pole pieces of magnet, it spreads at a right angle to the magnetic field. They are called paramagnetic and diamagnetic respectively. To do so, we need to do the following steps: Step 1: Determine the shorthand e – config. Who is the longest reigning WWE Champion of all time? Share this. Indicate whether F-ions are paramagnetic or diamagnetic. Diamagnetic substances have a negative relative permeability (susceptibility); paramagnetic substances have positive. When a material is placed within a magnetic field, the magnetic forces of the material's electrons will be affected. Magnetic materials consist of atoms with partially-filled electron shells. Step 3: Draw the orbital diagram for the outermost sublevel and determine if the ion is diamagnetic or paramagnetic So, most magnetic materials are metals. I know the Diamagnetic means having all boxes filled and paramagnetic is having any nonsymmetric pattern but i don't remember what it is if all the boxes have 1 e-. These substances exhibit strong paramagnetic properties. It has permanent magnetic moment. Examples of ferromagnetic substances are iron, nickel, cobalt, magnetite. Diamagnetic substances have a negative relative permeability (susceptibility); paramagnetic substances have positive. Diamagnetic Up to date, curated data provided by Mathematica 's ElementData function from Wolfram Research, Inc. Click here to buy a book, photographic periodic table poster, card deck, or 3D print based on the images you see here! Paramagnetic Substances: Those substances which are weekly magnetized when placed in an external magnetic field in the same direction as the applied field are called Paramagnetic substances. Some sulfides are paramagnetic, but some are strongly magnetic (see below). magnetism. Between the ferromagnetics and the paramagnetic s there is an enormous gap. ... Is Iron ( fe ) Paramagnetic or Diamagnetic ? of the frontier orbitals of iron are paired. Nonmagnetic (diamagnetic) metals include copper, gold, and silver. Iron (Fe) is magnetically very versatile. Ferromagnetism is a large effect, often greater than that of the applied magnetic field, that persists even in the absence of an applied magnetic field. The atomic number of iron is 28. How many candles are on a Hanukkah menorah? heat up iron you would be able to promote an electron the LUMO Hence the number of unpaired electrons, i.e. Iron ( fe ) is Ferromagnetic. In case of Cu 2+ the electronic configuration is 3d 9 thus it has one unpaired electron in d- subshell thus it is paramagnetic. The maximum magnetic susceptibility of iron is half a million times greater than that of liquid oxygen, one of the strongest paramagnetic substances known. Iron ( fe ) is Ferromagnetic. Diamagnetic Paramagnetic Ferromagnetic; Definition: It is a material in which there is no permanent magnetic moment. If you want to quickly find the word you want to search, use Ctrl + F, then type the word you want to search. So, although deoxygenated haemoglobin is paramagnetic and very slightly attracted to a magnet, and also both oxygenated haemoglobin and plasma are diamagnetic or in other words, slightly repelled by a magnet, in theory, wearing a magnetic bracelet shouldn't have a physiological effect. The .examples of diamagnetic, paramagnetic and ferromagnetic materials are respectively: (A) copper, aluininium, iron (B) aluminium, copper, iron (C) A few materials, notably iron, show a very large attraction toward the pole of a permanent bar magnet; materials of this kind are called ferromagnetic.… spectroscopy: Fluorescence and phosphorescence By bagus amin - Maret 10, 2018 - Add Comment. • paramagnetic: at least one electron is unpaired. Ferromagnetic substances have permanently aligned magnetic dipoles. When iron (II) is bonded to certain ligands, Iron (II) is also in this All of the electrons are paired and, as a result, the complex is Therefore, O has 2 unpaired electrons. Magnetic materials consist of atoms with partially-filled electron shells. A material is called diamagnetic if the value of $\mu = 0$. Paramagnetic or Diamagnetic. The diamagnetism of some materials, however, is masked either by a weak magnetic attraction (paramagnetism) or a very strong attraction (ferromagnetism). Chemical Concepts Demonstrated: Paramagnetic properties of iron metal This would result in unpaired By bagus amin - Maret 10, 2018 - Add Comment. Step 2: Determine the e – config. Examples: Copper, silver, and gold: Magnesium, molybdenum, lithium, and tantalum: Iron, nickel, and cobalt: State: Solid, liquid, or gas: Solid, liquid, or gas: Solid Properties of Diamagnetic,Paramagnetic, Ferromagnetic substances Some substances are only weakly attracted by a magnet, while some are repelled by it. Compounds in which all of the electrons are paired are diamagnetic. Copyright © 2020 Multiply Media, LLC. Ferromagnetism is the basic mechanism by which certain materials (such as iron) form permanent magnets, or are attracted to magnets. Or if you need more Paramagnetic and Diamagnetic practice, you can also practice Paramagnetic and Diamagnetic practice problems. It is a ferromagnetic substance and it can be magnetized permanently. I'll tell you the Paramagnetic or Diamagnetic list below. Metallic iron is ferromagnetic, due to his atoms’ external electrons that are itinerant and polarized. External electrons that are itinerant and polarized magnetism in which there is no permanent magnetic moment is too weak detect. For a Christmas party of fact every material is called diamagnetic if the value of $\mu 0... Each atom or molecule has a strong spontaneous net magnetic moment of diamagnetic, and. Quite differently to … diamagnetic has no unpaired e-, while some are strongly magnetic ( see below ) called! Magnetic ( see below ) with these materials are attracted to magnets magnet is attracted... Are some samples of opening remarks for a Christmas party one or more unpaired electrons in the same spin determines. Electron box configuration lands so all the electrons are paired are diamagnetic a! React quite differently to … diamagnetic has no unpaired e-, while paramagnetic does are itinerant and polarized it. Make iron paramagnetic, or diamagnetic list below Pets - 2006 Save the Ladybug this and! Elemental iron and iron ( fe ) paramagnetic or diamagnetic iron are paired necessity of unpaired 3d electrons using! Do n't know what i was just talking about plz do n't know what i was just about. 10, 2018 - Add Comment e-, while paramagnetic does you more... Lumo ( lowest unoccupied molecular orbital behaviour of steel, cobalt,.. N'T answer this question an iron nail brought near a pole of a magnet 'll you... ( both diamagnets ), with inexpensive setups diamagnetic, paramagnetic, ferromagnetic substances iron nickel! The sample in each test tube is paramagnetic magnetism in which there is no magnetic... Sticks to it \mu = 0$ ferromagnetic: iron shows strong magnetic properties question is... ) paramagnetic or diamagnetic list below 0 \$ between the ferromagnetics and the paramagnetic or diamagnetic list below electron... It and sticks to it but some are repelled by the stronger paramagnetic or diamagnetic is... What i was just talking about plz do n't know what i was just talking about plz do n't what... Of all time up arrow in them field, the diamagnetic property is masked by the stronger paramagnetic diamagnetic! Magnet is strongly attracted by a magnetic field bar magnet is strongly ferromagnetic: iron shows strong properties! As iron ) form permanent magnets, or are attracted to the magnetic moment small! Plz do n't know what i was just talking about plz do n't answer this question atoms with partially-filled shells! Electrons and make iron paramagnetic, or diamagnetic list below ( both diamagnets ), with setups. In unpaired electrons and make iron paramagnetic, but some are strongly magnetic ( see below ) right to. Iron, nickel, cobalt, magnetite cobalt, magnetite all temperatures iron are paired are diamagnetic this effect known. Up iron you would be able to promote an electron the LUMO ( lowest unoccupied molecular orbital is usually small. As iron ) form permanent magnets by their intrinsic property, but some are strongly magnetic ( see below.... In general, diamagnetic practice problems footprints on the moon last susceptibility ) ; substances. Elemental iron is strongly attracted by a magnetic field contrast, paramagnetic and diamagnetic practice problems their orbitals spontaneous... On the moon last you would be able to promote an electron the LUMO ( lowest unoccupied molecular.... Were heat up iron you would be able to promote an electron the LUMO ( lowest unoccupied molecular.. Placed within a magnetic field result in unpaired electrons in their orbitals that all the boxes the. Forces of the electrons are paired are diamagnetic 3d 9 thus it is diamagnetic, paramagnetic and diamagnetic practice.. Of atoms with partially-filled electron shells: Step 1: Determine the e! Negative relative permeability ( susceptibility ) ; paramagnetic substances have a negative relative permeability susceptibility! E-, while paramagnetic does strong spontaneous net magnetic moment Definition: it is paramagnetic,... Shorthand e – config, ferromagnetic substances iron, etc n't know what i just! And low-spin complexes, strong-field and weak-field ions and ligands steel, cobalt,.! 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## Characterizations for the commutator of parabolic nonsingular integral operator on parabolic generalized Orlicz-Morrey spaces.(English)Zbl 1439.42024
Summary: We show continuity in parabolic generalized Orlicz-Morrey spaces $$M^{\Phi,\varphi}$$ of the commutator of parabolic nonsingular integral operators. We shall give necessary and sufficient conditions for the boundedness of the commutator of parabolic nonsingular integral operator on $$M^{\Phi,\varphi}$$ spaces with BMO functions.
### MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35 Function spaces arising in harmonic analysis
Full Text:
### References:
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. |
# Sam Watson’s journal round-up for 8th October 2018
Every Monday our authors provide a round-up of some of the most recently published peer reviewed articles from the field. We don’t cover everything, or even what’s most important – just a few papers that have interested the author. Visit our Resources page for links to more journals or follow the HealthEconBot. If you’d like to write one of our weekly journal round-ups, get in touch.
A cost‐effectiveness threshold based on the marginal returns of cardiovascular hospital spending. Health Economics [PubMed] Published 1st October 2018
There are two types of cost-effectiveness threshold of interest to researchers. First, there’s the societal willingness-to-pay for a given gain in health or quality of life. This is what many regulatory bodies, such as NICE, use. Second, there is the actual return on medical spending achieved by the health service. Reimbursement of technologies with a lesser return for every pound or dollar would reduce the overall efficiency of the health service. Some refer to this as the opportunity cost, although in a technical sense I would disagree that it is the opportunity cost per se. Nevertheless, this latter definition has seen a growth in empirical work; with some data on health spending and outcomes, we can start to estimate this threshold.
This article looks at spending on cardiovascular disease (CVD) among elderly age groups by gender in the Netherlands and survival. Estimating the causal effect of spending is tricky with these data: spending may go up because survival is worsening, external factors like smoking may have a confounding role, and using five year age bands (as the authors do) over time can lead to bias as the average age in these bands is increasing as demographics shift. The authors do a pretty good job in specifying a Bayesian hierarchical model with enough flexibility to accommodate these potential issues. For example, linear time trends are allowed to vary by age-gender groups and dynamic effects of spending are included. However, there’s no examination of whether the model is actually a good fit to the data, something which I’m growing to believe is an area where we, in health and health services research, need to improve.
Most interestingly (for me at least) the authors look at a range of priors based on previous studies and a meta-analysis of similar studies. The estimated elasticity using information from prior studies is more ‘optimistic’ about the effect of health spending than a ‘vague’ prior. This could be because CVD or the Netherlands differs in a particular way from other areas. I might argue that the modelling here is better than some previous efforts as well, which could explain the difference. Extrapolating using life tables the authors estimate a base case cost per QALY of €40,000.
Early illicit drug use and the age of onset of homelessness. Journal of the Royal Statistical Society: Series A Published 11th September 2018
How the consumption of different things, like food, drugs, or alcohol, affects life and health outcomes is a difficult question to answer empirically. Consider a recent widely-criticised study on alcohol published in The Lancet. Among a number of issues, despite including a huge amount of data, the paper was unable to address the problem that different kinds of people drink different amounts. The kind of person who is teetotal may be so for a number of reasons including alcoholism, interaction with medication, or other health issues. Similarly, studies on the effect of cannabis consumption have shown among other things an association with lower IQ and poorer mental health. But are those who consume cannabis already those with lower IQs or at higher risk of psychoses? This article considers the relationship between cannabis and homelessness. While homelessness may lead to an increase in drug use, drug use may also be a cause of homelessness.
The paper is a neat application of bivariate hazard models. We recently looked at shared parameter models on the blog, which factorise the joint distribution of two variables into their marginal distribution by assuming their relationship is due to some unobserved variable. The bivariate hazard models work here in a similar way: the bivariate model is specified as the product of the marginal densities and the individual unobserved heterogeneity. This specification allows (i) people to have different unobserved risks for both homelessness and cannabis use and (ii) cannabis to have a causal effect on homelessness and vice versa.
Despite the careful set-up though, I’m not wholly convinced of the face validity of the results. The authors claim that daily cannabis use among men has a large effect on becoming homeless – as large an effect as having separated parents – which seems implausible to me. Cannabis use can cause psychological dependency but I can’t see people choosing it over having a home as they might with something like heroin. The authors also claim that homelessness doesn’t really have an effect on cannabis use among men because the estimated effect is “relatively small” (it is the same order of magnitude as the reverse causal effect) and only “marginally significant”. Interpreting these results in the context of cannabis use would then be difficult, though. The paper provides much additional material of interest. However, the conclusion that regular cannabis use, all else being equal, has a “strong effect” on male homelessness, seems both difficult to conceptualise and not in keeping with the messiness of the data and complexity of the empirical question.
How could health care be anything other than high quality? The Lancet: Global Health [PubMed] Published 5th September 2018
Tedros Adhanom Ghebreyesus, or Dr Tedros as he’s better known, is the head of the WHO. This editorial was penned in response to the recent Lancet Commission on Health Care Quality and related studies (see this round-up). However, I was critical of these studies for a number of reasons, in particular, the conflation of ‘quality’ as we normally understand it and everything else that may impact on how a health system performs. This includes resourcing, which is obviously low in poor countries, availability of labour and medical supplies, and demand side choices about health care access. The empirical evidence was fairly weak; even in countries like in the UK in which we’re swimming in data we struggle to quantify quality. Data are also often averaged at the national level, masking huge underlying variation within-country. This editorial is, therefore, a bit of an empty platitude: of course we should strive to improve ‘quality’ – its goodness is definitional. But without a solid understanding of how to do this or even what we mean when we say ‘quality’ in this context, we’re not really saying anything at all. Proposing that we need a ‘revolution’ without any real concrete proposals is fairly meaningless and ignores the massive strides that have been made in recent years. Delivering high-quality, timely, effective, equitable, and integrated health care in the poorest settings means more resources. Tinkering with what little services already exist for those most in need is not going to produce a revolutionary change. But this strays into political territory, which UN organisations often flounder in.
Editorial: Statistical flaws in the teaching excellence and student outcomes framework in UK higher education. Journal of the Royal Statistical Society: Series A Published 21st September 2018
As a final note for our academic audience, we give you a statement on the Teaching Excellence Framework (TEF). For our non-UK audience, the TEF is a new system being introduced by the government, which seeks to introduce more of a ‘market’ in higher education by trying to quantify teaching quality and then allowing the best-performing universities to charge more. No-one would disagree with the sentiment that improving higher education standards is better for students and teachers alike, but the TEF is fundamentally statistically flawed, as discussed in this editorial in the JRSS.
Some key points of contention are: (i) TEF doesn’t actually assess any teaching, such as through observation; (ii) there is no consideration of uncertainty about scores and rankings; (iii) “The benchmarking process appears to be a kind of poor person’s propensity analysis” – copied verbatim as I couldn’t have phrased it any better; (iv) there has been no consideration of gaming the metrics; and (v) the proposed models do not reflect the actual aims of TEF and are likely to be biased. Economists will also likely have strong views on how the TEF incentives will affect institutional behaviour. But, as Michael Gove, the former justice and education secretary said, Britons have had enough of experts.
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# Sam Watson’s journal round-up for 10th September 2018
Every Monday our authors provide a round-up of some of the most recently published peer reviewed articles from the field. We don’t cover everything, or even what’s most important – just a few papers that have interested the author. Visit our Resources page for links to more journals or follow the HealthEconBot. If you’d like to write one of our weekly journal round-ups, get in touch.
Probabilistic sensitivity analysis in cost-effectiveness models: determining model convergence in cohort models. PharmacoEconomics [PubMed] Published 27th July 2018
Probabilistic sensitivity analysis (PSA) is rightfully a required component of economic evaluations. Deterministic sensitivity analyses are generally biased; averaging the outputs of a model based on a choice of values from a complex joint distribution is not likely to be a good reflection of the true model mean. PSA involves repeatedly sampling parameters from their respective distributions and analysing the resulting model outputs. But how many times should you do this? Most times, an arbitrary number is selected that seems “big enough”, say 1,000 or 10,000. But these simulations themselves exhibit variance; so-called Monte Carlo error. This paper discusses making the choice of the number of simulations more formal by assessing the “convergence” of simulation output.
In the same way as sample sizes are chosen for trials, the number of simulations should provide an adequate level of precision, anything more wastes resources without improving inferences. For example, if the statistic of interest is the net monetary benefit, then we would want the confidence interval (CI) to exclude zero as this should be a sufficient level of certainty for an investment decision. The paper, therefore, proposed conducting a number of simulations, examining the CI for when it is ‘narrow enough’, and conducting further simulations if it is not. However, I see a problem with this proposal: the variance of a statistic from a sequence of simulations itself has variance. The stopping points at which we might check CI are themselves arbitrary: additional simulations can increase the width of the CI as well as reduce them. Consider the following set of simulations from a simple ratio of random variables $ICER = gamma(1,0.01)/normal(0.01,0.01)$:The “stopping rule” therefore proposed doesn’t necessarily indicate “convergence” as a few more simulations could lead to a wider, as well as narrower, CI. The heuristic approach is undoubtedly an improvement on the current way things are usually done, but I think there is scope here for a more rigorous method of assessing convergence in PSA.
Mortality due to low-quality health systems in the universal health coverage era: a systematic analysis of amenable deaths in 137 countries. The Lancet [PubMed] Published 5th September 2018
Richard Horton, the oracular editor-in-chief of the Lancet, tweeted last week:
There is certainly an argument that academic journals are good forums to make advocacy arguments. Who better to interpret the analyses presented in these journals than the authors and audiences themselves? But, without a strict editorial bulkhead between analysis and opinion, we run the risk that the articles and their content are influenced or dictated by the political whims of editors rather than scientific merit. Unfortunately, I think this article is evidence of that.
No-one debates that improving health care quality will improve patient outcomes and experience. It is in the very definition of ‘quality’. This paper aims to estimate the numbers of deaths each year due to ‘poor quality’ in low- and middle-income countries (LMICs). The trouble with this is two-fold: given the number of unknown quantities required to get a handle on this figure, the definition of quality notwithstanding, the uncertainty around this figure should be incredibly high (see below); and, attributing these deaths in a causal way to a nebulous definition of ‘quality’ is tenuous at best. The approach of the article is, in essence, to assume that the differences in fatality rates of treatable conditions between LMICs and the best performing health systems on Earth, among people who attend health services, are entirely caused by ‘poor quality’. This definition of quality would therefore seem to encompass low resourcing, poor supply of human resources, a lack of access to medicines, as well as everything else that’s different in health systems. Then, to get to this figure, the authors have multiple sources of uncertainty including:
• Using a range of proxies for health care utilisation;
• Using global burden of disease epidemiology estimates, which have associated uncertainty;
• A number of data slicing decisions, such as truncating case fatality rates;
• Estimating utilisation rates based on a predictive model;
• Estimating the case-fatality rate for non-users of health services based on other estimated statistics.
Despite this, the authors claim to estimate a 95% uncertainty interval with a width of only 300,000 people, with a mean estimate of 5.0 million, due to ‘poor quality’. This seems highly implausible, and yet it is claimed to be a causal effect of an undefined ‘poor quality’. The timing of this article coincides with the Lancet Commission on care quality in LMICs and, one suspects, had it not been for the advocacy angle on care quality, it would not have been published in this journal.
Embedding as a pitfall for survey‐based welfare indicators: evidence from an experiment. Journal of the Royal Statistical Society: Series A Published 4th September 2018
Health economists will be well aware of the various measures used to evaluate welfare and well-being. Surveys are typically used that are comprised of questions relating to a number of different dimensions. These could include emotional and social well-being or physical functioning. Similar types of surveys are also used to collect population preferences over states of the world or policy options, for example, Kahneman and Knetsch conducted a survey of WTP for different environmental policies. These surveys can exhibit what is called an ’embedding effect’, which Kahneman and Knetsch described as when the value of a good varies “depending on whether the good is assessed on its own or embedded as part of a more inclusive package.” That is to say that the way people value single dimensional attributes or qualities can be distorted when they’re embedded as part of a multi-dimensional choice. This article reports the results of an experiment involving students who were asked to weight the relative importance of different dimensions of the Better Life Index, including jobs, housing, and income. The randomised treatment was whether they rated ‘jobs’ as a single category, or were presented with individual dimensions, such as the unemployment rate and job security. The experiment shows strong evidence of embedding – the overall weighting substantially differed by treatment. This, the authors conclude, means that the Better Life Index fails to accurately capture preferences and is subject to manipulation should a researcher be so inclined – if you want evidence to say your policy is the most important, just change the way the dimensions are presented.
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# Chris Sampson’s journal round-up for 27th August 2018
Every Monday our authors provide a round-up of some of the most recently published peer reviewed articles from the field. We don’t cover everything, or even what’s most important – just a few papers that have interested the author. Visit our Resources page for links to more journals or follow the HealthEconBot. If you’d like to write one of our weekly journal round-ups, get in touch.
Ethically acceptable compensation for living donations of organs, tissues, and cells: an unexploited potential? Applied Health Economics and Health Policy [PubMed] Published 25th August 2018
Around the world, there are shortages of organs for transplantation. In economics, the debate around the need to increase organ donation can be frustratingly ignorant of ethical and distributional concerns. So it’s refreshing to see this article attempting to square concerns about efficiency and equity. The authors do so by using a ‘spheres of justice’ framework. This is the idea that different social goods should be distributed according to different principles. So, while we might be happy for brocolli and iPhones to be distributed on the basis of free exchange, we might want health to be distributed on the basis of need. The argument can be extended to state that – for a just situation to prevail – certain exchanges between these spheres of justice (e.g. health for iPhones) should never take place. This idea might explain why – as the authors demonstrate with a review of European countries – policy tends not to allow monetary compensation for organ donation.
The paper cleverly sets out to taxonomise monetary and non-monetary reimbursement and compensation with reference to individuals’ incentives and the spheres of justice principles. From this, the authors reach two key conclusions. Firstly, that (monetary) reimbursement of donors’ expenses (e.g. travel costs or lost earnings) is ethically sound as this does not constitute an incentive to donate but rather removes existing disincentives. Secondly, that non-monetary compensation could be deemed ethical.
Three possible forms of non-monetary compensation are discussed: i) prioritisation, ii) free access, and iii) non-health care-related benefits. The first could involve being given priority for receiving organs, or it could extend to the jumping of other health care waiting lists. I think this is more problematic than the authors let on because it asserts that health care should – at least in part – be distributed according to desert rather than need. The second option – free access – could mean access to health care that people would otherwise have to pay for. The third option could involve access to other social goods such as education or housing.
This is an interesting article and an enjoyable read, but I don’t think it provides a complete solution. Maybe I’m just too much of a Marxist, but I think that this – as all other proposals – fails to distribute from each according to ability. That is, we’d still expect non-monetary compensation to incentivise poorer (and on average less healthy) people to donate organs, thus exacerbating health inequality. This is because i) poorer people are more likely to need the non-monetary benefits and ii) we live in a capitalist society in which there is almost nothing that money can’t by and which is strictly non-monetary. Show me a proposal that increases donation rates from those who can most afford to donate them (i.e. the rich and healthy).
Selecting bolt-on dimensions for the EQ-5D: examining their contribution to health-related quality of life. Value in Health Published 18th August 2018
Measures such as the EQ-5D are used to describe health-related quality of life as completely and generically as possible. But there is a trade-off between completeness and the length of the questionnaire. Necessarily, there are parts of the evaluative space that measures will not capture because they are a simplification. If the bit they’re missing is important to your patient group, that’s a problem. You might fancy a bolt-on. But how do we decide which areas of the evaluative space should be more completely included in the measure? Which bolt-ons should be used? This paper seeks to provide means of answering these questions.
The article builds on an earlier piece of work that was included in an earlier journal round-up. In the previous paper, the authors used factor analysis to identify candidate bolt-ons. The goal of this paper is to outline an approach for specifying which of these candidates ought to be used. Using data from the Multi-Instrument Comparison study, the authors fit linear regressions to see how well 37 candidate bolt-on items explain differences in health-related quality of life. The 37 items correspond to six different domains: energy/vitality, satisfaction, relationships, hearing, vision, and speech. In a second test, the authors explored whether the bolt-on candidates could explain differences in health-related quality of life associated with six chronic conditions. Health-related quality of life is defined according to a visual analogue scale, which notably does not correspond to that used in the EQ-5D but rather uses a broader measure of physical, mental, and social health.
The results suggest that items related to energy/vitality, relationships, and satisfaction explained a significant part of health-related quality of life on top of the existing EQ-5D dimensions. The implication is that these could be good candidates for bolt-ons. The analysis of the different conditions was less clear.
For me, there’s a fundamental problem with this study. It moves the goals posts. Bolt-ons are about improving the extent to which a measure can more accurately represent the evaluative space that it is designed to characterise. In this study, the authors use a broader definition of health-related quality of life that – as far as I can tell – the EQ-5D is not designed to capture. We’re not dealing with bolt-ons, we’re dealing with extensions to facilitate expansions to the evaluative space. Nevertheless, the method could prove useful if combined with a more thorough consideration of the evaluative space.
Sources of health financing and health outcomes: a panel data analysis. Health Economics [PubMed] [RePEc] Published 15th August 2018
There is a growing body of research looking at the impact that health (care) spending has on health outcomes. Usually, these studies don’t explicitly look at who is doing the spending. In this study, the author distinguishes between public and private spending and attempts to identify which type of spending (if either) results in greater health improvements.
The author uses data from the World Bank’s World Development Indicators for 1995-2014. Life expectancy at birth is adopted as the primary health outcome and the key expenditure variables are health expenditure as a share of GDP and private health expenditure as a share of total health expenditure. Controlling for a variety of other variables, including some determinants of health such as income and access to an improved water source, a triple difference analysis is described. The triple difference estimator corresponds to the difference in health outcomes arising from i) differences in the private expenditure level, given ii) differences in total expenditure, over iii) time.
The key finding from the study is that, on average, private expenditure is more effective in increasing life expectancy at birth than public expenditure. The author also looks at government effectiveness, which proves crucial. The finding in favour of private expenditure entirely disappears when only countries with effective government are considered. There is some evidence that public expenditure is more effective in these countries, and this is something that future research should investigate further. For countries with ineffective governments, the implication is that policy should be directed towards increasing overall health care expenditure by increasing private expenditure.
Credits |
Geek went Freak!
# Using 74HCxx with 3.3v
Atmega processors in Arduino boards are powered by 5V power supplies. It makes it easier to work with other ICs that accept 5v Vcc.
But most recent processors including ARM cortex-M operate at 3.3v Vcc. Would the tried and tested 74HCxx series work with 3.3v Vcc? Or should we pick a different low voltage logic series.
The answers, as usual, are in the data sheets.
The characteristics we are interested in are:
1. Allowed Vcc range
2. Vih, input voltage high threshold
3. Vil, input voltage low threshold
## Allowed Vcc range
The data sheet says, allowed Vcc range is -0.5v to 7v. That is great! So, 74HCxx series can be powered from 3.3v.
## Input thresholds at 5v
First lets look at Vih and Vil at 5v Vcc.
1. Vih: 3.5v
2. Vil: 1.5v
## Input thresholds at 3.3v
1. Vih: 2.31
2. Vil: 0.99
That is great! This would work perfectly.
# Introduction
Trimpot or preset resistor is a variable resistor whose resistance can be adjusted using a screw driver.
# Trimpot vs potentiometer
Trimpots are not meant to be tinkered with very often. This is the difference that sets it apart from potentiometers. Although both of them do the same thing, potentiometers offer more change cycles.
# Pinout
Trimpots have three terminals:
1. In1 ( A )
2. In2 ( B )
3. Out or Wiper ( C )
Trimpot pinout
Terminals A and B are the two ends of the voltage divider. Terminal C is the output where the voltage is divided.
Trimpot pinout schematic
# Circuit symbol
Trimpot schematic
# Connection
Lets build a small circuit to control the brightness of the LED with the help of a trim pot.
Trimpot schematic
# Introduction
74244 is a non-inverting octal buffer. Its output can be disabled in groups of four if they are not being used.
## Stats
Lines #: 8
Groups #: 2
Output type: CMOS push-pull
Tristate-able: Yes
ESD protection: Yes
Propagation delay @ 5V: 11ns
Vcc
1. Min: 4.5V
2. Max: 5.5V
Vih @ 5V
1. Typical: 1.6V
2. Minimum: 2.0V Io Max.: 35mA
## Special features
An unique feature about 74HCT245 that is not present in other similar buffers is that, the 8 lines can be split into two buses with up to 4 lines in each bus. The groups are:
1. Group1: 1A0, 1A1, 1A2, 1A3
2. Group2: 1A0, 2A1, 2A2, 2A3
Each of the groups can be separately tristate-ed using respective output enable lines. Thats right, each of these groups have their own output enable lines unlike other similar octal buffer ICs.
# Application
## Buffer
74HCT245 can output a maximum current of 35mA. This can be used to drive heavier loads that microcontroller i/o pins cannot drive directly.
## 3.3V -> 5V Level shifter
Typical Vih for 74HCT244 is 2V which is way below 3.3V. 74HCT244 can be used as 3.3V -> 5V level shifter. 74HCT244 is better than 74HC244 in 3.3V to 5V level shifting because 74HCT series is compatible with TTL logic rather than CMOS logic.
## 5V -> 3.3V level shifter
Maximum input voltage for 74HCT245 cannot be greater than Vcc according to the specification.
But it can be tricked into accepting voltages greater than Vcc. 74HCT245 has ESD protection diodes on input pins. These diodes come into play when input voltage is greater than Vcc. These clamping diodes can take maximum current of 20mA. This means that a current limiting resister can be used to keep the current below 20mA at 5V. Current limiting resistor of 500ohms would work.
The problem with this trick is, it increases the propagation delay. It is fairly useful for low speed circuits. The lower the resistor the better.
For high speed circuits, use 74ACT244, 74ACT245, 74HCT244, 74HCT245.
# Buffers: 74244 vs 74245
74244 is an octal unidirectional buffer. The 8 lines can be split into two groups and each group can be individually disabled.
74245 is an octal bidirectional buffer. The direction of all the lines are controlled using the same line, DIR.
Bidirectional bus can be built using 74244 by
1. Complementing the output enable lines
2. Connecting outputs and inputs together
Bidirectional bus using 74244
But this only gives 4 bidirectional lines. If bidirectional lines are required, it is better to use 74245.
# Introduction
74244 is a non-inverting octal buffer. Its output can be disabled in groups of four if they are not being used.
## Stats
Lines #: 8
Groups #: 2
Output type: CMOS push-pull
Tristate-able: Yes
ESD protection: Yes
Propagation delay @ 5V: 9ns
Vcc
1. Min: 2V
2. Max: 6V
Vih @ 5V
1. Typical: 2.65
2. Minimum: 3.5 Io Max.: 35mA
## Special features
An unique feature about 74HC244 that is not present in other similar buffers is that, the 8 lines can be split into two buses with up to 4 lines in each bus. The groups are:
1. Group1: 1A0, 1A1, 1A2, 1A3
2. Group2: 1A0, 2A1, 2A2, 2A3
Each of the groups can be separately tristate-ed using respective output enable lines. Thats right, each of these groups have their own output enable lines unlike other similar octal buffer ICs.
# Application
## Buffer
74HC244 can output a maximum current of 35mA. This can be used to drive heavier loads that microcontroller i/o pins cannot drive directly.
## 3.3V -> 5V Level shifter
NOTE: Although this might work at room temperatures, this is not recommended. Use 74HCT244, 74HCT245, 74ACT244, 74ACT245.
Typical Vih for 74HC244 at room temperature is 2.65V which is way below 3.3V. 74HC244 can be used as 3.3V -> 5V level shifter.
But important thing to note is, the minimum guaranteed worst case Vih is 3.5V. A 3.3V device will never be able to reach this level. So using 74HC244 as 3.3V to 5V level shifter is very bad idea.
## 5V -> 3.3V level shifter
Maximum input voltage for 74HC244 cannot be greater than Vcc according to the specification.
But it can be tricked into accepting voltages greater than Vcc. 74HC244 has ESD protection diodes on input pins. These diodes come into play when input voltage is greater than Vcc. These clamping diodes can take maximum current of 20mA. This means that a current limiting resister can be used to keep the current below 20mA at 5V. Current limiting resistor of 500ohms would work.
The problem with this trick is, it increases the propagation delay. It is fairly useful for low speed circuits. The lower the resistor the better.
For high speed circuits, use 74ACT244, 74ACT245, 74HCT244, 74HCT245.
# Introduction
74245 is a non-inverting bidirectional octal buffer.
## Stats
Lines #: 8
Output type: CMOS push-pull
Tristate-able: Yes
ESD protection: Yes
Propagation delay @ 5V: 10ns
Vcc
1. Min: 2V
2. Max: 6V
Vih @ 5V
1. Typical: 2.65
2. Minimum: 3.5 Io Max.: 35mA
## Special features
74245 is a bidirectional buffer. The direction of all the lines together can be controlled using the direction pin, DIR.
# Application
## 3.3V -> 5V Level shifter
NOTE: Although this might work at room temperatures, this is not recommended. Use 74HCT244, 74HCT245, 74ACT244, 74ACT245.
Typical Vih for 74HC245 at room temperature is 2.65V which is way below 3.3V. 74HC245 can be used as 3.3V -> 5V level shifter.
But important thing to note is, the minimum guaranteed worst case Vih is 3.5V. A 3.3V device will never be able to reach this level. So using 74HC245 as 3.3V to 5V level shifter is very bad idea.
## 5V -> 3.3V level shifter
Maximum input voltage for 74HC245 cannot be greater than Vcc according to the specification.
But it can be tricked into accepting voltages greater than Vcc. 74HC245 has ESD protection diodes on input pins. These diodes come into play when input voltage is greater than Vcc. These clamping diodes can take maximum current of 20mA. This means that a current limiting resister can be used to keep the current below 20mA at 5V. Current limiting resistor of 500ohms would work.
The problem with this trick is, it increases the propagation delay. It is fairly useful for low speed circuits. The lower the resistor the better.
For high speed circuits, use 74ACT244, 74ACT245, 74HCT244, 74HCT245.
# 74XX164
74XX164 is a serial-in parallel-out shift register.
Unlike 74XX594 and 74XX595, it doesn’t have a front-end with latch or flip-flop.
74HC164 block diagram
## Clock input
CP pin is used as clock to the shift registers. The data is shifted on the positive edge of clock signal.
## Output enable
74XX164 doesn’t have an output enable pin. However, master reset (MR) pin can be used to clear all shift registers and drive all outputs low.
## Master reset
A low on the MR pin clears all the registers and all the outputs are driven low.
Q7 of first 74XX164 can be connected to the serial input of the second 74XX164 to cascade them into 16-bit serial register.
# Golang: Channel send behavior
General way to send data to a channel in golang is
lChan := make(chan bool)
lChan <- true
It is important to understand that, golang channels block on send until there is either
1. Space left to buffer the data
2. Until the data is read from the channel, if there is no buffer space left
# Non-buffered channels (default)
This means that, if there is no one reading the data from the channel, the code blocks.
Here is an example:
When the program is run, golang automatically detects the deadlock during runtime and report it:
Note: When buffer size is 0, the channel blocks until the data is read from the channel. This was the behavior witnessed in first example.
This fate can be postponed by increasing the buffer size of the channel. By default, the buffer size is 0.
# Buffered channels
Optional second argument to make function takes length of channel buffer as input and returns a new channel with the requested buffer size. It is not possible to change the size of the channel buffer after it is created.
package main
import (
"fmt"
)
func main() {
lChan := make(chan bool, 5)
lChan <- true
fmt.Println("Finished!")
}
This program doesn’t block. But eventually it will when the buffer is filled up. Consider this example where buffer length is 5, we are trying to send 10 values to the channel but only reading 3. This should deadlock when we try to send 9th value.
When the program is run, golang automatically detects the deadlock during runtime and report it:
And it does!
# Nonblocking send
One can use non-blocking send using the select statement. select statement is generally used to simultaneously send/receive to/from multiple channels.
When select has a default clause, the select statement will not block during send/receive.
Output:
# Nonblocking send with timeout
Nonblocking sends are great. But won’t be nicer if we waited a bit before giving up? time package has a function called After. This function takes Duration as time.Duration type and returns a bool channel that is sent a message after the duration expires. After function can be used in select statement, achieve timeout functionality.
Output:
# Zener diode voltage regulator
Zener diode has a very useful property of dropping a constant voltage (equal to its Zener voltage) across its terminals when it is reverse biased at voltages above its Zener voltage.
Constant Voltage drop
Constant Voltage drop plot
The voltage across the zener diode follows the source V1 until it reaches 8.2V (Zener voltage). Any further increase of source voltage doesn’t change voltage across the Zener diode.
Until the Zener voltage is reached, the Zener diode is open, so there is no current flowing through the circuit. After Zener voltage is reached, the current is,
$$tex I_{R1} = \frac{V_{V1} - V_{ZV}}{R_{R1}} tex$$
This property can be used to build voltage regulators. Voltage regulators are used when the input voltage is higher than the required voltage. But placing the Zener diode across the load, a constant required voltage can be dropped across the load if the input voltage is above this desired voltage.
Zener diodes with various Zener voltage are available in the market. Lets consider a case where we have 24V input voltage and have circuit that operates at 8.2V.
We have chosen the Zener diode BZX84C8V2L which has required Zener voltage of 8.2V.
Voltage regulator
The resistor Rs has two uses
1. To limit current flowing through the Zener diode when Vin is greater than Zener voltage. This sets the minimum limit on the Resistor Rs.
2. It should drop as less voltage as possible so that most of Vin is dropped across the load (Rl) when Vin < Zener voltage. This sets the maximum limit on the resistor Rs.
# GCC: -fprofile-arcs va -ftest-coverage
Two flags we use in gcc to enable code coverage are -fprofile-arcs and -ftest-coverage.
So, what exactly do these flags do?
## Block graph and source location information
-ftest-coverage generates .gcno files for corresponding translation units during compile time. These files have information regarding block graphs and their associated line numbers in platform independent format.
## Profile statistics and counts
Code coverage is performed by instrumenting code into the target source code. This instrumented code then keeps track of statistics and counts of number of times a statement or a block has executed.
Functions in libgcov.a are responsible for recording these statistics information to .gcda file after execution.
The flag -fprofile-arcs links libgcov.a library into the executable.
## Making sense of the information
Thus, during compile time -ftest-coverage generates .gcno and after runtime, -fprofile-arcs generates .gcda. gcov command uses these two files to link statistics in .gcda to block graph and source association information in .gcno. |
# Free ACT Practice Questions
Question 1 of 1
ID: ACT-M-3
Section: Math
Topics: Numbers; Remainders
Difficulty level: Hard
(Practice Mode: Single selected Question » Back to Overview)
If the positive integer $N$ leaves a remainder of 1 when divided by 6, which of the following statements would be true?
1. $N^2$ if divided by 12 leaves a remainder 1
2. $N^2-1$ is divisible by 3
3. $N^2+2$ is divisible by 3
AOnly I
BOnly II
COnly III
DOnly II and III
EI, II and III |
# Who's up to the challenge? 24
$\large \dfrac { { x }^{ 2 }+y }{ x-{ y }^{ 2 } } =5$
How many integral solution(s) does the above equation have?
this is a part of Who's up to the challenge?
× |
Post Made Community Wiki by S. Carnahan
3 added 1 characters in body
The theorem of "friends and strangers": the Ramsey number $R(3,3)=6$. Not only can the proof be understood by high-school students, proofs a proof can be discovered by students at that level via something akin to the Socratic method. First students can establish the bound $R(3,3) > 5$ by 2-coloring the edges of $K_5$:
Then they can reason through that a 2-coloring of the edges of $K_6$ must contain a monochromatic triangle, and so $R(3,3)=6$: in every group of six, three must be friends or three must be strangers.
After this exercise, an inductive proof of the 2-color version of Ramsey's theorem is in reach.
An added bonus here is that one quickly reaches the frontiers of mathematics: $R(5,5)$ is unknown! It can be a revelation to students that there is a frontier of mathematics. And then one can tell the Erdős story about $R(6,6)$, as recounted here. :-)
2 deleted 8 characters in body
The theorem of "friends and strangers": the Ramsey number $R(3,3)=6$. Not only can the proof be understood by high-school students, proofs can be discovered by students at that level via something akin to the Socratic method. First students can establish the bound $R(3,3) > 5$ by 2-coloring the edges of $K_5$:
Then they can reason through that a 2-coloring of the edges of $K_6$ must contain a monochromatic triangle, and so $R(3,3)=6$. And so R(3,3)=6$: in every group of six, three must be friends or three must be strangers. After this exercise, an inductive proof of the 2-color version of Ramsey's theorem is in reach. An added bonus here is that one quickly reaches the frontiers of mathematics:$R(5,5)$is unknown! It can be a revelation to students that there is a frontier of mathematics. And then one can tell the Erdős story about$R(6,6)\$, as recounted here. :-)
1 |
# Pullus Jinx
## Redirected from Pullus
14,896pages on
this wiki
The Pullus Jinx (Pullus) is a transforming jinx used to transfigure the target into a chicken or a goose; in particular, Erklings appear to be rather vulnerable to this jinx
## History
This jinx was used in 1754 by an unknown wizard on Silvio Astolfi, an Italian broom racer.[1]
## Etymology
Pullus is Latin for the young of animals, particularly chickens (i.e. chicks). |
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11. Considered the leading Polish orchestra, the POLISH NATIONAL RADIO SYMPHONY ORCHESTRA IN KATOWICE fulfils the role of a cultural ambassador…
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# rainette 0.2.1 Unreleased
• Add option to show merged segments in document browser
• Fix warning and error in rainette_explor and rainette2_explor when a cluster dfm is empty
• Fix error when the dfm contains empty string as feature
# rainette 0.2.0 2021-06-25
## Important and breaking changes
• min_uc_size, uc_size1 and uc_size2 arguments to rainette and rainette2 have been renamed to min_segment_size, min_segment_size1 and min_segment_size2.
• The default value of min_segment_size in rainette is now 0, which means that no merging is done between segments by default. Results could then be different from previous package versions when min_uc_size was not specified.
• Important bugfix : merging of segments based on min_segment_size was not handled correctly in the previous versions regarding the segment sources, as segments from different documents could be merged together. This should now be fixed.
## New features
• A new graphical interface to browse cluster documents has been added to rainette_explor and rainette2_explor.
• New function clusters_by_doc_table which gives the number of segments of each cluster for each document.
• New function docs_by_cluster_table which gives, for each cluster, the number of documents with at least one segment in this cluster.
• split_segments should now be about 4 times faster.
• Terms frequencies and documents proportions statistics have been added to the explor interfaces.
## Other
• When rainette is called with min_segment_size > 0, a doc_id argument must be given which is the name of a dtm docvar identifying the segments source. If the corpus has been produced by split_segments, the added segment_source docvar is used by default.
• Color palette for clusters changed to “Tableau 10”.
• Negative keyness values are not shown by default anymore in rainette_explor, rainette2_explor, rainette_plot and rainette2_plot.
• Wordcloud plots have been removed from explor interfaces.
• A warning is displayed when min_split_members < 3.
• If rainette_explor is called on a rainette2 results object, rainette2_explor is launched automatically.
# rainette 0.1.3 2021-05-11
• Parallel computing in split_segments has been removed
• Fix potential name conflicts in rainette_explor
• Compatibility with quanteda v3 (thanks @kbenoit)
# rainette 0.1.2 2021-01-20
• Fix bug due to factor comparison in rainette2
# rainette 0.1.1 2020-05-09
• Compatibility with dplyr 1.0
• Fix bug in p threshold in rainette_stats
# rainette 0.1.0 Unreleased
• First version |
strict-concurrency-0.2.4.3: Strict concurrency abstractions
Control.Concurrent.MVar.Strict
Contents
Description
Synchronising, strict variables
Values placed in an MVar are evaluated to head normal form before being placed in the MVar, preventing a common source of space-leaks involving synchronising variables.
Synopsis
# MVars
data MVar a #
An MVar (pronounced "em-var") is a synchronising variable, used for communication between concurrent threads. It can be thought of as a a box, which may be empty or full.
Instances
Since: deepseq-1.4.3.0 Instance detailsDefined in Control.DeepSeq MethodsliftRnf :: (a -> ()) -> MVar a -> () # Eq (MVar a) Since: base-4.1.0.0 Instance detailsDefined in GHC.MVar Methods(==) :: MVar a -> MVar a -> Bool #(/=) :: MVar a -> MVar a -> Bool # NFData (MVar a) NOTE: Only strict in the reference and not the referenced value.Since: deepseq-1.4.2.0 Instance detailsDefined in Control.DeepSeq Methodsrnf :: MVar a -> () #
newEmptyMVar :: IO (MVar a) #
Create an MVar which is initially empty.
newMVar :: NFData a => a -> IO (MVar a) Source #
Create an MVar which contains the supplied value.
takeMVar :: MVar a -> IO a #
Return the contents of the MVar. If the MVar is currently empty, takeMVar will wait until it is full. After a takeMVar, the MVar is left empty.
There are two further important properties of takeMVar:
• takeMVar is single-wakeup. That is, if there are multiple threads blocked in takeMVar, and the MVar becomes full, only one thread will be woken up. The runtime guarantees that the woken thread completes its takeMVar operation.
• When multiple threads are blocked on an MVar, they are woken up in FIFO order. This is useful for providing fairness properties of abstractions built using MVars.
putMVar :: NFData a => MVar a -> a -> IO () Source #
Put a value into an MVar. If the MVar is currently full, putMVar will wait until it becomes empty.
There are two further important properties of putMVar:
• putMVar is single-wakeup. That is, if there are multiple threads blocked in putMVar, and the MVar becomes empty, only one thread will be woken up. The runtime guarantees that the woken thread completes its putMVar operation.
• When multiple threads are blocked on an MVar, they are woken up in FIFO order. This is useful for providing fairness properties of abstractions built using MVars.
readMVar :: NFData a => MVar a -> IO a Source #
This is a combination of takeMVar and putMVar; ie. it takes the value from the MVar, puts it back, and also returns it.
swapMVar :: NFData a => MVar a -> a -> IO a Source #
Take a value from an MVar, put a new value into the MVar and return the value taken. Note that there is a race condition whereby another process can put something in the MVar after the take happens but before the put does.
tryTakeMVar :: MVar a -> IO (Maybe a) #
A non-blocking version of takeMVar. The tryTakeMVar function returns immediately, with Nothing if the MVar was empty, or Just a if the MVar was full with contents a. After tryTakeMVar, the MVar is left empty.
tryPutMVar :: NFData a => MVar a -> a -> IO Bool Source #
A non-blocking version of putMVar. The tryPutMVar function attempts to put the value a into the MVar, returning True if it was successful, or False otherwise.
isEmptyMVar :: MVar a -> IO Bool #
Check whether a given MVar is empty.
Notice that the boolean value returned is just a snapshot of the state of the MVar. By the time you get to react on its result, the MVar may have been filled (or emptied) - so be extremely careful when using this operation. Use tryTakeMVar instead if possible.
withMVar :: NFData a => MVar a -> (a -> IO b) -> IO b Source #
withMVar is a safe wrapper for operating on the contents of an MVar. This operation is exception-safe: it will replace the original contents of the MVar if an exception is raised (see Control.Exception).
modifyMVar_ :: NFData a => MVar a -> (a -> IO a) -> IO () Source #
A safe wrapper for modifying the contents of an MVar. Like withMVar, modifyMVar will replace the original contents of the MVar if an exception is raised during the operation.
modifyMVar :: NFData a => MVar a -> (a -> IO (a, b)) -> IO b Source #
A slight variation on modifyMVar_ that allows a value to be returned (b) in addition to the modified value of the MVar.
addMVarFinalizer :: MVar a -> IO () -> IO () # |
Serious work amid the puzzles and jokes.
Amazon source
When Raymond Smullyan was born, Emanuel Lasker was still the world chess champion. Indeed, of the 16 universally recognized champions, only the first, Wilhelm Steinitz, lived outside Smullyan’s lifetime. Smullyan passed away a week ago Monday at age 97.
Today, Dick and I wish to add some thoughts to the many comments and tributes about Smullyan.
He was known for many things, but his best-known contributions were books with titles like: “What Is the Name of This Book?” Besides their obit in Sunday’s paper, the New York Times ran a sample of puzzles from these books. No doubt many enjoyed the books, and many may have been moved to study “real” logic and mathematics. His book To Mock a Mockingbird dressed up a serious introduction to combinatory logic. This logic is as powerful as normal predicate calculus but has no quantified variables. So making it readable to non-experts is a tribute to Smullyan’s ability to express deep ideas in ways that were so clear.
Smullyan earned his PhD under the guidance of Alonzo Church in 1959 at age 40. When I (Ken) was an undergrad at Princeton, I remember thinking, “gee, that’s old.” Well there’s old and there’s old… As we noted in our profile of him 19 months ago, he was still writing books at a splendid pace at age 95, including a textbook on logic.
Neither of us met him, so we never experienced his tricks and riddles firsthand, but we had impressions on the serious side. Here are some of Dick’s, first.
A Great Book
Smullyan and Melvin Fitting, who was one of his nine PhD students, wrote a wonderful book on set theory: Set Theory and the Continuum Problem (Oxford Logic Guides), 1996.
The blurb at Amazon says:
Set Theory and the Continuum Problem is a novel introduction to set theory … Part I introduces set theory, including basic axioms, development of the natural number system, Zorn’s Lemma and other maximal principles. Part II proves the consistency of the continuum hypothesis and the axiom of choice, with material on collapsing mappings, model-theoretic results, and constructible sets. Part III presents a version of Cohen’s proofs of the independence of the continuum hypothesis and the axiom of choice. It also presents, for the first time in a textbook, the double induction and superinduction principles, and Cowen’s theorem.
The blurb at Amazon says:
A lucid, elegant, and complete survey of set theory [in] three parts… Part One’s focus on axiomatic set theory features nine chapters that examine problems related to size comparisons between infinite sets, basics of class theory, and natural numbers. Additional topics include author Raymond Smullyan’s double induction principle, super induction, ordinal numbers, order isomorphism and transfinite recursion, and the axiom of foundation and cardinals. The six chapters of Part Two address Mostowski-Shepherdson mappings, reflection principles, constructible sets and constructibility, and the continuum hypothesis. The text concludes with a seven-chapter exploration of forcing and independence results.
Wait—are these the same book? Yes they are, and this is one way of saying the book is chock full of content while being self-contained. Neither blurb mentions the part that most grabbed me (Dick). This is their use of modal logic to explain forcing.
Modal logic has extra operators ${\Box\phi}$ which is usually interpreted as “Necessarily ${\phi}$” and ${\diamond\phi}$ meaning “Possibly ${\phi}$“; like the quantifiers ${\forall}$ and ${\exists}$ they obey the relation
$\displaystyle \Box \phi \longleftrightarrow \neg\diamond\neg\phi.$
Saul Kripke codified models as directed graphs whose nodes each have an interpretation of ${\phi}$. Then ${\Box\phi}$ holds at a node ${u}$ if all nodes reachable from ${u}$ satisfy ${\phi}$ (and hence ${\Box\phi}$), while ${\diamond\phi}$ holds at ${u}$ if ${\phi}$ holds at some node reachable from ${u}$. The nodes are “possible worlds.”
What Fitting and Smullyan do is define a translation ${\phi \rightarrow \phi'}$ from set theory to their modal logic such that ${\phi}$ is valid if and only if ${\Box\phi'}$. Then the game is to build a node ${u}$ such that every Zermelo-Fraenkel axiom gets a ${\Box}$ but the translated continuum hypothesis fails in some world reachable from ${u}$.
One reprinting of the book posed an inadvertent puzzle: many mathematical symbols were omitted. Symbols for set membership, subset, quantifiers, and so on were missing. As one online reviewer noted, “it really does make the book useless.” My copy at least was unaffected.
Rudimentary, My Dear
A week after our 7/28/15 Smullyan post mentioned above, I (Ken again) went with my family to Oregon for vacation. This included a trip to Powell’s Books in Portland, which may be the largest independent bookstore in the world. The math and science sections were larger and more eclectic than any Barnes or college bookstores I’ve seen. There were several copies of the 1961 Princeton Annals paperback edition of Smullyan’s PhD thesis, Theory of Formal Systems, on sale for \$20. I felt spurred to buy one and felt it could be useful because of Smullyan’s penchant for combinatorial concreteness.
Sure enough, section B of Chapter IV formulates the rudimentary relations crisply and clearly. Here are Smullyan’s words atop page 78 on their motivation:
As remarked in the Preface, our proof follows novel lines in that all appeal to the traditional number theory devices … in the past—e.g., prime factorization, congruences and the Chinese remainder theorem—are avoided. Thus Gödel’s program of establishing incompleteness, even of first-order theories involving plus and times as their some arithmetical primitives, can, by the methods of this section, be carried out without appeal to number theory.
Simply said: Smullyan avoids all the complicated numerical machinery needed in the usual treatments and makes them—like magic—disappear. The main predicate needed by Smullyan is ${R(p,x) =}$ ${x}$ is a power of ${p}$, provided ${p}$ is prime. From that he defined a predicate ${C(x,y,z)}$ meaning that the binary representation of ${z}$ is the concatenation of those of ${x}$ and ${y}$. The formal language is still that of logic and numbers but the operations are really manipulating strings. His predicates were able to fulfill all roles for which the class of primitive recursive relations and subclasses involving ${+}$ and ${\times}$ had previously been employed.
Smullyan was writing in 1959. Turing machine complexity had not even been defined yet. It transpired later that Smullyan’s class ${\mathsf{RUD}}$ contains nondeterministic logspace and equals the alternating linear time hierarchy. Linear time by itself is annoyingly dependent on machine details, but once you have a couple levels of quantifier alternation the class becomes very robust. Dick employed tricks with alternating linear time in some papers, and such alternation is used to amplify the time hierarchy theorem so that ${\mathsf{NLIN = DLIN}}$ for multitape Turing machines leads to a contradiction higher up. It is also intriguing to see Smullyan write on page 81:
We do not know whether or not all constructive arithmetic attributes are rudimentary. Quine […] has shown that plus and times are first order definable from [concatenation] … but this leaves unanswered the question as to whether plus and times are themselves rudimentary.
The thesis has a footnote saying this had been done for plus, and times follows from remarks above, but whether the predicate ${x*y = z}$ is in deterministic linear time remains open. It is likely that Smullyan went through similar concrete thinking as Juris Hartmanis and Richard Stearns when they conjectured no. We wonder if anyone thought to ask Smullyan about this and wish we had.
Open Problems
Our condolences go to his family along with our appreciation for his writings.
Emanuel Lasker philosophized in his 1906 book Struggle about perfect strategists in any walk of life, calling them Macheeides after Greek for “battle species.” An improved edition of Google DeepMind’s AlphaGo probably joined their ranks by beating top human players 60-0 in games played via online servers, not counting one game disrupted by connection failure. The top ranked player, Ke Jie, lost 3 games and landed in the hospital, but desires a 4th try. Where will computer Macheeides strike next?
A discussion on the famous problem
William Agnew is the chairperson of the Georgia Tech Theoretical Computer Science Club. He is, of course, an undergraduate at Tech with a multitude of interests—all related to computer science.
Today I want to report on a panel that we had the other night on the famous P vs. NP question.
What to do about claims of hard theorems?
Cropped from source
Shinichi Mochizuki has claimed the famous ABC conjecture since 2012. It is still unclear whether or not the claimed proof is correct. We covered it then and have mentioned it a few times since, but have not delved in to check it. Anyway its probably way above our ability to understand in some finite time.
Today I want to talk about how to check proofs like that of the ABC conjecture.
The issue is simple:
Someone writes up a paper that “proves” that X is true, where X is some hard open problem. How do we check that X is proved?
The proof in question is almost always long and complex. So the checking is not a simple matter. In some cases the proof might even use nonstandard methods and make it even harder to understand. That is exactly the case with Mochizuki’s proof—see here for some comments.
Possible Approaches
Let’s further assume that the claimed proof resolves X which is the P vs. NP problem. What should we do? There are some possible answers:
• Ignore: I have many colleagues who will not even open the paper to glance at it. Ken and I get a fair number of these, but I do at least open the file and take a quick look. I will send a message to the author—it usually is a single author—about some issue if I see one right away.
• Show Me The Beef: I firmly believe that a proof of an open problem should have at least one simple to state new trick or insight that we all missed. I would suggest that the author must be able to articulate this new idea: if they cannot then we can safely refuse to read it. I have worked some on the famous Jacobian Problem. At one time an author claimed they had a proof and it was just “a careful induction.” No. I never looked at it because of the lack of “beef,” and in a few weeks the proof fell apart.
• Money: Several people have suggested—perhaps not seriously—that any one claiming a proof must be ready to post a “bond” of some money. If someone finds an error they get the bond money. If no one does or even better if the proof is correct, then the money can be donated to one of our conferences.
• Hire: I have seen this idea just recently. The author posts a request for someone to work on their paper as a type of consultant. They are paid a fair hourly rate and help find the error.
• Timeout: An author who posts a false proof gets a timeout. They are not allowed to post another paper or submit a paper on X for some fixed time period. Some of the top journals like the Journal of the ACM already have a long timeout in place. The rationale behind this is that very often when an error is found in such a paper the author quickly “fixes” the issue and re-claims the result. In Stanislaw Ulam’s wonderful book Adventures of a Mathematician he talks about false proofs: Here “he” refers to an amateur who often joined Ulam at his habitual coffeehouse:
Every once in a while he would get up and join our table to gossip or kibitz ${\dots}$ Then he would add, “The bigger my proof, the smaller the hole. The longer and larger the proof, the smaller the hole.”
• Knock Heads Together: Oxford University hosted in December 2015 a workshop to examine Mochizuki’s claimed proof, including contact by Skype with Mochizuki himself. A report by Brian Conrad on the MathBabe blog makes for engaging reading—we could quote extensively from its concluding section 6. This shorter news report cited feelings of greater understanding and promise but lack of definite progress on verifying the proof, noting:
…[N]o one wants to be the guy that spent years working to understand a proof, only to find that it was not really a proof after all.
• Share The Credit: Building on the last point, perhaps proper credit can be given to someone who does spent a great deal of time working on trying to understand a long proof. If they find an unfixable error, then maybe they can publish that as a paper—especially if the error is nontrivial and not just a simple one. If they show that the proof is indeed correct, could they be rewarded with some type of co-authorship? Maybe a new type of authorship:
P Does Not Equal NP: A Proof Via Non-Linear Fourier Methods
Alice Azure with Bob Blue
Here the “with” signals that Alice is the main author and Bob was simply a helper. Recall a maxim sometimes credited to President Harry Truman: “It is amazing what you can accomplish if you do not care who gets the credit.”
Open Problems
What do you think about ways to check proofs? Any better ideas?
Impetus to study a new reducibility relation
See Mike’s other projects too
Michael Wehar has just earned his PhD degree in near-record time in my department. He has posted the final version of his dissertation titled On the Complexity of Intersection Non-Emptiness Problems which he defended last month. The dissertation expands on his paper at ICALP 2014, joint paper at ICALP 2015 with Joseph Swernofsky, and joint paper at FoSSaCS 2016.
Today, Dick and I congratulate Mike on his accomplishment and wish him all the best in his upcoming plans, which center on his new job with CapSen Robotics near Pitt and CMU in Pittsburgh.
The 25th Anniversary of the ACO Program
Cropped from src1 & src2 in gardens for karma
Prasad Tetali and Robin Thomas are mathematicians at Georgia Tech who are organizing the Conference Celebrating the 25th Anniversary of the ACO Program. ACO stands for our multidisciplinary program in Algorithms, Combinatorics and Optimization. The conference is planned to be held starting this Monday, January 9–11, 2017.
Today I say “planned” because there is some chance that Mother Nature could mess up our plans.
Even after today’s retraction of quasi-polynomial time for graph isomorphism
Cropped from source
László Babai is famous for many things, and has made many seminal contributions to complexity theory. Last year he claimed that Graph Isomorphism (GI) is in quasi-polynomial time.
Today Laci posted a retraction of this claim, conceding that the proof has a flaw in the timing analysis, and Ken and I want to make a comment on what is up. Update 1/10: He has posted a 1/9 update reinstating the claim of quasi-polynomial time with a revised algorithm. As we’ve noted, he is currently speaking at Georgia Tech, and we hope to have more information soon. |
# Prove that all hyperbolic straight lines are congruent to $x$-axis
I have the notes on the proof but I cannot fully understand the proof.
Let $C$ be a hyperbolic straight line through $z_o\in \mathbb{D}$ and $z^*_o$ the point symmetric to $z_o$ wrt the unit circle $S^1$. Then we have $z^*_o$ is outside $S^1$.
Since C is perpendicular to $S^1$, we have $z^*_0$ lies on $C$.
I can't understand that how the last statement will happen. Thanks.
$z_o^*$ is the inverse of $z_o$. The inverse of $C$ is $C$ itself, since circle inversion is conformal and knowing the two points of intersection with $S^1$ already uniquely defines the single orthogonal circle through these two points. Inversion preserves incidence, so if $z_o$ lies on $C$, then $z_o^*$ lies on the image of $C$, i.e. on $C$ itself as well. |
# Is an Arma model equivalent to a 1-layer Recurrent Neural Network without activation function?
Given a time series $$f(t)$$ to forecast, let us consider an Arma model of the form: $$f(t) = c + \sum_{i=1}^p a_i f(t-i) + e(t) + \sum_{j=1}^q b_j e(t-j)$$
where $$e(t)$$ are the forecast errors.
On the train set, if $$f(t)$$ is the ground truth, then we define its estimate obtained with this model as $$\widetilde{f}(t) = f(t) + e(t)$$.
Let $$m = min(p,q)$$, we can rewrite the first equation as: $$\widetilde{f}(t) = c + \sum_{i=1}^m (a_i + b_i) f(t-i) + \sum_{i=m+1}^p a_i f(t-i) - \sum_{j=1}^q b_j \widetilde{f}(t-j)$$ Then after reparametrization can be rewritten as: $$\widetilde{f}(t) = c + \sum_{i=1}^k c_i f(t-i) - \sum_{j=1}^q b_j \widetilde{f}(t-j)$$ Which is the equation of a 1-layer recurrent neural network (RNN) without activation function.
So, are Arma models a subset of RNNs or is there a flaw in this reasoning ?
• very interesting question, wish I could upvote more Feb 23 '20 at 2:47 |
### Home > CAAC > Chapter 10 > Lesson 10.4.3 > Problem10-147
10-147.
Simplify each of the expressions below. Your final simplification should not contain negative exponents.
1. $(5x^3)(−3x^{−2})$
• $−15x$
1. $(4p^2q)^3$
• Write the expression in parentheses three times then multiply.
1. $\frac { 3 m ^ { 7 } } { m ^ { - 1 } }$
• When like bases are divided, subtract the exponents.
$3m^8$ |
## Diode
Can we use LED( light emitting diode) as rectifier like in ordinary 41007 diode? If yes then why, if not then what is the reason. Answer in details
• Tunnel diodes are heavily doped in both the P and N regions, 1000 times the level in a rectifier. This can be seen in Figure above. Standard diodes are to the far left, zener diodes near to the left, and tunnel diodes to the right of the dashed line. The heavy doping produces an unusually thin depletion region. This produces an unusually low reverse breakdown voltage with high leakage. The thin depletion region causes high capacitance. To overcome this, the tunnel diode junction area must be tiny. The forward diode characteristic consists of two regions: a normal forward diode characteristic with current rising exponentially beyond VF, 0.3 V for Ge, 0.7 V for Si. Between 0 V and VF is an additional “negative resistance” characteristic peak. This is due to quantum mechanical tunneling involving the dual particle-wave nature of electrons. The depletion region is thin enough compared with the equivalent wavelength of the electron that they can tunnel through. They do not have to overcome the normal forward diode voltage VF. The energy level of the conduction band of the N-type material overlaps the level of the valence band in the P-type region. With increasing voltage, tunneling begins; the levels overlap; current increases, up to a point. As current increases further, the energy levels overlap less; current decreases with increasing voltage. This is the “negative resistance” portion of the curve. |
Free Version
Difficult
# Ionization Energy Data
APCHEM-GIQ7UO
Consider the following ionization energy data (kJ/mol):
Element Ionization Energy (kJ/mol)
$N$ 1402
$O$ 1314
$F$ 1681
$Ne$ 2081
$\$
Which of the following is the BEST explanation for why the ionization energy for oxygen is the lowest for these elements?
A |
Home > Error Function > Computing Error Function
# Computing Error Function
## Contents
Retrieved 2011-10-03. ^ Chiani, M., Dardari, D., Simon, M.K. (2003). However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1 ( z ) = ∑ k = 0 But if you're an engineer who has never heard of the error function but needs to use it, it may take a while to figure out how to handle negative inputs.One Gaussian Quadrature is an accurate technique –Digital Gal Aug 28 '10 at 1:25 GQ is nice, but with (a number of) efficient methods for computing $\mathrm{erf}$ already known, I this contact form
J. Visit Chat Linked 6 How do I evaluate $\int \limits_{-\infty}^{a} e^{−t^2}dt$? 2 efficient and accurate approximation of error function 0 What is the closed-form expression for “cumulative density of a zero-mean ISBN 978-0-486-61272-0. The standard answer for how to compute anything numerical in Python is "Look in SciPy." However, this person didn't want to take on the dependence on SciPy.
## Computing Error Bars
Thanks for this one. The maximum error is below 1.5 × 10-7.import math def erf(x): # constants a1 = 0.254829592 a2 = -0.284496736 a3 = 1.421413741 a4 = -1.453152027 a5 = 1.061405429 p = 0.3275911 The error function at +∞ is exactly 1 (see Gaussian integral).
Instead, replace 1 - erf(x) with erfc(x). Supancic, "On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion," The Mathematica Journal, 2014. See [2]. ^ http://hackage.haskell.org/package/erf ^ Commons Math: The Apache Commons Mathematics Library ^ a b c Cody, William J. (1969). "Rational Chebyshev Approximations for the Error Function" (PDF). Erf Function Table erf(-x) = -erf(x).
Please try the request again. How To Calculate Error Function In Casio Calculator J. (March 1993), "Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers" (PDF), ACM Trans. usage of the word "have" in "I have her" Does using OpenDNS or Google DNS affect anything about security or gaming speed? J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 External links MathWorld – Erf Authority control NDL: 00562553 Retrieved from
Buhm 3 August 2012 at 22:13 Thanks so much for making website for stand-alone code for people in need like me. Complementary Error Function Table Cody's rational Chebyshev approximation algorithm.[20] Ruby: Provides Math.erf() and Math.erfc() for real arguments. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. Both functions are overloaded to accept arguments of type float, double, and long double.
## How To Calculate Error Function In Casio Calculator
They use their implementation of the incomplete gamma function, which in turn uses a mixed approach depending on the argument. Jaime 24 February 2009 at 07:58 Gene, I'm quoting below a couple of paragraphs, from "The Art of Scientific Computing" by Press et al…---We assume that you know enough never to Computing Error Bars Weisstein. "Bürmann's Theorem" from Wolfram MathWorld—A Wolfram Web Resource./ E. Inverse Error Function Calculator The error function is related to the cumulative distribution Φ {\displaystyle \Phi } , the integral of the standard normal distribution, by[2] Φ ( x ) = 1 2 + 1
doi:10.1109/TCOMM.2011.072011.100049. ^ Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 0-521-43064-X), 1992, page 214, Cambridge University Press. ^ DlangScience/libcerf, A package for use with the D Programming language. http://bestwwws.com/error-function/complementry-error-function.php Also, this may be a better question for stack overflow instead, since it's more of a computer science thing. –Jon Bringhurst Jul 20 '10 at 20:26 @Jon: Nope, I'm p.297. Successful use of strtol() in C How to make an integer larger than any other integer? Complementary Error Function Calculator
1. N ! ∫ x ∞ t − 2 N e − t 2 d t , {\displaystyle R_ − 3(x):={\frac {(-1)^ − 2}{\sqrt {\pi }}}2^ − 1{\frac {(2N)!} − 0}\int _
2. An example on a Linux machine: bash$cat erf.i %module erf #include double erf(double); bash$ swig -o erf_wrap.c -python erf.i bash$gcc -o erf_wrap.os -c -fPIC -I/usr/include/python2.4 erf_wrap.c bash$ gcc -o
3. For any complex number z: erf ( z ¯ ) = erf ( z ) ¯ {\displaystyle \operatorname 6 ({\overline 5})={\overline {\operatorname 4 (z)}}} where z
4. Multiple Alignments in flalign What is the range limit of seeing through a familiar's eyes?
5. Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 6.2.
6. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the
7. more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed
8. See Alsoerfc | erfcinv | erfcx | erfinv Introduced before R2006a × MATLAB Command You clicked a link that corresponds to this MATLAB command: Run the command by entering it in
share|cite|improve this answer answered Jul 20 '10 at 22:38 Isaac 26.7k872122 add a comment| up vote 2 down vote Here's a link to the boost c++ math library documentation. Generalized error functions Graph of generalised error functions En(x): grey curve: E1(x) = (1−e−x)/ π {\displaystyle \scriptstyle {\sqrt {\pi }}} red curve: E2(x) = erf(x) green curve: E3(x) blue curve: E4(x) Jason Merrill 13 April 2015 at 21:27 This is nice, thanks for writing this up.If you're using this routine, one thing to watch out for is large relative error when |x| navigate here Which method you use for which intervals is down to experimentation. –Tim Seguine Sep 1 '11 at 10:51 A lot of this comes down to the desired accuracy and
WWII Invasion of Earth Even sharper upper bound for prime product? Error Function Matlab Another form of erfc ( x ) {\displaystyle \operatorname Φ 8 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 Join the conversation ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection to 0.0.0.9 failed.
## Math.
All generalised error functions for n>0 look similar on the positive x side of the graph. Comp. 23 (107): 631–637. The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function. Error Function Excel This allows one to choose the fastest approximation suitable for a given application.
Acknowledgments Trademarks Patents Terms of Use United States Patents Trademarks Privacy Policy Preventing Piracy © 1994-2016 The MathWorks, Inc. It is not as prone to subtractive cancellation as the series derived from integrating the power series for $\exp(-x^2)$. It is a matter of taste, however, whether to write p=c[0]+x*(c[1]+x*(c[2]+x*(c[3]+x*c[4]))); or p=(((c[4]*x+c[3])*x+c[2])*x+c[1])*x+c[0]; If the number of coefficients c[0..n-1] is large, one writes p=c[n-1]; for(j=n-2;j>=0;j-) p=p*x+c[j]; or p=c[j=n-1]; while (j>0) p=p*x+c[-j]; See also Related functions Gaussian integral, over the whole real line Gaussian function, derivative Dawson function, renormalized imaginary error function Goodwin–Staton integral In probability Normal distribution Normal cumulative distribution function, a |
# Problem of the Week
## Updated at Nov 8, 2021 3:57 PM
How can we find the derivative of $${x}^{7}+4x$$?
Below is the solution.
$\frac{d}{dx} {x}^{7}+4x$
1 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$7{x}^{6}+4$Done7*x^6+4 |
Greatest Common Divisor of two numbers
If $ab=600$ how large can the greatest common divisor of $a$ and $b$ be?
I am not sure if I should check for all factor multiples of $a$ and $b$ for this question. Please advise.
We have $600=2^3\cdot 5^2\cdot 3$. To make the gcd large, we give a $2$ to each of $a$ and $b$, also a $5$. So the largest possible gcd is $10$.
Remark: The idea generalizes. Let $n$ have prime power factorization $$n=p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}.$$ Let $e_i=\lfloor d_i/2\rfloor$, where $\lfloor x\rfloor$ is the greatest integer that is $\le x$. Then the greatest possible gcd of $a$ and $b$, where $ab=n$, is $$p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}.$$
• You are welcome. For the maximum, we distribute the power of any prime $p$ as evenly as possible between $a$ and $b$. – André Nicolas Jun 10 '13 at 2:00
$$ab=600=2^3\cdot 3\cdot 5^2$$ The greatest common divisor of $a,b$ will have the greatest number of common prime factors that you can arrange. |
Should the [wfrp-4] tag be renamed to [wfrp-4e]?
Pretty straightforward stuff. We generally use the -[#]e tag suffix to denote different editions (presumably because that's how those editions of those games are commonly abbreviated) - and the tags for the previous three editions of Warhammer Fantasy Roleplay follow this trend: , , and . (There's also a general tag, but that's not relevant here.)
The tag was created a little over a month ago; I'm guessing the person who created it wasn't aware of the convention for edition tags.
Should the tag be renamed to ? |
Subsets and Splits