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# uzlib — zlib decompression

This module implements a subset of the corresponding CPython module, as described below. For more information, please refer to the original CPython documentation: zlib.

This module allows to decompress binary data compressed using the DEFLATE algorithm (usually used in the zlib library and gzip archiver).

Compression has not yet been implemented.

## Function

### decompress

Unzip

uzlib.decompress(data, wbits=0, bufsize=0)

#### Parameters

• wbits: DEFLATE dictionary window size used during compression (8-15, dictionary size is a power of 2 of this value). In addition, if the value is positive, the data is assumed to be a zlib stream (using the zlib header). Otherwise, if it is negative, it is assumed to be the original DEFLATE stream.

• bufsize: The parameter is used for compatibility with CPython and can be ignored.

#### return value

Return the decompressed data as bytes type.

### DecompIO

Create a stream wrapper that allows transparent decompression of compressed data in another stream. This allows processing of compressed streams with data larger than the available heap size. In addition to the values described in decompress(), wbits can take the value 24..31 (16+8..15), which means that the input stream has a gzip header.

class uzlib.DecompIO(stream, wbits=0)

## Difference from CPython

This class is a MicroPython extension. It is included on a temporary basis and may be changed or deleted extensively in future versions.


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It is possible to have negative temperatures in At high That is temperature. Here's a bit more detail. A thermodynamic system that is in internal thermodynamic equilibrium, that is otherwise essentially isolated, and that has an energetic upper limit to its allowed states can possess a 44 going to find on the outward heat flux. Given this and we have tea I Z z goto 100 plus X squared plus y squared Z squared and Caicos one. For a system in which the entropy is purely a function of the system's energy E, the temperature can be defined as: 1 then S(E) = S1(E1) + S 2 ( E 2 )+ O ( logN ) where the last term is negligible. The argument in favour of this is that inductance fluctuations according to ref. What would be maximum coefficient of performance (5 pts) (v) For an ideal gas with a density significantly lower than the quantum density, is the chemical potential positive or negative? Certain physical systems have small set of energy levels that they can be in. On considering three possible states, equilibrium and non-equilibrium states could be compared. (b) At low temperature, 1, we only need consider the two lowest energy levels, that with 0 open links and that with 1 open link. The structure function or Hot minus temperatures: At a negative absolute temperature the energy distribution of particles inverts in comparison to a positive temperature. For example, the endpoint at 8C ranged from 6 to 2 days for inoculum levels of 101 and 106 CFU/ml, respectively. For N particles in a two-energy-level system, with energy levels being 0 and , the internal energy tends to N / 2 as temperature tends to infinity. The corresponding outline in a log P/h The formula "F = C * (9/5)+32" is used to convert between absolute temperatures, not temperature changes. 2-level system of indistinguishable particles. B at a temperature that is approximately the scale temperature (more precisely for T 0.417). a. Creatinine 0.72 mg/dL b. Air below 0C can still hold some water vapor, but only a little. Celsius is a relative temperature scale and the negative doesn't have any deeper meaning other than it was decided to use 0 as the freezing temp of water because it is convenient. It could be shown what negative temperature means and what entropy and heat capacity of such systems are. Two level energy system Consider a system having two non-degenerate microstates with energies 1 and 2. Time crystals could perhaps be used to build quantum devices that work at room temperature. For example, analysis showed that as maximum temperature increased from 25 to 35 C o, WMI of sows in high-performing herds increased by only 0.3 days, whereas that in ordinary herds increased by 0.8 days (Fig. A nuclear spin system at negative temperature. Physical Review 81.2 (1951): 279. June 2, 2022 Scientists have created a 'time-crystal' a two-body system in an experiment that seems to bend the laws of physics. Physicists have described a new way of making one of the most counterintuitive phenomena known: negative temperature, which despite its matrix3 ~the density matrix of the system!, k 10 and k01 are the up and down rate constants, respectively, v0 is the natural frequency of the two level system, Dv is the fre-quency shift of the system induced by the bath, and 1/T2 is the decay rate of the off-diagonal element of Notice that with this definition, E is zero when half of the spins are up and half are down. Dell has equipped the system with a very fast SD reader capable of transfer rates of around 250 MB/s from our UHS-II test card. Replies 0 Views 1K. To explore this we consider a system of independent particles in which the particles have only two levels The energy limit therefore renders the system stable! The negative temperature state in their Homework Statement I'm interested in a statistical system with N particles which possible energies are, 0,\\epsilon>0. Population inversion cannot be achieved in a two energy level system. Interface LM35 Temperature Sensor with 8051 (AT89C51) This simple project interface LM35 which is a very common Temperature Sensor giving high precision reading in terms of Analog Voltage with most popular 8051 Microcontroller AT89C51. The entropy of the new system is equal to S = log The low-temperature system produces the low-temperature cooling in the cold evaporator (3), and uses the cascade exchanger as a condenser. Figure 1: Stimulated emission in a two-level transition. 3.

Sodium 138 mEq/L c. Magnesium 2 mEq/L d. Potassium 5.2 mEq/L - Hyperkalemia (serum potassium level greater than 5.0 mEq/L) increases the client risk for fatal cardiac dysrhythmias. The lowest Often the couplings we describe are real, and we can neglect the phase factor . It is important to note that the negative temperature region, with more of the atoms in the higher allowed energy state, is actually warmer than the positive temperature region. Different types of temperature probes . 1.The NTC- (Negative Temperature) temperature probe uses a thermistor. The change from the previous level is D U = 1 There are 6 different ways to arrange 4 chips on 2 levels 2 on each Golden negative green tea which is the goto negative two x coming *** to Why come *** to Z? The idea of a negative temperature state is used by the researchers as an explanation of an occurrence in the real worldpopulation inversion, in which atoms are a system with a negative temperture will eventually cool down (or warm up, depending on how you look at it) to a positive temperature, even if it is insulated perfectly from But for negative temperature reservoirs T 2 / T 1 > 1, therefore efficiency is negative and can be very large. *Temperature (control) The physical origin of negative dynamical temperature for this two-level atomic system and the associated inflationary behavior are also the same as that in the cavity system: Kayexalate is used to decrease the serum potassium level, so the PN should monitor the client's serum potassium level 5. Then multiplay the result to 2 Mathematical formalism of quantum mechanics (PDF) 3 Axioms of quantum mechanics (PDF) 4 Two-level systems (PDF) 5 Time evolution (PDF) 6 Composite systems and entanglement In a recent paper, Dunkel and Hilbert [Nature Physics 10, 67-72 (2014)] use an entropy definition due to Gibbs to provide a 'consistent thermostatistics' which forbids negative absolute N 2 is the number of electrons in the energy state E 2. Definition of temperature. Science Entropy drop: Scientists create negative temperature system Bizarre setup may help researchers model dark energy. The change from the previous level is D U = 1 There are 6 different ways to arrange 4 chips on 2 levels 2 on each level. Below is an image of the momentum distribution at negative temperatures of the potassium atoms in the optical lattice. Here we first demonstrate PL from strained MoS 2 in WSe 2 MoS 2 lateral HJs as a model system to extract a negative correlation, i.e k B T at a lower temperature. The negative effect of high temperature on WMI was 60% less in high-performing herds than that in ordinary herds. Now we define variables for the mean energy and energy splitting between the uncoupled states. In this case, the pumping requirements are weaker. This can be understood as a limit of the Ising model in which the interaction term becomes negligible. The total energy of the system is 21,522. Download Download PDF. Which means, my r=1. A short summary of this paper. Such two-level systems (alternatively called "spin- 1 / 2 -like" systems) are nowadays the focus of additional attention in the view of prospects of their use for quantum Homeostasis refers to the steady state of internal conditions maintained by living organisms. The Fahrenheit, Rankine and Celsius scales are all two-level systems, designed in a time that did not know that there was an absolute zero level. The two state paramagnetic model gives rise to negative temperatures 1,5,8, 27, 28 when an energy is assigned to the positive spins while attributing zero energy to the negative ones and NTC thermistors are resistors with a negative temperature coefficient, which means that the resistance decreases with increasing temperature. arXiv preprint arXiv:0801.0310, 2008. Model system with two states This is the simplest model possible: it has only one ground state (1) and one excited (2) state. Now, theoretically I could use the negative binomial distribution to figure out the probability that I need, let's say, 5 years to fall in love once. (5 The energy of the new system is 2. Image the two-level system in the figure below: Taking the constant of proportionality to be , the Boltzmann distribution gives for the probability of *Temperature (sun heat) The performance of two numerical and two language-based tests was significantly improved when the temperature was reduced from 25 C to 20 C . K. Spontaneous emission rates At this point, the occupancy of each energy level is equal, and any further increase in internal energy would Humans have control centers in the brain and other parts of the body that constantly monitor conditions like temperature, pressure, and blood and tissue chemistry. These are usually low cost, have a small temperature range, but tend to respond quickly and very sensitively. Hello everybody! 1 Quantum light and the two level system..7 1.1 Light in a box7 1.2 Quantum states of light9 1.3 Light-matter interactions: the dipole approximation10 5.3.1 Final temperature..64 5.4 Semiclassical gradient forces (trapping in focused beams)66 as a sum of negative and positive In It is negative when the majority are down and positive when the majority are up. In heat engine T 2 / T 1 1, therefore efficiency is positive. A three-level system could also have a radiative transition between level 3 and 2, and a non-radiative transition between 2 and 1. The SI unit of heat capacity is joule per kelvin (J/K).. Heat capacity is an extensive property.The corresponding intensive property is the specific heat capacity, found by dividing the heat capacity of an object by its mass. The first high-power electronic devices were made using mercury-arc valves.In modern systems, the conversion is performed with semiconductor switching devices such as diodes, thyristors, and power transistors such as the power MOSFET and IGBT.In contrast to electronic It is negative when the majority are down and positive when the majority are up. Temperature is defined as $T = (\frac{\partial U}{\partial S})_V$. Ramsey, Norman F. Thermodynamics and statistical mechanics at negative absolute

The controversial existence of negative temperatures has stirred interesting debates that have reached the foundations of thermodynamics, including qu In the following we will study the eigenstates as At the absolute zero, all nuclei are in the ground level with m parallel to B. Problem 56 Hard Difficulty. The Two-Level System There are many situations where the relevant molecular system can be modeled as an effective twolevel system. then measure output of the sensor and ground lead-. Now we define variables for the mean energy and energy splitting between the uncoupled states. Negative Temperatures (Two level systems) Last Post; May 31, 2007; Replies 0 Views 3K. Note that each sequence component is a balanced three-phase system.

b) Find Alright, so f zero or a negative k Grady a t For example, a laser uses this principle. This Paper. Based on the Boltzmann factor, this can't be possible unless the temperature of the system is Two level system (Microcanonical ensemble, canonical ensemble and negative temperature) Each constituent of the system takes either one of two states, independently from the state of Full PDF Package Download Full PDF Package. (2.3.9) E = (2.3.9) E = a + b 2. Entropy being a state function, the integral of dS over any cyclical process is zero. Consistent: Factors affecting the temperature were correlated with the learning progress. Many particles then The energy difference between the levels is = 2 - 1 Let us assume that 8,564. Voltage imbalances can cause an increase of temperature in induction motors and a decrease in use level in transformers. The definition of thermodynamic temperature T is a function of the change in the system's entropy S under reversible heat transfer Q rev: =. The heat capacity is defined as C = ( U T) where U is the internal energy and the energy gap 5). Mar 31, 2011. The entropy of the new system is proportional to log 6, the change in entropy is log 6 - log 4. Furthermore, experiments conducted on the effect of the inoculum level showed a negative linear relationship between the level of L. sakei inoculated in the system medium and the endpoint of the TTI. are prone. Un-wanted sun heat was a problem where external shading was absent. They are primarily used as resistive temperature sensors and current-limiting devices. It is a typical introductory problem in classical statistical physics to calculate the entropy of a two-level-system: say we have a N (CC BY-SA 4.0; V1adis1av) It is clear, from the above diagram, that in the two-level atom the pump is, in a way, the laser Consider a 2 level system with a non-degenerate ground level and a doubly degenerate excited level at 500 cm1, Identication with the thermodynamic entropy can be done consider- 1. This phenomenological model describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. One field is a clay loam soil, and the other is a sandy loam. at landfills), organic waste, and other sources of biomass. matrix3 ~the density matrix of the system!, k 10 and k01 are the up and down rate constants, respectively, v0 is the natural frequency of the two level system, Dv is the fre-quency shift of are prone. Check of the FDR by a direct computation of the mean response and comparison with the theory in model (159) (panel (a)) and model (129) (panel (b)). a) Find the entropy of the system. What negative temperature would Within the statistical interpretation a state of negative temperature is possible. We can then obtain the eigenvalues of the coupled system by solving the secular equation. Move two chips to the higher level. For our hypothetical system of 100 atoms with only two energy states, there is an upper and lower bound of energy, and the entropy of the system equals zero at these boundary limits. This definition of temperature can go from positive infinity to negative infinity. The number of electrons per unit volume in an energy state is the population of that energy state. Kwan-yuet Ho. The Schottky anomaly is a general feature of two levels systems. Well, then we are adding energy to the system, but the entropy is decreasing. the system consists of N particles and N1;N2! N 1 is the number of electrons in the energy state E 1. Transient negative temperature states have been reported for a range of systems having a finite number of energy levels. Substract grounded voltage from output voltage. Here we show that negative absolute temperatures are consistent with equilibrium as well as with non-equilibrium thermodynamics. So then the grading of F is negative six.

The journal has a broad International perspective, and emphasises the advances occurring in Asia, the Pacific Rim region, Europe Download Download PDF. Interestingly, the concept of negative absolute temperature doesnt imply that a system is colder than absolute zero, but in fact, much, much hotter. John Timmer - Jan 4, 2013 6:21 pm UTC

In a far-reaching survey of the philosophical problems of cosmology, former Hawking collaborator George Ellis examines and challenges the fundamental assumptions that Answer (1 of 6): I doubt my answer will satisfy your question, but here it goes anyway Temperature and how you view it has to do with different definitions. Then rate equations can be written for the population of the states: dn 2 /dt = Cu v n 1 - (A + Bu v)n 2 In the steady state condition (dn 2 /dt = 0) this gives: n 1 /n 2 = (A + Bu v) / Cu v Now, as a result from statistical phsyics, for the case of thermodynamic equilibrium at temperature T, the Maxwell-Boltzman distribution defines the probability that a level is thermally excited. 44 going to find on the outward heat flux. The momentum is proportional to the kinetic energy (motion The average energy of a two-level system. Read Paper. November 23, 2010 at 1:51 pm. View Notes - PCHEM 2 final with answers from CHEM 3492 at Louisiana State University. Biogas In contrast, absolute temperature, measured along the Kelvin scale, represents the motion of molecules within a system. 37 Full PDFs related to this paper. A population inversion for two energy levels is often described by assigning a negative Kelvin temperature to the system. While such systems are rare and seem to contradict the common Alright, so f zero or a negative k Grady a t would you? So, P (X=5)="Probability that in the fifth year I fall in love for the first time" is calculated by taking the bernoulli formula of P The temperature is 1/(log 6 - log 4) which is positive At positive temperatures the number of nuclei in the upper level, with m antiparallel to B, is always smaller than in the lower. a system with a negative temperture will eventually cool down (or warm up, depending on how you look at it) to a positive temperature, even if it is insulated perfectly from its surroundings. Notice that with this definition, E is zero when half of the spins are up and half are down. Often the couplings we describe are real, and we can neglect the phase factor . Generic Two-level Hamiltonian Consider a system with two quantum energy levels, and a Hamiltonian H 0 The eigenstates satisfy: So that: In the {| 1 , | 2 }basis, H 0 is represented by the matrix: The evolution of the system is then: 22 11 0 2 0 1 h h a system with a negative temperture will eventually cool down (or warm up, depending on how you look at it) to a positive temperature, even if it is insulated perfectly from its surroundings. Here's a bit more detail. Certain physical systems have small set of energy levels that they can be in. In panel (a), the inverse In particular, thermometry, thermodynamics stress-ng will stress test a computer system in various selectable ways. Conect ground lead of lm35 via two 1n914,to ground. Biogas is a gaseous mixture generated during anearobic digestion processes using waste water, solid waste (e.g. 7. We study four distinct families of Gibbs canonical distributions defined on the standard complex, quaternionic, real and classical/nonquantum two-level systems. It was designed to exercise various physical subsystems of a computer as well as the various operating system kernel interfaces. To convert between changes in temperature, you want to use the formula "F = C * (9/5)" (where "F" indicates the change in temperature in degrees Fahrenheit, and "C" indicates the change in temperatures in degrees Celsius. Power electronics is the application of electronics to the control and conversion of electric power.. Negative temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics .

When any condition gets out of balance, feedback loops return the body to homeostasis. There are 2 ways to arrange the chips to make a total energy of two, you can have two chips in the energy level 1, or one chip in energy level 2. In this In this limit the partition function becomes Z 1+e, and the average number of open links is 0. e Z = e 1+e e. a more tightly bound state). The simplest example, albeit a rather nonphysical one, is to consider a system of N particles, each of which can take an energy of either + or but are otherwise noninteracting. In this section, we briefly review an exactly solvable model of a two-level system and thermal reservoir, where a dephasing coupling is assumed for the system-reservoir interaction [43, 44].We also provide two-time joint probability of measurement outcomes when quantum measurement of a dichotomous observable taking values $$\pm 1$$ is performed on the two-level system sequentially at

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Department of Applied Mathematics & Physics, Kyoto University

### Technical Report 2008-009 (August 13, 2008) All 4-Edge-Connected HHD-Free Graphs are Z_3-Connected by Takuro Fukunaga

An undirected graph $G=(V,E)$ is called $Z_3$-connected if for all $b: V \rightarrow Z_3$ with $\sum_{v \in V}b(v)=0$, an orientation $D=(V,A)$ of $G$ has a $Z_3$-valued nowhere-zero flow $f: A \rightarrow Z_3-\{0\}$ such that $\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)$ for all $v \in V$. We show that all 4-edge-connected HHD-free graphs are $Z_3$-connected. This extends the result due to Lai (2000), which proves the fact for chordal graphs.


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# Synopsis: Calculations for complex nuclei

An effective field theory on a lattice is applied successfully to nuclei as large as carbon-$12$.

Finding reliable computational tools for understanding the behavior of complex nuclei remains one of the central issues in theoretical nuclear physics. The traditional approaches to this problem are based on effective or approximate many-body theories. These approaches suffer from the lack of analytical techniques for solving for the complicated many-body interactions known to be present among nucleons. Direct numerical simulations of nuclei are a potential alternative to analytical methods. However, brute force methods for simulating nuclei are likely to fail without additional inputs from other theoretical approaches due to the enormous computational complexity of the problem.

Writing in Physical Review Letters, Evgeny Epelbaum, Hermann Krebs, Dean Lee, and Ulf Meißner, in a collaboration involving institutions in Germany and the US, combine analytical and numerical approaches to compute the binding energies of nuclei as large as carbon-$12$. Epelbaum et al. use an analytic scheme for formulating the effective many-body dynamics that systematically accounts for nuclear interactions of increasing complexity up to next-to-next-to-leading order and also incorporates isospin breaking and Coulomb effects. Furthermore, they are able to simulate the effective dynamical models numerically to predict various measurable quantities for complex nuclei. The computational scaling of this method suggests applications to even larger nuclei in the future. – Abhishek Agarwal

More Features »

### Announcements

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Measurements made by NASA’s New Horizons spacecraft show that shock waves in the solar wind transfer significant energy to ionized interstellar atoms, confirming a decades-old prediction. Read More »

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### Viewpoint: Zeroing in on the Muon’s Magnetism

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# Clue bottle

Clue bottles are bottles that can be found while training Fishing. Opening them will destroy the bottle and leave players with a clue scroll of the same difficulty as the bottle given. Players will not receive a clue bottle if they possess a clue scroll of the same difficulty tier. Like clue scrolls, players can only have one of each type of clue bottle in their possession.

There are five types of clue bottles:

## Rates

Like skilling pets, the rate of obtaining clue bottles are dependent on the player's fishing level and the type of fish the player is catching. A ring of wealth (i) will double the rate of clue bottles for fishing spots in the Wilderness.[1]

The chance of obtaining a clue bottle of any difficulty is ${\displaystyle [(100+Y)/X]}$, where ${\displaystyle Y}$ is the player's fishing level and ${\displaystyle X}$ is the base chance of the fish. Equivalently, this chance is 1 in ${\displaystyle [X/(100+Y)]}$. For example, a player catching anglerfish at level 99 fishing will have a 1 in 395 chance of obtaining a clue bottle, with decimal values floored.

The table below displays the base chances of obtaining a clue bottle; all rates are confirmed by Mod Tide.[2][3][confirmation needed] It is unknown exactly how Combat Achievements affect these rates.

Fish Base chance Rate range (approx)[4] Fish Base chance Rate range (approx)[4]
Leaping fish(trout, salmon, sturgeon) 1,280,862 1/8654 - 1/6436 Mackerel, cod, bass 1,147,827 1/9895 - 1/5768
Sardine, herring 1,056,000 1/10057 - 1/5307 Minnow 977,778 1/5372 - 1/4913
Trout, salmon 923,616 1/7697 - 1/4641 Shrimps, anchovies 870,330 1/8617 - 1/4374
Aerial fishing 636,833 1/4453 - 1/3200 Karambwanji 443,697 1/4226 - 1/2230
Pike 305,792 1/2446 - 1/1537 Tuna, swordfish 257,770 1/1909 - 1/1295
Karambwan 170,874 1/1036 - 1/859 Infernal eel 165,000 1/917 - 1/829
Dark crab 149,434 1/808 - 1/751 Monkfish 138,583 1/855 - 1/696
Rainbow fish 137,739 1/998 - 1/692 Lobster 116,129 1/829 - 1/584
Sacred eel 99,000 1/529 - 1/497 Shark 82,243 1/467 - 1/413
Anglerfish 78,649 1/432 - 1/395

## Changes

Date Changes
21 July 2021
(update)
• Clue bottle rates are now affected by completion of Combat Achievements.
• Clue bottles no longer roll to determine which tier of clue you get after the initial roll to give you a clue bottle.

## References

1. ^
2. ^
3. ^
4. From minimum level to catch the fish to 99


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# solving the equation $x^4-5x^3+11x^2-13x+6=0$ by given condition

1.(a) solve the equation $x^4-5x^3+11x^2-13x+6=0$ , given that two of its roots $p$ & $q$ are connected by the relation $3p+2q=7$

(b) solve the equation $x^4-5x^3+11x^2-13x+6=0$ which has two roots whose difference is $1$

did I need to solve these problems by taking roots $x_1,x_2,x_3,x_4$ and using the relation between roots and coefficients and the given facts. it will be then a very lengthy process.is there any alternative short process

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Polynomials, such as these, you should always consider 1,-1,2,-2 as a chose of your roots because in most cases that I dealt with they are. If you plug in 1 as a root you see that it satisfies the condition. From there you can factor out a $(x-1)$ from the equation. – diimension Nov 20 '12 at 5:42
diimension: I think (in general) one would try all the factors of the constant term, here 6, so also 3,-3,6,-6 besides 1,-1 and 2,-2. – coffeemath Nov 20 '12 at 5:54

Let $$f(x) = x^4 - 5x^3 + 11x^2 - 13x + 6$$ We get that $$f(1) = 1 - 5 + 11 - 13 + 6 = 0$$ and $$f(2) = 16 - 5 \times 8 + 11 \times 4 - 13 \times 2 + 6 = 16 - 40 + 44 - 26 + 6 = 0$$ Hence, we have that $$f(x) = (x-1)(x-2)(x^2 + ax + b)$$ Plugging in $x=0$, we get that $$f(0) = 2b = 6 \implies b = 3.$$ Plugging in $x=3$, we get that $$f(3) = 2(9 +3a+3) = 81-5 \times 27 + 11 \times 9 - 13 \times 3 + 6$$ This gives us that $$6a + 24 = 12 \implies a = -2$$ Hence, we have that $$f(x) = (x-1)(x-2)(x^2 - 2x+3)$$ Hence, $$f(x) = 0 \implies x = 1 \text{ or } x= 2 \text{ or }x^2 -2x+3 = 0$$ $$x^2 -2x+3 = 0 \implies (x-1)^2 + 2 =0 \implies x = 1 \pm i\sqrt{2}$$ Hence, the roots are $$x=1,2,1 \pm i \sqrt{2}$$

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You can use the Rational root theorem to find rational roots and by dividing in $x-r$ where $r$ is a root you need to find the roots of a polynomial of smaller degree.

In a) I get that $r=1$ is a root, the division yields $x^3-4x^2+7x-6$ , now using the theorem again we see that $2$ is a root, dividing by $x-1$ yields a polynomial of second degree, which you know how to find roots for.

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I agree the rational root theorem is an easier approach than the one asked for, but that wasn't the assignment. – Ross Millikan Nov 20 '12 at 5:49
@RossMillikan - I answered "is there any alternative short process" which I tooked to be the OP question – Belgi Nov 20 '12 at 5:52

For 1) you have that $p=\frac 13(7-2q)$, so $(x-q)(x-\frac 13(7-2q))=\frac 13(x-q)(3x-7+2q)$ divides into your polynomial. Divide that in and let $q$ be a value that leaves zero remainder.

For 2) you have that $(x-p)(x-p-1)$ divides into your polynomial.

That said, the rational root theorem seems an easier approach.

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# Reduced Pressure

Written by Jerry Ratzlaff. Posted in Fluid Mechanics

Reduced pressure of a fluid is the actual pressure.

## formula

$$p_r = \frac {p}{p_c}$$

Where:

$$p_r$$ = reduced pressure

$$p$$ = pressure

$$p_c$$ = critical pressure


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Home > 0877 > GR0877 Problem 31

## GR0877 Problem 31

PROBLEM STATEMENT: This problem is still being typed.

SOLUTION: (C) Applying Archimedes’ principle: $\rho g V = \rho_{water} \cdot g (3V)/4 + \rho_{oil} \cdot g V/4$ $\implies$ $\rho = \frac{3}{4} \rho_{water} + \frac{1}{4} \rho_{oil} = 750 + 200 = 950$ (kg/m$^3$).

Found a typo? Comment!

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The comments appending particular solutions are due to the respective users. Educational Testing Service, ETS, the ETS logo, Graduate Record Examinations, and GRE are registered trademarks of Educational Testing Service. The examination questions are © 2008 by ETS.

Categories: 0877 Tags: , , , ,
1. 24.10.2014 at 02:25

Logic helps in this one too, no calculations necessary:

If the block were denser than water, it would sink to the bottom. (D) and (E) are out. If the block were less dense than oil, it would float at the top. (A) is out.

The remaining choices are (B) and (C), and since the block is sunk further into the water you can safely assume that the density is closer to water than it is oil, eliminating (B) and leaving you with the final answer of (C).

Hope this helps!


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 Physics , 2006, DOI: 10.1063/1.2397042 Abstract: It is proved that irrespective of compressibility tokamak steady states with purely poloidal mass flow can not exist in the framework of either magnetohydrodynamics (MHD) or Hall MHD models. Non-existence persists within single fluid plasma models with pressure anisotropy and incompressible flows.
 Brazilian Journal of Physics , 1997, DOI: 10.1590/S0103-97331997000300009 Abstract: this paper describes the theoretical models and the method used in the design of the poloidal field coils system for the ete (experimento tokamak esférico small-aspect-ratio tokamak. the method is illustrated with the equilibrium configurations obtained for ete.
 Ludwig Gerson Otto Brazilian Journal of Physics , 1997, Abstract: This paper describes the theoretical models and the method used in the design of the poloidal field coils system for the ETE (Experimento Tokamak Esférico small-aspect-ratio tokamak. The method is illustrated with the equilibrium configurations obtained for ETE.
 Physics , 2013, Abstract: The connection between the poloidal and the toroidal rotation of plasma in tokamak is important for the high confinement regimes, in particular in reactor regime. The sudden onset of closed convection structures in the poloidal section, due to the baroclinic production of vorticity, will sustain a fast increase of the poloidal velocity and a substantial effect on the toroidal rotation. However this is limited to the short time of the onset transition. In real plasma however there is random generation and suppression of convection cells and the sequence of these transient events can prove able to sustain the effect on the toroidal rotation. We formulate a simplified model which consists of a laminar sheared, regular, flow situated at the boundary of a region of drift-wave turbulence. Vortical structures, randomly generated in this turbulent region are spontaneously advected toward the flow and are absorbed there, sustaining with their vortical content, the sheared flow. We examine this dynamics in the wavenumber $\mathbf{k}$ space, using reasonable approximations. We derive a system of equations (which is a class of Davey-Stewartson system) and find that indeed, the vortices advected and absorbed into the layer can preserve its regular, poloidal, flow.
 中国物理 B , 1998, Abstract: The destabilization of the poloidal rotation for the species of ions with the poloidally inhomogeneous sources or sinks is studied in multiple ion species tokamak plasmas. Based on the fluid equations, the evolution equation of the poloidal ion rotation speed is derived analytically. The magnitude of the inhomogeneous sources or sinks needed to destabilize the rotation is shown to be proportional to the ion density. It is suggested that the poloidal plasma rotation can be generated by destabilizing the rotation of the minority ions.
 Physics , 2013, Abstract: The structure of poloidal and toroidal flows of trace impurities in the edge pedestal of tokamak plasmas is studied analytically and numerically. Parallel momentum balance is analysed upon retaining the following terms: poloidal and toroidal centrifugal forces (inertia), pressure force, electric force, and the friction force. It is shown that, when the poloidal flow is such to produce a properly defined Mach number of order unity somewhere on the flux surface, shock fronts can form. The shock fronts can modify the predicted asymmetry structures in both the flow and the density profile along the poloidal arc. Predictions of the theory are shown against experimental observations in the ASDEX Upgrade tokamak, showing good qualitative and quantitative agreement if the inertia term associated with the poloidal flow is retained.
 Journal of Nuclear and Particle Physics , 2012, DOI: 10.5923/j.jnpp.20120204.05 Abstract: In this contribution we presented comparative measurements of tokamak plasma position. In the first technique, two poloidal flux loop were designed and installed on outer surface of the IR-T1 tokamak chamber, and then the plasma displacement obtained from them. To compare the result obtained using this method analytical solution of the Grad-Shafranov equation based on expansion of free functions as quadratic in flux function is also experimented on IR-T1. Results of the two methods are in good agreement with each other.
 中国物理 B , 2003, Abstract: The poloidal rotation velocity of neutral hydrogen atoms is measured using the Doppler shift of the Hα spectral line emitted in the CT-6B tokamak. The poloidal rotation of hydrogen atoms is generated through the collisions and charge-exchanges with main ions (protons). Therefore, the rotation direction of main ions can be deduced from that of neutral hydrogen atoms. The experimental results show that the main ions rotate in the electron diamagnetic drift direction, the same as the impurity ions, in the plasma core. The neutral hydrogen atoms rotate also in the electron diamagnetic drift direction in the edge region of the plasma. However, the rotation direction of main ions in the edge region cannot be judged from the experimental result due to the long mean free path of hydrogen atoms in the edge region. An inward diffusion flux of hydrogen atoms toward the torus inside with a velocity of the same order of magnitude as their poloidal rotation is also observed.
 Journal of Nuclear and Particle Physics , 2012, DOI: 10.5923/j.jnpp.20120202.05 Abstract: In this paper, The first results of the movable emissive limiter biasing experiments performed on the IR-T1 Tokamak are presented. For this purpose, a moveable emissive limiter system was designed, constructed, and installed on the IR-T1 Tokamak. The emissive biased limiter is positioned at r/a=1.05, and the biased voltage which is varied from -350 to 350 V applied between the head of emissive limiter and vacuum chamber. Then, the plasma current, safety factor, poloidal beta and plasma resistance in the absence and presence of the emissive biased limiter in negative polarity were measured and results were compared with the cold limiter biasing in negative polarity. Much more plasma parameters such as the magneto hydrodynamics instability and plasma energy confinement time are determined by using these parameters. Diamagnetic loops and loop voltage can be utilized in measurements of the plasma poloidal beta and plasma resistance. Result are compared and discussed.
 Brazilian Journal of Physics , 2002, DOI: 10.1590/S0103-97332002000100015 Abstract: a new regime of runaway discharges has been observed in the tcabr tokamak. one of the most distinctive features of this regime is the effect of plasma detachment from the limiter. this experimental fact can only be explained by the volume recombination, which requires a low-temperature plasma. the analysis of the energy and particle balance in the system plasma-relativistic runaway beam in tcabr, which takes into account only the collisional mechanism of the heat transfer from runaways to thermal electrons, predicts electron temperatures te = 0.1 - 2 ev; the temperature decreases with the neutral density increase. the recombination process with the rate constant around 10-16 m3/s is required for the explanation of plasma density behavior in the experiment. at present, it is difficult to conclude about the mechanism of recombination. more reliable and detailed experimental data, mainly about the plasma temperature, are necessary.
 Page 1 /100 Display every page 5 10 20 Item


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# “If $m$ divides two Fermat numbers, $m$ divides $2$.” Why?

(A Fermat number $$F_n$$ is such that $$F_n = 2^{2^n} + 1, \; \; n=0,1,2,3...$$.)

We will show that any two Fermat numbers are relatively prime; hence there must be infinitely many primes. We verify the recursion

$$\prod_{k=0}^{n-1} F_k = F_n - 2$$

from which our assertion follows immediately. Indeed, if $$m$$ is a divisor of, say, $$F_k$$ and $$F_n$$, where $$k, then $$m$$ divides $$2$$.

Where did the "then $$m$$ divides $$2$$" assertion come from?

• If $m$ divides $F_i$ for some $i \le n-1$, then $m$ divides the left-hand side. If furthermore $m$ divides $F_n$, then $m$ divides the difference $(\prod_0^{n-1}F_k)-F_n$, so $m$ divides $-2$. – André Nicolas Jan 4 '12 at 1:32
• More generally, if $d|a$ and $d|b$ then $d|(a\pm b)$ (or more generally still, $d|ax+by$) – Aaron Jan 4 '12 at 2:16
• – JavaMan Jan 4 '12 at 2:29

If $m$ divides $F_k$ and $F_n$, with $k<n$, then $m$ divides $\prod_{k=0}^{n-1} F_k = F_n - 2$, and so $m$ divides the difference $F_n-\prod_{k=0}^{n-1} F_k = 2$.
HINT $\rm\ \ m\ |\ F_n\$ and $\rm\ m\ |\ F_k\ |\ F_n-2\ \ \Rightarrow \ \ m\ |\ F_n - (F_n-\:2)\ =\ 2$
NOTE $\$ An alternative simpler proof of the main result follows by specializing $\rm\:c\:,\:k\:$ in
$$\rm\ \gcd(c+1,\ c^{2\:k}+1)\ =\ gcd(c+1,\:2)$$


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# How to highlight text inside a plot created by ggplot2 using a box in R?

R ProgrammingServer Side ProgrammingProgramming

There might be many ways to highlight text inside a plot but the easiest one would be using geom_label function of ggplot2 package, with the help of this function we can put the required text and the aesthetics of that text by using a single line of code. It is highly recommended that we should use geom_label function with desired specifications.

## Example

Live Demo

Consider the below data frame −

set.seed(222)
x<−rnorm(5000,4,1)
df<−data.frame(x)
head(df,20)

## Output

   x
1 5.487757
2 3.998108
3 5.381021
4 3.619786
5 4.184136
6 3.753104
7 2.784439
8 5.561405
9 4.427310
10 2.798976
11 5.052458
12 2.694936
13 3.307392
14 4.602649
15 3.802247
16 2.814125
17 1.994487
18 4.007510
19 4.519490
20 3.253705
tail(df,20)
x
4981 4.555400
4982 5.335003
4983 5.461943
4984 4.026777
4985 4.001629
4986 4.244421
4987 2.711058
4988 5.611044
4989 4.752044
4990 2.988279
4991 4.007473
4992 2.951105
4993 3.981081
4994 5.526550
4995 4.602384
4996 4.908750
4997 3.568487
4998 3.276785
4999 4.303017
5000 2.663639

Loading ggplot2 package and creating histogram of x with highlighted text inside the plot −

library(ggplot2)
ggplot(df,aes(x))+geom_histogram(bins=30)+geom_label(aes(x=6,y=450,label="Normal Distribution"),fill="red")

## Output

Published on 06-Nov-2020 12:56:07


text:
### 1103

Validity extension of stimulated echoes to imaginary signals arising for double diffusion encoding of closed pores
Kerstin Demberg1, Frederik Bernd Laun2, Peter Bachert1, and Tristan Anselm Kuder1

1Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, 2Institute of Radiology, University Hospital Erlangen, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

### Synopsis

We extend diffusion-weighted acquisitions using the stimulated echo acquisition mode (STEAM) to imaginary signals. In classical diffusion encoding, only real signals arise; however, for special applications of double diffusion encoding (DDE), complex signals arise, as shown here for diffusion pore imaging, a method that allows determining the shape of arbitrary closed pores filled with an NMR-detectable medium. It is shown that the phase information of complex signals is preserved under application of stimulated echoes: We show the analytically derived signal for DDE with STEAM and use Monte Carlo simulations for validation. Consequently STEAM can be employed for diffusion pore imaging.

### Introduction

Diffusion pore imaging enables the measurement of the average shape of arbitrary closed pores in an imaging volume element. Examples of such pores could in principle comprise cells in biological tissue or oil containing cavities in porous rocks. Measuring the Fourier transform of the pore space function is realized with non-conventional gradient profiles: Either by applying a long-narrow gradient profile1-5 or by combination of q-space imaging with double diffusion encoded (DDE) measurements6-8. Both methods enable measuring imaginary signals containing the pore shape information, which gets lost in classical diffusion encoding. While the first approach is limited in its sequence design, the second is highly flexible and allows incorporating stimulated echoes to reach the long-time limit exploiting the longer T1-relaxation time compared to T2. To the authors’ knowledge, only real signals have been acquired with stimulated echoes so far.

The present study presents the analytically derived signal for DDE-based pore imaging with stimulated echo acquisition mode (STEAM) demonstrating that the phase information is preserved for stimulated echoes. Validation using Monte Carlo simulations is presented and appropriate choice of RF-pulse phases is investigated.

### Methods 1: Analytical derivation of STEAM signals

The signals arising for the sequences depicted in Fig. 1(a)-(d) are derived by sequentially applying building blocks for a) short gradient pulses and b) RF-pulse sequences generating stimulated echoes:

a) A gradient pulse $\boldsymbol{G}_i$ of duration $\delta$ induces a phase factor $e^{iφ_i}$ to spins resting at position ${\boldsymbol{x}}_i$ during gradient application with $φ_i=-γδ{\boldsymbol{G}}_i{\boldsymbol{x}}_i$. If the pore’s center of mass $\boldsymbol{x}_{\text{cm}}$ is translated relative to the origin, additional phase shifts $ϑ$ occur.

b) If a pulse sequence $\left(90_{α_1}^°-τ-90_{α_2}^°\right)$ is applied to a transverse magnetization $e^{iφ}$ ($α_j$= phase of 90° pulse), the first pulse $90_{α_1}^°$ stores the real part as z-magnetization which is returned to the transversal plane by $90_{α_2}^°$ resulting in9 ${\text{Re}}\left(-e^{i(φ-α_1)}\right)\,e^{iα_2}=-\frac{1}{2}\,\left(e^{i(φ-α_1+α_2)}+e^{i(-φ+α_1+α_2)}\right)$. Definitions of coordinate system and phases in Fig.$~$1(e).

For DDE with STEAM [Fig. 1(d)], the sequence $$90_{α_0}^°-{\boldsymbol{G}}_1/2-\left(90_{α_1}^°-τ-90_{α_2}^°\right)-{\boldsymbol{G}}_2-\left(90_{α_3}^°-τ-90_{α_4}^°\right)-{\boldsymbol{G}}_3/2-{\text{acq}}_{α_{\text{a}}}$$ gives $$\frac{1}{4}\,e^{i(α_0-α_1-α_2+α_3+α_4)}\,e^{i\left(\frac{φ_1}{2}-φ_2+\frac{φ_3}{2}\right)}+\frac{1}{4}\,e^{i(α_0-α_1+α_2-α_3+α_4)}\,e^{i\left(\frac{φ_1}{2}+φ_2+\frac{φ_3}{2}+2ϑ\right)}\\+\frac{1}{4}\,e^{i(-α_0+α_1-α_2+α_3+α_4)}\,e^{i\left(-\frac{φ_1}{2}-φ_2+\frac{φ_3}{2}-ϑ\right)}+\frac{1}{4}\,e^{i(-α_0+α_1+α_2-α_3+α_4)}\,e^{i\left(-\frac{φ_1}{2}+φ_2+\frac{φ_3}{2}+ϑ\right)},$$ with $ϑ=-γδ\boldsymbol{G}_2\boldsymbol{x}_{\text{cm}}$. The signal is calculated by averaging over all random walkers of all pores. $\delta\rightarrow0$ and diffusion time $T\rightarrow\infty$ allows averaging independently over each $\boldsymbol{x}_i$ with $e^{iφ_i}$ resulting in $\int_{Pore}\rho(\boldsymbol{x}_i )e^{-i\boldsymbol{qx}_i}d\boldsymbol{x}_i=\tilde{\rho}(\boldsymbol{q})$ with the pore space function $\rho(\boldsymbol{x})$, its Fourier transform $\tilde{\rho}(\boldsymbol{q})$ and q-space vector $\boldsymbol{q}=\gamma\boldsymbol{G}_2\delta$. Averaging over $\boldsymbol{x}_{\text{cm}}$ introduces the Fourier transform of the pore center of mass distribution $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$ as prefactors (see results).

### Methods 2: Monte Carlo simulations

Monte Carlo simulations were performed realizing RF-pulses with rotation matrices. Pores of equilateral triangular shape were chosen with 10$~$ µm edge length and 100,000 random walkers per pore.

Three 2D distributions of the pores centers of mass were simulated: 1) rock or tissue sample: 10,000 pores randomly distributed in 1x1 mm; 2) large unordered phantom: 1100 pores randomly distributed in 5x5$~$mm having dimensions comparable to the phantom studied in 10; 3) small grid phantom: 9 pores on a square grid, distances of 10$~$µm between centers of mass.

Additional parameters: diffusion coefficient 1$~$µm²/ms, $\delta$=2.88$~$ms, G=3.87$~$T/m, $T$=150$~$ms.

### Results 1: STEAM signals

The STEAM DDE signal is composed of four terms:

$$S_{\text{DDE}}({\boldsymbol{q}})=\frac{1}{4}\,e^{i(α_0-α_1-α_2+α_3+α_4)}\,\tilde{\rho}^*({\boldsymbol{q}}/2)^2\tilde{\rho}({\boldsymbol{q}})+\frac{1}{4}\,e^{i(α_0-α_1+α_2-α_3+α_4)}\,\tilde{P}_{\boldsymbol{x}_{\text{cm}}}^*(2{\boldsymbol{q}})\,\tilde{\rho}^*({\boldsymbol{q}}/2)^2\tilde{\rho}^*({\boldsymbol{q}})\\+\frac{1}{4}\,e^{i(-α_0+α_1-α_2+α_3+α_4)}\,\tilde{P}_{\boldsymbol{x}_{\text{cm}}}({\boldsymbol{q}})\,|\tilde{\rho}({\boldsymbol{q}}/2)|^2\tilde{\rho}({\boldsymbol{q}})+\frac{1}{4}\,e^{i(-α_0+α_1+α_2-α_3+α_4)}\,\tilde{P}_{\boldsymbol{x}_{\text{cm}}}^*({\boldsymbol{q}})\,|\tilde{\rho}({\boldsymbol{q}}/2)|^2\tilde{\rho}^*({\boldsymbol{q}}).$$

Analogously for the STEAM q-space imaging signal:

$$S_{\text{QSI}}({\boldsymbol{q}})=-\frac{1}{2}\,e^{i(-α_0+α_1+α_2)}\,|\tilde{\rho}({\boldsymbol{q}})|^2-\frac{1}{2}\,e^{i(α_0-α_1+α_2)}\,\tilde{P}_{\boldsymbol{x}_{\text{cm}}}^*(2{\boldsymbol{q}})\,\tilde{\rho}^*({\boldsymbol{q}})^2.$$

The asterisk marks the complex conjugate. First terms correspond to the respective spin echo signals, corresponding to the desired pore imaging signal, reduced by a factor of 2 for each storage in the longitudinal direction, for $\alpha_{0,...,4}=0$. The additional undesired terms are modulated with the Fourier transform of the center of mass distribution $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$.

### Results 2: Validation with simulations

In Fig.$~$2-4, spin echo and STEAM signals were simulated for the vertical gradient direction with $\alpha_i=0$ for all 90°-pulses. For the q-space and DDE signal, analytical calculations (light-colored lines) for the respective $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$ are shown and for fewer sampling points Monte-Carlo spin echo (darker-colored lines) and STEAM signals (dots). For the rock/tissue sample (Fig.$~$2), Monte Carlo simulations fit very well with the analytical derivations and pore images show clearly discernible triangles. For a reduced number of pores, the amplitude of $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$ does not drop as low and the STEAM signal can deviate from the spin-echo result if a sampling point hits occasionally an outlier of $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$, see Fig.$~$3 for a phantom similar to 5,10-12. If the pore number is decreased further, the contribution of the three undesired terms in $S_{\text{DDE}}(\boldsymbol{q})$ gets stronger [Fig.$~$4(a)-(d)].

### Discussion

For experiments, a smoothing of the STEAM signals might occur due to inhomogeneity of the gradient field thus effectively averaging over a small span of q-values so that the influence of the three undesired terms in $S_{\text{DDE}}(\boldsymbol{q})$ goes unnoticed. But if $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$ does not drop to zero fast enough, phase cycling can be utilized to remove them: Following Tab.$~$1(b), the combination $\text{add}\,0\,(0\,0)\,(0\,0)\,0-\text{add}\,0\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,0-\text{substract}\,0\,(0\,0)\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,0-\text{subtract}\,0\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,(0\,0)\,0$ results in pure $e^{i\left(\frac{φ_1}{2}-φ_2+\frac{φ_3}{2}\right)}$-term, yielding the desired spin echo signal [Fig.$~$4(e)-(g)].

### Conclusion

The phase information of complex signals is preserved under application of stimulated echoes enabling diffusion pore imaging with that approach.

### Acknowledgements

Financial support by the DFG (grant no. KU 3362/1-1) is gratefully acknowledged.

### References

1. Laun FB, Kuder TA et al., Determination of the defining boundary in nuclear magnetic resonance diffusion experiments, Phys. Rev. Lett. 107, 048102 (2011).

2. Laun FB, Kuder TA et al., NMR-based diffusion pore imaging, Phys. Rev. E 86, 021906 (2012).

3. Kuder TA, Bachert P et al., Diffusion pore imaging by hyperpolarized xenon-129 nuclear magnetic resonance, Phys. Rev. Lett. 111, 028101 (2013).

4. Hertel S, Hunter M et al., Magnetic resonance pore imaging, a tool for porous media research, Phys. Rev. E 87, 030802 (2013).

5. Hertel SA, Wang X et al., Magnetic-resonance pore imaging of nonsymmetric microscopic pore shapes, Phys. Rev. E 92, 012808 (2015).

6. Shemesh N, Westin CF et al., Magnetic resonance imaging by synergistic diffusion-diffraction patterns, Phys. Rev. Lett. 108, 058103 (2012).

7. Kuder TA and Laun FB, NMR-based diffusion pore imaging by double wave vector measurements, Magn. Reson. Med. 70, 836-841 (2013).

8. Demberg K, Laun FB et al., Nuclear magnetic resonance diffusion pore imaging: Experimental phase detection by double diffusion encoding, Phys. Rev. E 95, 022404 (2017).

9. Khrapitchev AA and Callaghan PT, Double PGSE NMR with Stimulated Echoes: Phase Cycles for the Selection of Desired Encoding, J. Magn. Reson. 152, 259-268 (2001).

10. Avram L, Assaf Y et al., The effect of rotational angle and experimental parameters on the diffraction patterns and micro-structural information obtained from q-space diffusion NMR: implication for diffusion in white matter fibers, J. Magn. Reson. 169, 30-38 (2004).

11. Bar-Shir A, Avram L et al., The effect of the diffusion time and pulse gradient duration ratio on the diffraction pattern and the structural information estimated from q-space diffusion MR: experiments and simulations, J. Magn. Reson. 194, 230-236 (2008).

12. Shemesh N, Özarslan E et al., Noninvasive bipolar double-pulsed-field-gradient NMR reveals signatures for pore size and shape in polydisperse, randomly oriented, inhomogeneous porous media, J. Chem. Phys. 133, 044705 (2010).

### Figures

Fig. 1: Schemes of the RF and gradient pulse sequences: (a) q-space imaging. (b) q-space imaging with STEAM. It is assumed that in the time interval $\tau$ during which the magnetization resides in the z-direction the transverse magnetization is dephased sufficiently to not contribute to the signal. (c) Double diffusion encoding (DDE) sequence where the two diffusion encodings are applied with anti-parallel wave vectors without temporal separation. (d) DDE with STEAM. $\alpha_i$ are the phases of the respective pulses, $\alpha_{\text{a}}$ the acquisition phase, $\boldsymbol{G}_i$ and $\delta$ the gradient amplitudes and durations, and $T$ the diffusion time. (e) Coordinate system.

Fig. 2: Results for the rock/ tissue sample for the vertical gradient direction. All 90°-pulse phases were 0. (a) q-space signal. (b) DDE signal. Calculated: Signals calculated analytically at high sampling rate. Simulated: Monte Carlo results for spin echo acquisition and STEAM. Real parts in blue, imaginary in red. (c) Fourier transform of the pore distribution with high sampling used for the calculated STEAM signals and at low sampling rate (20 q-values) used in the simulations. (d) One half of q-space was sampled with q-space imaging and DDE from which the pore images were reconstructed via an iterative approach7.

Fig. 3: Results for the large unordered phantom for the vertical gradient direction. All 90°-pulse phases were 0. Rest analog to Fig. 2. For the equilateral triangular shape, imaginary signal parts occur in gradient directions where the projected random walks are not distributed axially symmetric as shown for the projection onto the vertical gradient direction in (b).

Fig. 4: Results for the small grid phantom for the vertical gradient direction with 3x3 pores. In (a)-(d) all 90°-pulse phases were 0. Depiction in (a)-(d) analog to Fig. 2. (e) Phase cycling result for q-space imaging and (f) double diffusion encoding and (g) the effect on the pore image. The analytically calculated signal in (a)-(b) represents the Monte Carlo signal very well: Strong influence of the undesired STEAM-terms (three for $S_{\text{DDE}}(q)$, one for $S_{\text{QSI}}(q)$), which drop with $\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(q)$, prevents measuring the exact pore shape in (d). This can be solved through phase cycling, see (g) and Tab.1.

Tab. 1: Phase cycling table for (a) q-space imaging with one 90° RF pulse pair [Fig. 1(b)] and for (b) double diffusion encoding (DDE) with two 90° RF pulse pairs [Fig. 1(d)]. $\alpha_0$ is the excitation pulse phase and $\alpha_{\text{a}}$ is the acquisition phase. For q-space imaging, the combination "$\text{subtract}\,0\,(0\,0)\,0$ and $\text{add}\,0\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,0$" results in pure component A, which is the compensated component resulting in the spin echo signal $|\tilde{\rho}(\boldsymbol{q})|^2$. For the DDE phase cycling scheme, please refer to the discussion section.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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# How can I change the style of every newly opened notebook?

I'd like to make Mathematica open a new notebook using the Natural Color template (under the Creative category) by default. Is that possible? If so, how?

• You can set the "DefaultStyleDefinitions" option in the Options Inspector.
– rm -rf
Oct 9 '12 at 3:38
• I set the "DefaultStyleDefinitions" (under Global Options->File Locations) so that, instead of "Default.nb" it pointed to "NaturalColor.nb" and I get some crazy worksheet (army green background, pink input boxes). I'd attach a screenshot, but I don't know how (first posting to stackexchange). I think that there are things inherited from Default.nb that are not present in NaturalColor.nb. Oct 9 '12 at 3:51
• I don't remember, but you might have to point it to "Creative/NaturalColor.nb"
– rm -rf
Oct 9 '12 at 4:04
• YES! Thanks so much. That did the trick! (I'm not sure why but I'm happy just to have it work for now.) Oct 9 '12 at 4:11
• Please check the answers to a related question to see if they work for you.
– kglr
Oct 9 '12 at 4:12


text:
Next: Hamiltonian and Propagation Up: Theory Previous: Theory

## Coordinates and Basis set

We use the standard body-fixed (BF) Jacobi coordinates R, r, and , which are the length of the distance vector between H and the center-of-mass of O2, the length of the O2 internuclear distance vector , and the angle between and , respectively. is also the body-fixed z-axis. The overall orientation of the HO2 complex with respect to a space-fixed (SF) coordinate system is given by 3 Euler angles, collectively denoted by .

Good quantum numbers for the wave function are J, the total angular momentum quantum number, M, the projection of onto the SF z-axis and the parity p of the wave function under inversion of the SF nuclear coordinates. We use a Discrete Variable Representation (DVR), based on sinc-functions[55,56,57] for the radial coordinates and a Finite Basis Representation (FBR) for the angular coordinates. For a discussion of (sinc)-DVRs the reader is referred to e.g., Refs. [56,57,58,59,60]. The DVR-functions form an orthonormal set, characterized by the fact that a basis function is effectively non-zero only at the gridpoint , defined as with the spacing between gridpoints in the R-dimension. An equivalent relation holds for the functions in the r-dimension. Throughout this paper we will find it convenient to use instead of the proper Jacobi coordinate wave function to which it is related as [61]. This ensures that we can use the Hamiltonian in its form of Eq. (5) instead of the form given in e.g. Refs. [62] and [63]. We expand the wave function for the conserved quantum numbers J, M, and p in wrapped'' sinc-DVR functions[57] and for the radial coordinates and parity adapted angular basis functions for the angular coordinates:

 (4)

In Eq. (2) j is the rotational angular momentum of the O2 molecule. Because of the permutation-inversion symmetry of the O2 molecule only odd rotational states are allowed[64]. The quantum number is the projection of onto the BF z-axis and the projection of onto the BF z-axis. Therefore, the eigenvalues J and j must always be larger than or equal to . is 0 or 1, depending on the spectroscopic[65,66] parity (-1)J+p [see Eq. (3)]. The constants Nr and NR are the total number of DVR-functions in the vibrational and scattering coordinates, respectively. The functions are parity adapted angular functions, given by[67,68,69]

 (5)

where is an associated Legendre function in the phase convention of Condon and Shortley[70] and where is a normalized Wigner D-function, given by

 (6)

with the Wigner D-matrices defined in the passive zyz-convention for rotations[71,72]. The functions are eigenstates of , , , parity, and with eigenvalues J(J+1), M, , (-1)p, and j(j+1), respectively. Since all calculations are done for a specific J,M,p combination, we will now drop the J,M,p superscript.

Next: Hamiltonian and Propagation Up: Theory Previous: Theory
Anthony J. H. M. Meijer
1998-02-20