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I was not amused during my recent spell of unemployment, to be awarded just £87.40 a week in housing benefit. My rent is £600.00 a month – not exactly an overwhelmingly expensive amount in London.
But apparently the government decrees that I should be able to find a room in a house for £378.73 a month. In London. Yep. I haven’t paid that kind of rental amount since 2003.
However, as I am over 35, if I were living in a one-bedroom flat, then I would get up to £800.00 a month. Logical, isn’t it?
So because I am doing the cost-saving procedure of renting a room in a shared house (albeit with nobody else living here), I get penalised.
I got my strop on and wrote a complaint.
******
Dear Sir/Madam
I would like to raise a complaint about the exceptionally low level of housing benefit support provided for unemployed persons such as myself.
I am a 37-year old male, living in a shared house with rent at £600pm.
I have been advised that I will receive just £87.40 per week. This comes to £378.73 per month – leaving me with a shortfall of £221.27 each month.
This I can cover with my jobseeker’s allowance which comes to £316.77 per month.
My total income per month is therefore £695.50. My rent is £600.00 and my mobile phone bill is £12.00 a month.
This leaves me with a grand total of £83.50 per month for food and toiletries, etc.
Previously when I was unemployed for 6 months in 2011/2012, I received £600.00 per month for a flat that I was living in. I am advised that because I have taken the cost-saving option of house-sharing, that I am now penalised. I was advised that if I had my own more-expensive flat, then my rent would have been covered.
This makes no sense. Why would you fund a higher one-bedroom flat rental cost, but not the lower cost of renting a room in a shared house?
It does seem that I am being unfairly penalised – I doubt that any other groups of benefit claimants receive such small amounts as £83.50 per month to live on.
Now, I will survive until I get a job – and I will get a job. I am no layabout – I spend 8-10 hours almost every single day looking for work and upskilling myself.
There is also an upside to my current relative destitution, and that is that I have lost 5kg in weight in the last two months due to my budgetary situation and having to restrict my food intake at times.
There are constantly stories of the huge benefits that some get, from large councils houses (remember Bob Crowe getting a council house despite earning £100k a year?), to above-inflation increases in pensions, vast arrays of people receiving benefits that they don’t need such as middle-income people receiving child tax credits and rich pensioners with their free TV licenses, winter fuel allowances, etc.
It seems the whole system is against someone like myself; no children, middle-aged, stable, balanced, male, temporarily out of work. I have worked for all but 9 months of the last 17 years since university, contributing large volumes of tax towards the aforementioned unfair benefits, yet I cannot receive enough to cover my rent.
I will be working easily for the next 30 years, and as I am changing career into development/programming, I expect that as my earning potential hopefully increases significantly, I will be paying ever-larger sums of tax.
I know that I won’t get anything in return to this letter other than platitudes and your blaming of “government austerity”. It does seem immaterially unfair that I am having to suffer due to the reckless overspending of the Labour government and the horrendous deficit that they left.
If you know anyone who requires a junior web developer, or a website making, please let me know. | http://worldofwinfield.co.uk/complaint-harrow-council-housing-benefi/ |
Dogs Are Pregnant for how long before giving birth?
Dogs Are Pregnant for how long before giving birth?
Like with humans, we separate the dog gestation period into three trimesters, each lasting about 21 days.
While there are some outwards signs of pregnancy in dogs, it is difficult to determine whether a dog is pregnant without veterinary diagnostics, especially in the early stages, as there are several medical conditions that result in symptoms that are similar those that appear during pregnancy.
Here’s what you should know about dog pregnancy.
The Dog Gestation And Labor Period
For a female dog to get pregnant, she must first be in heat.
In dogs who haven’t had spay surgery, this happens about once every six months, and the heat cycle lasts for 18 to 21 days. A female dog will be receptive to males starting at about 9 days into the cycle, and they may get pregnant any time over the course of the next three to eleven days. Breeders keep track of these cycles and run tests to determine the optimum time for breeding.
When a dog becomes pregnant, the embryos start to travel through the uterine horn at around day seven. By day 16, the embryos embed in the uterine lining, and by day 22, the fetuses begin to form. From about day 28 to day 30, a vet will be able to see the heartbeats of the fetuses with an ultrasound.
The puppies’ eyelids start to form around day 32. Toes form around day 35, claws come in around day 40, and the coat and skeleton come in by around day 45.
After day 50, a veterinarian can perform an x-ray to see the puppies’ skeletons and form an accurate count of how many to expect in the litter.
By around day 58, the puppies should be completely formed.
The mother dog will start looking for a place to nest and give birth. Labor should begin within the next three to four days. Labor happens in three stages and a veterinarian or someone with experience should supervise, as complications can occur.
The first stage of labor lasts from about twelve to 24 hours. During this time, contractions begin in the uterus, but there may be no outward signs of contractions yet. The mother may seem restless, refuse to eat, vomit, pant, or show other signs that labor has begun.
The mother dog gives birth to her puppies during the second stage of labor, which can take up to 24 hours. Usually the mother dog births a puppy every 30 to 60 minutes, but they should not take more than two hours each. It is helpful to rely on a vet’s x-rays to know the number of expected puppies so it is clear when stage two is complete.
The third stage of labor happens when the placenta appears, and it will likely occur at around the same time as stage two. Stage three is complete when the last of the placentas have been delivered, and it should be finished shortly after stage two has ended.
Is My Dog Pregnant? What Are the Symptoms of Pregnancy in Dogs?
In the earliest stages of a dog’s pregnancy, there will be very few outward signs. You may notice some weight gain, but there are several reasons a dog might gain weight that aren’t related to pregnancy.
Noticeable symptoms of pregnancy usually don’t appear until the third or fourth week. During this time, some dogs suffer from morning sickness, tiredness, or lack of appetite.
Again, other medical conditions result in similar symptoms, so it is important to consult a veterinarian if you suspect your dog is pregnant. Dogs who vomit due to pregnancy should be fed small meals throughout the day, rather than two larger meals.
Between the 25th and 28th day of pregnancy, a vet will be able to feel the belly to determine if puppies are on the way. Only a professional veterinarian should do this. If you try to feel for puppies on your own, you could harm the fetuses or cause a miscarriage.
Around day 40, the belly will start to expand.
The nipples may get darker and start to swell. It is normal for some milky fluid to be discharged from the nipples, and it shouldn’t cause you concern. You can check with your veterinarian if anything seems out of the ordinary.
During the final stages of pregnancy, the belly may start to sway when the mother dog walks. Around two weeks before she gives birth, you’ll probably be able to see and feel the puppies moving inside the mother’s belly.
For someone who is not a professional or does not have experience with pregnant dogs, the symptoms of pregnancy can be confused with other conditions, even at the later stages. Your veterinarian will be able to run several diagnostics to determine if your dog is pregnant for sure, and you should rely on their professional medical advice.
Veterinary Diagnostics To Tell If A Dog Is Pregnant
Because symptoms of dog pregnancy can mimic signs of other medical conditions, it is important to have your veterinarian run diagnostics if you suspect your dog is pregnant.
Your veterinarian will be able to perform some tests at several stages of the pregnancy and give you advice on how to care for and feed your pregnant dog.
It is important to note that many medications and supplements are not recommended for pregnant dogs and may harm puppies, so make sure your vet is aware of anything you give your dog regularly so they can let you know what is safe.
By around day 28 of pregnancy, your veterinarian will be able to perform abdominal palpitations to determine if your dog is pregnant, and they can instruct you on how to feel for yourself. You should not attempt to do this on your own, as you can easily harm the fetuses or cause a miscarriage.
At this time, the fetuses will feel like small golf balls or grapes.
Between days 28 to 35, your vet can run an ultrasound and detect the heartbeats of the puppies. They will be able to provide an estimate of how many puppies to expect in the litter, but they will be able to give a more accurate reading once they can perform an x-ray later on in the pregnancy.
After around day 30, your vet will be able to give your dog a blood test to detect the hormone relaxin. Dogs’ bodies only release this hormone during pregnancy, so detecting it will accurately determine that your dog is pregnant.
It is only around day 45 to 55 that a vet can perform an x-ray to see the skeletons of the puppies and accurately determine the size of the litter. The closer this is done to the end of the pregnancy, the more accurate the count will be. | https://scholarsark.com/question/dogs-are-pregnant-for-how-long-before-giving-birth/ |
So I got this question:Example 3: When a football is kicked with a vertical speed of 20m/s, its height, h metres, after t seconds is given by the formula:
How long after the kick is the football at a height of 15m?
Solution
Substitute 15 for h in the formula:
This produces a quadratic equation. The solution of the equation gives the times when the height is 15m.
Collect all the terms on one side of the equation.
Either
or
The football is at a height of 15m twice: first on the way up, 1 s after the kick, then on the way down, 3 s after the kick.
A baseball is hit with a vertical speed of 31.3m/s. Use the formula in the modelling box following Example 3. How long after being hit is the ball at a height of 25m? Give your answer to 1 decimal place.
How exactly do I form the equation? I got it but I don't think I'm on the right track. Thanks a bunch.
Substitute u = 31.3 (NOT 25). Substitute h = 25. Solve the resulting quadratic equation (it won't factorise in the nice way your example did. I suggest using the quadratic formula or using a graphics calculator)
For the second time for which the object is at 25 m you have reset your clock to start at 0 s when the ball is at its maximum height. Thus when you find the time it takes for the ball to get back to 25 m, you need to add the time it took for the ball to get to its maximum height.
Do that and you will get the 5.3 s. | http://mathhelpforum.com/algebra/25738-solving-quadratic-equations.html |
Many times when you are on the go, you are headed somewhere that requires you to be there by a certain time. Today, I’m going to teach you how to use your journey as an opportunity for some mental maths!
Ease on Down the Road
What to Do
- A litre of gas costs $1.90 a litre. What does it cost for 5 litres? 10 litres? 15 litres? 20 litres? What is an easy way to figure this out? How can you estimate the cost by rounding the cost per litre?
- The speed limit is 65 kilometers per hour. How far will you go in 1 hour? Two hours? Three hours? How long will it take to go 500 kilometers?
Use a calculator to check your answers.
Parent Pointer: An important algebra concept is finding relationships between two quantities such as kilometers per hour or cost per litre. | https://conceptualthinkers.net/more-tips-to-help-your-child-engage-in-maths-part-14/ |
Audio Version of Every Day Hope Message:
Your dreams are not lost, lacking or dormant. They are not gone, empty or expired. And even if you’ve forgotten what it feels like to dream, to embrace the barometers of your soul, your dreams can still come alive when you ask them in.
Dreams ignite your purpose. They bring you to life, and there is no better representation of who you are at your core than the visions you hold dearest.
Don’t worry about where you are so much; think more on where you’re going and where you want to be.
How many dreams sit on your back burner? What hopes are you procrastinating on? Why are you waiting to check them off your bucket list? Stop putting off the plans you have for yourself.
There will never be a perfect time or place to shoot for your dreams. But the opportunity to get started is always there, you just have to take it. Make it *your* time; make your dreams a reality. If left unattended and allowed to languish, dreams will never come true. Choose to reinvigorate your passion and ignite your soul; move tomorrow to today and let your dreams come alive. It’s never too late. | https://hopecourageinspiration.com/2014/08/21/ignite-your-purpose/ |
1. Introduction {#sec1-materials-12-03808}
===============
The shallow shell is an open shell with a small curvature and radius of curvature compared with various shell parameters (i.e., length and width). With the development research into composite materials, composite laminated shallow shells are widely applied in some modern engineering practice with a high level of intensity and rigidity, for instance, petroleum equipment, aerospace equipment, and marine equipment. It is worth noting that the composite laminated shallow shells are typically operated under complicated environmental conditions and subjected to complex boundary conditions. So, it is particular importance to fully investigate the free vibration characteristics of composite laminated shallow shells with non-classical boundary conditions.
Through many years of hard work by research scholars, some shell theories have been summarized, such as classical shell theory (CST) \[[@B1-materials-12-03808],[@B2-materials-12-03808],[@B3-materials-12-03808]\], first-order shear deformation shell theory (FSDT) \[[@B4-materials-12-03808],[@B5-materials-12-03808]\], and high-order shell theory (HST) \[[@B6-materials-12-03808],[@B7-materials-12-03808],[@B8-materials-12-03808],[@B9-materials-12-03808],[@B10-materials-12-03808]\]. CST is the basic shell theory and is known as the simplest equivalent single layer, which is based on the Kirchhoff--Love hypothesis. To analyze the complex shell structure, some shell theories were developed along with some assumptions, such as Reissner--Naghdi's shell theory and Donner--Mushtari's theory. A more detailed description of these theories can be found in the research by Reddy \[[@B11-materials-12-03808]\], Leissa \[[@B12-materials-12-03808]\], and Qatu \[[@B13-materials-12-03808]\]. The main application area is thin shell structures. To analyze the thick shell, CST ignores the effect of transverse shear deflection, causing the calculation of natural frequencies to be inaccurate. To improve the influential impact of transverse shear deformation, FSDT is conducted. HST can attenuate the dependence of FSDT on shear correction factors; however, there is a large amount of calculation in the study of the high-order stress resultant force. Simultaneously, many remarkable researchers have investigated the composite laminated shallow shell in recent years and published some excellent papers. Ye et al. \[[@B14-materials-12-03808]\] investigated the free vibration characteristics of the composite laminated shallow shell under general elastic boundary conditions. The closed form auxiliary functions are used to transform the displacement variables into standard Fourier cosine series. Kurpa et al. \[[@B15-materials-12-03808]\] extended the R-function method to investigate the composite laminated shallow shells on an arbitrary planform by FSDT. Fazzolari and Carrera E \[[@B16-materials-12-03808]\] conducted the Ritz formulation and Carrera unified formulation to investigate the composite laminated doubly-curved anisotropic shell, and the free vibration response is discussed. Awrejcewicz et al. \[[@B17-materials-12-03808]\] proposed R-functions theory and the spline-approximation to study the bending performance of the composite shallow shell with a static loading boundary condition. Tran et al. \[[@B18-materials-12-03808]\] presented a static feature of the cross-ply composite hyperbolic shell panels on Winkler--Pasternak elastic foundation, and the smeared stiffeners technique was adopted. Biswal et al. \[[@B19-materials-12-03808]\] discussed the free vibration characteristic of composite shells consisting of woven fiber glass/epoxy with hygrothermal environments. The FSDT and quadratic eight-noded isoparametric element are adopted to study the free vibration characteristics under elevated temperatures and moisture concentrations conditions. Garcia et al. \[[@B20-materials-12-03808]\] investigated the effect of polycaprolactone nanofibers on the dynamic behavior of glass fiber reinforced polymer composites. Garcia et al. \[[@B21-materials-12-03808]\] investigated the influence of the inclusion of nylon nanofibers on the global dynamic behaviour of glass fibre reinforced polymer (GFRP)composite laminates. Shao et al. \[[@B22-materials-12-03808]\] conducted the enhanced reverberation-ray matrix (ERRM) method to investigate the transient response of the composite shallow shell. In these studies, the kinetic analysis of composite laminated shallow shell is proposed to free vibration, and many analytical and computational methods were developed.
These include the Ritz method \[[@B23-materials-12-03808],[@B24-materials-12-03808],[@B25-materials-12-03808],[@B26-materials-12-03808],[@B27-materials-12-03808]\], dynamic stiffness method \[[@B28-materials-12-03808]\], closed-form solution \[[@B29-materials-12-03808],[@B30-materials-12-03808],[@B31-materials-12-03808]\], boundary domain element method \[[@B32-materials-12-03808]\], Meshfree approach \[[@B33-materials-12-03808]\], Galerkin method \[[@B34-materials-12-03808],[@B35-materials-12-03808]\], and finite element method \[[@B36-materials-12-03808],[@B37-materials-12-03808],[@B38-materials-12-03808]\].
In recent years, the wave-based method (WBM) has been adopted to investigate the dynamic behavior of engineering structures in some applications. WBM was first proposed in the work of \[[@B39-materials-12-03808]\] to analyze the coupled vibro-acoustic systems and the steady-state dynamics characteristics of the system concerned. Deckers et al. \[[@B40-materials-12-03808]\] presented a literature review of WBM research for 15 years. With the research on structural vibration in recent years, WBM has been adopted in the dynamic analysis for some engineering structures, such as the dynamic characteristics of cylindrical shell structures, which many researchers have studied using WBM. Chen et al. \[[@B41-materials-12-03808]\] analyzed the free and force vibration characteristics of a cylindrical shell in discontinuity thickness form. Xie et al. \[[@B42-materials-12-03808]\] conducted WBM to study the free vibration and acoustic dynamic characteristics of underwater cylindrical shells with bulkheads. Wei et al. \[[@B43-materials-12-03808]\] investigated the non-uniform stiffener distribution of a cylindrical shell. At the same time, as many reinforcements and coupling structures are more common in engineering applications, the corresponding research is increasing, such as the cylindrical shell coupled elastically with annular plate and the ring stiffened cylindrical shell with frame ribs \[[@B44-materials-12-03808],[@B45-materials-12-03808]\]. Also, the free vibration characteristics of the composite laminated cylindrical shell have been investigated \[[@B46-materials-12-03808]\]. Therefore, it is meaningful to develop an effective method for the general processing ability of composite laminated shallow shells with general boundary conditions. According to the author's literature review of related topics, there has not been any published work with regard to the application of the presented method to analyze the free vibration characteristics of the composite laminated shallow shell with general boundary conditions.
For the first time, the wave-based method is adopted to study the free vibration characteristics for a composite laminated shallow shell with general boundary conditions. According to the relationship between the displacement vector and force resultants, the governing equation of composite shells is established by FSDT and CST. By converting the displacement variable into a wave function form and the boundary matrices, the total matrix is established. Solving the root of the total matrix determinant using the dichotomy method, the natural frequencies of composite laminated shallow shells are calculated. To verify the correctness of the solutions by the presented method, the comparisons of the current solutions with the results in represented literatures are shown. Furthermore, the influence of material parameters and geometric constants, such as length to radius ratios, length to thickness ratios, modulus ratios, and elastic restrained constants, are discussed in some numerical examples. The main purpose of this paper is to provide a relatively new method for analyzing the free vibration characteristics of composite laminated shallow shells, which provides a new direction for composite laminated structure analysis. When studying the vibration analysis of the composite laminated shallow shell with general boundary conditions, it is easier to obtain the total matrix, and the boundary conditions are easy to replace. The advantages of the presented method lie in its simplicity, low computational cost, and high precision.
2. Theoretical Formulations {#sec2-materials-12-03808}
===========================
2.1. The Description of Model {#sec2dot1-materials-12-03808}
-----------------------------
In [Figure 1](#materials-12-03808-f001){ref-type="fig"}a, the schematic diagrams of the composite laminated shallow shells under elastic restraint are shown. *L~x~*, *L~y~*, and *h* express the length, width, and thickness, respectively, of the composite laminated shallow shells. *R~x~* and *R~y~* indicated the principle curvature radii. In the middle surface of the model, a global coordinate (*o*-*xyz*) is established in the length, width, and thickness directions. For the *k*th layer of the composite shell, the distances of top and bottom surface to the middle surface are denoted as *Z~k~*~+1~ and *Z~k~*. For the elastic boundary conditions, there is one set of linear springs (*K~u~*, *K~v~*, and *K~w~*) and one pair of rotational springs (*K~ϕx~* and *K~ϕy~*), which set on two edges, *x* = 0 and *L~x~*. Through the changing of two pairs of elastic restrained springs, an arbitrary elastic boundary condition can be achieved. In [Figure 1](#materials-12-03808-f001){ref-type="fig"}b, with the changing of the principle curvature radii, the composite laminated shallow shells have various types, such as plate (i.e., *R~x~* = *R~y~* = ∞), cylindrical shell (i.e., *R~x~* = *R*, *R~y~* = ∞), spherical shell (i.e., *R~x~* = *R~y~* = *R*), and hyperbolic paraboloidal shell (i.e., *R~x~* = −*R~y~* = *R*).
2.2. First-Order Shear Deformation Shell Theory (FSDT) {#sec2dot2-materials-12-03808}
------------------------------------------------------
### 2.2.1. Kinematic Relations and Stress Resultants {#sec2dot2dot1-materials-12-03808}
This section is divided into subheadings. It should provide a concise and precise description of the experimental results and their interpretation, as well as the experimental conclusions that can be drawn.
According to the relationship between the displacement variables and rotation transverses of the composite shallow shells by FSDT, the displacement variables are shown as follows \[[@B13-materials-12-03808]\]:$$\begin{array}{l}
{u\left( {x,z,t} \right) = u_{0}\left( {x,t} \right) + z\phi_{x}\left( {x,t} \right)} \\
{v\left( {x,y,t} \right) = v_{0}\left( {x,y} \right) + z\phi_{y}\left( {x,y} \right)} \\
{w\left( {x,z,t} \right) = w\left( {x,t} \right)} \\
\end{array}$$ where *u*~0~, *v*~0~, and *w*~0~ are the displacements of the arbitrary point along the *x*, *y*, and *z* directions, respectively, in the middle surface. *ϕ~x~* and *ϕ~y~* are the *y* and *x* axes transverse rotations, respectively, and *t* is a time variable. The linear strain relationship between the change strain and curvature in the middle surface under the assumption of small deformation is given as follows:$$\begin{array}{l}
{\varepsilon_{xx} = \varepsilon_{xx}^{0} + z\varepsilon_{xx}^{1}} \\
{\varepsilon_{yy} = \varepsilon_{yy}^{0} + z\varepsilon_{yy}^{1}} \\
{\gamma_{xy} = \gamma_{xy}^{0} + z\gamma_{xy}^{1}} \\
{\gamma_{xz} = \gamma_{xz}^{0}} \\
{\gamma_{yz} = \gamma_{yz}^{0}} \\
\end{array}$$ where {$\varepsilon_{xx}^{0}$, $\varepsilon_{yy}^{0}$} are the normal strains of the middle surface, {$\gamma_{xy}^{0}$, $\gamma_{xz}^{0}$, $\gamma_{yz}^{0}$} are the shear stains, and {$\varepsilon_{xx}^{1}$, $\varepsilon_{yy}^{1}$, $\gamma_{xy}^{1}$} are the curvature and twisting changes of the middle surface. The detailed expressed formulations of the strains and changes are defined as follows:$$\begin{array}{ll}
{\varepsilon_{xx}^{0} = \frac{\partial u_{0}}{\partial x} + \frac{w_{0}}{R_{x}}} & {\varepsilon_{yy}^{0} = \frac{\partial v_{0}}{\partial y} + \frac{w_{0}}{R_{y}}} \\
{\varepsilon_{xx}^{1} = \frac{\partial\phi_{x}}{\partial x}} & {\varepsilon_{yy}^{1} = \frac{\partial\phi_{y}}{\partial y}} \\
{\gamma_{xy}^{0} = \frac{\partial v_{0}}{\partial x} + \frac{\partial u_{0}}{\partial y}} & {\gamma_{xy}^{1} = \frac{\partial\phi_{y}}{\partial x} + \frac{\partial\phi_{x}}{\partial y}} \\
{\gamma_{xz}^{0} = \frac{\partial w_{0}}{\partial x} - \frac{u_{0}}{R_{x}} + \phi_{x}} & {\gamma_{yz}^{0} = \frac{\partial w_{0}}{\partial y} - \frac{v_{0}}{R_{y}} + \phi_{y}} \\
\end{array}$$
The corresponding stresses expressed by the Hooke's law are as follows:$$\left\{ \begin{array}{l}
\sigma_{xx} \\
\sigma_{yy} \\
\tau_{xy} \\
\tau_{xz} \\
\tau_{yz} \\
\end{array} \right\} = \begin{bmatrix}
\overline{Q_{11}} & \overline{Q_{12}} & 0 & 0 & \overline{Q_{16}} \\
\overline{Q_{12}} & \overline{Q_{22}} & 0 & 0 & \overline{Q_{26}} \\
0 & 0 & \overline{Q_{44}} & \overline{Q_{45}} & 0 \\
0 & 0 & \overline{Q_{45}} & \overline{Q_{55}} & 0 \\
\overline{Q_{16}} & \overline{Q_{26}} & 0 & 0 & \overline{Q_{66}} \\
\end{bmatrix}\left\{ \begin{array}{l}
\varepsilon_{xx} \\
\varepsilon_{yy} \\
\gamma_{xy} \\
\gamma_{xz} \\
\gamma_{yz} \\
\end{array} \right\}$$ where $\overline{Q_{ij}}$ (*i*, *j* = 1,2,4,5,6) are the transform coefficients and depend on material parameters; and the constants *Q~ij~* (*i*,*j* = 1,2,4,5,6), which are associated with the strains and stresses, can be expressed as follows:$$\begin{matrix}
{Q_{11} = \frac{E_{1}}{1 - \mu_{12}\mu_{21}},Q_{12} = Q_{21} = \frac{\mu_{12}E_{2}}{1 - \mu_{12}\mu_{21}},Q_{22} = \frac{E_{2}}{1 - \mu_{12}\mu_{21}}} \\
{Q_{44} = G_{23},Q_{55} = G_{13},Q_{66} = G_{12}} \\
\end{matrix}$$ where *E*~1~, *E*~2~ are the Yong's moduli and *μ*~12~ and *μ*~21~ are the Poisson's ratios. By integrating the stresses and moments over the cross section and thickness, the relationship between the strains and curvature in the middle surface is given as follows:$$\begin{Bmatrix}
N_{xx} \\
N_{yy} \\
N_{xy} \\
M_{xx} \\
M_{yy} \\
M_{xy} \\
\end{Bmatrix} = \begin{bmatrix}
A_{11} & A_{12} & A_{16} & B_{11} & B_{12} & B_{16} \\
A_{12} & A_{22} & A_{26} & B_{12} & B_{22} & B_{26} \\
A_{16} & A_{26} & A_{66} & B_{16} & B_{26} & B_{66} \\
B_{11} & B_{12} & B_{16} & D_{11} & D_{12} & D_{16} \\
B_{12} & B_{22} & B_{26} & D_{12} & D_{22} & D_{26} \\
B_{16} & B_{26} & B_{66} & D_{16} & D_{26} & D_{66} \\
\end{bmatrix}\begin{Bmatrix}
\varepsilon_{xx}^{0} \\
\varepsilon_{yy}^{0} \\
\gamma_{xy}^{0} \\
\varepsilon_{xx}^{1} \\
\varepsilon_{yy}^{1} \\
\gamma_{xy}^{1} \\
\end{Bmatrix},\begin{Bmatrix}
Q_{y} \\
Q_{x} \\
\end{Bmatrix} = K_{c}\begin{bmatrix}
A_{44} & A_{45} \\
A_{45} & A_{55} \\
\end{bmatrix}\begin{Bmatrix}
\gamma_{yz}^{0} \\
\gamma_{xz}^{0} \\
\end{Bmatrix}$$ where {*N~xx~*, *N~xy~*, *N~yy~*} is the in-plane force resultant, {*M~xx~*, *M~yy~*, *M~xy~*} is the bending and twisting moment resultant, and {*Q~x~*, *Q~y~*} is the transverse shear force resultant. *K~c~* is the shear correction factor and the value is set as 5/6. Furthermore, the stretching stiffness coefficients, coupling stiffness coefficients, and bending stiffness coefficients are given as follows:$$A_{ij} = {\sum\limits_{k = 1}^{N}\overline{Q_{ij}}}\left( {Z_{k + 1} - Z_{k}} \right),B_{ij} = \frac{1}{2}{\sum\limits_{k = 1}^{N}\overline{Q_{ij}}}\left( {Z_{k + 1}^{2} - Z_{k}^{2}} \right),D_{ij} = \frac{1}{3}{\sum\limits_{k = 1}^{N}\overline{Q_{ij}}}\left( {Z_{k + 1}^{3} - Z_{k}^{3}} \right)$$ where *N* is the number of the layers. *Z~k~*~+1~ and *Z~k~* are the distance from the top surface and bottom surface, respectively, to the middle surface of the *k*th layer. For analysis of the general cross-ply composite laminates shallow shell, the transform coefficients $\overline{Q_{16}}$, $\overline{Q_{26}}$ and $\overline{Q_{45}}$ are zero. So, the corresponding stiffness coefficients will be vanished
### 2.2.2. Wave Function Solutions {#sec2dot2dot2-materials-12-03808}
The theoretical equations of the composite shell based on FSDT are given as follows \[[@B13-materials-12-03808]\]:$$\begin{matrix}
{\frac{\partial N_{xx}}{\partial x} + \frac{\partial N_{xy}}{\partial y} + \frac{Q_{x}}{R_{x}} = I_{0}\frac{\partial^{2}u_{0}}{\partial t^{2}} + I_{1}\frac{\partial^{2}\phi_{x}}{\partial t^{2}}} \\
{\frac{\partial N_{yy}}{\partial y} + \frac{\partial N_{xy}}{\partial x} + \frac{Q_{y}}{R_{y}} = I_{0}\frac{\partial^{2}v_{0}}{\partial t^{2}} + I_{1}\frac{\partial^{2}\phi_{y}}{\partial t^{2}}} \\
{\frac{N_{xx}}{R_{x}} + \frac{N_{yy}}{R_{y}} - \frac{\partial Q_{x}}{\partial x} - \frac{\partial Q_{y}}{\partial y} = - I_{0}\frac{\partial^{2}w_{0}}{\partial t^{2}}} \\
{\frac{\partial M_{xx}}{\partial x} + \frac{\partial M_{xy}}{\partial y} - Q_{x} = I_{1}\frac{\partial^{2}u_{0}}{\partial t^{2}} + I_{2}\frac{\partial^{2}\phi_{x}}{\partial t^{2}}} \\
{\frac{\partial M_{yy}}{\partial y} + \frac{\partial M_{xy}}{\partial x} - Q_{y} = I_{1}\frac{\partial^{2}v_{0}}{\partial t^{2}} + I_{2}\frac{\partial^{2}\phi_{y}}{\partial t^{2}}} \\
\end{matrix}$$ where *I~i~* (*i* = 0, 1, 2) are the inertia mass moments. Submitting Equations (3) and (6) into Equation (8), the force vector and moment resultants can be transformed as displacement variables. Furthermore, the theoretical equations are follows:$$\left\lbrack \begin{array}{lllll}
T_{11} & T_{12} & T_{13} & T_{14} & T_{15} \\
T_{21} & T_{22} & T_{23} & T_{24} & T_{25} \\
T_{31} & T_{32} & T_{33} & T_{34} & T_{35} \\
T_{41} & T_{42} & T_{43} & T_{44} & T_{45} \\
T_{51} & T_{52} & T_{53} & T_{54} & T_{55} \\
\end{array} \right\rbrack\begin{Bmatrix}
u_{0} \\
v_{0} \\
w_{0} \\
\phi_{x} \\
\phi_{y} \\
\end{Bmatrix} = \begin{Bmatrix}
0 \\
\end{Bmatrix}$$ where *T~ij~* (*i*,*j* = 1, 2, 3, 4, 5) are the operators of the matrix **T** in Equation (9), and are shown as follows:$$\begin{array}{l}
{T_{11} = A_{11\,}\frac{\partial^{2}}{\partial x^{2}} + A_{66}\,\frac{\partial^{2}}{\partial y^{2}} - \frac{K_{c}\,\,A_{55}}{R_{x}{}^{2}} - I_{0}\,\frac{\partial^{2}}{\partial t^{2}},T_{12} = \left( {A_{66} + A_{12}} \right)\frac{\partial^{2}}{\partial y\partial x},T_{13} = \left( {\frac{A_{12}}{R_{y}}\, + \frac{A_{55}\,K_{c} + A_{1}{}_{1}}{R_{x}}\,} \right)\frac{\partial}{\partial x}} \\
{T_{14} = B_{11\,}\frac{\partial^{2}}{\partial x^{2}} + B_{66}\,\,\frac{\partial^{2}}{\partial y^{2}} + \,\frac{K_{c}\,\,A_{5}5}{Rx} - I_{1\,}\frac{\partial^{2}}{\partial t^{2}},T_{15} = \left( {B_{12} + B_{6}{}_{6}} \right)\frac{\partial^{2}}{\partial y\partial x}} \\
{T_{21} = T_{12},T_{22} = A_{66}\,\frac{\partial^{2}}{\partial x^{2}} + A_{22}\,\frac{\partial^{2}}{\partial y^{2}} - \frac{K_{c}\,A_{4}4}{R_{y}{}^{2}} - I_{0\,}\frac{\partial^{2}}{\partial t^{2}},T_{23} = \left( {\frac{A_{44}\,K_{c} + A_{22}}{\,R_{y}} + \frac{A_{12}}{R_{x}\,}\,} \right)\frac{\partial}{\partial y}} \\
{T_{24} = \left( {B_{12} + B_{66}} \right)\frac{\partial^{2}}{\partial y\partial x},T_{25} = B_{66}\,\frac{\partial^{2}}{\partial x^{2}} + B_{2}{}_{2}\,\frac{\partial^{2}}{\partial y^{2}} + \,\frac{K_{c}\,A_{44}}{Ry} - I_{1}\,\frac{\partial^{2}}{\partial t^{2}}} \\
{T_{31} = T_{13},T_{23} = T_{32},T_{33} = - A_{55\,}K_{c}\frac{\partial^{2}}{\partial x^{2}} - A_{44}\,K_{c}\frac{\partial^{2}}{\partial y^{2}} + \left( {\,\frac{A_{11}}{R_{x}{}^{2}} + \frac{2\,A_{1}{}_{2}\,}{R_{x}R_{y}} + \frac{A_{22}}{R_{y}{}^{2}}\,} \right) + I_{0}\,\frac{\partial^{2}}{\partial t^{2}}} \\
{T_{34} = \left( {- A_{55}\,K_{c}\, + \frac{B_{12}}{R_{y}} + \frac{B_{11}}{R_{x}}} \right)\frac{\partial}{\partial x},T_{35} = - A_{44}\,K_{c}\, + \frac{B_{22}}{R_{y}} + \frac{B_{12}}{R_{x}}\,} \\
{T_{41} = T_{14},T_{42} = T_{24},T_{43} = T_{34},T_{44} = D_{11}\,\frac{\partial^{2}}{\partial x^{2}} + D_{66}\,\,\frac{\partial^{2}}{\partial y^{2}} - \,K_{c\,}A_{55} - I_{2}\,\frac{\partial^{2}}{\partial t^{2}}} \\
{T_{51} = T_{15},T_{52} = T_{25},T_{53} = T_{35},T_{54} = T_{45},T_{55} = D_{66}\,\frac{\partial^{2}}{\partial x^{2}} + D_{22}\,\frac{\partial^{2}}{\partial y^{2}} - K_{c}A_{44} - I_{2}\frac{\partial^{2}}{\partial t^{2}}} \\
\end{array}$$
For certain cross-ply composite laminated shallow shells under shear diaphragm boundary conditions, which are set at the opposite supports *y* = 0 and *Ly* (*u*~0~ = *w*~0~ = *ϕ~x~* = *N~yy~* = *M~yy~* = 0), the generalized displacement variables are transformed in the wave function form as follows:$$\begin{Bmatrix}
{u\left( {x,y,t} \right)} \\
{v\left( {x,y,t} \right)} \\
{w\left( {x,y,t} \right)} \\
{\phi_{x}\left( {x,y,t} \right)} \\
{\phi_{y}\left( {x,y,t} \right)} \\
\end{Bmatrix} = {\sum\limits_{n = 0}^{\infty}\begin{Bmatrix}
{U_{0}e^{ik_{n}x}\sin(K_{y}y)e^{- j\omega t}} \\
{V_{0}e^{ik_{n}x}\cos(K_{y}y)e^{- j\omega t}} \\
{W_{0}e^{ik_{n}x}\sin(K_{y}y)e^{- j\omega t}} \\
{\Phi_{x}e^{ik_{n}x}\sin(K_{y}y)e^{- j\omega t}} \\
{\Phi_{y}e^{ik_{n}x}\cos(K_{y}y)e^{- j\omega t}} \\
\end{Bmatrix}}$$ where *K~y~* = *n*π/*L~y~* is the *y* direction modal wave number and *k~n~* is the wave number in the x direction. *U*~0~, *V*~0~, *W*~0~, *Φ~x~*, and *Φ~y~* are the corresponding displacement amplitude variables of the *n*th mode for the composite laminated shallow shells.
Submitting the wave function solutions of the displacement variables into Equation (9), the governing equation can be obtained as follows:$$\left\lbrack \begin{array}{lllll}
L_{11} & L_{12} & L_{13} & L_{14} & L_{15} \\
L_{21} & L_{22} & L_{23} & L_{24} & L_{25} \\
L_{31} & L_{32} & L_{33} & L_{34} & L_{35} \\
L_{41} & L_{42} & L_{43} & L_{44} & L_{45} \\
L_{51} & L_{52} & L_{53} & L_{54} & L_{55} \\
\end{array} \right\rbrack\begin{Bmatrix}
U_{0} \\
V_{0} \\
W_{0} \\
\Phi_{x} \\
\Phi_{x} \\
\end{Bmatrix}$$ where *L~ij~* (*i*,*j* = 1, 2, 3, 4, 5) are the governing equation coefficients of Equation (12), given as follows:$$\begin{array}{l}
{L_{11} = - k_{n}{}^{2}A_{11} - \,\frac{A_{55}\,K_{c}}{R_{x}{}^{2}} - K_{y}{}^{2}A_{66} + I_{0}\omega^{2},L_{12} = - ik_{n}K_{y}\left( {A_{12}\, + A_{66}} \right),L_{13} = ik_{n}\,\left( {\frac{A_{11}}{R_{x}} + \frac{A_{12}}{R_{y}} + \frac{K_{c}A_{55}\,\,}{R_{x}}} \right)} \\
{L_{14} = \frac{A_{55}\,K_{c}}{R_{x}} - k_{n}{}^{2}B_{11} - K_{y}{}^{2}B_{6}{}_{6}\, + I_{1}\omega^{2},L_{15} = - ik_{n}K_{y}\left( {B_{12} + B_{66}} \right)} \\
{L_{21} = - L_{12},L_{22} = - K_{y}{}^{2}A_{22} - \,\frac{K_{c}A_{44}\,}{R_{y}{}^{2}} - k_{n}{}^{2}A_{66} + I_{0}\,\omega^{2},L_{23} = K_{y}\left( {\frac{A_{22} + A_{44}\,K_{c}\,}{\,R_{y}} + \frac{A_{12}}{R_{x}\,}} \right)} \\
{L_{24} = ik_{n}K_{y}\left( {B_{12} + B_{66}\,} \right),L_{25} = - K_{y}{}^{2}B_{22} - k_{n}{}^{2}B_{66} + \,\frac{K_{c}A_{44}\,}{\,R_{y}}\, + I_{1}\,\omega^{2}} \\
{L_{31} = L_{13},L_{32} = - L_{23},L_{33} = \frac{A_{11}\,}{R_{x}{}^{2}} + \,\frac{2\,A_{12}}{R_{x}R_{y}} + \frac{A_{22}}{R_{y}{}^{2}} + K_{c}\,k_{n}{}^{2}A_{55} + K_{c}K_{y}{}^{2}A_{44} - I_{0\,}\omega^{2}} \\
{L_{34} = i\,k_{n}\left( {\frac{B_{12}\,}{R_{y}} + \frac{B_{11}\,}{R_{x}} - K_{c\,}A_{55}\,} \right),L_{35} = K_{y}\left( {K_{c}\,A_{44} - \frac{\,B_{12}\,}{R_{x}} - \frac{B_{22}\,}{R_{y}}} \right)} \\
{L_{41} = - L_{14},L_{42} = L_{24},L_{43} = - L_{34},L_{44} = k_{n}{}^{2}D_{11} + K_{y}{}^{2}D_{66} + K_{c}A_{55} - I_{2}\omega^{2}} \\
{L_{51} = L_{15},L_{52} = - L_{25},L_{53} = L_{35},L_{54} = - L_{45},L_{55} = K_{y}{}^{2}D_{22} + \,k_{n}{}^{2}D_{66} + K_{c}A_{44} - I_{2}\,\omega^{2}} \\
\end{array}$$
The solutions of Equation (12) can be solved and the determinant of the matrix **T** equal to zero. The characteristics equation of axial wavenumber *k~n~* is shown as follows:$$\lambda_{10}k_{n}^{10} + \lambda_{8}k_{n}^{8} + \lambda_{6}k_{n}^{6} + \lambda_{4}k_{n}^{4} + \lambda_{2}k_{n}^{2} + \lambda_{0} = 0$$
There is a fifth-order equation of $k_{n}^{2}$ and *λ*~10~, *λ*~8~, *λ*~6~, *λ*~4~, *λ*~2~, and *λ*~0~ are the coefficients, which depend on the coefficient matrix **T**. There are ten characteristic axial wavenumbers to be obtained as ±*k~n~*~,1~, ±*k~n~*~,2~, ±*k~n~*~,3~, ±*k~n~*~,4~, and ±*k~n~*~,5~. Through the characteristic axial wavenumbers ±*k~n~*~,*i*~ (*i* = 1, 2, 3, 4, 5), the corresponding basic solution vector is defined as follows:$$\left\{ {\xi_{n,i},\eta_{n,i},1,\chi_{n,i},\psi_{n,i}} \right\}$$
The coefficients in Equation (15) are defined as follows:$$\xi_{n,i} = \left\lbrack \frac{\mathsf{\Omega}_{1}}{\mathsf{\Omega}} \right\rbrack_{k_{n} = \pm k_{n,i}},\eta_{n,i} = \left\lbrack \frac{\mathsf{\Omega}_{2}}{\mathsf{\Omega}} \right\rbrack_{k_{n} = \pm k_{n,i}},\chi_{n,i} = \left\lbrack \frac{\mathsf{\Omega}_{4}}{\mathsf{\Omega}} \right\rbrack_{k_{n} = \pm k_{n,i}},\psi_{n,i} = \left\lbrack \frac{\mathsf{\Omega}_{5}}{\mathsf{\Omega}} \right\rbrack_{k_{n} = \pm k_{n,i}}$$ where Ω and Ω*~i~* (*i* = 1, 2, 4, 5) are given as follows:$$\begin{array}{lll}
{\mathsf{\Omega} = \left| \begin{array}{llll}
L_{11} & L_{12} & L_{14} & L_{15} \\
L_{21} & L_{22} & L_{24} & L_{25} \\
L_{41} & L_{42} & L_{44} & L_{45} \\
L_{51} & L_{52} & L_{54} & L_{55} \\
\end{array} \right|_{k_{n} = \pm k_{n,i}}} & {\mathsf{\Omega}_{1} = \left| \begin{array}{llll}
{- L_{13}} & L_{12} & L_{14} & L_{15} \\
{- L_{23}} & L_{22} & L_{24} & L_{25} \\
{- L_{43}} & L_{42} & L_{44} & L_{45} \\
{- L_{53}} & L_{52} & L_{54} & L_{55} \\
\end{array} \right|_{k_{n} = \pm k_{n,i}}} & {\mathsf{\Omega}_{2} = \left| \begin{array}{llll}
L_{11} & {- L_{13}} & L_{14} & L_{15} \\
L_{21} & {- L_{23}} & L_{24} & L_{25} \\
L_{41} & {- L_{43}} & L_{44} & L_{45} \\
L_{51} & {- L_{53}} & L_{54} & L_{55} \\
\end{array} \right|_{k_{n} = \pm k_{n,i}}} \\
{\mathsf{\Omega}_{4} = \left| \begin{array}{llll}
L_{11} & L_{12} & {- L_{13}} & L_{15} \\
L_{21} & L_{22} & {- L_{23}} & L_{25} \\
L_{41} & L_{42} & {- L_{43}} & L_{45} \\
L_{51} & L_{52} & {- L_{53}} & L_{55} \\
\end{array} \right|_{k_{n} = \pm k_{n,i}}} & {\mathsf{\Omega}_{5} = \left| \begin{array}{llll}
L_{11} & L_{12} & L_{14} & {- L_{13}} \\
L_{21} & L_{22} & L_{24} & {- L_{23}} \\
L_{41} & L_{42} & L_{44} & {- L_{43}} \\
L_{51} & L_{52} & L_{54} & {- L_{53}} \\
\end{array} \right|_{k_{n} = \pm k_{n,i}}} & \\
\end{array}$$
On the basis of the generalized displacement variables being transformed in the wave function form in Equation (11), related to the basic solution vector in Equation (15), the generalized displacement variables are shown in the matrix form:$$\mathbf{\delta}_{n} = \mathbf{Y}_{n}\left( y \right)\mathbf{D}_{n}\mathbf{P}_{n}\left( x \right)\mathbf{W}_{n}$$ where **δ***~n~* = {*u*, *v*, *w*, *ϕ~x~*, *ϕ~y~*}*^T^* is the generalized displacement resultant; **Y***~n~*(*y*) is the modal matrix in the y direction; **D***~n~* is the coefficient matrix of the displacement resultant; **P***~n~*(*x*) is the axial wavenumber matrix; and **W***~n~* is the wave contribution factor resultant. The detailed expression of them is given as follows:$$\mathbf{Y}_{n}\left( y \right) = diag\left\{ {\sin\left( {K_{y}y} \right),\cos\left( {K_{y}y} \right),\sin\left( {K_{y}y} \right),\sin\left( {K_{y}y} \right),\cos\left( {K_{y}y} \right)} \right\}$$ $$\mathbf{D}_{n} = \begin{bmatrix}
\xi_{n,1} & \xi_{n,2} & \cdots & \xi_{n,ns - 1} & \xi_{n,ns} \\
\eta_{n,i} & \eta_{n,i} & \cdots & \eta_{n,ns - 1} & \eta_{n,ns} \\
1 & 1 & \cdots & 1 & 1 \\
\chi_{n,1} & \chi_{n,2} & \cdots & \chi_{n,ns} & \chi_{n,ns - 1} \\
\psi_{n,1} & \psi_{n,2} & \cdots & \psi_{n,ns} & \psi_{n,ns - 1} \\
\end{bmatrix}$$ $$\mathbf{P}_{n}\left( x \right) = diag\left\{ {e^{ik_{n,1}x},e^{ik_{n,2}x},\cdots,e^{ik_{n,ns - 1}x},e^{ik_{n,ns}x}} \right\}$$ $$\mathbf{W}_{n} = \left\{ {W_{n,1},W_{n,2},\cdots,W_{n,ns - 1},W_{n,ns}} \right\}^{T}$$ where *ns* is the number of characteristics roots of axial wavenumber in Equation (14). Also, the generalized force resultant **f***~n~* = {*N~xx~*, *N~xy~*, *Q~x~*, *M~xx~*, *M~xy~*}*^T^* can refer to the constitutive relationship in Equations (3) and (6), as follows:$$\mathbf{f}_{n} = \mathbf{Y}_{n}\left( y \right)\mathbf{F}_{n}\mathbf{P}_{n}\left( x \right)\mathbf{W}_{n}$$ where the coefficient matrix **F***~n~* of force resultant **f***~n~* is given as follows:$$\begin{array}{l}
{F_{n,1i} = ik_{n,i}A_{11}\,\xi_{n,i} - K_{y}A_{12}\eta_{n,i} + \frac{A_{11}\,}{R_{x}} + \frac{A_{12\,}}{\,R_{y}} + ik_{n,i}B_{11}\,\chi_{n,i}\, - K_{y}B_{12}\,\psi_{n,i}} \\
{F_{n,2i} = K_{y}A_{66}\xi_{n,i} + ik_{n,i}A_{66}\,\eta_{n,i} + K_{y}B_{66}\chi_{n,i} + i\,k_{n,i}B_{66}\,\psi_{n,i}} \\
{F_{n,3i} = K_{c}\,A_{55}\left( {ik_{n,i} + \,\chi_{n,i} - \frac{\xi_{n,i}}{R_{x}}} \right)} \\
{F_{n,4i} = ik_{n,i}B_{11}\,\,\xi_{n,i} - K_{y}B_{12}\,\eta_{n.i}\, + \frac{B_{11\,}}{R_{x}\,} + \frac{B_{12}\,}{R_{y}} + ik_{n,i}D_{11}\,\chi_{n,i}\, - K_{y}D_{12\,}\psi_{n,i}} \\
{F_{n,5i} = K_{y}B_{66}\,\xi_{n,i} + ik_{n,i}B66\eta_{n,i} + K_{y}D_{66}\,\chi_{n,i} + i\,k_{n,i}D_{66}\,\psi_{n,i}\,} \\
\end{array}$$
2.3. Classical Shell Theory (CST) {#sec2dot3-materials-12-03808}
---------------------------------
### 2.3.1. Kinematic Relations and Stress Resultants {#sec2dot3dot1-materials-12-03808}
For the integrity of the paper, the governing equations and wave function solutions in the CST are given. On the basis of the theoretical technique of FSDT, the governing equation can refer to CST by setting the slope of the rotation components *ϕ~x~* and *ϕ~y~* close to the transverse normal, as follows \[[@B12-materials-12-03808],[@B13-materials-12-03808]\]:$$\phi_{x} = \frac{u_{0}}{R_{x}} - \frac{\partial w_{0}}{\partial x},\phi_{y} = \frac{v_{0}}{R_{y}} - \frac{\partial w_{0}}{\partial y}$$
### 2.3.2. Wave Function Solutions {#sec2dot3dot2-materials-12-03808}
In CST, the shear deformation in the kinematics equation is negligible, and the in-plane displacement can be expressed as a linear change in the thickness direction of the shallow shell. So, the governing equation can be given as follows \[[@B13-materials-12-03808]\]:$$\begin{matrix}
{\frac{\partial N_{xx}}{\partial x} + \frac{\partial N_{xy}}{\partial y} + \frac{Q_{x}}{R_{x}} = I_{0}\frac{\partial^{2}u_{0}}{\partial t^{2}}} \\
{\frac{\partial N_{yy}}{\partial y} + \frac{\partial N_{xy}}{\partial x} + \frac{Q_{y}}{R_{y}} = I_{0}\frac{\partial^{2}v_{0}}{\partial t^{2}}} \\
{\frac{N_{xx}}{R_{x}} + \frac{N_{yy}}{R_{y}} - \left( {\frac{\partial Q_{x}}{\partial x} + \frac{\partial Q_{y}}{\partial y}} \right) = - I_{0}\frac{\partial^{2}w_{0}}{\partial t^{2}}} \\
\end{matrix}$$ where $$\begin{matrix}
{Q_{x} = \frac{\partial M_{xx}}{\partial x} + \frac{\partial M_{xy}}{\partial y}} \\
{Q_{y} = \frac{\partial M_{yy}}{\partial y} + \frac{\partial M_{xy}}{\partial x}} \\
\end{matrix}$$
Submitting the generalized displacement variables in CST, the governing equation can be expressed as follows:$$\begin{bmatrix}
{\widetilde{T}}_{11} & {\widetilde{T}}_{12} & {\widetilde{T}}_{13} \\
{\widetilde{T}}_{21} & {\widetilde{T}}_{22} & {\widetilde{T}}_{23} \\
{\widetilde{T}}_{31} & {\widetilde{T}}_{32} & {\widetilde{T}}_{33} \\
\end{bmatrix}\begin{Bmatrix}
u_{0} \\
v_{0} \\
w_{0} \\
\end{Bmatrix}$$ where ${\widetilde{T}}_{ij}$ (*i*,*j* = 1, 2, 3) are the operators, which are shown as follows:$$\begin{array}{l}
{{\widetilde{T}}_{11} = \left( {A_{11} + \frac{2B_{11}}{R_{x}} + \frac{D_{11}}{R_{x}{}^{2}}} \right)\frac{\partial^{2}}{\partial x^{2}} + \left( {A_{6}{}_{6} + \frac{2B_{66}\,\,}{R_{x}} + \frac{\,D_{66}}{R_{x}{}^{2}}} \right)\,\frac{\partial^{2}}{\partial y^{2}} - I_{0}\,\frac{\partial^{2}}{\partial t^{2}}} \\
{{\widetilde{T}}_{12} = \left( {A_{12} + A_{66} + \frac{B_{12} + B_{66}}{R_{y}} + \frac{B_{12} + B_{66}}{R_{x}} + \frac{D_{12} + D_{66}}{R_{x}R_{y}}} \right)\frac{\partial^{2}}{\partial y\partial x}} \\
{{\widetilde{T}}_{13} = - \left( {B_{11} + \frac{D_{11}}{R_{x}}} \right)\frac{\partial^{3}}{\partial x^{3}} + \left( {\frac{A_{12}\,}{R_{y}} + \,\frac{A_{11}}{R_{x}} + \frac{B_{12}}{R_{x}R_{y}} + \frac{B_{1}{}_{1}}{R_{x}{}^{2}}\,} \right)\frac{\partial}{\partial x} - \,\left( {B_{12} + 2\,B_{66} + \frac{D_{12} + 2\,D_{66}}{R_{x}}} \right)\frac{\partial^{3}}{\partial y^{2}\partial x}} \\
{{\widetilde{T}}_{21} = {\widetilde{T}}_{12},{\widetilde{T}}_{22} = \left( {A_{66} + \frac{2B_{66}\,}{\,R_{y}}\, + \,\frac{D_{66}}{\,R_{y}{}^{2}}} \right)\frac{\partial^{2}}{\partial x^{2}} + \,\left( {A_{22} + \frac{2B_{22}\,}{\,R_{y}} + \frac{D_{22}}{\,R_{y}{}^{2}}\,} \right)\,\frac{\partial^{2}}{\partial y^{2}} - I_{0}\,\frac{\partial^{2}}{\partial t^{2}}} \\
{{\widetilde{T}}_{23} = - \,\left( {B_{22\,} + \frac{D_{22}}{R_{y}}} \right)\frac{\partial^{3}}{\partial y^{3}} - \,\,\left( {2B_{66} + \,B_{12} + \frac{D_{12} + 2D_{66}}{R_{y}}} \right)\frac{\partial^{3}}{\partial y\partial x^{2}} + \left( {\,\frac{A_{12}}{R_{x}\,} + \,\frac{A_{22}}{R_{y}} + \frac{B_{12}}{R_{x}\,R_{y}} + \frac{B_{22}}{\,R_{y}{}^{2}}\,} \right)\frac{\partial}{\partial y}} \\
{{\widetilde{T}}_{31} = {\widetilde{T}}_{13},{\widetilde{T}}_{32} = {\widetilde{T}}_{23}} \\
\begin{array}{l}
{{\widetilde{T}}_{33} = D_{11}\,\frac{\partial^{4}}{\partial x^{4}} + D_{22}\frac{\partial^{4}}{\partial y^{4}} + 2\left( {D_{12} + 2\,D_{66}} \right)\frac{\partial^{4}}{\partial y^{2}\partial x^{2}} + \left( {- \,\frac{2\,B_{11}}{R_{x}} - \frac{2\,B_{12}\,}{R_{y}}} \right)\frac{\partial^{2}}{\partial x^{2}}} \\
{+ \left( {- \,\frac{2\,B_{12}}{R_{x}} - \frac{2\,B_{22}}{R_{y}}\,} \right)\frac{\partial^{2}}{\partial y^{2}} + \left( {\frac{A_{11}}{R_{x}{}^{2}} + \frac{2\,A_{12}}{R_{x\,}R_{y}}\, + \frac{A_{22}}{R_{y}{}^{2}}\,} \right) + I_{0}\,\frac{\partial^{2}}{\partial t^{2}}} \\
\end{array} \\
\end{array}$$
For the generalized displacement functions of cross-ply composite laminated shallow shell with shear diaphragm boundary conditions, which are set as opposite support edges *y* = 0 and *Ly* (*u*~0~ = *w*~0~ = *N~yy~* = *M~yy~* = 0), the displacement variables can be shown in the wave functions form:$$\begin{Bmatrix}
\end{Bmatrix} = {\sum\limits_{n = 0}^{\infty}\begin{Bmatrix}
{U_{0}e^{ik_{n}x}\sin\left( {K_{y}y} \right)e^{- j\omega t}} \\
{V_{0}e^{ik_{n}x}\cos\left( {K_{y}y} \right)e^{- j\omega t}} \\
{W_{0}e^{ik_{n}x}\sin\left( {K_{y}y} \right)e^{- j\omega t}} \\
{\Phi_{x}e^{ik_{n}x}\sin\left( {K_{y}y} \right)e^{- j\omega t}} \\
\end{Bmatrix}}$$
Submitting Equation (30) into Equation (28), the governing equation can transform into the matrix form as follows:$$\begin{bmatrix}
{\widetilde{L}}_{11} & {\widetilde{L}}_{12} & {\widetilde{L}}_{13} \\
{\widetilde{L}}_{21} & {\widetilde{L}}_{22} & {\widetilde{L}}_{23} \\
{\widetilde{L}}_{31} & {\widetilde{L}}_{32} & {\widetilde{L}}_{33} \\
\end{bmatrix}\begin{Bmatrix}
U_{0} \\
V_{0} \\
W_{0} \\
\end{Bmatrix}$$ where ${\widetilde{L}}_{ij}$ (*i*,*j* = 1, 2, 3) are the governing equation coefficients, as follows:$$\begin{array}{l}
{{\widetilde{L}}_{11} = - \,k_{n}{}^{2}\left( {A_{11} + \,\frac{2\,B_{11}}{R_{x}} + \,\frac{D_{11}}{R_{x}{}^{2}}} \right) - K_{y}{}^{2}\left( {A_{66} + \,\frac{2\,B_{66}}{R_{x}} + \frac{D_{66}}{R_{x}{}^{2}}} \right) + I_{0}\omega^{2}} \\
{{\widetilde{L}}_{12} = - ik_{n}K_{y}\left( {A_{12} + A_{66}\, + \frac{B_{66} + B_{12}\,}{R_{y}}\, + \frac{B_{66} + B_{12}}{R_{x}} + \frac{D_{12} + D_{66}}{R_{x}R_{y}}\,} \right)} \\
{{\widetilde{L}}_{13} = ik_{n}{}^{3}\left( {B_{11} + \frac{D_{11}\,}{R_{x}}} \right) + ik_{n}\left( {\frac{A_{11}\,}{R_{x}} + \,\frac{A_{12}}{R_{y}} + \frac{B_{11}}{R_{x}{}^{2}} + \frac{B_{12}}{R_{x}R_{y}}} \right) + ik_{n}K_{y}{}^{2}\left( {B_{12} + 2B_{66} + \,\frac{D_{12} + 2\,D_{66}}{R_{x}}} \right)} \\
{{\widetilde{L}}_{21} = - {\widetilde{L}}_{12},{\widetilde{L}}_{22} = - k_{n}{}^{2}\left( {A_{66} + \frac{2\,B_{66}}{R_{y}} + \,\frac{D_{66}}{R_{y}{}^{2}}\,} \right)\,\, - K_{y}{}^{2}\left( {A_{22} + \frac{2\,B_{22}}{R_{y}} + \frac{D_{22}\,}{R_{y}{}^{2}}} \right) + I_{0\,}\omega^{2}\,} \\
{{\widetilde{L}}_{23} = \,K_{y}{}^{3}\left( {B_{22} + \frac{D_{22}}{Ry}} \right)\, + k_{n}{}^{2}K_{y}\left( {2\,B_{66} + B_{12\,} + \,\frac{D_{12} + 2\,D_{66}}{\,R_{y}}\,} \right) + K_{y}\left( {\frac{A_{12}}{R_{x\,}} + \frac{A_{22}\,}{\,R_{y}} + \,\frac{B_{12}}{R_{x\,}R_{y}} + \,\frac{B_{22}}{R_{y}{}^{2}}\,} \right)} \\
\begin{array}{l}
{{\widetilde{L}}_{31} = {\widetilde{L}}_{13},{\widetilde{L}}_{32} = - {\widetilde{L}}_{23},{\widetilde{L}}_{33} = k_{n}{}^{4}D_{11} + K_{y}{}^{4}D_{22} + 2\,k_{n}{}^{2}K_{y}{}^{2}\left( {D_{12} + 2D_{66}} \right) + 2k_{n}{}^{2}\left( {\frac{\,B_{11}}{R_{x}} + \,\frac{B_{12}}{R_{y}}} \right)} \\
{+ 2\,K_{y}{}^{2}\left( {\frac{B_{12}}{R_{x}} + \,\frac{\,B_{22}}{R_{y}}} \right) + \left( {\frac{A_{11}}{R_{x}{}^{2}} + \,\frac{A_{22}}{R_{y}{}^{2}} + \,\frac{2\,A_{12}}{R_{x}R_{y}}} \right)\, - I_{0}\omega^{2}\,} \\
\end{array} \\
\end{array}$$
Next, the corresponding basic solution vector is set as {*ξ~n,i~, η~n,i~*, 1}*^T^*, and the detailed expression of the vector is given as follows:$$\xi_{n,i} = \left\lbrack \frac{\mathsf{\Omega}_{1}}{\mathsf{\Omega}} \right\rbrack_{k_{n} = \pm k_{n,i}},\eta_{n,i} = \left\lbrack \frac{\mathsf{\Omega}_{2}}{\mathsf{\Omega}} \right\rbrack_{k_{n} = \pm k_{n,i}}$$ where $$\mathsf{\Omega} = \left| \begin{matrix}
N_{11} & N_{12} \\
N_{21} & N_{22} \\
\end{matrix} \right|_{k_{n} = \pm k_{n,i}},\mathsf{\Omega}_{1} = \left| \begin{matrix}
{- N_{13}} & N_{12} \\
{- N_{23}} & N_{22} \\
\end{matrix} \right|_{k_{n} = \pm k_{n,i}},\mathsf{\Omega}_{2} = \left| \begin{matrix}
N_{11} & {- N_{13}} \\
N_{21} & {- N_{23}} \\
\end{matrix} \right|_{k_{n} = \pm k_{n,i}}$$
On the basis of the basic solution vector, the displacement resultant **δ**~n~ = {*u*, *v*, *w*, *ϕx*}*^T^* and force resultant **f***~n~* = {*N~xx~*,*N~xy~* + *M~xy~*/*R~y~*,*Q~x~* + ∂*M~xy~*~/~∂*~y~*,*M~xx~*}*^T^* are expressed as follows:$$\begin{array}{l}
{\mathbf{\delta}_{n} = \mathbf{Y}_{n}\left( y \right)\mathbf{D}_{n}\mathbf{P}_{n}\left( x \right)\mathbf{W}_{n}} \\
{\mathbf{f}_{n} = \mathbf{Y}_{n}\left( y \right)\mathbf{F}_{n}\mathbf{P}_{n}\left( x \right)\mathbf{W}_{n}} \\
\end{array}$$ where $$\mathbf{Y}_{n}\left( y \right) = diag\left\{ {\sin\left( {K_{y}y} \right),\cos\left( {K_{y}y} \right),\sin\left( {K_{y}y} \right),\sin\left( {K_{y}y} \right)} \right\}$$ $$\mathbf{D}_{n} = \begin{bmatrix}
\xi_{n,1} & \xi_{n,2} & \cdots & \xi_{n,ns - 1} & \xi_{n,ns} \\
\eta_{n,1} & \eta_{n,2} & \cdots & \eta_{n,ns - 1} & \eta_{n,ns} \\
1 & 1 & \cdots & 1 & 1 \\
{\frac{\xi_{n,1}}{R_{x}} - i\,k_{n,1}} & {\frac{\xi_{n,2}}{R_{x}} - i\,k_{n,2}} & \cdots & {\frac{\xi_{n,ns - 1}}{R_{x}} - i\,k_{n,ns - 1}} & {\frac{\xi_{n,ns}}{R_{x}} - i\,k_{n,ns}} \\
\end{bmatrix}$$ $$\mathbf{F}_{n} = \begin{bmatrix}
F_{n,11} & F_{n,12} & \cdots & F_{n,1ns - 1} & F_{n,1ns} \\
F_{n,21} & F_{n,22} & \cdots & F_{n,2ns - 1} & F_{n,2ns} \\
F_{n,31} & F_{n,32} & \cdots & F_{n,3ns - 1} & F_{n,3ns} \\
F_{n,41} & F_{n,42} & \cdots & F_{n,4ns - 1} & F_{n,4ns} \\
\end{bmatrix}$$ in which the coefficients *F~n~*~,*ji*~(*j* = 1--4,*i* = 1--*ns*) are given as follows:$$\begin{array}{l}
{F_{n,1i} = ik_{n,i}\left( {A_{11} + \,\frac{B_{11}}{R_{x}\,}\,} \right)\,\xi_{n,i} + K_{y}\left( {- A_{12} - \frac{B_{12}}{R_{y}}} \right)\eta_{n,i} + \frac{A_{11}}{R_{x}\,} + \frac{A_{12}}{\,R_{y}} + k_{n,i}{}^{2}B_{11} + K_{y}{}^{2}B_{12}} \\
{F_{n,2i} = K_{y}\left( {A_{66} + \,\frac{B_{66}}{R_{x}\,} + \,\frac{B_{66}}{\,R_{y}} + \frac{D_{66}}{R_{x}\,R_{y}}} \right)\,\xi_{n,i}\, + ik_{n,i}\left( {A_{66} + \frac{2B_{66}}{R_{y}} + \frac{D_{66}}{R_{y}{}^{2}}} \right)\,\eta_{n,i} + 2ik_{n,i}K_{y}\left( {- B_{66} - \frac{D_{66}}{R_{y}}} \right)\,} \\
\begin{array}{l}
{F_{n,3i} = - \left( {k_{n,i}{}^{2}\left( {B_{11} + \frac{D_{11}}{R_{x}\,}\,} \right)\, + 2K_{y}{}^{2}\left( {B_{66} + \,\frac{D_{66}}{R_{x}\,}} \right)} \right)\xi_{n,i}\, - ik_{n,i}K_{y}\left( {B_{12} + 2\,B_{66} + \frac{D_{12} + 2\,D_{66}\,}{R_{y}}} \right)\,\eta_{n,i}} \\
{+ ik_{n,i}{}^{3}D_{11} + ik_{n,i}K_{y}{}^{2}\left( {D_{12} + 4D_{66}} \right) + ik_{n,i}\left( {\frac{B_{11}}{R_{x}} + \frac{B_{12}}{R_{y}}} \right)\,} \\
\end{array} \\
{F_{n,4i} = ik_{n,i}\left( {B_{11}\, + \,\frac{D_{11}}{R_{x}\,}\,} \right)\xi_{n,i} - K_{y}\left( {B_{12} + \frac{D_{12}\,}{R_{y}}} \right)\eta_{n.i}\, + \,k_{n,i}{}^{2}D_{11}\,\, + K_{y}{}^{2}D_{12} + \frac{B_{11}}{R_{x}\,} + \,\frac{B_{12}}{\,R_{y}}} \\
\end{array}$$
2.4. Implementation of the WBM {#sec2dot4-materials-12-03808}
------------------------------
Through the introduction of the generalized displacement and force resultant, the final governing equations are assembled by the generalized displacement coefficient matrix, generalized force coefficient matrix, and boundary matrix. The final governing equation of the whole structure is defined as follows:$$\left\lbrack \mathbf{K} \right\rbrack\left\{ \mathbf{W} \right\} = \left\{ \mathbf{F} \right\}$$ where **F** is the external force vector and is related to the external situation; when analyzing the free vibration dynamic, the external force **F** should vanish. **W** = {**W**~1~, **W**~2~}*^T^* is the wave contribution factor resultant of the composite shell, and **W***~i~* = {*W~i~*~,1~, *W~i~*~,2~,..., *W~i~*~,*ns*~}*^T^*(*i* = 1, 2) is the wave contribution factor vector and is associated with the boundary conditions at *x* = 0 and *x* = *L*. **K** is the total matrix and the detailed expression of the matrix is shown as follows:$$\mathbf{K}_{2ns \times 2ns} = \begin{bmatrix}
{\mathbf{B}_{1}\left( 0 \right)} & \mathbf{0}_{\frac{1}{2}ns \times ns} \\
{\mathbf{D}_{n}\mathbf{P}_{n}\left( L \right)} & {- \mathbf{D}_{n}\mathbf{P}_{n}\left( 0 \right)} \\
{\mathbf{F}_{n}\mathbf{P}_{n}\left( L \right)} & {- \mathbf{F}_{n}\mathbf{P}_{n}\left( 0 \right)} \\
\mathbf{0}_{\frac{1}{2}ns \times ns} & {\mathbf{B}_{2}\left( 0 \right)} \\
\end{bmatrix}$$
For the classical boundary conditions, the boundary matrix can be shown as follows:$$\mathbf{B}_{1,2}\left( x \right) = \left( {\mathbf{T}_{\delta}\mathbf{D}_{n} + \mathbf{T}_{f}\mathbf{F}_{n}} \right)\mathbf{P}_{n}\left( x \right)$$ where **T***~δ~* and **T***~f~* are the transform matrix of boundary matrix, as follows:
Free edge (*F*):$$\begin{matrix}
{{FSDT}:\left\{ \begin{array}{l}
{\mathbf{T}_{\delta} = diag\left\{ {0,0,0,0,0} \right\}} \\
{\mathbf{T}_{f} = diag\left\{ {1,1,1,1,1} \right\}} \\
\end{array} \right.} \\
{{CST}:\left\{ \begin{array}{l}
{\mathbf{T}_{\delta} = diag\left\{ {0,0,0,0} \right\}} \\
{\mathbf{T}_{f} = diag\left\{ {1,1,1,1} \right\}} \\
\end{array} \right.} \\
\end{matrix}$$
Clamped edge (*C*):$$\begin{matrix}
{{FSDT}:\left\{ \begin{array}{l}
{\mathbf{T}_{\delta} = diag\left\{ {1,1,1,1,1} \right\}} \\
{\mathbf{T}_{f} = diag\left\{ {0,0,0,0,0} \right\}} \\
\end{array} \right.} \\
{{CST}:\left\{ \begin{array}{l}
{\mathbf{T}_{\delta} = diag\left\{ {1,1,1,1} \right\}} \\
{\mathbf{T}_{f} = diag\left\{ {0,0,0,0} \right\}} \\
\end{array} \right.} \\
\end{matrix}$$
Shear-diaphragm edge (*SD*):$$\begin{matrix}
{{FSDT}:\left\{ \begin{array}{l}
{\mathbf{T}_{\delta} = diag\left\{ {0,1,1,0,1} \right\}} \\
{\mathbf{T}_{f} = diag\left\{ {1,0,0,1,0} \right\}} \\
\end{array} \right.} \\
{{CST}:\left\{ \begin{array}{l}
{\mathbf{T}_{\delta} = diag\left\{ {0,1,1,0} \right\}} \\
{\mathbf{T}_{f} = diag\left\{ {1,0,0,1} \right\}} \\
\end{array} \right.} \\
\end{matrix}.$$
For the elastic boundary conditions, the boundary condition matrix **B**~1~(*x*) and **B**~2~(*x*) are given as follows:$$\mathbf{B}_{1,2}\left( x \right) = \left( {\mathbf{K}_{\delta}\mathbf{D}_{n} \pm \mathbf{F}_{n}} \right)\mathbf{P}_{n}\left( x \right),$$ where **K***~δ~* is the stiffness transform matrix and the detailed expression about it is as follows:
When the composite shell is under elastic restraint in the axial direction, the stiffness transform matrix **K***~δ~* is given as follows:$$\begin{matrix}
{{FSDT}:} \\
{{CST}:} \\
\end{matrix}\left\{ \begin{matrix}
{\mathbf{K}_{\delta} = diag\left\{ {K_{u},0,0,0,0} \right\}} \\
{\mathbf{K}_{\delta} = diag\left\{ {K_{u},0,0,0} \right\}} \\
\end{matrix} \right.$$ where {*K~u~*, *K~v~*, *K~w~* } are linear springs and {*K~ϕx~*, *K~ϕy~* } are rotational springs, which are set in various directions. When the other displacements are under elastic restraint, the stiffness transform matrix **K***~δ~* is given as follows:$$\begin{array}{ll}
{v:} & {\begin{array}{r}
{{FSDT}:} \\
{{CST}:} \\
\end{array}\left\{ \begin{array}{l}
{\mathbf{K}_{\delta} = diag\left\{ {0,K_{v},0,0,0} \right\}} \\
{\mathbf{K}_{\delta} = diag\left\{ {0,K_{v},0,0} \right\}} \\
\end{array} \right.} \\
{w:} & {\begin{array}{r}
{{FSDT}:} \\
{{CST}:} \\
\end{array}\left\{ \begin{array}{l}
{\mathbf{K}_{\delta} = diag\left\{ {0,0,K_{w},0,0} \right\}} \\
{\mathbf{K}_{\delta} = diag\left\{ {0,0,K_{w},0} \right\}} \\
\end{array} \right.} \\
{\phi_{x}:} & {\begin{array}{r}
{{FSDT}:} \\
{{CST}:} \\
\end{array}\left\{ \begin{array}{l}
{\mathbf{K}_{\delta} = diag\left\{ {0,0,0,K_{\phi x},0} \right\}} \\
{\mathbf{K}_{\delta} = diag\left\{ {0,0,0,K_{\phi x}} \right\}} \\
\end{array} \right.} \\
{\phi_{y}:} & {{{FSDT}:}\mathbf{K}_{\delta} = diag\left\{ {0,0,0,0,K_{\phi y}} \right\}} \\
\end{array}$$
Through the introduction of the boundary conditions **B**~1~(*x*) and **B**~2~(*x*), which include the classical and elastic boundary conditions, the total matrix **K** is established. When analyzing the free vibration characteristics, the external force vector F vanishes. When calculating the natural frequencies, a series of the total matrix determinant is obtained. Using the dichotomy method to search the zeros position of the total matrix determinant, the natural frequency will be obtained with each circumferential mode number *n*. Through the numerical dichotomy method when the sign changed, the location of the total matrix **K** determinant is calculated and the natural frequencies can be obtained. Furthermore, to analyze the free vibration characteristics of the composite laminated shallow shell with arbitrary boundary conditions, the shell structure is considered to be calculated as a whole model and the displacement variable solutions are set as infinite wave function forms; the convergence study of the truncated number does not need to be considered. Thus, the computational cost of the present approach is low.
3. Numerical Examples and Discussion {#sec3-materials-12-03808}
====================================
Through the description of the theory formulation with FSDT and CST, the free vibration characteristics of composite laminated shallow shell with arbitrary classical boundary conditions, elastic boundary conditions, and their combinations are analyzed by WBM. In this part, some numerical examples are listed to verify the correctness of the results by WBM through the comparison with the presented results. Also, some numerical examples are presented to study the influence of the material parameters and geometric constants on the natural frequencies of composite laminated shallow shells with general boundary conditions.
3.1. Composite Laminated Shallow Shell with Classical Boundary Conditions {#sec3dot1-materials-12-03808}
-------------------------------------------------------------------------
In this section, the free vibration characteristics of composite laminated shallow shells with arbitrary classical boundary conditions are concerned. Through the introduction of the boundary transform matrix **T***~δ~* and **T***~f~*, arbitrary classical boundary conditions can transform into boundary matrices **B**~1~(*x*) and **B**~2~(*x*) to investigate the free vibration characteristics of composite shallow shell with classical boundary conditions. In order to verify the correctness of the calculation by the presented method, some numerical examples are selected for verification. At the same time, the selected material parameters and geometric parameters are consistent with the examples in the comparative literatures.
First, the composite laminated shallow shell with full shear diagram boundary condition is concerned. In [Table 1](#materials-12-03808-t001){ref-type="table"} and [Table 2](#materials-12-03808-t002){ref-type="table"}, the fundamental frequency parameters $\mathsf{\Omega} = \omega L_{x}^{2}\sqrt{{\rho/E_{2}}h^{2}}$ for three type cross-ply composite laminated shallow shells (i.e., cylindrical shell, spherical shell, and hyperbolic paraboloidal shell) with various radius to length ratios *R~y~*/*L~y~* (i.e., *R~y~*/*L~y~* = 2, 5, 10) under Shear-diaphragm boundary condition (SD-SD) by FSDST and CST are presented. Three kinds of cross-ply type layered composite shells (i.e., \[0°/90°/90°/0°\], \[0°/90°\], and \[90°/0°\]) are concerned. The material parameters and geometric constants are given as follows: *L~x~* = 1 m, *L~y~*/*L~x~* = 1, *h*/*L~y~* = 0.01 and 0.1, *E*~2~ = 7 GPa, *E*~1~/*E*~2~ = 15, *G*~12~ = *G*~13~ = 0.5*E*~2~, *G*~23~ = 0.5*E*~2~, *μ*~12~ = 0.25, *ρ* = 1650 kg/m^3^. The presented results compare with the results by Qatu \[[@B13-materials-12-03808]\] and Shao et al. \[[@B22-materials-12-03808]\]. From [Table 1](#materials-12-03808-t001){ref-type="table"} and [Table 2](#materials-12-03808-t002){ref-type="table"}, the presented results by WBM match well with the results in the presented literatures. The maximum divergence is −4.61% with the situation **R~y~/L~y~** = −1 for the \[90°/0°\] cross-ply composite laminated paraboloidal shell. It is obvious that the errors in [Table 2](#materials-12-03808-t002){ref-type="table"} by CST are lower than the errors in [Table 1](#materials-12-03808-t001){ref-type="table"} by FSDT. Also, from [Table 1](#materials-12-03808-t001){ref-type="table"}, it can be found that, with the radius to length ratios *R~y~*/*L~y~* from 2 to 10 for the composite shallow shell with the lamination schemes 0°/90°/90°/0° and 0°/90°, the errors between the solutions by the presented method those of the the results in the literature by Quta are generally growing. This is caused by the curvature effect, which is not well predicted by shallow shell theory, thus full shell theory should be considered. Furthermore, when the parameter (*R~x~*/*R~y~*) decreases from 1 to −1, the fundamental frequency parameters for the composite laminates are lower. It can be observed that the fundamental frequency parameter Ω for the composite laminated spherical shell is higher than that for the cylindrical shell and hyperbolic paraboloidal shell.
In the next part, the fundamental frequency parameters Ω of a composite laminated plate with SD-SD boundary conditions are compared with the results by Qatu \[[@B13-materials-12-03808]\] in [Table 3](#materials-12-03808-t003){ref-type="table"} and [Table 4](#materials-12-03808-t004){ref-type="table"}. In [Table 3](#materials-12-03808-t003){ref-type="table"}, two types of layered cross-ply composite laminated plates (i.e., \[0°/90°\] and \[0°/90°/90°/0°\]) by FSDT and CST are investigated with various length to thickness ratios *L~y~*/*h* (i.e., *L~y~*/*h* = 5, 10, 20, and 100). The material constants and geometric parameters are set as follows: *L~x~* = 1 m, *L~y~*/*L~x~* = 1, *E*~2~ = 7 GPa, *E*~1~/*E*~2~ = 15, *G*~12~ = *G*~13~ = 0.5*E*~2~, *G*~23~ = 0.5*E*~2~, *μ*~12~ = 0.25, *ρ* = 1650 kg/m^3^. From [Table 3](#materials-12-03808-t003){ref-type="table"}, it is clearly seen that the results by the presented method agree well with the solutions in the presented literatures. Also, with the growing of the length to thickness ratios *L~y~*/*h*, the fundamental frequency parameters are decreased for the two types of layered cross-ply composite laminated plates by different theory. Particularly, the fundamental frequency parameter is basically unchanged with \[0°/90°/90°/0°\] cross-ply composite laminated plate by CST. Furthermore, three types of layered cross-ply composite laminated plates (i.e., \[0°/90°\], \[0°/90°/0°\], and \[0°/90°/90°/0°\]) with high modulus ratios under SD-SD boundary conditions are considered. With different shell theory, the presented results agree well with the solutions in the represented literature by Qatu \[[@B13-materials-12-03808]\].
In the next part, the fundamental frequency parameter Ω of the composite laminated shallow shell under classical combination boundary conditions is discussed. In [Table 5](#materials-12-03808-t005){ref-type="table"} and [Table 6](#materials-12-03808-t006){ref-type="table"}, the composite laminated shallow cylindrical shell and spherical shell with various classical combination boundary conditions (i.e., F-F, F-S, F-C, S-S, S-C, C-C) are investigated by FSDT and CST. Two types of layered lamination schemes (i.e., \[0°/90°\] and \[0°/90°/0°\]) and radius constants (i.e., *R* = 5, 20) are discussed. The material constants and geometric parameters are defined as follows: *L~x~* = 1 m, *L~y~*/*L~x~* = 1, *E*~2~ = 7 GPa, *E*~1~/*E*~2~ = 25, *G*~12~ = *G*~13~ = 0.5*E*~2~, *G*~23~ = 0.2*E*~2~, *μ*~12~ = 0.25, *ρ* = 1650 kg/m^3^. Also, the fundamental frequency parameters Ω are compared with the solutions in the represented literature by Qatu \[[@B13-materials-12-03808]\]. From the comparison of the results by the presented method and represented literature, it can be seen that the errors obtained by the two different methods are small. The maximum error of 3.62% appears in the situation with \[0°/90°/0°\] composite laminated cylindrical shell (FSDT, *R* = 5) with F-F boundary condition in [Table 5](#materials-12-03808-t005){ref-type="table"}. Furthermore, the maximum error is 3.71% in [Table 6](#materials-12-03808-t006){ref-type="table"} for the \[0°/90°\] composite laminated shallow spherical shell (CST, *R* = 20) with the F-F boundary condition. For various boundary conditions, the maximum parameters Ω appear when the composite shells have the C-C boundary condition. Simultaneously, the minimum frequency parameters emerge with F-F for several lamination schemes and shell theory. So, the composite laminated shallow shells with arbitrary classical combination boundary conditions by WBM can be verified through the presented numerical examples. In order to further investigate the free vibration characteristics of composite laminated shallow shells with arbitrary combination boundary conditions, some mode shapes (*n*, *m*) of the composite laminated cylindrical shell and spherical shell are shown in [Figure 2](#materials-12-03808-f002){ref-type="fig"} and [Figure 3](#materials-12-03808-f003){ref-type="fig"}, respectively.
In this section, the influence of the length to thickness ratio *L~x~*/*h* and length to radius ratio *L~x~*/*R~x~* on the fundamental frequency parameter Ω is discussed. In [Table 7](#materials-12-03808-t007){ref-type="table"}, [Table 8](#materials-12-03808-t008){ref-type="table"} and [Table 9](#materials-12-03808-t009){ref-type="table"}, the fundamental frequency parameter Ω for three types of the layered (i.e., \[0°/90°/90°/0°\], \[0°/90°\], and \[90°/0°\]) composite laminated shallow cylindrical shell, spherical shell, and hyperbolic paraboloidal shell with SD-SD by FSDT and CST is discussed. The material parameters and geometric constants are defined as follows: *L~x~* = 1 m, *L~y~*/*L~x~* = 1, *E*~2~ = 7 GPa, *E*~1~/*E*~2~ = 15, *G*~12~ = *G*~13~ = 0.5*E*~2~, *G*~23~ = 0.5*E*~2~, *μ*~12~ = 0.25, *ρ* = 1650 kg/m^3^. Especially with the composite laminated shallow shells with *L~x~*/*R~x~* = 0, the composite shells are transformed into the plate form. From [Table 7](#materials-12-03808-t007){ref-type="table"}, [Table 8](#materials-12-03808-t008){ref-type="table"} and [Table 9](#materials-12-03808-t009){ref-type="table"}, with the growing of length to radius ratio *L~x~*/*R~x~* (i.e., *L~x~*/*R~x~* = 0, 0.1, 0.2, and 0.5), the fundamental frequency parameters of the composite cylindrical and spherical shell generally grow for various length to thickness ratios *L~x~*/*h*, lamination schemes, and shell theories. Simultaneously, the fundamental frequency parameters of the composite laminated hyperbolic paraboloidal shell are generally decreased with the changing of the length to radius ratio *L~x~*/*R~x~*. It can be clearly seen that, for different laminated schemes and shell theories, when the length to thickness ratio *L~x~*/*h* = 0.01, the fundamental frequency parameter of various composite laminated shallow shells increases significantly. Relatively, when *L~x~*/*h* = 0.1, the frequency parameter increases a little and remains within a stable range. To further investigate the effect of the length to thickness ratio *L~x~*/*h* on frequency parameters Ω of the composite shallow shell, the variations of the frequency parameter Ω for composite shells with SD-SD boundary conditions, with respect to diverse length to radius ratios *L~x~*/*R~x~* and length to radius ratios *L~x~*/*h*, by FSDT and CST are shown in [Figure 4](#materials-12-03808-f004){ref-type="fig"} and [Figure 5](#materials-12-03808-f005){ref-type="fig"}. It can be seen that, for different laminated schemes, shallow shell structures, and shell theories, as the length to thickness ratio *L~x~*/*h* increases, the fundamental frequency parameters Ω gradually decrease. At the same time, it can be seen that, for the composite laminated hyperbolic paraboloidal shell, the variation of the fundamental frequency parameters Ω is small and the effect of the length to thickness ratio *L~x~*/*h* is not particularly obvious.
In the previous numerical example, the effect of geometric parameters on the fundamental frequency parameters is discussed. In this part, the influence of the material parameter on the frequency parameter is investigated. In [Table 10](#materials-12-03808-t010){ref-type="table"} and [Table 11](#materials-12-03808-t011){ref-type="table"}, the fundamental frequency parameter for composite laminated shallow spherical shells with various length to radius ratios *L~x~*/*R~x~*, modulus ratios *E*~1~/*E*~2~, and boundary conditions (i.e., SD-SD, F-F, and C-C) are discussed by FSDT and CST. The material parameters and geometric constants are given as follows: *L~x~* = 1 m, *L~y~*/*L~x~* = 1, *h*/*L~y~* = 0.1, *E*~2~ = 7 GPa, *G*~12~ = *G*~13~ = 0.5*E*~2~, *G*~23~ = 0.5*E*~2~, *μ*~12~ = 0.25, *ρ* = 1650 kg/m^3^. It can be clearly seen from [Table 10](#materials-12-03808-t010){ref-type="table"} and [Table 11](#materials-12-03808-t011){ref-type="table"} that the fundamental frequency parameter Ω generally grows with the changing of modulus ratios *E*~1~/*E*~2~ from 5 to 40. To further reflect the impact of modulus ratios *E*~1~/*E*~2~ on fundamental frequency parameters, the variations of the fundamental frequency parameter Ω for composite spherical shells with various boundary conditions with respect to multiple length to radius ratios *L~x~*/*R~x~* and modulus ratios *E*~1~*/E*~2~ by FSDT and CST are shown in [Figure 6](#materials-12-03808-f006){ref-type="fig"}. Therefore, it can be concluded that the modulus ratios *E*~1~*/E*~2~ has a significant effect on the fundamental frequency parameters of the composite spherical shell and plays a positive role. Different boundary conditions cause the stiffness matrix to change. For the free boundary condition, the determinant of the stiffness matrix increases with respect to the clamped boundary condition, and when the mass matrix remains unchanged, the natural frequency increases.
3.2. Composite Laminated Shallow Shell with Elastic Boundary Conditions {#sec3dot2-materials-12-03808}
-----------------------------------------------------------------------
The composite laminated shallow shell with elastic constraint is widely encountered in many engineering applications. So, analysis of the composite shallow shells with such an elastic boundary condition is necessary and significant. Therefore, in this section, the free vibration characteristics of the composite shallow shell with elastic boundary conditions are discussed.
In this section, the effect of the restrained springs on the frequency parameter of the certain cross-ply composite laminated shallow shells is discussed. The certain cross-ply layered \[0°/90°\] composite laminated shallow shells with S-elastic boundary conditions are concerned by FSDT. For the elastic restrained edge, there is only one set of spring component on one displacement or transverse rotational direction and the range of stiffness constants is defined as 10^0^--10^12^. The material parameters and geometric constants are defined as follows: *L~x~* = 1 m, *L~y~*/*L~x~* = 1, *R~x~*/*L~y~*, *E*~2~ = 7 GPa, *E*~1~/*E*~2~ = 25, *G*~12~ = *G*~13~ = 0.5*E*~2~, *G*~23~ = 0.2*E*~2~, *μ*~12~ = 0.25, *ρ* = 1650 kg/m^3^. In [Table 12](#materials-12-03808-t012){ref-type="table"}, [Table 13](#materials-12-03808-t013){ref-type="table"}, [Table 14](#materials-12-03808-t014){ref-type="table"}, [Table 15](#materials-12-03808-t015){ref-type="table"} and [Table 16](#materials-12-03808-t016){ref-type="table"}, the lowest two frequency parameter Ω for the composite shells with S-elastic boundary conditions by restrained spring components *K~u~*, *K~v~*, *K~w~*, *K~ϕx~*, and *K~ϕy~* for a certain circumferential number of *n* = 1 is calculated.
In [Table 12](#materials-12-03808-t012){ref-type="table"} and [Table 13](#materials-12-03808-t013){ref-type="table"}, when the certain cross-ply composite laminated shallow shells are only restrained in the direction of *u* and *v*, the frequency parameters generally increase with the various composite laminated shallow shell forms. Also, the increase in frequency parameters is small and basically remains within a stable range. Correspondingly, when *K~u~* = 10^6^, the frequency parameter starts to increase slightly, and when *K~u~* = 10^10^, the frequency parameter remains basically unchanged. When the composite laminated shallow shells are under the elastic restraint *K~w~* in [Table 14](#materials-12-03808-t014){ref-type="table"}, the frequency parameters Ω are generally decreased with the growing of the spring stiffness from 10^0^ to 10^12^. In particular, for the hyperbolic paraboloidal shell, the frequency parameter increases less than that of the other composite laminated shallow shell forms. For the effect of transverse rotational spring stiffness on the frequency parameter of composite laminated shallow shells, [Table 15](#materials-12-03808-t015){ref-type="table"} and [Table 16](#materials-12-03808-t016){ref-type="table"} show the changing rule of the frequency parameters with the growing of the stiffness constants for *K~ϕx~* and *K~ϕy~*. In general, as the stiffness constants *K~ϕx~* and *K~ϕy~* continue to increase, the frequency parameters corresponding to each structure tend to increase; at the same time, the main change region of the frequency parameter is between *K~ϕx,y~* = 10^4^--10^10^. However, when *K~ϕx,y~* = 10^7^, there will be some jitter in the frequency parameters, which suddenly increase or decrease. Therefore, as the elastic restrained stiffness constants in different directions increase, the frequency parameters of various composite shell forms are gradually increasing and have different change regions. Furthermore, the variations of the frequency parameter with the changing of the stiffness constants are shown in [Figure 7](#materials-12-03808-f007){ref-type="fig"}, [Figure 8](#materials-12-03808-f008){ref-type="fig"}, [Figure 9](#materials-12-03808-f009){ref-type="fig"}, [Figure 10](#materials-12-03808-f010){ref-type="fig"} and [Figure 11](#materials-12-03808-f011){ref-type="fig"}.
4. Conclusions {#sec4-materials-12-03808}
==============
A semi-analyzed method is conducted for the free vibration characteristics of composite laminated shallow shells with general boundary conditions, including classical boundary conditions, elastic boundary conditions, and their combinations. Through the relationship between the displacement vector and force resultants, the formulations are established related to classical shell theory (CST) and first-order shear deformation shell theory (FSDT). According to diverse boundary conditions, the boundary matrix and the total matrix of the composite shallow shell will be established. Through the dichotomy method to search the zeros position of the total matrix determinant, the natural frequency can be obtained. Correspondingly, some numerical examples are calculated and the conclusions can be summarized as follows:
First, by comparing the solutions by the presented method with some reported literature results, the correctness of the calculation for the free vibration characteristics of composite laminated shallow shells with classical boundary conditions, elastic boundary conditions, and their combinations can be proven.
Second, some numerical examples are extended to investigate the influence of material parameters and geometric constants, like length to radius ratios, length to thickness ratios, and modulus ratios, on the frequency parameter. It can concluded that different material and geometric parameters have different influence factors on frequency parameters. Simultaneously, changing laws obtained by various composite laminated shallow shell structures are not consistent.
Finally, the effect of boundary elastic restrained stiffness on the natural frequency parameters is discussed. By changing the value of the spring stiffness in different displacement directions and transverse rotation from 10^0^ to 10^12^, the variation of the frequency parameter with the elastic restrained spring stiffness constants is obtained. It can be seen from numerical analysis examples that the different elastic constants have a positive effect on the frequency parameters and have a certain effect on the increase of the frequency parameters. Simultaneously, the effect of each spring stiffness constant has its own influence range.
Methodology, D.S.; validation, D.H. and Q.W.; formal analysis, D.H.; investigation, D.H. and Q.W.; data curation, D.H. and Q.W.; writing---original draft preparation, D.H. and C.M.; writing---review and editing, Q.W. visualization, D.H.; supervision, D.S. and H.S.
This research was funded by the National Natural Science Foundation of China (Grant Nos. 51679056, 51705537 and 51875112), Innovation Driven Program of Central South University (Grant number: 2019CX006), and the Natural Science Foundation of Hunan Province of China (2018JJ3661). The authors also gratefully acknowledge the supports from State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2018-11).
The authors declare no conflict of interest.
######
(**a**): Geometric model of the composite laminated shallow shell with elastic restraint; (**b**): geometric model of the composite laminated shallow shell with various curvature types.
######
The mode shapes for the composite laminated shallow cylindrical shell with various boundary conditions. (**a**) C-C,(1,1); (**b**) C-C,(1,2); (**c**) C-C,(1,3); (**d**) F-C,(1,1); (**e**) F-C,(1,2); (**f**) F-C,(1,3); (**g**) F-F,(1,1); (**h**) F-F,(1,2); (**i**) F-F,(1,3); (**j**) F-S,(1,1); (**k**) F-S,(1,2); (**l**) F-S,(1,3); (**m**) S-C,(1,1); (**n**) S-C,(1,2); (**o**) S-C,(1,3).
######
The mode shapes for the composite laminated shallow spherical shell with various boundary conditions. (**a**) C-C,(1,1); (**b**) C-C,(1,2); (**c**) C-C,(1,3); (**d**) F-C,(1,1); (**e**) F-C,(1,2); (**f**) F-C,(1,3); (**g**) F-F,(1,1); (**h**) F-F,(1,2); (**i**) F-F,(1,3); (**j**) F-S,(1,1); (**k**) F-S,(1,2); (**l**) F-S,(1,3); (**m**) S-C,(1,1); (**n**) S-C,(1,2); (**o**) S-C,(1,3).
{#materials-12-03808-f004}
######
Variation laws of the fundamental frequency parameter Ω for composite laminated shallow shells with the SD-SD boundary conditions with respect to various length to radius ratios *L~x~*/*R~x~* and length to thickness ratios *L~x~*/*h* by classical shell theory (CST). (**a**) Cylindrical shell, 0°/90°/90°/0°; (**b**) cylindrical shell, 0°/90°; (**c**) cylindrical shell, 90°/0°; (**d**) spherical shell, 0°/90°/90°/0°; (**e**) spherical shell, 0°/90°; (**f**) spherical shell, 90°/0°; (**g**) hyperbolic paraboloidal, 0°/90°/90°/0°; (**h**) hyperbolic paraboloidal, 0°/90°; (**i**) hyperbolic paraboloidal, 90°/0°.
{#materials-12-03808-f006}
![Variation laws of the frequency parameter Ω for composite shallow \[0°/90°\] shells with various stiffness constant *K~u~* by FSDT. (**a**) *m* = 1; (**b**) *m* = 2.](materials-12-03808-g007){#materials-12-03808-f007}
![Variation laws of the frequency parameter Ω for composite shallow \[0°/90°\] shells with various stiffness constant *K~v~* by FSDT. (**a**) *m* = 1; (**b**) *m* = 2.](materials-12-03808-g008){#materials-12-03808-f008}
![Variation laws of the frequency parameter Ω for composite shallow \[0°/90°\] shells with various stiffness constant *K~w~* by FSDT. (**a**) *m* = 1; (**b**) *m* = 2.](materials-12-03808-g009){#materials-12-03808-f009}
![Variation laws of the frequency parameter Ω for composite shallow \[0°/90°\] shells with various stiffness constant *K~ϕx~* by FSDT. (**a**) *m* = 1; (**b**) *m* = 2.](materials-12-03808-g010){#materials-12-03808-f010}
![Variation laws of the frequency parameter Ω for composite shallow \[0°/90°\] shells with various stiffness constant *K~ϕy~* by FSDT. (**a**) *m* = 1; (**b**) *m* = 2.](materials-12-03808-g011){#materials-12-03808-f011}
materials-12-03808-t001_Table 1
######
The fundamental frequency parameter Ω for the composite shallow shell with the SD-SD boundary condition by first-order shear deformation shell theory (FSDT). WBM, wave-based method.
*R~x~/R~y~* *R~y~/L~y~* 0°/90°/90°/0°
------------- ------------- --------------- --------- --------- --------- --------
1 2 12.3093 12.5718 −2.09% 12.3633 −0.44%
5 11.1495 11.2522 −0.91% 11.2135 −0.57%
10 10.9672 11.0428 −0.69% 11.0329 −0.60%
0 2 11.2142 11.3342 −1.06% 11.2756 −0.54%
5 10.9562 11.0316 −0.68% 11.0217 −0.59%
10 10.9562 10.9867 −0.28% 10.9842 −0.25%
−1 2 10.3671 10.7031 −3.14% 10.4300 −0.60%
5 10.8169 10.9273 −1.01% 10.8826 −0.60%
10 10.8831 10.9605 −0.71% 10.9493 −0.60%
*R~x~/R~y~* *R~y~/L~y~* 0°/90°
1 2 10.0265 10.2492 −2.17% 10.0998 −0.73%
5 8.3845 8.5084 −1.46% 8.4783 −1.11%
10 8.1132 8.2190 −1.29% 8.2111 −1.19%
0 2 8.6166 8.7523 −1.55% 8.7075 −1.04%
5 8.1475 8.2445 −1.18% 8.2458 −1.19%
10 8.0644 8.1592 −1.16% 8.1636 −1.22%
−1 2 7.8626 8.0831 −2.73% 7.9596 −1.22%
5 8.0549 8.1538 −1.21% 8.1546 −1.22%
10 8.0537 8.1448 −1.12% 8.1534 −1.22%
*R~x~/R~y~* *R~y~/L~y~* 90°/0°
1 2 10.0261 10.2492 −2.18% 10.0998 −0.73%
5 8.3843 8.5084 −1.46% 8.4783 −1.11%
10 8.1131 8.2190 −1.29% 8.2111 −1.19%
0 2 8.3533 8.5784 −2.62% 8.4421 −1.05%
5 8.1474 8.1774 −0.37% 8.1448 0.03%
10 8.0644 8.1259 −0.76% 8.1136 −0.61%
−1 2 7.4152 7.7739 −4.61% 7.5071 −1.22%
5 7.8606 8.0223 −2.02% 7.9581 −1.22%
10 7.9554 8.0785 −1.52% 8.0540 −1.22%
materials-12-03808-t002_Table 2
######
The fundamental frequency parameter Ω for composite shallow shell with the SD-SD boundary condition by classical shell theory (CST) theory.
*R~x~/R~y~* *R~y~/L~y~* 0°/90°/90°/0°
------------- ------------- --------------- --------- ---------- ---------- --------
1 2 66.52832 66.5774 −0.07% 66.5285 0.00%
5 29.29062 29.309 −0.06% 29.2906 0.00%
10 18.12154 18.129 −0.04% 18.1215 0.00%
0 2 35.10566 35.1838 −0.22% 35.1622 −0.16%
5 18.08579 18.1107 −0.14% 18.1038 −0.10%
10 13.96209 13.9703 −0.06% 13.9681 −0.04%
−1 2 11.67421 11.9776 −2.60% 11.6742 0.00%
5 12.1784 12.2279 −0.41% 12.1783 0.00%
10 12.25251 12.2649 −0.10% 12.2525 0.00%
*R~x~/R~y~* *R~y~/L~y~* 0°/90°
1 2 65.98726 66.0139 −0.04% 65.98717 0.00%
5 27.95602 27.9666 −0.04% 27.95599 0.00%
10 15.85265 15.8573 −0.03% 15.85258 0.00%
0 2 28.16678 28.2471 −0.28% 28.16667 0.00%
5 15.84446 15.8484 −0.02% 15.81926 0.16%
10 10.85982 10.8616 −0.02% 10.85171 0.07%
−1 2 8.121357 8.37737 −3.06% 8.17396 −0.64%
5 8.487776 8.54161 −0.63% 8.51081 −0.27%
10 8.545494 8.56847 −0.27% 8.55691 −0.13%
*R~x~/R~y~* *R~y~/L~y~* 90°/0°
1 2 65.98726 66.0139 −0.04% 66.0139 −0.04%
5 27.95602 27.9666 −0.04% 27.9666 −0.04%
10 15.85265 15.8573 −0.03% 15.8573 −0.03%
0 2 27.692 27.827 −0.49% 27.69195 0.00%
5 15.81913 15.8342 −0.10% 15.84456 −0.16%
10 10.85163 10.8567 −0.05% 10.85977 −0.08%
−1 2 8.173862 8.34143 −2.05% 8.12143 0.64%
5 8.510863 8.52632 −0.18% 8.48796 0.27%
10 8.557038 8.55594 0.01% 8.54535 0.14%
materials-12-03808-t003_Table 3
######
The fundamental frequency parameter Ω for a composite plate with the SD-SD boundary condition with variety theory.
Lamination Theory
------------------- --------- --------- --------- --------- ---------- ---------- ---------- ----------
100 8.56394 8.55196 8.56847 8.56858 12.26147 12.26147 12.37733 12.27746
20 8.44807 8.44807 8.55811 8.55808 11.90100 11.90100 12.27733 12.27731
10 8.11956 8.11956 8.52569 8.52570 10.97163 10.97163 12.27733 12.27733
5 7.14661 7.14661 8.39526 8.39527 8.77840 8.77841 12.27733 12.27734
materials-12-03808-t004_Table 4
######
The fundamental frequency parameter Ω for a composite plate with the SD-SD boundary condition by variety theory.
Lamination Theory
------------------- --------------- --------- -------- ---------
100 9.6873 9.6873 9.696 9.6961
10 8.9001 8.9001 9.6436 9.6436
*L~y~*//*h* 0°/90°/0°
100 15.183 15.1834 15.228 15.2278
10 12.163 12.1629 15.228 15.2278
*L~y~*//*h* 0°/90°/90°/0°
100 15.184 15.1839 15.228 15.2278
10 12.226 12.2272 15.228 15.2278
materials-12-03808-t005_Table 5
######
The fundamental frequency parameter Ω for two types of the layered composite shallow cylindrical shell with various boundary conditions, theories, and radii.
Lamination Schemes Theory *R* Boundary Conditions
-------------------- ------------------------------------ ------------------------------------ ------------------------------------ --------------------- -------- -------- -------- -------- --------
0°/90° CST 20 Ref. \[[@B13-materials-12-03808]\] 6.128 6.489 7.008 9.56 12.136 15.757
WBM 6.147 6.376 7.257 9.633 12.236 15.895
Error 0.31% −1.74% 3.55% 0.77% 0.82% 0.88%
5 Ref. \[[@B13-materials-12-03808]\] 6.096 6.444 7.014 9.598 12.154 15.747
WBM 6.070 6.433 7.231 9.630 12.225 15.861
Error −0.42% −0.16% 3.10% 0.34% 0.58% 0.72%
FSDT 20 Ref. \[[@B13-materials-12-03808]\] 5.763 6.087 6.535 8.894 10.609 12.623
WBM 5.778 6.007 6.532 8.803 10.555 12.621
Error 0.26% −1.32% −0.05% −1.03% −0.50% −0.01%
5 Ref. \[[@B13-materials-12-03808]\] 5.716 6.030 6.524 8.931 10.647 12.663
WBM 5.850 5.994 6.502 8.826 10.582 12.651
Error 2.34% −0.59% −0.34% −1.17% −0.61% −0.10%
0°/90°/0° CST 20 Ref. \[[@B13-materials-12-03808]\] 3.902 4.484 6.866 15.106 22.557 32.091
WBM 3.966 4.501 6.891 15.229 22.214 32.385
Error 1.63% 0.39% 0.36% 0.82% −1.52% 0.92%
5 Ref. \[[@B13-materials-12-03808]\] 3.894 4.472 6.901 15.136 22.560 32.062
WBM 3.966 4.475 6.917 15.253 22.214 32.352
Error 1.84% 0.07% 0.23% 0.77% −1.53% 0.90%
FSDT 20 Ref. \[[@B13-materials-12-03808]\] 3.787 4.318 6.146 12.166 14.250 16.385
WBM 3.796 4.312 6.146 12.104 14.218 16.384
Error 0.23% −0.14% 0.00% −0.51% −0.22% 0.00%
5 Ref. \[[@B13-materials-12-03808]\] 3.773 4.301 6.176 12.212 14.284 16.408
WBM 3.910 4.392 6.170 12.148 14.250 16.406
Error 3.62% 2.11% −0.09% −0.53% −0.24% −0.01%
materials-12-03808-t006_Table 6
######
The fundamental frequency parameter Ω for two types of the layered composite shallow spherical shell with various boundary conditions, theories, and radii.
Lamination Schemes Theory *R* Boundary Conditions
-------------------- ------------------------------------ ------------------------------------ ------------------------------------ --------------------- -------- -------- -------- -------- --------
0°/90° CST 20 Ref. \[[@B13-materials-12-03808]\] 6.132 6.493 7.002 9.588 12.165 15.822
WBM 6.360 6.360 7.066 9.663 12.274 15.975
Error 3.71% −2.05% 0.91% 0.78% 0.90% 0.97%
5 Ref. \[[@B13-materials-12-03808]\] 6.162 6.51 6.971 9.903 12.465 16.82
WBM 6.250 6.482 6.940 9.945 12.560 16.990
Error 1.43% −0.43% −0.44% 0.42% 0.77% 1.01%
FSDT 20 Ref. \[[@B13-materials-12-03808]\] 5.768 6.093 6.535 8.922 10.64 12.713
WBM 5.764 6.076 6.532 8.833 10.590 12.714
Error −0.07% −0.27% −0.04% −1.00% −0.47% 0.01%
5 Ref. \[[@B13-materials-12-03808]\] 5.787 6.105 6.511 9.247 11.004 14.081
WBM 5.765 6.073 6.493 9.146 10.946 14.078
Error −0.38% −0.53% −0.27% −1.09% −0.53% −0.02%
0°/90°/0° CST 20 Ref. \[[@B13-materials-12-03808]\] 3.909 4.49 6.863 15.116 22.562 32.136
WBM 3.924 4.512 6.888 15.236 22.214 32.430
Error 0.38% 0.49% 0.36% 0.79% −1.54% 0.91%
5 Ref. \[[@B13-materials-12-03808]\] 4.009 4.562 6.861 15.29 22.64 32.785
WBM 4.010 4.599 6.909 15.356 22.215 33.051
Error 0.02% 0.81% 0.70% 0.43% −1.88% 0.81%
FSDT 20 Ref. \[[@B13-materials-12-03808]\] 3.794 4.325 6.146 12.178 14.264 16.487
WBM 3.794 4.319 6.146 12.114 14.231 16.486
Error 0.00% −0.14% −0.01% −0.52% −0.23% −0.01%
5 Ref. \[[@B13-materials-12-03808]\] 3.891 4.397 6.163 12.394 14.499 17.959
WBM 3.884 4.369 6.163 12.312 14.454 17.951
Error −0.18% −0.63% 0.00% −0.66% −0.31% −0.04%
materials-12-03808-t007_Table 7
######
The fundamental frequency parameter Ω for three types of layered composite shallow cylindrical shells for variety theories, length to thickness ratios *L~x~*/*h*, and length to radius ratios *L~x~*/*R~x~* with the SD-SD boundary condition.
*L~x~/h* *L~x~/R~x~* Lamination Schemes
---------- ------------- -------------------- --------- --------- --------- --------- --------
100 0 12.2531 12.2775 8.5520 8.5686 8.5520 8.5686
0.1 13.9407 13.9621 10.8462 10.8598 10.8382 10.8516
0.2 18.0687 18.0858 15.8342 15.8445 15.8090 15.8192
0.5 35.0954 35.1057 28.1209 28.1668 27.6433 27.6921
20 0 11.8617 12.2773 8.3908 8.5581 8.3908 8.5581
0.1 11.9222 12.3350 8.5016 8.6680 8.4735 8.6386
0.2 12.1011 12.2811 8.8000 8.9626 8.7410 8.9010
0.5 13.2636 13.6185 10.6002 10.7427 10.4142 10.5501
10 0 10.9053 12.2773 8.0202 8.5257 8.0202 8.5257
0.1 10.9125 12.2803 8.0644 8.5724 8.0149 8.5146
0.2 10.9338 12.2894 8.1475 8.6547 8.0476 8.5383
0.5 11.0809 12.3534 8.6166 9.1042 8.3533 8.8006
materials-12-03808-t008_Table 8
######
The fundamental frequency parameter Ω for three types of layered composite shallow spherical shells for variety theories, length to thickness ratios *L~x~*/*h*, and length to radius ratios *L~x~*/*R~x~* with the SD-SD boundary condition.
*L~x~/h* *L~x~/R~x~* Lamination Schemes
---------- ------------- -------------------- --------- --------- --------- --------- --------
100 0 12.2531 12.2775 8.5520 8.5686 8.5520 8.5686
0.1 18.1044 18.1215 15.8423 15.8526 15.8423 15.8526
0.2 29.2786 29.2906 27.9478 27.9560 27.9478 27.9560
0.5 66.5196 66.5283 65.9775 65.9873 65.9775 65.9873
20 0 11.8617 12.2773 8.3908 8.5581 8.3908 8.5581
0.1 12.1375 12.5436 8.7873 8.9489 8.7873 8.9489
0.2 12.9248 13.3057 9.8753 10.0240 9.8753 10.0240
0.5 17.3215 17.6007 15.3115 15.4258 15.3114 15.4258
10 0 10.9053 12.2773 8.0202 8.5257 8.0202 8.5257
0.1 10.9672 12.3284 8.1132 8.6132 8.1131 8.6132
0.2 11.1495 12.4794 8.3845 8.8692 8.3843 8.8692
0.5 12.3093 13.4531 10.0265 10.4333 10.0261 10.4333
materials-12-03808-t009_Table 9
######
The fundamental frequency parameter Ω for three types of layered composite shallow hyperbolic paraboloidal shells for variety theories, length to thickness ratios *L~x~*/*h*, and length to radius ratios *L~x~*/*R~x~* with the SD-SD boundary condition.
*L~x~/h* *L~x~/R~x~* Lamination Schemes
---------- ------------- -------------------- --------- -------- -------- -------- --------
100 0 12.2531 12.2775 8.5520 8.5686 8.5520 8.5686
0.1 12.2283 12.2525 8.5404 8.5570 8.5289 8.5454
0.2 12.1543 12.1783 8.4944 8.5109 8.4716 8.4877
0.5 11.6512 11.6742 8.1583 8.1739 8.1059 8.1213
20 0 11.8617 12.2773 8.3908 8.5581 8.3908 8.5581
0.1 11.8377 12.2524 8.4015 8.5698 8.3464 8.5120
0.2 11.7659 12.1778 8.3783 8.5466 8.2693 8.4325
0.5 11.2784 11.6712 8.1079 8.2715 7.8571 8.0092
10 0 10.9053 12.2773 8.0202 8.5257 8.0202 8.5257
0.1 10.8831 12.2519 8.0537 8.5664 7.9554 8.4514
0.2 10.8169 12.1761 8.0549 8.5724 7.8606 8.3452
0.5 10.3671 11.6619 7.8626 8.6024 7.4152 7.8559
materials-12-03808-t010_Table 10
######
The fundamental frequency parameter Ω for composite shallow \[0°/90°\] spherical shells for variety length to radius ratios *L~x~*/*R~x~*, modulus ratios *E*~1~/*E*~2~, and boundary conditions by FSDT.
Boundary Conditions *L~x~*/*R~x~* *E*~1~/*E*~2~
--------------------- --------------- --------------- --------- --------- --------- ---------
SD-SD 0.1 6.8768 7.5476 8.1132 9.0869 10.3019
0.2 7.1668 7.8326 8.3845 9.3295 11.1241
0.5 8.8937 9.5415 10.0265 10.8234 11.8112
F-F 0.1 3.9347 4.5645 5.0554 5.8596 6.8237
0.2 3.9482 4.5782 5.0664 5.8626 6.8150
0.5 4.0430 4.6793 5.1581 5.9229 6.8248
C-C 0.1 4.2046 5.0239 5.6852 6.7780 8.0950
0.2 5.0485 6.3327 7.3725 9.0778 11.1241
0.5 9.0303 12.0219 14.4032 18.2531 22.2144
materials-12-03808-t011_Table 11
######
The fundamental frequency parameter Ω for composite shallow \[0°/90°\] spherical shells for variety length to radius ratios *L~x~*/*R~x~*, modulus ratios *E*~1~/*E*~2~, and boundary conditions by CST.
Boundary Conditions *L~x~*/*R~x~* *E*~1~/*E*~2~
--------------------- --------------- --------------- --------- --------- --------- ---------
SD-SD 0.1 7.2637 7.9884 8.6201 9.7202 11.1634
0.2 7.5404 8.2592 8.8759 9.9450 11.3972
0.5 9.0807 9.8983 10.4389 11.3430 12.5272
C-C 0.1 4.2891 5.1468 5.8463 7.0198 8.4698
0.2 5.1199 6.4318 7.4987 9.2602 11.3972
0.5 9.0807 12.0872 14.4816 18.3556 22.2167
materials-12-03808-t012_Table 12
######
The frequency parameter Ω for composite shallow \[0°/90°\] shells with S-*K~u~* boundary conditions by FSDT.
Stiffness Plate Cylindrical Shell Spherical Shell Hyperbolic Paraboloidal
----------- -------- ------------------- ----------------- ------------------------- -------- --------- -------- --------
10^0^ 6.0999 11.8764 6.1713 11.9613 6.0843 12.1225 5.8619 7.0016
10^1^ 6.0999 11.8764 6.1713 11.9613 6.0843 12.1225 5.8619 7.0016
10^2^ 6.0999 11.8764 6.1713 11.9613 6.0843 12.1225 5.8619 7.0016
10^3^ 6.0999 11.8764 6.1713 11.9613 6.0843 12.1225 5.8619 7.0016
10^4^ 6.0999 11.8764 6.1713 11.9613 6.0843 12.1225 5.8619 7.0016
10^5^ 6.0999 11.8764 6.1713 11.9614 6.0843 12.1226 5.8619 7.0016
10^6^ 6.0999 11.8766 6.1716 11.9615 6.0847 12.1228 5.8619 7.0016
10^7^ 6.1000 11.8780 6.1739 11.9629 6.0884 12.1251 5.8619 7.0016
10^8^ 6.1009 11.8911 6.1960 11.9763 6.1241 12.1468 5.8600 7.0017
10^9^ 6.1070 11.9770 6.3472 12.0653 6.3634 12.2935 5.6539 7.0042
10^10^ 6.1187 12.1169 6.6216 12.2166 6.7778 12.5456 7.0567 7.4902
10^11^ 6.1223 12.1553 6.7039 12.2594 6.8965 12.6167 7.0752 7.3410
10^12^ 6.1228 12.1598 6.7137 12.2645 6.9106 12.6251 7.0754 7.3395
materials-12-03808-t013_Table 13
######
The frequency parameter Ω for composite shallow \[0°/90°\] shells with S-*K~v~* boundary conditions by FSDT.
----------- -------- ------------------- ----------------- ------------------------- -------- --------- -------- --------
10^5^ 6.0999 11.8764 6.1713 11.9613 6.0844 12.1226 5.8619 7.0016
10^6^ 6.1000 11.8765 6.1715 11.9614 6.0849 12.1228 5.8619 7.0016
10^7^ 6.1012 11.8772 6.1731 11.9621 6.0907 12.1251 5.8619 7.0016
10^8^ 6.1122 11.8844 6.1890 11.9686 6.1466 12.1476 5.8619 7.0016
10^9^ 6.2016 11.9437 6.3170 12.0234 6.5714 12.3419 5.8600 7.0016
10^10^ 6.4698 12.1403 6.6986 12.2091 7.5921 13.0667 5.9118 7.0022
10^11^ 6.6018 12.2486 6.8848 12.3139 7.9735 13.4990 5.9009 7.0021
10^12^ 6.6204 12.2645 6.9109 12.3294 8.0215 13.5632 5.9008 7.0021
materials-12-03808-t014_Table 14
######
The frequency parameter Ω for composite shallow \[0°/90°\] shells with S-*K~w~* boundary conditions by FSDT.
----------- -------- ------------------- ----------------- ------------------------- --------- --------- -------- --------
10^3^ 6.0999 11.8765 6.1713 11.9614 6.0843 12.1226 5.8619 7.0016
10^4^ 6.1002 11.8766 6.1716 11.9616 6.0846 12.1227 5.8619 7.0016
10^5^ 6.1031 11.8786 6.1745 11.9635 6.0876 12.1245 5.8619 7.0016
10^6^ 6.1315 11.8980 6.2027 11.9826 6.1172 12.1424 5.8605 7.0024
10^7^ 6.3925 12.0948 6.4630 12.1764 6.3902 12.3238 6.4573 7.0719
10^8^ 7.7073 14.0587 7.7845 14.1180 7.8101 14.1629 7.7352 9.1254
10^9^ 8.7004 19.8103 8.7935 19.8887 19.8721 21.5398 8.7320 9.1254
10^10^ 8.8380 21.1601 8.9336 21.2291 21.1699 21.6778 8.8733 9.1254
10^11^ 8.8521 21.2786 8.9478 21.3223 21.2528 21.7339 8.8433 9.1254
10^12^ 8.8535 21.2894 8.9493 21.3296 21.2599 21.7405 8.8434 9.1254
materials-12-03808-t015_Table 15
######
The frequency parameter Ω for composite shallow \[0°/90°\] shells with S-*K~ϕx~* boundary conditions by FSDT.
----------- -------- ------------------- ----------------- ------------------------- -------- --------- -------- --------
10^3^ 6.0999 11.8762 6.1713 11.9611 6.0843 12.1223 5.8619 7.0016
10^4^ 6.0997 11.8739 6.1711 11.9588 6.0840 12.1200 5.8619 7.0016
10^5^ 6.0974 11.8507 6.1688 11.9354 6.0812 12.0971 5.8619 7.0013
10^6^ 6.0711 11.5921 6.1418 11.6754 6.0485 11.8418 5.8619 6.9694
10^7^ 6.5494 9.0772 6.6282 9.1053 6.6639 9.0215 5.8621 9.1254
10^8^ 6.2722 14.4763 6.3469 14.5752 6.3025 14.7114 5.8621 8.9458
10^9^ 6.2620 14.2671 6.3365 14.3650 6.2894 14.5019 5.8621 8.9351
10^10^ 6.2611 14.2479 6.3356 14.3457 6.2882 14.4828 5.8621 8.9350
10^11^ 6.2610 14.2460 6.3355 14.3438 6.2881 14.4808 5.8621 8.9350
10^12^ 6.2609 14.2458 6.3355 14.3436 6.2880 14.4807 5.8621 8.9350
materials-12-03808-t016_Table 16
######
The frequency parameter Ω for composite shallow \[0°/90°\] shells with S-*K~ϕy~* boundary conditions by FSDT.
----------- -------- ------------------- ----------------- ------------------------- -------- --------- -------- --------
10^3^ 6.0997 11.8763 6.1711 11.9612 6.0841 12.1225 5.8619 7.0016
10^4^ 6.0977 11.8755 6.1691 11.9604 6.0820 12.1217 5.8619 7.0016
10^5^ 6.0780 11.8669 6.1485 11.9521 6.0611 12.1145 5.8610 7.0017
10^6^ 5.8434 11.7713 5.9050 11.8600 5.8139 12.0349 5.7743 7.0070
10^7^ 5.7759 9.0847 9.1053 9.2703 9.0215 9.4964 7.1004 9.1254
10^8^ 7.5204 12.8896 7.6516 12.9518 7.6426 12.9930 7.1247 9.4010
10^9^ 7.4432 12.8039 7.5710 12.8671 7.5546 12.9172 7.1250 9.3841
10^10^ 7.4360 12.7961 7.5634 12.8594 7.5463 12.9103 7.1250 9.3840
10^11^ 7.4352 12.7954 7.5627 12.8587 7.5455 12.9096 7.1250 9.3840
10^12^ 7.4352 12.7953 7.5626 12.8586 7.5454 12.9095 7.1250 9.3840
| |
The utility model discloses a high bar steel conveyor after long service life's multiple length is cut, including main part and one end and main part articulated flip, be equipped with half first groove in the main part, half groove of corresponding second is put with half first trench to last being equipped with of flip, flip closes first half groove and second half flute profile flip on of backin the main part and levels up the through -hole, it was through locking mechanism and main part lock solid after flip closed, this high bar steel conveyor after long service life's multiple length iscut, but the condition of observational level through -hole and interior rolled piece, and operate horizontal through -hole and interior rolled piece, especially horizontal through -hole damages theconveying equipment trouble that causes after high excellent multiple length is cut and when needing the maintenance, conveyor's after can cutting corresponding high excellent multiple length flip opens, study for a second time courses one has flunked the horizontal through -hole that is used for carrying the rolled piece after taking out the rolled piece. | |
Making capacity = 2.55 X breaking capacity (for all circuit breakers) = 2.55 X 2000 = 5100 MVA.
03․ Current chopping mainly occurs in
Air blast circuit breakers retain the same distinguish power irrespective of the magnitude of the current to be interrupted. When breaking low current with such breakers, powerful de-ionising effect of air blast causes. The current to fall abruptly to zero before the natural current zero is reached. This phenomenon is known as current chopping and result in the production of high voltage transient across the contact of the circuit breaker.
04․ Where air circuit breakers are used?
Air circuit breaker are used in DC circuits and AC circuits up to 12 kV. AC air circuit breaker are widely used in indoor type medium and low voltage switch gear.
05․ On what factor does the performance of a circuit breaker depend?
Circuit breaker performance depends upon breaking speed, critical length of breaking and also some other factors.
06․ Keeping in view the cost and overall effectiveness, the following circuit breaker is suited for capacitor bank switching
The arc time constant is the least (a few microsec) in vacuum CB as compared to other breaker types. The rapid building up of dielectric strength after final arc extinction ( 20 kV / microseconds ) is unique feature of VCBs. These arc, therefore ideally suited for capacitor switching.
07․ In a short circuit test circuit breaker, time to reach the peak re-striking voltage 60 microsecond & the peak re-striking voltage 120 kV. Determine average RRRV (Rate of rise re-striking voltage)?
08․ A CT is connected in ___________________ with the line.
A CT is connected in series with the line. PT is connected across the line. In CT number of turns is inversely proportional to current and in PT number of turns is inversely proportional to voltage.
09․ In relay coil which is used?
CT used in the relay coil for reducing heavy current of a power circuit to a suitable value of operate the relay. | https://www.electrical4u.com/electrical-mcq.php?subject=power-systems&page=48 |
- In a large mixing bowl, add the following ingredients: the beef, pork, bread crumbs, egg, salt, pepper, and ¼ cup of the spicy mustard.
- Using your hands like we did (or a large spoon if you don’t want to get your hands dirty), mix the ingredients together until they’re fully combined.
- Using a teaspoon measuring spoon, form the meatballs, rolling the meatballs in your hands if you need to. We actually shaped our individual meatballs using a heaping teaspoon (possibly closer to 1 ½ teaspoons) but what truly matters is that your meatballs are all equally sized. Divide the meatballs as equally as you can into 3 batches. Stick the first batch in for 10 minutes so the meatballs have a chance to firm up. The original recipe said you could stick the meatballs in for 10-15 minutes but we tried sticking a batch in for 15 minutes one time and the meatballs came out like meat ice cubes !
- Take a large skillet out and pour the oil in, setting the heat to medium-high. Wait for the oil to get hot before adding the first batch of meatballs to the skillet. (Stick the second batch of meatballs in the freezer after you put the first batch in the skillet.) Our oil actually got hot quicker than we planned so our meatballs only had 5 minutes to be in the freezer before getting put in the skillet. The meatballs actually held their shape so if you’re in a rush, you only need to keep your meatballs in the freezer for 5-6 minutes. Once your meatballs are in the skillet though, allow them to cook for somewhere between 5 to 7 minutes (5 minutes worked for us), turning them as they’re cooking so you can ensure they’re getting cooked equally all over (a nice crust on the meatballs is a bonus to the stirring of the meatballs). After the 5-7 minutes has passed, scoop the meatballs out and put them on a plate afterwards. Wait for the oil to get hot again before adding the second batch into the skillet. Stick the third batch in the freezer afterwards. Cook the meatballs, take them out, wait for the oil to get hot again and add the last batch in, cooking it and adding it to the pile of cooked meatballs after its 5 minutes of cooking is over with.
- Now that all the meatballs are cooked and out of the skillet, drain the oil from the skillet (the recipe never says to drain but there was just so much oil in the skillet, it only felt right to drain it). Pour the whiskey and beef broth in (to play it safe, we took the skillet over to the sink when pouring the whiskey in). Wait for the mixture to come to a boil* and then allow it to reduce for 5 minutes or so, just long enough to get the mixture a little bit thicker.
- Pour the Worcestershire sauce and the other ¼ cup of spicy mustard into the skillet, stirring to combine as well as you can. This should take a couple of minutes to achieve.
- Pour the heavy cream into the skillet, stirring as you do so until everything looks blended together.
- Now take all those meatballs you cooked earlier and add them back into the skillet.
- Bring the heat down to a simmer and allow everything to cook for 5-7 minutes or however long it takes for the meatballs to get cooked through completely. Honestly our meatballs were fully cooked during step 4 so we just let the dish simmer for the minimum time.
- Serve right away !
*When the whiskey starts getting heated up, all of its wonderfully sweet aromas start blooming and it’s just awesome. There’s no mistaking that sweet whiskey aroma !
This recipe was initially made as an appetizer (each one is only 1 ¼ inch in diameter) and you could eat them that way, but we eat this as an actual meal because they’re too delicious to eat just as a snack ! The meatballs themselves have a nice crust on the outside but the meat is still soft and tender on the inside and the sauce, man that’s where the flavor’s really at ! The tanginess of the mustard really comes through but there’s just a little bit of sweetness from the whiskey and the heavy cream just makes the sauce have a silky consistency. The sauce is so delicious that you’ll want to consume it like a soup instead !
We got this recipe from a Ree Drummond cookbook.
We weren’t paid in any way to mention French’s, Jack Daniels, Swanson, or Ree Drummond.
Take care everybody ! | https://sweetnsavorytherapy.com/2019/01/25/whiskey-mustard-meatballs/ |
8.
Written by José Santiago, Digital Content Specialist, Public Engagement , World Economic Forum. (n.d.). 15 quotes on climate change by world leaders. Retrieved December 04, 2017, from https://www.weforum.
org/agenda/2015/11/15-quotes-on-climate-change-by-world-leaders/7. Jacobson, M. Z., & Delucchi, M. A. (2011). Providing all global energy with wind, water, and solar power, Part I: Technologies, energy resources, quantities and areas of infrastructure, and materials.
Energy Policy, 39(3), 1154-1169. doi:10.1016/j.enpol.2010.11.0406.
Global Climate Change. (2014, June 02). Retrieved December 02, 2017, from https://climate.nasa.gov/5. Extreme Weather. (n.
d.). Retrieved November 11, 2017, from http://nca2014.
globalchange.gov/highlights/report-findings/extreme-weather#intro-section-24. Deng, B., Jia, B., & Zhang, Z. (2016). Dynamic Wireless Charging For Roadway-Powered Electric Vehicles: A Comprehensive Analysis And Design. Progress In Electromagnetics Research C, 69, 1-10.
doi:10.2528/pierc160711063. Delucchi, M. A., & Jacobson, M. Z.
(2011). Providing all global energy with wind, water, and solar power, Part II: Reliability, system and transmission costs, and policies. Energy Policy, 39(3), 1170-1190. doi:10.1016/j.enpol.
2010.11.0452.
Clean Cities Alternative Fuel Price Report. (2017, October). Retrieved December 02, 2017, from https://www.afdc.energy.gov/publications/search/keyword/?q=alternative fuel price report1.
Carbon dioxide concentration | NASA Global Climate Change. (2017, May 17). Retrieved November 20, 2017, from https://climate.nasa.gov/vital-signs/carbon-dioxide/Bibliography In terms of cost, availability, CO2 emissions, and general pollution, alternative fuels will be a significant improvement over what we currently use.
As seen in the previous tables, cost for natural and clean energy will continue to decrease and fossil energy cost will continue to rise. It will continue to rise because fossil fuels are not renewable and they are being depleted quickly. While the fossil fuels are becoming scarcely available, wind-solar-and water are everywhere; no matter how much we use today, the same amount will be available in the future. As far as CO2 emissions are concerned,of the prices are increasing.
conventional fuels. As indicated by Table 1, most delivered. Social cost is included for prices for the most used fuels. *This table shows the price per kWh*This table shows recent retailTable 1: Table 2: When compared to current fuel- gas, diesel, coal, etc.-the future cost of alternative fuels will be cheaper than fossil fuels. Alternative fuel prices are decreasing as fossil fuel prices are increasing as shown by Table 1 and Table 2 below.
According to Jacobson and Delucchi, in 2011 the total power required to satisfy all end users was approximately 12.5 trillion watts (TW). They predicted that by 2030 the world would require almost 17 TW in power. They also reported that wind energy worldwide can produce 1700 TW, water energy combined (tidal, hydroelectric, and wave) worldwide can produce 8.
3 TW, and solar energy can produce 6500 TW worldwide 7. These numbers clearly meet the power requirements. These fuels can be used to power EVs. As far as the abundance of roadways, roadway power would not be practical in all locations. It would perform best if it was implemented on major highways.
Installing and transitioning to these alternative methods of energy would be costly, but as demand increases the cost would decrease. A question that arises when discussing alternative fuels is: where will the energy come from? There are several ways we can produce clean energy by utilizing the planet’s natural and renewable resources. One resource is wind.
Wind turbines are powered by the speed of the wind blowing over the blades causing them to turn which rotates a mechanism attached to a generator, in turn producing electricity. Wind turbines can be used to power an individual home, wind farms can be connected to a grid to power a large area, or can be placed off-shore to harvest higher power produced by strong off shore winds. Most turbines can produce power from relatively low wind speeds, but faster wind = more power.
Also, the turbines have a fail-safe shut off when wind speeds are too high to prevent damage from storms. Another alternative fuel source is water. Water power can be harnessed in number of ways including tidal, hydroelectric, and wave. Solar power is another renewable energy source. Solar panels turn the sunlight into energy. When the aforementioned fuel sources are not feasible (cloudy and rainy locations-no solar power, locations with little to no wind, and/or locations that don’t have large bodies of flowing water- no water power) there is another method for electricity generation for vehicles: roadway power- using electromagnetic fields generated by roadways. This concept is still in its infancy.
A conceptual design and analysis created by researchers Deng, Jia, and Zhang explains how roadway power would function: powering coils would be buried beneath the road and electronic vehicles (EVs) would have a pickup coil installed to collect energy as the EV passes from one section to the next, giving EVs virtually limitless range 4. These examples are just a few of many alternative resources for electricity and fuel for vehicles. Additionally, all said examples are widely abundant across the globe.According to NASA, in 2005 the CO2 levels were at 378.21 parts per million (ppm), and the most recent measurement was taken in October of 2017 with levels rising to 407.06 ppm 1. The CO2 that we are producing comes from industrial factories, power plants, and burning fossil fuels like gasoline, coal, diesel, propane, etc.
So, what can we do to help reduce our impact on climate change? One way is to change the way we obtain and use power. We can greatly diminish the use of fossil fuels by switching to alternative power sources. Some examples for generating power include wind, solar, water, and even using the roadways to generate power. Due to the complexity and detail with these sources of energy, only wind will be discussed in depth with just a brief mention of the other types of energy. In this paper the following topics will be discussed: alternative fuel sources for vehicles, abundance, cost, availability, CO2 emissions, and general pollution.
The information presented in this paper was obtained from reputable sources including scientific journals, peer reviewed journals, and government owned websites, with minimal personal opinions/bias. “There’s one issue that will define the contours of this century more dramatically than any other, and that is the urgent threat of a changing climate.”—Former President of the United States, Barack Obama 8. Climate change is a very complex and many-sided topic, and it is gaining recognition in the scientific community. The world is getting warmer, the climate is changing. Most scientists agree that the climate is changing, however some scientists are in denial that human activity is the cause for the dramatically increasing changes.
According to NASA, ice is melting fast: Arctic ice is melting at a rate of 13.2% per decade and land ice (Greenland/ Antarctica) is disappearing at a rate of 127 Gigatons per year ±39 Gigatons 6. In addition to rapidly melting ice, among other consequences, the weather is wreaking havoc and the rate of serious weather-related disasters is steadily increasing as well. Instances of terrible heatwaves, heavy downpours, floods, hurricanes, winter storms, tornadoes, and other natural disasters are getting worse. For example, back in 2011, Oklahoma and Texas experienced a record breaking heatwave and drought: 100 days over 100°F. Those high temperatures for that length of time were the highest ever recorded since reliable record keeping began in 1895. This drought led to over $10 billion in agricultural losses according to the National Climate Assessment in 2014 5. These are just a few examples of how the climate is changing.
There are strong links to human activity pushing climate change forward, for example: the amount of CO2 in the atmosphere has dramatically risen over the past 70 years. As shown in the graph below (obtained from NASA.gov), the levels of CO2 have had a natural oscillation of highs and lows. However, these levels have skyrocketed since the 1950’s 1. Can We Save the Planet with Alternative Fuels? December 11, 2017Kayla BresnanGeneral Chemistry II Extra CreditCan We Save the Planet with Alternative Fuels? | https://housecleaningwestpalm.com/8-a-jacobson-m-z-2011-providing/ |
How to choose diamond circular saw blade and diamond segments?
Diamond circular saw blade is composed of alloy steel matrix and diamond segment Good diamond saw blades are better than ordinary diamond saw blades in stability and life Then how to choose diamond saw blades, this article will do a detail introduction
2022-07-07
Changes in cutting surfaces during diamond segment processing
Diamond segment in the process of cutting stone, the cutting surface of the segment is constantly changing, so which changes are good, which changes are not good? In this paper, the different cutting surfaces of the segment will be analyzed to find the cause and give a corresponding solution
2022-07-08
Five factors affecting product design of diamond segment
In the design and manufacturing process of diamond segment, there are many design directions, but there are two main directions, either the designed segment is sharp, or the segment is life-span So what factors determine the segment design direction? ? There are five main factors
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Arcing process before diamond segment welding
In the diamond segment welding process, the segment bottom and the saw blade base are welded, the arc of the segment bottom and the arc of the saw blade base teeth have some deviations If the welding is forced, the welding strength cannot be reached due to the small welding surface It is required that segments are likely to be dropped during processing, so segment arcing has become a process that must be mastered and implemented
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Application of diamond segment on granite saw blades
Diamond is the world s hardest material, according to the molding process is different, divided into synthetic diamond and natural diamond products, due to the natural diamond storage is small, expensive and so on, synthetic diamond has become the mainstream for industrial diamond products
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Three factors affecting diamond segment cutting granite
The diamond blade is used on the diamond circular saw blade During the cutting process, various problems will be encountered, such as slow cutting efficiency, fast blade consumption, unstable cutting of the blade, slippage of the blade, and inability to cut the blade The problems of moving stone, cutting offset of the diamond segment, etc , these are the problems that often occur when the diamond segment is sawing granite, so besides the problem of the diamond segment, are there other factors that are affected by it?
2022-07-08
The purpose and value of recycling of used diamond segments
At present, many companies and individuals are recycling diamond segments What is the purpose and significance of these segment recycling? What is the price of these segments for recycling? Why would anyone recycle this material? Many people don t understand This article will tell you where did the obsolete diamond segment finally flow?
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Characteristics and application of arc diamond segment
The arc diamond segment is a type of diamond segment often used in diamond circular saw blades From the shape point of view, the bottom and upper part of the arc segment are curved, and the entire segment is shaped like a circle Part of the runway interception, the arc-shaped segment is mainly used for circular saw blades with a diameter of 115-850mm
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How is the diamond segment produced?
Diamond segment production process is actually very cumbersome, especially special custom diamond segment, need to know the hardness of the customer cutting object, wear resistance after the formulation of the deployment, in order to carry out production, and the production process needs to be constantly adjusted to produce a better diamond segment product
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Matching choice of diamond segment and substrate for diamond circular saw blades
When the diamond segment is applied to the diamond saw blade, it is welded with the substrate to form a diamond circular saw blade, which is used to cut stone, ceramic tiles, concrete and other materials In the process of using diamond segment, according to the different matrix, it is matched with different segments, so what principle is the matching method based on? This article mainly introduces the structure of diamond circular saw blade and segment matching and the reason for matching
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Which is better for vacuum sintering and hot press sintering diamond segment? | https://www.linxingstone.com/index.php?m=search&c=index&a=tag&tag=151 |
Arrays
There are four kinds of arrays:
int * p;
These are simple pointers to data, analogous to C pointers. Pointers are provided for interfacing with C and for specialized systems work. There is no length associated with it, and so there is no way for the compiler or runtime to do bounds checking, etc., on it. Most conventional uses for pointers can be replaced with dynamic arrays, out and ref parameters, and reference types.
int [3] s;
These are analogous to C arrays. Static arrays are distinguished by having a length fixed at compile time.
The total size of a static array cannot exceed 16Mb. A dynamic array should be used instead for such large arrays.
A static array with a dimension of 0 is allowed, but no space is allocated for it. It's useful as the last member of a variable length struct, or as the degenerate case of a template expansion.
Static arrays are value types. Unlike in C and D version 1, static arrays are passed to functions by value. Static arrays can also be returned by functions.
int [] a;
Dynamic arrays consist of a length and a pointer to the array data. Multiple dynamic arrays can share all or parts of the array data.
Array Declarations
There are two ways to declare arrays, prefix and postfix. The prefix form is the preferred method, especially for non-trivial types.
Prefix Array Declarations
Prefix declarations appear before the identifier being declared and read right to left, so:
int [] a; int [4][3] b; int [][5] c; int *[]*[3] d; int []* e;
Postfix Array Declarations
Postfix declarations appear after the identifier being declared and read left to right. Each group lists equivalent declarations:
int [] a; int a[]; int [4][3] b; int [4] b[3]; int b[3][4]; int [][5] c; int [] c[5]; int c[5][]; int *[]*[3] d; int *[]* d[3]; int * (*d[3])[]; int []* e; int (*e)[];
Rationale: The postfix form matches the way arrays are declared in C and C++, and supporting this form provides an easy migration path for programmers used to it.
There are two broad kinds of operations to do on an array - affecting the handle to the array, and affecting the contents of the array. C only has operators to affect the handle. In D, both are accessible.
The handle to an array is specified by naming the array, as in p, s or a:
int * p; int [3] s; int [] a; int * q; int [3] t; int [] b; p = q; p = s.ptr; p = a.ptr; s = ...; a = p; a = s; a = b;
Slicing an array means to specify a subarray of it. An array slice does not copy the data, it is only another reference to it. For example:
int [10] a; int [] b; b = a[1..3]; foo(b[1]); a[2] = 3; foo(b[1]);
The [] is shorthand for a slice of the entire array. For example, the assignments to b:
int [10] a; int [] b; b = a; b = a[]; b = a[0 .. a.length];
are all semantically equivalent.
Slicing is not only handy for referring to parts of other arrays, but for converting pointers into bounds-checked arrays:
int * p; int [] b = p[0..8];
When the slice operator appears as the lvalue of an assignment expression, it means that the contents of the array are the target of the assignment rather than a reference to the array. Array copying happens when the lvalue is a slice, and the rvalue is an array of or pointer to the same type.
int [3] s; int [3] t; s[] = t; s[] = t[]; s[1..2] = t[0..1]; s[0..2] = t[1..3]; s[0..4] = t[0..4]; s[0..2] = t;
Overlapping copies are an error:
s[0..2] = s[1..3]; s[1..3] = s[0..2];
Disallowing overlapping makes it possible for more aggressive parallel code optimizations than possible with the serial semantics of C.
If a slice operator appears as the lvalue of an assignment expression, and the type of the rvalue is the same as the element type of the lvalue, then the lvalue's array contents are set to the rvalue.
int [3] s; int * p; s[] = 3; p[0..2] = 3;
The binary operator ~ is the cat operator. It is used to concatenate arrays:
int [] a; int [] b; int [] c; a = b ~ c;
Many languages overload the + operator to mean concatenation. This confusingly leads to, does:
"10" + 3 + 4
produce the number 17, the string "1034" or the string "107" as the result? It isn't obvious, and the language designers wind up carefully writing rules to disambiguate it - rules that get incorrectly implemented, overlooked, forgotten, and ignored. It's much better to have + mean addition, and a separate operator to be array concatenation.
Similarly, the ~= operator means append, as in:
a ~= b;
Concatenation always creates a copy of its operands, even if one of the operands is a 0 length array, so:
a = b; a = b ~ c[0..0];
Appending does not always create a copy, see setting dynamic array length for details.
Many array operations, also known as vector operations, can be expressed at a high level rather than as a loop. For example, the loop:
T[] a, b; ... for (size_t i = 0; i < a.length; i++) a[i] = b[i] + 4;
assigns to the elements of a the elements of b with 4 added to each. This can also be expressed in vector notation as:
T[] a, b; ... a[] = b[] + 4;
A vector operation is indicated by the slice operator appearing as the lvalue of an =, +=, -=, *=, /=, %=, ^=, &= or |= operator. The rvalue can be an expression consisting either of an array slice of the same length and type as the lvalue or an expression of the element type of the lvalue, in any combination. The operators supported for vector operations are the binary operators +, -, *, /, %, ^, & and |, and the unary operators - and ~.
The lvalue slice and any rvalue slices must not overlap. The vector assignment operators are evaluated right to left, and the other binary operators are evaluated left to right. All operands are evaluated exactly once, even if the array slice has zero elements in it.
The order in which the array elements are computed is implementation defined, and may even occur in parallel. An application must not depend on this order.
Implementation note: many of the more common vector operations are expected to take advantage of any vector math instructions available on the target computer.
int [3] abc; int [] def = [ 1, 2, 3 ]; void dibb( int * array) { array[2]; *(array + 2); } void diss( int [] array) { array[2]; *(array + 2); } void ditt( int [3] array) { array[2]; *(array + 2); }
Experienced FORTRAN numerics programmers know that multidimensional "rectangular" arrays for things like matrix operations are much faster than trying to access them via pointers to pointers resulting from "array of pointers to array" semantics. For example, the D syntax:
double [][] matrix;
declares matrix as an array of pointers to arrays. (Dynamic arrays are implemented as pointers to the array data.) Since the arrays can have varying sizes (being dynamically sized), this is sometimes called "jagged" arrays. Even worse for optimizing the code, the array rows can sometimes point to each other! Fortunately, D static arrays, while using the same syntax, are implemented as a fixed rectangular layout:
double [3][3] matrix;
declares a rectangular matrix with 3 rows and 3 columns, all contiguously in memory. In other languages, this would be called a multidimensional array and be declared as:
double matrix[3,3];
Within the [ ] of a static or a dynamic array, the symbol $ represents the length of the array.
int [4] foo; int [] bar = foo; int * p = &foo[0]; bar[] bar[0 .. 4] bar[0 .. $] bar[0 .. bar.length] p[0 .. $ ] bar[0]+$ bar[$ -1]
Static array properties are:
Static Array Properties Property Description .init Returns an array literal with each element of the literal being the .init property of the array element type. .sizeof Returns the array length multiplied by the number of bytes per array element. .length Returns the number of elements in the array. This is a fixed quantity for static arrays. It is of type size_t. .ptr Returns a pointer to the first element of the array. .dup Create a dynamic array of the same size and copy the contents of the array into it. .idup Create a dynamic array of the same size and copy the contents of the array into it. The copy is typed as being immutable. D 2.0 only .reverse Reverses in place the order of the elements in the array. Returns the array. .sort Sorts in place the order of the elements in the array. Returns the array.
Dynamic array properties are:
Dynamic Array Properties Property Description .init Returns null. .sizeof Returns the size of the dynamic array reference, which is 8 in 32-bit builds and 16 on 64-bit builds. .length Get/set number of elements in the array. It is of type size_t. .ptr Returns a pointer to the first element of the array. .dup Create a dynamic array of the same size and copy the contents of the array into it. .idup Create a dynamic array of the same size and copy the contents of the array into it. The copy is typed as being immutable. D 2.0 only .reverse Reverses in place the order of the elements in the array. Returns the array. .sort Sorts in place the order of the elements in the array. Returns the array.
For the .sort property to work on arrays of class objects, the class definition must define the function: int opCmp(Object) . This is used to determine the ordering of the class objects. Note that the parameter is of type Object , not the type of the class.
For the .sort property to work on arrays of structs or unions, the struct or union definition must define the function: int opCmp(ref const S) const . The type S is the type of the struct or union. This function will determine the sort ordering.
Examples:
int * p; int [3] s; int [] a; p.length; s.length; a.length; p.dup; s.dup; a.dup;
The .length property of a dynamic array can be set as the lvalue of an = operator:
array.length = 7;
This causes the array to be reallocated in place, and the existing contents copied over to the new array. If the new array length is shorter, the array is not reallocated, and no data is copied. It is equivalent to slicing the array:
array = array[0..7];
If the new array length is longer, the remainder is filled out with the default initializer.
To maximize efficiency, the runtime always tries to resize the array in place to avoid extra copying. It will always do a copy if the new size is larger and the array was not allocated via the new operator or resizing in place would overwrite valid data in the array.
char [] a = new char [20]; char [] b = a[0..10]; char [] c = a[10..20]; char [] d = a; b.length = 15; b[11] = 'x'; d.length = 1; d.length = 20; c.length = 12; c[5] = 'y'; a.length = 25; a[15] = 'z';
For example:
To guarantee copying behavior, use the .dup property to ensure a unique array that can be resized. Also, one may use the phobos .capacity property to determine how many elements can be appended to the array without reallocating.
These issues also apply to appending arrays with the ~= operator. Concatenation using the ~ operator is not affected since it always reallocates.
Resizing a dynamic array is a relatively expensive operation. So, while the following method of filling an array:
int [] array; while (1) { c = getinput(); if (!c) break ; ++array.length; array[array.length - 1] = c; }
will work, it will be inefficient. A more practical approach would be to minimize the number of resizes:
int [] array; array.length = 100; for (i = 0; ; i++) { c = getinput(); if (!c) break ; if (i == array.length) array.length *= 2; array[i] = c; } array.length = i;
Picking a good initial guess is an art, but you usually can pick a value covering 99% of the cases. For example, when gathering user input from the console - it's unlikely to be longer than 80.
Also, you may wish to utilize the phobos reserve function to pre-allocate array data to use with the append operator.
If the first parameter to a function is an array, the function can be called as if it were a property of the array:
int [] array; void foo( int [] a, int x); foo(array, 3); array.foo(3);
It is an error to index an array with an index that is less than 0 or greater than or equal to the array length. If an index is out of bounds, a RangeError exception is raised if detected at runtime, and an error if detected at compile time. A program may not rely on array bounds checking happening, for example, the following program is incorrect:
try { for (i = 0; ; i++) { array[i] = 5; } } catch (RangeError) { }
for (i = 0; i < array.length; i++) { array[i] = 5; }
The loop is correctly written:
Implementation Note: Compilers should attempt to detect array bounds errors at compile time, for example:
int [3] foo; int x = foo[3];
Insertion of array bounds checking code at runtime should be turned on and off with a compile time switch.
Pointers are initialized to null .
. Static array contents are initialized to the default initializer for the array element type.
Dynamic arrays are initialized to having 0 elements.
Associative arrays are initialized to having 0 elements.
Void initialization happens when the Initializer for an array is void. What it means is that no initialization is done, i.e. the contents of the array will be undefined. This is most useful as an efficiency optimization. Void initializations are an advanced technique and should only be used when profiling indicates that it matters.
Static initalizations are supplied by a list of array element values enclosed in [ ]. The values can be optionally preceded by an index and a :. If an index is not supplied, it is set to the previous index plus 1, or 0 if it is the first value.
int [3] a = [ 1:2, 3 ];
This is most handy when the array indices are given by enums:
enum Color { red, blue, green }; int value[Color.max + 1] = [ Color.blue :6, Color.green:2, Color.red :5 ];
These arrays are statically allocated when they appear in global scope. Otherwise, they need to be marked with const or static storage classes to make them statically allocated arrays.
A string is an array of characters. String literals are just an easy way to write character arrays. String literals are immutable (read only).
char [] str1 = "abc" ; char [] str2 = "abc" .dup; immutable ( char )[] str3 = "abc" ; immutable ( char )[] str4 = str1; immutable ( char )[] str5 = str1.idup;
The name string is aliased to immutable(char)[] , so the above declarations could be equivalently written as:
char [] str1 = "abc" ; char [] str2 = "abc" .dup; string str3 = "abc" ; string str4 = str1; string str5 = str1.idup;
char[] strings are in UTF-8 format. wchar[] strings are in UTF-16 format. dchar[] strings are in UTF-32 format.
Strings can be copied, compared, concatenated, and appended:
str1 = str2; if (str1 < str3) ... func(str3 ~ str4); str4 ~= str1;
with the obvious semantics. Any generated temporaries get cleaned up by the garbage collector (or by using alloca() ). Not only that, this works with any array not just a special String array.
A pointer to a char can be generated:
char * p = &str[3]; char * p = str;
Since strings, however, are not 0 terminated in D, when transferring a pointer to a string to C, add a terminating 0:
str ~= "\0" ;
or use the function std.string.toStringz .
The type of a string is determined by the semantic phase of compilation. The type is one of: char[], wchar[], dchar[], and is determined by implicit conversion rules. If there are two equally applicable implicit conversions, the result is an error. To disambiguate these cases, a cast or a postfix of c, w or d can be used:
cast ( immutable ( wchar ) []) "abc" "abc"w
String literals that do not have a postfix character and that have not been cast can be implicitly converted between string, wstring, and dstring as necessary.
char c; wchar w; dchar d; c = 'b'; w = 'b'; w = 'bc'; w = "b" [0]; w = "\r" [0]; d = 'd';
printf() is a C function and is not part of D. printf() will print C strings, which are 0 terminated. There are two ways to use printf() with D strings. The first is to add a terminating 0, and cast the result to a char*:
str ~= "\0" ; printf( "the string is '%s'
" , cast ( char *)str);
or:
import std.string; printf( "the string is '%s'
" , std.string.toStringz(str));
String literals already have a 0 appended to them, so can be used directly:
printf( "the string is '%s'
" , cast ( char *) "string literal" );
So, why does the first string literal to printf not need the cast? The first parameter is prototyped as a const(char)*, and a string literal can be implicitly cast to a const(char)*. The rest of the arguments to printf, however, are variadic (specified by ...), and a string literal is passed as a (length,pointer) combination to variadic parameters.
The second way is to use the precision specifier. The length comes first, followed by the pointer:
printf( "the string is '%.*s'
" , str.length, str.ptr);
The best way is to use std.stdio.writefln, which can handle D strings:
import std.stdio; writefln( "the string is '%s'" , str);
A pointer T* can be implicitly converted to one of the following:
void*
A static array T[dim] can be implicitly converted to one of the following:
T []
const( U )[]
void[]
A dynamic array T[] can be implicitly converted to one of the following:
const( U )[]
void[]
Where U is a base class of T.
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Hòn Mây Cát BàHòn Mây Cát Bà is an island in Haiphong. Hòn Mây Cát Bà is situated southeast of Hang Suối, northwest of Hòn Nghiên.
Localities in the Area
Cát Bà
TownPhoto: Binh Giang, Public domain.
Hòn Mây Cát Bà
- Type: Island
- Description: island in Vietnam
- Category: island
- Location: Haiphong, Northern Vietnam, Vietnam, Southeast Asia, Asia
Latitude20.7127° or 20° 42' 45.8" north
Longitude107.0647° or 107° 3' 52.8" east
Elevation31 metres (102 feet)
GeoNames ID8595241
Also Known As
- Cebuano: Hòn Mây Cát Bà
- Vietnamese: Hòn Mây Cát Bà
- Hon May Cat Ba
In the Area
Localities
- Cát Bà
- Hang Suối4 km northwest
- Đồng Khê Sâu7 km northwest
- Chân Châu8 km northwest
Landmarks
- Hòn NghiênIsland, 1½ km southeast
- Hòn Cát DứaIsland, 2½ km northeast
- Hòn Cát Đuôi RồngIsland, 2½ km west
- Hòn Cát Dứa ConIsland, 3 km northeast
Other Places
- Catba Sandy Beach ResortHotel, 5 km north
- Entrée ProfondeMarine channel, 6 km northeast
- Princes Hotel CatbaHotel, 9 km northwest
- Passe HenrietteMarine channel, 13 km northeast
Explore Your World
- Ilha da BoegaIsland, Portugal
- Channel RockIsland, Massachusetts, United States
- Gourd IslandIsland, Alaska, United States
- Turtle IslandIsland, Hancock County, Maine
- Ko RaetIsland, Thailand
- Cane Break IslandIsland, Georgia, United States
Popular Destinations in Northern Vietnam
Escape to a Random Place
|French PolynesiaPolynesia||NagasakiJapan|
|BatumiGeorgia||IpohMalaysia|
About Mapcarta. Thanks to Mapbox for providing amazing maps. Text is available under the CC BY-SA 4.0 license, excluding photos, directions and the map. Photo: Inkey, CC BY-SA 3.0. | https://mapcarta.com/29536160 |
Teenage Mutant Ninja Turtles III
Teenage Mutant Ninja Turtles III is a 1993 American martial arts superhero comedy film written and directed by Stuart Gillard. Based on the fictional superhero team the Teenage Mutant Ninja Turtles, it is the sequel to the film Teenage Mutant Ninja Turtles II: The Secret of the Ooze (1991) and is the final installment of the original trilogy. It was produced by Clearwater Holdings Ltd. and Golden Harvest. With the voices of Brian Tochi, Corey Feldman, Tim Kelleher, Robbie Rist, and James Murray. This was the last Teenage Mutant Ninja Turtles film released by New Line Cinema and released on VHS along with Columbia TriStar Home Video. Like the previous film, it was internationally distributed by 20th Century Fox.
|Teenage Mutant Ninja Turtles III|
Theatrical release poster
|Directed by||Stuart Gillard|
|Produced by|
|Written by||Stuart Gillard|
|Based on|
|Starring|
|Music by||John Du Prez|
|Cinematography||David Gurfinkel|
|Edited by|
Production
companies
|Distributed by|
Release date
Running time
|96 minutes|
|Country||United States|
|Language||English|
|Budget||$50 million|
|Box office||$42.2 million|
(United States)
With this film, the All Effects Company provided the animatronics, rather than Jim Henson's Creature Shop, which acted as the providers for the previous films. Despite being a moderate box office success, it is the lowest rated entry in the series.
Plot
In 1603, in feudal Japan, a young man is being chased by four samurai on horseback. As they go into the woods, a mysterious woman emerges from the underbrush and watches closely. However, the samurai eventually capture and take the youth, revealed to be a prince named Kenshin, with them.
In the present, two years after the events of the previous film with the defeat of The Shredder and The Foot Clan, April O'Neil has been shopping at the flea market in preparation for her upcoming vacation. She brings her friends gifts to cheer them up. Michelangelo is given an old lamp (the lampshade of which he wears as an impression of Elvis Presley in "Blue Hawaii"), Donatello is given a broken radio to fix, Leonardo is given a book on swords, and Raphael is to receive a fedora but, having stormed off earlier, he is never formally given it. For Splinter, she brings an ancient Japanese scepter. Back in the past, Kenshin is being scolded at by his father, Lord Norinaga, for disgracing their family name, but Kenshin argues that his father's desire for war is the true disgrace. Their argument is interrupted by Walker, an English trader who has come to supply Norinaga with added manpower and firearms, and Kenshin leaves his father's presence to brood alone in a temple. There, he finds the same scepter and reads the inscription: "Open Wide the Gates of Time".
In the present, April is looking at the scepter and it begins to light up. She is then sent back in time, while Kenshin takes her place; each wears what the other did. Upon arrival, April is accused of being a witch, but Walker deduces she has no power and has April put in prison to suffer. Back in the present, Kenshin is highly distressed upon seeing the turtles and calls them "kappa". After learning from Kenshin of the situation, the turtles decide to go back in time to get April. However, according to Donatello's calculations, they have to do it within 60 hours, otherwise the scepter's power will disappear due to the space-time continuum being out of sync. They bring in Casey Jones to watch over the lair and use the scepter to warp through time. When doing so, the turtles are replaced by four of Norinaga's Honor Guards and are confused at their new surroundings.
Back in time, the turtles awake on horseback and make a poor show of riding their steeds. During the confusion, Mikey (who is carrying the scepter) ends up riding off alone into the forest and gets ambushed by an unknown assailant. The others go to search for April at Norinaga's castle, where their identity as Honor Guards allows them cover in their search. After following Niles, one of Walker's thugs into the prison, the turtles rescue April and also free another prisoner named Whit (locked up for trying to start a mutiny against Walker, and who bears a striking resemblance to Casey), but their sloppy escape ends up leaving them all alone in the wilderness and without a clue where to go. Meanwhile, in the present, Kenshin is getting impatient and anticipates a fight from Casey. Casey instead introduces him and the Honor Guards to television hockey, which manages to calm them down for the time being.
Out in the woods, the turtles, April, and Whit are again attacked, this time by villagers mistaking them for Norinaga's forces. The attack stops when Mitsu, leader of the rebellion against Lord Norinaga, unmasks Raphael and sees that he looks just like one of her prisoners. The turtles realize that she is talking about Mikey and accompany Mitsu to her village. When they arrive, the village is being burned down by Walker's men. As the turtles help the villagers save it, Mikey is let out by a pair of clueless soldiers and joins in the fight. Walker is forced to retreat, but the fire continues to burn and has trapped a young boy named Yoshi inside a house. Michelangelo saves Yoshi from the fire, then Leonardo helps him recover by performing CPR.
As Walker continues bargaining with Lord Norinaga over buying guns in exchange for gold, the turtles spend some time in the village. Donatello decides to have a replica scepter made so they can get back home, while Michaelangelo teaches some of the people about pizza and later tries to console Mitsu about Kenshin, whom she is in love with. Raphael also gets in touch with his sensitive side through the child Yoshi, and teaches Yoshi how to control his temper. Back in the present, the Honor Guards from the past are quickly adjusting to life in the 20th Century, and Casey decides to challenge them to a hockey game. To Casey's dismay, the Honor Guards think hockey is about beating up each other. Meanwhile, Kenshin and Splinter fear that the ninja turtles will not return home in time before their sixty hours are up.
In the past, the replica scepter is completed, but an argument between Michelangelo and Raphael ends up breaking it. To make matters worse, Mitsu informs them that Lord Norinaga has agreed to purchase Walker's guns and will attack the village in the morning. However, when Raphael sneaks off to visit Yoshi, he is surprised to find the original scepter in the child's possession. The turtles are overjoyed to see it but are angry at Mitsu for hiding it and essentially forcing them to fight her war. However, Mitsu's grandfather clarifies that it was his idea to have the turtles fight in her place.
Suddenly, Whit betrays everybody and captures Mitsu, and the turtles return to Norinaga's palace to save her. After rescuing her, they are cornered by Norinaga and are made to fight waves of his soldiers. The turtles respond by freeing the prisoners in the palace, starting an all-out war on the palace grounds. After a while of fighting, Leo defeats Lord Norinaga in a heated sword duel, comedically finishing him by cutting his hair and then trapping him inside of a bell. Deciding to cut his losses, Walker takes the scepter and tries to escape to his boat. When cornered by the turtles at the dock, Walker throws the scepter into the air as a distraction. The turtles catch the scepter, while Whit (who reformed after Walker went back on a deal they had made) launches a fireball from a catapult at Walker and knocks him off the dock to his death.
The turtles are now ready to return to their own time, but Mikey says he would rather stay (in particular because he wanted to be with Mitsu). Raphael decides he wants to stay as well because he feels like the Turtles are appreciated in Japan unlike back home. The other turtles and April try to convince them otherwise until Kenshin activates the scepter and makes the decision harder. After a long debate (which included Mitsu telling Mikey to keep his promise about Kenshin returning to the past), Michelangelo reluctantly agrees to go home with his brothers, but just barely misses grabbing the scepter in time. The Honor Guards switch back with the Turtles (all except for Michelangelo). Fortunately, the last remaining Honor Guard Benkei activates the scepter and swaps places with Mikey just before the scepter burns out.
In the past, Norinaga admits surrender to Mitsu and Kenshin, and the two lovers share a tender reunion. Meanwhile, Michaelangelo is depressed over the thought of growing up, but Splinter cheers him up by performing the "lampshade Elvis" impression, and the rest of the turtles join in with a final dance number.
Cast
Live actors
- Paige Turco as April O'Neil
- Elias Koteas as Casey Jones / Whit
- Stuart Wilson as Walker
- John Aylward as Niles
- Sab Shimono as Lord Norinaga
- Vivian Wu as Mitsu
- Henry Hayashi as Kenshin
- Travis A. Moon as Yoshi
Voice cast
- Brian Tochi as Leonardo
- Corey Feldman as Donatello
- Tim Kelleher as Raphael
- Robbie Rist as Michaelangelo
- James Murray as Splinter
Rist and Tochi (who did the voices of Michaelangelo and Leonardo, respectively) are the only two voice actors to voice the same character throughout all three live-action TMNT movies. Corey Feldman returned as the voice of Donatello after being absent for the second movie.
Puppeteers
- Jim Martin as Leonardo (face performance)
- Mark Caso as Leonardo (in-suit performer)
- Rob Mills as Donatello (facial assistant)
- Jim Raposa as Donatello (in-suit performer)
- Noel MacNeal as Raphael (face performance)
- Matt Hill as Raphael (in-suit performer)
- Gord Robertson as Michelangelo (face performance)
- David Fraser as Michaelangelo (in-suit performer)
- James Murray as Splinter (principal puppetry)
- Lisa Sturz as Splinter (assisted puppetry)
- Tim Lawrence as Splinter (assisted puppetry)
Reception
Critical response
Reviews for the film have been largely negative by critics. Based on a sample of 30 reviews, the film holds a 23% rating on Rotten Tomatoes with the consensus "It's a case of one sequel too many for the heroes in a half shell, with a tired time-travel plot gimmick failing to save the franchise from rapidly diminishing returns." On Metacritic it has a score of 40 out of 100, based on reviews from 12 critics.
Michael Wilmington of The Los Angeles Times noted that distributors deliberately kept the film away from critics. Despite mild praise for the look of the film, Wilmington called the first film a fluke hit and called this third film "sequel hell". James Berardinelli gave it one out of four stars, citing that "any adults accompanying their kids will have to invent new and interesting ways to stay awake. Not only is this movie aimed at young children, the script could have been written by them." TV Guide gave it two out of four stars and said in their review, "If the time-travel gimmick has to be employed twice in a row then it's probably best to banish these characters to a retirement sewer", when commenting about a possible future film invoking time travel.
Box office
Teenage Mutant Ninja Turtles III debuted at No. 1 at the box office. Although it made twice its production budget, returns were barely more than half that of the previous film and roughly one-fifth that of the first; thus, it was considered a disappointment. There would not be another live-action adaptation until the 2014 reboot.
Home media
As with both of the previous films, the British PG version was censored due to usage of forbidden weapons (Michelangelo's nunchaku). For these scenes, alternate material was used. The cuts were waived for the DVD release. The German theatrical and video version was based on the censored UK cut; the DVD is uncut.
The film was released to VHS and Laserdisc in 1993.
The film has been released on two Blu-ray box sets with both of its predecessors.
References
- "Detail view of Movies Page". Afi.com. Retrieved 4 October 2017.
- "New Line to Reprise The Mutant Turtles". p. 10. Retrieved September 22, 2019 – via Adweek.
- "Teenage Mutant Ninja Turtles III". Rottentomatoes.com. Retrieved 4 October 2017.
- https://www.metacritic.com/movie/teenage-mutant-ninja-turtles-iii
- Michael Wilmington (March 22, 1993). "No Spark in Samurai-Style 'Ninja Turtles'". The Los Angeles Times. Archived from the original on 2012-10-11. Retrieved 2018-11-21.
- "Review: Teenage Mutant Ninja Turtles III". preview.reelviews.net. Retrieved 4 October 2017.
- "Teenage Mutant Ninja Turtles III". TVGuide.com. Retrieved 4 October 2017.
- "Weekend Box Office Ninja Turtles Capture Top Spot". The Los Angeles Times. Retrieved 2010-11-09.
- "Weekend Box Office Ninja Turtles' Are Still Power Dudes". The Los Angeles Times. Retrieved 2010-11-09.
- Wurm, Gerald. "Teenage Mutant Ninja Turtles III (Comparison: BBFC PG VHS - BBFC PG DVD) - Movie-Censorship.com". Movie-censorship.com. Retrieved 4 October 2017.
- "Teenage Mutant Ninja Turtles III". Worldcat. 1993. Retrieved 3 May 2015.
- "Teenage Mutant Ninja Turtles III". LDDB. 1993. Retrieved 10 September 2019. | https://zims-en.kiwix.campusafrica.gos.orange.com/wikipedia_en_all_nopic/A/Teenage_Mutant_Ninja_Turtles_III |
A Validated, Subject-Specific Finite Element Model for Predictions of Rotator Cuff Tear Propagation.
Rotator cuff tears are a significant clinical problem previously investigated by unvalidated computational models that either use simplified geometry or isotropic elastic material properties to represent the tendon. The objective of this study was to develop an experimentally validated, finite element model of supraspinatus tendon using specimen-specific geometry and inhomogeneous material properties to predict strains in intact supraspinatus tendon. Three-dimensional tendon surface strains were determined at 60°, 70°, and 90° of glenohumeral abduction for articular and bursal surfaces of supraspinatus tendon during cyclic loading to serve as validation data. A finite element model was developed using the tendon geometry and inhomogeneous material properties to predict surface strains for loading conditions mimicking experimental loading conditions. Experimental strains were directly compared with computational model predictions to validate the model. Overall, the model successfully predicted magnitudes of strains that were within the experimental repeatability of 3% strain of experimental measures on both surfaces of the tendon. Model predictions and experiments showed the largest strains to be located on the articular surface (~8% strain) between the middle and anterior edge of the tendon. Importantly, the reference configuration chosen to calculate strains had a significant effect on strain calculations, and therefore must be defined with an innovative optimization algorithm. This study establishes a rigorously validated, specimen-specific computational model using novel surface strain measurements for use in investigating the function of the supraspinatus tendon and to ultimately predict the propagation of supraspinatus tendon tears based on the tendon's mechanical environment.
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We had a very exciting week in May from the 16th to the 21st here at Earth Tipi! For more than a year I have been working with the Fruit Tree Planting Foundation to bring fruit orchards throughout the Pine Ridge Reservation. Funding was finally identified in October of 2012 and last month we were planting like maniacs! There are a total of five recipient sites that had a total of 200 trees planted plus an additional 100 trees given away to community members who volunteered at the plantings.
The first recipient was Red Cloud Indian School located in the town of Pine Ridge. At that location, 32 trees were planted and we had over 90 participants! We planted 41 trees at Lakota Hope followed by live entertainment and an arbor dedication. Next on the list was Thunder Valley where we were joined by Grow Permaculture who brought 500 pine trees to the reservation. They have been helping to plant trees as well as distributing pine trees to anyone who wants them!
Twenty trees were planted at Oglala Lakota College where the trees will be used for a study, however the event was cancelled due to rain. The fruit from those trees will be used for food canning and preserving workshops down the road! Last in the line up was Little Wound School. Little Wound s a massive school which serves more than 1000 students from preschool to 12th grade. We planted 100 trees there on May 21! | http://earthtipi.org/fruit-tree-planting/ |
German Supreme Court decides on organizer's liability for changing the departure of the return flight to an earlier time
The plaintiff's spouse had booked a one week package holiday to Turkey at EUR 369 per person for the plaintiff and himself. The return flight was scheduled to depart on June 1, 2009, 16:40 hrs. One day before, the organizer changed the departure time to 05:15 hrs and thus the plaintiff and her spouse were to be picked up at the hotel as early as 01:25 hrs. They therefore looked for an alternative return flight which they booked on their own and which departed at 14 hrs.
The plaintiff (to whom her spouse had assigned his claim) sued for reimbursment of the full package price (except EUR 70 for meals), compensation for the costs of the alternative flight and compensation for loss of holiday enjoyment.
The organizer referred to his general conditions of contract due to which
- he had reserved the right to alter the times of fligths as long as this would not affect the overall arrangement of the package;
- consumers were not allowed to assign any claims to other persons.
Both the first instance court and the appelate court only granted a small reduction of the package prize for the plaintiff herself.
Upon further appeal of the latter, the German Civil Supreme Court (BGH) held that the change of the departure time of the return flight constituted malperformance of the contract which entitled the plaintiff to claim for damages if either she had asked the organizer to provide an alternative flight before booking same on her own or such demand seamed unreasonable according to the circumstances . There was no justified interest on behalf of the organizer to interdict an assignment of such claims. However, as the change did not cause any significant detriment with regard to the holiday arrangement there was no claim for loss of holiday enjoyment.
As a result, the Supreme Court referred the matter back to the appelate court to complete the findings on a potential demand to the organizer and the actual costs of the alternative flight booked by the plaintiff. | https://iftta.org/content/german-supreme-court-decides-organizers-liability-changing-departure-return-flight-earlier-t |
My Mom didn't call them Apple Turnovers. She didn't call them fruit pies or apple pockets. She called them hand pies it's the first dessert I can remember her teaching me how to make. She used to fill them with any fruit she found at the local produce stand...peaches, apples, cherries, strawberries and blueberries. So good and so easy to make now that we can buy puffed pastry already made and ready to use.
Line baking sheet with parchment paper and preheat oven to 400 degrees.
Melt butter in a large skillet and add diced apples. Cook for about 3 minutes. Next add brown sugar, lemon juice, cinnamon, cornstarch, and nutmeg. Continue cooking until apples are tender and sauce has begun to thicken. Stir apples occasionally while cooking. Remove apple mixture from heat and add the vanilla. Set aside to cool.
Unfold pastry sheet on a lightly floured work surface. Smooth or roll out seems to make sure they do not separate. With a sharp knife or pizza cutter, cut each sheet into four squares.
Mix the egg white and water together to make an egg wash. Brush egg wash all along the edges of the pastry squares. Next spoon about 1/4 to 1/3 cup of the mixture into the center of the 1 pastry square leave about an inch on all sides. Fold pastry in half diagonally and pinch the edges with a fork to seal. Prick with fork for steam vents. Sprinkle a little course sugar on top if desired. Repeat with remaining squares.
Transfer to an baking sheet. Leave at least an inch between pies. Bake at 400 degrees for 14-16 minutes until the top is puffed and nicely browned. Remove turnovers from pan to a wire rack and cool 10 minutes.
Combine powdered sugar, milk, and vanilla. Mix to a smooth, thick consistency. Add more milk to thin, if necessary. Drizzle glaze over warm turnovers and serve. | http://www.welcome-home-blog.net/2014/10/apple-turnovers.html |
Rats
Adult male Sprague Dawley rats (n = 183; 350–400 g at arrival; Charles River, Quebec, Canada) were single-housed in temperature (24 ± 2 °C) and humidity controlled (55 ± 10%) animal facility rooms. The light:dark cycle was 12:12 h (light on at 7:00 a.m.). Food and water were available ad libitum unless stated otherwise. All procedures were approved by the Institutional Animal Care Committee and complied with the Canadian and National Institute of Health Guides for the Care and Use of Laboratory Rats (NIH Publication #20-23). All rats were behaviorally naïve prior to beginning the set-shift procedure. Following set-shifting testing, rats were allowed to return to their pre-set-shift body weights, prior to undergoing additional behavioral tests.
Surgery
Rats were anesthetized with a ketamine (80 mg/kg; Vetoquinol)/xylazine (6 mg/kg; Bayer) mixture (i.p.) and positioned in a stereotaxic apparatus (David Kopf Instruments). Post-surgical analgesia was ascertained with meloxicam (subcutaneous; 1 mg/kg; Boehringer Ingelheim). The scalp was incised, and burr holes were prepared above the PFC skull region. Two stainless steel guide cannulae (22 G, PlasticsOne) were implanted into the prelimbic PFC (coordinates: 3.1 mm anterior, 1.4 mm lateral from Bregma (10˚ angle), and 3.0 mm ventral to dura; [25]), and secured with anchoring screws and dental cement. Rats were allowed one-week recovery prior to commencement of experiments.
Drugs and injection procedure
(-)-Cannabidiol, NAD299 hydrochloride, AM251 and (+)-MK801 maleate were from Tocris and THC from Cayman Chemical (Ann Arbor, MI, USA). All drugs were dissolved in dimethyl sulfoxide (DMSO; Sigma-Aldrich) and diluted to final concentrations (final DMSO 5%) in saline containing 5% cremophor EL (Sigma-Aldrich). Before testing rats were handled for 5 min per day. For microinfusions, rats were gently restrained prior to insertion of the microinjectors. All microinfusions were 500 nl/hemisphere. Injectors were removed after 1 min and behavioral testing began 5 min later. The following drug concentrations (indicated later with subscript i.e. THC 100 ) were used (in ng/500 nl): THC (10, 50, 100 or 500), CBD (10, 100 or 500), NAD299 (100), AM251 (100 or 200) and (in μg/500 nl) MK801 (3 or 6). When two drugs were tested simultaneously they were injected as a co-mixture. To avoid possible sub-chronic effects of infused substances, any given treatment was administered only once. A seven-day rest period was allowed between the different behavioral tests to allow proper drug washout between tests.
Attentional set-shifting
Prior to training, rats were food restricted to ~85% of their free feeding body weight and familiarized with the reward cue (45 mg sucrose pellets; banana flavor; BioServ, USA). Set-shifting was conducted in an operant chamber (Med-associates, St Albans VT, USA) enclosed in a sound-attenuating box. The test chamber (30.5 × 24 × 21 cm) was equipped with a house light, two retractable levers, pellet receptacle and two cue lights (Fig. 1c). The chamber was computer controlled with customized software procedures (MED-PC IV, Med-Associates) adapted from [26] and [16].
Fig. 1 Histology and set-shift procedure. a Microphotograph of Cresyl violet stained coronal section containing guide canulae tracks for injections directly in the prefrontal cortex (PFC). b Schematic representation of the injection sites (circles) superimposed on the coronal plane of a rat brain (modified from [25]). Numbers on the schemes indicate distance from Bregma point. c Experimental timeline for the attentional set-shift procedure. The grey rectangles represent front panel of the operant box, circles correspond to the light cues (gray = light off; white = light on), white rectangles correspond to extended levers, the lever rewarded with sucrose pellet is indicated with plus sign above it; D1-D4 indicate different days of the procedure. Note that rats were injected with the drug solution on the last day of the procedure Full size image
Briefly, rats were placed in the chamber (house light on, one lever extended) and learned to associate each lever press with a food pellet reward. After 15 presses the lever was retracted, the alternate lever was inserted, and the rat needed to press it 15 times. Subsequently, the levers were randomly inserted into the chamber (15× each) until pressed by the rat. Next, the house light was switched off and a timed lever-pressing trial started (trial duration: 20 s). Trials began with illumination of the chamber and insertion of one lever for 10 s. Pressing the lever resulted in reward and a 4 s light cue. A failed trial (omission trial) resulted in lever retraction, no reward and turning off the house light. After 30 trials the rat was removed from the chamber. The next day the rat was exposed to the timed trials until a performance criterion of ≥85 presses/90 trials was achieved.
Following this, the rats’ preferred lever was determined based on seven sets of trials (set: initial trial + secondary trial/s). All trials began with chamber illumination and insertion of both levers for 10 s. During the initial trial, pressing any lever was rewarded while during the secondary trial/s only pressing the lever other than during initial trial. A set was considered completed when both levers were pressed. The lever with more presses on the initial trials was considered as ‘preferred’.
The next day rats were exposed to 100 visual-cue discrimination trials beginning with illumination of one cue light, which indicated the correct lever position. After 3 s the house light was switched on and both levers inserted for 10 s. Only a correct response was rewarded. The successful training criterion was 10 successive correct presses.
The next day the rat received a specific intra-PFC drug treatment and after 5 min 20 visual-cue discrimination trials were presented to test long-term memory retrieval and motoric functions. Immediately after, the response discrimination procedure began (set-shift = 10 successive correct presses; max 120 trials). Trials were similar to visual discrimination trials, except that the correct response was set to the non-preferred lever (see above) irrespectively of the cue light position.
The total number of trials to set-shift (excluding omissions) and the total number of errors were calculated. Perseverative (>5 errors/16 successive trials) and regressive errors (≤5 errors/16 successive trials) were counted when an error was made by following the light cue and never-reinforced errors when an error was made but the cue light indicated the correct lever.
Spontaneous alternation behavior
The Y-maze (black nonreflecting acrylic, 3 arms at 120˚ from each other; arm length: 50 cm; wall height: 40 cm) was located on the floor and dimly illuminated (40 lux). Following room acclimatization (1 h), intra-PFC-injected rats were placed at the end of one arm facing the wall, and allowed to explore the maze freely for 10 min. Alternation behavior was video recorded and the sequence and total number of arm entries (all paws in) was scored off-line. The alternation score was calculated as unique triplets/(total arm entries-2), where unique triplets = a consecutive entry to all three arms (i.e. ABC; Fig. 5a). Re-entries to the same arm (i.e. AA) and returns to previously visited arm (i.e. ABA) were scored separately.
Elevated plus maze test
The elevated plus maze (EPM) apparatus (black acrylic, 4 arms (10 × 50 cm) stemming from a 10 × 10 cm platform and forming a plus shape) was raised above the floor by 50 cm and was dimly illuminated (40 lux). Two opposite arms were enclosed with 40 cm high walls while other two arms were opened (except for 1 cm high ledge). Intra-PFC-injected rats were placed on the central platform (facing closed arm) and explored the maze for 10 min. Behavior was video recorded and analyzed offline (Behaview software; www.pmbogusz.net). The number of entries (all paws in) and the time spent in closed and open arms was scored.
The three-chambered social approach test
The social interaction apparatus consisted of a transparent acrylic chamber divided into three equal compartments separated with guillotine doors [15]. One day before testing rats were room acclimated for 30 min and subsequently habituated to the apparatus (5 min center + 8 min entire apparatus). Social interaction testing consisted of 2 phases. In phase 1 (social motivation test), following a 30 min acclimatization, rats received assigned intra-PFC microinfusions and placed in the central compartment (5 min; guillotine doors in place). Subsequently, wire enclosures were placed in the side compartments (one contained a stranger male rat) and the tested rat could explore the entire apparatus for 8 min. In phase 2 (social memory test), a new, novel unfamiliar rat was placed in the previously empty enclosure cage and the test rat could explore both chambers (containing the previously encountered rat or the new stranger rat) for 8 min. Behaviors were video recorded and analyzed offline. The total duration of exploratory bouts that the rat spent with a stranger vs. the empty enclosure (phase 1) was calculated as a sociability score = t stranger /(t stranger + t empty ). The time spent with the previously encountered vs. novel stranger rat in phase 2, was calculated as a social recognition score = t novel /(t novel + t familiar ).
Histology
At the end of experiments rats were overdosed with pentobarbital (Euthanyl) and decapitated. Brains were removed, fixed in 10% formalin and cryoprotected in 30% sucrose solution. Coronal sections (50 μm thick) were cut, mounted on glass slides and stained with Cresyl violet. Sections with the cannula tip locations were microphotographed and referred to a rat brain atlas [25].
Statistical analysis
| |
---
abstract: 'We have studied the ordering of the $q$-colours Potts model in two dimensions on a square lattice. On the basis of our observations we propose that if $q$ is large enough the system is not able to break global and local null magnetisation symmetries at zero temperature: when $q<4$ the system forms domains with a size proportional to the system size while for $q>4$ it relaxes towards a non-equilibrium phase with energy larger than the ground state energy, in agreement with the previous findings of De Oliveira et al. [@PetriEL; @Petri].'
author:
- |
M. IBÁÑEZ DE BERGANZA$^{\S}
$[^1], V. LORETO$^{\S}$ and A. PETRI$^{\ddag}$
title: Phase ordering and symmetries of the Potts model
---
Introduction
============
Phase ordering [@Lifshitz; @Bray] is one of the important topics in non-equilibrium statistical mechanics. For systems with two coexisting phases the situation is generally well understood from the analytic and numeric points of view [@Bray; @Gunton; @RefsIsing]. For the Potts model [@Wu] with a $q>2$-degenerated ground state, the situation is not so clear in general. Also in this case, an Allen-Cahn power-law regime [@Grest88] and dynamical scaling relations for the structure factor and for the distribution of domain sizes [@Lau; @Jeppesen; @Sire] have been predicted [@Bray] and found numerically [@Potts-numeric]. On the other hand, a singular behaviour of the Potts model has been recently observed by De Oliveira et. al. [@Petri; @PetriEL] when the degeneracy ($q$) is large enough: in the thermodynamic limit the model was shown to relax towards a ‘glassy’, disordered, phase with a non negligible density of defects when it is quenched at zero temperature.
This paper is devoted to discuss some ideas for understanding this glassy phase. In the next section we recall the phase and report new results on the ordering of the Potts model after a quench at low but finite temperature. The system equilibrates locally nucleating domains that eventually become of the size of system. In section 3 we study the $T=0$ case, and we interpret the impossibility to equilibrate in the thermodynamic limit as an impossibility of breaking the local zero magnetisation. Section 4 is to conclude and summarise the main results.
Dynamics with thermal fluctuations
==================================
De Oliveira et. al. [@PetriEL; @Petri] have recently observed an interesting slow relaxation in the dynamics of the Potts model after a quench from a disordered state to zero temperature, and the impossibility for the system to achieve the ground state in the $L \to \infty$ limit, being $L$ the linear size of the system. Let us briefly describe this effect. Given a lattice ${\mathbf}{L}$, in which each site $i \in {\mathbf}{L}$ can take $q$ equivalent values, or *colours*, the Potts model [@Wu] is defined by the Hamiltonian:
$$H=\frac{1}{2}\sum_{\{i,j\}}(1-\delta_{c_i,c_j})$$
where $\{i,j\}$ means $i \in {\mathbf}L$, $j$ is a neighbour of $i$, and where $c_i \in \{1 \cdots q\}$ is the colour of site $i$. When quenching at zero temperature a two-dimensional system on a square lattice using Glauber single spin-flip dynamics, systems with $q>4$ obey the Allen-Cahn law [@Bray] $e(t) \propto t^{-1/2}$ ($e$ being the energy per site) up to a time $\bar t$, increasing with $L$, when they get trapped in a blocked configuration, invariant with respect to the single spin-flip dynamics at zero temperature [@Spirin; @Godreche; @Derrida96][^2]. Inverting the usual order of the $t \to
\infty$ and $L \to \infty$ limits, i.e., assuming that the thermodynamic limit is taken before the infinite-time limit, and extrapolating to $t \to \infty$ the $e(t)$ data with the Allen-Cahn power-law $e(t) \propto t^{-1/2}$ [@PetriEL], one finds a positive energy $e^* \equiv e(t \to \infty)>0$ for $q>4$ and, in particular the data for $e^*$ is very well fitted by the expression $e^*=bL^{-1/2}+b'(q-4)^{1/2}$, $b,b'$ being real constants. This would imply that an infinite-size system with $q >
4$ relaxes after an infinite time towards a phase with stationary observables and positive energy, different from the ground state with zero energy, and this is the reason for which this out-of-equilibrium ‘phase’ is called ‘glassy phase’ in [@Petri]. A noticeable point is that the onset of this glass-like phase occurs in the two-dimensional Potts model for $q>4$, and this $q$-range coincides with the one in which the system presents a first-order phase transition [@Wu]. A justification of this fact is in progress [@ILP]. We now give a general characterisation of the equilibration of the system at finite temperature, and an interpretation of the non-equilibrium phase in terms of the symmetries of the problem in Section 3.
We have studied the dynamics after a quench at finite temperature, $T=0.1$, of the $7$ colours Potts model with periodic boundary conditions, being $T_c=\ln^{-1}(1+q^{1/2})=0.7730...$ the critical temperature of the model [@Wu]. In Fig. 1 we present the energy per site (on top) and the magnetisation (centre) with respect to $t^{-1/2}$ of a 2-d square lattice system with nearest-neighbours interaction and periodic boundary conditions. The magnetisation, $m$, is
$$m=\frac{q}{q-1} \big( \sum_c^q x_c^2 -1/q \big)$$
where $x_c$ is the fraction of colour $c$, $x_c=N_c/N$, $N_c$ being the number of sites in the system with colour $c$ and $N=L^2$ is the total number of sites. The magnetisation is zero when $x_c=1/q\ \forall c$, and one when all spins have the same colour. Initially the system is in an uncorrelated configuration at infinite temperature, with $e=2(q-1)/q$ and $m=0$. For $t<\tau$, systems with $q>4$ coarsen with equal proportion of all colours and with the expected power-law dependence of the energy on time [@Grest88] while, for $t>\tau$, finite-size systems equilibrate, breaking the $m=0$ symmetry and approaching gradually the ground state with $m=1$, $e=0$. The above description of the problem suggests that, for $q>4$ and for times larger than $\tau$, coalescence effects, not considered in the derivation of the Allen-Cahn law, become relevant in the dynamics, and allow the mean domain size, $\ell$ [@Bray], to grow up to the system size, $L$. The numerical results reported in Fig. 1 suggest that, for $T>0$, $\tau$ is not divergent in $L$, but constant above a certain $L$, i.e., that the “nucleation” needed to equilibrate the system is a local process no longer dependent on $L$. In fact, the $L=10^3$ and $L=500$ curves of Fig. 1 coincide within $\sigma_e^2(t,L)$, the variance of the distribution of energy values corresponding to different realisations of the quench (shown in error bars).
We observe from Fig. 1 that the $m=0$ symmetry is broken later in larger systems. As in [@Fialkowsky] for the continuous Ising model, the magnetisation is supposed to be zero for all times in the absence of finite-size effects, and in fact we find [@ILP] a similar scaling relation $m(t,L)=m(t/L^2,1)$ for $q=2,3$, and even a slower dependence of $m(t,L)$ on $L$ for $q>4$. One could ask why $\tau$ seems to approach a limit value with $L\to
\infty$, while the magnetisation at fixed times tends to zero in this limit. To answer this question we propose the following argument: systems with $q>4$ equilibrate leaving at $t=\tau$ the power-law dependence of the energy on time and forming domains that will eventually become system-sized, and $\tau$ seems to be a characteristictime of the model. It is this local nucleation the origin of the ordering, rather than the changes on the global colour fraction, which is a finite-size effect [@ILP]. In order to supply a quantitative support to this argument we define an order parameter, $\gamma$, accounting for the spatial ordering, which seems to characterise the dynamical ordering more than the global magnetisation, $m$ [@magnetization]. Let us define $\gamma$ as a distance from the $\mu({\mathbf}r)=0$, $\forall {\mathbf}r$, situation, where $\mu({\mathbf}r)$ is the magnetisation of a cell centred in ${\mathbf}r$ and of size $\lambda$, independent of $L$, in such a way that $\lambda/L\to 0$ in the thermodynamic limit. In particular,
$$\gamma\equiv\frac{1}{V} \int d{\mathbf}r\ \mu({\mathbf}r)=\frac{q}{q-1} \frac{1}{V} \int d{\mathbf}r\ \big( \sum_c^q {\phi({\mathbf}r)_c}^2 -1/q \big)
\rightarrow \frac{1}{L^2}\sum_{i\in {\mathbf}L}^L \mu \vert_{\mbox{\tiny
cell } i},$$
where $V$ is the volume of the general system and $\phi_c({\mathbf}r)\in [0:1]$ is the proportion of colour $c$ in the cell centered in ${\mathbf}r$. The expression at the right of the arrow in (3) is the definition of $\gamma$ in the lattice ${\mathbf}L$ and $\mbox{cell } i$ is a cell centred in the position of site $i$ [^3]. For $L \to \infty$, $\gamma$ so defined is zero in the completely uncorrelated configuration, and one in the ordered configuration, when all the sites have the same colour and $m=1$. Since $\mu \ge 0$, $\gamma$ is a distance, functional of $\mu({\mathbf}r)$, and can be used as an alternative order parameter accounting for the spatial ordering of the system. We see in Fig. 1 (bottom) that also $\gamma$ presents a power-law dependence on time. As expected by the argument exposed above, $\gamma$ seems not to depend on $L$ for large $L$: curves for $L=500$ and $10^3$ coincide within their standard deviation limits. Moreover, $\gamma$ seems to leave the power-law regime and converge to 1 at $\tau$.
$T=0$ dynamics and symmetries of the glassy phase
=================================================
At zero temperature, the situation is different: the above defined time, $\tau$, needed to leave the Allen-Cahn law is divergent with $L$, as described in [@PetriEL], and an infinite-size system is always supposed to follow the Allen-Cahn power-law. In larger and larger systems, it is less and less likely to find a blocked configuration or a path in phase space to the ground state. A measure of this fact can be seen in the value of the energy variance, $\sigma_e^2(t,L)$, which decreases with $L$ (see inset of Fig. 3). This means that in a large system it is less likely to find a deviation from the power-law regime, in which blocked dynamics [@Spirin] or breaking of the $m=0$ symmetry [@Fialkowsky] are not present. This explains at a qualitative level the fact that in large systems the magnetisation is broken later (and hence the system equilibrates later), but it does not address the fact that the energy of an infinite size system with $q>4$ converges to a positive value for $t \to \infty$. In other words, it does not explain why the term $e^*$ of the generalised Allen-Cahn law $e = a t^{-1/2}+e^*$ is different from zero for $q>4$. One could ask why systems with $q>4$ cannot converge to a zero energy phase in the limit $L \to \infty$, even respecting the $m=0$ symmetry, as the $q=2$, 3 cases do. We discuss this point in the following.
Numerical simulations show that in the latest stage of the coarsening at zero temperature, systems with $q=2$, 3 and with large enough $L$ (to respect the $m=0$ symmetry) present final configurations formed by domains of the size of system (Fig. 2, left) and $\ell(t)$, the mean domain size, is proportional to $L$ when $t \to \infty$. The energy, or the perimeter of the interface, is proportional to $L$, and thus, the energy per site is zero in the thermodynamic limit. In fact we have $e^*(q)=0$ for $q=2$, 3, as said before. On the other hand, for $q>4$, we observe that the system presents not only $m=0$, but also a local symmetry that is a local equal fraction of all colours. We define this local symmetry by introducing a certain scale, $\lambda$, with $L>\lambda>\ell$, such that the magnetisation is also zero in every cell of the system of size greater than $\lambda$ and *being $\lambda$ such that $\lambda/L\to 0$ for $L\to\infty$*. We propose, and we argue below, that the zero temperature dynamics cannot form domains of the size of the system for $q>4$, hence the perimeter of the interface (as well as the final number of domains) grows with $L^2$ and the energy per site of the glassy phase is not zero in the thermodynamic limit, while, in this limit, systems with $q=2,\ 3$ do not break the $m=0$ symmetry when forming domains of the size of the system but they can break the local $m=0$ symmetry. A numerical confirmation of this argument is that $\gamma$ converges to $\gamma^*=1$ in the $L,t \to \infty$ limits for $q=2,\ 3$, but $\gamma*<1$ for $q=7$: the fraction of the volume $V$ for which $\mu \ne 1$ is not negligible in the thermodynamic limit. In Fig. 3 we report the energy, magnetisation and order parameter of local ordering as a function of $t^{-1/2}$ for the $T=0$ dynamics in the $q=7$, 3 and 2 cases. It seems that $\gamma$ presents the same power-law dependence on time as the energy in the $q=7$ case. Inverting the order of the limits and extrapolating to $t \to \infty$ with a linear fit, as done for the energy in [@PetriEL], we have $\gamma^*=0.807\pm 0.005$, while for $q=2,3$, $\gamma$ converges to 1.
This local zero magnetisation for $q>4$ (see Fig 2, right) characterises the glassy phase also in the sense that it predicts a $q^{1/2}$ dependence of $e^*$, which coincides in the $q \to
\infty$ limit with the result $e^* \sim (q-4)^{1/2}$ of [@PetriEL]. Let us show this fact in the approximation of domains of identical size $\ell$ [@approximation]. In order to satisfy the local-$m=0$ condition there must be $nq$ domains inside each cell of size $\lambda$, $n$ being an integer. In $d$ dimensions it is $e\sim n_D\ \ell^{d-1}/L^d=\ell^{-1}$, where $n_D=L^d\ell^{-d}$ is the number of domains, and setting $\lambda^d/\ell^d=nq$ we have $e\sim \lambda^{-1} q^{1/d}$. When $q$ approaches 4, coalescence effects not considered in this argument become important, to which we attribute the slower dependence on $q$ of the energy, $e^* \sim (q-4)^{1/2}$.
As we have seen there are evidences that, in absence of thermal fluctuations, the Metropolis dynamics cannot break the local symmetry of colours when $q>4$. This fact and the $q>4$ limit can be explained (not yet in a rigorous fashion) as a consequence of an assumption on the glassy phase symmetries. To introduce it let us define the $q$ ‘coarse-grained’ fields $\phi_c({\mathbf}r) \in
[0,1]$, in analogy with [@Sire], as the fraction of colour $c$ in a cell centred in ${\mathbf}r$ and of size $\lambda$. Clearly $\sum_c^q \phi_c=1$ and the fields $\phi_c$ describe the configuration in a $\lambda$-dependent way. Given the fact that the Hamiltonian is invariant with respect to $S_q$, the group of permutations of the $q$ colours, we assume that the $m=0$ symmetric configuration corresponding to $L\to\infty$, $t \to
\infty$ is maximally ordered in such a way that any permutation of the colours, $p\in S_q$, is equivalent to a $p$-dependent global spatial transformation, $U_p$. i.e.:
$$\begin{aligned}
\phi_{p(c)}({\mathbf}r)=\phi_c(U_p{\mathbf}r) \hspace{0.5cm} \forall p \in
S_q\end{aligned}$$
If we assume the operator $U$ to be linear we have that $U: S_q
\to {\mathscr}{L}(\mathbb R^2)$ is a representation of $S_q$ in the vector space of linear operators in the plane, ${\mathscr}{L}(\mathbb
R^2)$. Since there is no representation of $S_q$ in ${\mathscr}{L}(\mathbb R^2)$ for $q>4$ different from the trivial representation ($U_p=\mathbb I$, the identity operator on $\mathbb
R^2$, $\forall p$), from the above assumption it follows the local-zero magnetisation of the glassy phase for $q>4$: setting $U_p=\mathbb I$ in (4) we obtain $\phi_{c'}({\mathbf}r)=\phi_c({\mathbf}r)$, that with the normalisation condition gives $\phi_{c}({\mathbf}r)=1/q$ $\forall {\mathbf}r,c$, i.e., the local $m=0$ symmetry in the cell of size $\lambda$. The assumption (4) can also ‘justify’ the spatial symmetries of configurations with $q \le 4$ in the $m=0$ regime: for $q=2$ the group $S_2$ admits a nontrivial representation in the plane, $\{\mathbb I,\alpha\}$, where $\mathbb I$ is the identity and $\alpha$ is the rotation of $\pi$ radians in the polar angle and, in fact, configurations with $q=2$ in the glassy phase seem to present this symmetry (see Fig. 2, left), and a similar $2\pi/3$ rotational symmetry seem to exist for $q=3$ configurations, even if it is often hindered by the two-preferred directions of the square lattice interaction [@Spirin] and by the formation of structures locally stable under the $T=0$ dynamics [@SafranI; @Lifshitz]. Moreover, in a general $d$-dimensional system, there exist no nontrivial representations of $S_q$ in $\mathbb R^d$ for $q \ge d+2$ [@groups]. This proposition seems to be true for all $q>4$, and we are working on the general proof. If verified, this consequence of the assumption (4) would coincide with the result by Lifshitz [@Lifshitz], who argued that a $d$-dimensional system quenched below its critical point does not necessarily equilibrate into an ordered state, in the presence of a ground state that is degenerated more than $d+1$ times.
Conclusions and further research
================================
We have studied the ordering dynamics of the 2-d Potts model in a square lattice with $q=2,3$ and 7 by use of single-spin flip dynamic Monte-Carlo simulations. At positive temperature, $T=0.1$, and for $q=7$ the system relaxes leaving the Allen-Cahn power law at a given time, which is independent on $L$ for large $L$. At zero temperature, in the $L \to \infty$ limit, systems with $q>4$ are not able to nucleate breaking the local symmetry, and hence they converge to a phase with nonzero energy. We give a quantitative support for this argument in the $q=7$ case with the help of an order parameter of local ordering that we define. The $q>4$ limit is presented as a consequence of a hypothesis on the symmetries of the glassy phase. For both temperatures the magnetisation at equal times decreases with increasing sizes and it is argued to be zero in the thermodynamic limit at any time [@Fialkowsky]. A generalisation of this study for different $q$ values and for larger $L$ is being performed [@ILP] together with an investigation of the relation between the ordering properties and the phase space structure.\
[99]{} I. M. Lifshitz, Zh. Eksp. Teor. Fiz. **42** (1962) (Sov. Phys. JEPT **15** (1962) 939).
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J. D. Gunton, M. San Miguel and P. Sahni, *Phase Transitions and Critical Phenomena*, edited by C. Domb and J. L. Lebowitz (Academic, London, 1983), Vol. 8, p. 267, and references therein.
See references 18–26 of [@Grest88]
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C. Jeppesen and O. G. Mouritsen, Phys. Rev. B [47]{} 14724 (1993).
C. Sire and S. N. Majumdar, Phys. Rev. Lett. [74]{} 004321 (1995).
See references 1–17 of [@Grest88].
M. J. de Oliveira, A. Petri, T. Tomé, Europhys. Lett. [65]{} 20 (2004).
M. J. de Oliveira, A. Petri and T. Tomé, Physica A [342]{} (1-2), 97-103 (2004); A. Petri, Braz. J. Phys [33]{} 521 (2003); M. J. de Oliveira, A. Petri, T. Tomé, cond-mat/0402310 (2004).
V. Spirin, P. L. Krapivsky and S. Redner, Phys. Rev. E [65]{} 016119 (2001).
C. Godrèche, J. M. Luck, J. Phys.: Condens. Matter [17]{} S2573 (2005), and references therein.
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M. Ibáñez, V. Loreto, A. Petri, to be published.
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The global magnetisation can be used to characterise the equilibrium of the Potts model, e.g. in a variational mean-field approximation in terms of $m$ in: P. M. Chaikin, P. Lubensky, *Principles of condensed matter physics*, Cambridge University Press, Cambridge, 1995.
The validity of the mentioned approximation is discussed in [@ILP].
S. A. Safran, P. S. Sahni, G. S. Grest, Phys. Rev. B [28]{} 2693 (1983).
P. S. Sahni, D. J. Slorovitz, G. S. Grest, M. P. Anderson and S. A. Safran, Phys. Rev. B [28]{} 2705 (1983).
The dimension $d$ of a given irreducible representation of $S_q$ is given by the Hook Formula (see, v.g., notes of C. Procesi in http://www.mat.uniroma1.it/people/procesi/tensor.pdf).
![Energy per site ($e$), magnetisation ($m$) and order parameter of local ordering ($\gamma$) with respect to $t^{-1/2}$ for the Potts model with $q=7$ after a quench to temperature $T=0.1$. Results are averaged over 200, 60 and 30 realisations (for $L=200,\ 500$ and $10^3$, respectively). Time is in MCS units. Error bars in the $y$-axes are the variances, $\sigma^2$, of $e$ and $\gamma$, corresponding to the average over different realisations of the quench (shown every 2000 MCS only). In larger systems the $m=0$ symmetry is broken later [@Fialkowsky]. $\sigma^2$ decreases with the size of the system at equal times. The blue line is a fit to the $L=10^3$ system energy in the range $t^{-1/2}\in [0.02,0.1]$, and the vertical dotted line is an estimation of the nucleation point, corresponding to the time at which $e$ differs from the fit more than $\sigma_e^2/2$. At that time, $\tau$, the system nucleates leaving the Allen-Cahn power law, and $\gamma$ increases towards 1.](emg_varT01.eps "fig:"){height="8.5cm"}\
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
{height="3.5cm"} {height="3.5cm"}
$q=2$ $q=7$
$\gamma=0.992$ $\gamma=0.775<\gamma^*$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Energy per site ($e$), magnetisation ($m$) and order parameter of local ordering ($\gamma$) with respect to $t^{-1/2}$ for the Potts model after a quench at zero temperature. The $q=7$, 3 and 2 cases are presented, and three different sizes for the $q=7$ case. Magnitudes are an average over 240, 60 and 40 realisations of the quench in systems with $L=200$, 500 and $10^3$, respectively. For the cases $q=2$ and 3 the energy converges to zero following the Allen-Cahn law, while in the $q=7$ case it converges to $e^*= 0.1192$ ([@PetriEL]). The global magnetisation decreases with $q$ at equal times and sizes, for all sizes and times studied. It also decreases with $L$ for all times [@Fialkowsky], and it is supposed to be zero in the thermodynamic limit. Even with this constraint, systems with $q<4$ can form system-sized domains with zero energy, and in fact they order spatially, and $\gamma(t \to \infty)\to 1$, while the zero-temperature dynamics cannot break the local zero magnetisation symmetry to form domains of the size of the system when $q>4$, as illustrated by the fact that $\gamma(t \to \infty)\to \gamma^*<1$ for $q=7$, and this is the reason for which $e^*>0$. The extrapolated line is a fit to the $L=10^3$ data for $0.005<t^{-1/2}<0.06$ MCS$^{-1/2}$. The fit coincides with the data within its variance limits. The equation of the fit is $\gamma_{\mbox{\tiny{fit}}}=\gamma^*+at^{-1/2}$ with $\gamma^*=0.8072$ and $a=-3.1756$. On inset we present the variance $\sigma_e^2$ of the energy, which is a decreasing quantity with $L$, for all times and for the three sizes studied.](emg_var.eps "fig:"){height="8.5cm"}\
[^1]: Telephone numbers and e-mails:\
+39 0649913450, [email protected] (corresponding author)\
+39 0649913437, [email protected]\
+39 0649934112, [email protected]
[^2]: There exists a fraction of realisations which fall to the ground state, but this fraction decreases very fast with $L$ and it is negligible in the thermodynamic limit.
[^3]: In the numerical construction of $\gamma$, we have taken the nearest 24 neighbours of each site for the definition of the mentioned cell.
| |
Remember Me Thursday
Good Morning friends,
Today is a very important day so without hesitation we decided that our blog should be up today and not tomorrow as it is Remember Me Thursday.
I have always rescued, and only believe in rescuing and that is why this day is extra special for me.
On this day the world will share and discuss the importance of pet adoption, and shine a light on all orphan pets waiting in shelters and rescues to be adopted . We all come from different countries, religions but one thing we all have in common is the fate of the pets in shelters so we need to be their voice as they do not have one.
Those orphan pets in the shelters need homes, they want a warm bed to sleep in, good food, toys but most of all they want to be loved by humans so every time one is adopted you are saving two lives, the pet and yours.
We can help by spreading the word, changing the minds of people to open their hearts also :
01. Adopt and open your house to an orphan pet
02. Go to Remember Me Thursday Events in your area
PLUS I think the most important of all is to show everyone all over the world that adoption is the only way to go as the adopted/rescued pets deserve a second chance.
We want to say a special thanks to Mike Arms, President and CEO of Helen Woodward Animal Center, for creating this day to draw the attention of the orphaned pets sitting in shelters and by lighting a candle will draw people to their fate of euthanasia. Thank you Helen Woodward Animal Center for permitting me to share this poem by Mike Arms.
So let’s all get together, light a candle and save the shelter pets and give them the good life they deserve,
Thank you with love from
Layla (my rescue from a shelter)
Happenings in the Layla Neighborhood
TGIF !!!!!!!!
Today in the Jewish Religion, and Mom being a Jewish Mom we start our New Year 5781 which means lots of yummy food so I am going to be very good all day to make sure I get some.
We are celebrating the Holiday at home as we are still on lock down, 6 months already so I am going to be doing some of the cooking here, not all as I am one person and Madam cannot eat some of the foods although she would love to.
Like most holidays in all religions there are traditional foods and I am going to share here the symbolic foods you will find on the table :
01. Apples and Honey : Honey symbolizes sweetness for the New Year and in the old days the Jews believed that apples held healing powers so we dip the apples into the honey.
02. Round Challah – Because the world is round and the round challah symbolizes the world.
03. Pomegranates – Symbolizes plentiful for the New Year
04. Fish Head – Symbolizes Head of the Year
05. My favorite at the end of the meal is coffee with Honey Cake
In the Temple you will hear the Shofar which is a rams horn blown like a trumpet on the New Year (and other holidays). It has four sounds which the Rabbi says and the Shofar is blown with a specific sound.
Our New Year starts at sunset and ends the following day at sunset but only outside Israel do they celebrate two nights.
So that is what we are celebrating this weekend.
Other than that what is happening in the Layla Neighborhood let’s think LOL. Layla and I spent some time in the park this week, we had one good air day which was beautiful so we sat outdoors enjoying it and she was so happy to be out and seeing some other people and of course doggy friends. A friend of mine has just rescued a dog who is just the cutest and we got to meet her.
I have started part time dog sitting for a really cute little dog who Layla really likes so she now has a friend visiting which is good for her. His name is Baxter and he is a rescue and just adorable. It is so great to be working again and hopefully more dogs will be needing dog sitters again.
On top of that I am finally doing something for me which I have wanted to do for years, I have started an online workshop to become a Domestic Violence Advocate, this means I will be finally able to give back and help those that are fortunate to get away like I did.
So as you can all see we are always busy here, life is good in our neighborhood and we want to wish everyone a Happy Healthy Sweet New Year.
With love from Layla
We Must Not Forget
Friday has arrived, and this Friday is a solemn day which we must not forget and make sure it never happens again.
Today is 9/11, when innocent people died just for being American, let us remember the countless lives lost and selfless acts by the brave responders on September 11, 2001 — both human and four-legged.
There were 300 specialized canine search and rescue teams in the days following the attack who worked tirelessly searching to save lives. These four-legged heroes shone in the face of this incredible tragedy.
I often feel that without dogs it would be harder to find survivors. They are amazing creatures and we are blessed to have them in our lives.
We must not forget and not let it happen again.
Fast forward to 2020 and what is happening all over the world is frightening and the fires on the West coast are horrific and an eye opener as to how we should work on climate change and start protecting our earth.
On Wednesday 9 of September I woke up to darkness, orange sky and nothing else. I looked outside thinking it was 4 in the morning but my clock said 8 and I could not believe it. San Francisco was in darkness all day – the air was bad, and it was really eerie. The smoke from the fires had settled in our city, we had no daylight that day and everyone just stayed indoors. You can see the photo of our park at 11 in the morning taken by a friend of mine and the one from out of my window to get an idea what we went through.
I also had to take Layla for her annual check up, all good but my major question from the vet was her not sleeping at nights, wanting to go out at 3 every morning. We came to the conclusion that she has turned night into day and day into night because we are so stuck inside because of the virus and the bad air lately. We joked that she should get chicken soup every night to make her sleep. He also feels she might be starting arthritis so we are trying Gaberpentin at night to see if that will settle her pain wise. I hope it will as we both slept last night and it was amazing.
So that is what is happening in the Layla neighborhood, just going day by day and seeing how much we can be outdoors when possible.
Please keep safe everyone, our prayers go out to those that are being evacuated from the fires and that all the animals will be safe.
With love from Layla
National Disaster Preparedness Month plus Give Away
TGIF !!!!!!!
Good morning all my friends and how are you all doing ?
Over here in the Layla neighborhood we are still indoors most of the time because of the smoke in the air and I told Mom it is getting boring BOL.
September is National Disaster Preparedness Month for everyone and although it is in June for pets I personally think doing it twice a year reminds us to make sure we are prepared.
This is such an important month as with what is happening all over the world we need to be prepared especially for our fur babies as they depend on us in every way to keep them safe.
We always remind everyone to check their emergency bags, make sure they are up to date and nothing has expired, check our pets microchips, documentation and that the bag is ready to be picked up and we can go.
While I was checking our bag I started thinking about things I felt were missing for Layla so we decided to do a review on some new products by Tropiclean to add to our bag.
I am always looking for natural products, no chemicals where possible and these are really amazing and happy to share with you all. I am going to confess that Layla is a Tropiclean dog LOL as I use most of their products on her as they are safe, and with her allergies work really well.
01. Flea and Tick Spot on Treatment, Layla is a roller in the park, nothing like rolling on the lawn and I always worry about fleas and hate all the chemical flea products on the market so I tried their flea and tick spot treatment which is made from Cedarwood oil, Peppermint oil and Almond Oil. Since I put it on her I have not seen any fleas and she is not scratching, phew.
Madam Roller : https://youtu.be/pRLmpF6cn5k
I also want to add that they have flea and tick prevention products for your house and yard.
02. Oxymed Soothing Pet Wipes, love these as they are easy to put in my purse or have in my emergency bag and they help with itching and hot spots. They are made with purified water, glycerol, mild coconut cleanser, oatmeal extract and fragrance.
03. Waterless Dog Shampoo, it is deep cleansing and no rinse formula. It smells of Berry and Coconut so Madam smells very fruity. The ingredients are purified water, mild cleanser, hydrolyzed plant protein, organic blend of while plum extract, cucumber extract, avena sativa oatmeal. It also does not affect spot on flea and tick treatment either. This is a great product to have in my emergency bag as it does not need water and for those that go camping I feel it is a must also.
04. Tangle Remover, this is important to have for those dogs that get matted. It is a no rinse formula and for Madam who hates water this is just perfect. Like the waterless dog shampoo it does not affect spot on flea or tick treatment.The ingredients are Purified water, hydrolyzed plant protein, organic blend of chamomile extract, kiwi extract, mallow extract, awapuhi extract, proprietary cationic emulsion, fragrance and vitamin E. Another one I will be putting in my emergency bag.
What I love about the Tropiclean products are they all made in the USA and are cruelty free. Plus for all cat parents they have fantastic cat products too.
Their link is : https://tropiclean.com/
I have been sponsored by Tropiclean and this Give Away is open for participants over the age of 18 and in the USA only.
Enter on the widget below :
:http://www.rafflecopter.com/rafl/display/7d1ec19c52/?
Layla woofs good luck to everyone entering as she has put her paw stamp of approval on all these products and now she is going back to bed.
Have a wonderful long weekend everyone Be safe with a woof and love from Layla
Rainbow Bridge Remembrance Day
TGIF !!!!!!!
Good morning friends, another week has gone by and this year is going so fast we cannot keep up.
The air quality has been really bad here in San Francisco because of the fires so we have not been out that much, short walks and I think both Layla and I need some real outdoors so tomorrow I might risk it a bit and take her to the park for an hour.
Today, 28 August is a very special but sad day although for me the memories make me smile too.
It is Rainbow Bridge Remembrance Day, a day that we remember all our pets that are no longer with us and are waiting over the Rainbow Bridge. So I am going to share with you some of the pets that are in my heart and have been part of my life.
I was blessed to grow up in a home with many different pets. We had Ceaser, a mutt, who to this day when I think back about her I chuckle as she was scared of heights and used to go up to the second floor of our house and then howl till my Mom carried her down. She was the size of a Boxer and heavy LOL.
We had in the sewing room/playroom budgies who I loved till they escaped out of the window and were not found, their names were Fritzie and Freddy. In that room were also our little white mouse called Pinky because of her tail and Hammy our Hamster who lived next door to Pinky, we used to close the room door and then let them run around free every day.
In our bedrooms we had Goldfish and two Terrapins, oy. My great aunt told me we had to feed the Terrapins bugs and worms from the garden and she used to collect them and bring them to me to feed them, they were called Ron and Eth after a British comedy.
When the fish and terrapins passed away we buried them under an apple tree we had in our garden and from that day on my Grandmother would not let us eat the apples.
When we moved to Israel we found on the streets a mutt who we called Whiskey, who became a princess. She was living with us during the 1973 war so one of us kids were responsible for carrying her downstairs to the bomb shelter every time a siren went off.
In those days there were no special regulations for dogs like today and when we left Israel and went to Europe, Whiskey flew back to South Africa where she was met at the airport by a Pet Hotel and stayed with them till we got back. I remember my Dad feeling bad that she had to be crated so he had a big big crate made for her although she was a Chi Mix. It was enormous.
Whiskey lived a very good life with us in South Africa, she would sit on a chair every morning and wait for her piece of cheese from my Father. I still chuckle over this one when going into the dining room and there she was on a chair.
When I went back to Israel I decided to get a cat, well I adopted a cat, thinking Sasha would be cute and small but Sasha grew to be a large cat but I loved him. He kept me amused all the time. A couple of years with Sasha and I rescued Shula, a Persian cat who was afraid of everything but after a couple of months she ruled the house. She was pregnant when I rescued her so decided to keep the kitten also, my Pichevkes (Hebrew for nick nacks). She was my baby and I adored her. All three cats used to get boiled fish every Friday night and were spoiled rotten.
Sasha passed away at the age of 14, Shula at the age of 11 and Pichevkas at the age of about 9.
I then came to the States where I adopted Baby, whom you all know about, she was my hero and ran away with me from the Domestic Violence Relationship, was the first to be in a DV shelter in San Francisco and kept me alive in so many ways. Baby passed away about a year after I left the shelters, she was 6 years old after she contracted IMHA which has no cure.
When she passed away a good friend of mine said to me that She had come into my life for a reason, she knew I was safe so she left me so that I could adopt a new dog.
That is when Layla who is my Boss came into my life and my life would be so empty without her.
I unfortunately do not have many photos but all the memories are etched in my brain, and today being Rainbow Bridge Remembrance Day I am smiling now thinking about all of them.
So let us all share some memories, thank all those pets over the Rainbow Bridge for being part of our lives, teaching us so much and I do believe one day we will all meet up again.
Be safe everyone,
With love and woofs from Layla
Praying for the fire victims
TGIF !!!!!!!!!!!!!!!
Good Morning friends,
Layla would like to apologize about not blogging last week but Mom decided to have a break from the laptop and actually it was really nice as we had lots of Mom and Me time together and nothing else.
This past week we have really been staying indoors because of the poor air quality from the fires burning in Northern California. It is so sad to see what is happening as there has been no rains and everything is dry. The fires were started by lightning which happened last Saturday night and it was a crazy night. So at the moment Mom is working sharing all information possible on our Facebook page to help save the animals, including the wildlife.
https://abc7news.com/wildfire-live-updates-cal-fire-to-give-update-czu-lightning-complex-fires/6381687/?fbclid=IwAR1faK1w8oHwi6sEwla09ePvnT0R129eonpyhxs4YmVFZJ1LTjiEUQ5h3sQ
This is the time that I say Facebook does something good as when a disaster happens someone starts a page with all the information and everyone posts, gets together to help out with trucks, cars and whatever else is needed. It is one way for the word to get out fast and I call them all heroes.
So let us all pray for all those in the path of the fires and that this nightmare will end soon.
We are reminding everyone today to make sure your emergency bag is ready, your microchip is up to date, and that you are ready to go if necessary.
Please be safe
With love from Layla
Helping Pets in need during the Lock Down
TGIF !!!!!!!!!!
Friday has arrived, first Friday in August and we decided to dedicate our blog this week to an amazing organization that has just started, a Free Pet Food Pantry in San Francisco, in one of the areas where the residents have been hit hard economically.
This organization is called the Latino Task Force. I am sure you will ask who they are ?
They formed as a clearinghouse of COVID-19 information and resources for the Mission District community. The group has more than three dozen community-based organizations, along with City Government partners, that are working together to meet the needs of immigrants, families, elders, homeless, LGBTQ, youth, and other Latino populations who are experiencing challenging times during the pandemic.
It’s an all volunteer organization that provides boxes of food to 7,000 people a week. SFDOG is working with the San Francisco SPCA to distribute dog and cat food as part of the services operating at the Mission Food Hub three days per week. They are recruiting volunteers to hand out food and raising funds to keep purchasing food for families to feed their pets.
You can see the links for the Latino Task Force and SFDOG below.
This is the link : https://www.ltfrespuestalatina.com/contact
This is the link : https://sfdog.org/blog/help-san-francisco-families-feed-their-pets
The SPCA and Pet Food Express (An amazing pet store where I shop all the time) contributed food to kick off the pantry. And they will purchase food through the SPCA at their corporate partner rate on an ongoing basis. They are at this moment testing the pantry on a three-month trial, which could be extended and expanded to other locations. As it stands with the Mission Food Hub, they can only distribute pet food that is shipped directly from the manufacturer.
I am putting it out here if you manufacture pet food/treats and would like to donate I am sure it will be welcome. Thank you.
Our pets are our family, I always joke that Layla eats better than me but then she is my furkid so there is no question about that, and as I am watching pets unfortunately being dumped at shelters as families cannot afford them at this moment it is a reminder that there are solutions, amazing people out there who are helping in whatever way possible and we must learn to reach out and ask for help.
So Layla and I want to thank these fantastic volunteers for all the amazing work you are doing, you are our heroes this week.
Do you have a free pet food bank in your city ?
To all our furfriends, please be safe, wear masks and remember you are not alone, we are unfortunately all in the same boat today.
With a woof and love from
Layla
My photo shoot
Good morning all our friends,
July is over, August is around the corner and we are still stuck inside. The weather has been typical SF weather, foggy, windy and cold but hoping this weekend there will be a warm up so we can do some park time, I need it, not sure of Layla.
As I told you a couple of weeks ago we were going to do a photo shoot in our park, Bossy becoming a model for a couple of hours. Well on July the 18th I met up with Debbie from “MisterDebs” to do it and it was really exciting and interesting. I had got Layla groomed and pretty for it too so off we went.
This is the first time I have done something like this so was really curious to see how Layla would react and behave and she really surprised me. Because it was late afternoon I took her dinner with me so instead of treats so she could eat her food and turned it into a sort of picnic in the park.
Debbie was amazing, patient with Layla and we had so much fun with the photoshoot. I loved that she did some of the photos in Black and White as they are really special. Towards the end unfortunately some idiot set off fireworks so Layla freaked out poor thing and I had to put her in her backpack to relax her and feel comfortable.
There are so many photos so I decided to turn this into a photo shoot blog to share with you and as there are so many I am going to put here my favorites, yes even one with me walking her home, what a concept.
You can visit her on her website :
What I have concluded is that I am a lousy photographer and need a better camera one day and want to thank Debbie for making these beautiful memories for me, they will be cherished.
Thank you Debbie for that great afternoon, the laughter, the fun and most of all the gift you gave me of taking the photographs.
We hope you enjoy them,
With a woof and love Layla
Quirky Layla’s week
TGIF !!!!
Good morning everyone,
The days are getting longer, the weeks are going fast and living in lock down is slowly taking its toll on me. That is one of the reasons I took a break from blogging last week also.
I lost all my work so am sitting here figuring out daily what to do and like many I think I am driving myself crazy. BUT thank goodness for Layla she is keeping me on my feet and amused especially as she is back to being her funny normal self.
Layla has some new quirks which are amusing but on the other hand not sure where she has picked them up or do they come with age ?
She now sits at the door barking when she wants to go out, telling me to get ready to go and does not let up till she sees the leash. This is a dog that has never barked unless someone knocked on the door and I have to laugh although it can be aggravating also. She has become a really bossy little girl.
I was very fortunate and a friend of mine, who grooms his dog did Layla for me last week so her fur is short and with the heat I am sure she is a lot more comfortable. It is not perfect but a lot lot better plus she got a bath.
Last Saturday we did a professional photo shoot in our dog park with a great photographer who took some amazing photos. Unfortunately towards the end some idiot decided to set off fireworks in the neighborhood and she freaked out but thank goodness I had her backpack with me so I put her in it and she started relaxing. As soon as I get the photos I will share with you as some of them are really adorable.
Our adventures have become nothing as such which is sad as I miss them and am sure she does also and like me getting bored of the same walks and dog park so I decided that as they are having a ShiTzhu meetup this Saturday in another park to take her. There will not be many people, the park is enormous so great for social distancing and for both of us it will be mentally good to be somewhere else for a couple of hours. Yes I will be wearing my mask and have hand sanitizer with me.
So as you can see we have slowed down here, our routine is not exciting and praying for it all to be safer so that we can start out adventures again.
Be safe everyone, have a great weekend
With a woof and love Layla
Life is back to normal
TGIF !!!!!!!!!
I do not know why I say this lately as every day seems to be Friday and cannot wait to get back to working / dog sitting which hopefully will happen soon.
Thank goodness the 4th of July is over, the fireworks were like a war zone here and I cannot believe how people went crazy, from 7 in the evening till 3 in the morning my studio was shaking from all the booms. It was totally insane and what a waste of money. Thank goodness Layla was on her meds as she slept through them all but on Sunday morning she did not want to go out I think from fear. They are still setting them off in my area but not as bad but still just insane.
Layla is off her meds, second day now, doing really well and back to being Miss Piss and Vinegar, bossing me around and keeping me on my toes. It is great and I cannot believe how she has healed although I do have meds in the house for an emergency and the hemp oil of course. I am not allowing her to jump on or off anything although have caught her here and there jumping on the bed. She is eating really well which is good and drinking too. So I am totally relaxing here now. It was a month of hell but I can put it behind me now.
On Monday I took her to the park which is good for both of us. She had a great time and even made a new friend, they followed each other all over the place and it was really cute. I am doing things slowly now she is off her meds plus carrying her backpack just in case I need to use it although she hopes she will be carried all the time LOL.
Our weekend plans are to spend outdoors as much as possible, just relaxing safe distancing in the park with mask on of course plus I go to the park at odd hours when I know there are less people.
So that is what is happening in the Layla Neighborhood at the moment , be safe everyone and have a wonderful quiet weekend, | https://laylaswoof.com/ |
Q:
Finding and counting strings using multiple index vectors
I have a character array (this can also be stored as a cell array if more useful) (list) and wish to tally the number of substring occurrences against two different indexes held in two separate variables type and ind.
list =
C C N N C U C N N N C N U N C N C
ind =
1 1 2 2 2 3 3 3 4 1 1 2 3 3 3 4 4
type =
15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16
No spaces exist in the character array - added for clarity.
Using the above example, the desired output would tally all instances of unique letters in list, for each ind and for each type - creating three columns (for C/N/U), each with 4 rows (for each ind) - per type. This is done using the order in which the entries in each array appear.
Desired output of above example (the labels are added for clarity only):
Type 15 Type 16
Ind C N U C N U
1 2 0 0 1 1 0
2 1 2 0 0 1 0
3 1 1 1 1 1 1
4 0 1 0 1 1 0
I am only aware of how to do this with a single index (using unique, full and sparse).
How can I bet go about doing this with a dual index?
A:
One possibility could be to transform your letters to doubles by substracting e.g. -64 to map the number 3 to the letter C.
Then you can use unique with 'rows' and 'stable', to get the following result:
list = char('CCNNCUCNNNCNUNCNC')
ind = [1 1 2 2 2 3 3 3 4 1 1 2 3 3 3 4 4]
type = [15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16]
data = [type(:) ind(:) (list(:) - 64)]
[a,~,c] = unique(data,'rows','stable')
occ = accumarray(c,ones(size(c)),[],@numel)
output = [a, occ]
output =
15 1 3 2
15 2 14 2
15 2 3 1
15 3 21 1
15 3 3 1
15 3 14 1
15 4 14 1
16 1 14 1
16 1 3 1
16 2 14 1
16 3 21 1
16 3 14 1
16 3 3 1
16 4 14 1
16 4 3 1
If you have the Statistics Toolbox you should consider using grpstats.
If you don't mind a mind twisting output then crosstab is the far easiest solution:
output = crosstab(type(:),ind(:),list(:)-64)
%// type in downwards, ind to the right
output(:,:,1) = %// 'C'
2 1 1 0
1 0 1 1
output(:,:,2) = %// 'N'
0 2 1 1
1 1 1 1
output(:,:,3) = %// 'U'
0 0 1 0
0 0 1 0
The following one liner looks close like your desired output:
output2 = reshape(crosstab(ind(:),list(:)-64,type(:)),4,[],1)
output2 =
2 0 0 1 1 0
1 2 0 0 1 0
1 1 1 1 1 1
0 1 0 1 1 0
Also in this toolbox, you can find the tabulate function which offers another option in combination with accumarray:
[~,~,c] = unique([type(:) ind(:)],'rows','stable')
output = accumarray(c(:),list(:),[],@(x) {tabulate(x)} )
Which also allows the following output:
d = unique([type(:) ind(:) list(:)-64],'rows','stable')
output2 = [num2cell(d(:,[1,2])) vertcat(output{:})]
output2 =
[15] [1] 'C' [2] [ 100]
[15] [2] 'N' [2] [66.6667]
[15] [2] 'C' [1] [33.3333]
[15] [3] 'U' [1] [33.3333]
[15] [3] 'C' [1] [33.3333]
[15] [3] 'N' [1] [33.3333]
[15] [4] 'N' [1] [ 100]
[16] [1] 'N' [1] [ 50]
[16] [1] 'C' [1] [ 50]
[16] [2] 'N' [1] [ 100]
[16] [3] 'U' [1] [33.3333]
[16] [3] 'N' [1] [33.3333]
[16] [3] 'C' [1] [33.3333]
[16] [4] 'N' [1] [ 50]
[16] [4] 'C' [1] [ 50]
| |
This is a recipe I make at Christmastime or for tea parties. Our girls adore them! It's also a great way to use up those leftover Christmas candy canes! Enjoy!
2 egg whites
1/8 teaspoon salt
1/8 teaspoon cream of tartar
1/2 cup sugar
2 peppermint candy canes, crushed
In large mixing bowl, combine egg whites, salt, and cream of tartar. Mix together, adding sugar one tablespoon at a time. Beat on high for 5-7 minutes until stiff peaks form. Drop by teaspoonfuls onto foil-lined baking sheet. Sprinkle with crushed peppermint candies. Bake at 225 degrees for 90 minutes. Turn off heat, and leave candy in oven with door ajar for one hour to cool. Makes 3 dozen.
Click here for a printable version of this recipe. | https://www.countryatheartrecipes.com/2014/01/peppermint-meringues.html |
A few readers have had the question of what does financial independence actually mean and how do you know if you can retire early. We are out camping with the kids and enjoying the beautiful BC and Alberta scenario for a few weeks. I can’t really do any calculation specific posts due to time constraints, so I thought I would do a quick post to cover the FIRE topic.
FIRE/ financial independence typically implies you have enough income producing assets that produce cash flows which are sufficient to cover your family expenditures in perpetuity. This may seem like a difficult amount of assets to obtain and also to predict what is needed in the future. There are some general rules to follow that help to frame the picture.
3-4% Safe Withdrawal Rate (SWR)
Traditional retirement calculators will often show that a 5-6% withdrawal rate will almost always provide a reliable 30 year withdrawal period from your savings. This implies that if you retired at age 65, your nest egg should last till about 95 years old, then run out. If you die before then, you will have an estate to leave to your family. But how much can you take from your portfolio each year if you are retiring way before 65 years old?
The SWR rule implies that if you sell between 3-4% of a portfolio/year which is invested in a well structured portfolio, you should be able to continually draw this amount without significantly impeding the total portfolio value indefinitely. This is because the value of that portfolio will be going up via general growth in the 6-8 % range/year long term over that same withdrawal period. Inflation cost increases in the 3-4% /year range which leaves 3-4%/year available to spend every year.
Another way to look at the SWR would be based on your annual expenditures. At a 4% withdrawal rate, you would need exactly 25 years worth of current yearly living expenses to be financially independent. At 3% it would be 33.3 years worth.
You can quickly tell that it is your after tax spending that will create widely different demands on the portfolio size you need to achieve financial independence (FI). If you have a frugal lifestyle of $50,000/year, you need $1.25 to $1.665 million for FI. A spending range of $100,000/year will need $2.5 to $3.33 million and $150,000/ year needs $3.75 to $5 million/year for FI.
All doctors and professionals can achieve a $2-3 million portfolio within 15-20 years of work with the basic financial principles we have been discussing. 10-15 years to financial independence is achievable for most professionals that apply aggressively all the savings and investing techniques we will discuss.
Cost of Living Inputs Change in FIRE – You Need Less Than You Think
There could be multiple costs that adjust as soon as you have achieved financial independence. The reductions include:
1. No Insurance Cost – once you have a huge nest egg, life insurance is of little purpose. The main point of life insurance is to cover the expenses for your dependants if you are dead and can’t earn an income. Your financial independence portfolio already does that job quite well by itself. Disability insurance doesn’t do much for you if you are no longer working. My family saves about $3000/year now that we don’t carry life insurance or excess disability insurance beyond the basic plan we get through our medical association benefits.
2. Eliminate Child Care Cost – Our family would save close to $30,000 if we no longer worked since we have a nanny.
3. Debt servicing (mortgage or other) are often low when financially independent.
4. No Professional Expenses (College Licensing, Professional Education and Malpractice Insurance) – about $15,000/year for both of us.
5. No Ongoing Allocation to Retirement Savings Required – once independence is achieved, ongoing savings via TFSA/ RRSP etc aren’t required. I still allocate for RESP $2500/child contributions since that is a $500/child from the government as a match. Stopping RRSP/TFSA contributions would save us $20,000/year that we currently contribute to both/year (I will address why we don’t maximize the RRSP fully in the RRSP post – it comes down to incorporation and tax efficiency)
6. Reduced Income Taxes – there are taxes when selling investments, but these will be far less than during your standard income tax years – more on this below. Remember that you won’t need to replace your full working income in order to retire.
There are some potential increases in spending at early retirement that may include:
1. More spending on travel or hobbies during the excess free time
2. Education for children during university years
3. Investment into a new endeavour for professional purposes such as starting a new business or project.
Overall, most individuals that have experienced early financial independence find that their expenses lower significantly from their working years. In my family’s particular case, stopping working during FIRE would correlate to about $50,000 in reduced personal after tax expenses. To spend $50,000 beyond your basic living costs personally, you would need about $80 to 90K pre tax income assuming 35-45% personal tax brackets. Let’s review a bit about how personal taxation can change in early retirement.
Reduced Taxes and Optimal Corporate Withdrawal
A huge advantage of having a professional corporation with a significant amount of assets is the ability to control your corporate withdrawal process. This is a complicated topic that will be discussed further in the corporation posts. A big reason why our family can be in a financial independence setting is because of the financial flexibility that a corporation provides. Generally speaking, it is easier to quickly build assets in a corporation, because you pay less initial corporate tax (often around 14-18% depending on which province you are in) than if you paid out your income earned all personally (35-50% tax brackets).
A huge advantage in the Canadian taxation system relates to dividend income. Dividend only income can be received personally with very little taxation. If you can keep your expenses reasonable, a significant amount of your savings can end up in your pocket.
Dividends from a private controlled Canadian corporation are called ineligible dividends. These would include most privately controlled small business corporations which include most professional corporations like medical, dental, legal or accounting corporations. In the province of BC, you can receive $50,000 of only ineligible dividends and pay about $3000 in tax. This is similar in many provinces and I will update the post with a chart once back home.
So a couple can receive $100,000 in ineligible dividends and pay about $6000 in tax. This leaves $94,000 to spend after tax. As you take more ineligible dividend, the tax percentage does increase gradually.
This compares to eligible dividends which are received by investors from publicly traded Canadian corporations. Examples would include Bell, Royal Bank or other large companies that pay a portion of their yearly earnings out as dividends to their shareholders. The dividend yield is the percent of the total dividend payments/year received divided by the total purchase price/share that an investor spent for the stock. If you received $3 on a stock that cost $100/share, the dividend yield is 3/100 or 3%.
If you receive only eligible dividends as your personal income from an investment portfolio, you can receive about $50,000 TAX FREE. The only problem here is you need around $1.66 million in eligible Canadian stocks that yield 3% to receive $50,000 in dividend income yearly or $3.32 million to get $100,000 tax free as a couple.
See this Financial Post article by Jonathan Chevreau for more details on the exact tax reasons why this is possible.
My Family’s Case For Financial Independence
Our family currently needs over $200,000 pre tax income to maintain personal living expenses of around $150,000/year after tax. In FIRE, we will be able to reduce personal spending by about $50,000 mainly from reduced child care costs and no need for ongoing personal savings. Luckily, our current spending already has a generous travel allotment of around $25,000/ year which we don’t think would change much while the kids are still in school.
So with around $100,000/ year via ineligible dividend payments from our large corporate portfolio savings, we would be set for financial independence based on the 25-33x yearly SWR. As mentioned before, we both currently work an amount that allows us to enjoy a balance between our young family and our professional life. Recently it has felt great to give up a few shifts to facilitate travel and experiences without feeling any concern for the missed income. It is nice to feel the flexibility to adjust our work to whatever the future holds. This trip with my wife and kids has been a blast and we look forward to many more over the years to come! | https://financiallyfreemd.com/2017/07/12/financial-independence-fire-financially-independent-retire-early/ |
How do you grow vervain seeds?
Take a look at the similar writing assignments
How do you grow vervain seeds?
Plant blue vervain seeds directly outdoors in late autumn. Cold temperatures break the dormancy of the seeds so they are ready to germinate in spring. Cultivate the soil lightly and remove weeds. Sprinkle the seeds over the surface of the soil, then use a rake to cover the seeds no more than 1/8 inch (3 ml.)
Is Russian sage easy to grow?
Russian sage is easy to grow and cold hardy to USDA zones 5 to 9. It grows best in warm climates and tolerates clay or average soils, as long as the drainage is good, but they need full sun to produce lots of flowers and sturdy stems that won't flop over as they grow taller.
Is Russian sage from Russia?
Russian sage produces small blue flowers on long spikes. The genus was named by the Russian botanist Karelin about 1840 after B. A. Perovski, the Turkestani governor of the Russian province of Orenburg – but the plant is not native to Russia or a sage (Salvia).
Does Russian Sage attract bees?
Russian Sage It also grows well in poor or rocky soil, making it an excellent choice for almost any landscape. It blooms nearly all summer, so bees love to visit it time and time again. It is loved equally as well by hummingbirds and butterflies.
Can you grow Russian sage indoors?
If you are growing a Russian sage plant from seed, start indoors in early spring and transplant the seedlings outdoors after the danger of frost has passed.
Should Russian sage be deadheaded?
Remove the top half of the stems if the plant stops blooming in summer. This encourages new growth and a fresh flush of flowers. Propagate Russian sage plants by dividing the clumps or taking cuttings in spring. Dividing the clumps every four to six years reinvigorates the plants and helps to control their spread.
Should you cut back Russian sage?
Wait until late winter or early spring to cut back Russian sage, butterfly bush and pest-free perennials. ... Cut your Russian sage back to about 4 inches above the ground. If your plants tend to flop during the summer you can cut them a second time. Prune the plants back halfway once they reach 12 inches.
How do you make Russian sage tea?
A perennial plant with purple flowers, Russian sage not only looks great in a garden, but it also has a variety of medicinal benefits. Steep sage leaves in hot water for 15-30 minutes to create a medicinal tea that can help relieve stomach pain and indigestion.
Is Russian sage a good cut flower?
Blue mealy sage can be cut, put in a vase with no water and last for months. Yarrow in white, red or yellow make excellent cut flowers and can also be dried. ... Foliage plants such as purple fountain grass, Russian sage and artemisia, grow well in our area and add a degree of stylishness to fresh flower arrangements.
Does Russian Sage repel mosquitoes?
Both Russian Sage and Lemon Verbena are known for cooking, but they have distinct smells that keep mosquitoes away. ... The Dusty Miller plant has nice foliage color that helps repel the mosquito, according to Gaskins. Lemongrass smells just like lemon when the foliage is brushed.
Read also
- Is Hope Mikaelson stronger than Klaus?
- What does vervain symbolize?
- What herbs can you dehydrate?
- What is the scientific name for Verbena?
- Can you eat verbena?
- What does verbena taste like?
- Is blue vervain the same as vervain?
- Is Wolfsbane a vervain?
- How many sessions of acupuncture do you need for fertility?
- Is Damon Elena's dad?
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- Can you Reroot an African violet?
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- Do African violets need direct sunlight? | https://green-infos.com/library/lecture/read/10740-how-do-you-grow-vervain-seeds |
Need help with my Biology question – I’m studying for my class.
Note: Here are the questions and there is a filed attached below to look at and answer these questions based on the PDF file I have provided.
Question 4. Now, write the generalized equation for the calculation of the total magnification. Your equation should use words, not numbers.
Question 5. Click on the shadowy e and drag it down toward the bottom of the circle (known as the Field of View or FOV), which direction does the stage move? Look at the slide mounted on the microscope while you are doing this.
Question 6. What happens to the stage when you click and drag the e to the top of the FOV?
Question 7. The virtual person who placed this e on the stage, placed it so that the slide label is facing upward and able to be read. However, how does the e look through the FOV?
Question 8. Slide the Course Focus Adjustment knob to the right until the letter e is clear. Then move the Fine Focus slider until it is even clearer. What happens to the amount of light?
Question 9. What do you have to do as you move from a lower magnification objective to a higher one? On a real microscope, would you open or close the iris diaphragm?
Question10. What is the low to high rule? Look for the answers to these in the video: Microscope Basics
Question 11. Why can you see only part of an image? An alternative way of looking at this is: why does a dark moon appears on one side? Look for the answers to these in the video: Common Microscope
Question 12. Why can’t I find the same part of the specimen once I move to a higher magnification? Look for the answers to these in the video: Common Microscope Mistakes:
Question 13. What is the FOV (in millimeters) at low-power of your microscope? Write the units (mm) after the number.
FOVlow = _______ mm
Question 14. How many microns are there in one millimeter? (Hint: which is smaller, micron or millimeter? Therefore, there would be many __________ in one __________.)
1 mm = ______ µm
Question 15. What is the FOV at low power in microns?
FOVlow = _____ µm
Question 16. Calculate the FOV of the high power field in microns (µm). Show your work.
Question 17. Convert the diameter of your field of view you just calculated for high magnification (answer in the previous question) into millimeters.
Question 18. How did the FOV change when you moved from low power to high power?
Question 19. Is the relationship between magnification and FOV direct or inverse? Explain.
Question 20. Describe how these terms are related:
Question 21. Cheek cells have been described as having a modified “fried egg” shape: flat except for the small lump at the nucleus. Can you tell that the cells are flat? Look for cells overlapping one another. Also, the edges of some cells may be “folded” back. Draw a few cheek cells, and label the cell membrane, nucleus and cytoplasm for one of them. (You may need to look these parts up in your textbook.)
Question 22. Recall the calculation for the FOV at high power. Estimate the size of a single cheek cell (at its widest point). One way to look at this is to estimate how many of these cells it would take to cross the entire diameter of the field of light. Then divide the FOV by the number of cells. Assume in the youtube video that you are seeing the entire FOV.
Width of single cheek cell = FOVhigh . = ______ µm
Number of cells across diameter
= _______________________ = ______µm
Question 23. Onion cells (and plant cells, in general) look a bit more like a brick wall. The cells may overlapping one another, depending on whether you accidentally folded the membrane peel. Draw a several onion cells, and label the cell wall, cell membrane, nucleus and cytoplasm for one of them. (You may need to look these parts up in your textbook.)
Question 24. Why do you think that the onion cell appears blocky, rather than rounded like the animal cell?
Question 25. Why do you think there is no evidence of chloroplasts in these onion plant cells? (Hint: Where does an onion with these types of leaves grow: above or below ground? Think about when you leave an onion on your kitchen counter after a while. What happens?)
Question 26. Recall the calculation for the FOV at high power in a previous question. Estimate the size of a single onion cell (along its longest edge).
Length of single onion = FOV . = ___________µm
Number of cells across diameter
= = ___________µm
Question 27. Draw a Paramecium, showing any visible internal structures. Label cell membrane. You may or may not see evidence of the cilia on the cell membrane. Label where the cilia would be.
Question 28. Can you see movement inside the Paramecium? Does there appear to be a single location where the yeasts enter the cell? This is called the oral groove. Label this on the drawing you did for the question above.
Question 29. The Congo Red dye changes from red to blue color when the solution is quite acidic (pH less than 3). Is there any evidence of color change as the yeast cells are consumed?
Question 30. Consider what a plant needs to grow and how different substances are moved from one part of a plant to another. List 3 key factors that a plant needs to grow.
Question 31. Above is a diagram showing the orientation of the cross section and the longitudinal section as the slices were made from the very young corn plant. Draw a few of the red cells as they appear in each section. Label the red and green cells as the vascular bundles.
(a) longitudinal section: (b) cross section:
Question 32. Which of these 3 factors you listed in Question 30 are the vascular bundle cells you observed involved? (If you are not sure, read in your textbook!)
Question 33. Describe what you think is the function of the cell you have drawn above, and explain how the cell’s shape (structure) helps with its function.
Question 34. Draw one adipose cell. On your drawing, label the nucleus, cell membrane, and the area where fat is stored.
Question 35. Remember that adipose cells are spheres and you are only observing two dimensions. Because the adipose tissue on the slide has been sliced very thin, the cell you drew in Question 34 may only be a slice across the middle of the whole cell. Imagine a slice from a tomato. Now imagine how your diagram relates to the entire cell and try to draw the whole cell below.
Question 36. The adipose cell is a beautiful example of structure and function. Its structure is basically a sphere, which of all possible shapes gives the maximum volume for the amount of surface material used (cell membrane). Describe how this feature makes the cell well-suited for its function.
Question 37. Draw one cell with cilia:
Question 38. Epithelial tissues are generally those which cover and protect other tissues. Ciliated epithelial cells are found on inside surfaces of certain tubes in the body, such as the respiratory airways and the oviducts. In these locations, ciliated cells are specialized for moving fluids.
- In the respiratory tract, cilia “beat” in a coordinated manner toward the mouth. Suggest a valuable result of this behavior. | https://pointerwriters.com/bio-120-grossmont-college-experiment-5-the-microscope-the-cell-questions/ |
A Professional Development Model for High School Teachers in Taiwan
The main purpose of the study was to construct a professional development model for senior high school teachers. The teacher evaluation program has been put in practice now in Taiwan. Based on the results of teacher evaluation, a professional development model was needed consequently. This paper is based on the results of a three year research program. Teacher evaluation indicators were ascertained, and several evaluation tools were constructed. A prototype model of teacher evaluation was also constructed and verified. This study then start to survey the need of professional development for high school teachers, and based on the needs to generate a professional development model for teachers. Several results were concluded, such as supporting system is needed so that teachers can find the possible resource to improve their teaching. A fair evaluation evaluation standard system is also important to the professional development activity. | http://i08.cgpublisher.com/proposals/929/index_html |
Planet Earth Facts
This article has been inspired by a book called The Big Countdown: Seven Quintillion, Five hundred Quadrillion Grains of Sand on Planet Earth. Its author is Paul Rockett. It is full of pictures, infographics and amazing planet Earth facts. It was published in 2014 so I imagine that most of the facts are still correct.
Planet Surface
70.8% of the earth’s surface is water.
29.2% of the earth’s surface is land.
The Earth’s Atmosphere
This book takes great delight in using enormous numbers. It tells us about the Earth’s atmosphere is that it is made up of atoms as is everything around us. Apparently, some scientists think that there are about 200 tredecillion atoms in the atmosphere. One tredecillion has 42 zeros in it!
We learn about the 5 layers of the Earth’s atmosphere. They are the troposphere, the stratosphere, the mesosphere, thermosphere and exosphere. The outer layer of the exosphere is 8000 kilometres away, well not exactly because there isn’t a specific edge. It just drifts off into outer space.
One of the facts I found interesting, is that an estimated 40 tonnes of meteors crash into the middle layer every single day. Luckily for us, they burnt out before getting any closer to the Earth.
7 Quintillion, 500 Quadrillion Grains Of Sand On The Planet Earth
7 Quintillion, 500 Quadrillion Grains Of Sand On The Planet Earth is part of the title of the book and is also a chapter title.
This is obviously an estimate. There is no way anyone could count all the grains of sand on the Earth. Sand comes in different sizes anything from 0.06 of a millimetre to 2 mm. To come up with the number 7 quintillion 500 quadrillion, scientist calculated how many grains of sand would fit into a teaspoon and then they multiplied the number of teaspoons they thought they were in all the beaches in the world and all the deserts in the world. How accurate do you think they might have been?
There are some amazingly long beaches in the world, according to this book the longest one in the world is Praia do Cassino beach in Brazil which is just over 250 km long. The next couple of longest beaches are Cox’s Bazar in Bangladesh which is just over 240 km long and Padre Island Texas where the beach is about 230 km long. After that come to beaches which are both called Ninety Mile Beach one of them is in New Zealand and one of them is in Australia. The one that is in Australia is actually slightly longer and the one that is in New Zealand is actually about 88 miles long not 90 as its name suggests.
The largest sun sand castle in the world was built in America and was nearly as tall as 3 double decker buses.
Deserts make up about 9.5% of the world surface, however, only about 20% of the deserts in the world are covered by sand. The others are covered with rocks and pebbles and different types of soils.
People
We now have over 7 billion people living on the Earth. This is twice as many as they were 50 years ago. In the next 50 years it is estimated that we will have over 9 billion people living on the Earth. Nearly 90 babies are born every 20 seconds.
The Earth’s population is spread over 7 continents – Asia, Africa, North America, South America, Europe, Australasia and Antarctica.
Asia has the largest land mass and also the largest population with over 4 billion people living there, most of those in China.
Australasia has the fewest people living there. Nobody lives there permanently but about 4000 scientists come To live and work there each year.
Water
As we’ve said most of the Earth is covered with water, 70.8% of its surface in fact. 68.3% of the earth’s surface is covered with saltwater and 2.5% of the earth’s surface is covered with freshwater. However, about 41% of the known species of fish are only found in freshwater.
The largest fish that is found in salt water is the whale shark which can grow up to 12 metres in length and its mouth is 1 and 1/2 metres wide!
The largest freshwater fish is the Beluga sturgeon this can live in both freshwater and saltwater and it can measure up to 5 metres long.
Sea sponges are a type of animal life, scientists reckon that they have probably been around for over 760 trillion years.
Most of us will have heard of the longest rivers in the world the very longest is the river Nile in Africa which is about 6,650 km long after that is the Amazon in South America, the Yangtze in Asia and the Mississippi in North America.
The largest waterfalls in the world are the Angel Falls in Venezuela Which has a height of 979 m, and then Tugela in South Africa and then or Utigord in Norway.
Volcanoes
Children often enjoy learning about volcanoes at school. There are three types of volcanoes – Composite volcanoes, Cinder Cone volcanoes, and Shield volcanoes.
Volcanoes can also be classified as active dormant or extinct.
Active means it’s erupted in the last 10000 years. Dormant volcanoes, are those which have not erupted but they might erupt again, and Extinct volcanoes are those which are not expected to ever erupt again. | https://tutor-your-child.com/category/geography |
Way Out West is a multi-generational jazz collective including new faces Flo Moore and Tom Millar alongside the legendary Tony Kinsey and many other musicians including Kate Williams, Tim Whitehead, Pete Hurt, Emily Saunders, Chris Biscoe, Tony Woods, Dave Jones, Nette Robinson, Vasilis Xenopoulos, Mike Outram, Gary Willcox and Mick Sexton.
The collective’s experience embraces most of the history of jazz, and this concert will reflect the many ways in which the imagination can be used to create new music.
Way Out West to unikalna organizacja zrzeszająca muzyków i kompozytorów od lat 21 do 85 reprezentujących wszystkie gatunki jazzu od beebopu, poprzez mainstream, free jazz, muzykę latynowską, fusion i nowoczesny jazz. W ciągu 10 lat istnienia zespół zagrał ponad 400 koncertów. W Jazz Café POSK występuje raz w miesiącu prezentując muzyków wybranych z najlepszych londyńskich zespołów. | http://www.jazzcafeposk.org/event/wow-nov-15/ |
If you are struggling to think of a middle name for your daughter Libby, then this blog post is just for you! Middle names can be difficult to come up with, especially if they are not traditional.
This list contains 121 different middle names that would work great with Libby. Check them out and see which one suits her the best!
What Does The Name Libby Mean?
Libby is predominantly an English female name, a pet name for Elizabeth derived from the Hebrew Elisheva, which means ‘God is my oath‘ or the more modern name, which means ‘my heart.’
People believe that the name of their children can have an impact on their future success in life. It can also have a bearing on a child, what their name means.
As a result, people often choose names based on their meanings, believing that the meaning of the child’s name may reflect the nature of the child or influence the child later on.
How Do You Pronounce Libby?
Libby is pronounced as /LIB ee/.
Famous People Called Libby
- Libby Davies (born 1953), Canadian member of parliament
- Libby Gill (born 1954), American motivational writer, speaker and coach
- Libby Gleeson (born 1950), Australian writer
- Libby Fischer Hellmann, American crime fiction writer
- Libby Lane (born 1966), British Anglican bishop
- Libby Larsen (born 1950), American classical composer
- Libby Morris (born 1930), Canadian comic actress
- Libby Munro (born 1981), Australian actress
- Libby Potter, British reporter
- Libby Rees (born 1995), British author
How Common is the Name Libby?
The first name Libby was given to a child in February 7th, 1873, according to the Social Security Administration’s birth record database.
Libby is the 1,910th most prevalent name in the history of the world. Between 1880 and 2019, the Social Security Administration recognized 16,654 babies born in the United States with the first name Libby. In 2009, 355 newborns were given the name Libby, which was the highest known use of the name.
Libby was the 1153rd most popular girl’s name in the United States. There were 202 newborn girls called Libby in 2020.
How to Choose the Perfect Name to go With Libby
The most appropriate middle name for Libby is the one that blends nicely with her first and last names. For example, choose the middle names you want for Libby from the list below and jot them down on a piece of paper.
Say the names out loud with Libby at the start. For example, Libby Ann. Remove any names that are strange, bizarre, or difficult to pronounce.
Step 2 should be repeated, but this time include the last name and listen to how it sounds. Delete any further names that don’t sound suitable.
Examine the initials of the remaining names and identify the ones that work particularly well together. Make certain that it does not sound absurd or harsh. Consider looking for ones that can be given nicknames when they are grouped together.
By the time you are finished with steps 1-4, you will have cut down the selection to a manageable amount of options that will assist you in deciding on the ideal middle name for Libby.
Don’t panic, most people have a hard time finding a name they love, it’s rare it comes to the first thing and often can take months of poring over name lists and finding that perfect middle to suit the first name and last name. Keep saying the list of baby names along with the first and middle names out loud until you find the one you love the most.
121 Best Middle Names for Libby
We’ve pulled together the best girl’s names that go well with the name Libby. Here’s our list:
First Name for Middle Name Libby
If you have already decided on Libby as the middle name you might be looking for the perfect first name as Libby makes a great middle name. We have our top 10 here!
10 Classic Girl Names That Go Well With Libby
10 Modern Names That Go With Libby
10 Unusual Girl Names That Go With Libby
European Names That Match the Name Libby
If you love European names or even have European ancestry we’ve compiled the best Irish, French, Spanish, Italian, German and Polish names to go with Libby. We’ve kept these spelt in their original authentic way, but you can always update the spelling if you prefer.
7 Irish Girl Names That Go with Libby
If you like Irish girl names check out our post.
7 French Names to Go with Libby
7 Spanish Names That Go Well with Libby
7 Italian Names for Libby
7 German Names for Libby
7 Polish Names for Libby
Names of Certain Length
If you have a long or short surname, you might want to choose middle names with just 1, 2 or 3 syllables, we’ve got the best of each to help you choose the best middle name for your child.
7 Names for Libby with 1 Syllable
7 Names for Libby with 2 Syllables
7 Names for Libby with 3 Syllables
Names for Libby with the Same Initial
7 Names for Libby Starting with A Vowel
Nicknames for Libby
Libby is often given the following nicknames:
- Libs
- Lib
- Libster
Different Ways to Spell Libby
- Libbie
- Liby
- Libbi
- Libi
Similar Names Like Libby
- Millie
- Lottie
- Gracie
- Evie
- Kit
- Emmy
- Gwen
- Dottie
Looking for a sibling name for Libby? We’ve compiled a list of fantastic good middle names for Lilith.
Sibling Names for Libby
If you already have a child named Libby, you might be looking for some great sister or brother names that go with it. We’ve compiled some classic names that go together really well with Libby.
Sister Names for Libby
Perhaps you are expecting a baby girl, here are some great baby names that match Libby. We’ve chosen a selection of popular girls’ names and modern names that match.
Brother Names for Libby
If you are looking for matching boy names to go with Libby, and your family name, then this list of boys names might match!
What are middle names for?
Giving a child a middle name is a tradition that dates back to the medieval era. Back then, people would give their children multiple names in order to confuse evil spirits.
They would also use middle names in order to honor important people in their lives. Middle names are still given today for these reasons and many others.
Some parents choose to give their children middle names based on names they like. Other parents choose middle names because they have a special meaning to them. Whatever the reason, it is clear that there are many benefits to giving your child a middle name.
Often people can make the middle name a nickname later on or even you can put the initials together to form a new name, for example, Libby Abigail could become LA!
Research has also indicated that having middle names improves people’s chances of getting a better job in the future! According to a study published in the European Journal of Social Psychology, people who have middle names are believed to have a high social position and to be intellectually superior.
Take Away For Baby Name Libby
Middle names are a great way to honor family members and create meaningful connections with children, both now and in the future. This article examined 121 of the most popular middle name options that would be perfect for your daughter named Libby.
We also provided you with some tips on how to choose an appropriate middle name so it blends nicely with her first and last names as well as what other nicknames she may go. | https://mommyandlove.com/middle-names-for-libby/ |
This country-specific Q&A gives a pragmatic overview of the law and practice of insurance & reinsurance law in the Switzerland.
It addresses topics such as contract regulation, licensing, penalties, policyholder protection, alternative dispute resolution as well as personal insight and opinion as to the future of the insurance market over the next five years. Downdload here.
- COVID-19: UK Supreme Court dismissed appeals by insurers in the FCA test cases on business interruption insurance and strenghtens the position of the insured
London, 15th January 2021: UK Suprme Court dismisses appeals y the insurers in the FCA test case against business interrutption insurers and strenghtens the position of the the policyholders. Here you could find the press release and the full judgement of the Supreme Court.
- Federal Council brings revised Insurance Contract Act into force
During its meeting on 11 November 2020, the Federal Council brought the revised Insurance Policies Act into force with effect from 1 January 2022.
We are happy to assist you with the required implementaion of any changes to your insurance products.
- Federal Council adopts dispatch on partial revision of Insurance Oversight Act
On 21 October 2020, the Federal Council adopted the dispatch on a partial revision of the Insurance Oversight Act (IOA). Please find further Information here.
- Reachability of our firm from 16th March 2020
Dear Clients and Business Partners
We are fully committed to you. Due to the current situation, we have decided that our employees and we will work from our home offices until further notice to protect our and other's health. As usual, you can reach us by email and phone.
However, there could be delays in receiving letters and parcels. We kindly ask you to refrain from sending us letters and to replace them with emails - as far as possible.
Thank you for your understanding.
Stay healthy!
Chris Bell, Sara Andrea Behrend, Ulrike Mönnich
- "Principles of of Reinsurance Contract Law" (PRICL) published
- Leander D Loacker appointed Professor of law at University of Zurich
We, the partners of mbh-attorneys-at-law, congratulate Prof Dr Leander D Loacker, who is Of-counsel to the firm, on his appointment as full Professor of Law at the University of Zurich. Leander succeeds Prof Dr Anton K. Schnyder from 1st August 2018 as head of the chair of Private and Business Law, Private International Law, International Civil Procedure and Comparative Law. We wish Leander every success and all the best in his new position and look forward to continuing to cooperate successfully with him.
Details and contact in our firm
- mbh in the top 20 Insurance law firms in Switzerland 2018
mbh ATTORNEYS AT LAW – Mönnich, Bell & Behrend KLG two years after its foundation inthe top 20 of Swiss law firms in insurance law, according to the business magazine Bilanz. We are grateful and thank our clients and colleagues for their recommendations. Please find the ranking here:
- Sara Andrea Behrend joins mbh as new partner
mbh Attorneys at law are delighted to announce that Sara Andrea Behrend has joined us as a partner as of 1st October 2017.
Sara Andrea Behrend qualified as an attorney at law in 2010 and specialises in transport and insurance law as well as aviation law. In addition, Sara advises and assists her clients generally; helping them with a range of problems from employment law to contract wordings, to competition law.
Sara is a private pilot and has a number of other interests, including taking part in triathlons.
Her contact details are:
T: +41 44 55 21 687
M: +41 76 391 06 03
Email: [email protected]
You will find Sara's full CV here
- Federal Council opens consulting proceedings on a partial review of the Swiss insurance contract law
The Fedreal Council has opend the consulting proceedings on a partial review of the Swiss insurance contract law on 6 July 2016. You will find further information and documentation here. Please do not hesitate to contact us if you should need further advise on this.
- Event: Judicial control of general insurance conditions
On Tuesday 19 April 2016, the AIDA-Swiss Chapter held an event on the topic "Judicial control of general insurance conditions according to Art 8 of the Swiss Unfair Competition Act (UWG)" in Zurich. We are grateful for the lively interest of all participants and are pleased to share with you documentation in respect mbh's contribution for download.
- IDD - New European Directive on Insurance Distribution
Directive (EU) 2016/97 of the European Parliament and of the Council of 20 January 2016 on insurance distribution has been released. Please find an article by Helmut Heiss and Ulrike Mönnich on the new directive in HAVE/REAS 2016 (Issue 1) p. 25 to 34.
- PRICL
mbh appointed special advisors to Principles of Reinsurance Contract Law (PRICL), for the development of arbitration clauses.
PRICL is a project established in order to provide a uniform frame of reference and uniform legal terminology for reinsurance contracts. It is led by Professors Helmut Heiss and Anton Snyder at the University Of Zürich.
PRICL
- PAC RE 5-AT v AmTrust North America Inc (AmTrust) No CV-14-131-BLG-CSO
This was a declaratory judgment action arising out of a demand for arbitration by AmTrust that had been sent to Pacific Re Inc. ("Pacific Re") and to Pac Re 5-At ("Cell 5"), a protected cell of Pacific Re.
A judicial declaration was sought that under Montana law Cell 5, and only Cell 5, was the proper party to the arbitration proceedings.
Cell 5 is a protected cell, but not an incorporated cell. The issue arose because Cell 5 was not a separate legal entity, de jure, but had many of the attributes of such an entity de facto.
It was held that under the relevant Montana statute, although a protected cell has many of the attributes of independence from the Protected Cell Company, it does not itself have the capacity to sue and be sued and remained a part of the Protected Cell Company, which can sue or be sued on the protected cell's behalf, in the same manner as it can enter agreements on the behalf of the protected cell. Therefore, Pacific Re was properly before the arbitration tribunal and would be bound by the results of the arbitration.
- Brit UW Limited v F&B Trenchless Solutions Limited (“FBTS”) (2015 EWHC) 2237(Comm)
Brit UW Limited, acting for Brit Syndicate 2987 ("Brit"), sought a declaration that it had validly avoided a contractor's combined liability policy that it had entered into with the Defendant ("the Policy"). The avoidance was based on alleged material misrepresentation and failure to disclose material information prior to the conclusion of the Policy.
The case involved a lengthy consideration of the factual background and a counter allegation that the Plaintiff had affirmed the Policy
The judge found for Brit and held that the Policy had been validly avoided.
Link
- 2016: Founding of mbh in Zurich
mbh became active in Zürich as of 1st January 2016. It was founded on the principles that we should provide the services that our clients want and need and bring an international approach to the practice of law. We have brought together a group of people from different countries and cultures, all with the same belief; it is not about us, but about you, our clients. | https://www.mbh-law.ch/en/news |
Coconut Octopus Interesting Facts
What type of animal is a coconut octopus?
A coconut octopus is a type of medium-sized octopus that inhabits the sandy and muddy seabeds of the western Pacific.
What class of animal does a coconut octopus belong to?
Amphioctopus marginatus, known as a coconut octopus, belongs to the class of cephalopods. Cephalopoda is one of the most complex classes in terms of behavior and morphology in the Mollusca phylum. The literal meaning of the word 'cephalopoda' is 'head-foot' and these animals possess the most complex brains among all invertebrates.
How many coconut octopuses are there in the world?
The exact number of coconut octopuses is not known. However, this species is not critically endangered.
Where does a coconut octopus live?
A coconut octopus lives in the tropical waters of the western Pacific and the Indian Ocean. Its indigenous range extends from southern Japan to Australia, and New Guinea. Coconut octopuses are also seen in South Africa.
What is a coconut octopus's habitat?
A coconut octopus falls into the category of tropical water species. Shallow coastal water is their habitat of choice and they spend most of their time on muddy or sandy sea floors near the shoreline. A preference for calm water as a habitat is also displayed by this species. A coconut octopus is often found in lagoons, bays and other inlets. Sometimes their habitat includes deeper waters up to a depth of 600 ft (183 m). A coconut octopus sometimes finds shelter inside empty coconut shells, clam shells or stays buried in the sand with only its eyes visible.
Who do coconut octopuses live with?
A coconut octopus or veined octopus usually lives in solitude throughout its life except during the mating period.
How long does a coconut octopus live?
A coconut octopus has a longer lifespan than other smaller octopus species. A coconut octopus lives for three to five years. Males die within weeks, if not days. However, females live for months or years. They die generally after the hatching of their eggs as they abstain from eating during the entire brooding period.
How do they reproduce?
A coconut octopus attains sexual maturity at the age of one to two years. In warm water, the spermatophore is injected from one of the tentacles of a male to the mantle portion of a female. Mating activity is rapid mainly because a female octopus is prone to devour a male. Male coconut octopuses sometimes disguise themselves as females to save their lives.
A female octopus lays approximately 100,000 eggs after 11 months. The size of each of the eggs is less than 0.2 in (6 mm). The eggs are protected in a safe corner or crevice and a female coconut octopus looks after them until they hatch.
What is their conservation status?
IUCN has not evaluated the conservation status of a coconut octopus. Predators like great white sharks, viper fish and bigger octopuses hunt them in the sea. Seafood lovers also enjoy eating coconut octopus like other shell fish.
Coconut Octopus Fun Facts
What do coconut octopuses look like?
The body of a coconut octopus is reddish-brown with veins running across its body. They have a yellow siphon utilized both for locomotion and for respiration purposes. Their tentacles are white and blue and the edges of their arms have a dark tinge. A white shade is visible under their eyes in a trapezoid-shaped area. These octopuses can blend in with their surroundings using camouflage techniques.
How cute are they?
There can be differences of opinion regarding the cuteness of an octopus. A coconut octopus or a veined octopus can appear gross to some because of its tentacles. However, some people consider them adorable and a bit naughty as well.
How do they communicate?
A coconut octopus, or Amphioctopus marginatus, is a solitary creature that lies on its own on the bottom of the sea using a shell as a protective cover. Octopuses are rarely seen in contact with others of their kind except for mating. They use their eight arms to communicate and their body color may change during this interaction. Coconut octopuses sometimes interact with divers by latching on to them and testing their tools. They are often seen taking colorful plastic items from divers especially ones that resemble a shell.
How big is a coconut octopus?
A coconut octopus is approximately 3 in (8 cm) long if we consider just its main body. Their length can go up to 6 in (15 cm) if you include their tentacles. They fall into the category of medium sized octopuses.
How fast can coconut octopuses move?
A coconut octopus or veined octopus moves slowly as it carries coconut shells and clam shells with it for protection from other predators in the ocean. Stilt walking is the name given to this locomotive technique. Some predators may have enough strength to break their portable lair but most predators in the ocean are not intelligent enough to do so. When in danger, they create streams of water by funneling water through their siphon to propel it go faster.
How much does a coconut octopus weigh?
The bodyweight of a coconut octopus or veined octopus is between 7-9 oz (200-250 g).
What are their male and female names of the species?
A female octopus is called a 'hen'. Male octopuses have a specialized arm called a hectocotylus to inject sperm into the mantle region of females during mating.
What would you call a baby coconut octopus?
There is no particular name for a baby coconut octopus. A baby octopus is usually called a 'larvae'.
What do they eat?
A coconut octopus eats small invertebrates such as crabs, shrimp and, clams. Their beak is hard enough to break the shell of crustaceans. They are also able to prey on small fish in the ocean. A coconut octopus searches for prey and grabs it with their tentacles when they come too near. The prey is then brought to their mouths and perforated by their strong beaks.
Are they dangerous?
A coconut octopus is a venomous octopus but the venom is not dangerous to divers in the sea.
Would they make a good pet?
Octopuses are intelligent animals and they can be good pets if they acclimatize to their captive environment. A 55 gal (250 l) aquarium is required and a second tank is needed for holding the equipment when filtering the water. Food for a coconut octopus is also expensive. These octopuses display hunting behavior and prefer live prey in the same way they can find in the ocean.
Kidadl Advisory: All pets should only be bought from a reputable source. It is recommended that as a potential pet owner you carry out your own research prior to deciding on your pet of choice. Being a pet owner is very rewarding but it also involves commitment, time and money. Ensure that your pet choice complies with the legislation in your state and/or country. You must never take animals from the wild or disturb their habitat. Please check that the pet you are considering buying is not an endangered species, or listed on the CITES list, and has not been taken from the wild for the pet trade.
Did you know...
A coconut octopus is one of the rarest creatures that exhibit bipedal locomotion. Their movement resembles the walking of humans. While it is walking, this octopus carries a coconut shell or another protective item over its head with its other arms. Their bipedal abilities have evolved to adapt to a marine environment.
These little animals are also very selective about the tools they use. They are often seen picking up attractive plastic items from the sand, cleaning them with a careful spray of water jets and using them as shields.
Why does a coconut octopus glow?
No form of bioluminescence is produced by a coconut octopus. The sharp contrast between its darker body and white suckers may have a fluorescent effect. In reality, a coconut octopus does not glow.
How did the coconut octopus get its name?
A coconut octopus is one of the few invertebrates that are intelligent enough to use tools. Researchers from Melbourne Museum in Australia first noticed this type of behavior in this species. These octopuses were often noticed making a shelter in discarded, hollowed-out coconut half-shells lying on the sand of the ocean floor. Other than coconut shells, they also use clam shells and even plastic items as shields to protect themselves from predators. They are also known as a veined octopus due to the presence of veins in their main body.
Here at Kidadl, we have carefully created lots of interesting family-friendly animal facts for everyone to discover! Learn more about some other arthropods including blue crab, or jellyfish.
You can even occupy yourself at home by drawing one on our giant octopus coloring pages. | https://kidadl.com/facts/animals/coconut-octopus-facts |
BACKGROUND OF THE INVENTION
Field of the Invention
Related Arts
SUMMARY OF THE INVENTION
DESCRIPTION OF THE PREFERRED EMBODIMENT
Experiment 1
Action for Disintegrating Gallstones
Experiment 2
Experiment 3
Experiment 4
Experiment 5
Example 1
The present invention relates to an agent for disintegrating gallstones, and more particularly to an agent for disintegrating gallstones containing a combination of sodium or potassium hydrogen carbonate and N-acetylcysteine as its effective components.
The statistical study shows that gallstone carriers reach about 9 % of the population in Japan. In 1989, the cholecystolithiasis patients were reported to be 270,000, most of whom were subjected to surgical lithectomy. The gallstones are classified into cholesterol gallstones and pigment gallstones (bilirubin calcium and black-pigment stones) based on the component thereof. Studies on an agent for dissolving gallstones have been performed for a long time, resulting in the development of formulations of bile acid such as ursodeoxycholic acid and chenodeoxycholic acid for dissolving choresterol gallstones. Clinical tests have proved that such formulations are effective for gallstone dissolution. However, actual effect for gallstone dissolution given by such formulations depends on the size of the stone and cholecyst function, whereby the patient should be dosed for a long period such as a half- year to a year or more. A suitable agent for dissolving pigment stones has not yet been developed in spite of various studies.
The inventor has found out from studies on composition of the gallstones that mucopolysaccharides and mucoproteins (hereinafter abbreviated to muco-substances) play an important role as a component for binding the crystalline cholesterol or bilirubin. On the other hand, SH-compounds including N-acetylcysteine have been known to have action for splitting off the S-S bond of muco-substances. In particular, it has been reported that N-acetylcysteine accelerates dissolution of cholesterol gallstones [see Scand. J. Gastroent., 24, 373-380 (1989); Gastroenterology., 98, 454-463 (1990)]. However, this document reports that the dissolution of cholesterol gallstones takes about 10 hours to several weeks, and further that the absolute effect is small. EP-A-0 318 773 discloses a pharmaceutical composition for dissolution of gallstones comprising N-Acetylcysteine and a borate/sodium carbonate buffer.
On the other hand, the effective action of alkali metal carbonates for disintegrating gallstones has not been known.
Under such circumstance, it is desired to develop an agent which can easily destroy gallstones to make their small pieces which can be removed without surgical lithectomy.
The invention provides an aqueous solution comprising sodium or potassium hydrogen carbonate in combination with N-acetylcysteine.
The concentration of sodium or potassium hydrogen carbonate in the aqueous solution is 5 to 20 % (W/V) (represented by weight per volume hereinafter), for example, 8 %, 10 % or 12 %.
Suitably, N-acetylcysteine is contained in the aqueous solution at the concentration of 10 to 25 %, for example, 10 %, 15 % or 20 %.
Suitable medium for the aqueous solution is distilled water, particularly sterile water.
The aqueous solution as the agent for disintegrating gallstones of the present invention is observed to have pH 7 to 10, preferably 7.5 to 9. The aqueous solution may be adjusted by adding an acid or an alkali when it does not exhibit the above mentioned pH range with only effective components.
The dose of the agent for disintegrating gallstones according to the present invention varies with the age of patient, level of disease or the like, but generally 1 to 100 ml per one time for an adult. The administration methods include the following: a method in which the agent is infused or perfused into a bile duct or gallbladder through a tube which is left at the surgery; a method in which the agent is infused into a bile duct through a T-tube inserted into the bile duct at choledocholithotomy; or a method in which the agent is retrogressively infused into a gallbladder or bile duct from duodenal mamilla part under endoscopy.
N-Acetylcysteine used for the present invention exhibits remarkably high safety, since it is clinically used for human being as an agent for protecting liver or an expectorant. The sodium hydrogen carbonate, which is representative of the alkali metal hydrogen carbonate, is clinically employed as ulcer-preventive agent. Accordingly, the agent for disintegrating gallstones of the present invention exhibits high safety.
The present invention will be specifically explained hereinafter with reference to the Experiments and Examples. The percentage hereinbelow represents W/V %.
Table 1
Test Solution
Number of Pieces of Stones with Time
0
5
10
15
20
25
30
35
40 (min.)
1
1
1
14
24
47
113
129
143
174
2
1
1
13
52
256
332
389
423
450
3
1
41
58
123
180
185
203
230
315
4
1
1
1
1
1
1
1
1
1
Gallstones (54.9 ± 6.8 mg) nearest in shape, material and size were obtained from a single patient for this Experiment. The obtained gallstones were black-pigment stones having a rough structure and containing a number of muco-substances. 20 % of N-acetylcysteine were respectively added to distilled water and 10 % sodium hydrogen carbonate to obtain respective test solutions 1 and 2. On the other hand, a test solution 3 containing only 10 % sodium hydrogen carbonate and a test solution 4 (control) containing only distilled water were prepared. One stone of the obtained gallstones was put into each test solution for observing the number of a piece of the stones by taking pictures thereof every five minutes. Table 1 shows the degree of the gallstone disintegration.
As is apparent from Table 1, gallstone disintegration was observed in test solutions 1 to 3 except for distilled water (control test solution 4). In particular, gallstone disintegration was remarkably observed in the test solution 2 in which N-acetylcysteine was added to 10 % sodium hydrogen carbonate compared to the test solutions 1 and 3.
Table 2
Test Solution
Number of Pieces of Stones with Time
5
10
15
20
25
30
35
40 (min.)
5
22
43
68
84
103
112
126
166
6
1
1
1
1
4
8
11
15
Under the same conditions as in the Experiment 1, gallstone disintegratione was observed using a test solution 5 containing 10 % potassium hydrogen carbonate and a control test solution 6 containing 10 % sodium hydrogen oxide. The results were shown in Table 2.
Gallstone disintegration was rarely observed in the control test solution 6.
Table 3
Test Solution
Number of Pieces of Stones with Time
5
10
15
20
25
30
35
40 (min.)
7
6
36
49
88
137
425
465
480
8
14
96
140
268
280
315
340
390
Under the same conditions as in the Experiment 1, the Experiment 3 was performed using a test solution 7 in which 20 % of N-acetylcysteine was added to 10 % sodium hydrogen carbonate (pH 7.5) and a control test solution 8 in which 20 % N-acetylcysteine was added to 10 % potassium hydrogen carbonate (pH 7.5) for observing the difference caused by the kind of the alkali metal hydrogen carbonate. The results were shown in Table 3.
Table 4
Test Solution
Number of Pieces of Stones with Time
5
10
15
20
25
30
35
40 (min.)
9 (pH 6.0)
1
2
4
8
14
16
16
48
10 (pH 7.5)
6
36
49
88
137
425
465
480
11 (pH 8.2)
120
138
169
286
306
328
415
488
Under the same conditions as in the Experiment 1, gallstone disintegration was observed using test solutions 9 to 11 in which 10 % sodium hydrogen carbonate was added to 20 % N-acetylcysteine solution to find out the effect of pH on gallstone disintegration. Each of the test solutions 9 to 11 varies in pH. Specifically, the test solution 9 is pH 6.0, the test solution 10 is pH 7.5 and the test solution 11 is pH 8.2. The results were shown in Table 4.
The function of gallstone disintegration was remarkably reduced in the solution of at least pH 6.0 compared to the other solutions of different pH but of the same effective components.
Table 5
Test Solution
Number of Pieces of Stones with Time
5
10
15
20
25
30
35
40 (min.)
12
1
5
11
16
18
28
35
38
13
25
50
58
63
73
89
136
145
3
41
58
123
180
185
203
230
315
Under the same conditions as in the Experiment 1, gallstone disintegration was observed using test solutions 12, 13 and 3 of 1 %, 5 % and 10 % sodium hydrogen carbonate respectively to find out the effect of concentration of the sodium hydrogen carbonate. The results were shown in Table 5.
The effects on gallstone disintegration were great in the cases of 5 % and 10 % sodium hydrogen carbonate, particularly 10 % sodium hydrogen carbonate.
N-Acetylcysteine
200.0 mg
Sodium hydrogen carbonate
100.0 mg
Distilled water
Total
300.0 mg (per 1ml)
Injection for local administration was prepared by mixing the following compositions in accordance with a known method.
The effective components of an agent according to the present invention possess action for rapidly disintegrating gallstones, thereby useful for therapy of cholecystitis. | |
A ventilator is an automatic mechanical device designed to provide all or part of the work the body must produce to move gas into and out of the lungs. The act of moving air into and out of the lungs is called breathing, or, more formally, ventilation.
Background on Ventilation
During breathing, a volume of air is inhaled through the airways (mouth and/or nose, pharynx, larynx, trachea, and bronchial tree) into millions of tiny gas exchange sacs (the alveoli) deep within the lungs. There it mixes with the carbon dioxide-rich gas coming from the blood. It is then exhaled back through the same airways to the atmosphere. Normally this cyclic pattern repeats at a breathing rate, or frequency, of about 12 breaths a minute (breaths/min) when we are at rest (a higher resting rate for infants and children). The breathing rate increases when we exercise or become excited.
Gas exchange is the function of the lungs that is required to supply oxygen to the blood for distribution to the cells of the body, and to remove carbon dioxide from the blood that the blood has collected from the cells of the body. Gas exchange in the lungs occurs only in the smallest airways and the alveoli. It does not take place in the airways (conducting airways) that carry the gas from the atmosphere to these terminal regions. The size (volume) of these conducting airways is called the anatomical “dead space” because it does not participate directly in gas exchange between the gas space in the lungs and the blood. Gas is carried through the conducting airways by a process called “convection”. Gas is exchanged between the pulmonary gas space and the blood by a process called “diffusion”.
One of the major factors determining whether breathing is producing enough gas exchange to keep a person alive is the ‘ventilation’ the breathing is producing. Ventilation is expressed as the volume of gas entering, or leaving, the lungs in a given amount of time. It can be calculated by multiplying the volume of gas, either inhaled or exhaled during a breath (called the tidal volume), times the breathing rate (e.g., 0.5 Liters x 12 breaths/min = 6 L/min).
Therefore, if we were to develop a machine to help a person breathe, or to take over his or her breathing altogether, it would have to be able to produce a tidal volume and a breathing rate which, when multiplied together, produce enough ventilation, but not too much ventilation, to supply the gas exchange needs of the body. During normal breathing the body selects a combination of a tidal volume that is large enough to clear the dead space and add fresh gas to the alveoli, and a breathing rate that assures the correct amount of ventilation is produced. However, as it turns out, it is possible, using specialized equipment, to keep a person alive with breathing rates that range from zero (steady flow into and out of the lungs) up to frequencies in the 100’s of breaths per minute. Over this frequency range, convection and diffusion take part to a greater or lesser extent in distributing the inhaled gas within the lungs. As the frequency rates are increased, the tidal volumes that produce the required ventilation get smaller and smaller.
We will consider two classes of ventilators here: those that produce breathing patterns that mimic the way we normally breathe (i.e., at rates our bodies produce during our usual living activities: 12 – 25 breaths/min for children and adults; 30 – 40 breaths/min for infants) – these are called conventional ventilators; and those that produce breathing patterns at frequencies much higher than we would or could voluntarily produce for breathing – called high frequency ventilators.
There are two sets of forces that can cause the lungs and chest wall to expand: the forces produced when the muscles of respiration (diaphragm, inspiratory intercostal, and accessory muscles) contract, and the force produced by the difference between the pressure at the airway opening (mouth and nose) and the pressure on the outer surface of the chest wall. Normally, the respiratory muscles do the work needed to expand the chest wall, decreasing the pressure on the outside of the lungs so that they expand, which in turn enlarges the air space within the lungs, and draws air into the lungs. The difference between the pressure at the airway opening and the pressure on the chest wall surface usually does not play a role in this activity because, both of these locations being exposed to the same pressure (atmospheric), this difference is zero. However, when the respiratory muscles are unable to do the work required for ventilation, either or both of these two pressures can be manipulated to produce breathing movements.
It is not difficult to visualize that, if the pressure at the mouth and nose of an individual were increased while the pressure surrounding the rest of the person’s body remained at atmospheric, the person’s chest would expand as air is literally forced into the lungs. Likewise, if the pressure on the person’s body surface were lowered as the pressure at the person’s open mouth and nose remained at atmospheric, then again the pressure at the mouth would be greater than that on the body surface and air would be forced into the lungs. Thus, we have two approaches that can be used to mechanically ventilate the lungs: apply positive pressure (relative to atmospheric) to the airway opening – devices that do this are called positive pressure ventilators; or, apply negative pressure (relative to atmospheric) to the body surface (at least the rib cage and abdomen) – such devices are called negative pressure ventilators.
Mechanical Ventilators
The simplest mechanical device we could devise to assist a person’s breathing would be a hand-driven, syringe-type pump that is fitted to the person’s mouth and nose using a mask. A variation of this is the self-inflating, elastic breathing bag. Both of these require one-way valve arrangements to cause air to flow from the device into the lungs when the device is compressed, and out from the lungs to the atmosphere as the device is expanded. Also, it can be appreciated that such arrangements are not automatic, requiring an operator to supply the energy to push the gas into the lungs through the mouth and nose.
Automating the ventilator so that continual operator intervention is not needed for safe, desired operation requires 1) a stable attachment (interface) of the device to the patient, 2) a source of energy to drive the device, 3) a control system to make it perform appropriately, and 4) a means of monitoring the performance of the device and the condition of the patient.
- Patient Interface. Positive Pressure Ventilators: The ventilator delivers gas to the patient through a set of flexible tubes called a patient circuit. Depending on the design of the ventilator, this circuit can have one or two tubes. The circuit connects the ventilator to either an endotracheal or tracheostomy tube that extends into the patient’s throat (causing this arrangement to be called invasive ventilation), or a mask covering the mouth and nose or just the nose (referred to as noninvasive ventilation). Each of these connections to the patient may have a balloon cuff associated with it to provide a seal – either inside the trachea for the tracheal tubes or around the mouth and nose for the masks. Negative Pressure Ventilators:The patient is placed inside a chamber with his or her head extending outside the chamber. The chamber may encase the entire body except for the head (e.g., iron lung), or it may enclose just the rib cage and abdomen (cuirass). It is sealed to the body where the body extends outside the chamber. Although it is not generally necessary, the patient may have an endotracheal or tracheostomy tube in place.
- Power Sources. Positive Pressure Ventilators are typically powered by electricity or compressed gas. Electricity is used to run compressors of various types. These provide compressed air both for motive power as well as air for breathing. More commonly, however, the power to expand the lungs is supplied by compressed gas from tanks, or from wall outlets in the hospital. The ventilator is generally connected to separate sources of compressed air and compressed oxygen. This permits the delivery of a range of oxygen concentrations to support the needs of sick patients. Because compressed gas has all moisture removed, the gas delivered to the patient must be warmed and humidified in order to avoid drying out the lung tissue. A humidifier placed in the patient circuit does this. A humidifier is especially needed when an endotracheal or tracheostomy tube is used since these cover or bypass, respectively, the warm, moist tissues inside of the nose and mouth and prevent the natural heating and humidification of the inspired gas. Negative Pressure Ventilators are usually powered by electricity used to run a vacuum pump that periodically evacuates the chamber to produce the required negative pressure. Humidification is not needed if an endotracheal tube is not used. Oxygen enriched inspired air can be provided as needed via a breathing mask.
- Control System. A control system assures that the breathing pattern produced by the ventilator is the one intended by the patient’s caregiver. This requires the setting of control parameters such as the size of the breath, how fast and how often it is brought in and let out, and how much effort, if any, the patient must exert to signal the ventilator to start a breath. If the patient can control the timing and size of the breath, it is called a spontaneous breath. Otherwise, it is called a mandatory breath. A particular pattern of spontaneous and mandatory breaths is referred to as a mode of ventilation. Numerous modes, with a variety of names, have been developed to make ventilators produce breathing patterns that coordinate the machine’s activity with the needs of the patient.
- Monitors. Most ventilators have at least a pressure monitor (measuring airway pressure for positive pressure ventilators, or chamber pressure for negative pressure ventilators) to gauge the size of the breath and whether or not the patient is properly connected to the ventilator. Many positive pressure ventilators have sophisticated pressure, volume and flow sensors that produce signals both to control the ventilator’s output (via feedback in the ventilator’s control system) and to provide displays (with alarms) of how the ventilator and patient are interacting. Clinicians use such displays to follow the patient’s condition and to adjust the ventilator settings.
Conventional Ventilators
The vast majority of ventilators used in the world provide conventional ventilation. This employs breathing patterns that approximate those produced by a normal spontaneously breathing person. Tidal volumes are large enough to clear the anatomical dead space during inspiration and the breathing rates are in the range of normal rates. Gas transport in the airways is dominated by convective flow and mixing in the alveoli occurs by molecular diffusion. This class of ventilator is used in the ICU, for patient transport, for home care and in the operating room. It is used on patients of all ages from neonate to adult.
High Frequency Ventilators
It has been known for several decades that it is possible to adequately ventilate the lungs with tidal volumes smaller than the anatomic dead space using breathing frequencies much higher than those at which a person normally breathes. This is actually a common occurrence of which we may not be fully aware. Dogs do not sweat. They regulate their temperature when they are hot by panting as you probably know. When a dog pants he takes very shallow, very fast, quickly repeated breaths. The size of these panting breaths is much smaller that the animal’s anatomical dead space, especially in dogs with long necks. Yet, the dog feels no worse for this type of breathing (at least all the dogs interviewed for this article).
Devices have been developed to produce high frequency, low amplitude breaths. These are generally used on patients with respiratory distress syndrome (lungs will not expand properly). These are most often neonates whose lungs have not fully developed, but can also be older patients whose lungs have been injured. High frequency ventilators are also used on patients that have lungs that leak air. The very low tidal volumes produced put less stress on fragile lungs that may not be able to withstand the stretch required for a normal tidal volume.
There are two main types of high frequency ventilator: high frequency jet ventilators (HFJV) and high frequency oscillatory ventilators (HFOV). The HFJV directs a high frequency pulsed jet of gas into the trachea from a thin tube within an endotracheal or tracheostomy tube. This pulsed flow entrains air from inside the tube and directs it toward the bronchi. The HFOV uses a piston arrangement that moves back and forth rapidly to oscillate (vibrate lengthwise) the gas in the patient’s breathing circuit and airways. Both of these techniques cause air to reach the alveoli and carbon dioxide to leave the lungs by enhancing mixing and diffusion in the airways. Convection plays a minor role in gas transport with these ventilators while various forms of enhanced diffusion predominate.
Although high frequency devices that drive the pressure on the chest wall have been developed, most high frequency ventilators in use today are applied to the airway opening.
In future articles, the authors will explore topics such as how ventilators work, the controls and monitors that can be available on a ventilator, interpretation of graphical displays of ventilatory variables, as well as various clinical aspects of ventilator use. | https://ventworld.org/what-is-a-ventilator-part-1/ |
The COVID-19 pandemic and its dire consequences have hit Europe and the entire world with full strength. Putting to the test every person, family and community, the present crisis has exposed the vulnerabilities and apparent certainties of our politics, economics and societies.
Nevertheless, these trying times are also allowing us to re-discover our common humanity as brothers and sisters. We think of the many people who are sowing hope every day by exercising charity and solidarity. We would like to pray with deep gratitude for all those who serve their fellow human beings with empathy and warmth by supporting them selflessly: medical doctors, nursing staff, providers of basic services, law and order forces – and persons involved in pastoral care. We wish to pray for all the people who are suffering during this crisis - in particular the sick, the elderly, the poor, the excluded and children experiencing family instability. We also remember all those who passed away in our prayers.
Furthermore, we gratefully welcome the many individual and collective initiatives that are reinventing new forms of solidarity and new ways of sharing beyond the necessary social distancing.
We gratefully commend the numerous policy actions of mutual support and encourage the political decision-makers in the European Union and its Member States, to continue acting in a determined, transparent, empathic and democratic way. We pray for wisdom and strength among leaders at both national and European level. This is the time for all of us to demonstrate our joint commitment to the European project and to common European values of solidarity and unity, instead of capitulating to fear and nationalism.
Concrete expressions of this our shared European responsibility could, for example, be burden-sharing in the care for the sick, a facilitated exchange of medical materials, creative measures alleviating social, economic and financial shocks, as well as reinforced international cooperation and humanitarian assistance to support weaker health systems in needy regions of the world.
The Lenten journey towards Easter is in the substance of Christian faith. Let us regard this time of trial also as a time of grace and hope. Let us remain united and make our closeness felt to all, especially those in need. | https://mtceurope.org/en/our-actions/news-from-europe/342-ceccomececoviden.html |
It is known that, in a case in which an imaging object having a size of several millimeters is observed by using a microscope having a focal depth of several tens of micrometers, or in a case in which an imaging object having a depth in the optical-axis direction is observed, an omnifocal image, which is focused in substantially the entire region thereof, is generated by using a plurality of images (hereinafter referred to as the Z-stack image) captured while shifting the focal position in the optical-axis direction, and by combining pixels of images having maximum focusing degrees at individual pixel positions (for example, see Patent Literature 1).
In Patent Literature 1, in a case in which the focusing degree at each position has two or more peak values, these peak values are detected as a plurality of candidate values, and a plurality of omnifocal images are generated by utilizing a specific number of Z-stack images for each of the detected candidate values.
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How Long Can Tires Typically Last?
Car owners often ask how long their vehicle’s tires will last before they would have to replace them. A general consensus is tires should be inspected regularly. They for sure need to be replaced after 10 years even if they are not worn.
You can determine the age of your tires by looking at the code on the sidewall. The 4-digit number is the date code that identifies the age of the tire. The first two numbers indicate the week of the year it was made (out of 52 weeks per year), and the second two numbers represent the year.
To find out how long a set of tires would last on your car, you will have to calculate the number of miles you drive each year. According to the Federal Highway Administration, the average American drives between 14,000 and 15,000 miles a year. For a used car, start with the mileage reading when you first got the vehicle. Then compare that with the advertised warranty of the make and model of the tires.
Obviously, rough driving on a car will shorten the life of your tires. In the winter, you can switch to snow tires for better traction, stability and stopping power.
Here are basic maintenance tips to ensure the condition of their tires are always excellent on the road.
Tread
According to the National Highway Traffic Safety Administration (NHTSA), a tire should be changed once its tread is worn down to 2/32 of an inch. The easiest way to check the tread of a tire is to place a penny upside down in the tread groove of the tire. If you can see the top of Lincoln’s head, it means the treads are shallow and worn and the tire needs to be replaced.
Pressure
Check your vehicle manufacturer’s recommended level for tire pressure. This can be found in the vehicle’s doorjamb or owner’s manual. Keep a handheld tire pressure gauge in your car so you can regularly check your tire pressure. It is best to check tire pressure when the tires are cold.
Rotation
Rotating your tires can help prolong tire life. The U.S. Tire Manufacturers Association (USTMA) recommends tires be rotated every 5,000 to 8,000 miles.
Balancing
Vibration in your steering wheel, floorboard, or seat means your tires need balancing. Balancing all four wheels reduces wear on your tires and creates a smoother ride. Auto experts say a good rule of thumb is to have your tires rebalanced after every 12,000 miles or every other time your tires are rotated.
Wheel Alignment
A wheel alignment helps ensure all four tires are correctly angled with each other and the road. If the wheels are misaligned, the driver will experience a drop in handling capability. Follow the recommended interval in the car owner's manual or have wheel alignment checked once or twice a year.
When in the market for a used car, stop by Carousel Preowned. Carousel Preowned serving Iowa City, Cedar Rapids, Davenport, North Liberty, Marion, and Coralville, IA, is proud to be an automotive leader in our community. We do our best to ensure your complete satisfaction every time you step into our car dealership. This is why we offer the widest selection of used and pre-owned cars, trucks, and SUVs and provide true ease of purchase in Iowa. | https://www.carouseliowacity.com/how-long-can-tires-typically-last |
Prairie Sunset False Sunflower has masses of beautiful gold daisy flowers with brick red centers at the ends of the stems from early summer to late fall, which are most effective when planted in groupings. The flowers are excellent for cutting. Its attractive serrated oval leaves remain forest green in color with distinctive deep purple veins throughout the season. The fruit is not ornamentally significant. The deep purple stems can be quite attractive.
Prairie Sunset False Sunflower is an herbaceous perennial with an upright spreading habit of growth. Its medium texture blends into the garden, but can always be balanced by a couple of finer or coarser plants for an effective composition.
Prairie Sunset False Sunflower will grow to be about 4 feet tall at maturity, with a spread of 3 feet. When grown in masses or used as a bedding plant, individual plants should be spaced approximately 30 inches apart. It tends to be leggy, with a typical clearance of 2 feet from the ground, and should be underplanted with lower-growing perennials. The flower stalks can be weak and so it may require staking in exposed sites or excessively rich soils. It grows at a fast rate, and under ideal conditions can be expected to live for approximately 10 years. | http://plants.pecksgreenthumb.com/12100002/Plant/2392/Prairie_Sunset_False_Sunflower |
The remote video installation involves, in addition to the projector and the projection surface, a cache that allows filtering a vertical strip only. On the projection surface only this vertical band appears, the rest of the video image being projected on the cache. The projector and the cover are visible from the spectators, who can see the device, and watch on the cache the rest of the video image, blurred since the focus is made on the projection suface a few meters further. The video is a 2 minute loop. The last picture is on the first one in an exact continuity. We see a moving vertical line, which is the line bounded by two skins rubbing from top to bottom. The movement is repetitive and infinite.
Distance: It is the distance that separates the projected video from its origin, the effective distance between the lens of the projector and the location where one has chosen to see it and where it will be sharp. This is the distance canceled between two skins glued. Distance: di-stance, it is also two stanzas, two strophes of the same song. The hole of the lock by which we observe works here upside down. Abstract and tactile, the image is a line between two skins, two lights, two colors, two movements. | http://www.florencegirardeau.org/new/en/a-di-stance-2/ |
Australia rounds off 118-run win over SAfrica in 1st test
DURBAN, South Africa — Australia needed less than 20 minutes of the final day to take the one more wicket it needed to finish off a 118-run win in the series-opening test against South Africa.
Josh Hazlewood trapped Quinton de Kock lbw for 83 and Australia bowled South Africa out for 298 in its second innings for a commanding win Monday and the early advantage in the four-test series.
Australia had set South Africa a formidable target of 417 to win the test at Kingsmead, or to bat for two days for a draw.
The subdued finish to the test in a stadium that was nearly empty of fans was in contrast to a fiery fourth day's play on Sunday. The aggression between the teams appeared to have spilled over then with leaked security camera footage showing a confrontation between Australia's David Warner and South Africa's Quinton de Kock in the players' tunnel.
The International Cricket Council is investigating the incident.
By Gerald Imray, The Associated Press
Australia rounds off 118-run win over SAfrica in 1st test
Sports03:43 AM
DURBAN, South Africa — Australia needed less than 20 minutes of the final day to take the one more wicket it needed to finish off a 118-run win in the series-opening test against South Africa.
Josh Hazlewood trapped Quinton de Kock lbw for 83 and Australia bowled South Africa out for 298 in its second innings for a commanding win Monday and the early advantage in the four-test series.
Australia had set South Africa a formidable target of 417 to win the test at Kingsmead, or to bat for two days for a draw.
The subdued finish to the test in a stadium that was nearly empty of fans was in contrast to a fiery fourth day's play on Sunday. The aggression between the teams appeared to have spilled over then with leaked security camera footage showing a confrontation between Australia's David Warner and South Africa's Quinton de Kock in the players' tunnel.
The International Cricket Council is investigating the incident.
By Gerald Imray, The Associated Press
Top Stories
Australia rounds off 118-run win over SAfrica in 1st test
Sports03:43 AM
DURBAN, South Africa — Australia needed less than 20 minutes of the final day to take the one more wicket it needed to finish off a 118-run win in the series-opening test against South Africa.
Josh Hazlewood trapped Quinton de Kock lbw for 83 and Australia bowled South Africa out for 298 in its second innings for a commanding win Monday and the early advantage in the four-test series.
Australia had set South Africa a formidable target of 417 to win the test at Kingsmead, or to bat for two days for a draw.
The subdued finish to the test in a stadium that was nearly empty of fans was in contrast to a fiery fourth day's play on Sunday. The aggression between the teams appeared to have spilled over then with leaked security camera footage showing a confrontation between Australia's David Warner and South Africa's Quinton de Kock in the players' tunnel. | |
Can using a pendulum too much damage your energy field?
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Hello,
I have recently bought my first pendulum. It's moonstone, with a slightly pointed end and a small ball on the end of the chain, if that makes any difference to my question.
The second day I aquired it, I left it out in the sun for the whole day, in order to cleanze it.
After that, I used it quite a bit, and worked very well for me, and seemed very responsive to my questions. It would move really fast, and in great big swings.
After that, I put it away but I probably spent a good three hours just checking it out on the very first day. I noticed the next day that I felt tired, and when I tried my pendulum again, it barely swung at all. It's been about five days now, and I've been very cautious on how much I use it. But it still continues to barely swing at all, and I still feel slightly tired.
In the book I had on pendulums, I was told that it didn't matter how much you use it. Now I'm not so sure. And I've been reading on the internet about them potentially damaging energy fields. I really hope this is not the case. My questions are: Do you think that this is possible? And if so, how could a damaged energy field effect me, and how can I fix it?
I've heard that it is best not to use tarot cards a lot at first, because it might damage your energy.
I was told by a psychic, that I shouldn't use tarot cards more than once a week, and only on the same day. I've only been focusing on pendulums at this point, but I'm worried that this might be the case with them too.
Thanks in advance.
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Ginny --- I cannot believe no one has given you their opinion yet. Then let me be the first:
In all my years with the Pendulum and Tarot...but especially the Pendulum....I have never read or heard anything about it damaging someone's "energy field"! The pendulum is taping in to your own subconscious which is connected like a web to the universe and certainly beyond the veil that divides the bodily people from Spirit...and, yes...a universal energy field! Which is how psychics tap into the future. But using it will not damage your own. You can use it all you want but.....know that earth bound spirits of a lower vibration can enter into play and make a temporary negative impact. ....ASK your pendulum personally, before each reading session, if it needs cleansing, protection (prayer or blessings of protection) which could be what is happening or happened to you. I am hopeful that by now you and your 'Pendy' are better. Next, I will post is something I just wrote earlier today for some people using the Pendulum. But for now I want to tell you that the best cleansing is a combination of three things.....
1. A soaking and swishing around in salt water. Once you swish it a bit..let it soak for 10- 30 mins or more. Rinse with clear water if any metal is involved, ie. the chain or brass pendulum.
2. Let it sun bathe a while if its a good day.
3. Syy a protecting prayaer asking for only total protection from any negative influences by surrounding you and your pendulum in a white light of love and peace and truth.
It's also a good idea, as you probably know, not to let anyone else handle it. Store it away in a small pouch or box of some type. I prefer a black pouch as I store my cards in a black cloth.
Before each reading session..again.....ASK it if it needs cleansing...if it feels strong enough to answer your questions today and if that response is still a weak little yes then tell it to show you....prove it is strong enough by strong responses and literally you will watch the force grow immediately if its fine.
From what you said about bold responses from you early on....then you are six sensory connected already and have saved yourself the long periods of time it takes some people to achieve success with the pendulum. It worked that way for me, too...but with a background of years of study with and of Spirit. When I use "Spirit" I mean all forms collective consciousness....I had immediate strong responses as well.
One last comment.....the only thing that may waiver is when you ask questions about the future. Often times....those questions can be answered correctly but you can never take anything about the future to the bank. The reason is FREE WILL. If another person is involved....free will can change things...even things that were that person's intention can change. We have no control over someone elses' impact on a situation. It is good to check frequently on something that was foretold to be sure things are still going as you were told. Sometimes...even at the last minute...it can change.
I hope I have been helpful. We all go on what we have studied and what we have experienced so others may not agree with me. I will come back with what I wrote earlier as I said but by the time I finished with this it was almost all retold. I will do it just in case something was not hit upon.
Good luck to you, my friend....
Love, Light and Peace... | https://forums.tarot.com/topic/1658/can-using-a-pendulum-too-much-damage-your-energy-field |
March 4th is written 3/4. It was exactly 34 days before WrestleMania, which will be held at MetLife Stadium in New Jersey, the state Bundy was born in:
Converting to weeks, we find it’s 4 weeks, 6 days before WrestleMania:
Bundy’s death fell on a date with Full numerology of 46
The WrestleMania Riddle
Undoubtedly, Bundy’s biggest moment as a professional wrestler was at WrestleMania 2, where he battled Hulk Hogan in a Steel Cage match in the main event.
That event fell on April 7th, 1986 – the only WrestleMania to ever occur on a Monday. That means it was exactly 33 years to the day before this year’s WrestleMania event, also on April 7th:
33 is a number of Ritual human sacrifice
Curiously, Bundy’s real name Chris Pallies sums to 395, which looks a lot like 895…
The film King Kong came out in the year ’33
The film’s New York release date can be written 3/2/33
News of King Kong Bundy’s death did not break until March 5th, which falls a span of 31415 days after the film King Kong‘s initial release:
It was 120 days before the next total solar eclipse
Measuring instead to the most recent total solar eclipse, we find it’s been a span of 197 days since its anniversary:
Christopher Alan Pallies died a span of exactly 1 year, 28 weeks after that eclipse:
He died exactly 1717 weeks after he Main-Evented WrestleMania 2:
The date he died had 17 numerology:
Cathy Lee Crosby
The guest ring announcer for WrestleMania 2 was Cathy Lee Crosby
139 is the 34th Prime number
Her initials are C.C. = 3-3
Think about 34 and 33 in light of how he died on 3/4, 34 days before a WrestleMania that will be exactly 33 years after the one he headlined.
Crosby was born on 12/2 and as of the date of this news, is 27,122 days old:
She was the ring announcer for Bundy’s Steel Cage Match
Bundy was 221 weeks younger than his opponent, Hulk Hogan:
Consider how the number 353 is 35 both forwards and backwards (Super Bowl 53 was played just two months before WrestleMania 35)
The inverse of 35 is 53
Bundy’s death made news on 5/3
Hulk Hogan was born in the year ’53
The 53rd Prime number is 241
53 is the 16th Prime number
Crucifixion Code
It’s WrestleMania 35 coming up next month. Thirty-five has gematria of 1331, the ultimate solar eclipse number.
Thirty-five sums to 61 in Reduction
Bundy died at age 61
His real name was Christopher
On the date of WrestleMania 35, Bundy would be 61 years, 151 days old:
His last name, Pallies, matches Jesus in Ordinal:
The date he died is written internationally as 4/3
Sacrifice / Kill Code
His WrestleMania 2 opponent Hulk Hogan also has 43 gematria:
Check out how long it’s been since his birthday:
WrestleMania falls on the 97th day of the year, syncing up with his real full name:
He fought Hulk Hogan in the Main Event of WrestleMania 2
Bundy was billed at 6’4″
News of his death broke on the 64th day of the year:
The 64th Prime number is 311
Bundy was born on the 311th day of the year: | https://gematrinator.com/blog/2019/03/05/king-kong-bundy-dead-61 |
Retail premises for sale in 146 Commercial Road, Bournemouth, Dorset BH2
* Calls to this number will be recorded for quality, compliance and training purposes.
Property features
- Shop and maisonette
- All fully let
- Total income £46,800 p.a.
Property description
Situation and description
The property occupies a prominent position on Commercial Road in a busy parade of shops. It is within walking distance of Bournemouth Town Centre.
The premises consist of a commercial shop, currently being used as a takeaway plus a four bedroom maisonette on the upper floor which have been fitted out to a high standard.
Accommodation
Shop
Ground floor shop
Max Width: 9’11” (3.0 m)
Max Depth: 30’5” (12.3 m)
Sales area: 462.84 (43 sq.m.)
Living Accommodation
Stairs to landing with access to:
Bedroom 1: 14’5” x 13’3” (4.41m x 4.03m)
Bedroom 2: 11’9” x 6’6” (3.58m x 2.04m)
Both bedrooms have en-suite shower rooms
Kitchen: 8’10” x 7’4”
Further stairs to second floor landing with access to:
Bedroom 3: 15’0” x 13’6” (4.57m x 4.11m)
Bedroom 4: 10’5” x 13’6” (3.18m x 4.11m)
Both bedrooms have en-suite shower rooms
Tenure
Shop is currently let at a rental of £19,200 payable monthly in advance on a contracted out lease from 11th August 2017 and expiring on 31st July 2036, with rent reviews on 1st August 2019,2022,2025,2028,2031,2034. There is a rent deposit of £8,000.
Maisonette currently being let on a room by room basis earning £27,600 p.a. Including all bills.
Total income from letting: £46,800 p.a. Minus bills for residential.
The current owner would consider taking a 5 year lease from any prospective purchaser on the upper parts (residential) only at a rental of £19,200 p.a.
Planning
A3/A5 Café/Takeaway
C3 Dwelling House
Change of use of ground floor premises from cafe (Class A3) to cafe (Class A3) and takeaway (Class A5) use was granted 20th June 2014.Ref. No. F.
Trading hours are 8.00am to11:00pm.
Legal fees
The incoming tenant will be responsible for their own legal fees.
EPC Rating E
Viewing and further details
By arrangement with Ellis and Partners through whom all negotiations are to be conducted.
Tel: Website:
Property info
* Sizes listed are approximate. Please contact the agent to confirm actual size.
For more information about this property, please contact
Ellis and Partners, BH1 on +44 1202 060263 * (local rate)
Disclaimer
Property descriptions and related information displayed on this page, with the exclusion of Running Costs data, are marketing materials provided by Ellis and Partners, and do not constitute property particulars. Please contact Ellis and Partners for full details and further information. The Running Costs data displayed on this page are provided by PrimeLocation to give an indication of potential running costs based on various data sources. PrimeLocation does not warrant or accept any responsibility for the accuracy or completeness of the property descriptions, related information or Running Costs data provided here. | https://www.primelocation.com/for-sale/commercial/details/49728846/ |
We are going to change the day for the tailgate dive this week from Tuesday to Thursday the 25th. Jim "Hoover" Granger has a conflict on Tuesday night. We are planning on diving at Long Lake west of the airport in Detroit Lakes. We will leave the dive center at 6:00 and meet at the public access at 6:15. Bring a friend along and enjoy the dive and cookout. Some of you regulars from last year we haven't seen yet this year and hope you can also join us.
Friday the 26th will be the Rescue Diver Classroom so give us a call if you
need any information about the class.
Just a reminder that we are an authorized Sherwood Scuba dealer and can take
care of all your service needs. We still have some excellent closeouts and
consignments items left in stock. We still have one Al 80cf tank and a number of
72 steels left for sale.
We still have about 35+ medallions left in the area lakes that have not been found yet so start searching and win some nice prizes. The big medallion is still out there so stop in and get the latest clues.
Following is the next in a series of articles on free diving by Fred Johnson.
Breath Hold Training
Being able to hold your breath for long periods of time has more to do with carbon dioxide tolerance than lack of oxygen. You can build CO2 tolerance easily and quickly with training. Like climbing a tall ladder to paint for the first time... at first you're holding on for dear life, and soon you're at the top bouncing the ladder over to the side to reach a little farther. A quick way to experience a build up of C02 is to
exhale all of your breath and then hold it. The urge to breath will come quickly. Doing this often is a quick way to build up CO2 tolerance and will also give you an idea of just what your body can take while you are under the water. Being as comfortable as possible with that urge to breath will have great impacts on the lengths of time you can spend at depth. I'll have more technical information on breath holding techniques in my next article.
Last of Equipment Updates:
1st Piece of Updated Equipment: Mask
2nd Piece of Updated Equipment: Fins
3rd Piece of Updated Equipment: Wetsuit
Following: One of the secrets to a deep dive without expelling a lot of energy.
For a long time I used 2 styles of wetsuits for freediving. One was your basic water-ski, jet ski wetsuit and the other an older 7mm scuba diving suit for when the water was very cold. Both, I thought, were very good. They kept me warm enough, allowed me to spend a lot of time in the water each diving session and didn't seem too restrictive in filling my lungs and movement. I was very wrong on the amount of restriction and I also did not realize how much resistance an exterior lined wetsuit created while trying to dive.
I first used a freediving specific wetsuit while in Vancouver with world record freediving trainer Kirk Krack. I found out on my first dive what a difference a wetsuit makes. This style of suit is the ultimate in stretchabilty and low resistance. I was like a greased pig out of water with a professional apnea wetsuit on and knew then that one of these was going to be needed in the near future.
Apnea suits come in many different styles and thicknesses, just as scuba suits do and are made up of varying qualities of neoprene, just as scuba suits are. It's the quality of the neoprene and the lack of a lining whether outside or inside that makes for a good apnea suit. You also hear some of the same terms: open cell, closed cell, gold lining, lycra, jersey, nylon, farmer john, jacket and pants, etc. Most of the serious
freediving suits are a closed cell neoprene outside and open cell neoprene inside. This makes for a difficult suit to get into as you have to lube it with a mix of cream rinse and water in order for it to slide on, and you may well have to lube it with at least water to remove it.
The wetsuit without a lining of any kind has to be treated with care as fingernails, toenails, and other snags can tear a suit easily. Fortunately, I have found my 5mm apnea suit to be fairly tough compared to the 3mm suit I have and the 3mm I wore in Vancouver. I have worn the 5mm suit scuba diving with no problem. In fact, my 5mm apnea suit is now my choice of suit for scuba diving, it has no zippers, an integrated
hood, and would be what you might call a semi-dry suit as it lets in almost no water. I would be very hesitant to wear my 3mm suit scuba diving as you can just sense the fragility of it.
I've been discussing unlined suits as the ultimate in freediving suits but I cannot discount the newly developed linings that are on the market. Lined suits are nice in that they do not require lubrication to put on. Superstretch interior linings such as I have in my 3mm suit are, except possibly by the professionals, indistinguishable as far as stretchability from the unlined suit and with a closed cell exterior just as "greased". The lined suit does let in a little more water then the unlined suit but is still extremely warm with the lack of zipper and integrated hood. Apnea suit manufacturers have different make up of suits to choose from, some even have a lining sandwiched between the neoprene to add to the strength. You would have to weigh your style of diving and the abuse you think you might be giving the suit as you consider a suit.
One thing is for sure, combined with streamlined weights and a rubber weight
belt, it's amazing how you glide through the water with a completely closed cell
exterior wetsuit and how much less restrictive good quality neoprene is when
taking a deep breath or finning down to the depths. | http://tri-statediving.com/email/Edition12.htm |
The utility model discloses a can let revolving stage exceed 360 degrees safe limiting device, including spacing outer bucket, stopper, the solid fixed ring of stopper, spacing interior bucket, deep groove ball bearing, bearing clamp plate, locking screw, set screw. The utility model discloses an advantage that can let the revolving stage exceed 360 degrees safe limiting device can satisfy the motion key element and gyration limit function of power, the utility model discloses an adopt rigid connection between each part of can let revolving stage exceed 360 degrees safe limiting device, simple structure and effective has guaranteed safe limiting device limit function's reliability, greatly reduced the environment age to safe limiting device limit function's influence with the part. And the utility model discloses a this kind of the structure that can let the revolving stage exceed 360 degrees safe limiting device has adopted technique design all linked with one another, can carry out repeated upgrading to realize revolving stage 360x n (n=2.3.4.... ) the degree angle spacing. | |
A report in the Daily Mail has suggested that should Leicester dig their heels in regarding Demarai Gray’s future, the likes of Tottenham, Everton and Liverpool may be forced to pay £22 million if they hope to sign the winger.
The Mail reports that Gray is somewhat frustrated with life at the King Power Stadium, having struggled for game-time for the Foxes. And while he is yet to make a decision on his future, the Mail suggests that the Premier League champions are determined to keep hold of the winger who they signed from Birmingham City in 2016.
However, it appears that the decision on Gray’s future may be taken out of Leicester’s hands as, according to the Mail, the 20-year-old’s contract includes the eight-figure release clause – though the Foxes have denied the presence of such a clause.
Although Gray has clubs such as Tottenham and Everton chasing his signature, it would perhaps be a surprise if any of the trio were willing to pay such a figure during the upcoming summer transfer window.
The wide-man is undoubtedly an extremely talented player with a tremendous amount of potential, but he is yet to establish himself in a Leicester side that has struggled for much of the campaign, and thus, spending £22 million on him would perhaps be something of a gamble at this stage.
And with Gray seemingly unlikely to sign a contract extension currently – given his frustration and the fact that his current deal has three years left to run (via Mail) – it will surely do little harm to wait and see if Gray shows more signs of being a player worth such a substantial outlay next season before pursuing a deal.
In any event, if the Reds’ interest is real, then it just shows yet again that the club is planning to persist with the failed ‘buy ’em young’ policy, which continues to yield zero results for the club. Liverpool need the finished article in several positions, not more players with potential to come good in 4-5 years. | https://liverpooltransfernews.net/jurgen-klopp-pushing-to-sign-explosive-22m-title-winner/ |
The highly adaptable euryhaline species are able to endure a wide range of salt levels , according to The National Biological Information Infrastructure (NBII). Even these fish cannot transfer from freshwater to salt without a period of adjustment. As the two types of water mix, a salinity gradient is created, starting low at the river mouth and increasing towards the sea. Sometimes to treat an infection, or parasite, saltwater fish can be dipped in freshwater for a few hours. Post navigation Mystery snail: care, lifespan, breeding and tank mates- … Salinity (Amount Of Salt) One major factor that separates fish is salt. Any old how. Two that I immediately thought of were the saltwater crocodile and the bull shark. In addition to fish, there are several invertebrates that live in brackish water. Your email address will not be published. "Fishes" is used when one is discussing multiple types (species, genera, whatever), as in, "a red fish and a blue fish … Osmosis is the movement of liquid molecules through a semipermeable membrane (like the thin film inside of an egg ) from a low concentrated solute to a high concentrated solute. A saltwater tank is usually 1.024. Saltwater fish can live in salt water since they have always lived in it and they are able to get it out of their selves. Fish are categorized according Vertebrate & Fish Evolution Why can some fish live in freshwater, some in salt water, and some in both? If you fish brackish water, remember to wash your tackle carefully afterward. Now … it depends how salty your water is. Vertebrate & Fish Evolution Why can some fish live in freshwater, some in salt water, and some in both? Like a fish out of salt water. Brackish water is a mix of salty and fresh water found at estuaries where rivers join the ocean. Saltwater fish can live comfortably in saltwater that is 1.018 indefinately. Brackish Fish They generally inhabit shallower waters near shore and feed most actively at night. The advantage of fishing in brackish water is that you have an opportunity to catch a variety of saltwater and freshwater game fish species in the same area. It is similar to how you would make saltwater, it’s all about controlling the salinity. Most of these fish can only be kept in brackish habitats. This would slow the metabolic rate of the fish, thus slowing the rate at which its cells lose water to the surrounding high salinity seawater. "Fishes" has a particular use among ichthyologists and fisheries scientists. These are created when salt-tolerant trees (halophytes) grow in tropical intertidal zones. They are well known for their ability to shoot jets of water from their mouths. Since these setups are rare, your tank would be unique. Some fish, like bluegills, catfish, and largemouth bass live in freshwater. Brackish fish are very hardy fish for this very reason. Below we describe just a few of the many species available for you to choose from. This species is an Asian cichlid, of which there aren’t many. A large tank is needed for them to display their natural behaviors. That is why they dehydrate so fast and die. The aquarium should be as large as you can afford, as brackish water fish require more room than freshwater ones. Most eels live in freshwater, but American eels are different. Many freshwater species are very well adapted to fresh water, so much so that their bodies cannot withstand a transfer into salt water. For the bluegill to survive even for several minutes in high salinity conditions, the water would have to be very cool. Only saltwater fish drink water. The invention also provides a salt-water fish that can live in water that contains substantially less salt than its natural habitat. Bridges or weirs can be great places to try catching brackish water fish species such as snook or largemouth bass. All information, content, materials on this site, or obtained from a website to which the site is linked are provided to you “as is” without warranty of any kind either express or implied. The River Thames flowing through London is a classic river estuary. Preferably, the salt-water fish can live in fresh water or water having a salt content of about 1.001 to about 1.003. "Fish" can be singular or plural. If their preference is for saltwater, they can’t be kept in brackish water. Must be kept in aquarium with very low Nitrite and Nitrate levels to be kept successfully. These fish are called bluegill because of the bluish tinge around their gills area. We got a koi pond in our backyard and look at redoing it, the pond currently has 2 small koi fish. For a freshwater fish, the opposite happens if it is tossed in saltwater. These are only a few examples of brackish fish. Now, to answer whether milk fish freshwater or saltwater marine species then young milkfish is like to live in tropical water and they usually found in freshwaters rivers or brackish, wetlands, and swamps while mature or adults like to swim in ocean water or saltwater. This is usually around 10 grams of marine salt per liter of water, but you should work out the specific amount for your fish’s needs. They can even travel on land in short distances to get back to the sea as well. Saltwater fish can live in salt water since they have always lived in it and they are able to get it out of their selves. Freshwater denizens, such as channel catfish, largemouth bass and bullheads often live alongside saltwater species, such as sheepshead, snook and tarpon during portions of the year. We help you get all the tips you need to enjoy the water. The problem with this long term is that fish drink by osmosis. These eating habits make them best suited for more experienced aquarists. Largemouth, Striper & Channel Catfish like other Brackish fish species have a higher tolerance level of salinity & adapt to it where others can’t. We called them brackish water fish in the hobby. Anyone can look after brackish water fish. Your adorable goldfish is a stenohaline fish, preferring its freshwater habitat with very little salt. Betta are strictly freshwater fish that live in the Thailand rice paddies. Some species of fish mate in brackish waters, leaving the young to grow up there. Freshwater: Low salt concentration (less than 0.05%) Made up to 1% of all planet’s water bodies Found at: Streams, rivers, ponds and lakes. Brackish water fish live in salt water but are not considered salt water fish. While it is hard to say specifically how long freshwater fish such as bluegill would live in salty water, certainly, it is not long. Once the saltwater fish drink the salt for hydration, their kidneys pump the excess salt into their urine so they can get rid of it. Anglers can either choose to keep or release their catch. Most sucker fish though will tend to prefer fresh water conditions. The brackish world is one that should be explored more in captivity. They are largely unsuitable as pet or ornamental species, although historically lampreys were kept in saltwater pools as food fish. However, the high degree of salt tolerance exhibited by certain species, ongoing research to improve salinity tolerance, and various production studies conducted in many areas indicate that these fishes can be commercially produced in seawater. Much brackish water fish are quite distinctive which makes them very desirable. You must buy some marine salt. The green crab (Carcinus maenas) is an example of a euryhaline invertebrate that can live in salt and brackish water. They might survive for a couple of hours in very cool salty water because it would slow down their metabolic rate. This makeup most of their diet. Once the tank has been set up, caring for them is similar to caring for any other fish. Turtle Grass Shoots. Complex root systems help them to deal with high salt levels and wave action. Bluegill is a freshwater fish, so it can survive for a short time in very cool saltwater, longer in cool brackish water but it cannot live beyond a couple of hours. Can koi fish live in a tank with salt water? Specifically, brackish water has a salinity level of 0.05-3%, anything lower than 0.05% is freshwater, and anything higher than 3% is saltwater. Brackish water fish are typically found in estuaries where rivers meet the sea, creating habitats that are a mix of fresh and saltwater. Guppies can tolerate up to 150% of seawater salinity. We’ll explain to you what brackish water actually is before discussing some of the fish you could keep if you were to start a new tank. Since its cells are less concentrated with solute (salt), the salty seawater is the region of high concentration. Unfortunately, these fish are seen less often in aquariums because brackish water setups are rare. Aquarium Size Almost any size aquarium is suitable for brackish fish. Of course there are many others (salmon that return to fresh water streams to propagate). By definition, brackish water is simply water with a higher salt content than fresh and less than salt water. A brackish biotope tank is an aquarium that mimics a natural brackish environment not just in terms of water chemistry but in the tank inhabitants (plants and fish) as well. A reel that lasts for years without cleaning in freshwater will rust up beyond repair after one trip to saltwater if it gets inside. Brackish water habitats boast a vast array of fish that cannot be found in freshwater or saltwater. wrong water. You can then add it to your aquarium. Have you kept brackish water fish before? It is a good thing because the trout population will be high and the fact that you are advised... Anglers Crate was created to give you the ultimate fishing resource right at your fingertips! So you can keep your guppies in saltwater, provided you take sufficient care to acclimate them to the saltwater. different chemical compositions. Yes, guppies can live in saltwater. As these fish grow, you will need to gradually increase the salinity of their environment. This type of fish is special in so many ways. It is important to feed them well, Green Spot Puffers need high quality frozen crustaceans, like mollusks. People actually put about 1 tablespoon per 5-7 gallons of aquarium water just for disinfection. Brackish water condition commonly occurs when fresh water meets seawater. Largemouth Bass is found in all waters from fresh to brackish (a mix of fresh & saltwater) water that is tidal slow-moving rivers creeks, or streams. Brackish waterfalls in between. They are some of the brightest brackish water fish available. Due to the adult size of many brackish fish, bigger is better. Whether you’re a beginner or a seasoned hobbyist, find quality aquatic life when you shop LiveAquaria®. They can look quite scary, but they’re mostly peaceful towards other fish. This is a shame because many brackish water species are beautiful and fascinating, but don’t get the attention they deserve. We will discuss this in detail later. It is not known whether they would adapt, so chances are that they would eventually die. How Can I Make my Aquarium Water Brackish? There are many species of pufferfish available in captivity, Green Spot Puffers display some of the most personality. Mudskippers are very entertaining to watch. Thus, as they lose water from the low salinity conditions of the body to the high salinity of the ocean, they won’t drink to replace what they lose. Osmosis is the movement of liquid molecules through a semipermeable membrane (like the thin film inside of an egg ) from a low concentrated solute to a high concentrated solute. But it was not surprising to find catfish among the catch made by fishers who go out into the seas in search of bigger fish. They spend time in brackish estuaries between moving from river to sea. Most are restricted to one environment because they cannot change the way they regulate this salinity, but some species spend periods of their life in both environments. Saltwater catfish are not as large as their freshwater cousins, and rarely weigh much more than 3 pounds. Brackish water paludarium could be a good idea. The only exception is brine, which is extremely salty water with salinity over 5%. This is different to table salt, which is not appropriate. Peeing is their survival instinct in such a salty environment. Freshwater fish such as bluegill do not drink water, but they urinate for purposes of osmoregulation. It is also used as bait for larger game fish such as bass. These bottom-dwelling fish stay around the mouths of rivers where they can feed on algae and small invertebrates, which they catch by sifting through the substrate. Mudskippers can breathe air from their gill chambers and through their skin, which helps them live as amphibians. No, definitely not. The reason for being confused with perch is that they both belong to the same Sunfish family, which collectively has 27 species. A brackish biotope tank is an aquarium that mimics a natural brackish environment not just in terms of water chemistry but in the tank inhabitants (plants and fish) as well. Natural Habitat for Brackish Fish. In fact, some fishermen have caught bluegill in brackish water using shrimp as bait. Can Catfish Live in Saltwater and Sea? Starting a brackish aquarium could be a great way to switch from more common setups, but they can be suitable for beginners too. Leave the bumblebee gobies in brackish water, and get a similar fish, but a saltwater one, for this tank. Guppies are very colorful, lively, and extremely fun to watch fish, and they adapt to a variety of conditions. Mollies are livebearing fish that are easy to find, keep, inexpensive and small. Sooner than later, they would be dehydrated and die. Fish that can tolerate and adapt to fluctuation in salt levels are called Euryhaline fishes. They can work a tolerance to the salt however, so yes it is possible. BRACKISH WATER BAITS AND LURES. They need their own specific environment, so can usually only be kept with each other. The appearance of this plant is quite simple, but it is really fun to watch it. A Brackish water fish is a fish that survives in water that is part saltwater and part freshwater. Brackish water is mostly found where freshwater meets seawater. It is best to keep them in the same conditions that they were born in, a change in salinity could lower their chances of survival. They tend to beg for food so overfeeding can be a problem. Most of the fish don’t feed this plant, so there is no problem in keeping this in a fish tank. Yes, tilapia can live in saltwater; however, growers will primarily need to select the proper tilapia variety to raise in this type of surrounding. It is a hardy plant but it needs calcium to grow. And to have a clear insight into this topic, we will explain water salinity and evaluate the tolerance capacity of the different tilapia species against brackish waters. We’re thrilled to have you as part of our community. Molly Fish are very versatile, they are commonly kept in freshwater and brackish setups, some people even keep them in marine tanks. If you want a pufferfish that isn’t as aggressive, you could try Figure 8 Puffers. Feeding them can be difficult because they look for food out of the water. They would then move to saltwater as they mature into adults. LOL yes they can live in saltwater just not very long that is, I'd guess about 30 seconds . In the wild, this would be used to hit insects, to make them fall into the water for eating. Brackish fish species have a higher tolerance for varying levels of water salinity. William A. Wurts he various species of fish found in oceans, lakes, rivers and streams have evolved over millions of years and have adapted to their preferred environments over long periods of time. However, the smaller the aquarium, the smaller and fewer the fish you can put in it. However, most fish species can only survive in one or the other based on their salinity tolerance, or how much salt their bodies can handle. Long term exposure to the wrong salinity can drastically shorten a fish’s lifespan if it doesn’t kill them within a few days. Thus, water would be drained out of the cells of the freshwater fish to flow to the surrounding seawater. If you already have fish in the tank, slowly add the water while continuously using the hydrometer. They are most happy in shoals of six or more, so this drives the size of their tank even larger considering they can reach 12 inches. The Ultimate Hermit Crab Care Guide: Habitat, Food And Much More…, How To Take Care Of A Box Turtle – Ultimate Breed Guide List, 15+ Best Freshwater Shrimps For Aquariums, Eastern Box Turtle Complete Care Guide: Diet, Habitat And More….
can brackish fish live in saltwater
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This study examined the influence of aerobic exercise on the cardiorespiratory endurance and skeletal muscle of metabolic syndrome. An experimental research method was adopted, sampling community health service center people participating in health examination, 15 middle-aged men with waist circumference greater than 90cm (age 49.11±3.32) as subjects. Participants received 60-min aerobic exercise sessions two times a week for 10 weeks (20 sessions in total). The research tool uses a body composition analyzer (In Body) to detect skeletal muscle and basal metabolic rate; a three-minute cardiorespiratory endurance test (Harvard Step Test) is used to understand the subject’s cardiorespiratory endurance index after aerobic training. The results of the study found that weight loss, skeletal muscle rate t = -6.58*, and basal metabolic rate t = -5.77* all improved, and the cardiopulmonary endurance index increased from 52.64 to 58.31, from “poor” to “average” “within. The study concluded that aerobic exercise can consume more deep fat, reduce the risk of suffering from metabolic syndrome, help improve cardiorespiratory endurance and skeletal muscles, and achieve the results of rehabilitation and health improvement.
Keywords: Metabolic Syndrome; Cardiorespiratory Endurance; Aerobic Exercise; Skeletal Muscle Rate; Basal Metabolic Rate
Metabolic syndrome (MS) is a group of metabolic diseases
that appear in the same person. Its main metabolic abnormalities
include obesity, dyslipidemia, hyperglycemia, hypertension, insulin
resistance or glucose intolerance and other risk factors [1-4], one
person At the same time, as long as there are more than three risk
factors, it can be presumed to be metabolic syndrome , that is, MS
is not a disease but a warning sign of the body [6,7], it is also a “predisease
state” in which the body begins to experience metabolic
abnormalities. MS is an aggregation of risk factors that increase
the incidence of cardiovascular events and diabetes mellitus (DM).
Population aging is accompanied by higher prevalence of MS
[8,9]. The prevalence of MS increases with age, with about 40% of
people older than 60 years meeting the criteria . Now days MS
can no longer be considered a disease of only adult populations.
Alarmingly, MS and DM are increasingly prevalent in the pediatric
population, again in parallel with a rise in obesity . Most middleaged
people have abdominal obesity, and thus constitute a high-risk
group for MS. Statistics have indicated that people with visceral
obesity have a 50% chance of developing MS . Middle-aged
people who belong to the high-risk group of MS can easily lead to
chronic diseases such as diabetes, heart disease, and hypertension
if they do not control their diet and exercise [13-15].
General body composition is composed of body fat mass, body fat percentage, skeletal muscle rate (SMR) and basal metabolic rate (BMR) . Among them, the MS is closely related to its own BMR , and because the BMR is positively correlated with SMR growth and exercise . One of the most critical elements of physical fitness is cardiorespiratory endurance. Relying on the kinetic functions of the heart and lungs, cardiorespiratory endurance refers to the body’s ability to continue supplying energy to the human circulatory system and muscles over extended periods . There had been positive reports [19,21] about how advanced cardiorespiratory endurance not only enables one to engage in aerobic exercises, such as walking and jogging for a longer time. From the above literature, we could see the feasibility of this study. After a long period of aerobic exercise, people with metabolic syndrome and generally healthy people should be able to improve the growth rate of skeletal muscle and improve cardiorespiratory endurance [22,25]. This was also the focus of this research. In addition to dietary control, frequent exercise is the best method for staving off MS, where daily exercise invigorates the body . Evidence has indicated that aerobic exercise is an effective method of improving cardiorespiratory endurance, aerobic exercise can significantly improve health [27,30]. Relevant studies have indicated that starting from 30 years old, lack of exercise is the primary driver of aging-related loss of muscle mass [31,32]. Some other studies have indicated that exercise can enhance bone density, reduce body fat, enhance metabolism, and prevent chronic diseases [33,34]. Aerobic exercise can strengthen the muscles surrounding and supporting the joints can help maintain a good body shape and enhance the integrity of the joints, thereby helping to prevent injuries [35,36]. Aerobic exercise will strengthen the skeletal muscles and help the bones to stay strong. Just like your brain, skeletal muscles need to be exercised to maintain muscle strength . Some MS become obese, muscle strength deteriorates, physical vitality decreases, and even chronic diseases are caused by lack of exercise . MS is a symptom produced by modern society and civilization, and because aerobic exercise has a positive effect on the physical composition of individuals, this study uses middle-aged people with MS as the research object, and uses aerobic exercise to understand the cardiorespiratory endurance and skeletal muscles of MS.
An experimental research method was adopted, in cooperation with the community health service center, and implemented in the school gym. The subjects were people who participated in the health checkup at the community health service center. Middleaged men with a waist circumference greater than 90cm were the sample objects, a total of 15 people were sampled (age = 47.89 ± 6.24 years). Participants received 60-min aerobic exercise training sessions two times a week for 10 weeks (20 sessions in total). All training and testing are performed by the research team. This study did not involve personal privacy and strictly adhered to research ethics. As for the subjects’ psychological symptoms, disease history, family factors and other potential variables, they were listed as control variables. Aerobic training courses are shown in Table 1.
A body composition analyzer (InBody 230) was used to measure SMR, and BMR. The mechanism underlying the InBody analyzer is the method of bioelectrical impedance analysis, which utilizes the impedance of current flow; specifically, the lower the conductivity of the muscles, blood, body fat, and skin, the higher the impedance is . Take a 3-minute test of cardiorespiratory endurance (Harvard step test) to learn about the subjects’ cardiorespiratory endurance index after aerobic training. Use a 35cm high step, 96 beats per minute metronome, a total of three minutes of operation, after completing the test, measure 1 minute to 1.5 minutes, 2 minutes to 2.5 minutes, 3 minutes to 3.5 minutes, three 30-second wrist pulse rates. The cardiorespiratory endurance index score is then determined by the following equations. Cardiorespiratory Endurance Index = (100 x test duration in seconds) divided by (2 x sum of heart beats in the recovery periods). And consider the male cardiorespiratory endurance index norm , as shown in Table 2.
Pre-test and post-test data were obtained and analyzed in SPSS (version 23.0). Descriptive statistics (specifically, the mean and standard deviation) were used to summarize the participants’ characteristics, and t tests were used to analyze after aerobic training changes in cardiorespiratory endurance (CRE), SMR and BMR.
The results of this study are divided into two parts: First, Descriptive statistics of SMR, BMR, CRE before and after aerobic exercise; Second, Difference analysis of SMR, BMR, CRE before and after aerobic exercise for subjects.
In this study, 15 men (age 49.11±3.32) were the subjects. These subjects were middle-aged men with waist circumference greater than 90 cm. According to the data in Table 3, the weight of the subjects was overweight or obese, and the cardiorespiratory endurance did not reach the standard average value. The data in Table 4 shows that the average weight of the subjects decreased significantly after aerobic exercise training, while the skeletal muscle, basal metabolism, and cardiopulmonary endurance were significantly improved.
Pre-test results: The average weight of the subjects was 84.47 kg, the SMR was 29.30%, which was slightly too low, the BMR was 1585 kcal/day, and the cardiorespiratory endurance index was 52.64, which was a poor state.
Post-test results: The average weight of the subjects was 78.67 kg, the SMR was 34.98% above the normal range (32~34% of the normal range), and the BMR was 1623 kcal/day, which was in the normal range (the male is 1400~1700) Card) , cardiorespiratory endurance index (cardiorespiratory endurance index) of 58.31 belongs to average state. According to the above data, aerobic exercise can increase the body’s BMR [43,44], and improve the effect of cardiopulmonary endurance [45,46], and the BMR is positively correlated with skeletal muscle . Increasing skeletal muscle can increase the BMR, which not only helps burn calories and avoid weight gain, so the metabolic rate is low, and the risk of weight gain is low higher.
Table 5 shows the difference analysis of SMR, BMR, CRE before and after aerobic exercise, all of which have significant differences. In terms of weight: t = 7.89* reached a significant level of .05, which means that after the aerobic exercise course, the average weight dropped from 84.47 kg to 78.67 kg. In terms of skeletal muscle rate: t = -6.58* reached a significant level of .05, and the average skeletal muscle rate increased from 29.30% to 34.98%. In terms of basal metabolic rate: t = -5.77* reached a significant level of .05. The average basal metabolic rate increased from 1585 kcal/day to 1623 kcal/day, with an average daily increase of 38 calories. Cardiorespiratory endurance index (CRE index): t = -18.96* reached a significant level of .05. After the subjects undergo aerobic exercise, the cardiorespiratory endurance index increased from 52.64 to 58.31, that is, from “poor” to “average” range (see Table 2). Based on the above results, the subjects significantly increased their cardiorespiratory endurance index and skeletal muscle rate after ten weeks of aerobic exercise training.
This study focused on improvements to cardiorespiratory endurance, skeletal muscle and basal metabolism from middleaged people with metabolic syndrome of aerobic exercise.
In general, the public is often interested in studies reporting that certain types of physical activity can lead to weight loss . The present study’s findings revealed the following. In general, aerobic exercise does not immediately affect all middle-aged people with metabolic syndrome. In particular, aerobic exercise effect on skeletal muscle and basal metabolism are relatively weak, thus, a longer aerobic exercise period may be required for improvements in BMR to be notable. This is consistent with findings in the literature. For example, a study noted that although basal metabolic rate gradually decreases with age, an individual can enhance their metabolic function as long as they maintain favorable exercise habits . According to research reports, metabolic syndrome has become a precursor to chronic diseases in middle-aged people (at least 3 of the following: abdominal adiposity, low HDL cholesterol, high triglycerides, hypertension, and impaired fasting glucose) . With the successful conquest of communicable infectious diseases in most of the world, this new non-communicable disease (NCD) has become the major health hazard of modern world . Many studies have confirmed that exercise can prevent the occurrence of metabolic syndrome early, because exercise is a cost-effective intervention to both prevent and reduce the impact of the metabolic syndrome [52,55].
“Exercise is Medicine, Exercise is the best medicine”, aerobic exercise is an important item for regular exercise. Cardiopulmonary function is the most important fitness in physical fitness, and it is also closely related to cardiovascular disease risk factors. Therefore, in regular exercise, aerobic exercise can be listed as an important item, and it must also improve and maintain good cardiopulmonary function, which can not only promote health and prevent diseases disease and can improve the quality of life. Although aerobic exercise and CRE are often used interchangeably, it is important to recognize that they are different; aerobic exercise is a behavior and CRE is an attribute. CRE is improved by aerobic exercise, but it is influenced by other factors, including genetics. Some studies have reported that a 20‐week supervised aerobic exercise training reduced metabolic syndrome prevalence by 31% . Nevertheless, this experiment most middle-aged people with metabolic syndrome will improve CRE by following the aerobic exercise. A few prospective studies have revealed that aerobic exercise and cardiorespiratory endurance are predictors of metabolic syndrome incidence [57,58]. One clinical intervention studies have shown that regular aerobic exercise clearly improved risk factors for metabolic syndrome in obese people [59,60]. Studies have also shown that confirmed the improvement of metabolic syndrome with increased cardiorespiratory endurance [61,62]. Cardiopulmonary endurance is an objective indicator of aerobic exercise patterns, which is negatively correlated with the incidence of metabolic syndrome [63,64]. Laaksonen, et al . reported that half and two-thirds of adult men had a 47% and 75% reduction in the probability of developing metabolic syndrome in the results of the maximum oxygen uptake experiment.
The skeletal muscle mass comprises approximately 40%
of total body mass and is the primary source of insulin-mediated
glucose uptake and fatty acid oxidation. The aerobic exercise evokes
adaptation in skeletal muscle in a multitude of nerve stimulation,
the functional response to which is determined by training volume,
mode of training, intensity and frequency. With persistent aerobic
exercise exposure, there is mitochondrial biogenesis, fast-toslow
fiber-type transformation, changes in substrate metabolism,
and angiogenesis. Aerobic exercise brings additional benefits
to energy expenditure from its ability to develop and maintain
the functions of muscle mass and BMR [65,67]. Because their
physiological structures differ, men have greater muscle mass and
less body fat [Tomlinson, et al. 2016]. Studies have observed that
aerobic exercise training reduced body fat but slightly increased
body weight [47,69], because body fat is reduced faster than the
proliferation of skeletal muscle, only decreases body weight slightly.
Many studies have also confirmed that aerobic exercise training can
also increase the proportion of lean muscle mass, strengthen the
skeletal muscles, and enhance metabolic capacity [68,70]. Studies
have indicated that SMR is positively correlated with BMR and
lean muscle mass, which, in turn, are positively associated with
calories burnt in a day . In the absence of exercise in adults,
muscle mass begins to decline after the age of 30, and bone mass
is also lost . Studies have shown that low muscle mass is a risk
factor for low bone density, so even low-intensity aerobic exercise
will have a positive effect on preventing skeletal muscle loss .
On the other hand, although bone and muscle mass will change
with weight, some patients with abnormal metabolic function,
the increase in weight is not accompanied by the increase in bone
and muscle mass, which means that osteopenia type obesity will
occur osteosarcopenic obesity , and complicating metabolic
syndrome [75,76]. Additionally, some studies have confirmed that
aerobic training can increase the consumption of deep abdominal
fat, which reduces the risks of diabetes and metabolic syndrome
[54,68]. The total weight of men measured in this study belongs to
the obese group, and the abdomen is larger than 90cm. In order to
avoid excessive abdominal fat accumulation, exercise training can
be used to increase muscle mass and increase the basal metabolic
rate, this finding is consistent with others in the literature [29,79].
The damage caused by metabolic syndrome to the human body should not be underestimated, it worsens one’s health condition in the short term and can even produce chronic diseases. This study confirmed that aerobic exercise training is an effective way to improve the cardiorespiratory endurance and skeletal muscles of middle-aged people with metabolic syndrome. Many people incorrectly believe that they cannot control their own bodily functions, but many studies have confirmed that aerobic exercise can improve cardiorespiratory endurance and skeletal muscle, which means that bodily functions can be controlled. While it is clear that exercise is important, the mechanistic pathways behind exercise-induced benefits on metabolic syndrome are still being identified. Further, aerobic exercise will improve cardiorespiratory endurance and skeletal muscle function of metabolic syndrome which can act in conjunction with exercise programs, and for metabolic syndrome individuals whom are unable or unwilling to exercise to amplify the beneficial effects of exercise. These data together emphasize the importance of aerobic exercise to prevents the development of metabolic syndrome and promotes recovery and improved health in patients with cardiorespiratory endurance and skeletal muscle.
We thank to the participants and professionals involved in this study. The study was designed by H.W-Y., and H.H.; data were collected and analyzed by H.W-Y.; data interpretation and manuscript preparation were undertaken by H.W-Y., and H.H.; literature analysis by H.W-Y.; collection of funds by H.H. All authors have read and agreed to the published version of the manuscript. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Bio chemistryUniversity of Texas Medical Branch, USA
Department of Criminal JusticeLiberty University, USA
Department of PsychiatryUniversity of Kentucky, USA
Department of MedicineGally International Biomedical Research & Consulting LLC, USA
Department of Urbanisation and AgriculturalMontreal university, USA
Oral & Maxillofacial PathologyNew York University, USA
Gastroenterology and HepatologyUniversity of Alabama, UK
Department of MedicineUniversities of Bradford, UK
OncologyCirculogene Theranostics, England
Radiation ChemistryNational University of Mexico, USA
Analytical ChemistryWentworth Institute of Technology, USA
Minimally Invasive SurgeryMercer University school of Medicine, USA
Pediatric DentistryUniversity of Athens , Greece
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**Ji Sun**
Department of Technical Physics, Peking University, Beijing, P.R. China
[**Abstract**]{}
The well-known physical equivalence drawn from hole theory is applied in this article. The author suggests to replace, in the part of Feynman diagram which cannot be fixed by experiments, each fermion field operator, and hence fermion propagator, by pairs of equivalent fermion field operators and propagators. The formulation of this article thus yields additional terms which reveal characteristic effects that have not been explored previously; such characteristic effects lead to the appearence of logarithmic running terms and that finite radiative corrections are directly obtained in calculations.
PACS numbers: 03.70; 11.10.-z; 11.10.Gh; 12.20.Ds.
Published in Hadronic Journal 21 (1998) 583-612.
[**About the Author**]{}
[**Ji Sun**]{} (1921-1997), one of the influential physicists in China, has been dedicated in research and education of physics for half of a century. He graduated from the Department of Physics, Shanghai Jiaotong University, Shanghai, China in 1947. Since then, he had engaged in research and education in quantum machenics and particle physics in high educational institutions of China like Nankai University, Tsinghua University and Peking University. He was one of the founder and leading researcher of the Department of Technical Physics in Peking University. In the last ten years of his life, he had struggled with prostate cancer. The tremendous pain and even the high level paralysis caused by the disease did not prevent him from pursuring scientific truth until the last minute of his life. His dedication to physics and scientific world would be remembered by generations to come.
Introduction
============
Ultraviolet divergence is an important problem in QED. Investigation on this problem, to expose more characteristics of the divergence, may be helpful in the developments of the quantum field theory [@pap1]. This article is an attempt on this problem. In this article, we start from the well-known and well-established physical equivalence drawn from Dirac’s hole theory [@pap2], which gives also pairs of physically equivalent fermion propagators. In general, as measurements or observations fix only one of a physically equivalent pair, the physical equivalence from the hole theory thus gives nothing new, so such discussion is superfluous. However, there are certain parts of Feynman diagrams in QED processes in which the fermion fields or fermion propagators cannot be fixed experimentally ( e.g. fermion propagators in a self energy loop). Both of the two equivalent propagators are equally probable to happen; thus the physical equivalence might lead to additional term or terms. It will be shown in section 3 that the new additional terms coming from physical equivalence really reveal new characteristic physical effects which are closely related with ultraviolet divergences and can yield finite radiative corrections and hence finite results in direct calculations of QED processes.
It might be quite interesting to note that one of the striking characteristics of the ultraviolet divergence, i.e. the appearence of, for example, logarithmic running of QED coupling constant with scale, which has been verified in precise electroweak measurements, is also given by the formulation of this article, as will be shown in Sec. 3; the logarithmic running terms really appear in this article; however, there are, meanwhile, really more such logarithmic running terms with different charges. Such logarithmic terms will combine to give finite radiative corrections.
As the first paper of this work, main focus is given to the fundamental assumption and formulation, together with their foundations. Here as illustration, one QED process, the vertex, is calculated; other QED processes will be given in subsequent papers.
Fundamental Assumption and the Foundation of the Formulation
============================================================
Preliminary discussions on hole theory
--------------------------------------
In order to propose the fundamental assumption, we need first to investigate the hole theory [@pap2]. Before the investigation on hole theory, we review first the characterisitics of tensors formed by fermion and boson field operators under the transformation $x \to -x$, given by refs. [@pap4] and [@pap5]; their relevant results are rearranged[^1] in the form:
Under the transformation $x_{\mu} \to -x_{\mu}$[^2]:
$${{{\rm}}fermions~ fields:} ~ T \to -T, ~~ S \to S.
\label{eq1a}$$
$${{{\rm}}bosons~ fields:} ~ T \to T, ~~ S \to -S.
\label{eq1b}$$ $T$, $S$ in (\[eq1a\]), ( in (\[eq1b\])) are tensors formed by fermion (boson) field operators. $T$ represents tensors of even rank, including scalars, skew symmetric and symmetric tensors of second rank such as energy momentum tensor, etc. $S$ represents tensors of odd rank, including vectors such as the charge current density vector, etc.
Now, as a preliminary to the fundamental assumption, we discuss the physical equivalences drawn from hole theory.
We begin our discussion by [*reexpressing the well known equivalences drawn from the hole theory*]{}:
$$\begin{aligned}
& b_{-\vec{p},r=3,4} \equiv d^{\dag}_{\vec{p},r=1,2};
& b^{\dag}_{-\vec{p},r=3,4} \equiv d_{\vec{p},r=1,2};
\nonumber \\
& b_{\vec{p},r=1,2} \equiv d^{\dag}_{-\vec{p},r=3,4};
& b^{\dag}_{\vec{p},r=1,2} \equiv d_{-\vec{p},r=3,4}
\label{eq2} \end{aligned}$$ The equivalences may also be written as $$\sum_{\vec{p},r=3,4} b_{-\vec{p},r} u^{(r)}(-\vec{p})=
\sum_{\vec{p},r=1,2} d^{\dag}_{p,r} v^{(r)}(\vec{p})$$ etc. Here $b$ ($d$) refer to $-e$ ($+e$) fermions, $\equiv$ denotes physical equivalence, the both sides of which express the same physical entity or process.
Now we combine the equivalences (\[eq2\]) with (\[eq1a\]). As from (\[eq1a\]) that for fermions, $p(\vec{p},E)$ reverses its sign on reversing $x(\vec{x},t)$, it is reasonable to associate $p(\vec{p},E)$ with $x(\vec{x},t)$, (and hence $-p(-\vec{p},-E)$ with $-x (-\vec{x},-t)$) to agree with experimental facts. We obtain immediately a time dependent expression of the equivalences drawn from hole theory. The example $b_{-\vec{p},r=3,4} \equiv d^{\dag}_{\vec{p},r=1,2}$ is now expressed as the equivalence between the processes (a) (b) in Fig. 1; namely, there are two equivalent processes: (a) an electron with charge $-e$, momentum $-\vec{p}$, energy $-E$ propagating in $-t$ sense, denoted here and here after as $(-e,-\vec{p},-E,-t)$, is annihilated at $-x(-\vec{x},-t)$. ( $-x(-\vec{x},-t)$ is just the space-time point $x(\vec{x},t)$ viewed in the frame $-x$.) (b) an electron with $(+e, \vec{p}, E, t)$ is created at $x(\vec{x},t)$. (a) and (b) are two expressions of a same physical process. All other equivalences in (\[eq2\]) can be reexpressed similarly.
The equivalences drawn from hole theory also lead directly to pairs of equivalent electron propagators as depicted in Fig. 2, provided the equivalences hold at both $x_1$ and $x_2$. Thus there are two physically equivalent propagators in Fig. 2: Fig. 2 (a) is an electron with $-e,\vec{p},E$, propagating from $x_1$ to $x_2$ in $+t$ sense, denoted as $(-e,\vec{p},E,t)$; while Fig. 2 (b) is an electron with $+e,-\vec{p},-E$ propagating from $x'_2=-x_2$ to $x'_1=-x_1$, in $-t$ sense., denoted as $(+e,-\vec{p},-E,-t)$. Mathematical forms of Fig. 2 (a), (b) will be given below.
There are thus equivalences between $(-e,\vec{p},E,t)$ and $(+e,-\vec{p},-E,-t)$ processes. Relations between the equivalent fermion pairs (a), (b). both of Fig. 1 and of Fig. 2 are: (a)$\to$(b) and (b)$\to$(a) will occur under the simultaneous reflections ($x_{\mu} \to -x_{\mu}$) and ($Q \to -Q$). Note that $p_{\mu} \to -p_{\mu}$ is just a consequence of $x_{\mu} \to -x_{\mu}$ by (\[eq1a\]); and ($Q \to -Q$) is a consequence of transpose, which interchanges the initial and final states (see below, 2.2.1) and 2)).
The pair (a),(b) of Fig. 2, for example, may be considered as one fermion propagator, which is (a) or $(\vec{p}, E; \vec{x}, t)$, if it is regarded as $-e$ propagator; while it is (b) or $(-\vec{p}, -E; -\vec{x},
-t)$, if it is regarded as $+e$ propagator. Thus it follows that the axes of reference [*frames*]{} of [*physical equivalent*]{} $-e$ and $+e$ fermion fields (particle and antiparticle) are opposite in sense.
Physically equivalent pairs of fermion field operators
------------------------------------------------------
1\) The definition
The well known equivalences drawn from Dirac’s hole theory can be formulated or generalized as: “ [*To each fermion field operator $\Psi(x)$ there is a $\Psi^R(x)$, which is physically equivalent to $\Psi(x)$, defined as $$\Psi^R(x)=R \Psi(x) R^{-1}=\Psi'^{T}(x)
\label{eq3}$$ where $R=r\,tr$, $r$ being the reflection ($x \to -x$) operator, which bring $\Psi(x)$ into $\Psi'(x)=O \Psi(-x)$, $O$ being a matrix, $tr$ being the transpose operator, which bring $\Psi(x)$ into $\Psi^T(x)$, with the initial and final states interchanged.* ]{}”
2\) Several notes:
i\) The operator $O$ given here is a matrix, which is proved, under the requirement of invariance of Dirac equation, etc, as $O=\xi \gamma_5$. ($\xi= \pm1, \pm i$) [@pap5]. Thus,
$$\Psi^R(x)=\xi \gamma_5 \Psi^T(-x)
\label{eq3a}$$ which shows that the transformation of $R$ is just the same as that of joint operation $CPT$.
ii\) Eq. (\[eq3a\]) shows that the charge of $\Psi^R(x)$ should be $-Q$ if that of $\Psi(x)$ is $Q$, since they are connected by the transformation $CPT$. Eq. (\[eq3\]) gives directly that $\Psi^R(x)$ is the fermion proceeding in the sense of time $-t$, if $\Psi(x)$ is that proceeding in the sense of time $+t$; the opposite sense of time gives the interchange of creation and annihilation operators. Thus, e.g., the creation of $-e$ charge of $\Psi(x)$ is turned into annihilation of $-e$, or creation of $+e$ charge of $\Psi^R(x)$; therefore $\Psi^R(x)$ and $\Psi(x)$ are opposite in charge. This is just the physical equivalence drawn from hole theory.
The transformation (\[eq3\]) or (\[eq3a\]) is called “[**strong reflection**]{}” by Pauli [@pap3] (see also [@pap5]).
iii\) Thus, if we take $\Psi(x)$ as a $Q=-e$ fermion with $+p(+E)$ proceeding in the sense of $+t$, i.e., $\Psi_{-e,p,+E,+t}$, $\Psi^R(x)$ will then be $\Psi^R_{+e,-p,-E,-t}$ (see also Eq. (\[eq1a\])). The physical equivalence between $\Psi_{-e,p,+E,+t}$ and $\Psi^R_{+e,-p,-E,-t}$ is just the equivalences in Eq. (\[eq2\]) (e.g. $b^{\dag}_{\vec{p},r=1,2} \equiv d_{-\vec{p},r=3,4}$ etc., $b$, $d$ being operators of $-e$, $+e$ respectively, remember that the transpose makes $\Psi^R$ to proceed in the sense $-t$.).
iv\) The fact that $-E, -t$ fermions are closely related to the $+E,+t$ fermions with opposite charge was already given by Feynman [@pap6]. However, the problem is treated here in a somewhat different point of view; in this article, [*all processes of fermion field operators and of products of them always proceed in the order of increasing time in its “own frame of reference”*]{} (namely the frame that the $t$-axis is directed in $+t$ ($-t$) sense for the fermion $\Psi_{+t}$ ($\Psi{-t}$) ); $-t$ [*arises only when a process is viewed in other frame of reference*]{}. This explains also why “a particle travelling from $x_1$ to $x_2$ is the same as an antiparticle travelling from $x_2$ to $x_1$.” (see also Fig.2).
The transpose operation in (\[eq3\]) and (\[eq3a\]) keeps that all processes proceed in the order of increasing of time.
3\) It is significant to note that it has been proved that the transformation of $R$ (Eq. (\[eq3\])) [*leaves both field equations and commutation relations invariant [@pap7]*]{}. Also it has been proved that, if the electromagnetic fields $A_{\mu}$ is transformed by $R$ simultaneously, i.e.,
$$A_{\mu}^{R}(x) = R A_{\mu}(x) R^{-1}=A'^{T}_{\mu}(x)
\label{eq3'}$$ the transformation of $R$ [*leaves both field equations and commutation relations of fermion and boson fields invariant*]{}. Moreover, [*the fundamental equations of quantum electrodynamics are invariant*]{} under the transformation of $R$ [@pap7].
Therefore the transformation of $R$ is consistently defined for full interacting quantum field theory.
The Fundamental Assumption
--------------------------
1\) Before proposing the fundamental assumption, it might be significant to notice that QED processes consist of two possible cases (or parts of Feynman diagram): (a). observable case (or parts), which can be directly observed or determined by measurements, (e.g. external fermion or boson lines). (b). unobservable case (or parts), which cannot be directly determined by measurements or observations (e.g. fermion propagators in the loops of self energy processes).
In case (a), the observable case, although there are equivalent expressions $\Psi$ and $\Psi^R$, a fermion field or a physical process is always uniquely expressed by only one expression fixed by measurement (e.g. $\Psi$, if it is expressed with $+E, +t$ which agrees with measurement condition). Everything is as usual, nothing new can be given by the physical equivalence defined in Eq. (\[eq3\]).
In case (b), the unobservable case, a fermion field operator or physical process cannot be determined by measurements or observations. [*No measurement can determine which expression should be taken*]{}. The [*two equivalent expressions*]{}, e.g. $\Psi$ and $\Psi^R$, are then [*equally probable*]{}.
It should be noted that: the case (a) is the ordinary case, in which each $C$, $P$, $T$ reflection can be operated separately; while in case (b), although the transformation $R$ satisfies $CPT$ theorem, single reflections $P$, $T$, $C$ cannot be performed separately in general, since no observation or measurement can be performed in case (b); hence a single reflection, e.g., $T$, cannot be defined. Only case (a) has been considered nowadays; while case (b) has never been investigated so far (case (b) appears only in some intermediate steps.).
In case (b), the physically equivalent fermion field $\Psi(x)$ and $\Psi^R(x)$, which satify the same quantum field equation with e.m. interaction and commutation relation, are observed under different conditions (e.g. $\Psi(x)$ is measured or viewed in the frame of reference $+x (+\vec{x}, +t)$, $Q= -e$; while $\Psi^R(x)$ is viewed in the frame $-x (-\vec{x}, -t)$, $Q=+e$). In each measurement, we can [*only*]{} measure either $\Psi(x)$ or $\Psi^R(x)$, [*but not both*]{}; so it is suitable to apply, as in quantum mechanics, the [*superposition principle*]{}. As no measurement can determine which of $\Psi$ and $\Psi^R$ is more probable than the other; so $\Psi$ and $\Psi^R$ should appear with equal probability; we arrive thus naturally at the [*fundamental assumption*]{}.
2\) Fundamental assumption:
In the unobservable case (case (b)), a fermion field operator, which is $\Psi(x)$ in the observable case (case (a)), is expressed as
$$\frac{1}{\sqrt{2}} \left[ \Psi(x)+\Psi^R(x) \right]
\label{eq4}$$ [*where $\Psi^R(x)=R \Psi(x) R^{-1}$ is defined in Eq. (\[eq3\]) and the statements in that paragraph. This is the fundamental assumption of this article.*]{}
It should be noted that, in the case (a), $\Psi^R(x)$ is automatically turned into $\Psi(x)$ which is expressed in the frame of reference $+x (+\vec{x}, +t)$. This is due to that $\Psi(x)$, $\Psi^R(x)$ are just two expressions of the same $-e$ fermion field viewed from frame of reference $+x (+\vec{x}, +t)$, $Q=-e$ and $-x (-\vec{x}, -t)$, $Q=+e$, respectively; in other words, $\Psi^R(x)$ is just $\Psi(x)$ viewed in the frame $-x (-\vec{x}, -t)$, $Q=+e$. The case (a) is, e.g., to observe $\Psi^R(x)$, which should be transformed to the frame $+x (\vec{x}, t)$, i.e. into $\Psi(x)$, as $$R \Psi^R(x) R^{-1} \left | O \right>=\Psi(x) \left|O \right>
\label{eq5}$$ $\left | O \right>$ is the observable state which is fixed in the frame $+x (+\vec{x}, +t)$.
The transformation (\[eq5\]) between two expressions of the same field leaves the fermion field unchanged. Transformation (\[eq5\]) is just twice operation of transformation $R$ on $\Psi(x)$.
As $\Psi(x)$ and $\Psi^R(x)$ in expression (\[eq4\]) should be measured in two different measurements, if we pass from case (b) to case (a), the norm becomes, by (\[eq5\]) and (\[eq4\]), $\frac{1}{2}\left[ |\Psi(x)|^2+|\Psi^R(x)|^2 \right]=
|\Psi(x)|^2$. Thus, each fermion field operator is well connected between unobservable and observable parts of a Feynman diagram; this [*guarantees the conservation of fermion number*]{} and everything of fermion fields.
In the case (b), as $\Psi(x)$ and $\Psi^R(x)$ are equally probable, the complete set of fermion field operators should be extended to that given by (A1) and (A1’) in Appendix A.
The $S$-matrix formulation
--------------------------
The formulation of this article is the same as the $S$-matrix formulation of conventional QED, i.e. $$\begin{aligned}
&S=&\sum_n S^{(n)}
\nonumber \\
&S^{(n)}=&\frac{(-i)^n}{n!}\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_{n}
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_{n-1}
\cdots
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_{1}
\nonumber \\
&& T \left[ {{{\bf}}H}_I(x_n) {{{\bf}}H}_I(x_{n-1})\cdots
{{{\bf}}H}_I(x_1) \right]
\label{eq6}\end{aligned}$$ [*except only that in the unobservable case (case (b)),*]{} $\Psi(x)$ [*in each*]{} ${{{\bf}}H}_I(x)=-i e{\overline{\Psi}}(x)\gamma_{\mu}\Psi(x)
A_{\mu}(x)$ [*is replaced by*]{} $\frac{1}{\sqrt{2}}
\left[\Psi(x)+\Psi^R(x) \right]$ [*given in (\[eq4\]). $A_{\mu}(x)$ multiplied with $\Psi^R(x)$ is correspondingly transformed by $R$, ((\[eq3’\])), since $\Psi$ interacts with $A$ at same space-time points (while in case (a) nothing is changed*]{}).
In (\[eq6\]), ${{{\bf}}H}_I(x)$ denote interaction Hamiltonian densities. The chronological operator $T$ is defined as usual. ( i.e. all factors in the bracket after $T$ are arranged so that the time of the factors are increasing from right to left; a factor $\delta_p$ is multiplied where $p$ is the number of permutations of fermion field operators to bring them into chronological order).
In case (b), due to the replacement of $\Psi(x)$ by $\frac{1}{\sqrt{2}}
\left[\Psi(x)+\Psi^R(x) \right]$, each ${{{\bf}}H}_I(x)$ in Eq. (\[eq6\]) is then replaced by, $\frac{1}{2}\left[{{{\bf}}H}_I(x) +
{{{\bf}}H}_I^R(x) \right]$, ${{{\bf}}H}_I^R(x) = -i e{\overline{\Psi}}^R(x)\gamma_{\mu}\Psi^R(x)
A_{\mu}^R(x)$.
Actually ${{{\bf}}H}_I(x)$ becomes $\frac{i e}{2} \left[ {\overline{\Psi}}(x)\gamma_{\mu}\Psi(x) A_{\mu}(x)+
{\overline{\Psi}}^R(x)\gamma_{\mu}\Psi^R(x) A_{\mu}\right]$ on replacing $\Psi(x)$ by $\frac{1}{\sqrt{2}}\left[\Psi(x)+\Psi^R(x) \right]$; all other terms that might appear in the direct replacement are excluded by the requirement that every term in ${{{\bf}}H}_I(x)$ should be an electromagnetic interaction term which is a product of various fields at the same space-time point. Therefore, in case (b), ${{{\bf}}H}_I(x)$ is replaced by $\frac{1}{2}\left[{{{\bf}}H}_I(x) +
{{{\bf}}H}_I^R(x) \right]$. Also, if we pass from case (b) to case (a), as each $\Psi^R(x)$ is automatically turned into $\Psi(x)$, $\frac{1}{2}\left[{{{\bf}}H}_I(x) +
{{{\bf}}H}_I^R(x) \right]$ is turned into ${{{\bf}}H}_I(x)$ automatically.
The product of $\frac{1}{2}\left[{{{\bf}}H}_I(x) +
{{{\bf}}H}_I^R(x) \right]$ is arranged by the chronological operator $T$ into two products ( the product of ${{{\bf}}H}_I(x)$ and ${{{\bf}}H}^R_I(x)$). It may be interesting to note that the order of factors in the product of ${{{\bf}}H}_I(x)$ is automatically reversed.
Summarization of the formulation of this article
------------------------------------------------
I). The formulation of this article differs from conventional QED only in one point, namely:
*In the unobservable case, (or parts of a Feynman diagram; the case (b)), each fermion field operator $\Psi(x)$ is replaced by $\frac{1}{\sqrt{2}}\left[\Psi(x)+\Psi^R(x) \right]$, (the fundamental assumption).*
Everything except this point is the same as conventional QED and hence need not be qualified or discussed.
II). The critical point of the replacement stated in I) is the transformation of $R$ drawn from hole theory ((\[eq3\]) and (\[eq3’\])).
1). It has been proved that the fundamental equations of quantum electrodynamics are invariant under the transformation $R$ [@pap7]. [*Therefore the transformation of $R$ of this article leaves the fundamental equations of QED invariant, and hence is consistently defined for full interacting quantum field theory.*]{}
2). [*Causality is satisfied*]{}, which can be seen directly [*by the chronological operator $T$ in (\[eq6\])*]{}. Also, [*all processes*]{} defined above [*proceed in the sense of increasing time*]{}; $-t$ arises [*only when*]{} a process [*is viewed in other frame*]{} of reference. Furthermore, if only observable states defined above, such as $\Psi(x)$, are considered, there appears only $+t$.
3\) [*Unitarity is satisfied.*]{}
i\) The transformation $R$ defined in this article is itself unitary. Actually, the transformation $R$ satisfies $CPT$ theorem. Therefore the formulation of this article satisfies unitarity.
ii\) The fermion(s) (and fermion propagator(s)) in each $S^{(n)}$ in (\[eq6\]) [*are the same as in*]{} $S^{(n)}$ of the corresponding conventional $S$-matrix theory, throughout each process except that $\Psi(x)$ is replaced by $\frac{1}{\sqrt{2}}\left[\Psi(x)+\Psi^R(x) \right]$ in the observable case (case(b)).
a\) In the expression $\frac{1}{\sqrt{2}}\left[\Psi(x)+\Psi^R(x) \right]$ the sum of probabilities of $\Psi(x)$ and $\Psi^R(x)$ , which cannot be measured simultaneously, is equal to unity.
b\) The two expressions $\Psi(x)$ and $\Psi^R(x)$ of the same fermion field in Eq. (\[eq4\]) can only be measured in two different measurements; so if we pass from case (b) to case (a), $\Psi^R(x)$ should be transformed into $\Psi(x)$, as given by Eq. (\[eq5\]). So the norm of $\frac{1}{\sqrt{2}}\left[\Psi(x)+\Psi^R(x) \right]$ is then $\frac{1}{2}\left[|\Psi(x)|^2+|\Psi^R(x)|^2
\right]=|\Psi(x)|^2$ which is just that of $\Psi(x)$ in the case (a). Therefore, each fermion field operator is well connected between unobservable and observable parts of a Feynman diagram. This guarrantees the unitarity and conservation of quantum number and everything of fermion fields.
4\) The transformation $R$ is also called “strong reflection” by Pauli [@pap3] (see also [@pap5]), which satisfies $CPT$ theorem. However, it should be noted that separate $P$, $T$,... reflections, which is defined only in observable case (case (a)), is not defined in the unopbservable case (case (b)).
Consequences
============
The fundamental assumption and the formulation given in Sec. 2 lead directly to the following consequences:
1\. [*Two equivalent pairs of fermion propagators.*]{}
We have already given in Sec. 2 that the occurrence of two physically equivalent expressions of each fermion field operator leads to the occurrence of:
\(1) two equivalent expressions of each fermion propagator. This is a direct consequence of the physical equivalence drawn from hole theory, as shown in Fig. 2. There are in general two equivalent propagators for each fermion propagator in Feynman diagram. In the case when a propagator can be determined by measurement (the observable case (a)), the two equivalent propagators will automatically become identical.
\(2) two equivalent pairs of fermion propagators.
If we include the unobservable case (b), the appearence of pairs of physically equivalent fermion field operators extends the complete set of fermion field operators to (A1), (A1’) given in Appendix A. Such extended complete set of fermion field operators can constitute four fermion propagators, i.e. [*two equivalent pairs of fermion propagators*]{}. Namely, the propagator(A) formed by (A1a),(A1’a) and its equivalent (B) by (A1b),(A1’b); and similarly (C) and (D) formed respectively by (A1c),(A1’c) and (A1d),(A1’d), as given in Appendix A. The results of Appendix A, the propagators (A),(B),(C),(D), are written here:
$$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{-i \hat{p}+m}{p^2+m^2}
e^{i p(x_2-x_1)}
\label{eqA}$$ $$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{i \hat{p}-m}{p^2+m^2}
e^{i p(x_2-x_1)}
\label{eqB}$$ $$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{-i \hat{p}-m}{p^2+m^2}
e^{i p(x_2-x_1)}
\label{eqC}$$ $$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{i \hat{p}+m}{p^2+m^2}
e^{i p(x_2-x_1)}
\label{eqD}$$
2\. Occurrence of characteristic effects in the calculations of typical divergence problems in QED.
From the statement in 1, in the unobservable case (b), the formulation Eq. (\[eq6\]) gives directly, for each fermion propagator in the Feynman diagram of a QED process:
\(1) a sum of two propagators of an equivalent pair each multiplied with its interacting e.m. field operator.
\(2) two equivalent pairs of fermion propagators (\[eqA\]), (\[eqB\]) and (\[eqC\]), (\[eqD\]), each pair occurs with equal probability.
\(1) and (2) give two significant characteristic effects which will be discussed in and after the illustrative examples of QED process given below. In order to illustrate the fundamental assumption and the formulation of the article, typical divergence processes have been calculated. As the calculations are somewhat lengthy, we choose only one of them, the vertex, here to show the ability of the formulation of this article; other processes will be given in subsequent papers.
[**The vertex**]{} [@pap8]
The $S$-matrix element of the vertex of an electron incoming at $x_1$ with $(-e,\vec{p}_1,E_1)$ outgoing at $x_2$ with $(-e,\vec{p}_2,E_2)$ and interacting with the external field $A_{\mu}^e(x_3)=A^e_{\mu} e^{i q x_3}$ at $x_3$, is $\left<f \right|S^{(3)}\left|i\right>$. The $S^{(3)}$ in Eq. (\[eq6\]) has been rewritten, in Appendix B.,1., as
$$\begin{aligned}
&&S^{(3)}=-\frac{e^3}{8}\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_2
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_3
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_1
T \{
\nonumber \\
&&\big({\overline{\Psi}}(x_2)\hat{A}^{\cdot}(x_2)
\Psi^{{{{\bf}}\cdot}}(x_2){\overline{\Psi}}^{{{{\bf}}\cdot}}(x_3)
+{\overline{\Psi}}^R(x_2)\hat{A}^{R \cdot \cdot}(x_2)
\Psi^{R {{{\bf}}\cdot \cdot}}(x_2){\overline{\Psi}}^{R {{{\bf}}\cdot \cdot}}(x_3)
\big)\hat{A}_{\mu}^{e}(x_3)
\nonumber \\
&&\big(\Psi^{{{{\bf}}\cdot \cdot \cdot}}
(x'_3){\overline{\Psi}}^{{{{\bf}}\cdot \cdot \cdot}}(x'_1)
\hat{A}^{\cdot}(x_1)
\Psi(x'_1)
+
\Psi^{R {{{\bf}}\cdot \cdot \cdot \cdot}}
(x'_3){\overline{\Psi}}^{R {{{\bf}}\cdot \cdot \cdot \cdot}}(x'_1)
\hat{A}^{R \cdot \cdot}(x_1)
\Psi^R(x'_1)
\big)
\}
\label{eq7}\end{aligned}$$
The terms which are zero in the matrix element $\left<f\right|S^{(3)}\left|i\right>$ have not been written, where $\left|i\right>=b^{+}_{\vec{p}_1,r_1}\left|\,\right>_0$, $\left|f\right>=b^{+}_{\vec{p}_2,r_2}\left|\,\right>_0$; the factor $\frac{1}{3!}$ is omitted since only one of the $3!$ possible figures is taken. The space-time variables in (\[eq7\]) are only $x_2$, $x_3$, $x_1$; $x'_3$, $x'_1$ in (\[eq7\]) are just $\pm x_3$, $\pm x_1$, the sign is $+$ or $-$ according to the fermion propagators taken (see Appendix A.2).
The external field $\hat{A}^e_{\mu}(x)$ in (\[eq7\]) is equally well be multiplied to the second square bracket; namely written as $\hat{A}^e_{\mu}(x'_3)$ instead of $\hat{A}^e_{\mu}(x_3)$.
As pairs of fermion propagators in the two square brackets in (\[eq7\]) should be taken over all possible pairs, (A),(B) and (C),(D), there are thus four cases:
I\) . (A),(B) in both square brackets.
II). (A),(B) in the first, (C),(D) in the second square bracket .
III). (C),(D) in the first,(A),(B) in the second square bracket.
IV). (C),(D) in both square brackets.
As it can be easily shown that (III), (IV) give the same value as (I), (II); so it is only necessary to calculate (I), (II) with the result times 2. So Eq. (\[eq7\]) can be written as $$S^{(3)}=-\frac{e^3}{4}
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_2
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_3
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_1
T(\{I\}+\{II\})$$ Where
$$\begin{aligned}
&{I} =&\{\left[ \big({\overline{\Psi}}(x_2)\hat{A}^{\cdot}(x_2)(A)_{x_2,x_3}
+{\overline{\Psi}}^R(x_2)\hat{A}^{R \cdot
\cdot}(x_2)(B)_{x_2,x_3}\big)\right] \hat{A}^e_{\mu}(x_3)
\nonumber \\
&&\left[\big(
(A)_{x'_3,x'_1}\hat{A}^{\cdot}(x'_1)\Psi(x'_1)
+(B)_{x'_3,x'_1}\hat{A}^{R \cdot \cdot}(x'_1)\Psi^R(x'_1)
\big)\right]
\}
\label{eq8'1a}\end{aligned}$$
$$\begin{aligned}
&{II}=&\{
\left[ \big({\overline{\Psi}}(x_2)\hat{A}^{\cdot}(x_2)(A)_{x_2,x_3}
(C)_{x'_3,x'_1}\hat{A}^{\cdot}(x'_1)\Psi(x'_1)
+(D)_{x'_3,x'_1}\hat{A}^{R \cdot \cdot}(x'_1)\Psi^R(x'_1)
\big)\right]
\}
\label{eq8'2a}\end{aligned}$$ $(A)_{x_2,x_3}$ being $\Psi^{\cdot}(x_2){\overline{\Psi}}^{\cdot}(x_3)$ with propagator (A); similarly for propagators (B),(C),(D).
[I]{} and [II]{} are equally written as:
$$\begin{aligned}
&{I} =&\{\left[ \big({\overline{\Psi}}(x_2)\hat{A}^{\cdot}(x_2)(A)_{x_2,x_3}
+{\overline{\Psi}}^R(x_2)\hat{A}^{R \cdot
\cdot}(x_2)(B)_{x_2,x_3}\big)\right] {\overline{A}}^e_{\mu}(x'_3)
\}
\label{eq8'1b}\end{aligned}$$
\}
\label{eq8'2b}\end{aligned}$$
[III]{} and [IV]{} are defined similarly, hence,
$$\begin{aligned}
&\left<f\right| S^{(3)} \left|i\right>=&
\frac{-i e^3}{4} {\overline{u}}^{(r_2)}(\vec{p}_2)
\int\frac{{{{\rm}}d}^4 k}{k^2+\lambda^2}\gamma_{\nu}
\nonumber \\
&&[\big\{
\frac{-i(\hat{p}_2-\hat{k}_2)+m}{(p_2-k)^2+m^2}\gamma_{\mu}
\frac{-i(\hat{p}_1-\hat{k}_2)+m}{(p_1-k)^2+m^2}
\nonumber \\
&&+
\frac{i(\hat{p}_2+\hat{k}_2)-m}{(p_2+k)^2+m^2}\gamma_{\mu}
\frac{i(\hat{p}_1+\hat{k}_2)-m}{(p_1+k)^2+m^2}
\big\}
\delta^4(p_1-p_2+q)
&&+
\delta^4(p_1-p_2)]
\nonumber \\
&&
\gamma_{\nu} a_{\mu}(q) u^{(r_1)}(\vec{p}_1)
\label{eq8}\end{aligned}$$
The factor $\delta^4(p_1-p_2+q)$ in the first term of (\[eq8\]), is different from the corresponding factor $\delta^4(p_1-p_2)$ in the second.
In the first term of (\[eq8\]), which is the integral of [I]{}, (\[eq8’1a\]) or (\[eq8’1b\]), in which the propagators in both square brackets are all (A), (B); so $x'_3=x_3$. Hence (\[eq8’1a\]) = (\[eq8’1b\]) as required.
While the second term of (\[eq8\]), which is the integral of (II), (\[eq8’2a\]) or (\[eq8’2b\]), in which the propagators in the first square bracket are (A), (B); while those in the second square bracket are (C), (D). So $x'_3=-x_3$ (see Appendix A). However, it is required that (\[eq8’2a\]) [*has to be equal to*]{} (\[eq8’2b\]) for all values of $x_3$; namely it is required that $\hat{A}^e_{\mu}(x_3)=\hat{A}^e_{\mu}(x'_3)$, i.e., it is required that $e^{i q x_3}=e^{-i q x_3}$; as $x_3$ [*runs over all space-time points, so it is required*]{} $q=0$[^3].
The calculation of (\[eq8\]) is given in Appendix B. The final is $$\Lambda^{(2)}_{\mu f} (p_1,p_2;q)=-\frac{\alpha}{\pi}
\big\{\left[(\frac{q^2}{3m^2}\ln \frac{m}{\lambda}-
\frac{q^2}{8 m^2}\right]\gamma_{\mu}
-\frac{i}{8 m} (\gamma_{\mu}\hat{q}-\hat{q}\gamma_{\mu})
\big\}.
\label{eq9}$$ to order $q^2$, which is the same as that obtained in the renormalization theory.
[**About the characteristic effects**]{}
The two terms in the square bracket of (\[eq8\]) means that we should take the average of [I]{} and [II]{} which appear with equal probability. In order to take an insight into Eq. (\[eq8\]) consider for a moment the fictitious processes in which there is only [I]{} or [II]{} alone, i.e. $$S^{(3)}=-\frac{e^3}{4}\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_2
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_3
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_1
T\{I\}$$$$S^{(3)}=-\frac{e^3}{4}\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_2
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_3
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_1
T\{II\}$$ We have, from (A4), (A4I), (A4II) in Appendix B, $$\begin{aligned}
&&\big(\left<f\right| S^{(3)} \left|i\right>\big)_I=
\frac{-i e^3}{2} {\overline{u}}^{(r_2)}(\vec{p}_2)
\int {{{\rm}}d}^4 k \int_0^1 {{{\rm}}d} x \int_0^x {{{\rm}}d} y
\nonumber \\
&&[\frac{\{(2-2x-x^2)m^2-\frac{k^2}{2}+(1-x+y)(1-y)q^2\}\gamma_{\mu}
}
{\{k^2+m^2x^2+q^2y(x-y)+\lambda^2(1-x)\}^3}
\nonumber \\
&&+\frac{im q_{\mu}(1+x)(2 y-x) + mx (1-x) \sigma_{\mu\nu}q_{\nu}}
{\{k^2+m^2x^2+q^2y(x-y)+\lambda^2(1-x)\}^3}
]a_{\mu}(q) u^{(r_1)}(p_1)
\nonumber\end{aligned}$$ $$\begin{aligned}
&&\big(\left<f\right| S^{(3)} \left|i\right>\big)_{II}=
\frac{-i e^3}{2} {\overline{u}}^{(r_2)}(\vec{p}_2)
\int {{{\rm}}d}^4 k \int_0^1 {{{\rm}}d} x \int_0^x {{{\rm}}d} y
\nonumber \\
&&\frac{\{-(2-2x+x^2)m^2+\frac{k^2}{2}\}\gamma_{\mu}}
{\{k^2+m^2x^2+\lambda^2(1-x)\}^3}
a_{\mu}(q) u^{(r_1)}(p_1)
\nonumber\end{aligned}$$
We see at once that there are logarithmic running terms in both fictitious processes $\big(\left<f\right| S^{(3)} \left|i\right>\big)_I$ and $\big(\left<f\right| S^{(3)} \left|i\right>\big)_{II}$ (due to the $\pm \frac{k^2}{2}$ terms in the numerators of both integrands.). This shows why the conventional formulation of QED, which includes $\big(\left<f\right| S^{(3)} \left|i\right>\big)_I$ only, gives logarithmic running terms. While in the formulation of this article, althrough there appear also logarithmic terms, the $\pm \frac{k^2}{2}$ terms in the numerators of (\[eqA4I\]), and (\[eqA4II\]) conceled each other before integration over ${{{\rm}}d}^4 k$; thus there is no divergence in all processes of this article; the finite radiative corrections are obtained directly.
We see that the first pair of propagators in $\{I\}$ and $\{II\}$ are the same, they differ only in the second pair, (A), (B) in $\{I\}$, while (C), (D) in $\{II\}$. As (A),(B) and (C),(D) are propagators $(-e,\vec{p},E,\vec{x},t)$, $(+e,-\vec{p},-E,-\vec{x},-t)$ and $(-e,-\vec{p},-E,-\vec{x},-t)$, $(+e,\vec{p},E,\vec{x},t)$ respectively, so, e.g., the propagators $(\vec{p},E,\vec{x},t)$ in (A),(B) and in (C),(D) are respectively (A) and (D) which are [*opposite*]{} in charge. Similarly, the two propagators $(-\vec{p},-E,-\vec{x},-t)$ are also [*opposite*]{} in charge. So the [*charge content of the pair (A),(B) is different from that of (C),(D).* ]{}
Therefore the terms $\{I\}$ and $\{II\}$ in (\[eq8\]) are really coexisting physical processes with different charges. So it is not surprising that we can measure logarithmic running of , e.g., coupling constant , while the two coexisting processes $\{I\}$ and $\{II\}$ have the effect of concellation with each other at very large $k$ or very small distance (see the $\pm \frac{k^2}{2}$ terms in the numerators of (\[eqA4I\]) and (\[eqA4II\])) which renders the final results finite.
So far we have discussed, in the process of the vertex, the characteristic effect coming from the averaging of coexisting $\{I\}$ and $\{II\}$ processes. There is another characteristic effect coming from replacing a fermion propagator interacting with e.m. (photon) field by two propagators of an equivalent pair each multiplied with its interacting e.m. field operator. This effect will cancel the logarithmic divergence in the self energy of free electrons; while for non-free electrons, we obtain, on combining this effect with the averaging of coexisting processes $\{I\}$ and $\{II\}$ discussed above, finite radiative corrections. These will be given by the processes in subsequent papers.
Conclusion and Discussions
==========================
1\. Based on the fundamental assumption, $\Psi(x)$ is replaced by $$\frac{1}{\sqrt{2}}\left[\Psi(x)+\Psi^R(x) \right]$$ in case (b), the formulation of this article leads, in the case (b), to that:
1\) every propagator in Feynman diagram is replaced by two propagators of an equivalent pair, each multiplied with its interacting e.m. field operator.
2\) there occur two pairs of fermion propagators, (A), (B) and (C), (D), each with equal probability.
Each of 1), 2) gives a characteristic effect which is closely related to the ultraviolet divergence.
2\. As given above, we have obtained, on one hand, the appearence of logarithmic running terms; while on the other, the finite results of calculations. This is due to that, there appear, in this article, logarithmic running terms with different charges ( see the statements near the end of Sec. 3), so the cancellation of logarithmic terms at very large values of k, or very small distance, is a very natural physical effect. The cancellation of logarithmic terms takes place before the final integration over $k$; so no ultraviolet divergences appear finally in our calculations; finite radiative corrections are obtained in direct calculations. It might be possible that, according to the author, the appearence of ultraviolet divergence might be due to the negligence of the case (b).
3\. Since the calculations of each process are somewhat lengthy, only one of them can be given here, Actually the author have calculated the three typical divergence problems, i.e., the self energy of electron, the self energy of photon (vavuum polarization), and the vertex. Preliminary results show that all the radiative corrections are the same as those obtained in conventional renormalization treatments. Such works will be given in subsequent papers.
The three typical divergences are the source of all ultraviolet divergences in QED; so it might be expected that in the treatment of this article, the results of all higher order diagrams may be all the same as those given by the conventional renormalization treatment. However, such works, which are somewhat lengthy, can only be given in a series of subsequent papers.
[**ACKNOWLEDGMENTS**]{} The author would like to express his profound thank to Dr. Sining Sun and Dr. Peixin Liu for assisting in manuscript preparation in many respects.
Appendix A
==========
1.The complete set of fermion field operators
Since the unobservable case (b) is included in this article, the expansion of $\Psi$ and of ${\overline{\Psi}}$ should be written, to include explicitly physically equivalent fermion field operators, as
$$\Psi=\frac{1}{\sqrt{2}}(\Psi_{-e,p,x}+\Psi_{+e,-p,-x}+\Psi_{-e,-p,-x}
+\Psi_{+e,p,x})
\label{eqA1}$$ where,
$$\Psi_{-e,p,x}=\sum_{p,r=1,2}b_r(\vec{p})u^{(r)}(\vec{p})e^{i p x}
\label{eqA1a}$$
$$\Psi_{+e,-p,-x}=\sum_{p,r=3,4}d_r^+(-\vec{p})v^{(r)}(-\vec{p})e^{i p x}
\label{eqA1b}$$
$$\Psi_{-e,-p,-x}=\sum_{p,r=3,4}b_r(-\vec{p})u^{(r)}(-\vec{p})e^{i p x}
\label{eqA1c}$$
$$\Psi_{+e,p,x}=\sum_{p,r=1,2}d_r^+(\vec{p})v^{(r)}(\vec{p})e^{i p x}
\label{eqA1d}$$
$${\overline{\Psi}}=\frac{1}{\sqrt{2}}
({\overline{\Psi}}_{-e,p,x}+{\overline{\Psi}}_{+e,-p,-x}+{\overline{\Psi}}_{-e,-p,-x}
+{\overline{\Psi}}_{+e,p,x})
\label{eqA1'}$$ where,
$${\overline{\Psi}}_{-e,p,x}=\sum_{p,r=1,2}b^+_r(\vec{p})
{\overline{u}}^{(r)}(\vec{p})e^{-i p x}
\label{eqA1'a}$$
$${\overline{\Psi}}_{+e,-p,-x}=\sum_{p,r=3,4}d_r(-\vec{p}){\overline{v}}^{(r)}(-\vec{p})
e^{-i p x}
\label{eqA1'b}$$
$${\overline{\Psi}}_{-e,-p,-x}=\sum_{p,r=3,4}b_r^+(-\vec{p}){\overline{u}}^{(r)}(-\vec{p})
e^{-i p x}
\label{eqA1'c}$$
$${\overline{\Psi}}_{+e,p,x}=\sum_{p,r=1,2}d_r(\vec{p}){\overline{v}}^{(r)}(\vec{p})
e^{-i p x}
\label{eqA1'd}$$
2\. Fermion propagators.
There can be formed, from the complete set of fermion field operators, (A1) and (A1’), two equivalent pairs of fermion propagators, i.e. the propagators formed by (A1a), (A1’a) and by (A1c),(A1’c) together with their respective equivalents formed by (A1b), (A1’b) and by (A1d), (A1’d). Now we derive the mathematical forms of the four fermion propagators:
We derive first the mathematical form of the fermion propagator formed by (A1a),(A1’a) and its equivalent ( by (A1b),(A1’b)), i.e. Fig. 2 (a) and (b). The propagator (a), the electron propagator with $(-e,\vec{p},E,t)$ has already been given in the usual way as the matrix element of $T\left(\Psi(x_2){\overline{\Psi}}(x_1)\right)$ between vacuum states, where $\Psi(x)=\sum_{p,r=1,2}b_{p,r}u^{(r)}(\vec{p})\exp (i px)$, ($x_{02} > x_{01}$), as
$$_0\left<\right|
T\left(\Psi(x_2){\overline{\Psi}}(x_1)\right)
\left|\right>_0=
\frac{1}{(2\pi)^3}\int {{{\rm}}d}^3 p \frac{-i\hat{p}+m}{2E}
e^{i p(x_2-x_1)}
\label{eqA2a}$$
(\[eqA2a\]) is the propagator $(-e,\vec{p},E,\vec{x},t)$ in its own frame of reference, called the frame of $-e$; it is also the propagator $(+e,\vec{p},E,\vec{x},t)$ in the frame of $+e$, since (\[eqA2a\]) is independent of sign of $Q$. To obtain the propagator (b) of Fig. 2, we may write first the propagator $(+e,-\vec{p},-E,-\vec{x},-t)$ in the frame of $+e$ by operating $r$ ($x_{\mu} \to - x_{\mu}$ on (\[eqA2a\]), as
$$\begin{aligned}
&&_0\left<\right|
T\left(\Psi^r(x_2){\overline{\Psi}}^r(x_1)\right)
\left|\right>_0
=-
_0\left<\right|
T\left({\overline{\Psi}}^r(x_1)\Psi^r(x_2)\right)
\left|\right>_0
\nonumber \\
&&=\frac{1}{(2\pi)^3}\int {{{\rm}}d}^3 p \frac{-i\hat{p}-m}{2E}
e^{i p(x_2-x_1)}
\label{eqA2b'}\end{aligned}$$
Mathematical forms of an equivalent pair of propagators ( i.e. an equivalent pair of expressions of one propagator) should be written in same frame of reference since they have to be calculated together. So we have to write the propagator (b) as $(+e,-\vec{p},-E,-\vec{x},-t)$ in the same frame as (a), i.e. in the frame of $-e$. As the axes of frames of reference of physically equivalent $-e$ and $+e$ fermion fields are opposite in sense, as stated at the end of Sec. 2.1, we have only to replace $x$ in (\[eqA2b’\]) by $-x$, and hence $p$ by $-p$ according to (\[eq1a\]), which leads to the required propagator $(+e,-p,-E,-x,-t)$ in the frame of $-e$, as
$$_0\left<\right|
T\left(\Psi^R(x_2){\overline{\Psi}}^R(x_1)\right)
\left|\right>_0=
-\frac{1}{(2\pi)^3}\int {{{\rm}}d}^3 p \frac{-i\hat{p}+m}{2E}
e^{i p(x_2-x_1)}
\label{eqA2b}$$
Note that for a fermion field operator, e.g., the equivalence field $\Psi^R(x)$ of $\Psi(x)$ is just $\Psi(x)$ when viewed from $-e$ frame $(+x,+t)$; however, the equivalent propagator (\[eqA2b\]) is not exactly (\[eqA2a\]) when viewed in the frame of $-e$; but differ in a minus sign. This is due to that a propagator consists of two operators, one production and one annihilation operator. If the frame of reference is changed, the order of the two operators should also be changed, and hence a minus sign is brought in. (Notice that there are now two operations, the transpose operation and the operation of chronological operator $T$.) The pair of equivalent propagators (\[eqA2a\]), (\[eqA2b\]) should, as in usual field theory, be written in four dimensional form, denoted as (A),(B),
$$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{-i \hat{p}+m}{p^2+m^2}
e^{i p(x_2-x_1)}
$$ $$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{i \hat{p}-m}{p^2+m^2}
e^{i p(x_2-x_1)}
$$
We proceed next to propagator formed by (A1c),(A1’c) and its equivalent ( by (A1d),(A1’d)). This equivalent pair of fermion propagators, called (C),(D), can be more quickly obtained from (A),(B) through $x_{\mu} \to -x_{\mu}$ as: $$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{-i \hat{p}-m}{p^2+m^2}
e^{i p(x_2-x_1)}
$$ $$\frac{i}{(2\pi)^4}\int {{{\rm}}d}^4 p \frac{i \hat{p}+m}{p^2+m^2}
e^{i p(x_2-x_1)}
$$
Each of (A),(B),(C),(D) includes, in the usual way, two three dimensional propagators; e.g. (A) includes: $(-e,\vec{p},E)$ for $x_{02} > x_{01}$, and $(+e,\vec{p},E)$ for $x_{01} > x_{02}$. Similarly for (B),(C),(D). The space-time variable of propagator (A2a) is chosen as $x$, which is also that of (A). In this way the space-time variables of (A),(B),(C),(D) are $x$, $-x$, $-x$, $x$ respectively .
(C),(D) should be considered as independent propagators, although they could be obtained from (A),(B) through ($x_{\mu} \to -x_{\mu}$), since we are not permitted to make the transformation ($x_{\mu} \to -x_{\mu}$), e.g., on (A) alone, which is only a part of Feynman diagram of a whole QED process. There are altogether two equivalent pairs of propagators (A),(B) and (C),(D). The four propagators (A),(B),(C),(D) given here are all written in the frame of $-e$.
Appendix B
==========
1\. The step from $S^{(3)}$ in (\[eq6\]) to (\[eq7\]).
We write first, by (\[eq6\])
$$\begin{aligned}
S^{(3)}&=&-\frac{e^3}{2\sqrt{2}}
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_2
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_3
\int_{-\infty}^{\infty}{{{\rm}}d}^4 x_1
T\big\{
\nonumber \\
&&\left[{\overline{\Psi}}(x_2)\gamma_{\mu}
\Psi^{{{{\bf}}\cdot}}(x_2)
A^{\cdot}_{\mu}(x_2)
+{\overline{\Psi}}^R(x_2)\gamma_{\mu}
\Psi^{R {{{\bf}}\cdot \cdot}}(x_2)
A^{R \cdot \cdot}_{\mu}(x_2)\right]
\nonumber \\
&&\left[{\overline{\Psi}}^{{{{\bf}}\cdot}}(x_3)\gamma_{\mu}
\Psi^{{{{\bf}}\cdot \cdot \cdot}}(x_3)
A^{e}_{\mu}(x_3)
+{\overline{\Psi}}^{R {{{\bf}}\cdot \cdot}}(x_3)\gamma_{\mu}
\Psi^{R {{{\bf}}\cdot \cdot \cdot \cdot}}(x_3)
A^{e R}_{\mu}(x_3)\right]
\nonumber \\
&&\left[{\overline{\Psi}}^{{{{\bf}}\cdot \cdot \cdot}}(x_1)\gamma_{\mu}
\Psi(x_1)
A^{\cdot}_{\mu}(x_1)
+{\overline{\Psi}}^{R {{{\bf}}\cdot \cdot \cdot \cdot}}(x_1)\gamma_{\mu}
\Psi^{R }(x_1)
A^{R \cdot \cdot}_{\mu}(x_1)\right]
\big\}
\label{eqA3}\end{aligned}$$
The external field $A^e_{\mu}(x)$ is a classical e.m. field, which can be observed or measured directly; there are no physical equivalence, no creation or annihilation of photons; so $A^{e R}_{\mu}(x)$ is just $A^{e}_{\mu}(x)$, and hence can be extracted from the square bracket. (\[eqA3\]) is then written as (\[eq7\]). (If $CPT$ operations are applied, it can be seen that the twice minus sign cancelled out, leaving $A^{e R}_{\mu}(x)=A^{e}_{\mu}(x)$.)
We have written the space-time variable as in the first square bracket in (\[eq8\]), while as $x$ in the second. This is due to that the pair of propagator in each square bracket may be either taken as (A),(B) or as (C),(D). For example, if (A),(B) are in both square brackets (case I)), $x'=x$; if (A),(B) in the first while (C),(D) in the second ( case II)) $x'=-x$.
2\. The steps from (\[eq8\]) to (\[eq9\]).
Eq. (\[eq8\]) is
&&+
&&+
&&
$$
Start now from (\[eq8\]). We need only to calculate the first terms in each curly bracket in (\[eq8\]), called (\[eq8\])$_a$; the second terms in each curly bracket can be obtained from (\[eq8\])$_a$ through $k \to -k$. It is readily proved that the second terms give the same results as the first, i.e. (\[eq8\])$_a$; so the result of (\[eq8\]) is twice that of (\[eq8\])$_a$.
As the numerator and the denominator of the first term in the first curly bracket $$\gamma_{\nu}\frac{-i(\hat{p}_2-\hat{k}_2)+m}{(p_2-k)^2+m^2}\gamma_{\mu}
\frac{-i(\hat{p}_1-\hat{k}_2)+m}{(p_1-k)^2+m^2}
\gamma_{\nu}
\delta^4(p_1-p_2+q)
$$ are respectively $\{2m^2\gamma_{\mu}-\hat{q}\gamma_{\mu}\hat{q}
+\hat{k}\gamma_{\mu}\hat{k}-\hat{k}\gamma_{\mu}\hat{q}
+2 i m k_{\mu}\}$ and $$\frac{1}{(k^2+\lambda^2)(k^2-2 p_1 k)(k^2- 2 p_2 k)}$$.
Similarly, those in the second curly bracket are respectively $\{-2m^2\gamma_{\mu}-\hat{k}\gamma_{\mu}\hat{k}
+i m (\gamma_{mu}\hat{k}-\hat{k}\gamma_{\mu}\}$ and $(k^2+\lambda^2)[(p_2-k)^2+m^2][(p_1-k)^2+m^2]$.
We may write (\[eq8\])$_a$ as
$$\big(\left<f\right| S^{(3)} \left|i\right>\big)_a=
\frac{-i e^3}{2} {\overline{u}}^{(r_2)}(\vec{p}_2)
\int {{{\rm}}d}^4 k [(I)_a+(II)_a] a_{\mu}(q)u^{(r_1)}(p_1)
\label{eqA4}$$ where
$$\begin{aligned}
&(I)_a=\frac{2\{2m^2\gamma_{\mu}-\hat{q}\gamma_{\mu}\hat{q}
+\hat{k}\gamma_{\mu}\hat{k}-\hat{k}\gamma_{\mu}\hat{q}
+2 i m k_{\mu}\}}
{(k^2+\lambda^2)[(p_2-k)^2+m^2][(p_1-k)^2+m^2]}
\delta^4(p_1-p_2+q)
\nonumber \\
&=4\int_0^1{{{\rm}}d} x \int_0^1 {{{\rm}}d} y
\nonumber \\
&\frac{\{(2-2x-x^2)m^2-\frac{k^2}{2}+(1-x+y)(1-y)q^2\}\gamma_{\mu}
+im q_{\mu}(1+x)(2 y-x) + mx (1-x) \sigma_{\mu\nu}q_{\nu}}
{\{k^2+m^2x^2+q^2y(x-y)+\lambda^2(1-x)\}^3}
\label{eqA4I}\end{aligned}$$ by usual calculation; here, $\sigma_{\mu \nu}$
$$\begin{aligned}
&(II)_a=\frac{2\{-2 m^2 \gamma_{\mu}-\hat{k}\gamma_{\mu}\hat{k}
+i m (\gamma_{\mu}\hat{k}-\hat{k}\gamma_{mu})\}}
{(k^2+\lambda^2)[(p_2-k)^2+m^2][(p_1-k)^2+m^2]}
\delta^4(p_1-p_2)
\nonumber \\
&=4\int_0^1 {{{\rm}}d} x \int_0^1 {{{\rm}}d} y
\frac{\{-(2-2x+x^2)m^2+\frac{k^2}{2}\}\gamma_{\mu}}
{\{k^2+m^2x^2+\lambda^2(1-x)\}^3}
\label{eqA4II}\end{aligned}$$ Here, in the case (II), $x'_1=-x_1$, the corresponding $p_1$ should be turned into $-p_1$, (see (\[eq1a\])), Dirac Eq. for $-p(-\vec{p},-E)$ case should be $(-i \hat{p}+m)u=0$ in place of $(i \hat{p} +m) u=0$ for $p(\vec{p},E)$ case.
$$\begin{aligned}
&&(\ref{eq8})_a=-ie^3{\overline{u}}^{(r_2)}(\vec{p}_2)
\int {{{\rm}}d}^4 k \int_0^1 {{{\rm}}d} x
\int_0^1 {{{\rm}}d} y
\nonumber \\
&&\{
\frac{\gamma_{\mu}(1-x+y)(1-y)q^2+mx(1-x)\sigma_{\mu\nu}q_{\nu}}
{(k^2+l^2)^3}
+\gamma_{\mu}[(2-2 x-x^2)m^2-\frac{k^2}{2}]
\nonumber \\
&&[\frac{1}{(k^2+l^2)^3}-\frac{1}{(k^2+l_0^2)^3}]
-\frac{2 m^2 x^2}{(k^2+l^2_0)^3}\gamma_{\mu}\}
\label{eqA6} \end{aligned}$$ where, $l^2=m^2x^2+q^2y(x-y)+\lambda^2(1-x)$, $l_0^2=m^2x^2+\lambda^2(1-x)$, the Lorentz condition $q_{\mu}a_{\mu}(q)=0$ has been used.
Here in (\[eqA6\]), there are only the first terms in each curly bracket in (\[eq8\]) , so for the whole (\[eq8\]), we have $$\begin{aligned}
&&(\ref{eq8})=-2ie^3{\overline{u}}^{(r_2)}(\vec{p}_2)
&&\{
\label{eqA7} \end{aligned}$$ The last term is just $-\pi^2 i \gamma{\mu}$, a first order vertex. The other terms in (\[eqA7\]), denoted as (\[eq8\]’), can be, in the case of $q<<m$, expanded in power series in $q$. We have then, to the order of $q^2$, $$\begin{aligned}
(\ref{eq8}')&\cong& \pi^2 e^3 \int_0^1 x {{{\rm}}d} x
\{\frac{q^2x^2\gamma_{\mu}}{6 l_0^2}
-\frac{q^2m^2x^2}{6 l_0^4}(2-2 x-x^2)\gamma_{\mu}
\nonumber \\
&&
+\frac{q^2}{l_0^2}(1-x+\frac{x^2}{6})\gamma_{\mu}
+\frac{m}{l^2_0}x(1-x)\sigma_{\mu\nu}q_{\nu}\}
\label{eqA8}\end{aligned}$$ On using the well known integrals $\int_0^1\frac{{{{\rm}}d} x}{l_0^2}$, $\int_0^1\frac{x {{{\rm}}d} x}{l_0^2}$, $\int_0^1\frac{x^{n \geq 2}{{{\rm}}d} x}{l_0^2}$; $\int_0^1\frac{{{{\rm}}d} x}{l_0^4}$, $\int_0^1\frac{x {{{\rm}}d} x}{l_0^4}$, $\int_0^1\frac{x^{n \geq 2}{{{\rm}}d} x}{l_0^4}$ and taking $\lambda \to 0$, we have, by ordinary calculations, $$\Lambda^{(2)}_{\mu f}(p_1,p_2;q)=-\frac{\alpha}{\pi}
\{(\frac{q^2}{3 m^2}\ln \frac{m}{\lambda}-\frac{q^2}{8 m^2})
\gamma_{\mu}-\frac{i}{8 m}(\gamma_{\mu}\hat{q}-\hat{q}\gamma_{\mu})\}
\label{eqA9}$$ to the order $q^2$, which is just Eq. (\[eq9\]).
[99]{}
F.J. Dyson, Phys. Rev. 75 (1949) 486; 1736.
F.J. Dyson, Phys. Rev. 85 (1952) 631.
A. Salam, Phys. Rev. 82 (1951) 217.
J.C. Ward, Phys. Rev. 78 (1950) 182; 84 (1951) 497.
P.M.A. Dirac, Proc. Roy. Soc. (London) A126 (1930) 360.
W. Pauli, Niels Bohr and the Development of Physics (Pergamon Press, London, 1955), p. 30.
W. Pauli, Phys. Rev. 58 (1940) 716.
A.I.Akhiezer, and V.B. Berestetskii, Quantum Electrodynamics (Interscience Publishers, New York, 1965), p. 237 - p.251.
R.P. Feynman: The Theory of Positrons, Phys. Rev. 76 (1949) 749.
A. I. Akhiecer and V. B. Berestetskii, Quantum Electrodynamics (Russian Edition, 1959), Ch. IV, §22, 4. and Ch. III, §21.
T. Kinoshita, Quantum Electrodynamics (World Scientific, Singapore, 1990), and the references quoted therein.
[^1]: (\[eq1a\]),(\[eq1b\]) can be obtained from relevant results in refs. [@pap4], [@pap5]: Consider for example the case of boson fields $U$ which can be divided into two classes $U^+$ and $U^-$; From the paper of Pauli, the fundamental homogeneous linear equation of the fields, which should be in the typical form by the requirement of invariance against proper Lorentz group, as
$$\sum k U^+=\sum U^-, ~~~ \sum k U^-=\sum U^+
\label{eq1P}$$
There are altogether two possible substitutions keeping Eq. (\[eq1P\]) invariant: $$k_l \to - k_l (k_l=-i\frac{\partial}{\partial x_l}, l=1,2,3,4.),
U^+ \to U^+, U^- \to - U^-.
\label{eq2P}$$ (as $U^*$ belong to same class as $U$; for simplicity, we use $U$ to represent both $U$ and $U^*$ here and below. ) and
$$k_i \to -k_i, U^+ \to - U^+, U^- \to U^-
\label{eq2P'}$$ As the propagation vector $k_i$ belongs to $U^-$ class, so only (\[eq2P\]) among the two substitutions (\[eq2P\]), (\[eq2P’\]) is consistent; which may also be considered as: when $k_i \to - k_i$, if Eq. (\[eq1P\]) is required to remain invariant, then $U^+ \to U^+$, $U^- \to -U^-$; and hence $T \to T$, $S \to -S$. As the invariance of field equation (\[eq1P\]) is a fundamental requirement, so we may always write: If $x_{\mu} \to -x_{\mu}$, then $T \to T$, $S \to -S$. which is (\[eq1b\]). Similarly for (\[eq1a\]).
[^2]: Fermion fields here mean free fermion fields; non-free fields may include a boson term, e.g. $p + e/c\, A$. The fermion field $p$ and the boson field term $A$ obeys (\[eq1a\]) and (\[eq1b\]) respectively.
[^3]: Actually, $q=0$ is a significant physical condition; since as boson fields behaves differently from fermion fields under the reflection $x \to -x$ as given by (\[eq1a\]), (\[eq1b\]), the factor $e^{i p' x}$ of a fermion propagator with $p'=p-e/c\,A$ is unchanged under $x \to -x$ if $A=0$ is assumed, since $e^{i(-b)(-x)}=e^{i p x}$ by (\[eq1a\]).
| |
This level occupies the map slot MAP06. For other maps which occupy this slot, see Category:MAP06.
|This article about a map is a stub. Please help the Doom Wiki by adding to it.|
MAP06: The Asylum is the second map of Welcome to Hell. It was designed by Paul Schmitz and uses the music track "Evil Incarnate" (from Doom II's MAP31) by Robert Prince.
Contents
- 1 Walkthrough
- 2 Areas / screenshots
- 3 Speedrunning
- 4 Statistics
- 5 Technical information
- 6 Inspiration and development
- 7 Trivia
- 8 See also
- 9 Sources
- 10 External links
Walkthrough
Letters in italics refer to marked spots on the map. Sector, thing, and linedef numbers in boldface are secrets which count toward the end-of-level tally.
Essentials
Other points of interest
Secrets
- The south and west of the large circular room containing the exit contain alcoves with imps. After killing the imps, shoot the back walls of these alcoves. These will open paths to the north and south (in order to be able to access the northern path, it is necessary to drop down to the small pit by the evil eye in the east of the exit room, open the secret door to the south of the eye, then press the switch – pressing the switch then revealed allows access back to normal territory, with a switch en route that needs to be pressed). The north and south paths both lead to dark rooms with arrows on the floors – these point to switches which open up nearby teleporters. The other side of the switches from these teleporters are secret doors leading to rooms with octagonal platforms, both of which count as secrets. The northern one contains some ammo, a berserk and an armor. (sector 468)
- The southern room (see #1) contains a soul sphere and a megaarmor. (sector 474)
- The teleporters (see #1) take you to areas (visible from the early part of the level) with several rocky podiums in nukage. It is possible to jump over these podiums and through nukage waterfalls. There are secret walls ahead which lower on pressing, ultimately leading to teleporters taking you to further platforms above damaging floors. (sector 105)
- See #3. (sector 107)
- Each platform (see #3) has a trap containing some cacodemons – the southern one can be jumped into, leading to a computer area map. (sector 482)
- To leave these areas (see #3), jump onto the damaging floors, and go through the sections of wall with slight kinks in them – 30 seconds after stepping off the incoming teleporters, these will open, leading to teleporters back to the first groups of podiums (it is possible to jump back to the central cross). The pillars in the north and south of the central cross will have lowered – on stepping onto these pillars, doors will open, allowing access to further rooms with monsters. (sector 378)
- See #6. (sector 408)
Bugs
Demo files
Areas / screenshots
Speedrunning
Routes and tricks
Current records
The records for the map at the Doom Speed Demo Archive are:
|Run||Time||Player||Date||File||Notes|
|UV speed||0:55.37||vdgg||2021-06-04||whl6-055.zip|
|NM speed|
|UV max||8:58.09||vdgg||2021-06-04||whl6-858.zip|
|NM 100S|
|UV -fast|
|UV -respawn|
|UV Tyson|
|UV pacifist|
The data was last verified in its entirety on December 1, 2021.
Statistics
Map data
|Things||285|
|Vertices||5210*|
|Linedefs||4283|
|Sidedefs||6574|
|Sectors||495|
Things
This level contains the following numbers of things per skill level: | https://doomwiki.org/wiki/MAP06:_The_Asylum_(Welcome_to_Hell) |
The invention provides a ground-sealed lotus root yellow rice wine. The ground-sealed lotus root yellow rice wine is prepared by the following formula and steps: (1) soaking glutinous rice in clear water for 6-12 hours, steaming to be cooked, soft and free from hard cores, and cooling down to 30-40 DEG C; (2) peeling and crushing lotus root, steaming to be cooked and soft, and cooling down to 30-40 DEG C; (3) peeling oranges and smashing orange slices; (4) uniformly stirring the steamed and cooked glutinous rice, lotus root and oranges at the weight part ratio of the steamed and cooked glutinous rice to the lotus root to the oranges being 100:(50-60):(10-20), adding 6-10 parts by weight of distiller's yeast, stirring, standing under a quilt for 36-38 hours; (5) adding 3-3.5 parts by weight of wheat koji into the obtained fermentation broth, putting into a cylinder for saccharification, adding 250-260 parts by weight of water at 25-30 DEG C, sealing the cylinder for 6-10 days, squeezing, and storing for 90 days to obtain the end product. The orange and lotus root compounded raw materials are all selected from green and healthy natural food materials, and due to complex fermentation, the nutrient substances of the yellow rice wine, the oranges and lotus root are fully fused, so that the drinking value of the ground-sealed yellow rice wine is improved. | |
Texas & Pacific #610, Big Steam in the Lone Star State
There is an old saying that ‘Everything is bigger in Texas’. When it came to steam locomotives, the Texas & Pacific (T&P) Railway proved this fact by utilizing a ‘new’ steam locomotive design that featured a 2-10-4 wheel arrangement, which became known as the ‘Texas type’. Although the railroad company utilized many Texas type steam locomotives during the first half of the 20th century, the only remaining locomotive of the group is Texas & Pacific #610.
Brief History of Texas & Pacific #610
Texas & Pacific #610 was built in 1927 by Lima Locomotive Works. While another railroad (ATSF) briefly experimented with a 2-10-4 wheel arrangement nearly a decade earlier, they determined that the new design didn’t meet their needs and did not move beyond the initial trials. A few years later, T&P experimented with the design and eventually ordered several from Lima. As the case for most locomotive designs, as the first company to implement the design, T&P was given the opportunity to nickname the type and elected to nickname it after the railroad (and state).
Thesw locomotives were built to be workhorses and #610 did just that. The locomotive used oil for fuel and her 63″ diameter driving wheels helped her produce a tractive effort of more than 100,000 lbf with her booster. After serving the railway for more than two decades, T&P #610 saw her revenue career end in 1951.
While the majority of Texas type locomotives were scrapped after retirement, T&P #610 was placed on static display in Fort Worth. She remained static for another two decades before being selected as a leader of the American Freedom Train in 1976 for America’s bicentennial celebration. Although the locomotive needed significant work that fell well behind schedule, she was able to be restored in time to lead the AFT across Texas.
Following the American Freedom Train tour, the locomotive was leased to the Southern Railway to be part of their steam excursion roster. T&P #610 was operated by Southern from 1977-1981; but the program eventually needed more power than the locomotive could provide, so T&P #610 made her way back to Texas for static display.
T&P #610 wasn’t the only locomotive to lead the American Freedom Train. She was joined by Reading #2101 and Southern Pacific #4449 for the patriotic tour.
Current Status
Today T&P #610 is on static display at the Texas State Railroad in Palestine, which is about two hours southeast of Dallas. While the railroad remains active with various excursions, it is highly unlikely that #610 will return to service given the amount of effort that would be required to make her operational again and the limited use she would likely see. But, as the largest preserved non-articulated steam locomotive built by Lima still remaining, T&P #610 will likely remain on display as a reminder of the steam era in Texas.
If you are interested in learning more about the Texas State Railroad and their roster of equipment, including #610, be sure to visit the TSR website. | https://steamgiants.com/survivors/on-display/texas-pacific-610/ |
FIELD
BACKGROUND
SUMMARY
DESCRIPTION
The present disclosure relates to electrically conductive coating materials, electrically conductive coating systems, and methods including the same.
Aerospace vehicles, such as aircraft and/or spacecraft, often may be coated, covered, and/or painted with a dielectric layer, or coating. The dielectric layer may protect an underlying material of the aerospace vehicle and/or may provide a desired coloration, appearance, and/or aesthetic for the aerospace vehicle.
Aerospace vehicles often travel at relatively high speeds, which may cause a static charge to build up on an exposed surface of the dielectric layer. This static charge may be detrimental to the dielectric layer. Thus, there exists a need for improved electrically conductive coating materials, for electrically conductive coating systems, and/or for methods that include and/or utilize the electrically conductive coating materials and/or the electrically conductive coating systems.
Electrically conductive coating materials, electrically conductive coating systems, and methods including the same are disclosed herein. The electrically conductive coating systems include an electrically conductive base layer and a dielectric layer that extend across the electrically conductive base layer and has an average thickness. The systems further include a plurality of electrically conductive elements that is embedded within the dielectric layer. Each of the plurality of electrically conductive elements is defined by an elongate body that has a nonlinear conformation and is in electrical contact with the electrically conductive base layer. In addition, each of the plurality of electrically conductive elements is shaped such that, regardless of an orientation of a given electrically conductive element within the dielectric layer, the given electrically conductive element projects from the electrically conductive base layer at least 80% of the average thickness of the dielectric layer.
The electrically conductive coating materials include a liquid dielectric that is configured to be applied to an electrically conductive base layer to define a dielectric layer that extends across the electrically conductive base layer and has an average thickness. The electrically conductive coating materials also include a plurality of electrically conductive elements that is suspended within the liquid dielectric. Each of the plurality of electrically conductive elements is defined by an elongate body that has a nonlinear conformation. In addition, each of the plurality of electrically conductive elements is shaped such that, regardless of an orientation of a given electrically conductive element within the dielectric layer, the given electrically conductive element extends across at least 80% of the average thickness of the dielectric layer.
The methods include applying an electrically conductive coating material to an electrically conductive base layer and curing the electrically conductive coating material to define the electrically conductive coating system. The electrically conductive coating material includes a liquid dielectric and, subsequent to the curing, the liquid dielectric defines a dielectric layer that extends across the electrically conductive base layer and has an average thickness. A plurality of electrically conductive elements is suspended within the liquid dielectric during the applying and embedded within the dielectric layer subsequent to the curing. Each of the plurality of electrically conductive elements is shaped such that, regardless of an orientation of a given electrically conductive element within the dielectric layer, the given electrically conductive element extends across at least 80% of the average thickness of the dielectric layer.
FIGS. 1-12
FIGS. 1-12
FIGS. 1-12
FIGS. 1-12
FIGS. 1-12
FIGS. 1-12
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provide examples of electrically conductive coating materials and/or of electrically conductive coating systems , according to the present disclosure, of aerospace vehicles that may include and/or utilize electrically conductive coating materials and/or electrically conductive coating systems , and/or of methods of utilizing electrically conductive coating materials and/or of forming electrically conductive coating systems . Elements that serve a similar, or at least substantially similar, purpose are labeled with like numbers in each of , and these elements may not be discussed in detail herein with reference to each of . Similarly, all elements may not be labeled in each of , but reference numerals associated therewith may be utilized herein for consistency. Elements, components, and/or features that are discussed herein with reference to one or more of may be included in and/or utilized with any of without departing from the scope of the present disclosure.
In general, elements that are likely to be included in a given (i.e., a particular) embodiment are illustrated in solid lines, while elements that are optional to a given embodiment are illustrated in dashed lines. However, elements that are shown in solid lines are not essential to all embodiments, and an element shown in solid lines may be omitted from a given embodiment without departing from the scope of the present disclosure.
FIG. 1
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is a profile view of an aerospace vehicle , in the form of an aircraft , that may include and/or utilize electrically conductive coating materials and/or the electrically conductive coating systems according to the present disclosure. Aircraft includes a plurality of components, such as a fuselage , wings , and/or an empennage . These components may be covered by a skin and together may form and/or define an external surface of aircraft . At least a portion of skin and/or of external surface may be covered, coated, and/or defined by electrically conductive coating system . As examples, at least a portion of skin and/or of external surface of fuselage , wings , and/or empennage may be covered and/or defined by electrically conductive coating system .
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As discussed herein, aerospace vehicle may be configured for flight at relatively high speeds, and this high-speed flight may cause a static charge to be generated and/or to build up on external surface . If permitted to build up above certain levels, potentials, and/or voltages, this static charge may be detrimental to the performance of aerospace vehicle . However, electrically conductive coating systems , according to the present disclosure, may be configured to distribute, dissipate, shunt, and/or ground the static charge.
FIG. 2
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is a schematic representation of an electrically conductive coating material according to the present disclosure. Electrically conductive coating material also may be referred to herein as a coating material , a paint , a coating , and/or a material . Material may be configured to coat, cover, and/or protect an electrically conductive base layer , as illustrated in . As illustrated in , material includes a liquid dielectric and a plurality of electrically conductive elements . Electrically conductive elements are suspended within liquid dielectric and are discussed in more detail herein.
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Electrically conductive coating material , including liquid dielectric and/or electrically conductive elements thereof, may be applied to electrically conductive base layer in any suitable manner. As examples, material may be applied via spraying, brushing, and/or flowing. As an additional example, material may be cast as a film separately from electrically conductive base layer and subsequently brought into contact with and/or adhered to electrically conductive base layer . Application of electrically conductive coating material to electrically conductive base layer is discussed in more detail herein with reference to methods of .
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Liquid dielectric may be configured to be applied to electrically conductive base layer to form, create, and/or define a dielectric layer that extends across electrically conductive base layer and that has and/or defines an average thickness , as also illustrated in . As examples, liquid dielectric may be configured to cure, dry, polymerize, gel, and/or solidify on electrically conductive base layer to define dielectric layer . Examples of liquid dielectric include any suitable liquid dielectric material, liquid polymeric material, uncured paint, and/or uncured epoxy. Liquid dielectric also may be referred to herein as a precursor resin .
FIG. 3
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is a schematic representation of an electrically conductive coating system according to the present disclosure. Electrically conductive coating system also may be referred to herein as a coating system and/or as a system . System includes an electrically conductive base layer and a dielectric layer that extends across, coats, at least partially encapsulates, and/or covers electrically conductive base layer and/or at least a portion of at least one surface thereof. In addition, dielectric layer has and/or defines an average thickness . System further includes a plurality of electrically conductive elements . Electrically conductive elements also may be referred to herein as conductive elements and/or as elements . Elements are embedded within dielectric layer , are in electrical contact with electrically conductive base layer , and are discussed in more detail herein. As discussed in more detail herein, electrically conductive coating system may be formed and/or defined by applying electrically conductive coating material of to electrically conductive base layer .
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Dielectric layer may include and/or be any suitable dielectric, or electrically insulating, layer, coating, and/or cover that may extend across electrically conductive base layer and/or that may define average thickness . Dielectric layer may have and/or define an exposed surface , and average thickness may be defined between exposed surface and electrically conducive base layer , or a surface thereof. As an example, average thickness may be defined as a volume of dielectric layer divided by an area of exposed surface . Average thickness also may be referred to herein as a mean thickness of dielectric layer .
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Average thickness may have any suitable value. As examples, average thickness may be at least 5 micrometers, at least 10 micrometers, at least 15 micrometers, at least 20 micrometers, at least 30 micrometers, at least 40 micrometers, at least 50 micrometers, at least 75 micrometers, at least 100 micrometers, at least 150 micrometers, at least 200 micrometers, at least 250 micrometers, at least 500 micrometers, at least 1000 micrometers, at least 1500 micrometers, and/or at least 2000 micrometers. Additionally or alternatively, average thickness also may be less than 3000 micrometers, less than 2500 micrometers, less than 2000 micrometers, less than 1500 micrometers, less than 1000 micrometers, less than 750 micrometers, less than 500 micrometers, less than 250 micrometers, less than 200 micrometers, less than 150 micrometers, less than 100 micrometers, and/or less than 50 micrometers.
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As discussed, dielectric layer may include and/or be an electrically insulating layer. As such, dielectric layer may have and/or define less than a threshold electrical conductivity. Examples of the threshold electrical conductivity include electrical conductivities of less than 10Siemens/meter (S/m), less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, less than 10S/m, and/or less than 10S/m.
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Dielectric layer may be formed from and/or may include any suitable material and/or materials. As examples, dielectric layer may include and/or be a cured, gelled, and/or solidified dielectric material, polymeric material, paint, and/or epoxy.
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Electrically conductive base layer may include and/or be any suitable structure that may support dielectric layer and/or that may directly and/or indirectly electrically contact elements . As an example, electrically conductive base layer may form a portion of a substructure . As another example, electrically conductive base layer may cover and/or coat substructure . Under these conditions, electrically conductive base layer also may be referred to herein as an electrically conductive coating , an electrically conductive layer , and/or an electrically conductive film .
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An example of substructure includes a skin of an aircraft. When substructure includes the skin of the aircraft, the skin of the aircraft may include and/or be a metallic skin, an aluminum skin, and/or a composite skin. Examples of materials that may be included in and/or comprise electrically conductive base layer include any suitable conductive material, electrically conductive material, metal, gold, silver, platinum, aluminum, tungsten, carbon fiber, and/or conductive polymer.
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It is within the scope of the present disclosure that electrically conductive base layer may be maintained at, or near, a predetermined and/or specified electrical potential. As an example, electrically conductive base layer may be grounded, such as to a body of an aerospace vehicle that includes and/or utilizes system .
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As discussed, electrically conductive base layer may be formed from an electrically conductive material. As such, electrically conductive base layer may have and/or define at least a threshold electrical conductivity. Examples of the threshold electrical conductivity include threshold electrical conductivities of at least 10 S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, and/or at least 10S/m.
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As discussed, elements are in electrical contact with electrically conductive base layer . As an example, elements may be in direct electrical contact, and/or may be in direct physical contact with electrically conductive base layer . As another example, elements may touch electrically conductive base layer . As yet another example, elements may be in indirect electrical contact with electrically conductive base layer , such as when a thin film of the dielectric material that comprises dielectric layer extends between elements and electrically conductive base layer . Under these conditions, a thickness of the film may be such that at least a threshold electrical conductivity exists between a given element and electrically conductive base layer .
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FIG. 3
Elements are shaped such that, regardless of an orientation of a given element , the given element projects and/or extends from electrically conductive base layer to within a threshold fraction of average thickness of dielectric layer . Additionally or alternatively, the given element also may be shaped such that the given element extends across at least the threshold fraction of average thickness . This is illustrated in dashed lines in , with each element in system extending from electrically conductive base layer at least a threshold minimum fraction of average thickness and less than a threshold maximum fraction of average thickness . In the systems and methods according to the present disclosure, each element within system may have and/or define a maximum height above electrically conductive base layer that is within the threshold fraction of average thickness , that is greater than threshold minimum fraction , and/or that is less than threshold maximum fraction .
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Examples of the threshold fraction of average thickness include a substantial fraction of average thickness and/or a majority of average thickness . More specific examples of the threshold fraction of average thickness , or of threshold minimum fraction , include threshold fractions of at least 50%, at least 60%, at least 70%, at least 80%, at least 90%, at least 100%, and/or at least 110% of the average thickness of the dielectric layer. Additionally more specific examples of the threshold fraction of average thickness , or of threshold maximum fraction , include fractions of less than 200%, less than 175%, less than 150%, less than 140%, less than 130%, less than 120%, less than 110%, less than 100%, less than 90%, and/or less than 80% of the average thickness of the dielectric layer.
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FIG. 3
It is within the scope of the present disclosure that elements may extend from electrically conductive base layer to exposed surface of dielectric layer and/or may physically contact exposed surface . As an example, and as indicated in at , one or more element may project from and/or penetrate through exposed surface of dielectric layer . Under these conditions, elements may extend from surface of electrically conductive base layer a distance that is greater than average thickness . Stated another way, at least a portion of one or more elements may extend beyond dielectric layer , may be exposed to atmosphere, and/or may be exposed to an ambient environment that surrounds system .
FIG. 3
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Additionally or alternatively, and as indicated in at , one or more elements may end at exposed surface , may terminate within exposed surface , and/or may form a portion of exposed surface . Under these conditions, elements may extend from surface of electrically conductive base layer a distance that is equal, or at least substantially equal, to average thickness .
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FIG. 3
It is also within the scope of the present disclosure that elements may not contact, penetrate, and/or extend through exposed surface of dielectric layer , at least immediately subsequent to formation of system . This is illustrated in at . Under these conditions, elements may extend from surface of electrically conductive base layer a distance that is less than average thickness .
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FIGS. 2-3
FIG. 3
FIGS. 4-6
Electrically conductive elements disclosed herein, such as those of , are formed and/or defined by an elongate body that has a nonlinear conformation. The nonlinear conformation is selected (i.e., the electrically conductive element is shaped) such that each electrically conductive element projects from electrically conductive base layer to within the threshold fraction of average thickness , as discussed. This is illustrated generally in by the plurality of electrically conductive elements having at least substantially the same conformation but different orientations on electrically conductive base layer and all extending from surface of electrically conductive base layer to between threshold minimum fraction and threshold maximum fraction . This is also illustrated more specifically in , which are discussed in more detail herein.
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FIGS. 4-11
The nonlinear conformation may include any suitable shape and/or conformation, and different electrically conductive elements may have and/or define different nonlinear conformations. As an example, the nonlinear conformation may include at least one arcuate region. As another example, the nonlinear conformation may include at least one linear region. As additional examples, the nonlinear conformation may include one or more of a pyramidal shape, a conic shape, a coil, a helix, a spiral, a bent circular ring, a lobed structure, at least one loop, and/or at least one enclosed region. More specific examples of the nonlinear conformation are illustrated in and discussed in more detail herein.
50
As used herein, the phrase “nonlinear conformation” may include any conformation that is not arranged, entirely, in a straight line. As such, electrically conductive bodies may include one or more linear segments, or regions; however, these linear segments, when present, generally are not collinear and/or are connected by nonlinear segments, or regions.
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Elongate body may have and/or define at least a threshold aspect ratio. The threshold aspect ratio may be defined as a characteristic cross-sectional dimension, or transverse cross-sectional dimension, of elongate body divided by a length, or an extended length, of elongate body . Examples of the threshold aspect ratio include threshold aspect ratios of at least 10, at least 25, at least 50, at least 100, at least 250, at least 500, at least 1000, at least 2500, at least 5000, or at least 10000.
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When elongate body has a circular, or at least substantially circular, transverse cross-sectional shape, the characteristic cross-sectional dimension may be a diameter, or transverse diameter, thereof. When elongate body has a non-circular transverse cross-sectional shape, the characteristic cross-sectional dimension may be an effective cross-sectional diameter, or effective transverse cross-sectional diameter, thereof. The effective cross-sectional diameter may be defined as a diameter of a circle that has the same area as a transverse cross-sectional area of elongate body .
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FIG. 3
The extended length of elongate body may be defined as the length of elongate body were elongate body deformed into a straight line. As an example, and as illustrated in , elongate body may include a first end and a spaced-apart second end . When elongate body includes first end and second end , the length of elongate body may be defined as a total length from first end to second end as measured along elongate body .
FIGS. 9-11
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As another example, and as illustrated in and discussed in more detail herein, elongate body may define a continuous loop, a closed loop, and/or a ring. When elongate body defines the continuous loop and/or ring, the length of elongate body may be defined as a total distance from a given point back to given point , as measured along elongate body .
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It is within the scope of the present disclosure that the length, or extended length, of elongate body may have any suitable value. As examples, the length of elongate body may be greater than average thickness of dielectric layer and/or may be at least 5, at least 10, at least 20, at least 30, at least 40, at least 50, at least 75, and/or at least 100 times larger than average thickness . It is also within the scope of the present disclosure that the transverse cross-sectional diameter, or effective transverse cross-sectional diameter, of elongate body may have any suitable value. As examples, the transverse cross-sectional diameter of elongate body may be less than 50%, less than 40%, less than 30%, less than 20%, less than 10%, less than 5%, and/or less than 1% of average thickness of dielectric layer . More specific examples of the transverse cross-sectional diameter include transverse cross-sectional diameters of less than 250 micrometers, less than 200 micrometers, less than 150 micrometers, less than 100 micrometers, less than 50 micrometers, less than 25 micrometers, less than 10 micrometers, and/or less than 5 micrometers.
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Elongate body may be constructed and/or defined in any suitable manner. As an example, elongate body may include and/or be a conductive wire that may be bent and/or otherwise formed to define the nonlinear conformation. As another example, elongate body may include and/or be a conductive foil that may be cut, die-cut, laser cut, and/or bent to define the nonlinear conformation.
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It is within the scope of the present disclosure that elongate body may have and/or define a constant, or at least substantially constant, transverse cross-sectional shape and/or transverse cross-sectional area across at least a threshold fraction of the length thereof. Examples of the threshold fraction of the length include threshold fractions of at least 10%, at least 20%, at least 30%, at least 40%, at least 50%, at least 60%, at least 70%, at least 80%, at least 90%, at least 95%, and/or at least 99% of the length.
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Elongate body also may include any suitable material and/or materials. As examples, elongate body may include, or be formed from, a conductive material, an electrically conductive material, a metal, gold, silver, platinum, aluminum, tungsten, a carbon fiber, and/or a conductive polymer. Elongate body also may be referred to herein as an elongate conductive body .
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As discussed, electrically conductive element and/or elongate body thereof may be formed from an electrically conductive material. As such, electrically conductive element and/or elongate body may have and/or define at least a threshold electrical conductivity. Examples of the threshold electrical conductivity include threshold electrical conductivities of at least 10 S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, at least 10S/m, and/or at least 10S/m.
FIG. 3
FIG. 3
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With reference to , electrically conductive elements may have and/or define any suitable relative orientation, placement, spacing, and/or average spacing within electrically conductive coating system and/or within dielectric layer thereof. As an example, electrically conductive elements may be spaced-apart within dielectric layer . As another example, electrically conductive elements may have and/or define an areal density of at least 0.01 per square centimeter (cm), at least 0.05 per cm, at least 0.1 per cm, at least 0.15 per cm, at least 0.2 per cm, at least 0.3 per cm, at least 0.4 per cm, at least 0.5 per cm, at least 0.6 per cm, at least 0.7 per cm, at least 0.8 per cm, at least 0.9 per cm, and/or at least 1 per cm. Additionally or alternatively, the areal density may be less than 5 per cm, less than 4 per cm, less than 3 per cm, less than 2 per cm, less than 1 per cm, less than 0.8 per cm, less than 0.6 per cm, less than 0.4 per cm, and/or less than 0.2 per cm. may not be drawn to scale.
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As discussed, electrically conductive elements of electrically conductive coating materials and/or of electrically conductive coating systems , according to the present disclosure, may be defined by an elongate body that has a nonlinear conformation. Such a configuration may permit electrically conductive elements to extend between electrically conductive base layer to exposed surface , to dissipate static charge, and/or to conduct static charge away from exposed surface of dielectric layer and into electrically conductive base layer . In addition, the conformation of electrically conductive elements may permit such static charge dissipation without the added weight and/or aesthetic impact that may be caused by utilizing large, substantially spherical, conductive particles to dissipate the static charge. As an example, a mass, or weight, of electrically conductive elements may be substantially less than a mass, or weight, of a spherical conductive particle that might be utilized to dissipate the static charge. As another example, electrically conductive elements may be much more difficult to visually detect and/or observe when compared to the spherical conductive particles.
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With the above discussion in mind, electrically conductive elements may be described herein as defining an effective volume and an actual volume , with the actual volume being less than, or less than a threshold fraction of, the effective volume. Examples of the threshold fraction of the effective volume include threshold fractions of less than 50%, less than 40%, less than 30%, less than 25%, less than 20%, less than 15%, less than 10%, less than 5%, and/or less than 1%. This is in contrast to the above-described spherical conductive particles, where the effective volume and the actual volume may be substantially the same.
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As used herein, the term “effective volume ” may be a smallest volume of a standard three-dimensional geometric shape that completely surrounds a given electrically conductive element . Examples of the standard three-dimensional geometric shape include a sphere, a cylinder, a cube, a rectangular solid, a cone, a conic section, a triangular solid, and/or a pyramidal shape.
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As used herein, the term “actual volume ” may refer to a volume of elongate body . As an example, and when elongate body defines a constant, or at least substantially constant, transverse cross-sectional area, the actual volume of elongate body may be the transverse cross-sectional area multiplied by the length of elongate body . As another example, the actual volume of elongate body may be equal to a volume of fluid that is displaced by elongate body when elongate body is submerged within the fluid.
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FIG. 4
FIG. 5
FIG. 4
FIG. 6
FIGS. 4-5
FIGS. 4-6
FIG. 4
As discussed, elements are shaped such that, regardless of the orientation of a given element within a given dielectric layer , the given element projects from electrically conductive base layer to within a threshold fraction of an average thickness of dielectric layer and/or extends across the threshold fraction of average thickness . With this in mind, is a side view of an electrically conductive element , according to the present disclosure, in a first orientation, while is a side view of the electrically conductive element of in a second orientation, and is a side view of the electrically conductive element of in a third orientation. The electrically conductive element of has a helical conformation that defines a helix height and a helix diameter , as illustrated in . In the illustrated example, helix height is greater than helix diameter ; however, this is not required.
FIGS. 4-6
FIGS. 4-6
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As illustrated in , electrically conductive element may be embedded within a dielectric layer that has an average thickness . Dielectric layer may extend across an electrically conductive base layer . illustrate two different average thicknesses , a relatively thicker average thickness A, which is indicated in dashed lines, and a relatively thinner average thickness B, which is indicated in dash-dot lines.
FIGS. 4-6
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illustrate that, regardless of the orientation of electrically conductive element within dielectric layer and/or relative to electrically conductive base layer , electrically conductive element is shaped to project from electrically conductive base layer at least a threshold fraction of average thickness . In the example of the relatively thinner average thickness , which is illustrated in dash-dot lines, helix diameter is greater than or equal to average thickness and electrically conductive element extends at least to an exposed surface of dielectric layer for all orientations of electrically conductive element within dielectric layer and/or relative to electrically conductive base layer .
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FIG. 4
FIGS. 5-6
In the example of the relatively thicker average thickness , which is illustrated in dashed lines, there are certain orientations of electrically conductive element within dielectric layer and/or relative to electrically conductive base layer in which electrically conductive element does not extend to exposed surface , as illustrated in , and other orientations in which electrically conductive element does extend to and/or through exposed surface , as illustrated in . Regardless of the exact value for average thickness , the dimensions of electrically conductive element may be selected such that electrically conductive element always projects from electrically conductive base layer to within the threshold fraction of average thickness and/or always extends across the threshold fraction of average thickness , as discussed herein.
FIGS. 7-11
FIGS. 7-11
FIGS. 2-3
FIGS. 7-11
FIGS. 2-3
FIGS. 2-3
FIGS. 7-11
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provide examples of more specific conformations for electrically conductive elements that may form a portion of electrically conductive coating materials and/or of electrically conductive coating systems according to the present disclosure. Electrically conductive elements of may include and/or be more specific examples of electrically conductive elements of , and any of the structures, functions, and/or features of electrically conductive elements of may be included in and/or utilized with electrically conductive elements of without departing from the scope of the present disclosure. Similarly, any of the structures, functions, and/or features of electrically conductive elements of may be included in and/or utilized with electrically conductive elements of without departing from the scope of the present disclosure.
FIG. 7
FIG. 7
FIGS. 4-6
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illustrates an electrically conductive element that has a helical conformation . Helical conformation of may be at least substantially similar to helical conformation of and may define a helix height and a helix diameter . Helical conformation also may be referred to herein as a coil conformation and/or as a spring-shaped conformation .
FIG. 7
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further illustrates, in dashed lines, that an effective volume of electrically conductive elements that exhibit helical conformation may be approximated by a cylinder. The cylinder may have a diameter that is equal to helix diameter and a height that is equal to helix height .
FIG. 8
FIG. 8
FIG. 8
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illustrates an electrically conductive element that has a pyramidal conformation . In the example of , electrically conductive element may be formed from an elongate body in the form of a wire that is bent at two locations, with a first bend being defined, at least substantially, in an X-Y plane and a second bend being defined, at least substantially, in a Y-Z plane. As an example, electrically conductive element of may be bent at a 90 degree angle in both the X-Y plane and in the Y-Z plane; however, this specific angle is not required.
FIG. 8
FIGS. 4-7
FIGS. 3-6
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further illustrates, in dashed lines, that an effective volume of electrically conductive elements that exhibit pyramidal conformation may be approximated by a pyramidal shape and/or by a triangular pyramid. Similar to helical conformation of , the dimensions of pyramidal conformation may be selected such that electrically conductive element always projects from electrically conductive base layer to within the threshold fraction of average thickness and/or always extends across the threshold fraction of average thickness , as discussed herein and illustrated in .
FIG. 9
FIG. 9
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illustrates an electrically conductive element that has a bent circular ring conformation . In the example of , electrically conductive element may be formed from an elongate body in the form of a wire and/or foil that defines a circular ring and that is bent as illustrated.
FIG. 9
FIGS. 4-7
FIGS. 3-6
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further illustrates, in dashed lines, that an effective volume of electrically conductive elements that exhibit bent circular ring conformation may be approximated by a spherical shape. Similar to helical conformation of , the dimensions of bent circular ring conformation may be selected such that electrically conductive element always projects from electrically conductive base layer to within the threshold fraction of average thickness and/or always extends across the threshold fraction of average thickness , as discussed herein and illustrated in .
FIGS. 10-11
FIG. 10
FIG. 11
FIG. 11
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illustrate an electrically conductive element in the form of a lobed structure . is a top view of lobed structure , while is a side view of lobed structure . Lobed structure also may be referred to herein as being dumbbell-shaped, as being figure 8-shaped, and/or as including two linked nodes. Lobed structure may be cut from a foil, such as via die-cutting and/or laser-cutting, and bent as illustrated in .
FIG. 11
FIGS. 4-7
FIGS. 3-6
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further illustrates, in dashed lines, that an effective volume of electrically conductive elements that exhibit lobed structure may be approximated by a spherical shape. Similar to helical conformation of , the dimensions of lobed structure may be selected such that electrically conductive element always projects from electrically conductive base layer to within the threshold fraction of average thickness and/or always extends across the threshold fraction of average thickness , as discussed herein and illustrated in .
FIGS. 7-9 and 11
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As illustrated in , an actual volume of electrically conductive element may be substantially less than effective volume . Examples of the relationship between actual volume and effective volume are disclosed herein.
FIG. 12
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is flowchart depicting methods of forming an electrically conductive coating system according to the present disclosure. Methods include applying an electrically conductive coating material at and curing the electrically conductive coating material at . Methods further may include removing a portion of a dielectric layer at .
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Applying the electrically conductive coating material at may include applying the electrically conductive coating material to an electrically conductive base layer. Examples of the electrically conductive coating material are disclosed herein with reference to electrically conductive coating material . The applying at may be accomplished in any suitable manner. As examples, the applying at may include spraying the electrically conductive coating material onto the electrically conductive base layer, brushing the electrically conductive coating material onto the electrically conductive base layer, flowing the electrically conductive coating material onto the electrically conductive base layer, and/or casting the electrically conductive coating material as a film and subsequently adhering the film to the electrically conductive base layer.
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Curing the electrically conductive coating material at may include curing to form and/or define an electrically conductive coating system and/or to cure a dielectric liquid of the electrically conductive coating material such that the dielectric liquid forms and/or defines the dielectric layer. Examples of the electrically conductive coating system are disclosed herein with reference to electrically conductive coating system . The curing at may be accomplished in any suitable manner. As examples, the curing at may include heating the electrically conductive coating material, polymerizing at least a portion of the electrically conductive coating material, and/or drying the electrically conductive coating material.
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Removing the portion of the dielectric layer at may include removing at least a portion of the dielectric layer to expose a plurality of electrically conductive elements and/or to flatten and/or planarize an exposed surface of the dielectric layer. This may include removing at least a portion of the exposed surface of the dielectric layer, decreasing an average thickness of the dielectric layer, and/or modifying the electrically conductive coating system such that the plurality of electrically conductive elements extends through the exposed surface and/or forms a portion of the exposed surface.
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As a more specific example, and subsequent to the applying at , surface energy forces may cause a portion of a liquid dielectric within the electrically conductive coating material to wick up, wick around, and/or at least partially coat and/or cover a region of the plurality of electrically conductive elements that otherwise would extend through the exposed surface. Then, subsequent to the curing at , a corresponding portion of the dielectric layer also may at least partially coat and/or cover the region of the plurality of electrically conductive elements. Under these conditions, the removing at may include removing the portion of the dielectric layer that previously had wicked up, wicked around, and/or at least partially coated and/or covered the region of the plurality of electrically conductive elements. As another more specific example, and subsequent to the applying at and the curing at , the upper surface of the dielectric layer may be uneven, and the removing at may include polishing, smoothing, and/or planarizing the dielectric layer.
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It is within the scope of the present disclosure that the removing at may be performed at any suitable time and/or with any suitable sequence during methods . As an example, the removing at may be performed subsequent to the applying at and/or subsequent to the curing at .
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It is also within the scope of the present disclosure that the removing at may be performed in any suitable manner. As examples, the removing at may include polishing the electrically conductive coating system, sanding the electrically conductive coating system, and/or buffing the electrically conductive coating system.
Examples of inventive subject matter according to the present disclosure are described in the following enumerated paragraphs:
A1. An electrically conductive coating material for coating an electrically conductive base layer, the electrically conductive coating material comprising:
a liquid dielectric configured to be applied to the electrically conductive base layer to define a dielectric layer that extends across the electrically conductive base layer and has an average thickness; and
a plurality of electrically conductive elements suspended within the liquid dielectric, wherein each of the plurality of electrically conductive elements:
(i) is defined by an elongate body that has a nonlinear conformation; and
(ii) is shaped such that, regardless of an orientation of a given electrically conductive element of the plurality of electrically conductive elements within the dielectric layer, the given electrically conductive element extends across at least a threshold fraction of the average thickness of the dielectric layer.
A2. The electrically conductive coating material of paragraph A1, wherein the liquid dielectric is configured to cure on the electrically conductive base layer to define the dielectric layer.
A3. The electrically conductive coating material of any of paragraphs A1-A2, wherein the liquid dielectric is configured to solidify to define the dielectric layer.
A4. The electrically conductive coating material of any of paragraphs A1-A3, wherein the liquid dielectric includes at least one of a polymeric material, a paint, and an uncured epoxy.
B1. An electrically conductive coating system, comprising:
an electrically conductive base layer;
a dielectric layer that extends across the electrically conductive base layer and has an average thickness; and
a plurality of electrically conductive elements embedded within the dielectric layer, wherein each of the plurality of electrically conductive elements:
(i) is defined by an elongate body that has a nonlinear conformation;
(ii) is in electrical contact with the electrically conductive base layer; and
(iii) is shaped such that, regardless of an orientation of a given electrically conductive element of the plurality of electrically conductive elements within the dielectric layer, the given electrically conductive element at least one of (a) projects from the electrically conductive base layer to within a threshold fraction of the average thickness of the dielectric layer and (b) extends across at least the threshold fraction of the average thickness of the dielectric layer.
B2. The electrically conductive coating system of paragraph B1, wherein each of the plurality of electrically conductive elements is in direct electrical contact with the electrically conductive base layer.
B3. The electrically conductive coating system of any of paragraphs B1-B2, wherein each of the plurality of electrically conductive elements is in direct physical contact with the electrically conductive base layer.
B4. The electrically conductive coating system of any of paragraphs B1-B3, wherein each of the plurality of electrically conductive elements touches the electrically conductive base layer.
B5. The electrically conductive coating system of any of paragraphs B1-B4, wherein each of the plurality of electrically conductive elements has a maximum height above the electrically conductive base layer that is within the threshold fraction of the average thickness of the dielectric layer.
B6. The electrically conductive coating system of any of paragraphs B1-B5, wherein each of the plurality of electrically conductive elements extends from the electrically conductive base layer at least the threshold fraction of the average thickness of the dielectric layer.
C1. The electrically conductive coating material of any of paragraphs A1-A4 or the electrically conductive coating system of any of paragraphs B1-B6, wherein the elongate body has an aspect ratio of at least 10, at least 25, at least 50, at least 100, at least 250, at least 500, at least 1000, at least 2500, at least 5000, or at least 10000.
C2. The electrically conductive coating material of any of paragraphs A1-A4 or C1 or the electrically conductive coating system of any of paragraphs B1-C1, wherein the elongate body has a length that is at least 5, at least 10, at least 20, at least 30, at least 40, at least 50, at least 75, or at least 100 times the average thickness of the dielectric layer.
C3. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C2 or the electrically conductive coating system of any of paragraphs B1-C2, wherein the elongate body has an effective transverse cross-sectional diameter that is at least one of less than 50%, less than 40%, less than 30%, less than 20%, less than 10%, less than 5%, or less than 1% of the average thickness of the dielectric layer.
C4. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C3 or the electrically conductive coating system of any of paragraphs B1-C3, wherein the elongate body has an/the effective transverse cross-sectional diameter that is at least one of less than 250 micrometers, less than 200 micrometers, less than 150 micrometers, less than 100 micrometers, less than 50 micrometers, less than 25 micrometers, less than 10 micrometers, or less than 5 micrometers.
C5. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C4 or the electrically conductive coating system of any of paragraphs B1-C4, wherein the elongate body extends between a first end and a spaced-apart second end.
C6. The electrically conductive coating material of any of paragraphs A1-A4 or C1-05 or the electrically conductive coating system of any of paragraphs B1-05, wherein the elongate body defines at least one closed loop.
C7. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C6 or the electrically conductive coating system of any of paragraphs B1-C6, wherein the elongate body is a conductive wire.
C8. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C7 or the electrically conductive coating system of any of paragraphs B1-C7, wherein the elongate body is a conductive foil.
C9. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C8 or the electrically conductive coating system of any of paragraphs B1-C8, wherein the elongate body has a length, and further wherein the elongate body has an at least substantially constant transverse cross-sectional shape across at least a threshold fraction of the length.
C10. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C9 or the electrically conductive coating system of any of paragraphs B1-C9, wherein the elongate body has a/the length, and further wherein the elongate body has an at least substantially constant transverse cross-sectional area across at least a/the threshold fraction of the length.
C11. The electrically conductive coating material of paragraph C10 or the electrically conductive coating system of paragraph C10, wherein the threshold fraction of the length is at least one of at least 10%, at least 20%, at least 30%, at least 40%, at least 50%, at least 60%, at least 70%, at least 80%, at least 90%, at least 95%, or at least 99% of the length.
C12. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C11 or the electrically conductive coating system of any of paragraphs B1-C11, wherein the elongate body is formed from at least one of an electrically conductive material, a metal, gold, silver, platinum, aluminum, tungsten, a carbon fiber, and a conductive polymer.
C13. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C12 or the electrically conductive coating system of any of paragraphs B1-C12, wherein the plurality of electrically conductive elements is spaced-apart within the dielectric layer.
C14. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C13 or the electrically conductive coating system of any of paragraphs B1-C13, wherein the plurality of electrically conductive elements has an areal density of at least one of:
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(i) at least 0.01 per square centimeter (cm), at least 0.05 per cm, at least 0.1 per cm, at least 0.15 per cm, at least 0.2 per cm, at least 0.3 per cm, at least 0.4 per cm, at least 0.5 per cm, at least 0.6 per cm, at least 0.7 per cm, at least 0.8 per cm, at least 0.9 per cm, or at least 1 per cm; and
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(ii) less than 5 per cm, less than 4 per cm, less than 3 per cm, less than 2 per cm, less than 1 per cm, less than 0.8 per cm, less than 0.6 per cm, less than 0.4 per cm, or less than 0.2 per cm.
C15. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C14 or the electrically conductive coating system of any of paragraphs B1-C14, wherein the elongate body is an elongate conductive body.
C16. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C15 or the electrically conductive coating system of any of paragraphs B1-C15, wherein each of the plurality of electrically conductive elements defines an effective volume and an actual volume, and further wherein the actual volume is less than a threshold fraction of the effective volume.
C17. The electrically conductive coating material of paragraph C16 or the electrically conductive coating system of paragraph C16, wherein the threshold fraction of the effective volume is less than 50%, less than 40%, less than 30%, less than 25%, less than 20%, less than 15%, less than 10%, less than 5%, or less than 1%.
C18. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C17 or the electrically conductive coating system of any of paragraphs B1-C17, wherein the elongate body includes at least one arcuate region.
C19. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C18 or the electrically conductive coating system of any of paragraphs B1-C18, wherein the elongate body includes at least one linear region.
C20. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C19 or the electrically conductive coating system of any of paragraphs B1-C19, wherein the nonlinear conformation defines at least one of:
(i) a pyramidal shape;
(ii) a conic shape;
(iii) a coil;
(iv) a helix;
(v) a spiral;
(vi) a bent circular ring;
(vii) a lobed structure;
(viii) at least one loop; and
(ix) at least one enclosed region.
C21. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C20 or the electrically conductive coating system of any of paragraphs B1-C20, wherein the threshold fraction of the average thickness is at least one of:
(i) at least 50%, at least 60%, at least 70%, at least 80%, at least 90%, at least 100%, or at least 110% of the average thickness of the dielectric layer; and
(ii) less than 200%, less than 175%, less than 150%, less than 140%, less than 130%, less than 120%, less than 110%, less than 100%, less than 90%, or less than 80% of the average thickness of the dielectric layer.
C22. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C21 or the electrically conductive coating system of any of paragraphs B1-C21, wherein the threshold fraction of the average thickness is at least one of a substantial fraction of the average thickness and a majority of the average thickness.
C23. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C22 or the electrically conductive coating system of any of paragraphs B1-C22, wherein the electrically conductive base layer is maintained at a predetermined electrical potential.
C24. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C23 or the electrically conductive coating system of any of paragraphs B1-C23, wherein the electrically conductive base layer is grounded.
C25. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C24 or the electrically conductive coating system of any of paragraphs B1-C24, wherein the electrically conductive base layer forms a portion of a substructure.
C26. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C25 or the electrically conductive coating system of any of paragraphs B1-C25, wherein the electrically conductive base layer covers a/the substructure.
C27. The electrically conductive coating material of any of paragraphs C25-C26 or the electrically conductive coating system of any of paragraphs C25-C26, wherein the substructure includes a skin of an aircraft, and optionally wherein the electrically conductive coating system includes the substructure.
C28. The electrically conductive coating material of paragraph C27 or the electrically conductive coating system of paragraph C27, wherein the substructure is an aluminum skin of an aircraft.
C29. The electrically conductive coating material of any of paragraphs C27-C28 or the electrically conductive coating system of any of paragraphs C27-C28, wherein the substructure is a composite skin of an aircraft.
C30. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C29 or the electrically conductive coating system of any of paragraphs B1-C29, wherein the electrically conductive base layer is formed from at least one of a/the electrically conductive material, a/the metal, gold, silver, platinum, aluminum, tungsten, a/the carbon fiber, and a/the conductive polymer.
C31. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C30 or the electrically conductive coating system of any of paragraphs B1-C30, wherein the dielectric layer has an exposed surface.
C32. The electrically conductive coating material of paragraph C31 or the electrically conductive coating system of paragraph C31, wherein at least one of:
(i) the average thickness of the dielectric layer is defined between the exposed surface and the electrically conductive base layer;
(ii) the plurality of electrically conductive elements projects from the exposed surface;
(iii) the plurality of electrically conductive elements forms a portion of the exposed surface;
(iv) the plurality of electrically conductive elements physically contacts the exposed surface;
(v) the plurality of electrically conductive elements penetrates through the exposed surface; and
(vi) the plurality of electrically conductive elements extends from the electrically conductive base layer and to the exposed surface.
C33. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C32 or the electrically conductive coating system of any of paragraphs B1-C32, wherein the average thickness of the dielectric layer is at least one of:
(i) at least 5 micrometers, at least 10 micrometers, at least 15 micrometers, at least 20 micrometers, at least 30 micrometers, at least 40 micrometers, at least 50 micrometers, at least 75 micrometers, at least 100 micrometers, at least 150 micrometers, at least 200 micrometers, at least 250 micrometers, at least 500 micrometers, at least 1000 micrometers, at least 1500 micrometers, or at least 2000 micrometers; and
(ii) less than 3000 micrometers, less than 2500 micrometers, less than 2000 micrometers, less than 1500 micrometers, less than 1000 micrometers, less than 750 micrometers, less than 500 micrometers, less than 250 micrometers, less than 200 micrometers, less than 150 micrometers, less than 100 micrometers, or less than 50 micrometers.
C34. The electrically conductive coating material of any of paragraphs A1-A4 or C1-C33 or the electrically conductive coating system of any of paragraphs B1-C33, wherein a/the electrically conductive coating material, which includes a/the liquid dielectric and the plurality of electrically conductive elements, is configured to be applied to the electrically conductive base layer via spraying to form the dielectric layer.
D1. A method of forming an electrically conductive coating system, the method comprising:
applying an electrically conductive coating material to an electrically conductive base layer; and
curing the electrically conductive coating material to define the electrically conductive coating system.
D2. The method of paragraph D1, wherein the electrically conductive coating material includes the electrically conductive coating material of any of paragraphs A1-A4 or C1-C34.
D3. The method of any of paragraphs D1-D2, wherein the electrically conductive coating system includes the electrically conductive coating system of any of paragraphs B1-C34.
D4. The method of any of paragraphs D1-D3, wherein the electrically conductive coating system includes a/the dielectric layer and a/the plurality of electrically conductive elements, and further wherein the method includes removing at least a portion of the dielectric layer to expose the plurality of electrically conductive elements.
D5. The method of paragraph D4, wherein the removing includes at least one of polishing the electrically conductive coating system, sanding the electrically conductive coating system, and buffing the electrically conductive coating system.
D6. The method of any of paragraphs D1-D5, wherein the applying includes spraying the electrically conductive coating material onto the electrically conductive base layer.
As used herein, the terms “selective” and “selectively,” when modifying an action, movement, configuration, or other activity of one or more components or characteristics of a system, mean that the specific action, movement, configuration, or other activity is a direct or indirect result of user manipulation of an aspect of, or one or more components of, the system.
As used herein, the terms “adapted” and “configured” mean that the element, component, or other subject matter is designed and/or intended to perform a given function. Thus, the use of the terms “adapted” and “configured” should not be construed to mean that a given element, component, or other subject matter is simply “capable of” performing a given function but that the element, component, and/or other subject matter is specifically selected, created, implemented, utilized, programmed, and/or designed for the purpose of performing the function. It is also within the scope of the present disclosure that elements, components, and/or other recited subject matter that is recited as being adapted to perform a particular function may additionally or alternatively be described as being configured to perform that function, and vice versa. Similarly, subject matter that is recited as being configured to perform a particular function may additionally or alternatively be described as being operative to perform that function.
The various disclosed elements of systems and steps of methods disclosed herein are not required to all systems and methods according to the present disclosure, and the present disclosure includes all novel and non-obvious combinations and subcombinations of the various elements and steps disclosed herein. Moreover, one or more of the various elements and steps disclosed herein may define independent inventive subject matter that is separate and apart from the whole of a disclosed system or method. Accordingly, such inventive subject matter is not required to be associated with the specific systems and methods that are expressly disclosed herein, and such inventive subject matter may find utility in systems and/or methods that are not expressly disclosed herein.
As used herein, the phrase, “for example,” the phrase, “as an example,” and/or simply the term “example,” when used with reference to one or more components, features, details, structures, embodiments, and/or methods according to the present disclosure, are intended to convey that the described component, feature, detail, structure, embodiment, and/or method is an illustrative, non-exclusive example of components, features, details, structures, embodiments, and/or methods according to the present disclosure. Thus, the described component, feature, detail, structure, embodiment, and/or method is not intended to be limiting, required, or exclusive/exhaustive; and other components, features, details, structures, embodiments, and/or methods, including structurally and/or functionally similar and/or equivalent components, features, details, structures, embodiments, and/or methods, are also within the scope of the present disclosure.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a profile view of an aerospace vehicle, in the form of an aircraft, that may include and/or utilize electrically conductive coating materials and/or electrically conductive coating systems according to the present disclosure.
FIG. 2
is a schematic representation of an electrically conductive coating material according to the present disclosure.
FIG. 3
is a schematic representation of an electrically conductive coating system according to the present disclosure.
FIG. 4
is a side view of an electrically conductive element, according to the present disclosure, in a first orientation.
FIG. 5
FIG. 4
is a side view of the electrically conductive element of in a second orientation.
FIG. 6
FIGS. 4-5
is a side view of the electrically conductive element of in a third orientation.
FIG. 7
FIGS. 4-6
illustrates an effective volume and an actual volume of the electrically conductive element of .
FIG. 8
illustrates another electrically conductive element according to the present disclosure.
FIG. 9
illustrates another electrically conductive element according to the present disclosure.
FIG. 10
is a top view illustrating another electrically conducive element according to the present disclosure.
FIG. 11
FIG. 10
is a side view of the electrically conductive element of .
FIG. 12
is flowchart depicting methods of forming an electrically conductive coating system according to the present disclosure. | |
This Nonprovisional application claims priority under 35 U.S.C. §119(a) on Patent Application No. 2007-175649 filed in Japan on Jul. 3, 2007, the entire contents of which are hereby incorporated by reference.
1. Field of the Invention
The present invention relates to a solid-state image capturing device having a color filter provided above each light receiving section provided on a surface of a semiconductor substrate, and configured with semiconductor elements for performing photoelectric conversion on image light from a subject to capture the image light; a manufacturing method for the solid-state image capturing device; a color filter; a forming method for the color filter; a liquid crystal display apparatus using the color filter; and an electronic information device, such as a digital camera (e.g., digital video camera and digital still camera), an image input camera, a scanner, a facsimile machine and a camera-equipped cell phone device, having the solid-state image capturing device, which is manufactured by the manufacturing method of the solid-state image capturing device, as an image input device in an image capturing section of the electronic information device, or an electronic information device having the liquid crystal display apparatus as a display section.
2. Description of the Related Art
Conventionally, it is known that a solid-state image capturing device capable of capturing a color image has a structure where color filters for optically separating incident light are provided above the respective corresponding light receiving sections provided on a surface of a semiconductor substrate.
FIG. 8
A specific structure of a solid-state image capturing device having such color filters will be explained in detail with reference to .
FIG. 8
is a longitudinal cross sectional view showing an exemplary essential structure of a conventional solid-state image capturing device.
FIG. 8
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In , a conventional solid-state image capturing device has a plurality of light receiving sections for performing photoelectric conversion on incident light, formed in a matrix on a surface layer of a semiconductor substrate . On a semiconductor substrate , gate film composed of polysilicon is formed adjacent to the light receiving section so as to read out and transfer photoelectrically converted signal charges by each light receiving section . Further, on a substrate area other than each light receiving section , that is, on the gate film , for example, a shading film is formed via interlayer insulation film, so that light will not enter. The shading film is opened above each light receiving section . An interlayer insulation film is formed on the semiconductor substrate , in which the light receiving sections are formed, and the shading film for the purpose of electric insulation.
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Further, on the interlayer insulation film , an interlayer insulation film is formed on the interlayer insulation film , so that the unevenness resulted from the shape of a base of the interlayer insulation film is improved and color filters can be formed on the resultant flat surface. The interlayer insulation film is composed of a transparent acrylic material and the like in order to improve adhesiveness to the color filters and improve optical transmittance.
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FIG. 9
On the interlayer insulation film , a plurality of color filters , and are arranged and formed in a corresponding manner to respective light receiving sections . Desirably, the color filters , and are formed checkerwise in order as shown in , for example.
FIG. 9
FIG. 9
FIG. 8
FIG. 9
FIG. 8
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is a plan view hypothetically showing a desirable formation of color filters shown together with light receiving sections . A-A′ in corresponds to the left side of , and B-B′ in corresponds to the right side of the .
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On the color filters , and , an interlayer insulation film is formed for the purpose of protecting the color filters , and and planarizing the base surface of microlenses before forming the microlenses . On the interlayer insulation film , microlenses are formed in a corresponding manner to respective light receiving section for the purpose of refracting incident light and condensing light that enters towards the shading film , which forms ineffective areas, into the light receiving section . Due to the microlenses , the amount of light that enters the light receiving sections increases, improving the light receiving sensitivity.
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Conventionally, the formation of the color filters , and are performed as follows.
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First, a photosensitive color filter material (material for the color filter ) is applied on the interlayer insulation film up to a desired thickness by a spin coating method and the like. The applied color filter material is exposed by an exposing device so that a pattern will be left only in desired areas among areas corresponding to the positions of respective light receiving sections . A process using a developing solution is performed to pattern the color filter
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Next, a photosensitive color filter material (material for the color filter ) is applied on the interlayer insulation film up to a desired thickness by a spin coating method and the like. The applied color filter material is exposed by an exposing device so that a pattern will be left only on another desired areas among areas corresponding to the positions of respective light receiving sections . A process using a developing solution is performed to pattern the color filter . Similarly, a color filter is further patterned on areas that remain after the formation of the color filters and among areas corresponding to the positions of respective light receiving sections .
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FIG. 8
A photosensitive material, including a pigment and a dye, is used as a color filter material, and desired color filters , and are formed on desired light receiving sections using a photolithography technique. The color filters , and have a plurality of colors. In the case of , the color filters , and are formed as a green color filter (G; green color), a blue color filter (B; blue color) and a red color filter (R; red color) respectively.
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FIG. 10
FIG. 10
FIG. 8
At this point, owing to the resolutions of the color filters , and , corner portions (edge portion) of rectangular color filters , and have blunt edges as shown in . Further, due to the alignment accuracy during the patterning, a gap (interval C) as shown in or an overlapping portion D at the upper and lower end portions as shown in is formed between adjacent color filters , and
Reference 1, for example, proposes a method for forming a color filter that forms a color filter using a dye containing negative type curing composite that enables an excellent rectangular pattern formation. According to such a conventional method for forming a color filter, it is said that a color filter having an excellent rectangular pattern formation (accurate rectangle color filter having edges) can be formed with good cost-effectiveness.
Further, Reference 2, for example, proposes a method for forming a color filter, in which a newly developed photopolymerization initiator is mixed with a photosensitive colored composite for color filters. According to such a method for forming a color filter, it is said that the sensitivity of the photosensitive colored composite is significantly increased, so that sufficient curing with a small amount of exposure is possible, and further that an accurate pattern can be formed because the photosensitive colored composite has an excellent pattern forming capability.
Further, Reference 3, for example, proposes a method for forming a color filter, in which a color filter forming area in a base interlayer insulation film is processed in a concave form, and an edge portion of the color filter is formed into a thin film. According to such a method for forming a color filter, it is said that the color filter in a thin film form has an excellent pattern forming capability, so that an accurate pattern can be formed.
Further, Reference 4, for example, proposes a method for forming a color filter, in which a red alignment mark is formed concurrently with the formation of red color filters of all the color filters, and an alignment is performed with a red color alignment light by the red alignment mark to pattern a blue resist material concurrently with the formation of blue filters of all the color filters. According to such a method for forming a color filter, it is said that high contrast can be obtained at the formation of the blue filter due to the red alignment mark that does not absorb the red alignment light, so that the alignment accuracy of the lithography technique can be improved and an accurate color filter processing can be achieved.
Reference 1: Japanese Laid-Open Publication No. 2005-274967
Reference 2: Japanese Laid-Open Publication No. 2005-202252
Reference 3: Japanese Laid-Open Publication No. 2005-123225
Reference 4: Japanese Laid-Open Publication No. 2003-215321
The conventional techniques described above, however, have the following problems.
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As described above, the color filter material includes a pigment or a dye. Consequently, the resolution of the color filter is not as good as that of a resist used for a manufacturing process of a common semiconductor apparatus. For this reason, the rectangular formation collapses at the corner portion (edge portion) of a rectangle pattern of the color filters , and , resulting in the gap C in between the adjacent color filters , and . If the gap C exists, white light, which passes through the gap C and does not pass through the color filters , and , enters. The white light then reflects multiply inside the interlayer insulation films and , or enters directly above the light receiving section . As a result, the color reproducibility deteriorates upon capturing an image and the color turns weak, causing inferiority that is too white. As discussed, if the gap C exists, in between the color filters , and , optical transmittance of the color filters , and substantially increases, deteriorating the light separation characteristics of the device.
In order to minimize the gap C, References 1 to 4 described above try to improve the color filter material to improve the resolution and light receiving sensitivity and control the collapsing of the formation, so that the gap C described above can be reduced. Further, References 1 to 4 try to improve the base pattern to improve the process accuracy, and try to improve the alignment accuracy to limit the gap C.
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However, one cell size including one light receiving section is reduced with the achievement of the high pixel density and the miniaturization of chips. Even the resolution is improved by introducing a micromachining technique, such improvement on the resolution by the micromachining technique will be insufficient by the reducing of cell size, resulting in a relatively constant gap size. Therefore, the deteriorating of the light separation characteristics of the device always occurs. Similarly, with regard to the improvement on the processing accuracy of the base and the improvement on the alignment accuracy, such improvement effect is canceled out due to the reducing of the cell size.
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On the other hand, even if a mask pattern is contrived in a lithography technique so that the color filter becomes extremely closer to a rectangle shape after patterning, an overlapping portion D of the adjacent color filters , and is caused as shown in FIG. owing to a limit on the alignment accuracy. If light condensed by the microlens passes through an area with the overlapping portion D, desirable separation of colors into single colors of the respective color filters , and cannot be performed, resulting in mixed color characteristics and therefore deterioration in light separation characteristics of the device. Moreover, the difference in the filter thickness of the color filter layers owing to the overlapping portion D of the adjacent color filters , and deteriorates the evenness of the interlayer insulation film . This affects the formation of the microlens above, changing the condensing amount of light into the light receiving section and making the resultant image uneven.
The present invention is intended to solve the conventional problems described above. The objective of the present invention is to provide a manufacturing method of a solid-state image capturing device. The manufacturing method enables to manufacture a solid-state image capturing device having good light separation characteristics, without having to improving the resolution of the color filter material or the characteristics of the light receiving sensitivity, without processing the base film of the color filter, by eliminating the gap in between the adjacent color filters, and by forming a color filter without causing the overlapping of adjacent color filters making the best of the limit of the alignment accuracy. The objective of the present invention is to further provide a solid-state image capturing device made by the manufacturing method according to the present invention; a color filter; a forming method for the color filter; a liquid crystal display apparatus having the color filter; and an electronic information device having the solid-state image capturing device used in an image capturing section or an electronic information device having the liquid crystal display apparatus as a display section.
A manufacturing method for a solid-state image capturing device having a plurality of light receiving sections for performing photoelectric conversion on incident light, provided on a surface side of a semiconductor substrate; and color filters of different colors for optically separating the incident light, provided above the plurality of corresponding light receiving sections according to the present invention includes: a color filter patterning step of patterning the color filters in such a manner that a predetermined interval is formed between adjacent color filter materials of different colors; and a color filter forming step of heat treating and fluidizing the patterned color filter materials and further curing the color filter materials to form color filters of different colors, thereby achieving the objective described above.
Preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the color filter material is melted by the heat treating and is widened by gravity along a substrate plane direction so as to fill the predetermined interval with the color filter materials.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the color filter forming step performs the heat treating either individually for each color after patterning the color filter materials, or simultaneously for a plurality of colors after successively patterning the plurality of color filter materials of different colors, or simultaneously for two colors of the plurality of the colors.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, when the heat treating is performed individually for the each color, the color filter material is melted and widened up to a predetermined color filter area and the predetermined interval between the color filter material and an adjacent color filter material is completely filled.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, when the heat treating is performed either simultaneously for the plurality of colors or simultaneously for two colors of the plurality of colors, the color filter materials of the plurality of colors or the two colors are simultaneously melted by the heat treating, and the color filter area is widened until the predetermined interval is completely filled with the color filter materials.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the heat treating is performed at between 150 degrees Celsius and 165 degrees Celsius for two to five minutes.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, polystyrene resin added with a pigment or dye is used for one or more of the color filter materials.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the polystyrene resin includes a resist material including greater than or equal to 1 wt percent and less than or equal to 10 wt percent of polyhydroxy styrene.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, greater than or equal to 1.5 wt percent and less than or equal to 3.5 wt percent of an yellow pigment and greater than or equal to 6.0 wt percent and less than or equal to 8.0 wt percent of a green pigment mixed with each other is used as the pigment and a material including the pigment is used for a green color filter; a material including greater than or equal to 5.0 wt percent and less than or equal to 10.0 wt percent of a red pigment is used for a red color filter; and greater than or equal to 4.0 wt percent and less than or equal to 8.0 wt percent of copper phthalocyanine blue having α, β, or ε crystalline form is used as the pigment, and a material including the pigment is used for a blue color filter.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, a heat bridge agent for curing is added to the color filter material.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, both of epoxy compounds and melanin resin are added as the heat bridge agent to the color filter material.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, greater than or equal to 0.1 wt percent or and less than or equal to 2.0 wt percent of either the epoxy compounds or the melanin resin is added as the heat bridge agent.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, in the color filter patterning step, the predetermined interval is set so that there will be no gap after melting, due to the characteristic of the color filter materials and the melting amount of the color filter materials at the time of heat treating.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, in the color filter patterning step, the predetermined interval is set so that adjacent color filter materials will not over lap with each other by setting the amount of the position shifting due to the alignment accuracy to the limit value.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the predetermined interval is set greater than or equal to 0.1 μm and less than or equal to 0.2 μm in the color filter patterning step.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the color filter patterning step includes: a first color filter patterning step of applying a desired thickness of a photosensitive first color filter material on an interlayer insulation film provided on the semiconductor substrate, exposing the first color filter material by an exposing device so that a pattern will be left only on a desired area among areas corresponding to the plurality of light receiving sections, and performing a process using a developing solution on the first color filter material to pattern a first color filter; a second color filter patterning step of applying a desired thickness of a photosensitive second color filter material on the interlayer insulation film, exposing the second color filter material by the exposing device so that a pattern will be left on another desired area different from the desired area among areas corresponding to the plurality of light receiving sections, and performing a process using a developing solution on the second color filter material to pattern a second color filter; and a third color filter patterning step of applying a desired thickness of a photosensitive third color filter material on the interlayer insulation film, exposing the third color filter material by the exposing device so that a pattern will be left on the rest of the area among areas corresponding to the plurality of light receiving sections, and performing a process using a developing solution on the third color filter material to pattern a third color filter.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the first color filter is a green color filter, the second color filter is either a red or blue color filter, and the third color filter is the other color of the red or blue color filter.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the color filters of different colors have either three primary colors of green, red and blue or four colors using complementary colors of yellow, magenta and cyan, and green.
Still preferably, in a manufacturing method for a solid-state image capturing device according to the present invention, the color filters of different colors form Bayer array.
A solid-state image capturing device manufactured by the manufacturing method of the solid-state image capturing device according to the present invention includes a plurality of light receiving sections for performing photoelectric conversion on incident light, provided on a surface side of a semiconductor substrate; and color filters of different colors for optically separating the incident light, provided above the plurality of corresponding light receiving sections, in which the color filters of different colors are formed in such a manner that there is no gap in between adjacent color filters and that adjacent color filters do not overlap with each other, thereby achieving the objective described above.
A forming method for a color filter according to the present invention includes: a color filter patterning step of patterning the color filters in such a manner that a predetermined interval is formed between adjacent color filter materials of different colors; and a color filter forming step of heat treating and fluidizing the patterned color filter materials and further curing the color filter materials to form color filters of different colors, thereby achieving the objective described above.
Preferably, in a forming method for a color filter according to the present invention, the thickness of the color filter materials is set in such a manner that the predetermined interval is filled at the time of the heat treating by melting the color filter materials and that the color filter materials do not overlap with each other.
Still preferably, in a forming method for a color filter according to the present invention, the color filter material is melted by the heat treating and is widened by gravity along a substrate plane direction so as to fill the predetermined interval with the color filter materials.
A color filter is formed by the forming method for a color filter according to the present invention, thereby achieving the objective described above.
A liquid display apparatus has a color filter formed by the forming method for a color filter according to the present invention, the color filter being provided on one of the two substrates putting liquid crystal material in between, thereby achieving the objective described above.
An electric information device according to the present invention includes the solid-state image capturing device manufactured by the manufacturing method for a solid-state image capturing device according to the present invention, as an image input device equipped in an image capturing section, thereby achieving the objective described above.
An electric information device according to the present invention has the liquid crystal display apparatus according to the present invention in a display section, thereby achieving the objective described above.
The functions of the present invention having the structures described above will be described hereinafter.
According to the present invention, color filter material is patterned on a semiconductor substrate in such a manner that a predetermined interval is formed in between adjacent color filter materials of different colors, heat treatment is performed on the color filter material to fluidize it, and further, the color filter material is cured. At the time of the patterning, the interval between the color filter materials is set, taking the characteristics of the color filter materials and the melting amount of the color filter materials at the heat treatment into consideration, so that there will be no gap after melting. Further, considering the amount of the position shifting due to the alignment accuracy, the interval is set such that the color filters will not overlap with each other after melting. With the method as described above, the color filters are formed in such a manner that the gap in between the color filters is filled, there is no gap between adjacent color filters, and further, the adjacent color filters do not overlap with each other in the end portions. As a result, a solid-state image capturing device with an excellent light separation characteristics is manufactured.
As described above, according to the present invention, in forming the color filters above the corresponding light receiving sections provided on the semiconductor substrate, the gaps between adjacent color filters are filled and the adjacent color filters do not overlap with each other. Therefore, it is achievable to manufacture a solid-state image capturing device having excellent light separation characteristics. Further, the color filter material used for the present invention does not have to be a material having high resolution or high sensitivity characteristic that requires improvement on micromachining technology as the cell size is reduced. Further, because the color filter material is melted and fluidized by heat treatment, the material can be processed with an equipment that only has alignment accuracy of one previous generation. Further, there is no need of processing a base interlayer insulation film in order to compensate for processing accuracy for the color filter. As a result, a man hour and manufacturing cost for the solid-state image capturing device can be significantly reduced.
These and other advantages of the present invention will become apparent to those skilled in the art upon reading and understanding the following detailed description with reference to the accompanying figures.
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semiconductor substrate
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light receiving section (photodiode)
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polysilicon gate film (gate electrode film)
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shading film
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, , interlayer insulation film
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′, ′, ′ color filter before heat treatment (patterned color filter material)
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microlens
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CCD image sensor
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A CMOS image sensor
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electronic information device
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solid-state image capturing apparatus
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memory section
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display section
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liquid crystal display apparatus
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communication section
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image output section
Hereinafter, a case where an embodiment of the manufacturing method of the solid-state image capturing device according to the present invention is applied for a CCD image sensor, will be described in detail with reference to accompanying figures. Note that the manufacturing method of the solid-state image capturing device according to the present invention is applicable to not only a CCD image sensor, but a CMOS image sensor.
FIG. 1
is a longitudinal cross sectional view showing an exemplary essential structure of a CCD image sensor related to the embodiment of the solid-state image capturing device according to the present invention.
FIG. 1
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In , a CCD image sensor according to the embodiment includes a plurality of light receiving sections formed in a matrix on a surface layer of a semiconductor substrate . The light receiving sections are formed of a plurality of photodiodes that perform photoelectric conversion on incident light. On the semiconductor substrate , a polysilicon gate film (gate electrode film) is formed for reading out and transferring signal charges, which are photoelectrically converted by each light receiving section . Further, a shading film , which is made of tungsten, aluminum and the like, is formed on the gate film via an interlayer insulation film, so that light does not enter the substrate area other than each light receiving section , that is, such as the gate film . The shading film is formed in such a manner to avoid being formed over each light receiving section . An interlayer insulation film is formed on the semiconductor substrate having the light receiving sections and the shading film formed thereon, for the purpose of electrical insulation.
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On the interlayer insulation film , an interlayer insulation film is further formed so that unevenness resulted from a base shape formed on the interlayer insulation film is improved and color filters are formable on the planarized surface. The insulation film is composed of a transparent acrylic material and the like in order to improve adhesiveness to the color filters and improve optical transmittance. In recent years, there is such a case where an inner microlens (inner-layer microlens) is provided between the upper interlayer insulation film and the lower interlayer insulation film for the purpose of improving condensing efficiency by the microlens .
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FIG. 1
On the planarized interlayer insulation film , a plurality of color filters , and are formed in a corresponding manner to respective light receiving sections in order to optically separate incident light. The color filters , and are of different colors. In a case of , the color filters , and are formed with primary color filters of green (G; green color), blue (B; blue color) and red (R; red color) respectively. The color filters , and for the respective colors form Bayer array, where the green color filters are alternately arranged checkerwise.
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On the color filters , and , an interlayer insulation film is formed for the purpose of protecting the color filters , and and planarizing above the surfaces of the color filters , and prior to forming microlenses . On the interlayer insulation film , microlenses are formed in a corresponding manner with respective light receiving section for the purpose of refracting incident light and condensing light that enters the shading film , which forms ineffective areas, into the light receiving section . Due to the condensing of light by the microlenses , the amount of light that enters the light receiving section increases, improving the light receiving sensitivity.
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FIGS. 2 to 7
With the structure described above, a manufacturing method of the CCD image sensor according to the embodiment of the present invention will be described hereinafter in detail with reference to .
FIGS. 2
FIGS. 3
FIGS. 2
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, and are longitudinal cross sectional views for successively explaining color filter forming steps ( to ) respectively with respect to the manufacturing method of the CCD image sensor of the embodiment. , and are plan views corresponding to respective steps of , and . Since forming steps of the light receiving section , the gate film , the shading film , the interlayer insulation films to and the microlens are the same as those in the manufacturing method of a conventionally used solid-state image capturing device, the explanation for such steps will be omitted herein, and the forming steps of the color filters , and only will be explained.
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FIGS. 2 and 3
FIG. 3
As shown in a patterning step () of a color filter ′ in , the interlayer insulation film is planarized, and subsequently, color filter material is patterned to form the color filters ′ that have not yet been heat treated. First, a photosensitive color filter material is applied on the interlayer insulation film up to a desired thickness (about 0.5 μm, for example) by a spin coating method and the like. The applied color filter material is exposed by an exposing device so that a pattern will be left only on a desired island area (checkerwise) corresponding to the position of the light receiving section in such a manner that the pattern is left on every other area (light receiving area). A developing process using a developing solution is performed to pattern a predetermined form (island from and checkerwise form in ).
Further, in order to secure reliability, a heat bridge agent, such as epoxy compounds and melanin resin, may be added to the color filter material. For example, greater than or equal to 0.1 wt percent (percent by weight) and less than or equal to 2.0 wt percent (percent by weight) of both of epoxy compounds and melanin resin can be added to the color filter material. Both of the epoxy compounds and melanin resin are used. The melanin resin is the primary heat bridge agent. Although the epoxy compounds also serve for bridge, the bridge speed can be controlled by the epoxy compounds. Such a mixture of the epoxy compounds and the melanin resin enables itself to be melted by heat and then cured.
As described above, the heat bridge agent is added to the color filter material and the color filter material is melted by heating to form a desired form. Subsequently, the color filter material is bridged not to be affected by a post-process such as heat treatment.
FIG. 2
FIG. 3
This state is shown as a cross sectional view in and as a top view in .
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FIGS. 4 and 5
Next, as shown in a patterning step () for a color filters ′ and ′ in respectively, the color filter ′ is formed on the interlayer insulation film by patterning the color filter material, and further, the color filter ′ is formed by patterning the color filter material.
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The color filter ′, whose color is different from that of the color filter ′, is patterned similarly to the case for the color filter ′. At this point, the color filter ′ is patterned such that a predetermined interval is created between the color filter ′ and the color filter ′ of a different color. The interval between the color filter ′ and the color filter ′ is set to be an interval C, taking the characteristics of the color filter materials, the melting amount of the color filter materials during the heat treatment, and the least amount of the materials needed in order not to have an interval (gap C) after melting into consideration, so that there will be no gap after melting. Further, considering the amount of the position shifting due to the alignment accuracy, the interval (gap C) is set such that the color filter and the color filter will not overlap with each other after melting.
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For example, it is preferable to set the interval between the color filter ′ and the color filter ′ in the range of greater than or equal to 0.1 μm and less than or equal to 0.2 μm (or greater than 0.1 μm and less than or equal to 0.2 μm). If the interval between the color filter materials is smaller than 0.1 μm, the alignment accuracy will be less than 0.07 μm to 0.09 μm and the color filter materials will overlap with each other before melting. In addition, if the interval between the color filter materials is greater than 0.2 μm, the interval between the color filter materials will not be filled after melting.
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FIG. 4
FIG. 5
Subsequently, the color filter ′, whose color is different from those of the color filter ′ and the color filter ′, is patterned. At this point, the interval between the color filter materials is set similarly to the positional relationship between the color filter ′ and the color filter ′. This state is shown as a cross sectional view in and as a top view in .
A material having properties of melting by heat treatment and extending in a transverse direction by gravity is used as the color filter material. For example, a pigment or dye is added to polystyrene resin, which has a low melting point, as the color filter material, resulting in the color filter material having both characteristics to be a color filter and photosensitivity function.
A resist material including greater than or equal to 1 wt percent and less than or equal to 10 wt percent (percent by weight) of polyhydroxy styrene, for example, is used as the polystyrene resin described above. In addition, greater than or equal to 1.5 wt percent and less than or equal to 3.5 wt percent of C. I. Pigment yellow 150 (color filter number) or C. I. Pigment yellow 138 (color filter number) and greater than or equal to 6.0 wt percent and less than or equal to 8.0 wt percent of C. I. Pigment green 7 (color filter number) or C. I. Pigment green 36 (color filter number) mixed with each other is used as the pigment, and a material including the pigment is used for the green color filter. Further, greater than or equal to 5.0 wt percent and less than or equal to 10.0 wt percent of C. I. Pigment red 254 (color filter number) is used as the pigment, and a material including the pigment is used for the red color filter. Further, greater than or equal to 4.0 wt percent and less than or equal to 8.0 wt percent of copper phthalocyanine blue having α, β, or ε crystalline form is used as the pigment, and a material including the pigment is used for the blue color filter. The color filter numbers are standardized numbers used for color materials of pigments, which define respective colors with their own transmissivities.
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FIGS. 6 and 7
Next, as shown in a forming step () of the color filters , and in , a heat treatment is performed for four minutes at 160 degrees Celsius using a hot plate, for example, so as to melt and fluidize the color filters , and . As a result, gaps C in between the color filters , and are completely filled, so that the color filters , and that have no gaps are formed, and they are melted, fluidized and then cured.
FIG. 6
FIG. 7
This state is shown as a cross sectional view in and as a top view in . In this case, the color filter material is cured by heat bridge.
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With the structure above, according to the embodiment of the present invention, the interval between the color filters is set to be an interval, taking the characteristic of the color filter materials and the melting amount of the color filter materials at the heat treatment into consideration, so that there will be no gap after melting. Further, considering the amount of the position shifting due to the alignment accuracy, the interval C is set such that the end portions of the color filter , and will not overlap with one another after melting. As described above, the color filters , and are patterned on the semiconductor substrate such that a predetermined interval C is set between the adjacent color filters of different colors. The color filters , and are then heat treated to fluidize and cure them. As a result, there is no need to improve the resolution or light receiving sensitivity of the color filter material or there is no need to process the base film, and the gap C or the overlapping portion D between the adjacent color filters , and can be eliminated, which enables the color filters , and having excellent light separation characteristics to be formed. Accordingly, the light receiving sensitivity characteristics of the light receiving sections can be improved.
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Although the case where the color filters , and of three primary colors (R, G, B) are provided has been described in the embodiment described above, this does not necessarily have to be the only case. The colors can also be four colors of three complementary colors (yellow, cyan, and magenta) and green, which is a primary color, besides the three primary colors (R, G, B). If signal processing methods or devices are different, any number of colors, (e.g., four or more colors) can be set.
Further, although all the color filter materials are patterned and they are then melted by heat treatment in the embodiment described above, the color filter materials may be melted by heat treatment in the middle of patterning the plurality of color filter materials in accordance with the interval between the color filter materials and the abilities of the color filter materials.
More specifically, the heat treatment can be performed on the color filter materials either individually for each color after the patterning of the color filter materials, or simultaneously for a plurality of colors after the patterning of the plurality of color filter materials, or simultaneously for two colors of the plurality of the colors. For example, when the heat treatment is performed individually for each color, the color filter materials can be melted and expanded up to a predetermined color filter area, and a predetermined interval C between this color filter material and an adjacent color filter material can be completely filled. In addition, when the heat treatment is performed simultaneously for the plurality of colors (three colors herein) as described above (or it may be two colors of the plurality of colors), all the color filter materials of a plurality of colors (three colors herein) can be simultaneously melted by heat treatment, and the area of the color filters can be expanded until the predetermined interval C is completely filled with the color filter materials.
Further, although the heat treatment is performed at 160 degrees Celsius for four minutes in the embodiment described above, the heat treatment does not necessarily have to be performed thus. The heat treatment may be performed in the temperature range of 150 degrees Celsius to 165 degrees Celsius for two to five minutes. If the temperature is lower than 150 degrees Celsius and the heating time is as short as two minutes, it is difficult for the color filter material to fluidize. If the temperature exceeds 165 degrees Celsius and the heating time exceeds five minutes, it may influence the color filter material in an undesired way.
Further, although the manufacturing method for the solid-state image capturing device has been described in the embodiment described above, the method does not necessarily have to be for the solid-state image capturing device. A forming method for the color filter used for the manufacturing method of the solid-state image capturing device may also be applied to a manufacturing method for a liquid crystal display apparatus, which has color filters formed by the forming method for the color filter and provided on one of the two substrates putting liquid crystal material inbetween. In such a case, a forming method for a color filter may include a color filter patterning step of patterning a color filter in such a manner that a predetermined interval is formed between adjacent color filter materials of different colors; and a color filter forming step of heat treating and fluidizing the patterned color filter material and further curing the color filter material to form a color filter of each color.
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Further, although not explained in the embodiment described above, an electronic information device will be described hereinafter. The electronic information device, such as a digital camera (e.g., digital video camera and digital still camera), an image input camera (e.g., monitoring camera, a door intercom camera, a car-mounted camera, a camera for television telephone and a camera for cell phone), a scanner, a facsimile machine and a camera-equipped cell phone device, has an image capturing section equipped with the CCD image sensor or a CMOS image sensor A, for which the manufacturing method for the solid-state image capturing device according to the present invention is applied, according to the embodiment of the present invention described above as an image input device.
FIG. 11
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is a block diagram showing an exemplary diagrammatic structure of an electronic information device including the solid-state image capturing apparatus including the CCD image sensor according to the embodiment of the present invention or the CMOS image sensor A for which the manufacturing method for the solid-state image capturing device according to the present invention is applied, in an image capturing section.
FIG. 11
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In , the electronic information device according to the present invention includes a solid-state image capturing apparatus for performing a variety of signal processing on an image capturing signal from the image sensor according to the embodiment or the CMOS image sensor A, for which the manufacturing method for the solid-state image capturing device according to the present invention is applied, so that high-quality color image data can be obtained. Further, the electronic information device includes: a memory section (e.g., recording media) for data-recording a high-quality color image data obtained by the solid-state image capturing apparatus after a predetermined signal process is performed on the image data for recording; a display section (e.g., liquid crystal display device) for displaying this color image data from the solid-state image capturing apparatus on a display screen (e.g., liquid crystal display screen) after a predetermined signal process is performed for display; a communication section (e.g., transmitting and receiving device) for communicating this color image data from the solid-state image capturing apparatus after a predetermined signal process is performed on the image data for communication; and an image output section for printing (typing out) and outputting (printing out) this color image data from the solid-state image capturing apparatus . Further, the electronic information device may also include at least any of: the memory section , the display section , the communication section , and the image output section (e.g., a printer), other than the solid-state image capturing apparatus . In addition, a liquid crystal display apparatus, which has color filters formed by the forming method for the color filter and provided on one of the two substrates putting liquid crystal material in between, can be applied to the display section . Further, other than the electronic information device , an electronic information device having a liquid crystal display apparatus, which has color filters formed by the forming method for the color filter and provided on one of the two substrates putting liquid crystal material in between, can be considered.
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Therefore, according to the embodiments other than the ones described above, based on color image signals from the solid-state image capturing apparatus , the electronic information device of the present invention is capable of displaying the color image signals on a display screen finely, printing out (outputting) the color image signals finely by an image output section , communicating the color image signals finely as communication data via wire or radio, storing the color image signals finely by performing a predetermined data compression process on the memory section , and performing various data processes finely.
As described above, the present invention is exemplified by the use of its preferred embodiment. However, the present invention should not be interpreted solely based on the embodiment described above. It is understood that the scope of the present invention should be interpreted solely based on the claims. It is also understood that those skilled in the art can implement equivalent scope of technology, based on the description of the present invention and common knowledge from the description of the detailed preferred embodiment of the present invention. Furthermore, it is understood that any patent, any patent application and any references cited in the present specification should be incorporated by reference in the present specification in the same manner as the contents are specifically described therein.
The present invention can be applied to the field of a solid-state image capturing device having a color filter provided above each light receiving section provided on a surface of a semiconductor substrate, and configured with semiconductor elements for performing photoelectric conversion on image light from a subject to capture an image light; the manufacturing method of the solid-state image capturing device; and an electronic information device, such as a digital camera (e.g., digital video camera and digital still camera), an image input camera, a scanner, a facsimile machine and a camera-equipped cell phone device, having the solid-state image capturing device as an image input device in an image capturing section of the electronic information device. In forming the color filters above the corresponding light receiving sections provided on the semiconductor substrate, the gaps between adjacent color filters are filled and the adjacent color filters do not overlap with each other. Therefore, it is achievable to manufacture a solid-state image capturing device having excellent light separation characteristics. Further, the color filter material used for the present invention does not have to be a material having high resolution or high sensitivity characteristic that requires improvement on micromachining technology as the cell size is reduced. Further, because the color filter material is melted and fluidized by heat treatment, the material can be processed with an equipment that only has alignment accuracy of one previous generation. Further, there is no need of processing a base interlayer insulation film in order to compensate for processing accuracy for the color filter. As a result, a man hour and manufacturing cost for the solid-state image capturing device can be significantly reduced.
Various other modifications will be apparent to and can be readily made by those skilled in the art without departing from the scope and spirit of this invention. Accordingly, it is not intended that the scope of the claims appended hereto be limited to the description as set forth herein, but rather that the claims be broadly construed.
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DESCRIPTION OF THE PREFERRED EMBODIMENTS
INDUSTRIAL APPLICABILITY
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a longitudinal cross sectional view showing an exemplary essential structure of a CCD image sensor related to one embodiment of the solid-state image capturing device according to the present invention.
FIG. 2
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is a longitudinal cross sectional view of an essential portion of a CCD image sensor for explaining a patterning step for a color filter ′ before heat treatment in a manufacturing method for a CCD image sensor according to one embodiment of the present invention.
FIG. 3
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is a plan view of an essential portion of a CCD image sensor for explaining a patterning step for a color filter ′ before heat treatment in a manufacturing method for a CCD image sensor according to one embodiment of the present invention.
FIG. 4
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is a longitudinal cross sectional view of an essential portion of a CCD image sensor for explaining a patterning step for color filters ′ and ′ before heat treatment in a manufacturing method for a CCD image sensor according to one embodiment of the present invention.
FIG. 5
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is a plan view of an essential portion of a CCD image sensor for explaining a patterning step for color filters ′ and ′ before heat treatment in a manufacturing method for a CCD image sensor according to one embodiment of the present invention.
FIG. 6
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is a longitudinal cross sectional view of an essential portion of a CCD image sensor for explaining a forming step for color filters ′, ′ and ′ after heat treatment in a manufacturing method for a CCD image sensor according to one embodiment of the present invention.
FIG. 7
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is a plan view of an essential portion of a CCD image sensor for explaining a forming step for color filters ′, ′ and ′ after heat treatment in a manufacturing method for a CCD image sensor according to one embodiment of the present invention.
FIG. 8
is a longitudinal cross sectional view showing an exemplary essential structure of a conventional solid-state image capturing device.
FIG. 9
FIG. 8
is a plan view of an essential portion of color filters for hypothetically showing a desirable forming state of color filters together with light receiving sections in the solid-state image capturing device in .
FIG. 10
FIG. 8
is a plan view of an essential portion of color filters for showing actual forming state of color filters together with light receiving sections in the solid-state image capturing device in .
FIG. 11
is a block diagram showing an exemplary diagrammatic structure of an electronic information device including the solid-state image capturing apparatus including the CCD image sensor according to the embodiment of the present invention or the CMOS image sensor for which the manufacturing method for the solid-state image capturing device according to the present invention is applied, in an image capturing section. | |
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Lessons are selected to provide support where each student needs it most, and specially-trained GCSE maths tutors adapt the pitch and pace of each lesson. This ensures a personalised revision programme that raises grades and boosts confidence. | https://thirdspacelearning.com/resources/gcse-maths/factor-trees-worksheets/ |
About Operation Theater Suite Course:
An Operation Theater Suite is the “heart” of any major surgical hospital. An operating theatre, operating room, surgery suite or a surgery center is a room within a hospital within which surgical and other operations are carried out. Through Operation Theater Suite course we are going to discuss the various considerations while planning an operation theatre, functioning of an operation theatre including scheduling, administration, staffing etc.
The aim of a well-functioning operation theatre is to provide the maximum benefit for maximum number of patients arriving to the operation theatre. Both the present as well as future needs should be kept in mind while planning. This training strives to help the individuals undertaking this course understand the same. This training will help teach the individuals to understand how to maximize operational efficiency at the facility, i.e. to maximize the number of surgical cases that can be done on a given day while minimizing the required resources and related costs.
This course aims at providing insight about the importance of Operation theatres, history if surgery and operation theatres, purpose of surgery and its various types. We will look at the various points to be considered for an efficient operation theatre plan. Further we would also consider the location consideration for an OT planning. Apart from this a lot more topics will be discussed to understand the overview of how an OT functions.
Pre-Requisites for Operation Theater Suite Course:
- Basic computer Knowledge and English Knowledge
- Passion to learn
Operation Theater Suite Course Objectives:
- At preparing learners to manage operation theatres smoothly by developing well defined Protocols and Policies of an OT.
- At exposing Learners to understand the Planning considerations and determining the ideal layout as per the requirement.
- At understanding quality concepts in regards to providing medical and nursing care in OT
Target Audience for Operation Theater Suite Course: | https://www.educba.com/course/operation-theater-suite-course/ |
Men's Wearhouse Files For Bankruptcy
As we previewed on month ago in 'Work-From-Home'-Epidemic Set To Bankrupt Suit-Sellers, "I Guarantee It", on Monday the retail wreck continued on Sunday when Tailored Brands, the owner of Men's Wearhouse filed for bankruptcy, adding to a list of brick-and-mortar retailers that have succumbed to the economic fallout from the COVID-19 crisis.
The retailer filed for Chapter 11 bankruptcy in the U.S. Bankruptcy Court for the Southern District of Texas. Tailored Brands said in a statement that it has entered into a restructuring agreement with more than 75% of its senior lenders, and that could reduce the company’s debt by at least $630 million. The company also said it has received commitments for $500 million in debtor-in-possession financing from its existing lenders. In the court filing, the company listed both its assets and liabilities in the range of $1 billion to $10 billion.
The Houston, Texas-based retailer, which was already struggling with competition from fast-fashion brands and a shift to online shopping before the pandemic, said it will continue to build on its previously announced plans to reduce its corporate workforce by 20% and shut as many as 500 stores.
The company’s four retail brands, including Moores Clothing for Men and K&G Fashion Superstore, will continue to operate through the process. It employs 18,000 workers and operates 1,274 retail and apparel rental stores in the U.S. and 125 in Canada, according to court documents. | http://www.whatreallyhappened.com/zh-hans/node/865408 |
Born in: Winchester, Virginia, U.S.
Famous as: Actor
Height: 5'7" (170 cm), 5'7" Males
Spouse/Ex-: Madeleine Stowe (m. 1982)
father: Peter Michael Benben Sr.
mother: Gloria Patricia
children: May Theodora Benben
U.S. State: Virginia
Recommended For You
Brian Benben is an American actor who is best known for his role in the television series ‘Dream On.’ Owning to his rugged good looks and serious demeanor, he often portrays doctors in television shows. Benben has worked in several television shows and films since his acting debut as a stage actor in off-off-Broadway productions. After establishing himself as a brilliant stage actor with performances in plays like ‘Wild Oats’ and ‘The Tooth of Crime,’ he later joined the television industry. His television acting career started way back in the early 1980s and he has several years of experience under his belt. He has done many important and challenging roles in his career which earned him a lot of praise. Benben’s popular works include the series ‘Dream On’, ‘Private Practice’ and ‘Kingpin’. Coming to his films, he played a supporting role in the science fiction action film ‘Dark Angel’ in which he portrayed the character of Special Agent Arwood 'Larry' Smith.
Recommended Lists:
- Brian Benben started his professional career at a young age as a stage actor. He appeared in several theater shows including ‘Wild Oats’, ‘The Overcoat’ ‘The Tooth of Crime’, ‘A Moon for the Misbegotten’, ‘Gossip’, and ‘A Midsummer Night's Dream’. After learning his trade in the theater, Brian moved into television and appeared in a very challenging role, portraying the fictionalized version of the real-life mobster Meyer Lansky as ‘Michael Lasker’ in the popular television show ‘The Gangster Chronicles’.His portrayal of the notorious character earned him a lot of critical praise and gave him the first taste of some serious popularity. Brian Benben’s other challenging roles include playing a homosexual lover in the TV-movie drama ‘American Playhouse: Family Business’ in 1983 and portraying the controversial character of California senator, Tom Hayden, who was convicted along with several others on the charges of inciting riots that disrupted the 1968 Democratic National Convention, in ‘Conspiracy: The Trial of the Chicago 8’.His versatile acting capabilities landed him many roles on television. He portrayed the main character of ‘Dr. Mark Doyle’ in the show ‘Kay O'Brien’ in 1986. He then portrayed the lead character of ‘Martin Tupper’ in the HBO television series ‘Dream On’ between 1990 and 1996 and played ‘Dr. Heywood Klein’ in the 2003 series ‘Kingpin’.In 2008, he accepted the lead role of ‘Dr. Sheldon Wallace’ in ‘Private Practice’. He appeared in the second and third season in a recurring role and later emerged as a lead character in the fourth season. Between 1981 and 1994, Brian Benben also appeared in some movies including ‘Gangster Wars’, ‘Clean and Sober’, ‘Dark Angel’, and ‘Radioland Murders’.Recommended Lists: | https://www.thefamouspeople.com/profiles/brian-benben-34825.php |
Xiaomi Mi 11X Pro has been launched officially. The smartphone comes packed with a 6.67 inches (16.94 cm)AMOLED Display and offers 1080 x 2400 pixels screen resolution. It is powered by Qualcomm Snapdragon 888 processor. It is backed by a 4520 mAh battery. The phone runs on Android v11. On the camera front, the smartphone comes with a Single camera setup consisting of a 108MP + 8MP + 5MP primary sensor. On the front, the smartphone includes a 20 MP selfie shooter. The Xiaomi Mi 11X Pro comes with up to 8 GB RAM and 128 GB of internal storage,2G,3G,4G,5G LTE. Finger Print sensor Side. and USB Type-C port. | https://www.bgr.in/gadgets/mobiles/xiaomi-mi-11x-pro-price-in-india-1048790/ |
Early Spring!
This article is Sponsored by:
Dear Janet,
I'm starting to divide my perennials - wow, how they grow. Do I throw out the center of the sedum when I divide it?
I'm digging up perennials that have gotten too big and splitting them into smaller pieces but I wonder if there are others I should divide even earlier to keep them in the best shape.
M.G.
Dear M.G.,
Perennials bloom best and are most resistant to disease and insect problems when they are not crowded. So it's a good idea to divide whenever a plant reaches the maximum size your garden allows.
Although you can divide whenever you have the time to do it, April is an excellent time for this job, and so is September. At those times we can be fairly certain that replanted pieces will have at least a month of cool air and warm soil, perfect conditions for establishing new roots.
As a general rule, divide perennials with running roots every two years, and throw out the oldest, center parts. Bee balm, some artemisias and yarrows are in this group.
Divide any clump--forming perennial such as tall sedum 'Autumn Joy' when you see it is producing smaller stems at its center than on the edges. Those central, crowded stems bear fewer, smaller flowers, are more susceptible to pest problems and often need help standing up. Take a look at your hostas, daylilies, Siberian irises and ornamental grasses now as new growth starts and you will see that even those still in tight clumps need division to free up the central stems.
Divide every plant you lift into at least four pieces. Break off the oldest, center bit of each piece and compost it. Re-set only one division in the original space. Give the rest to friends, start new gardens with them, or put them on the compost.
Check around for disposal sites for emerald ash-borer infested wood.
Be sure to call around locally to look for a site to dispose of emerald ash-borer infested wood. These infested trees should not leave your local area. Search on line to get ideas of places close to you. If you wish to dispose of them yourself then find a professional to talk you through the steps to be sure that the infested would does not contribte to harming other trees.
"For shame!" To those tree care companies...
...who are trying to cash in on the concern of ash tree owners by claiming to have an "Emerald Ash Borer Solution." Although such companies can defend their ads as true because their treatments involve a water-soluble insecticide as a possible preventive to EAB infestation, it is most certainly not a cure for already infested trees or an answer to the problem as a whole.
If those companies are the experts they claim to be and someone you should work with on your trees, they will honestly admit at least three things:
- That there is no saving an ash tree already showing significant damage.
- That insecticides cannot reach borers already inside an ash tree.
- That insecticide treatments aimed at keeping the borers out of a tree not yet infested cannot be guaranteed and must be repeated every year.
If they are thinking of the long term interests of their clients within the six-county EAB quarantine area they will recommend that you remove infested ash trees as soon as possible. They will also urge their clients to start thinking about planting replacement trees near still-healthy ashes, because they have heard and understood Forestry Department, DNR and Department of Agriculture experts who project that every ash tree within that area will probably become infested during the next three years.
to more than 120 volunteers who have spent three days learning about ash trees and emerald ash borers, and being tested to become EAB Town Crier and Scouts. They did all this so they can volunteer many more days over the next two years to walk their communities and provide information directly to ash tree owners. I am so proud to be working with all of you!
to those who use the words "done" and "garden" in the same sentence, as in "When will this garden of yours be done?" You don't understand the concept, because a garden is never "done." It is a living, changing thing and all the more beautiful because of those characteristics. Those in the know, like W.S. say, "A garden is a process, not a product, and I intend to keep on growing all my life!" | https://gardenatoz.org/whats-up/ensemble-editions/early-spring/grow-510 |
The keyboard on the iPhone is split into several different modes, including letters, numbers and symbols. There are different functional keys on each keyboard mode, including a shift key for the letters. You may already know that you can press the shift key then a letter key to capitalize that letter.
But this can be tedious if you are typing a lot of capital letters, so you might be looking for a way to turn on caps lock so that you can type in all capital letters. Fortunately this is an option on the device, and our guide below will show you how to use it.
How to Type in All Capital Letters on the iPhone 6 Plus
These steps were performed on an iPhone 6 Plus in iOS 8.1.2, but these steps will also work with other iPhone models and in other versions of iOS.
You can read more about the iOS 8 keyboard here.
Step 1: Open an app that uses the keyboard, such as Messages.
Step 2: Double-tap the up arrow at the left side of the keyboard. You will know that caps lock is enabled when there is a horizontal line under the arrow, as in the image below.
You will then be able to type in all capital letters until you either touch the up arrow button again, or switch to the number keyboard.
Have you seen people with word suggestions above their iPhone keyboard and wondered how you could have that too? This article will show you how to enable word suggestions in iOS 8.
Matthew Burleigh has been writing tech tutorials since 2008. His writing has appeared on dozens of different websites and been read over 50 million times.
After receiving his Bachelor’s and Master’s degrees in Computer Science he spent several years working in IT management for small businesses. However, he now works full time writing content online and creating websites.
His main writing topics include iPhones, Microsoft Office, Google Apps, Android, and Photoshop, but he has also written about many other tech topics as well.
Disclaimer: Most of the pages on the internet include affiliate links, including some on this site. | https://www.solveyourtech.com/use-caps-lock-iphone/ |
What is a 'Discounted Cash Flow (DCF)' and how do the Calculations Work?
A discounted cash flow (DCF) is a valuation method used to estimate the attractiveness of an investment opportunity. DCF analysis uses future free cash flow projections and discounts them to arrive at a present value estimate, which is used to evaluate the potential for investment. If the value arrived at through DCF analysis is higher than the current cost of the investment, the opportunity may be a good one.
In simple terms; future predicted incomes and drawing backwards to figure out what a current value is. While the calculations involved are complex, the purpose of DCF analysis is simply to estimate the money an investor would receive from an investment, adjusted for the time value of money.
Calculated as:
DCF is also known as the Discounted Cash Flows Model.
So what does that formula mean? Let’s break it down.
The time value of money is the assumption that a dollar today is worth more than a dollar tomorrow. For example, assuming 5% annual interest, $1.00 in a savings account will be worth $1.05 in a year. How DCF works in this example: we must consider $1.05 a year from now to be worth $1.00 today. When it comes to assessing the future value of investments, it is common to use the weighted average cost of capital (WACC) (This will be a separate article) as the discount rate.
For a hypothetical Company X, we would apply DCF analysis by first estimating the firm's future cash flow growth. We would start by determining the company's trailing twelve month (ttm) free cash flow (FCF), equal to that period's operating cash flow minus capital expenditures.
Say that Company X's ttm FCF is 50 Milllion. We would compare this figure to previous years' cash flows in order to estimate a rate of growth. It is also important to consider the source of this growth. Are sales increasing? Are costs declining? These factors will inform assessments of the growth rate's sustainability.
Say that you estimate that Company X's cash flow will grow by 10% in the first two years, then 5% in the following three. After a few years, you may apply a long-term cash flow growth rate, representing an assumption of annual growth from that point on.
For this example, we will say that Company X's is 3%. You will then calculate a WACC; say it comes out to 8%. The terminal value, or long-term valuation the company's growth approaches, is calculated using the Gordon Growth Model (Another whole article):
Terminal value = projected cash flow for final year (1 + long-term growth rate) / (discount rate - long-term growth rate)
Now you can estimate the cash flow for each period, including the terminal value:
Year 1 = 50 * 1.10 55
Year 2 = 55 * 1.10 60.5
Year 3 = 60.5 * 1.05 63.53
Year 4 = 63.53 * 1.05 66.70
Year 5 = 66.70 * 1.05 70.04
Terminal value = 70.04 (1.03) / (0.08 - 0.03) 1,442.75
Finally, to calculate Company X's discounted cash flow, you add each of these projected cash flows, adjusting them for present value using the WACC:
DCFCompany X = (55 / 1.081) + (60.5 / 1.082) + (63.53 / 1.083) + (66.70 / 1.084) + (70.04 / 1.085) + (1,442.75 / 1.085) = 1231.83
Have I lost you? And this is the easy version! Feel free to ask me more and I’m happy to go into detail.
$1.23 billion is our estimate of Company X's present enterprise value. If the company has net debt, this needs to be subtracted. The result is an estimate of the company's fair equity value. If our estimate is higher than the current asking price, we might consider Company X a good investment.
Discounted cash flow models are powerful, but they are only as good as their imports. The key is the effectiveness of the information going into the equation. Small changes in inputs can result in large changes in the estimated value of a company, and every assumption has the potential to erode the estimate's accuracy. | http://bestbuyproperties.ca/What%20is%20a%20Discounted%20Cash%20Flow%20and%20how%20do%20the%20Calculations%20Work.html |
From April 2023, the Corporation Tax rate will rise for companies with profits of more than £50,000, following the Chancellor’s announcement at his Spring 2021 Budget.
However, the new higher rate of Corporation Tax will not be the same for all companies and will instead be tied to their profits.
Companies generating profits of £250,000 or more will see their Corporation Tax rates rise from the current 19 per cent to 25 per cent.
Meanwhile, those with profits between the £50,000 and £250,000 thresholds will receive marginal relief, which means that their effective rate of Corporation Tax will increase with their profits to a maximum of 25 per cent.
The marginal relief fraction is set at 3/200. The amount of marginal relief is found by multiplying the fraction by the difference between the company’s profits and the upper profits limit of £250,000.
For example, if a company has taxable profits of £100,000, they would be entitled to marginal relief of £2,250 (3/200 x (£250,000 – £100,000)). This means that in this example, marginal relief gives an effective rate of Corporation Tax of 22.75 per cent.
The new Corporation Tax thresholds are adjusted for companies with accounting periods that are shorter than 12 months and where a company has associated companies.
Companies with profits of less than £50,000 will continue to pay Corporation Tax at 19 per cent under the new small profits rate (SPR).
The reforms are complex and require careful calculations based on various criteria.
Given that the rate of tax a business pays will be based on their profits, there may be new opportunities to minimise liabilities through careful tax planning, but it is important that strategies are put in place well in advance of the new rates being launched. | https://www.appleaccounting.co.uk/time-to-prepare-for-corporation-tax-changes/ |
---
abstract: 'We prove that, for $n=3$ and 4, the minimal nonabelian finite factor group of the outer automorphism group ${\rm Out} \, F_n$ of a free group of rank $n$ is the linear group ${\rm PSL}_n(\Z_2)$ (conjecturally, this may remain true for arbitrary rank $n > 2$). We also discuss some computational results on low index subgroups of ${\rm Aut} \, F_n$ and ${\rm Out} \, F_n$, for $n = 3$ and 4, using presentations of these groups.'
author:
- 'Mattia Mecchia and Bruno P. Zimmermann'
title: On minimal finite factor groups of outer automorphism groups of free groups
---
Introduction
============
It is shown in [@Z] that the minimal nontrivial finite quotient (nontrivial factor group of smallest possible order) of the mapping class group $\M_g$ of a closed orientable surface of genus $g$ is the symplectic group ${\rm PSp}_{2g}(\Z_2)$, for $g = 3$ and 4; in fact, this may remain true for arbitrary genus $g \ge 3$. Since $\M_g$ is perfect for $g \ge 3$, such a minimal nontrivial finite quotient is a nonabelian simple group. There are canonical projections onto symplectic groups $$\M_g \to {\rm Sp}_{2g}(\Bbb Z) \to {\rm Sp}_{2g}(\Z_p)
\to {\rm PSp}_{2g}(\Z_p),$$ and the projective symplectic groups ${\rm
PSp}_{2g}(\Z_p)$ are simple if $p$ is prime and $g \ge 3$. It is a consequence of the congruence subgroup property for the symplectic groups ${\rm Sp}_{2g}(\Bbb Z)$ that their finite simple quotients are exactly the finite projective symplectic groups ${\rm PSp}_{2g}(\Z_p)$ (see [@Z]). But also for the mapping class groups $\M_g$, all known finite quotients seem to be strongly connected to the symplectic groups, and it would be interesting to know what other finite simple groups can occur (see [@T] for a computational approach for genus two and three, and [@MR] for comments on the congruence subgroup property).
In the present note, we consider the outer automorphism group ${\rm Out} \,
F_n$ of a free group $F_n$ of rank $n$. It is well-known that ${\rm Out} \, F_2
\cong {\rm GL}_2(\Z) \cong \D_{12} *_{\D_4}\D_8$ (a free product with amalgamation of two dihedral groups of orders 12 and 8), and we shall assume in the following that $n \ge 3$; then the abelianization of ${\rm Out} \, F_n$ has order two. There is a canonical projection of ${\rm Out} \, F_n$ onto ${\rm
GL}_n(\Z)$, and we consider also the preimage of ${\rm SL}_n(\Z)$ in ${\rm
Out} \, F_n$ which we denote by ${\rm SOut} \, F_n$ (the unique subgroup of index two of ${\rm Out} \, F_n$). It is well-known that ${\rm SOut} \, F_n$ is a perfect group (see [@Ge] for a presentation), so the minimal nontrivial quotient will be again a nonabelian simple group. There are projections $${\rm SOut} \, F_n \to {\rm SL}_n(\Bbb Z) \to {\rm SL}_n(\Z_p) \to {\rm
PSL}_n(\Z_p),$$
and the finite linear groups ${\rm PSL}_n(\Z_p)$ are simple if $p$ is prime. It is a consequence of the congruence subgroup property for the linear group ${\rm SL}_n(\Z)$ that the finite simple quotients of ${\rm SL}_n(\Z)$ are exactly the finite projective linear groups ${\rm PSL}_n(\Z_p)$, $p$ prime ([@Z]).
Our main result is the following:
[**Proposition.**]{} [*For $n=3$ and 4, the minimal nontrivial finite quotient of ${\rm SOut} \, F_n$, and also the minimal nonabelian finite quotient of ${\rm Out}
\, F_n$, is the linear group ${\rm PSL}_n(\Z_2)$.*]{}
We note that ${\rm PSL}_3(\Z_2) \cong {\rm PSL}_2(\Z_7)$ is the unique simple group of order 168, and that ${\rm PSL}_4(\Z_2)$ is isomorphic to the alternating group $\A_8$, of order 20160. Conjecturally, the Proposition remains true for arbitrary $n \ge 3$. Concerning other finite simple groups, it is shown in [@Gi] that infinitely many alternating groups occur as quotients of ${\rm Out} \, F_n$.
As a consequence of the Proposition we have also the following:
[**Corollary.**]{} [*The minimal index of a proper subgroup of ${\rm SOut}
\, F_4$, and also of a proper subgroup of ${\rm Out} \, F_4$ different from ${\rm SOut} \, F_4$, is eight (the minimal index of a proper subgroup of ${\rm PSL}_4(\Z_2) \cong \A_8$).*]{}
For $n = 3$ this minimal index should be seven (the minimal index of a proper subgroup of ${\rm PSL}_3(\Z_2) \cong {\rm PSL}_2(\Z_7)$) but for the moment we cannot exclude index six by the present methods; we verified this, however, by computational methods (GAP), see section 3 for some comments.
Proof of the Proposition and the Corollary
==========================================
We denote by ${\rm Aut} \, F_n$ the automorphism group of the free group $F_n$ and by ${\rm SAut} \, F_n$ its subgroup of index two which is the preimage of ${\rm SL}_n(\Z)$ under the canonical projection of ${\rm Aut} \, F_n$ onto ${\rm GL}_n(\Z)$. Fixing a free generating set of $F_n$, inversions and permutations of generators generate a subgroup (Weyl group) $W_n \cong (\Z_2)^n \rtimes \S_n$ of ${\rm Aut} \, F_n$; let $SW_n$ denote $W_n \cap
\, {\rm SAut} \, F_n$, with $SW_n \cong (\Z_2)^{n-1} \rtimes \S_n$.
We note that, by results in [@WZ], $W_n$ is the finite subgroup of maximal possible order of both ${\rm Aut} \, F_n$ and ${\rm Out} \, F_n$, for $n \ge 3$, unique up to conjugation if $n > 3$ (for $n=3$ there is one other subgroup of maximal possible order 48).
Let $\Delta$ denote the central element of $W_n$ inverting all generators; note that $\Delta$ is in $SW_n$ if and only if $n$ is even.
The proof of the Proposition is based on the following:
[**Lemma**]{} ([@BV Prop. 3.1]).
*Let $n \ge 3$ and $\phi$ be a homomorphism from ${\rm SAut} \, F_n$ to a group $G$. If the restriction of $\phi$ to $SW_n$ has nontrivial kernel $K$ then one of the following holds:*
i) $n$ is even, $K = \langle \Delta \rangle$ and $\phi$ factors through ${\rm PSL}_n(\Z)$;
ii) $K$ is the intersection of $SW_n$ with the subgroup $(\Z_2)^n$ of $W_n$ generated by all inversions and the image of $\phi$ is isomorphic to ${\rm PSL}_n(\Z_2)$, or
iii) $\phi$ is the trivial map.
[*Proof of the Proposition.*]{} We consider the case $n = 3$ first. Since ${\rm SOut} \, F_3$ is perfect, a minimal nontrivial finite quotient of ${\rm
SOut} \, F_3$ is a nonabelian simple group. The only nonabelian simple group with an order smaller than the order 168 of the linear group ${\rm PSL}_3(
\Z_2) \cong {\rm PSL}_2(\Z_7)$ is the alternating group $\A_5$, of order 60 (see [@C] for information about the finite simple groups). Since the order 24 of $SW_3$ does not divide the order of $\A_5$, by the Lemma every homomorphism from ${\rm SOut} \, F_3$ to $\A_5$ is trivial, hence the minimal possibility for a nontrivial finite quotient of ${\rm SOut} \, F_3$ is the linear group ${\rm PSL}_3(\Z_2) \cong {\rm PSL}_2(\Z_7)$, the unique simple group of order 168.
As for ${\rm Out} \, F_3$, if the finite nonabelian group $G$ is a quotient of ${\rm Out} \, F_3$ then the image of ${\rm SOut} \, F_3$ has index one or two in $G$ and order at least 168. Since ${\rm Out} \, F_3$ surjects onto ${\rm PSL}_3(\Z_2) = {\rm PGL}_3( \Z_2)$, of order 168, this is again the minimal possibility for $G$.
We come now to the proof of the Proposition for $n=4$. Suppose that $\phi: {\rm SOut}
\, F_4 \to G$ is a nontrivial homomorphism onto a finite simple group $G$ of order less than the order 20160 of ${\rm PSL}_4( \Z_2) \cong \A_8$. The simple groups of order less than 20160 are the following (see [@C]):
- the alternating groups $\A_d$ of degrees $d=5, 6$ or 7;
- the linear groups ${\rm PSL}_2(\Z_p) = {\rm L}_2(p)$, for the primes $p=7$, 11, 13, 17, 19, 23, 29 and 31;
- the linear groups ${\rm PSL}_2(q) = {\rm L}_2(q)$, for the prime powers $q=8$, 9, 16, 25 and 27 (over the finite fields of the corresponding orders);
- the linear group ${\rm PSL}_3( \Z_3) = {\rm L}_3(3)$;
- the unitary group ${\rm PSU}_3( \Z_3) = {\rm U}_3(3)$;
- the Mathieu group ${\rm M}_{11}$.
Since the order of none of these groups is divided by the order $2^6 \cdot 3$ of $SW_4$, the restriction of $\phi$ to $SW_4$ has nontrivial kernel and the Lemma applies. Since the cases ii) and iii) of the Lemma are excluded by the hypotheses, we are necessarily in case i). Now also case i) may be excluded by appealing to the fact that the finite quotients of ${\rm
PSL}_4(\Z)$ are exactly the groups ${\rm PSL}_4(\Z_p)$, $p$ prime ([@Z Theorem 1]); however also the following more direct argument, in the spirit of the previous ones, applies. In case i) of the Lemma, the kernel of the restriction of $\phi$ to $SW_4$ is the subgroup of order two generated by the central involution $\Delta$ of $SW_4$. Hence the factor group $SW_4/\langle
\Delta \rangle$, of order $2^5 \cdot 3$, embeds into $G$. However none of the above groups has such a subgroup: considering orders again, the only candidates which remain are ${\rm U}_3(3)$ and ${\rm L}_2(31)$, but both do not have a subgroup isomorphic to $SW_4/\langle \Delta \rangle$.
There is one other simple group of order 20160 not isomorphic to ${\rm
PSL}_4(\Z_2) \cong \A_8$, the linear group ${\rm PSL}_3(4) = {\rm L}_3(4)$. Considering the maximal subgroups of ${\rm L}_3(4)$, it is easy to see that $G = {\rm L}_3(4)$ has no subgroup isomorphic to $SW_4$, so the Lemma applies again leaving us with case i); finally, also case i) is excluded since ${\rm L}_3(4)$ has no subgroup isomorphic to $SW_4/\langle \Delta \rangle$ (or by appealing again to [@Z Theorem 1]).
This completes the proof of the Proposition for ${\rm SOut} \, F_4$, and hence also for ${\rm Out} \, F_4$.
[*Proof of the Corollary.*]{} Both ${\rm SOut} \, F_4$ and ${\rm Out} \, F_4$ have subgroups of index eight since both admit surjections onto the alternating group $\A_8 \cong {\rm PSL}_4(\Z_2)$. By the proof of the Proposition, ${\rm
SOut} \, F_4$ does not admit a nontrivial homomorphism to an alternating groups $\A_d$ of degree $d < 8$, hence ${\rm SOut} \, F_4$ has no proper subgroup of index less than eight. Similarly, the same holds for ${\rm Out} \, F_4$ which does not admit a nontrivial homomorphism to a symmetric group $\S_d$ of degree $d < 8$.
Comments on computations
========================
We employed the low index subgroup procedure of the computer algebra system GAP in order to find the smallest indices of proper subgroups of ${\rm SAut} \,
F_3$ and ${\rm SOut} \, F_3$. We used the 4-generator presentation of ${\rm
Aut} \, F_3$ in [@MaKS section 3.5, Corollary N1] (note that some of the relations apply only for $n > 3$; see also [@CM section 7.3], and [@MC], [@NN section 6], section 6\] for presentations of ${\rm Out} \, F_3$), and created by GAP the unique subgroups ${\rm SAut} \, F_3$ and ${\rm SOut} \, F_3$ of index two. We found that the three smallest indices of a proper subgroup of ${\rm SAut} \,
F_3$ and ${\rm SOut} \, F_3$ are 7, 8 and 13; the factor groups of the cores of the corresponding subgroups (the largest normal subgroup contained in a subgroup) are ${\rm PSL}_3(\Z_2) = {\rm GL}_3(\Z_2)$ for indices 7 and 8, a semidirect product $(\Z_2)^3 \rtimes {\rm GL}_3(\Z_2)$ for index 8, and ${\rm
PSL}_3(\Z_3)$ for index 13.
Remark. Concerning the smallest indices of proper subgroups of ${\rm SL}_3(\Bbb
Z)$, index 8 is now missing and there remain only the indices 7 and 13 (see [@MaKS section 3.5] for a presentation of ${\rm GL}_r(\Bbb Z)$). Computing the abelianization of the core of the index 8 subgroup of ${\rm SOut} \, F_3$ (a normal subgroup of index 1344, with quotient $(\Z_2)^3 \rtimes {\rm
GL}_3(\Z_2)$), we found the free abelian group $\Bbb Z^{14}$; so this gives an explicit example of a finite index subgroup of ${\rm
SOut} \, F_3$ with infinite abelianization. We note that the existence of such subgroups was known since by [@MC], ${\rm Out} \, F_3$ is virtually residually torsion-free nilpotent; on the other hand, no such examples seem to be known for rank $n > 3$ (see [@T] for the corresponding situation for mapping class groups).
We then went on to compute that the three minimal indices of subgroups of ${\rm SAut} \, F_4$ are 8, 15 and 16; the factor groups of the cores of the corresponding subgroups are ${\rm PSL}_4(\Z_2) = {\rm
GL}_4(\Z_2) \cong \A_8$ for indices 8 and 15 and, for index 16, a semidirect product $(\Z_2)^4 \rtimes {\rm GL}_4(\Z_2)$ of order 322560; at present we don’t know if the abelianization of the core in the case of index 16 is finite or infinite. On the other hand, the minimal indices of subgroups of ${\rm SOut} \, F_4$ are 8 and 15, and index 16 is now missing. We note that each ${\rm SAut} \, F_r$ admits a surjection onto a finite group with a normal subgroup $(\Z_n)^r$ and factor group ${\rm GL}_r(\Bbb Z_n)$ (by dividing out first the kernel of the natural projection of ${\rm Inn} \, F_r \cong F_r$ onto $(\Z_n)^r$, then projecting onto ${\rm GL}_r(\Bbb Z_n)$).
Employing in addition the quotient group procedure of GAP, we verified also that the three smallest simple factor groups of ${\rm SAut} \, F_3$ and ${\rm
SOut} \, F_3$ are, as expected, the groups ${\rm PSL}_3(\Z_2)$ of order 168, ${\rm PSL}_3(\Z_3)$ of order 5616, and ${\rm PSL}_3(\Z_5)$ of order 372000 (we note e.g. that, by the Lemma, the first Janko group does not occur since it has no subgroup isomorphic to $SW_3 \cong S_4$).
We note that these computations can be slightly extended but that already the suspected minimal index 31 of a proper subgroup of ${\rm SOut} \, F_5$ as well as its suspected minimal nonabelian quotient ${\rm PSL}_5(\Z_2)$ (of order 9.999.360, with a subgroup of index 31) appear quite large for such computations so we didn’t pursue this further.
[99]{}
M.R. Bridson, K. Vogtmann, [*Actions of automorphism groups of free groups on homology spheres and acyclic manifolds.*]{} arXiv:0803.2062
J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, [*Atlas of Finite Groups.*]{} Oxford University Press 1985
H.S.M. Coxeter, W.O.J. Moser, [*Generators and Relations for Discrete Groups.*]{} Fourth edition Springer 1980
S.M. Gersten, [*A presentation for the special automorphism group of a free group.*]{} J. Pure Appl. Algebra 33, 269 - 279 (1984)
R. Gilman, [*Finite quotients of the automorphism group of a free group.*]{} Can. J. Math. 29, 541 - 551 (1977)
W. Magnus, A. Karrass, D. Solitar, [*Combinatorial Group Theory.*]{} Second revised edition Dover 1976
J. McCool, [*A faithful polynomial representation of ${\rm Out}(F_3)$.*]{} Math. Proc. Camb. Phil. Soc. 106, 207 - 213 (1989)
D.B. McReynolds, [*The congruence subgroup problem for braid groups: Thurston’s proof.*]{} arXiv:0901.4663
B.H. Neumann, H. Neumann, [*Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen.*]{} Math. Nachr. 4, 106 - 125 (1951)
F. Taherkhani, [*The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups.*]{} Experimental Math. 9, 261 - 274 (2000)
S. Wang, B. Zimmermann, [*The maximum order finite groups of outer automorphisms of free groups.*]{} Math. Z. 216, 83-87 (1994)
B. Zimmermann, [*On minimal finite quotients of mapping class groups.*]{} arXiv:0803.3244 (to appear in Rocky Mountain J. Math.)
| |
First we realize that odd gears rotate clockwise and even gears rotate counter-clockwise. If i is odd we change the bi value: bi :=(ai-bi) mod ai. From now we can assume that every gear rotates clockwise at the speed 1 tooth per second.
Let d:=GCD(a1, a2), GCD stands for greatest common divisor. If b2-b1 is not divisible by d then the solution doesn't exist. Else we find such n1 and n2 that a1*n1- a2*n2=d. These numbers can be found using the extended Euclidean algorithm.
Now let l:=(b2-b1)/d. Then a1*l*n1- a2*l*n2= b2-b1 is a solution, but not necessary the best. Let h:=LCM(a1, a2) -- this is the first nonzero time when the states of both gears are 0 (LCM is least common multiple). So this is the period of the movement of the two gears. Therefore a1*l* n1- a2*l* n2 +k*h-k*h =a1*(l*n1+ k*h/a1)- a2* (l*n2+k*h/a2) is also a solution.
But we want to find the smallest time when gears meet, so l*n1+k*h/a1 should be as small as possible. Let m1:=l*n1+ k*h/a1 and a1*m1+ b1 is the first time when gears meet. We know that a1*m1+ b1 is less than h= LCM(a1, a2) and the gears met only once until time h. Clearly m1 is nonnegative and less than h/1, so m1 :=(l*n1) mod (h/a1).
The first two gears meet at LCM(a1, a2)*n + a1* m1+ b1 for every nonnegative integer n. The third gear has to satisfy equation x=a3*n3+ b3=LCM(a1, a2)*n + a1* m1+ b1. We can use the same algorithm to find minimum n and n3 as before.
So after we have find the time when first k gears meet we can find the time when first k+1 gears meet. For the first k gears the time is a*n+b and for the k+1-th gear the time is ak+1*nk+1+bk+1. We can use the same algorithm as before to find minimal n and nk+1. So complexity of our algorithm is O(N*M), because for each gear we calculate some numbers. Euclidean algorithm works in O(M) where M is number of digits of numbers. | https://ipsc.ksp.sk/2005/real/solutions/g.html |
The Boston Redevelopment Authority’s board of directors approved seven new projects at this month's meeting. In total, the projects represent a combined investment of $258.5 million and will generate over 530 construction jobs, yielding 586 new residential units in the city.
Below are summaries of the newly approved projects.
Address: 89 Waumbeck Street
Neighborhood: Roxbury
Land Sq. Feet: 44,000 sq ft
Building Size: 34,000 sq ft
Uses: Residential
Residential Units: 15
The sites will be developed as part of Mayor Walsh’s Neighborhood Homes Initiative (NHI), a product of Housing A Changing City: Boston 2030, the Walsh administration’s comprehensive housing plan. NHI uses City-owned land and funding as a resource to create affordable and mixed-income homeownership opportunities for a range of homebuyers. In turn, this project will feature 18 residential units, 12 of which will be designated as affordable.
Upon completion, the new homes will be priced to attract a mix of moderate, middle and market rate buyers. Prices for the new affordable homes will range from $250,000 - $400,000 and will be affordable to households with a combined income of $60,000 - $100,000. The affordable moderate and middle-income homes will have a 50-year resale restriction to provide affordability for future generations of homebuyers.
Address: 232 Old Colony Avenue
Neighborhood: South Boston
Land Sq. Feet: 9,383 sq ft
Building Size: 37,384 sq ft
Uses: Ownership / Residential / Retail
This project site is located approximately five blocks northeast of Andrew Square, within walking distance of the MBTA Red Line Andrew Station. The developer, 232 Old Colony Avenue LP, has proposed to construct 24 condominium units and 3 on-site affordable units, commercial/retail restaurant space and 29 parking spaces.
The project will include a mix of 16 two-bedroom units and eight one-bedroom units within a 5-story structure. A 2,855 square foot restaurant space will also be included on the ground floor.
Address: 201 South Huntington Avenue
Neighborhood: Jamaica Plain
Land Sq. Feet: 84,626 sq ft
Building Size: 169,000 sq ft
The BRA Board’s approval will allow the 167-unit project to move forward as planned. Of the 167 residential units, 110 will be located in the renovated Goddard House with 57 units in the new building and 22 of the total units on-site will be designated affordable.
In addition to the renovation of Goddard House and the construction of a new building, this project will include 180 spaces allotted toward indoor and outdoor bicycle parking, a combination of covered and surface vehicle parking to the tune of 83 spaces, and open space improvements that will include a landscape buffer in front of the building with new street trees to improve the pedestrian realm.
Neighborhood: Mission Hill
Land Sq. Feet: 12,000 sq ft
Building Size: 45,500 sq ft
Uses: Residential
Walter Huntington, LLC received approval for the development of a new five- and six-story mixed-use building located at 35 South Huntington Avenue. The building will be comprised of 38 residential units with 5 IDP units, 7,080 square feet of ground-floor commercial/ retail restaurant space, and 26 parking spaces located in the building’s underground garage.
Address: 125 Guest Street
Neighborhood: Brighton
Land Sq. Feet: 81,665 sq ft
Building Size: 311,000 sq ft
Uses: Rental / Residential / Retail
The approval of this primarily residential building will yield the construction of 295 rental units, 38 of which will be on-site IDP units. In order to meet a variety of market demands, 125 Guest Street will feature studio, one-, two-, and three-bedroom units. This phase of the project will also include 155 parking spaces and 300 bicycle spaces dedicated to residents.
The ground-floor of the building will provide 16,000 square feet of retail space, while the remaining area of approximately 39,000 square feet will be residential amenity spaces, such as a building lobby, mail room, fitness room, conference room, loading dock, mechanical, vehicle, and bike parking.
Address: 150 Camden St
Neighborhood: Roxbury
Land Sq. Feet: 15,500 sq ft
Building Size: 49,305 sq ft
Uses: Rental / Residential
The Board’s approval of the Amended and Restated Development Plan for the Douglass Park expansion project will result in 44 multifamily residential rental units with 8 IDP units. The project site, located at 150 Camden Street in the South End, is on a portion of Parcel 16.
During the public testimony portion of the hearing, one community-member noted that in its earliest stages, the project was met with skepticism, but as it has moved forward, residents and stakeholders have come to embrace the opportunities that this development will bring to their neighborhood.
Harvard University Life Lab- Second Ammendment
Neighborhood: Allston
Land Sq. Feet: 93,794 sq ft
Building Size: 14,750 sq ft
Uses: Education / OtherResearch
The Board’s approval of the Second Amendment to the IMP for Harvard University’s Allston campus, includes the construction of this Harvard Life Lab. The Life Lab represents a new, but temporary two-story building that will provide both wet lab and co-working space for small Harvard-related scientific start-ups. In addition, the Life Lab will include general and dedicated lab spaces for up to 50 individuals, including a faculty member-in-residence, conference and support spaces to promote wet lab community connections. This project, which is considered a pilot by the university, aims to build on the success of the adjacent Harvard Innovation Lab. | https://www.elevatedboston.com/blog/posts/2016/03/22/bra-board-approves-259-million-in-new-projects-at-march-meeting/ |
2.1: Pencils on A Plot
- Measure your pencil to the nearest $\frac14$-inch. Then, plot your measurement on the class dot plot.
- What is the difference between the longest and shortest pencil lengths in the class?
- What is the most common pencil length?
- Find the difference in lengths between the most common length and the shortest pencil. | https://im.openupresources.org/6/students/8/2.html |
Everyone loves a cheese & onion sandwich, but how can you make it better? Easy, make it a cheese & onion toasted sandwich! For this use either 1/2 a small onion, or 1/4 medium onion, any more and the onion doesn’t rend to soften that much.
Directions
- Heat up the sandwich toaster onto its highest setting
- Butter 1 side of each piece of bread
- Lay the cheese slices evenly across one of the pieces of bread
- Sprinkle the diced onion evenly over the cheese
- Lay the other piece of bread on top, and pop into the toaster
- Enjoy!
As per usual with all toasted sandwiches, the buttered side of the bread should always be on the outside of the sandwich. This helps the bread turn into an amazing crispy golden brown colour. | http://www.cheesetoastie.co.uk/classic-toasties/cheese-onion-toastie/ |
files, printing all in parallel, one per column. Synopsis:
pr [option]… [file]…
By default, a 5-line header is printed at each page: two blank lines;
a line with the date, the file name, and the page count; and two more
blank lines. A footer of five blank lines is also printed.
The default page_length is 66
lines. The default number of text lines is therefore 56.
The text line of the header takes the form
‘date string page’, with spaces inserted around
string so that the line takes up the full page_width. Here,
date is the date (see the -D or --date-format
option for details), string is the centered header string, and
page identifies the page number. The
LC_MESSAGES locale
category affects the spelling of page; in the default C locale, it
is ‘Page number’ where number is the decimal page
number.
Form feeds in the input cause page breaks in the output. Multiple form feeds produce empty pages.
Columns are of equal width, separated by an optional string (default is ‘space’). For multicolumn output, lines will always be truncated to page_width (default 72), unless you use the -J option. For single column output no line truncation occurs by default. Use -W option to truncate lines in that case.
The program accepts the following options. Also see Common options.
Begin printing with page first_page and stop with last_page. Missing ‘:last_page’ implies end of file. While estimating the number of skipped pages each form feed in the input file results in a new page. Page counting with and without ‘+first_page’ is identical. By default, counting starts with the first page of input file (not first page printed). Line numbering may be altered by -N option.
With each single file, produce column columns of output (default is 1) and print columns down, unless -a is used. The column width is automatically decreased as column increases; unless you use the -W/-w option to increase page_width as well. This option might well cause some lines to be truncated. The number of lines in the columns on each page are balanced. The options -e and -i are on for multiple text-column output. Together with -J option column alignment and line truncation is turned off. Lines of full length are joined in a free field format and -S option may set field separators. -column may not be used with -m option.
With each single file, print columns across rather than down. The -column option must be given with column greater than one. If a line is too long to fit in a column, it is truncated.
Print control characters using hat notation (e.g., ‘^G’); print other nonprinting characters in octal backslash notation. By default, nonprinting characters are not changed.
Double space the output.
Format header dates using format, using the same conventions as for the command ‘date +format’. See date invocation. Except for directives, which start with ‘%’, characters in format are printed unchanged. You can use this option to specify an arbitrary string in place of the header date, e.g., --date-format="Monday morning".
The default date format is ‘%Y-%m-%d %H:%M’ (for example,
‘2020-07-09 23:59’);
but if the
POSIXLY_CORRECT environment variable is set
and the
LC_TIME locale category specifies the POSIX
locale, the default is ‘%b %e %H:%M %Y’ (for example,
‘Jul 9 23:59 2020’.
Timestamps are listed according to the time zone rules specified by
the
TZ environment variable, or by the system default rules if
TZ is not set. See Specifying the Time Zone
with
TZ in The GNU C Library Reference Manual.
Expand tabs to spaces on input. Optional argument in-tabchar is the input tab character (default is the TAB character). Second optional argument in-tabwidth is the input tab character’s width (default is 8).
Use a form feed instead of newlines to separate output pages. This does not alter the default page length of 66 lines.
Replace the file name in the header with the centered string header. When using the shell, header should be quoted and should be separated from -h by a space.
Replace spaces with tabs on output. Optional argument out-tabchar is the output tab character (default is the TAB character). Second optional argument out-tabwidth is the output tab character’s width (default is 8).
Merge lines of full length. Used together with the column options -column, -a -column or -m. Turns off -W/-w line truncation; no column alignment used; may be used with --sep-string[=string]. -J has been introduced (together with -W and --sep-string) to disentangle the old (POSIX-compliant) options -w and -s along with the three column options.
Set the page length to page_length (default 66) lines, including the lines of the header [and the footer]. If page_length is less than or equal to 10, the header and footer are omitted, as if the -t option had been given.
Merge and print all files in parallel, one in each column. If a line is too long to fit in a column, it is truncated, unless the -J option is used. --sep-string[=string] may be used. Empty pages in some files (form feeds set) produce empty columns, still marked by string. The result is a continuous line numbering and column marking throughout the whole merged file. Completely empty merged pages show no separators or line numbers. The default header becomes ‘date page’ with spaces inserted in the middle; this may be used with the -h or --header option to fill up the middle blank part.
Provide digits digit line numbering (default for digits is 5). With multicolumn output the number occupies the first digits column positions of each text column or only each line of -m output. With single column output the number precedes each line just as -m does. Default counting of the line numbers starts with the first line of the input file (not the first line printed, compare the --page option and -N option). Optional argument number-separator is the character appended to the line number to separate it from the text followed. The default separator is the TAB character. In a strict sense a TAB is always printed with single column output only. The TAB width varies with the TAB position, e.g., with the left margin specified by -o option. With multicolumn output priority is given to ‘equal width of output columns’ (a POSIX specification). The TAB width is fixed to the value of the first column and does not change with different values of left margin. That means a fixed number of spaces is always printed in the place of the number-separator TAB. The tabification depends upon the output position.
Start line counting with the number line_number at first line of first page printed (in most cases not the first line of the input file).
Indent each line with a margin margin spaces wide (default is zero). The total page width is the size of the margin plus the page_width set with the -W/-w option. A limited overflow may occur with numbered single column output (compare -n option).
Do not print a warning message when an argument file cannot be opened. (The exit status will still be nonzero, however.)
Separate columns by a single character char. The default for char is the TAB character without -w and ‘no character’ with -w. Without -s the default separator ‘space’ is set. -s[char] turns off line truncation of all three column options (-COLUMN|-a -COLUMN|-m) unless -w is set. This is a POSIX-compliant formulation.
Use string to separate output columns. The -S option doesn’t
affect the -W/-w option, unlike the -s option which does. It
does not affect line truncation or column alignment.
Without -S, and with -J,
pr uses the default output
separator, TAB.
Without -S or -J,
pr uses a ‘space’
(same as -S" ").
If no ‘string’ argument is specified, ‘""’ is assumed.
Do not print the usual header [and footer] on each page, and do not fill out the bottom of pages (with blank lines or a form feed). No page structure is produced, but form feeds set in the input files are retained. The predefined pagination is not changed. -t or -T may be useful together with other options; e.g.: -t -e4, expand TAB characters in the input file to 4 spaces but don’t make any other changes. Use of -t overrides -h.
Do not print header [and footer]. In addition eliminate all form feeds set in the input files.
Print nonprinting characters in octal backslash notation.
Set page width to page_width characters for multiple text-column output only (default for page_width is 72). The specified page_width is rounded down so that columns have equal width. -s[CHAR] turns off the default page width and any line truncation and column alignment. Lines of full length are merged, regardless of the column options set. No page_width setting is possible with single column output. A POSIX-compliant formulation.
Set the page width to page_width characters, honored with and without a column option. With a column option, the specified page_width is rounded down so that columns have equal width. Text lines are truncated, unless -J is used. Together with one of the three column options (-column, -a -column or -m) column alignment is always used. The separator options -S or -s don’t disable the -W option. Default is 72 characters. Without -W page_width and without any of the column options NO line truncation is used (defined to keep downward compatibility and to meet most frequent tasks). That’s equivalent to -W 72 -J. The header line is never truncated.
An exit status of zero indicates success, and a nonzero value indicates failure. | https://www.gnu.org/software/coreutils/manual/html_node/pr-invocation.html#pr-invocation |
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF THE INVENTION
1. Field of the Invention
This invention relates to encryption, and more particularly to methods of fast RC4™-like encryption.
2. Description of the Related Art
Many modern data processing systems require the use of encryption algorithms to ensure secure transfer of data. One such encryption algorithm is known as the RC4™ encryption algorithm. The RC4™ encryption algorithm is a very widely used method of encryption, due to both its small size and fast speed. Algorithms similar to RC4™ have also been implemented. RC4™ and similar encryption algorithms are known as stream ciphers. Each iteration of the encryption process produces a number of bits that is exclusive-OR'ed (XOR'ed) with a number of plaintext bits in order to produce the cipher text data. In the case of the RC4™ encryption algorithm, the number of bits produced each iteration is 8 (i.e. one byte). Decryption of a cipher data to a plaintext data may be similar to encryption.
8
Encryption using algorithms such as RC4™ involves a permutation array having 256 elements for 8-bit encryption (embodiments of similar algorithms that encrypt a different number of bits and thus have different array sizes are also possible). This array may also be referred to as the ‘S’ array. Thus, for 8-bit encryption, the array size is 256 elements (2=256). The array may store a permutation of the values between 0 and 255, with each location holding one of the values. For this embodiment, all additions are performed in mod 256. That is, if a result of any sum is greater than 255, the value of 256 is subtracted from the sum enough times so that the result is less than 256 but not negative. Two byte indices, ‘i’ and ‘j’ are also maintained for indexing elements in the array. The array may be initialized with a 256-byte key, K, using the following procedure:
Procedure 1
1.1: for i = 0 to 255, S[i] = i
1.2: j = 0
1.3: for i = 0 to 255 do the following:
1.3.1: A = S[i]
1.3.2: j = j + A + K[i]
1.3.3: B = S[j]
1.3.4: S[i] = B
1.3.5: S[j] = A
1.4: i = 0
1.5: j = 0
Procedure 1 initializes the S array for array permutation and then initializes both i and j to zero. The initialized array may then be used for encryption operations. Table 1 below illustrates an example of array initializing based on the key sequence (i.e. values of K[i]) 1, 2, 4, 2, 7, 6, 3, 5. It should be noted that for the initialization procedure, the value of K[i] is a value that is input into the procedure, not a value that must be obtained from the S array. For the sake of simplicity, the table shown here is limited to 8 3-bit “bytes” instead of 256.
TABLE 1
S
S swap
key
0
1
2
3
4
5
6
7
i
A
j
B
0
1
2
3
4
5
6
7
initialize
0
0
1
2
3
4
5
6
7
1
0
1
2
3
4
5
6
7
0
0
1
1
1
0
2
3
4
5
6
7
2
1
0
2
3
4
5
6
7
1
0
3
3
1
3
2
0
4
5
6
7
4
1
3
2
0
4
5
6
7
2
2
1
3
1
2
3
0
4
5
6
7
2
1
2
3
0
4
5
6
7
3
0
3
0
1
2
3
0
4
5
6
7
7
1
2
3
0
4
5
6
7
4
4
6
6
1
2
3
0
6
5
4
7
6
1
2
3
0
6
5
4
7
5
5
1
2
1
5
3
0
6
2
4
7
3
1
5
3
0
6
2
4
7
6
4
0
1
4
5
3
0
6
2
1
7
5
4
5
3
0
6
2
1
7
7
7
4
6
4
5
3
0
7
2
1
6
end initialize
0
0
4
5
3
0
7
2
1
6
As can be seen from examining the progress of the array initialization, the index ‘i’ progresses sequentially through each value, while the index ‘j’ is dependent on previous values of itself, A (which is S[i]) and K[i] and thus does not progress through any set sequence. In fact, all subsequent values produced in the loop of 1.3 are dependent upon the value of A, and thus retrieving A from the array becomes a critical step in the process of initialization (this is also true of the encryption process, as will be shown below). The final line (labeled ‘end initialize’) is the initialized array that can be used to begin encryption operations.
Once the array has been initialized, encryption can be performed using an encryption procedure, shown below as Procedure 2.
Procedure 2
2.1: i = i + 1
2.2: A = S[i]
2.3: j = j + A
2.4: B = S[j]
2.5: S[i] = B
2.6: S[j] = A
2.7: g = A + B
2.8: V = S[g]
2.9: result = V XOR (the next byte to be encrypted)
Table 2 below illustrates the progression of Procedure 2 in for steps 2.1 through 2.8 in generating a value V that may be XORed with the data to be encrypted. As with Table 1, the example of Table 2 is limited to eight 3-bit “bytes” for the purposes of clarity, instead of 256 8-bit bytes, and the additions are performed in mod 8. It is noted that since all additions are module additions, after i has obtained its maximum value, the next value of i is zero.
TABLE 2
S
S swap
0
1
2
3
4
5
6
7
i
A
j
B
0
1
2
3
4
5
6
7
g
V
end initialize
0
0
4
5
3
0
7
2
1
6
4
5
3
0
7
2
1
6
1
5
5
2
4
2
3
0
7
5
1
6
7
6
4
2
3
0
7
5
1
6
2
3
0
4
3
2
4
0
7
5
1
6
7
6
3
2
4
0
7
5
1
6
3
0
0
3
0
2
4
3
7
5
1
6
3
3
0
2
4
3
7
5
1
6
4
7
7
6
0
2
4
3
6
5
1
7
5
5
0
2
4
3
6
5
1
7
5
5
4
6
0
2
4
3
5
6
1
7
3
3
0
2
4
3
5
6
1
7
6
1
5
6
0
2
4
3
5
1
6
7
7
7
0
2
4
3
5
1
6
7
7
7
4
5
0
2
4
3
7
1
6
5
4
7
0
2
4
3
7
1
6
5
0
0
4
7
7
2
4
3
0
1
6
5
7
5
7
2
4
3
0
1
6
5
1
2
6
6
7
6
4
3
0
1
2
5
0
7
7
6
4
3
0
1
2
5
2
4
2
4
7
6
4
3
0
1
2
5
0
7
7
6
4
3
0
1
2
5
3
3
5
1
7
6
4
1
0
3
2
5
4
0
7
6
4
1
0
3
2
5
4
0
5
3
7
6
4
1
3
0
2
5
3
1
7
6
4
1
3
0
2
5
5
0
5
0
7
6
4
1
3
0
2
5
0
7
Each of the values of V produced during the performing of the procedure may be XOR'ed with a byte of plaintext data in order to encrypt it.
8
FIG. 1
The RC4™ and related encryption algorithms may be implemented in either hardware or software. In one hardware embodiment, each element of the permutation array may be input to a multiplexer. Several levels of multiplexers may be cascaded if necessary. The value of ‘A’ needed for each of the subsequent operations in the initialization or encryption (or decryption) procedures may be obtained through the cascaded multiplexers. For example, using 4-to-1 multiplexers, four levels of multiplexers may be used in order to obtain the value of ‘A’. Each multiplexer may receive two bits as select inputs, with a total of 8 select inputs for the entire array (i.e. 2=256) which represent the index ‘i’. The multiplexers that receive elements of the array as inputs may receive the two least significant bits (1:0), while the next level of multiplexers receives the next two least significant bits (3:2) and so on. Using these multiplexers, a value of ‘A’ may be retrieved from the array. is a block diagram illustrating a circuit using cascaded multiplexers in order to fetch a value from the array. Similar groups of cascaded multiplexers may be used for retrieving values of B and V.
Retrieving the value of A and thus the values of B and V are major factors in determining the amount of time elapsed in performing an encryption (or decryption) operation. Thus, any delays in retrieving these values can adversely impact the efficiency RC4™ and related encryption algorithms.
A method and apparatus for encrypting information is disclosed. In one embodiment, a method for encrypting information includes obtaining a value A from an array having a plurality of values and determining a value B based on the value A in a first pipeline stage. In a second pipeline stage, a value V may be determined from the value A and the value B. The value V may then be exclusive ORed (XORed) with a data value that forms a portion of the information being encrypted. A first logic unit may included the first pipeline stage, while a second logic unit may include the second pipeline stage. The method and apparatus may apply to the initialization of the array and to encryption of information. The array may be stored in a plurality of flip-flops in one embodiment, or may be stored in one or more register files in another embodiment. Embodiments using other types of storage technology are also possible and contemplated. In addition to being used for encrypting information, the method and apparatus may also be used for decrypting information. The first and/or second pipeline stages may also be divided into substages in order to implement pipelines having a greater number of stages, e.g., 3-stage pipelines, 4-stage pipelines, and so on.
While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that the drawings and description thereto are not intended to limit the invention to the particular form disclosed, but, on the contrary, the invention is to cover all modifications, equivalents, and alternatives falling with the spirit and scope of the present invention as defined by the appended claims.
Array Generation with Shifting
FIG. 2
200
Turning now to , a flow diagram of one embodiment of a method of initializing an array for an encryption algorithm using array shifting is shown. In the embodiment shown, method corresponds with Procedure 3 shown below.
Procedure 3
3.1: for i = 0 to 255, S[i] = i
3.2: j = 1
3.3: for i = 0 to 255 do the following:
3.3.1: A = S[0]
3.3.2: j = j + A + K[i] − 1
3.3.3: B = S[j]
3.3.4: S[0] = B
3.3.5: S[j] = A
3.3.6: shift the S array (i.e., for all k, S[k] <− S[k+1])
3.4: shift the S array (i.e., for all k, S[k] <− S[k+1])
3.5: i = 0
3.6: j = 1.
200
202
204
206
208
210
256
256
Method begins with each location in the S array being set to it's own index value (S[i]=i) as performed in the loop of items , , and . Once the loop is completed, index values i is set to zero while index value while j is set to one (3.2 of the procedure, item of the flowchart). Each of these values is used to determine the position of a value in the array that is to be read when the array is being initialized (or, as will be discussed below, when the array is being used for encryption. Following the setting of the index values, the initialization procedure enters a loop (3.3). For this particular embodiment, the array includes positions, and thus the execution of the loop includes iterations. Embodiments where the array is larger or smaller are possible and contemplated.
212
214
To begin the execution of the second loop, a value A is obtained from the first position in the array, S[0] (3.3, item ). In this particular embodiment, the value A is read from the first array position for each iteration of the initialization. After reading the value A from the array, the method calculates a value of the index value j (3.3.2, item ). In this embodiment, the value j is calculated by the equation j=j+A+K[i]−1. The value of K[i] is a key value received from a key sequence used to initialize the array. In this embodiment, since the value of A is read from S[0] for each iteration, the value of 1 is subtracted during the calculation of j.
th
216
218
Once the index value j is calculated, a second value B is read from the jposition of the array, S[j] (3.3.3, item ). After obtaining the value B a swap operation may be performed (3.3.4, 3.3.5, item ). The swap operation involves writing the value of B into the S[0] position of the array and the value of A into the S[j] position. Thus, in this embodiment, the value of A is obtained from the first array position for each iteration and the value B is written into the first array position for each iteration.
220
After the swap operation is complete, the array is shifted, with each value of S[k] being overwritten by the value of S[k+1] (3.3.6, item ). In other words, the value present in the S[3] array position is written into the S[2] position, the value in the S[2] position is written into the S[1] position, the value in the S[1] position is written into the S[0] position, and so on. It should be noted that the value in the S[0] position is written into the last array position (S[255] in this embodiment) during the shift operation.
222
224
226
228
If the index value has not reached its upper limit (item ), the execution of the loop continues, with the index value i being incremented (item ). Once the method has gone through all of its iterations, execution of the loop is terminated and another array shift operation takes place (3.4, item ). The array shift operation is identical to the one that took place during the execution of the loop. Also, the index values i and j are set to zero and one, respectively, following termination of loop execution (3.6, item ).
Table 3 illustrates the initialization of an array using Procedure 3 described above. For the purposes of simplicity, the array shown here has been restricted to 8 elements, although the basic principles still apply.
TABLE 3
S
S swap
S shift
key
0
1
2
3
4
5
6
7
i
A
j
B
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
initialize
1
0
1
2
3
4
5
6
7
1
0
1
2
3
4
5
6
7
0
0
1
1
1
0
2
3
4
5
6
7
0
2
3
4
5
6
7
1
2
0
2
3
4
5
6
7
1
1
0
2
3
3
2
0
4
5
6
7
1
2
0
4
5
6
7
1
3
4
2
0
4
5
6
7
1
3
2
2
7
3
3
0
4
5
6
7
1
2
0
4
5
6
7
1
2
3
2
0
4
5
6
7
1
2
3
3
0
0
0
0
4
5
6
7
1
2
3
4
5
6
7
1
2
3
0
7
4
5
6
7
1
2
3
0
4
4
2
6
6
5
4
7
1
2
3
0
5
4
7
1
2
3
0
6
6
5
4
7
1
2
3
0
6
5
5
4
2
2
4
7
1
5
3
0
6
4
7
1
5
3
0
6
2
3
4
7
1
5
3
0
6
2
6
4
2
1
1
7
4
5
3
0
6
2
7
4
5
3
0
6
2
1
5
7
4
5
3
0
6
2
1
7
7
5
6
6
4
5
3
0
7
2
1
4
5
3
0
7
2
1
6
end initialize
0
1
5
3
0
7
2
1
6
4
The initialization of the array illustrated in Table 3 is performed using a key sequence of 1, 2, 4, 2, 7, 6, 3, 5 (these are the values of K[i]). The initialization for this example progresses through eight iterations, with each iteration producing an S array, an S-swap array (due to the swap operation in 3.3.4 and 3.3.5) and an S-shift array resulting from the shift operation. In 3.4, the array is shifted one extra time.
Hardware for performing the initialization procedure may include a 256-to-1 multiplexer for this embodiment (or in general, an N-to-1 multiplexer, wherein N is the number of elements in the array) for reading the value B from S[j]. Similarly, each array location may be associated with a comparator for the storing of the value A, which may be performed by sending A to each array position and comparing the value of j with the location indexes. When a match is found with between j and a given location index, the value of A is written into the corresponding location.
FIG. 3
300
Moving now to , a flow diagram of one embodiment of a method of generating an encryption byte using array shifting is shown. For the embodiment shown, method corresponds to Procedure 4 shown below.
Procedure 4
4.1:
A = S[0]
4.2:
j = j + A − 1
4.3:
B = S[j]
4.4:
S[0] = B
4.5:
S[j] = A
4.6:
i = i + 1
4.7:
g = A + B − i
4.8:
V = S[g]
4.9:
result = K XOR (the next byte to be encrypted)
4.10:
shift the S array (i.e., for all k, S[k] <− S[k + 1])
300
302
304
306
308
312
Method begins with the reading of the value A from the first position of the array, S[0] (4.2, item ). As with the initialization procedure discussed above, the value of A is read from the first array position for each iteration of the encryption performed by Procedure 4. After reading value A from the array, the index value j is calculated using the equation j=j+A −1 (4.2, item ). Using the calculated value of index value j, the value B is determined by reading the S[j] position of the array (4.3, item ). A swap operation is then performed in 4.4 and 4.5 (item ) by writing the value A to S[j] and value B to S[0]. In 4.6 (item ), index value i is incremented.
314
316
318
320
322
th
In 4.7 (item ), the index value g is calculated using the equation g=A+B −i. Using the index value g, the value V is read from the gposition of the array, S[g] (4.8, item ). After reading the value of V from the array, a data byte is encrypted by XORing it with V (4.9, item ). It is noted that in this embodiment, the values of A, B, V, and the amount of data encrypted in one iteration are each one byte (8 bits) in length. However, embodiments may be implemented using larger or smaller blocks of data. After the encryption is performed in this embodiment, the S array is shifted (4.10, item ) so that the next value of A to be read is in the S[0] position for the next iteration, if any. A determination of whether any more data bytes are to be encrypted in item .
Table 4 below illustrates the generation of the V values for the encryption of Procedure 4, using the 8-element array initialized using Procedure 3.
TABLE 4
S
S swap
S shift
0
1
2
3
4
5
6
7
A
j
B
i
0
1
2
3
4
5
6
7
g
V
0
1
2
3
4
5
6
7
end initialize
1
0
5
3
0
7
2
1
6
4
5
3
0
7
2
1
6
4
5
4
2
1
2
3
0
7
5
1
6
4
6
6
3
0
7
5
1
6
4
2
3
0
7
5
1
6
4
2
3
6
4
2
4
0
7
5
1
6
3
2
5
6
4
0
7
5
1
6
3
2
0
7
5
1
6
3
2
4
0
5
3
3
3
7
5
1
6
0
2
4
0
3
7
5
1
6
0
2
4
3
7
5
1
6
0
2
4
3
7
3
6
4
6
5
1
7
0
2
4
3
1
5
5
1
7
0
2
4
3
6
5
1
7
0
2
4
3
6
5
7
6
5
6
1
7
0
2
4
3
5
6
3
1
7
0
2
4
3
5
6
1
7
0
2
4
3
5
6
1
7
6
6
6
7
0
2
4
3
5
1
1
7
7
0
2
4
3
5
1
6
7
0
2
4
3
5
1
6
7
5
5
7
5
0
2
4
3
7
1
6
5
7
0
2
4
3
7
1
6
5
0
2
4
3
7
1
6
5
0
4
7
0
7
2
4
3
0
1
6
5
7
5
2
4
3
0
1
6
5
7
2
4
3
0
1
6
5
7
2
5
6
1
6
4
3
0
1
2
5
7
7
7
4
3
0
1
2
5
7
6
4
3
0
1
2
5
7
6
4
0
4
2
4
3
0
1
2
5
7
6
6
7
3
0
1
2
5
7
6
4
3
0
1
2
5
7
6
4
3
2
1
3
1
0
3
2
5
7
6
4
1
0
0
3
2
5
7
6
4
1
0
3
2
5
7
6
4
1
0
1
3
4
3
0
2
5
7
6
4
1
7
1
0
2
5
7
6
4
1
3
0
2
5
7
6
4
1
3
0
0
0
5
0
2
5
7
6
4
1
3
3
7
2
5
7
6
4
1
3
0
FIG. 4
FIG. 4
400
is a flow diagram of another embodiment of initializing an array for an encryption algorithm using array shifting. Method of is associated with Procedure 5 below.
Procedure 5
5.1: for i = 0 to 255, S[i] = i
5.2: j = 1
5.3: for i = 0 to 255 do the following:
5.3.1: A = S[0]
5.3.2: j = j + A + K[i] − 1
5.3.3: B = S[j]
5.3.4: shift the S array (i.e., for all k, S[k] <− S[k+1])
5.3.5: S[255] = B
5.3.6: S[j−1] = A
5.4: shift the S array (i.e., for all k, S[k] <− S[k+1])
5.5: i = 1
5.6: j = 0
Procedure 5 is similar to array initialization Procedure 3 discussed above. However, instead of shifting the array at the end of the loop as in Procedure 3, the array is shifted after determining the value B. In general, the array shifting may be done any time within the loop. However, changing the point in the procedure when the array is shifted may alter some of its steps. In the case of Procedure 5, the steps prior to shifting the array are identical to those of Procedure 3 (e.g., A is read from the first array position S[0], index value j is computed in the same manner, etc.). The swap operation in Procedure 5 is performed by assigning the value B to the last position in the array, S[255], and assigning the value of A to the S[j−1] position of the array. This is because that the locations where A and B are stored have moved due to the shifting of the array just after obtaining B. The shift operation places the value of A in S[255] and the value of B is shifted from the S[j] position to the S[j−1] position. Thus, the swap is conducted by exchanging the values in these positions.
426
Procedure 5 also differs from Procedure 3 in that the index value i is set to one while the index value j is reset to zero in the procedure's final steps. The setting of i=1 in 5.5 (item ) may simplify the computation in 6.8 of Procedure 6 (discussed below).
Table 5 illustrates the initialization using an exemplary 8-element array for the sake of simplicity.
TABLE 5
S
S shift
S swap
key
0
1
2
3
4
5
6
7
i
A
j
B
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
initialize
1
0
1
2
3
4
5
6
7
1
0
1
2
3
4
5
6
7
0
0
1
1
1
2
3
4
5
6
7
0
0
2
3
4
5
6
7
1
2
0
2
3
4
5
6
7
1
1
0
2
3
2
3
4
5
6
7
1
0
2
0
4
5
6
7
1
3
4
2
0
4
5
6
7
1
3
2
2
7
3
0
4
5
6
7
1
3
2
0
4
5
6
7
1
2
3
2
0
4
5
6
7
1
2
3
3
0
0
0
4
5
6
7
1
2
3
0
4
5
6
7
1
2
3
0
7
4
5
6
7
1
2
3
0
4
4
2
6
5
6
7
1
2
3
0
4
5
4
7
1
2
3
0
6
6
5
4
7
1
2
3
0
6
5
5
4
2
4
7
1
2
3
0
6
5
4
7
1
5
3
0
6
2
3
4
7
1
5
3
0
6
2
6
4
2
1
7
1
5
3
0
6
2
4
7
4
5
3
0
6
2
1
5
7
4
5
3
0
6
2
1
7
7
5
6
4
5
3
0
6
2
1
7
4
5
3
0
7
2
1
6
end initialize
1
0
5
3
0
7
2
1
6
4
FIG. 5
FIG. 5
500
Turning now to , a flow diagram of another embodiment of a method of generating an encryption byte using array shifting is shown. Method shown in is associated with the encryption algorithm of Procedure 6 shown below.
Procedure 6
6.1: A = S[0]
6.2: j = j + A − 1
6.3: B = S[j]
6.4: shift the S array (i.e., for all k, S[k] <− S[k+1])
6.5: S[255] = B
6.6: S[j−1] = A
6.7: i = i + 1
6.8: g = A + B − i
6.9: V = S[g]
6.10: result = V XOR (the next byte to be encrypted)
508
514
516
518
520
522
502
Procedure 6 may be used to perform encryption utilizing an array generated according to Procedure 5 above. Procedure 6 is similar to Procedure 5 in that the shifting operation (6.4, item ) occurs just after the fetching of value B from the array. As such, the swap operation is performed in the same manner as in Procedure 5, with value B being written into the S[255] position and the value A being written into the S[j−1] position. The remainder of the procedure includes the incrementing of index value i (6.7, item ), calculating index value g (6.8, item ), reading a value of V from the g position of the array (6.9, item ), and XORing V with a byte of data to be encrypted (6.10, item ). In item , a determination is made as to whether more data bytes are to be encrypted, and if so, the method returns to item (corresponding to 6.1 of the procedure). It should be noted that while the maximum value of i in this embodiment is 255, this does not imply that the procedure is capable of incrementing only 256 bytes of information. When the index value i reaches its maximum value in an iteration, the addition of 1 in the modulo system results in the value of zero and thus encryption may continue until all desired information is encrypted.
Table 6 below illustrates the generation of the V values for encryption performed by Procedure 6 using the array initialized in the example associated with Procedure 5.
TABLE 6
S
S shift
S swap
0
1
2
3
4
5
6
7
A
j
B
i
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
g
V
end initialize
0
1
5
3
0
7
2
1
6
4
5
3
0
7
2
1
6
4
5
4
2
2
3
0
7
2
1
6
4
5
3
0
7
5
1
6
4
2
5
6
3
0
7
5
1
6
4
2
3
6
4
3
0
7
5
1
6
4
2
3
0
7
5
1
6
3
2
4
4
6
0
7
5
1
6
3
2
4
0
5
3
4
7
5
1
6
3
2
4
0
7
5
1
6
0
2
4
3
7
3
7
5
1
6
0
2
4
3
7
3
6
5
5
1
6
0
2
4
3
7
5
1
7
0
2
4
3
6
0
5
5
1
7
0
2
4
3
6
5
7
6
6
1
7
0
2
4
3
6
5
1
7
0
2
4
3
5
6
5
3
1
7
0
2
4
3
5
6
1
7
6
7
7
0
2
4
3
5
6
1
7
0
2
4
3
5
1
6
0
7
7
0
2
4
3
5
1
6
7
5
5
0
0
2
4
3
5
1
6
7
0
2
4
3
7
1
6
5
4
7
0
2
4
3
7
1
6
5
0
4
7
1
2
4
3
7
1
6
5
0
2
4
3
0
1
6
5
7
6
5
2
4
3
0
1
6
5
7
2
5
6
2
4
3
0
1
6
5
7
2
4
3
0
1
2
5
7
6
6
7
4
3
0
1
2
5
7
6
4
0
4
3
3
0
1
2
5
7
6
4
3
0
1
2
5
7
6
4
5
7
3
0
1
2
5
7
6
4
3
2
1
4
0
1
2
5
7
6
4
3
0
3
2
5
7
6
4
1
0
0
0
3
2
5
7
6
4
1
0
1
3
5
3
2
5
7
6
4
1
0
0
2
5
7
6
4
1
3
6
1
0
2
5
7
6
4
1
3
0
0
0
6
2
5
7
6
4
1
3
0
2
5
7
6
4
1
3
0
2
7
FIG. 6
FIG. 6
600
is a flow diagram of another embodiment of initializing an array for an encryption algorithm using array shifting. Method shown in is associated with Procedure 7 below. Although similar to the above array initialization procedures, Procedure 7 differs in that the shifting of the array occurs before the reading of the value B.
Procedure 7
7.1: for i = 0 to 255, S[i] = i−1
7.2: j = 1
7.3: for i = 0 to 255 do the following:
7.3.1: A = S[1]
7.3.2: j = j + A + K[i] − 1
7.3.3: shift the S array (i.e., for all k, S[k] <− S[k+1])
7.3.4: B = S[j]
7.3.5: S[0] = B
7.3.6: S[j] = A
7.4: shift the S array (i.e., for all k, S[k] <− S[k+1])
7.5: i = 0
7.6: j = 0
Table 7 below illustrates the performance of Procedure 7 using an exemplary 8-element array.
TABLE 7
S
S shift
S swap
key
0
1
2
3
4
5
6
7
i
A
j
0
1
2
3
4
5
6
7
B
0
1
2
3
4
5
6
7
initialize
1
7
0
1
2
3
4
5
6
1
7
0
1
2
3
4
5
6
0
0
1
0
1
2
3
4
5
6
7
1
1
0
2
3
4
5
6
7
2
1
0
2
3
4
5
6
7
1
0
2
0
2
3
4
5
6
7
1
3
3
2
0
4
5
6
7
1
4
3
2
0
4
5
6
7
1
2
2
7
2
0
4
5
6
7
1
3
3
3
0
4
5
6
7
1
2
2
3
0
4
5
6
7
1
2
3
0
0
0
4
5
6
7
1
2
3
0
0
4
5
6
7
1
2
3
7
0
4
5
6
7
1
2
3
4
4
2
4
5
6
7
1
2
3
0
6
6
5
4
7
1
2
3
0
6
6
5
4
7
1
2
3
0
5
5
4
5
4
7
1
2
3
0
6
2
2
4
7
1
5
3
0
6
3
2
4
7
1
5
3
0
6
6
4
2
4
7
1
5
3
0
6
2
1
1
7
4
5
3
0
6
2
5
1
7
4
5
3
0
6
2
7
7
5
7
4
5
3
0
6
2
1
6
6
4
5
3
0
7
2
1
end initialize
0
0
4
5
3
0
7
2
1
6
612
612
614
Since the array is shifted after determining the index value j but prior to reading the value B from the S[j] location in the array, the value A is read from the S[1] position in this embodiment (7.3.1, item ). Thus, when the swap operation occurs in 7.3.5 and 7.3.6 (items and , respectively), B can be written into the S[0] position and A can be written into the S[j] position. This ensures that the correct values of A and B will be read for subsequent iterations of the loop.
626
As with the procedures discussed above, an extra shift occurs after exiting the loop (7.4, item ). Also, index values i and j are reset to zero after exiting the loop.
FIG. 7
700
Moving now to , a flow diagram of another embodiment of a method of generating an encryption byte using array shifting is shown. The encryption procedure of method is associated with Procedure 8 shown below.
Procedure 8
8.1. A = S[1]
8.2. j = j + A − 1
8.3. shift the S array (i.e., for all k, S[k] <− S[k+1])
8.4. B = S[j]
8.5. S[0] = B
8.6. S[j] = A
8.7. i = i + 1
8.8. g = A + B − i
8.9. V = S[g]
8.10. result = V XOR (the next byte to be encrypted)
706
704
Procedure 8 is associated with array initialization Procedure 7, and thus the shifting of the array (8.3, item ) occurs just after calculating the index value j (8.2, item ). An example of the generation of the V values using Procedure 8 using the array initialized in the example associated with Procedure 7 is shown below in Table 8.
TABLE 8
S
S shift
S swap
0
1
2
3
4
5
6
7
A
j
0
1
2
3
4
5
6
7
B
i
0
1
2
3
4
5
6
7
g
V
end initialize
0
4
5
3
0
7
2
1
6
0
4
5
3
0
7
2
1
6
5
4
5
3
0
7
2
1
6
4
2
1
2
3
0
7
5
1
6
4
6
6
2
3
0
7
5
1
6
4
3
6
3
0
7
5
1
6
4
2
4
2
4
0
7
5
1
6
3
2
5
6
4
0
7
5
1
6
3
2
0
5
0
7
5
1
6
3
2
4
3
3
3
7
5
1
6
0
2
4
0
3
3
7
5
1
6
0
2
4
7
3
7
5
1
6
0
2
4
3
6
4
6
5
1
7
0
2
4
3
1
5
6
5
1
7
0
2
4
3
5
7
5
1
7
0
2
4
3
6
6
5
6
1
7
0
2
4
3
5
6
3
6
1
7
0
2
4
3
5
1
7
1
7
0
2
4
3
5
6
6
6
6
7
0
2
4
3
5
1
1
7
6
7
0
2
4
3
5
1
7
5
7
0
2
4
3
5
1
6
5
7
5
0
2
4
3
7
1
6
5
7
5
0
2
4
3
7
1
6
0
4
0
2
4
3
7
1
6
5
7
0
7
2
4
3
0
1
6
5
7
5
7
2
4
3
0
1
6
5
2
5
2
4
3
0
1
6
5
7
6
1
6
4
3
0
1
2
5
7
7
7
6
4
3
0
1
2
5
7
4
0
4
3
0
1
2
5
7
6
4
2
4
3
0
1
2
5
7
6
6
7
4
3
0
1
2
5
7
6
3
2
3
0
1
2
5
7
6
4
1
3
1
0
3
2
5
7
6
4
1
0
1
0
3
2
5
7
6
4
0
1
0
3
2
5
7
6
4
1
3
4
3
0
2
5
7
6
4
1
7
1
3
0
2
5
7
6
4
1
0
0
0
2
5
7
6
4
1
3
0
5
0
2
5
7
6
4
1
3
3
7
In general, a variety of embodiments of the procedures discussed above are possible. As previously noted, the array may be shifted at any time during the procedure providing the appropriate modifications are made to ensure that A and B are read from (and written into) the correct location. Furthermore, the shifting of the array during the execution of these procedures allows value A to be read from the same array location with each iteration. Reading A from the same location each iteration may significantly reduce the amount of delay present in comparison to embodiments where A may be read from a different location in each iteration. Furthermore, reading A from the same location each iteration may allow for the elimination of some circuitry, such as an array of multiplexers forming an N-to-1 multiplexer (where N is the number of array elements, e.g., 256) may also be eliminated.
Pipelining Using Array Shifting
FIG. 1
The procedures described above that involve array shifting may be implemented in a pipeline (although array shifting is not necessarily required to implement a pipeline). For each iteration of the encryption procedure, there are three values that are obtained from the array: A, B, and V. For one type of hardware implementation, an N-to-1 multiplexer may be used to obtain a value from an arbitrary position in the array, where N is the number of elements (e.g. 256). An N-to-1 multiplexer may be implemented using an array of multiplexers, such as that shown in . However, in embodiments wherein array shifting occurs, A may be read from the same array position for each iteration of the procedure being performed. Thus, N-to-1 multiplexers may be required only for the reading of the values B and V.
FIGS. 8
FIG. 8
FIG. 9
FIG. 10
FIG. 10
9
10
, and illustrate embodiments of hardware implementations that may be used to pipeline the initialization and encryption procedures. is associated with all shifting array embodiments. is associated with initialization Procedure 5 and encryption Procedure 6. During initialization, is associated with Procedure 7 up through step 7.3.4 and then with Procedure 5 from step 5.3.3. During encryption, is associated with Procedure 8 up through step 8.4 and then with Procedure 6 from step 6.3. However, it should be noted that embodiments based on the other initialization and encryption procedures are also possible and contemplated (including those disclosed herein), and may be realized with modifications to the embodiments shown.
FIG. 8
800
805
810
Turning now to , a block diagram of one embodiment of a circuit used as a building block in creating a pipeline for an encryption algorithm is shown. In the embodiment shown array element circuit (AEC) includes multiplexer and flip-flop . Although single-bit implementations of the multiplexer and flip-flop are shown here, the circuit may be considered to be a multi-bit implementation having the bit-width of a value stored in an array element (e.g. 8 bits). Alternatively, the diagram may be viewed as being associated one of a plurality of bits in a given bit position.
805
805
810
810
FIG. 4
In the embodiment shown, multiplexer is a 4-to-1 multiplexer with inputs for S[k], S[k+1], A, and k. A value of k is selected only on the first clock cycle of an initialization procedure, and implements the entire first loop of since it is applied simultaneously to all positions of the S array. The output R[k] of multiplexer is coupled to the input of flip-flop . Flip-flop is one of many different types of storage devices that may be used to store values for an array.
810
The value of S[k] is selected during the encryption procedure when there is no data to encrypt for a particular iteration. Thus, the S[k] input to the multiplexer is coupled to the output of flip-flop .
805
810
During shift operations, the value of S[k+1] is selected. When this selection is made, multiplexer will allow a value stored in the next element of the array to propagate through to flip-flop . For example, when this selection is made, the value stored in S[3] is written into S[2], the value stored (prior to being overwritten by the value from S[3]) in S[2] is written to S[1], and so forth. Therefore, selecting of the S[k+1] input allows the shift operation to take place.
800
800
The selection of the A input allows the value of A to be written into the array location during the swap operation. In this particular embodiment, A is written to S[j−1] location. The embodiment in which AEC is implemented may function by sending value A to the multiplexer select inputs for each element in the array while sending the value of (j−1) to a comparator (not shown) in each AEC . When the comparator finds a match with the array element corresponding the value of (j−1), the value of A may be selected by the multiplexer inputs, thereby allowing it to propagate through to be stored in the flip-flops.
As previously noted, the embodiment shown herein may be associated with Procedures 5 and 6, and thus the value of B is always written into the S[255] position. However, the embodiment shown herein may be modified for other ones of the procedures presented herein. For example, the A input to the multiplexer may be coupled to allow either the values of A or B to propagate through for embodiments where B may be written to any one of the array elements during the swap operation.
FIG. 9
FIG. 9
Moving to , a block diagram of one embodiment of a pipelining circuit for an encryption algorithm utilizing array shifting is shown. The embodiment shown here in is associated with Procedures 5 and 6, although alternate embodiments designed to work with other procedures including those disclosed herein are possible and contemplated.
900
800
800
800
800
800
910
Encryption circuit includes a plurality of AECs , one for each element of the array. Each AEC may include multiplexers for selection, flip-flops for storage, and may also include comparison logic for comparing a received index value with the index value of that particular array position. An output of each of the AECs s is fed back to an input so that the value of that array position can be maintained for any iteration wherein no data value is to be encrypted (e.g. when S[k] is selected as discussed above). A majority of the elements also have an input that is coupled to the output of a next element in the array. This allows the shifting operation to take place when the S[k+1] input is selected as discussed above. The first element of the array [k=0] may be coupled to an input of each of the AECs . This may allow the value of A to be written into an AEC that corresponds with the [j−1] element of the array during the swap operation. Alternatively, the value of A may be received from add/fetch unit #2 () as will be discussed in further detail below.
900
910
910
Encryption circuit includes two add/fetch units that may perform similar functions. Each add/fetch unit may make up one or more pipeline stages. For example, add/fetch unit #1 may comprise the first pipeline stage for two-stage pipelines, or the first two pipeline stages for the 4-stage pipelines. Add/fetch unit #2 may comprise the 2nd pipeline stage for 2-stage embodiments or the 3rd and 4th stages for 4-stage embodiments.
1
0
Add/fetch unit #1 is coupled to receive the output provided by each of the AEC's of the array, and thus may include an array of multiplexers for elements 0-255. Add/fetch unit is coupled to read value A from the AEC , calculate index value j, and subsequently fetch value B from the S[j] position of the array. Add fetch unit #1 is also coupled to receive a key sequence through the K[i] input during the initialization process In addition, add/fetch unit is configured to generate a ‘shift’ signal which initiates the shifting.
920
Add/fetch unit #2 is coupled to receive the values A and B from add/fetch unit #1. In this particular embodiment, add/fetch unit #2 is coupled to write B into S[255] and A into S[j−1] when during a swap operation. In other embodiments based on different initialization and encryption procedures, add/fetch unit #2 may be coupled to write values A and B to locations other than S[j−1] and S[255], respectively. Add/fetch unit #2 also increments index value i and calculates index value g. Upon calculating the index value g, add/fetch unit #2 may fetch the value V[g] from the array. V[g] is then provided as an output to XOR unit , where it is XOR'ed with the data byte to be encrypted for that iteration.
930
930
930
800
930
800
800
930
Select logic is coupled to receive the calculated index value j from add fetch unit #1, and in turn calculate [j−1] to determine which element A is to be written into during a swap operation. Select logic is also coupled to receive a shift signal from add/fetch unit. When asserted, the shift signal invokes a shift of the array, and thus select logic provides signals to the select inputs of multiplexers in AEC's in order to cause the values of each S[k+1] to be stored in the S[k] position (with the exception of S[255], which stored the value previously held in S[0]). In one embodiment, select logic may provide signals separately to each AEC in order to allow elements to be written into individually (as when A is written to the S[j−1] position during a swap). Alternatively, each AEC may include comparison logic that allows the select signals to propagate to the multiplexer select inputs, and may thus allow an array element to be addressed individually such that A (and/or B in some embodiments) can be written to a desired array position for a swap operation. Select logic may also receive a signal associated with the information to be encrypted indicating that no data is to be encrypted on a given iteration.
900
910
Each of the various components of encryption circuit is coupled to receive a clock signal. In embodiments wherein a 2-stage pipeline is implemented, the functions of each of the add/fetch units may be performed in a single clock cycle. For embodiments implementing a 4-stage pipeline, the functions of each add/fetch unit may be performed in two clock cycles. It is also noted that other embodiments implementing pipelines other than the 2- and 4-stage pipelines discussed here.
FIGS. 8 and 9
Tables 9A and 9B shown below illustrate the operations of a two stage pipeline in accordance with Procedures 5 and 6 and the circuit embodiments shown in . It should be noted that the values of K in the first pipeline stage are used for the initialization procedure, and will have a value of zero during the encryption procedure.
TABLE 9A
iteration 1
iteration 2
iteration 3
pipeline
have j<sub>0</sub>
have j<sub>1</sub>
have j<sub>2</sub>
stage 1
A<sub>1 </sub>= S<sub>0</sub>
A<sub>2 </sub>= S<sub>0</sub>
A<sub>3 </sub>= S<sub>0</sub>
j<sub>1 </sub>= j<sub>0 </sub>+ A<sub>1 </sub>+
j<sub>2 </sub>= j<sub>1 </sub>+ A<sub>2 </sub>+
j<sub>3 </sub>= j<sub>2 </sub>+ A<sub>3 </sub>+
K<sub>0 </sub>− 1
K<sub>1 </sub>− 1
K<sub>2 </sub>− 1
B<sub>1 </sub>= S[j<sub>1</sub>]
B<sub>2 </sub>= S[j<sub>2</sub>]
B<sub>3 </sub>= S[j<sub>3</sub>]
for k = 1
for k = 1
for k = 1
through 255
through 255
through 255
R<sub>k </sub>=
R<sub>k </sub>=
R<sub>k </sub>=
B if k = 255
B if k = 255
B if k = 255
A if k = j<sub>1 </sub>− 1
A if k = j<sub>2 </sub>− 1
A if k = j<sub>3 </sub>− 1
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
pipeline
i<sub>1 </sub>= i<sub>0 </sub>+ 1
i<sub>2 </sub>= i<sub>1 </sub>+ 1
stage 2
g<sub>1 </sub>= A<sub>1 </sub>+
g<sub>2 </sub>= A<sub>2 </sub>+
B<sub>1 </sub>− i<sub>1</sub>
B<sub>2 </sub>− i<sub>2</sub>
V<sub>1 </sub>= S[g<sub>1</sub>]
V<sub>2 </sub>= S[g<sub>2</sub>]
result<sub>1 </sub>= V<sub>1 </sub>XOR
result<sub>2 </sub>= V<sub>2 </sub>XOR
the next byte
the next byte
for encryption
for encryption
TABLE 9B
iteration 4
iteration 5
iteration 6
pipeline
have j<sub>3</sub>
have j<sub>4</sub>
have j<sub>5</sub>
stage 1
A<sub>4 </sub>= S<sub>0</sub>
A<sub>5 </sub>= S<sub>0</sub>
A<sub>6 </sub>= S<sub>0</sub>
j<sub>4 </sub>= j<sub>3 </sub>+
j<sub>5 </sub>= j<sub>4 </sub>+
j<sub>6 </sub>= j<sub>5 </sub>+
A<sub>4 </sub>+ K<sub>3 </sub>− 1
A<sub>5 </sub>+ K<sub>4 </sub>− 1
A<sub>6 </sub>+ K<sub>5 </sub>− 1
B<sub>4 </sub>= S[j<sub>4</sub>]
B<sub>5 </sub>= S[j<sub>5</sub>]
B<sub>6 </sub>= S[j<sub>6</sub>]
for k = 1
for k = 1
for k = 1
through 255
through 255
through 255
R<sub>k </sub>=
R<sub>k </sub>=
R<sub>k </sub>=
B if k = 255
B if k = 255
B if k = 255
A if k = j<sub>4 </sub>− 1
A if k = j<sub>5 </sub>− 1
A if k = j<sub>6 </sub>− 1
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
pipeline
i<sub>3 </sub>= i<sub>2 </sub>+ 1
i<sub>4 </sub>= i<sub>3 </sub>+ 1
i<sub>5 </sub>= i<sub>4 </sub>+ 1
stage 2
g<sub>3 </sub>= A<sub>3 </sub>+
g<sub>4 </sub>= A<sub>4 </sub>+
g<sub>5 </sub>= A<sub>5 </sub>+
B<sub>3 </sub>− i<sub>3</sub>
B<sub>4 </sub>− i<sub>4</sub>
B<sub>5 </sub>− i<sub>5</sub>
V<sub>3 </sub>= S[g<sub>3</sub>]
V<sub>4 </sub>= S[g<sub>4</sub>]
V<sub>5 </sub>= S[g<sub>5</sub>]
result<sub>3 </sub>= V<sub>3 </sub>XOR
result<sub>4 </sub>= V<sub>4 </sub>XOR
result<sub>5 </sub>= V<sub>5 </sub>XOR
the next byte
the next byte
the next byte
for encryption
for encryption
for encryption
Since the embodiments associated with the example illustrated in Tables 9A and 9B are 2-stage pipelines, a byte of data may (assuming a data byte is sent for encryption each clock cycle) be encrypted on each clock cycle starting with the second iteration (i.e. when the pipeline is full). Table 10 below illustrates both an initialization and encryption of the pipelined embodiment of Tables 9A and 9B.
TABLE 10
1st pipeline
2nd pipeline
S array
R values
key
0
1
2
3
4
5
6
7
A
j
B
0
1
2
3
4
5
6
7
A
B
i
g
V
initialize
1
0
1
2
3
4
5
6
7
1
0
1
2
3
4
5
6
7
0
1
1
0
2
3
4
5
6
7
1
2
0
2
3
4
5
6
7
1
0
2
3
2
0
4
5
6
7
1
3
4
2
0
4
5
6
7
1
3
2
7
3
0
4
5
6
7
1
2
3
2
0
4
5
6
7
1
2
3
0
0
0
4
5
6
7
1
2
3
0
7
4
5
6
7
1
2
3
0
4
2
6
5
4
7
1
2
3
0
6
6
5
4
7
1
2
3
0
6
5
4
2
4
7
1
5
3
0
6
2
3
4
7
1
5
3
0
6
2
4
2
1
7
4
5
3
0
6
2
1
5
7
4
5
3
0
6
2
1
7
5
6
4
5
3
0
7
2
1
6
end initialize
4
5
3
0
7
2
1
6
0
5
3
0
7
2
1
6
4
0
extra clock
5
3
0
7
2
1
6
4
5
4
2
3
0
7
5
1
6
4
2
1
3
0
7
5
1
6
4
2
3
6
4
4
0
7
5
1
6
3
2
5
2
2
5
6
0
7
5
1
6
3
2
4
0
5
3
7
5
1
6
0
2
4
3
3
4
3
4
6
7
5
1
6
0
2
4
3
7
3
6
5
1
7
0
2
4
3
6
0
3
4
7
3
5
1
7
0
2
4
3
6
5
7
6
1
7
0
2
4
3
5
6
7
6
5
0
5
1
7
0
2
4
3
5
6
1
7
6
7
0
2
4
3
5
1
6
5
6
6
5
3
7
0
2
4
3
5
1
6
7
5
5
0
2
4
3
7
1
6
5
1
6
7
0
7
0
2
4
3
7
1
6
5
0
4
7
2
4
3
0
1
6
5
7
7
5
0
4
7
2
4
3
0
1
6
5
7
2
5
6
4
3
0
1
2
5
7
6
0
7
1
6
5
4
3
0
1
2
5
7
6
4
0
4
3
0
1
2
5
7
6
4
2
6
2
6
7
3
0
1
2
5
7
6
4
3
2
1
0
3
2
5
7
6
4
1
4
4
3
5
7
0
3
2
5
7
6
4
1
0
1
3
0
2
5
7
6
4
1
3
3
1
4
0
0
0
2
5
7
6
4
1
3
0
0
0
2
5
7
6
4
1
3
0
0
3
5
6
1
It should be noted that the example shown in Table 10 is based upon an 8-element array as are the previous examples illustrating various procedures disclosed within. Also note than an extra clock cycle is present in order to fill the pipeline. Therefore, index value i is initially set to zero and incremented to one during the extra clock cycle.
FIG. 10
FIG. 10
1050
is a schematic diagram of an exemplary embodiment of a circuit that may be used for pipelining an encryption algorithm. In particular, the circuit illustrated in is configured for a 4-stage pipeline and an 8-element array (for the purposes of clarity), and illustrates one possible way a circuit may be implemented for performing the initialization and encryption procedures.
Tables 11A and 11B shown below illustrate the operation of a 4-stage pipeline. This illustration combines Procedure 5 with Procedure 7 and Procedure 6 with Procedure 8. For initialization, the first pipeline stage uses Procedure 7 through step 7.3.2. On the clock edge at the end of the first pipeline stage the array is shifted as in step 7.3.3. Since the shifting of the pipeline occurs on the clock edge, it may be thought of as happening “between” the pipeline stages. Step 7.3.4 is the same as step 5.3.3 and is performed in the second pipeline stage. Steps 5.3.4, 5.3.5 and 5.3.6 are accomplished on the next clock edge.
FIG. 1
th
For encryption or decryption, the first pipeline stage uses Procedure 8 up through step 8.2. On the clock edge at the end of the first pipeline stage, the array is shifted as in step 8.3. Step 8.4 is the same as step 6.3 and is performed in the second pipeline stage. Steps 6.4, 6.5, and 6.6 are accomplished on the next clock edge. Steps 6.7 and 6.8 are performed in the third pipeline stage. Step 6.9 involves a cascade of multiplexers as shown in and may be in the third pipeline stage, the fourth pipeline stage, or split with the early part of the cascade in the third pipeline stage and the later part in the fourth pipeline stage. If some or all of step 6.9 is executed in the fourth pipeline stage, then step 6.10 is also executed in the fourth pipeline stage (or in some embodiments, a 5pipeline stage). Finally, if step 6.9 is entirely in the third pipeline stage, then step 6.10 may be executed in either the third pipeline stage or the fourth pipeline stage. If step 6.10 is executed in the third pipeline stage, then there is no fourth pipeline stage but the encryption result is available late in the third clock cycle. If it is executed in a fourth pipeline stage, the encryption result is available early in the fourth clock cycle. Tables 11A and 11B show the case where the cascade of multiplexer to obtain V[g], step 6.9, are in the third pipeline stage and the XORing of the value of V[g] with the data to be encrypted, step 6. 10, is in the fourth pipeline stage.
TABLE 11A
iteration 1
iteration 2
iteration 3
pipeline
have j<sub>0</sub>
have j<sub>1</sub>
have j<sub>2</sub>
stage 1
A<sub>1 </sub>= S<sub>1</sub>
if j<sub>1 </sub>= 1
if j<sub>2 </sub>= 1
j<sub>1 </sub>= j<sub>0 </sub>+
then A<sub>2 </sub>= A<sub>1</sub>
then A<sub>3 </sub>= A<sub>2</sub>
A<sub>1 </sub>+ K<sub>0 </sub>− 1
else A<sub>2 </sub>= S<sub>1</sub>
else A<sub>3 </sub>= S<sub>1 </sub>
j<sub>2 </sub>= j<sub>1 </sub>+
j<sub>3 </sub>= j<sub>2 </sub>+
A<sub>2 </sub>+ K<sub>1 </sub>− 1
A<sub>3 </sub>+ K<sub>2 </sub>− 1
pipeline
B<sub>1 </sub>= S[j<sub>1</sub>]
B<sub>2 </sub>= S[j<sub>2</sub>]
stage 2
for k = 1
for k = 1
through 255
through 255
R<sub>k </sub>=
R<sub>k </sub>=
B<sub>1 </sub>if k = 255
B<sub>2 </sub>if k = 255
A<sub>1 </sub>if k =
A<sub>2 </sub>if k = j<sub>2 </sub>− 1
j<sub>1 </sub>− 1
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
pipeline
i<sub>1 </sub>= i<sub>0 </sub>+ 1
stage 3
g<sub>1 </sub>= A<sub>1 </sub>+
B<sub>1 </sub>− i<sub>0</sub>
pipeline
stage 4
TABLE 11B
iteration 4
iteration 5
iteration 6
pipeline
have j<sub>3</sub>
have j<sub>4 </sub>if j<sub>4 </sub>= 1
have j<sub>5</sub>
stage 1
if j<sub>3 </sub>= 1
then A<sub>5 </sub>= A<sub>4</sub>
if j<sub>5 </sub>= 1
then A<sub>4 </sub>= A<sub>3</sub>
else A<sub>5 </sub>= S<sub>1</sub>
then A<sub>6 </sub>= A<sub>5</sub>
else A<sub>4 </sub>= S<sub>1</sub>
j<sub>5 </sub>= j<sub>4 </sub>+
else A<sub>6 </sub>= S<sub>1</sub>
j<sub>4 </sub>= j<sub>3 </sub>+ A<sub>4 </sub>+ K<sub>3 </sub>− 1
A<sub>5 </sub>+ K<sub>4 </sub>− 1
j<sub>6 </sub>= j<sub>5 </sub>+
A<sub>6 </sub>+ K<sub>5 </sub>− 1
pipeline
B<sub>3 </sub>= S[j<sub>3</sub>]
B<sub>4 </sub>= S[j<sub>4</sub>]
B<sub>5 </sub>= S[j<sub>5</sub>]
stage 2
for k = 1
for k = 1
for k = 1
through 255
through 255
through 255
R<sub>k </sub>=
R<sub>k </sub>= B<sub>4</sub>
Rk =
B<sub>3 </sub>if k = 255
if k = 255
B<sub>5 </sub>if k = 255
A<sub>3 </sub>if k =
A<sub>5 </sub>if k =
A<sub>5 </sub>if k =
j<sub>3 </sub>− 1
j<sub>5 </sub>− 1
j<sub>5 </sub>− 1
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
S<sub>k + 1 </sub>otherwise
pipeline
i<sub>2 </sub>= i<sub>1 </sub>+ 1
i<sub>3 </sub>= i<sub>2 </sub>+ 1
i4 = i<sub>3 </sub>+ 1
stage 3
g<sub>2 </sub>= A<sub>2 </sub>+
g<sub>3 </sub>= A<sub>3 </sub>+
g<sub>4 </sub>= A<sub>4 </sub>+
B<sub>2 </sub>− i<sub>1</sub>
B<sub>3 </sub>− i<sub>2</sub>
B<sub>4 </sub>− i<sub>3</sub>
pipeline
V<sub>1 </sub>=
V<sub>2 </sub>=
V<sub>3 </sub>=
stage 4
A<sub>2 </sub>if g<sub>1 </sub>= 255
A<sub>3 </sub>if g<sub>2 </sub>= 255
A<sub>4 </sub>if g<sub>3 </sub>= 255
B<sub>2 </sub>if g<sub>1 </sub>+
B<sub>3 </sub>if g<sub>2 </sub>+
B<sub>4 </sub>if g<sub>3 </sub>+
1 = j<sub>2</sub>
1 = j<sub>3</sub>
1 = j<sub>4</sub>
S[g<sub>1</sub>] otherwise
S[g<sub>2</sub>] otherwise
S[g<sub>3</sub>] otherwise
result<sub>1 </sub>= V<sub>1 </sub>XOR
result<sub>2 </sub>= V<sub>2 </sub>XOR
result<sub>3 </sub>= V<sub>3 </sub>XOR
the next byte
the next byte
the next byte
for encryption
for encryption
for encryption
As shown above, Tables 11A and 11B illustrate how operations may be separated in order to form a 4-stage pipeline. Since the pipeline has 4 stages, an actual encryption does not take place until the fourth iteration. However, an encryption may be performed with each iteration beginning with the fourth (providing that data is provided for each iteration). If each pipeline stage is configured to perform its respective operations within one clock cycle, then one encryption for each clock cycle may occur.
Table 12 below further illustrates the operation of a 4-stage pipeline. It is noted that in the example shown that index value i is initially set to −1 and three extra clock cycles are added in order to fill the pipeline. Thus, encryption begins on the first clock cycle following extra clock #3.
TABLE 12
1st pipeline
2nd pipeline
pipelines
S array
R values
3rd
4<sup>th</sup>
key
0
1
2
3
4
5
6
7
X
A
j
A
j
B
0
1
2
3
4
5
6
7
A
B
i
g
g
V
initialize
0
1
0
1
2
3
4
5
6
7
1
0
1
2
3
4
5
6
7
1
0
2
0
1
1
0
2
3
4
5
6
7
1
2
0
2
3
4
5
6
7
1
2
2
7
0
2
3
2
0
4
5
6
7
1
3
4
2
0
4
5
6
7
1
3
0
0
0
2
7
3
0
4
5
6
7
1
2
3
2
0
4
5
6
7
1
2
3
4
4
2
0
0
0
4
5
6
7
1
2
3
0
7
4
5
6
7
1
2
3
0
5
5
4
4
2
6
5
4
7
1
2
3
0
6
6
5
4
7
1
2
3
0
6
4
4
2
5
4
2
4
7
1
5
3
0
6
2
3
4
7
1
5
3
0
6
2
7
7
5
4
2
1
7
4
5
3
0
6
2
1
5
7
4
5
3
0
6
2
1
set j = 0
7
5
6
4
5
3
0
7
2
1
6
set i = −1
extra clock #1
4
5
3
0
7
2
1
6
5
4
0
5
3
0
7
2
1
6
4
0
extra clock #2
5
3
0
7
2
1
6
4
3
3
6
5
4
2
3
0
7
5
1
6
4
2
1
extra clock #3
3
0
7
5
1
6
4
2
0
0
5
3
6
4
4
0
7
5
1
6
3
2
5
2
2
4
0
7
5
1
6
3
2
4
7
7
3
0
5
3
7
5
1
6
0
2
4
3
3
4
3
3
4
6
7
5
1
6
0
2
4
3
5
5
7
7
3
6
5
1
7
0
2
4
3
6
0
3
5
6
3
6
5
1
7
0
2
4
3
6
1
1
7
5
7
6
1
7
0
2
4
3
5
6
7
6
6
7
6
3
1
7
0
2
4
3
5
6
7
7
5
1
7
6
7
0
2
4
3
5
1
6
5
6
7
4
7
5
7
0
2
4
3
5
1
6
0
0
4
7
5
5
0
2
4
3
7
1
6
5
1
6
0
7
4
3
0
2
4
3
7
1
6
5
2
2
5
0
4
7
2
4
3
0
1
6
5
7
7
5
1
3
7
7
2
4
3
0
1
6
5
7
4
4
0
2
5
6
4
3
0
1
2
5
7
6
0
7
2
5
3
7
4
3
0
1
2
5
7
6
3
3
2
4
0
4
3
0
1
2
5
7
6
4
2
6
3
5
5
5
3
0
1
2
5
7
6
4
0
0
1
3
2
1
0
3
2
5
7
6
4
1
4
4
4
4
5
7
0
3
2
5
7
6
4
1
3
0
0
0
1
3
0
2
5
7
6
4
1
3
3
1
5
7
4
7
0
2
5
7
6
4
1
3
2
2
1
0
0
0
2
5
7
6
4
1
3
0
0
3
6
5
7
0
Pipelining without Shifted Arrays
The examples above are directed to pipelining when a procedure involving array shifting is used for the initialization and encryption. The array shifting is performed in order to allow a data value (typically A) to be read from the same location of the array for each iteration. However, using the storage elements of a register file or other memory device, pipelining of the encryption and initialization procedures can be accomplished without performing the array shifting.
FIG. 11A
FIG. 9
1100
1105
1105
1105
1105
1105
1105
1105
is a block diagram of another embodiment of a pipelining circuit for an encryption algorithm. Encryption circuit , instead of using the array element circuits of the embodiment shown in , stores array values in a register unit having one or more register files. Register unit may store the entire array. Each register address within register unit may be associated with one array position. Register unit may also be configured for multiple simultaneous reads. In one embodiment, the register files of register unit include 3 read ports, and thus the contents of 3 different array addresses may be read at any given time. The register files of register unit may also include multiple write ports. In the embodiment shown, each register file of register unit includes two write ports, allowing the simultaneous writing of two locations. Thus, the register files can simultaneously support three read operations and two write operation, thereby allowing operations to be conducted in parallel for a pipelined implementation. Using a register file with three read ports and two write ports, a 3- or 4-stage pipeline may be implemented.
FIG. 11B
FIG. 10B
1105
1105
1105
shows one example of a register unit . In the embodiment shown, register unit includes four register files each having 64 entries (for 256 entries total). Thus, register unit may store an array having 256 elements with each array element corresponding to an entry in a register file. Also shown in are the three read ports and two write ports previously discussed.
FIG. 11A
1110
1105
1105
1110
1110
Returning to , fetch/add unit performs the functions for a first pipeline stage. Fetch/add unit may provide the index i as an address to register unit . Responding to the address input, register unit may return the value of A to fetch/add unit from the S[i] position of the array. Fetch/add unit also includes arithmetic circuitry for calculating the index value j.
1115
1110
1105
1115
1105
1115
Fetch/swap unit may receive the calculated index value j and the value A from fetch/add unit . The received index value j is then provided to register unit in order to obtain the value B from the S[j] position of the array. Fetch/swap unit also performs the swap operation by forwarding the value A to an address indicated by index value j (S[j]) and value B to an address indicated by index value i (S[i]) in register unit . The index value g is also calculated by fetch/swap unit .
1120
1115
1120
1105
1125
1125
Fetch/encrypt unit is coupled to receive the index value g from fetch swap unit . The index value g is provided by fetch/encrypt unit to register unit , which returns value V from the S[g] position of the array. The value V retrieved from the array is then provided to XOR unit , where it is XORed with a block of data (e.g. one byte) to be encrypted. XOR unit may occur in either the third clock cycle for a three-clock pipeline implementation of a fourth clock cycle for a four-clock pipeline implementation. In a three-clock pipeline implementation, the encryption result is available near the end of the third clock cycle, wherein in a four clock-pipeline, the encryption result is available early in the fourth clock cycle. The time of arrival within the clock cycle of the data byte (being encrypted) may be an important factor in deciding whether to implement a three-clock or a four-clock pipeline.
FIGS. 9
10
As with the previously described pipeline embodiments (, , tables 9-12), the operation of each pipeline stage occurs simultaneously with respect to the operations of the other pipeline stages. Thus, in this embodiment, the reading of values A, B, and V each occur simultaneously once the pipeline is full.
1
2
2
3
3
Table 13 below illustrates the operation for one embodiment of encryption using a 3-stage pipeline. In the embodiment shown, it is assumed that the operations of each pipeline stage occur in a single clock cycle. Thus, beginning with the third clock cycle, when the pipeline is full, an encryption of a data byte may occur with each successive clock cycle. It is also noted that, as in the other pipelined embodiments discussed herein, operations in each pipeline stage are simultaneous with respect to each other. For example, in the third clock cycle, the value of Vbeing XORed with a data byte occurs simultaneously with the operations of stage 2 of the pipeline (reading B, performing the swap operation, and calculating g), which in turn are simultaneous to operations occurring in stage 1 of the pipeline (reading A, calculating j).
TABLE 13
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 5
Iteration 6
Stage 1
A<sub>1 </sub>= S[i<sub>1</sub>]
A<sub>2 </sub>= S[i<sub>2</sub>]
A<sub>3 </sub>= S[i<sub>3</sub>]
A<sub>4 </sub>= S[i<sub>4</sub>]
A<sub>5 </sub>= S[i<sub>5</sub>]
A<sub>6 </sub>= S[i<sub>6</sub>]
j<sub>1 </sub>= j<sub>0 </sub>+ A<sub>1</sub>
j<sub>2 </sub>= j<sub>1 </sub>+ A<sub>2</sub>
j<sub>3 </sub>= j<sub>2 </sub>+ A<sub>3</sub>
j<sub>4 </sub>= j<sub>3 </sub>+ A<sub>4</sub>
j<sub>5 </sub>= j<sub>4 </sub>+ A<sub>5</sub>
j<sub>6 </sub>= j<sub>5 </sub>+ A<sub>6 </sub>
Stage 2
B<sub>1 </sub>= S[j<sub>1</sub>]
B<sub>2 </sub>= S[j<sub>2</sub>]
B<sub>3 </sub>= S[j<sub>3</sub>]
B<sub>4 </sub>= S[j<sub>4</sub>]
B<sub>5 </sub>= S[j<sub>5</sub>]
S[i<sub>1</sub>] = B<sub>1</sub>
S[i<sub>2</sub>] = B<sub>2</sub>
S[i<sub>3</sub>] = B<sub>3</sub>
S[i<sub>4</sub>] = B<sub>4</sub>
S[i<sub>5</sub>] = B<sub>5</sub>
S[j<sub>1</sub>] = A<sub>1</sub>
S[j<sub>2</sub>] = A<sub>2</sub>
S[j<sub>3</sub>] = A<sub>3</sub>
S[j<sub>4</sub>] = A<sub>4</sub>
S[j<sub>5</sub>] = A<sub>5</sub>
g<sub>1 </sub>= A<sub>1 </sub>+ B<sub>1</sub>
g<sub>2 </sub>= A<sub>2 </sub>+ B<sub>2</sub>
g<sub>3 </sub>= A<sub>3 </sub>+ B<sub>3</sub>
g<sub>4 </sub>= A<sub>4 </sub>+ B<sub>4</sub>
g<sub>5 </sub>= A<sub>5 </sub>+ B<sub>5</sub>
Stage 3
V<sub>1 </sub>= S[g<sub>1</sub>]
V<sub>2 </sub>= S[g<sub>2</sub>]
V<sub>3 </sub>= S[g<sub>3</sub>]
V<sub>4 </sub>= S[g<sub>4</sub>]
V<sub>1 </sub>XOR
V<sub>2 </sub>XOR
V<sub>3 </sub>XOR
V<sub>4 </sub>XOR
byte
byte
byte
byte
Although the various embodiments of the method and apparatus described above with respect to the encryption of information, it should be noted that these same embodiments may also be used for the decryption of information. Furthermore, while the various method embodiments have been described herein as being performed using hardware, these same methods may be implemented using software as well.
While the present invention has been described with reference to particular embodiments, it will be understood that the embodiments are illustrative and that the invention scope is not so limited. Any variations, modifications, additions, and improvements to the embodiments described are possible. These variations, modifications, additions, and improvements may fall within the scope of the inventions as detailed within the following claims.
BRIEF DESCRIPTION OF THE DRAWINGS
Other aspects of the invention will become apparent upon reading the following detailed description and upon reference to the accompanying drawings in which:
FIG. 1
(Prior Art) is a block diagram illustrating the cascading of multiplexers in order to retrieve a value from an array for one embodiment of an encryption apparatus;
FIG. 2
is a flow diagram of one embodiment of a method of initializing an array for an encryption algorithm using array shifting;
FIG. 3
is a flow diagram of one embodiment of a method of generating an encryption byte using array shifting;
FIG. 4
is a flow diagram of another embodiment of initializing an array for an encryption algorithm using array shifting;
FIG. 5
is a flow diagram of another embodiment of a method of generating an encryption byte using array shifting;
FIG. 6
is a flow diagram of another embodiment of initializing an array for an encryption algorithm using array shifting;
FIG. 7
is a flow diagram of another embodiment of a method of generating an encryption byte using array shifting;
FIG. 8
is a block diagram of one embodiment of a circuit used as a building block in creating a pipeline for an encryption algorithm;
FIG. 9
is a block diagram of one embodiment of a pipelining circuit for an encryption algorithm utilizing array shifting;
FIG. 10
is a schematic diagram of an exemplary embodiment of a circuit that may be used for pipelining an encryption algorithm;
FIG. 11A
is a block diagram of another embodiment of a pipelining circuit for an encryption algorithm; and
FIG. 11B
FIG. 10A
is a block diagram of one embodiment of a register file that may be used with the embodiment discussed in . | |
Q:
Maximum Likelihood Estimator of scaled beta
I want to find the MLE of $X = \theta{Y}$, where $\theta > 0$ and $Y \sim \mathrm{Beta}(2,1)$. The density for $X$ is given by $$f_{\theta}(x) = \frac{2x}{\theta^{2}}$$
on $[0,\theta]$.
It has been a bit problematic finding the MLE here (for fixed $x \in [0, \theta]$, doesn't seem to be any value $\hat{\theta}$ that maximizes this function). I'd appreciate any help with this.
A:
If you have just a single observation, $X$, then $\theta$ is necessarily $\ge X$, so the decreasing function $\theta\mapsto2x/\theta^2$ has as its domain the interval $[X,\infty)$. And there is indeed a value of $\theta$ in that domain where $\theta\mapsto2x/\theta^2$ assumes its maximum value.
If there are $n$ i.i.d. observations $X_1,\ldots,X_n$, then the domain is $\left[\max\{X_1,\ldots,X_n\},\infty\right)$.
| |
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
BRIEF DESCRIPTION OF THE DRAWINGS
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention relates to a geomagnetic sensor and an azimuth calculation method medium thereof. More particularly, the present invention relates to a geomagnetic sensor precisely calculating a pitch angle and a roll angle according to a region using 3-axis acceleration sensors to calculate an azimuth and an azimuth calculation method and medium thereof.
A geomagnetic sensor is a device measuring the intensity and direction of geomagnetism, which a human can not feel, and more particularly, a sensor measuring the geomagnetism using a flux gate is called a flux gate geometric sensor.
The flux gate geometric sensor is a device measuring the intensity and direction of an external magnetic field by using a high permeability material such as permalloy as a magnetic core, adding excitation magnetic field through a coil winding the magnetic core, and measuring second harmonic component proportional to the external magnetic field generated according to magnetic saturation of the magnetic core and non-linear magnetic features. The flux gate geometric sensor was developed in the late 1930s and is good in sensitivity, cost-effective and miniaturized, compared with many kinds of other geomagnetic sensors.
Especially, as a micro electro mechanical system (MEMS) technology is nowadays developing, a micro flux gate sensor with low power consumption can be equipped in various portable electronic devices including a cell phone, personal digital assistant (PDA) and laptop computer using the technology.
Meanwhile, the geomagnetic sensor generally uses a 2 or 3 axis flux gate. When an azimuth is measured using the geomagnetic sensor, if the geomagnetic sensor is tilted, the azimuth can be miscalculated. Accordingly, an algorithm compensating the azimuth using a tilt angle, that is, a pitch angle and a roll angle is generally performed. Therefore, a conventional geomagnetic sensor calculates the pitch angle and the roll angle using 2-axis acceleration sensors to compensate the azimuth. In this case, the range of measuring a tilt is limited to ±90°.
-1
In addition, the conventional geomagnetic sensor calculates the pitch angle and the roll angle by applying a function of sin () to a value of X axis and a value of Y axis of the acceleration sensor. However, if the tilt is over 60°, the tilt of a signal gets flat because of the nature of a sine function. In this case, if the resolving power of an analog to digital converter (ADC) converting an output value of the acceleration sensor into a digital value is not high enough, a tilt angle can not precisely be acquired. Specifically, if the tilt is over +90°, for example, if the tilt is 120°, +60° instead of +120° is recognized.
So, if an error occurs in the calculated pitch angle value and roll angle value, azimuth compensation does not work well so that the azimuth itself is miscalculated.
According to the invention, there is provided a geomagnetic sensor, including a geomagnetic measurement module including flux gates of X, Y and Z axes mutually crossing at right angles, a tilt measurement module including acceleration sensors of X, Y and Z axes mutually crossing at right angles, a tilt calculator primarily calculating a pitch angle and a roll angle using output values of each acceleration sensors of the X and Y axes, and performing a second calculation by adjusting at least one of the primarily calculated pitch angle and roll angle using an output value of the acceleration sensor of the Z axis, and a controller calculating an azimuth using the readjusted pitch angle and roll angle and an output value of the geomagnetic measurement module.
The tilt measurement module may normalize the output values of each acceleration sensor of the X, Y and Z axes into values of a preset range, and transmit the normalized values to the tilt calculator.
X
Y
Z
Z
The tilt calculator may primarily calculate θ θ, and <emiff1_type1/> using the acceleration sensors of X, Y and Z axes.
X
Z
X
Z
X
X
Y
Z
Y
X
X
Z
X
Z
X
X
Y
Z
Y
X
The tilt calculator can perform the second calculation in a manner that when the θ is between 0° and 45°, if the θ is 0° or more, the θ becomes the pitch angle, or if the θ is under 0°, 180°- θ becomes the pitch angle, when the θ is 45° or more, if <emiff1_type1/> is under 45°, 90°- θ becomes the pitch angle, or if the <emiff1_type1/> is 45° or more, the θ becomes the pitch angle, when the θ is between -45° and 0°, if the θ is 0° or more, the θ becomes the pitch angle, or if the θ is under 0°, -180°- θ becomes the pitch angle, or when the θ is under -45°, if the <emiff1_type1/> is under 45°, θ-90° becomes the pitch angle, or if the <emiff1_type1/> is 45° or more, the θ becomes the pitch angle.
Y
Z
Z
Z
Y
Y
X
Z
X
Y
Y
Z
Y
Z
Y
Y
X
Z
X
Y
Meanwhile, the tilt calculator may perform the second calculation in a manner that when the <emiff1_type1/> is between 0° and 45°, if the <emiff1_type1/> is 0° or more, the <emiff1_type1/> becomes the roll angle, or if the <emiff1_type1/> is under 0°, 180°-<emiff1_type1/> becomes the roll angle, when the <emiff1_type1/> is 45° or more, if the <emiff1_type1/> is under 45°, 90°- <emiff1_type1/> becomes the roll angle, or if the θ is 45° or more, the <emiff1_type1/> becomes the roll angle, when the <emiff1_type1/> is between - 45° and 0°, if the <emiff1_type1/> is 0° or more, the <emiff1_type1/> becomes the roll angle, or if the <emiff1_type1/> is under 0°, -180°- <emiff1_type1/> becomes the roll angle, or when the <emiff1_type1/> is under -45°, if the θ is under 45°, <emiff1_type1/> -90° becomes the roll angle, or if the θ is 45° or more, the <emiff1_type1/> becomes the roll angle.
The geomagnetic measurement module may normalize the output values of each flux gate of the X, Y and Z axes into values of a preset range, and provides the normalized values to the controller.
The controller calculates the azimuth by applying the normalized output values of each flux gate of the X, Y and Z and the readjusted pitch angle and roll angle to a predetermined equation.
A method of calculating an azimuth according to the invention includes (a) calculating output values of flux gates of X, Y and Z axes using the flux gates of the X, Y and Z axes mutually crossing at right angles, (b) primarily calculating a pitch angle and a roll angle using acceleration sensors of X, Y and Z axes mutually crossing at right angles, (c) adjusting at least one of the primarily calculated pitch angle and roll angle using an output value of the acceleration sensor of the Z axis, and (d) calculating the azimuth using the readjusted pitch angle and roll angle and an output value of a geomagnetic measurement module.
In the step of (b), the output values of each acceleration sensor of the X, Y and Z axes may be normalized into values of a preset range using predetermined equations.
X
Y
Z
Z
In the step of (b), θ <emiff1_type1/> θ, and <emiff1_type1/> may primarily be calculated using the acceleration sensors of X, Y and Z axes.
X
Z
X
Z
X
X
Y
Z
Y
X
X
Z
X
Z
X
X
Y
Z
Y
X
In the step of (c), the pitch angle may be readjusted in a manner that when the θ is between 0° and 45°, if the θ is 0° or more, the θ becomes the pitch angle, or if the θ is under 0°, 180°- θ becomes the pitch angle, when the θ is 45° or more, if <emiff1_type1/> is under 45°, 90°- θ becomes the pitch angle, or if the <emiff1_type1/> is 45° or more, the θ becomes the pitch angle, when the θ is between -45° and 0°, if the θ is 0° or more, the θ becomes the pitch angle, or if the θ is under 0°, -180°- θ becomes the pitch angle, or when the θ is under -45°, if the <emiff1_type1/> is under 45°, θ-90° becomes the pitch angle, or if the <emiff1_type1/> is 45° or more, the θ becomes the pitch angle.
Y
Z
Z
Z
Y
Y
X
Z
X
Y
Y
Z
Y
Z
Y
Y
X
Z
X
Y
In the step of (c), the roll angle may be readjusted in a manner that when the <emiff1_type1/> is between 0° and 45°, if the <emiff1_type1/> is 0° or more, the <emiff1_type1/> becomes the roll angle, or if the <emiff1_type1/> is under 0°, 180°-<emiff1_type1/> becomes the roll angle, when the <emiff1_type1/> is 45° or more, if the <emiff1_type1/> is under 45°, 90°- <emiff1_type1/> becomes the roll angle, or if the θ is 45° or more, the <emiff1_type1/> becomes the roll angle, when the <emiff1_type1/> is between -45° and 0°, if the <emiff1_type1/> is 0° or more, the <emiff1_type1/> becomes the roll angle, or if the <emiff1_type1/> is under 0°, -180°- <emiff1_type1/> becomes the roll angle, or when the <emiff1_type1/> is under -45°, if the θ is under 45°, <emiff1_type1/>-90° becomes the roll angle, or if the θ is 45° or more, the <emiff1_type1/> becomes the roll angle.
Meanwhile, in the step of (a), the output values of each flux gate of the X, Y and Z axes can be normalized into values of a preset range using predetermined equations.
In addition, in the step of (d), the azimuth can be calculated by applying the normalized output values of each flux gate of the X, Y and Z and the readjusted pitch angle and roll angle to predetermined equation:
The method of the invention can be implemented as a computer program.
The invention thus provides a geomagnetic sensor, which primarily calculates a pitch angle and a roll angle using 3-axis acceleration sensors, calculates the precise pitch angle and roll angle by readjusting the pitch angle and the roll angle using an output value of acceleration sensor of Z axis, and precisely compensates an azimuth using the pitch angle and the roll angle, and an azimuth calculation method thereof. The invention thus aims to address the above problems and/or disadvantages.
Additional aspects and/or advantages of the invention will be set forth in part in the description which follows and, in part, will be apparent from the description, or may be learned by practice of the invention.
FIG. 1 is a block diagram showing the configuration of a geomagnetic sensor according to an exemplary embodiment of the present invention;
FIG. 2 is a block diagram showing an example of the configuration of the tilt measurement module used in the geomagnetic sensor of FIG. 1;
FIG. 3 shows an example of 3-axis location of the geomagnetic measurement module and the tilt measurement module in the geomagnetic sensor of FIG. 1;
FIGs. 4 and 5 show a readjusted region of a pitch angle and a roll angle, respectively;
FIG. 6 is a flow chart showing an azimuth calculation method according to an exemplary embodiment of the present invention;
FIG. 7 is a flow chart showing an example of a pitch angle calculation method used in the azimuth calculation method of FIG. 6; and
FIG. 8 is a flow chart showing an example of a roll angle calculation method used in the azimuth calculation method of FIG. 6.
These and/or other aspects, features, and advantages of the invention will become apparent and more readily appreciated from the following description of exemplary embodiments, taken in conjunction with the accompanying drawings of which:
Reference will now be made in detail to exemplary embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the like elements throughout. Exemplary embodiments are described below to explain the present invention by referring to the figures.
Hereinafter, an exemplary embodiment of the present invention will be described in detail with reference to the accompanying drawing figures.
FIG. 1 is a block diagram showing the configuration of a geomagnetic sensor according to an exemplary embodiment of the present invention. According to FIG. 1, the geomagnetic sensor 100 includes a geomagnetic measurement module 110, a tilt measurement module 120, a tilt calculator 130 and a controller 140.
The geomagnetic measurement module 110 outputs a voltage value corresponding to an external geomagnetism. In detail, the geomagnetic measurement module 110 can include flux gates of X, Y and Z axes mutually crossing at right angles. Accordingly, an output value corresponding to the geomagnetism can be obtained by supplying an electric signal to each axis flux gate.
The tilt measurement module 120 outputs a voltage value corresponding to the tilt of the main body of the geomagnetic sensor 100. Specifically, the tilt measurement module 120 has acceleration sensors of X, Y and Z axes mutually crossing at right angles. Accordingly, an output value of acceleration sensor of each axis corresponding to the tilt can be obtained by supplying an electric signal to each axis acceleration sensor.
FIG. 2 is a block diagram showing an example of the configuration of the tilt measurement module 120. According to FIG. 2, the tilt measurement module 120 includes 3-axis acceleration sensors 122, a signal processor 123, a tilt measurement controller 124 and a memory 125.
The 3-axis acceleration sensors 122 consist of acceleration sensors of X, Y and Z axes mutually crossing at right angles.
The signal processor 123 converts output values of each acceleration sensor of X, Y and Z axes into a digital value, and transmits the digital value to the tilt measurement controller 124.
Equation
1
AX
norm
=
AX
raw
AX
offset
-
AX
Scale
AY
norm
=
AY
raw
AY
offset
-
AY
Scale
AZ
norm
=
AZ
raw
AZ
offset
-
AZ
Scale
The tilt measurement controller 124 normalizes the output values of each acceleration sensor of X, Y and Z axes received from the signal processor 123 by mapping it with a value of a preset range. The normalization ranges from -1 to +1. The normalization can be performed as below.
norm
norm
norm
raw
raw
raw
offset
offset
offset
Scale
Scale
Scale
In Equation1, AX, AY and AZ are the normalized output value of each acceleration sensor of X, Y and Z axes respectively, AX, AY and AZ are the real output value of each acceleration sensor of X, Y and Z axes respectively, AX, AY and AZ are the preset offset value of each acceleration sensor of X, Y and Z axes respectively, and AX, AY and AZ are the preset scale value of each acceleration sensor of X, Y and Z axes respectively.
norm
norm
norm
norm
norm
norm
norm
norm
norm
norm
norm
norm
As described above, to map the output values of each acceleration sensor of X, Y and Z axes from -1 to +1, if AX, AY and AZ are over +1, AX, AY and AZ may be fixed to +1, and if AX, AY and AZ are under-1, AX, AY and AZ may be fixed to -1.
For the offset value and scale value of each acceleration sensor, a value used in a previous normalization process can be stored in the memory 125 and be read for use.
Equation
2
AX
offset
AX
Scale
=
,
=
AX
max
AX
min
+
2
AX
max
AX
min
-
2
AY
offset
AY
Scale
=
,
=
AY
max
AY
min
+
2
AY
max
AY
min
-
2
AZ
offset
AZ
Scale
=
,
=
AZ
max
AZ
min
+
2
AZ
max
AZ
min
-
2
Alternatively, the offset value and scale value can be calculated as below.
max
max
max
raw
raw
raw
min
min
min
raw
raw
raw
max
max
max
min
min
min
raw
raw
raw
max
max
max
min
min
min
In Equation 2, AX, AY and AZ are the maximum value of AX, AY and AZ respectively, and AX, AY and AZ are the minimum value of AX, AY and AZ respectively. AX, AY, AZ, AX, AY and AZ can be generated by selecting the maximum and minimum values among AX, AY and AZ measured by rotating the geomagnetic sensor 100 at least once in a preparation step before azimuth measurement, and be stored in the memory 125. Accordingly, AX, AY, AZ, AX, AY and AZ can be read to use in the normalization process.
Equation
3
θ
X
=
sin
-
1
AX
norm
ϕ
Y
=
sin
-
1
AY
norm
θ
X
/
cos
θ
Z
=
sin
-
1
AZ
mod
wherein
,
=
AZ
mod
AZ
norm
cos
ϕ
Y
ϕ
Z
AZ
mod
=
{
,
=
}
sin
-
1
AZ
mod
θ
X
/
cos
wherein
AZ
norm
cos
θ
X
Returning to FIG. 1, the tilt calculator 130 receives the normalized output value of each acceleration sensor of X, Y and Z axes from the tilt measurement module 120, and primarily calculates the pitch angle and the roll angle. The pitch angle and roll angle are primarily calculated as follows.
X
Y
Z
Z
In Equation 3, θ is the pitch angle calculated using the acceleration sensor of X axis, <emiff1_type1/> is the roll angle calculated using the acceleration sensor of Y axis, θ is the pitch angle calculated using the acceleration sensor of Z axis, and <emiff1_type1/> is the roll angle calculated using the acceleration sensor of Z axis.
Y
X
X
Y
X
Y
Meanwhile, in Equation 3, cos<emiff1_type1/> and cosθ are located in the denominator. Therefore, if θ or <emiff1_type1/> is 90°, the denominator becomes 0 so that an error occurs. To prevent this, θ and <emiff1_type1/> of 90° are replaced with 89° or 91°.
norm
norm
norm
mod
AX, AY, AZ, and AZ are saturated not to be over the range of ±1. That is, to apply to Equation 3, over +1 is fixed to +1 and under -1 is fixed to - 1.
Z
norm
-1
Meanwhile, Equation 3 varies according to an exemplary embodiment. That is, θ can be calculated using cos(AZ).
X
Y
Z
Z
X
Z
Y
Z
The tilt calculator 130 performs the secondary calculation to finalize the pitch angle and roll angle according to the size of the primarily calculated θ, <emiff1_type1/>, θ and <emiff1_type1/>. In detail, the tilt calculator 130 calculates the pitch angle by combination of θ and θ, and calculates the roll angle by combination of <emiff1_type1/> and <emiff1_type1/>.
X
X
Z
Z
X
Z
X
Z
X
Z
Z
-1
First, to calculate the pitch angle, the tilt calculator 130 determines if θ is between 0° and 45°. If θ is between 0° and 45°, it is determined if θ is 0° or more. If θ is 0° or more, θ becomes the pitch angle. However, if θ is under 0°, 180°- θ becomes the pitch angle. As described above, in the case of the function of sin(), the resolving power decreases on the basis of 45°. If the acceleration sensor of X axis is tilted on the basis of Y axis, Z axis becomes tilted. If the acceleration sensor of X axis is tilted by 45° or more, θ becomes lower than 0° so that 180°- θ becomes a precise pitch angle. According to this theory, the secondary calculation for the pitch angle and roll angle can be performed using the pitch angle θ and roll angle <emiff1_type1/> measured by the acceleration sensor of Z axis.
X
Y
Y
X
Y
Z
Meanwhile, if θ is 45° or more, it is determined if <emiff1_type1/> is 45° or more. If <emiff1_type1/> is 45° or more, θ becomes the pitch angle. Or, if <emiff1_type1/> is under 45°, 90°- θ becomes the pitch angle.
X
Z
Z
X
Z
X
If θ is between -45° and 0°, it is determined if θ is 0° or more. If θ is 0° or more, θ becomes the pitch angle. Or, if θ is under 0°, -180°- θ becomes the pitch angle.
X
Y
Y
X
Y
Z
If θ is under -45°, it is determined if <emiff1_type1/> is 45° or more. If <emiff1_type1/> is 45° or more, θ becomes the pitch angle. However, if <emiff1_type1/> is under 45°, θ-90° becomes the pitch angle. In this method, the pitch angle can finally be determined.
Y
Y
Z
Z
Z
Z
Y
Next, to calculate the roll angle, first it is determined if <emiff1_type1/> is between 0° and 45°. If <emiff1_type1/> is between 0° and 45°, it is determined if <emiff1_type1/> is 0° or more. If <emiff1_type1/> is 0° or more, <emiff1_type1/> becomes the roll angle. However, if <emiff1_type1/> is under 0°, 180°-<emiff1_type1/> becomes the roll angle.
Y
X
X
Y
X
Z
Meanwhile, if <emiff1_type1/> is 45° or more, it is determined if θ is 45° or more. If θ is 45° or more, <emiff1_type1/> becomes the roll angle. However, if <emiff1_type1/> is under 45°, 90°- <emiff1_type1/> becomes the roll angle.
Y
Z
Z
Y
Z
Y
Meanwhile, if <emiff1_type1/> is between -45° and 0°, it is determined if <emiff1_type1/> is 0° or more. If <emiff1_type1/> is 0° or more, <emiff1_type1/> becomes the roll angle. However, if <emiff1_type1/> is under 0°, -180°- <emiff1_type1/> becomes the roll angle.
Y
X
X
Y
X
Z
Furthermore, if <emiff1_type1/> is under -45°, it is determined if θ is 45° or more. If θ is 45° or more, <emiff1_type1/> becomes the roll angle. However, if θ is under 45°, <emiff1_type1/> -90° becomes the roll angle. In this method, the roll angle can finally be determined.
Equation
4
X
norm
=
X
raw
X
offset
-
X
Scale
Y
norm
=
Y
raw
Y
offset
-
Y
Scale
Z
norm
=
Z
raw
Z
offset
-
Z
Scale
norm
norm
norm
raw
raw
raw
offset
offset
offset
Scale
Scale
Scale
The controller 140 can calculate an azimuth using the secondarily calculated and readjusted pitch angle and roll angle. To calculate the azimuth, the geomagnetic measurement module 110 normalizes the output values of each flux gate of X, Y and Z axes using equations below to transmit to the controller 140. where X, Y and Z are the normalized output value of each flux gate of X, Y and Z axes respectively, X, Y and Z are the real output value of each flux gate of X, Y and Z axes respectively, X, Y and Z are the preset offset value of each flux gate of X, Y and Z axes respectively, and X, Y and Z are the preset scale value of each flux gate of X, Y and Z axes respectively.
offset
offset
offset
Scale
Scale
Scale
X, Y, Z, X, Y and Z can be stored in the geomagnetic measurement module 110's own memory (not shown), or can directly be calculated using an equation of the same form of Equation 2. As the detailed configuration of the geomagnetic measurement module 110 is conventional and is similar to that of the tilt measurement module 120 in FIG. 2, a drawing and description of the geomagnetic measurement module 110 is omitted.
Equation
5
ψ
=
tan
-
1
Y
norm
Z
norm
*
-
*
cosϕ
sinϕ
X
norm
Y
norm
Z
norm
*
-
*
*
-
*
*
cos
θ
sin
θ
sinϕ
sin
θ
cosϕ
The controller 140 applies the readjusted pitch angle and roll angle and the normalized output value of each flux gate of X, Y and Z axes to the below equation to calculate the azimuth.
norm
norm
norm
In Equation 5, X, Y and Z are the normalized output value of each flux gate of X, Y and Z axes respectively, θ is the pitch angle, and <emiff1_type1/> is the roll angle. Equation 5 is an equation corresponding to when a value of Z axis vertical to a horizontal plane is set to a negative number. In other words, Equation 5 is effective in a case where the 3-axis geomagnetic sensor 100 is horizontally located on the earth surface of the 3-axis Northern Hemisphere as in FIG. 3 when the 3-axis geomagnetic sensor 100 is located as in FIG. 3. In this case, the normalized value of Z axis is obtained as a negative value.
Meanwhile, signs in Equations 3 and 5 change according to the location of the axes in the 3-axis flux gates of the geomagnetic measurement module 110 and the 3-axis acceleration sensors of the tilt measurement module 120. The signs in Equations 3 and 5, change in a case where the pitch angle is greater than 90° when the azimuth is calculated. For example, if the pitch angle is 120°, signals of X and Y axes of the signals of the geomagnetic sensor 100 change their signs, and the acceleration sensor changes θ=θ-180 to apply to the equations.
FIG. 3 shows an example of 3-axis location in the 3-axis flux gates and 3-axis acceleration sensors. According to FIG. 3, each X axis of the geomagnetic measurement module 110 and the tilt measurement module 120 in the geomagnetic sensor 100 is located in the forward direction of the geomagnetic sensor 100, each Y axis is located in the direction perpendicular to each X axis on the same flat where the geomagnetic sensor 100 is located, and each Z axis is located in the upward direction of the geomagnetic sensor 100, crossing with X and Y axes at right angles. Equations 3 and 5 are applied when 3 axes are located as in FIG. 3.
X
FIG. 4 is a graph entirely showing the region of the pitch angle. The region of the pitch angle is divided into 4 large regions P1 ~ P4 according to the size of θ, and each large region is divided into 2 small regions a and b.
X
Z
X
X
Z
X
Accordingly, if it is 0°≤ θ <45° and θ ≥ 0°, the region of the pitch angle is recognized as an a region of P1, that is, P1-a, and θ becomes the pitch angle. However, if it is 0°≤ θ <45° and θ < 0°, the region of the pitch angle is recognized as P1-b and 180°- θ becomes the pitch angle.
X
Z
X
X
Z
Z
If it is θ ≥ 45° and θ ≥ 0°, the region of the pitch angle is recognized as P2-a and θ becomes the pitch angle. However, if it is θ ≥ 45° and θ < 0°, the region of the pitch angle is recognized as P2-b and 90°- θ becomes the pitch angle.
X
Z
X
X
Z
X
If it is -45° ≤ θ < 0° and θ < 0°, the region of the pitch angle is recognized as P3-a and -180°- θ becomes the pitch angle. However, if it is -45° ≤ θ < 0° and θ ≥ 0°, the region of the pitch angle is recognized as P3-b and θ becomes the pitch angle.
X
Z
X
Z
Y
X
Y
Z
X
Z
Y
X
Y
Z
Meanwhile, if it is θ < -45°, the region of the pitch angle is recognized as the large region of P4. In this case, the small region is determined according to the size of θ. That is, if it is θ < -45° and θ < 0°, the region of the pitch angle is recognized as P4-a. If it is <emiff1_type1/>≥ 45° in the region of P4-a, θ becomes the pitch angle, or if it is <emiff1_type1/>< 45° in the region of P4-a, θ - 90° becomes the pitch angle. However, if it is θ < -45° and θ≥ 0°, the region of the pitch angle is recognized as P4-b. If it is <emiff1_type1/> ≥ 45° in the region of P4-b, θ becomes the pitch angle, or if it is <emiff1_type1/>< 45° in the region of P4-b, θ - 90° becomes the pitch angle.
Y
FIG. 5 is a graph entirely showing the region of the roll angle. The region of the roll angle is divided into 4 large regions R1 ~ R4 according to the size of <emiff1_type1/>, and each large region is divided into 2 small regions a and b.
Y
Z
Y
Y
Z
Y
Accordingly, if it is 0°≤ <emiff1_type1/> <45° and <emiff1_type1/> ≥ 0°, the region of the roll angle is recognized as an a region of R1, that is, R1-a, and <emiff1_type1/> becomes the roll angle. However, if it is 0°≤ <emiff1_type1/> <45° and <emiff1_type1/> < 0°, the region of the roll angle is recognized as R1-b and 180°- <emiff1_type1/> becomes the roll angle.
Y
Z
Y
Y
Z
Y
If it is <emiff1_type1/> ≥ 45° and <emiff1_type1/> ≥ 0°, the region of the roll angle is recognized as R2-a and <emiff1_type1/> becomes the roll angle. However, if it is <emiff1_type1/> ≥ 45° and <emiff1_type1/> < 0°, the region of the roll angle is recognized as R2-b and 90°- <emiff1_type1/> becomes the roll angle.
Y
Z
Y
Y
Z
Y
If it is -45° ≤ <emiff1_type1/> < 0° and <emiff1_type1/> < 0°, the region of the roll angle is recognized as R3-a and -180°- <emiff1_type1/> becomes the roll angle. However, if it is -45° ≤ <emiff1_type1/> < 0° and <emiff1_type1/> ≥ 0°, the region of the roll angle is recognized as R3-b and <emiff1_type1/> becomes the roll angle.
Y
Z
Y
Z
X
Y
X
Z
Meanwhile, if it is <emiff1_type1/> < -45°, the region of the roll angle is recognized as the large region of R4. In this case, the small region is determined according to the size of <emiff1_type1/>. That is, if it is <emiff1_type1/> < -45° and <emiff1_type1/> < 0°, the region of the roll angle is recognized as R4-a. If it is θ ≥ 45° in the region of R4-a, <emiff1_type1/> becomes the roll angle, or if it is θ < 45° in the region of R4-a, <emiff1_type1/>- 90° becomes the roll angle.
Y
Z
X
Y
X
Z
Meanwhile, if it is <emiff1_type1/> < -45° and <emiff1_type1/> ≥ 0°, the region of the roll angle is recognized as R4-b. If it is θ ≥ 45° in the region of R4-b, <emiff1_type1/> becomes the roll angle, or if it is θ < 45° in the region of R4-b, <emiff1_type1/> - 90° becomes the roll angle.
Referring to FIGs. 4 and 5, the pitch angle and roll angle can be measured in the range of ±180°.
X
Y
Z
Z
FIG. 6 is a flow chart showing an azimuth calculation method according to an exemplary embodiment of the present invention. According to FIG. 6, the output values of each flux gate of X, Y and Z axes is calculated and normalized (S610), and the pitch angle and roll angle are primarily calculated using each acceleration sensor of X, Y and Z axes (S620). Specifically, θ, <emiff1_type1/>, θ and <emiff1_type1/> are primarily calculated.
X
Y
Z
Z
Subsequently, the second calculation is performed to readjust θ, <emiff1_type1/>, θ and <emiff1_type1/> using the output value of the acceleration sensor of Z axis (S630).
As a result of the second calculation, if the pitch angle and roll angle are finalized, the azimuth is calculated using the readjusted pitch angle and roll angle and the normalized output value of each flux gate of X, Y and Z axes (S640). The azimuth can be calculated using Equation 5.
FIG. 7 is a flow chart specifically describing a pitch angle calculation method used in the azimuth calculation method of FIG. 6. According to FIG. 7, first the output values of each acceleration sensor of X, Y and Z axes is normalized by mapping it with a value of a predetermined range (S710). The normalization can be performed using Equation 1.
X
Y
Z
Z
X
X
Z
Z
X
Z
X
As a result, if θ, <emiff1_type1/>, θ and <emiff1_type1/> are primarily calculated (S715), it is determined if θ is between 0° and 45° (S720). If θ is between 0° and 45°, it is determined if θ is 0° or more (S725). If θ is 0° or more, θ becomes the pitch angle (S730). However, if θ is under 0°, 180°- θ becomes the pitch angle (S735).
X
X
X
Y
Y
Z
Y
X
Meanwhile, if θ is not between 0° and 45°, it is determined if θ is 45° or more (S740). As a result, if θ is 45° or more, it is determined if <emiff1_type1/> is under 45° (S745). If <emiff1_type1/> is under 45°, 90°- θ becomes the pitch angle (S750). Or, if <emiff1_type1/> is 45° or more, θ becomes the pitch angle (S755).
X
X
X
Z
Z
X
Z
X
Meanwhile, if θ is neither between 0° and 45° nor 45° or more, it is determined if θ is between -45° and 0° (S760). If θ is between -45° and 0°, it is determined if θ is 0° or more (S765). If θ is 0° or more, θ becomes the pitch angle (S770). Or, if θ is under 0°, -180°- θ becomes the pitch angle (S775).
X
X
Y
Y
Z
Y
X
Meanwhile, if θ is not between -45° and 0°, either, it is determined if θ is under -45° (S780). Also, it is determined if <emiff1_type1/> is under 45° (S785). If <emiff1_type1/> is under 45°, θ-90° becomes the pitch angle (S790). Or, if <emiff1_type1/> is 45° or more, θ becomes the pitch angle (S795).
FIG. 8 is a flow chart specifically describing a roll angle calculation method used in the azimuth calculation method of FIG. 6. According to FIG. 8, first the output values of each acceleration sensor of X, Y and Z axes is normalized by mapping it with a value of a predetermined range (S810). The normalization can be performed using Equation 1.
X
Y
Z
Z
Y
Y
Z
Z
Y
Z
Y
As a result, if θ, <emiff1_type1/>, θ and <emiff1_type1/> are primarily calculated (S815), it is determined if <emiff1_type1/> is between 0° and 45° (S820). If <emiff1_type1/> is between 0° and 45°, it is determined if <emiff1_type1/> is 0° or more (S825). If <emiff1_type1/> is 0° or more, <emiff1_type1/> becomes the roll angle (S830). However, if <emiff1_type1/> is under 0°, 180°- <emiff1_type1/> becomes the roll angle (S835).
Y
Y
Y
X
X
Z
X
Y
Meanwhile, if <emiff1_type1/> is not between 0° and 45°, it is determined if <emiff1_type1/> is 45° or more (S840). As a result, if <emiff1_type1/> is 45° or more, it is determined if θ is under 45° (S845). If θ is under 45°, 90°- <emiff1_type1/> becomes the roll angle (S850). Or, if θ is 45° or more, <emiff1_type1/> becomes the roll angle (S855).
Y
Y
Y
Z
Z
Y
Z
Y
Meanwhile, if <emiff1_type1/> is neither between 0° and 45° nor 45° or more, it is determined if <emiff1_type1/> is between -45° and 0° (S860). If <emiff1_type1/> is between -45° and 0°, it is determined if <emiff1_type1/> is 0° or more (S865). If <emiff1_type1/> is 0° or more, <emiff1_type1/> becomes the roll angle (S870). Or, if <emiff1_type1/> is under 0°, -180°- <emiff1_type1/> becomes the roll angle (S875).
Y
Y
X
X
Z
X
Y
Meanwhile, if <emiff1_type1/> is not between -45° and 0°, either, it is determined if <emiff1_type1/> is under -45° (S880). Also, it is determined if θ is under 45° (S885). If θ is under 45°, <emiff1_type1/> -90° becomes the roll angle (S890). Or, if θ is 45° or more, <emiff1_type1/> becomes the roll angle (S895).
Therefore, the pitch angle and roll angle are precisely calculated also in a region where the resolving power of an inverse tangent (ARCSIN) function decreases so that the azimuth can be compensated.
As can be appreciated from the above description, the pitch angle and roll angle are precisely calculated using 3-axis acceleration sensor. That is, the pitch angle and roll angle are primarily calculated using each acceleration sensor of X and Y axes, and the pitch angle and roll angle are readjusted by the second calculation using an output value of acceleration sensor of Z axis. Accordingly, even when the resolving power of the ADC is not high enough, the pitch angle and roll angle can be precisely calculated. Consequently, the azimuth can precisely be calculated. In addition, the range of measuring the pitch angle and roll angle is extended from ±90° to ±180°.
In addition to the above-described exemplary embodiments, exemplary embodiments of the present invention can also be implemented by executing computer readable code/instructions in/on a medium, e.g., a computer readable medium. The medium can correspond to any medium/media permitting the storing and/or transmission of the computer readable code.
The computer readable code/instructions can be recorded/transferred in/on a medium in a variety of ways, with examples of the medium including magnetic storage media (e.g., floppy disks, hard disks, magnetic tapes, etc.), optical recording media (e.g., CD-ROMs, or DVDs), magneto-optical media (e.g., floptical disks), hardware storage devices (e.g., read only memory media, random access memory media, flash memories, etc.) and storage/transmission media such as carrier waves transmitting signals, which may include instructions, data structures, etc. Examples of storage/transmission media may include wired and/or wireless transmission (such as transmission through the Internet). Examples of wired storage/transmission media may include optical wires and metallic wires. The medium/media may also be a distributed network, so that the computer readable code/instructions is stored/transferred and executed in a distributed fashion. The computer readable code/instructions may be executed by one or more processors.
Although a few exemplary embodiments of the present invention have been shown and described, it would be appreciated by those skilled in the art that changes may be made in these exemplary embodiments without departing from the scope of the invention, as defined in the claims. | |
The invention discloses an automatic feces character identification method in combination of an image color, a HOG (Histogram of Oriented Gradient) and an SVM, which belongs to the fields of digital image processing and machine learning, and particularly relates to an automatic feces character identification method which combines the color information of an image, HOG features of the image and theSVM. The method comprises steps: the HOG features of a feces image are extracted firstly, wherein the features are formed through calculating and counting the size of a gradient histogram in an imagelocal area; the HOG features of the image are then used to train the SVM to obtain a classifier model; the color information of the image is then extracted; after the color information of the image is analyzed and processed, the feces character is judged preliminarily; the HOG features of a feces image which is preliminarily judged to be unknown are then extracted; the HOG features are inputted to the classifier mode to obtain a result; and the type of the feces character is obtained according to the size relationship between the result and zero. Compared with the original feces character identification method, the method of the invention has the advantages of high efficiency, low cost and high automatic degree. | |
So after some testing I concluded that this was about one second. Does anyone know the actual number that this should be?
make the number bigger or put it in a while loop that repeats for a certain amount of time to get a more accurate estimate, not just one second, but more like 60 seconds so small differences become more noticeable.
If you start this code and then press enter right when you start a separate timer and then press anything else at your desired time it will give you a good number. Obviously off because of human error but worth a shot.
I like For( loops better - they were made to do this kind of thing.
However, a For( loop works a little different from Repeat, so the time will be different.
Get a real stopwatch and start this program at the same time you start the stopwatch. At any time (the longer the better accuracy), stop your stopwatch and press any button but [On] on your calculator. Write down both the number shown on the calculator and the time on your stopwatch. Then press [On] to quit the program.
Divide the number from your stopwatch by the number your calculator output. That's about how many seconds it takes for it to loop.
Repeat the process many times, and find the average of all of them. Then go back to the code I gave you and replace the last argument 1 in the For( with that average. When you run your program again, it should loop about once every second. | http://tibasicdev.wikidot.com/forum/t-460609/stopwatches |
Q:
What does $\sum\limits_{i\neq j}$ mean?
I have seen similar questions (perhaps too similar, I´m sorry if so) but none of them do too much of a good job on defining what an expression such as $$\sum_{i\neq j}^n(x_iy_j)$$ means exactly. What set of numbers am I adding up? What does it mean to have two variables on the running variables under the sum? Is there another notation that might clear things up for me?
A:
In words, sum $x_iy_j$ for all allowed values of $i$ and $j$ except for $i = j$. As an example, assuming that $i$ and $j$ take values 1, 2 and 3, the full unrestricted sum would have nine terms:
\begin{align}
\sum_{i,j} x_iy_j =\\
& x_1 y_1 + x_1 y_2 + x_1 y_3\\
+& x_2 y_1 + x_2 y_2 + x_2 y_3 \\
+& x_3 y_1 + x_3 y_2 + x_3 y_3
\end{align}
whereas the restricted sum in your question would have only six terms: the diagonal terms where the two subscripts have the same value would not be present:
\begin{align}
\sum_{i \ne j} x_iy_j =\\
& &x_1 y_2 &+x_1 y_3\\
+& x_2 y_1 & &+ x_2 y_3 \\
+& x_3 y_1 &+ x_3 y_2 &
\end{align}
A:
In general, a summation of the form
$$\sum_{P(i)}x_i$$
where $i$ is some form of index and $P$ is a proposition means the sum of $x_i$ over all possible $i$ for which $P(i)$ is true. (This is of course not the only use of the summation notation, but it is a very common one.) For instance,
$$\sum_{i\in \{1,2,\dots,n\}}x_i$$
means $x_1+x_2+\dots+x_n$. Similarly,
$$\sum_{i\neq j}x_iy_j$$
means the sum over all $i,j\in\mathbb Z_{>0}$ satisfying $i\neq j$. Here, since there is an upper bound of $n$, presumably the meaning of $$\sum_{i\neq j}^nx_iy_j$$is the sum of $x_iy_j$ over all $i,j$ in the set $\{1,2,\dots,n\}$ satisfying $i\neq j$. The specific meaning could differ slightly depending on context: for instance it could be possible that $0$ is a permitted value for $i$ and $j$. This is however the general idea, and to confirm that this is the meaning in this particular case, you can often derive that from the context in which you encounter this symbol.
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Steven Fogarty scored his first professional hat trick, including two shorthanded goals, and had an assist, and Chris Nell made 29 saves in his first AHL action of the season Sunday at the Dunkin’ Donuts Center Providence, in a 4-1 Wolf Pack win over the Providence Bruins.
The matchup was the second half of a home-and-home between the two clubs, after the Bruins had defeated the Wolf Pack 3-2 in overtime Saturday night at the XL Center.
Tim Gettinger added a goal and an assist for Hartford, and Vinni Lettieri had two assists. Gemel Smith scored the only Providence goal, and Dan Vladar made 25 saves in the Bruin net.
“We played on the right side of the puck, and I thought we were able to execute plays and play the game with good pace,” Wolf Pack head coach Keith McCambridge said. “All those, combined with good work ethic, gave us a chance to win that game.”
The Wolf Pack had a strong first period, outshooting Providence 14-6, and went ahead on a power-play goal by Gettinger at 16:29. With Trent Frederic off for high-sticking, Fogarty tipped the puck to Sean Day, who moved behind the Bruin net before handing off to Gettinger. He scored on a wraparound, slipping the puck between Vladar’s left skate and the goal post.
The Bruins had a 15-7 shots advantage in the second frame, but the Wolf Pack had the only two goals of the period, both by Fogarty, to increase their lead to 3-0.
Fogarty scored the Wolf Pack’s second shorthanded goal in as many games at 3:50, on Providence’s first power play of the game. Fogarty centered the puck from the right-wing side for Lettieri, who was headed hard towards the net, and it deflected off of a Providence defender and in behind Vladar.
Fogarty clicked again at 9:57, on a tap-in set up by Lettieri. After Brandon Crawley kept the puck in the Bruin zone on the left side, Fogarty fed the puck to Lettieri in the right-wing circle. He immediately whipped it back to Fogarty at the left side of the goal mouth, and he was easily able to put the puck into the open side.
Smith ended Nell’s shutout bid at the 12-minute mark of the third period, on a rebound of a shot from the left point by Connor Clifton. That would be the only puck to get past Nell, who joined the Wolf Pack Wednesday from the Maine Mariners of the ECHL and was making his first AHL appearance since March 17 of last season.
“Tough building to play in, but I thought Chris did an excellent job for us,” said McCambridge. “When there were some breakdowns, he was in position to make saves, and was a big part of our penalty kill having success as well.”
The Wolf Pack got that goal back less than two minutes later, at 13:57, with their second shorthanded goal in three man-disadvantages. Ryan Lindgren broke up a Paul Carey pass in the Wolf Pack zone and carried down the left side the Providence end, before sliding a pass across the slot to Gettinger. He poked it on goal, and Fogarty was able to wedge it out from underneath Vladar, in a goalmouth scramble, and knock it into the net.
That completed the Wolf Pack’s first hat trick in over two years, and Fogarty’s first career four-goal game as a pro.
“Steven has all the details that you want players to have in his position,” McCambridge said of Fogarty. “And I’m real happy that he was able to get rewarded for that.”
The Wolf Pack’s next game is back home at the XL Center this Wednesday night, February 27, a 7:00 PM contest vs. the Hershey Bears. That is another chance to take advantage of the Wolf Pack’s “Click It or Ticket Family Value Pack”, which includes two tickets, two hot dogs and two sodas, all for just $40.
Tickets for all 2018-19 Wolf Pack home games are on sale now at the Agera Energy Ticket Office at the XL Center, on-line at www.hartfordwolfpack.com and by phone at (877) 522-8499. Tickets purchased in advance for kids 12 or younger start at just $13 each, and all tickets will have a $3 day-of-game increase.
Season ticket information for the Wolf Pack’s 2018-19 AHL season can be found on-line at www.hartfordwolfpack.com. To speak with a representative about all of the Wolf Pack’s many attractive ticketing options, call (855) 762-6451, or click here to request more info.
Hartford Wolf Pack 4 at Providence Bruins 1
Sunday, February 24, 2019 - Dunkin' Donuts Center
Hartford 1 2 1 - 4
Providence 0 0 1 - 1
1st Period-1, Hartford, Gettinger 12 (Day, Fogarty), 16:29 (PP). Penalties-Frederic Pro (high-sticking), 14:53; Blidh Pro (interference), 18:08; Carey Pro (tripping), 18:25.
2nd Period-2, Hartford, Fogarty 15 (Lettieri), 3:50 (SH). 3, Hartford, Fogarty 16 (Lettieri, Crawley), 9:57. Penalties-Crawley Hfd (boarding), 3:19; Beleskey Hfd (tripping), 18:25.
3rd Period-4, Providence, Smith 12 (Clifton, McNeill), 12:00. 5, Hartford, Fogarty 17 (Lindgren, Gettinger), 13:57 (SH). Penalties-Raddysh Hfd (hooking), 12:32; Fogarty Hfd (roughing), 13:57; Lindgren Hfd (roughing), 13:57; Clifton Pro (roughing), 13:57; Szwarz Pro (roughing), 13:57; Gettinger Hfd (cross-checking), 14:55; Crawley Hfd (interference), 19:07.
Shots on Goal-Hartford 14-7-8-29. Providence 6-15-9-30.
Power Play Opportunities-Hartford 1 / 3; Providence 0 / 5.
Goalies-Hartford, Nell 1-0-0 (30 shots-29 saves). Providence, Vladar 8-11-4 (29 shots-25 saves).
A-8,932
Referees-Ben O'Quinn (27), Andrew Bruggeman (22).
Linesmen-Chris Leavitt (65), Jared Waitt (60). | http://www.hartfordwolfpack.com/news/detail/fogartys-hat-trick-four-points-lead-pack-to-victory-in-providence |
Lemosho Route is a fresher route on Mount Kilimanjaro that approaches from the west. It is a tough and long route, but one that is preferred by most trustworthy Kilimanjaro outfitters due to its smaller crowds, scenic diversity and high victory rates.
Tour Program
Highlight
Day1: Arusha (1400m) – Londorossi Gate (2100m) – Mti Mkubwa camp (2750m) Hiking time: 3 hours
Day2: Lemosho Route 6 days–Mti Mkubwa Camp (2750m)–Shira 2 Camp (3840m)
Hiking time: 6.5 hours
Day3: Lemosho Route 6 days–Shira(3840m)–Lava Tower(4630m)–Barranco Camp (3950m)
Hiking time: 7 hours
Day4: Lemosho Route 6 days–Barranco Camp(3950m)–Barafu Camp(4550m) Hiking time: 7 hours
Day5: Lemosho Route 6 days–Summit Attempt Barafu camp(4550m)–Uhuru Peak (5895m)–Mweka (3100m)
Hiking time: 8 hours to reach Uhuru Peak, 7-8 hours to descend to Mweka
Day6: Lemosho Route 6 days–Mweka camp(3100m)–Mweka Gate(1980m) Hiking time: 3 hours
Map
Full Itinerary
Day 1 Arusha (1400m) – Londorossi Gate (2100m) – Mti Mkubwa camp (2750m)
Hiking time: 3 hours
Habitat: Montane forest
After eating breakfast at your hotel, your guide will brief you on the day. You will drive two hours from Arusha (1400m) to Londorossi Park Gate (2100m). afternoon lunch break and arrive at Mti Mkubwa (“Big Tree”) Campsite (2750m) in the early evening. The porters, who arrive at the campsite before the clients, will set up your tent and boil water for drinking and washing. The chef will prepare a snack then dinner for the clients. At nighttime, mountain temperatures may drop to freezing so be prepared!
Day 2 Lemosho Route 6 days – Mti Mkubwa Camp (2750m) – Shira 2 Camp (3840m)
Hiking time: 6.5 hours
Habitat: Moorland
After an early morning breakfast, you will begin your ascent out of the rainforest and into the heather moorland zone. You will cross many streams and walk over a plateau that leads to Shira 2 Camp (3840m). At this campsite, you will be next to a stream and have a spectacular view of the Western Breach and its glaciers in the East. Similar to the first night, You will enjoy evening snacks then dinner prepared by our chef. Be prepared for a cold night as temperatures drop below freezing at this exposed camp.
Day 3 Lemosho Route 6 days – Shira (3840m) – Lava Tower (4630m) – Barranco Camp (3950m)
Hiking time: 7 hours
Distance: Approximately 15 kilometers
Habitat: Semi desert
Following an early morning breakfast, you will leave the moorland environment and enter the semi desert and rocky landscape. After 5 hours of walking east, you will be come face to face with the Lava Tower (4630m).. At this point of the hike, it is normal for hikers to start feeling the effects from the altitude including headaches and shortness of breath. Following lunch, you will descend from Lava Tower (4630m) to the Barranco Campsite (3950m). The 680m descent gives hikers a huge advantage to allow their bodies to adjust to the conditions of high altitude. The descent to camp takes around 2 hours to reach. It is located in a valley below the Breach and Great Barranco Wall (“Breakfast Wall”). Drinking and washing water and dinner will be served as hikers view the sun setting.
Day 4 Lemosho Route 6 days – Barranco Camp (3950m) – Barafu Camp (4550m)
Hiking time: 7 hours
Distance: Approximately 13 kilometers
Habitat: Alpine desert
After an early morning breakfast, it is now time to conquer the Great Barranco Wall! Although it may look intimating at first glance, The trail then winds up and down in the Karanga Valley and intersects with the Mweka Route, which is the trail used to descend on the final two days. As you continue hiking for an hour, you will reach Barafu Hut. The word “barafu” in Swahili means “ice” and this camp is located on a rocky, exposed ridge. Tents will be exposed to wind and rocks so it is important for hikers to familiarize themselves with the campsite before dark. An early dinner will be served so hikers can rest before attempting the summit the same night. Your guide will brief you in detail on how to prepare for summit night. Get to sleep by 19:00!
Day 5 Lemosho Route 6 days – Summit Attempt Barafu camp (4550m) – Uhuru Peak (5895m) – Mweka (3100m)
Hiking time: 8 hours to reach Uhuru Peak, 7-8 hours to descend to Mweka
Distance: Approximately 7 kilometers ascent and 23 kilometers descent
Habitat: Stone scree and ice-capped summit
Your guide will wake you around 23:30 for tea and biscuits. You will then begin your summit attempt. The route heads northwest and you will ascend over stone scree. During the ascent, many hikers feel that this is the most mentally and physically challenging part of the climb. In about 6 hours, you will reach Stella Point (5740m), located on the crater rim. After enjoying the magnificent sunrise, you will continue ascending for about 2 hours on a snow-covered trail to Uhuru Peak (5895m). Reaching the summit of Mt. Kilimanjaro is a lifetime accomplishment! You will be able to spend a short time on the summit taking photographs and drinking tea before the descent to Barafu begins. The hike down to Barafu Camp takes about 3 hours.
At camp, you will rest and enjoy a hot lunch in the sun. After eating, you will continue descending down to Mweka Hut (3100m). The Mweka Trail will lead you through the scree and rocks to the moorland and eventually into the rain forest. Mweka Camp (3100m) is located in the upper rain forest, so fog and rain should be expected. You will have a dinner, wash, and rest soundly at camp.
Day 6 Lemosho Route 6 days – Mweka camp (3100m) – Mweka Gate (1980m)
Hiking time: 3 hours
Distance: Approximately 15 kilometers
Habitat: Forest
Following a well-deserved breakfast, your staff will have a big celebration full of dancing and singing. It is here on the mountain that you will present your tips to the guide, assistant guides, chef(s), and porters. After celebrating, you will descend for three hours back to Mweka Gate. The National Park requires all hikers to sign their names to receive certificates of completion. Hikers who reached Stella Point (5740m) receive green certificates and hikers who reached Uhuru Peak (5895m) receive gold certificates. After receiving certificates, hikers will descend into the Mweka village for 1 hour (3 kilometers). You will be served a hot lunch then you will drive back to Arusha for long overdue showers and more celebrations.
Inclusive
Camping safari according to the itinerary
Transportation in a 4×4 safari vehicle
Professional, English-speaking guide
Professional, English-speaking safari cook
Overnight stays in safari tents
Camping equipment (tents, sleeping mats, chairs, tables etc.)
Meals according to the itinerary
Mineral water
All mentioned activities
All national park fees
Flying Doctors insurance (AMREF) during the safari
Exclusive
Sleeping bag
Flight
Optional activities
Alcoholic and soft drinks
Visa fees
Tips
Personal spending money for souvenirs etc.
Travel insurance
Good tips attempting the Summit Mt Kilimanjaro
Choose the right route for your Kilimanjaro climb One of the biggest problems on Kilimanjaro is people trying to go up the mountain way too quickly.
Mentally prepare, depending on your level of comfort with the outdoors, you may need to prepare yourself for life on the mountain.
Make sure you have the right clothing and boots for Ascending and descending
Hydration is very important while climbing Kilimanjaro at higher levels of altitude, your body will dehydrate much quicker than it will at sea level, and you will have to make sure you are drinking plenty of water.
Don’t be afraid of little Headache, one of the greatest causes of headaches on the mountain is due to dehydration, so drinking water can greatly help to eliminate or lessen your headaches.
Slow and steady wins the race. When taking on a challenge like Kilimanjaro, or any long trek/climb, you have to remember that it is a marathon, not a sprint. Acclimatization to the low levels of oxygen in the mountains requires you to take your time, to slowly get your body used to lack of oxygen.
Bring some summit treats, there is always plenty of food to choose from on the mountain, but when your tummy is having a hard time with the altitude, those familiar snacks may be all you can get down.
New or extra batteries, replace your head lamp and camera batteries with new ones on your summit night.
Mountain water, the stream water high on the mountain Kilimanjaro has been tested and has been found to be fit for drinking. However, we recommend that you be on the safe-side and use water purification tablets before drinking.
Wet wipes, there is no washing water at Barafu, Kibo and Arrow Glacier camps. Wet Wipes are very useful
Stay out of the sun whenever possible, the second way of protecting yourself is to always wear your goofy looking, yet extremely important, sun hat.
Ladies…No make-up here is necessary!
Remember you are on holidays, enjoy yourself! The most important thing to remember is that you are on vacation, relax, have an open mind, and enjoy yourself! | https://gurugurusafaris.com/en/kilimanjaro/lemosho-route-6-days |
Swordfish is an extension of X-Wing, where instead of 2x2 square of candidates, we consider 3x3 square of candidates.
Similarly to X-Wing, the Swordfish is formed either by rows or columns and the eliminations occur in columns or rows, respectively.
In this figure, the Swordfish cells are marked with blue and we only consider a candidate X. The Swordfish is fixed by columns, i.e. in columns 3, 5 and 8 there are no other candidates X. There are more candidates X, marked in the rows which contain the swordfish, namely, the small Xs in rows B, C and G. All the small Xs can be eliminated but let us see why.
Suppose we place any of the X candidates in the Swordfish configuration (the blue cells), for example
B3. The scratched candidates can be eliminated. This leaves an X-Wing, formed by cells C5, C8, G5 and
G8, and which eliminates the remaining small Xs in the corresponding rows C and G.
You can check that no matter which blue cell we choose to place the X, all small Xs in the rows will be eliminated. Similarly to X-Wing (and to other methods) we do not know, where the blue X candidates are, but they surely be in some three out of the nine cells - one in each column.
Swordfish rarely occurs in its perfect form, i.e. there rarely are exactly nine cells. Some cells may already have a value, but the important feature of the structure is that the existing cells with candidate X are placed in a 3x3 grid.
The first example is a 2-3-2 Swordfish, that looks similarly to the figure above.
The candidate 5 in the dark blue cells are arranged in columns - every column contains
2 or 3 candidates from the dark blue cells and they are the only ones in the corresponding column.
Since the Swordfish is arranged in columns, we can eliminate all candidates in the corresponding rows.
To see, again, why this is the case, suppose that we place the candidate 5 in B3. This eliminates the candidate 5 on row B and the dark blue cell candidate on C3. This leaves an X-Wing with cells C5, C8, G5 and G8 (same as before), which eliminates the remaining candidates, marked in red.
Let us quickly see another example, where the Swordfish cells are arranged in rows. The Swordfish is formed by rows C, F and I, and there are no other 8s apart from the ones in the dark blue cells. The eliminations occur in columns 3 and 8. We are still looking for a perfect Swordfish and as soon as one pops up, we will include it in this article. | https://sudokuapp.com/learn/hard/swordfish |
Euclidean distance between two vectors (single row matrix)
I have two vectors (single row matrices). Assume that we already know the length len.
A = [ x1 x2 x3 x4 x5 .... ] B = [ y1 y2 y3 y4 y5 .... ]
To calculate Euclidean distance between them what is the fastest method. My first attempt is:
diff = A - B sum = 0 for column = 1:len sum += diff(1, column)^2 distance = sqrt(sum)
I have loop through this methods millions of times. So, I am looking for something which is fast and correct. Note that I am not using MATLAB and don't have pdist2 API available.
Answers
diff = A - B; distance = sqrt(diff * diff');
or
distance = norm(A - B);
[val idx] = sort(sum(abs(Ti-Qi)./(1+Ti+Qi)));
or
[val idx] = sort(sqrt(sum((Ti-Qi).^2)));
Val is the value and idx is the original index value of the column being sorted after applying Euclidean distance. (Matlab Code)
To add to @kol answer, | http://www.brokencontrollers.com/faq/13368597.shtml |
The panel will be asked to judge the three entries as a “body of work.” The judges will score each case on a scale of 0 to 10.
The judges will be asked to base their evaluations on creativity, strategy and targeting, as well as integration with the overall campaign and results that are attributable to the digital efforts. The digital component should be the campaign’s driving force. Work that is uniquely digital, rather than material that could appear in other media but used in a digital campaign, will be given preference.
Judges will be looking for both versatility and consistency, and will be asked to assess the overall impact of the digital efforts across all platforms. They will also evaluate the digital executions based on how visually compelling and user friendly they are, and whether they’re unique enough to stand out. You should keep this in mind when preparing your submission.
The panel will consist of digital experts from the client community, as well as agency professionals.
The judging panel will not meet as a group. The submissions will be delivered to them individually and they will review the work on their own.
The judges will fill out an online score sheet supplied by strategy. Their responses will be confidential.
The winner will be selected on the basis of a straight numerical tally. Each entry will be given two aggregate scores – strategic and creative – that will be combined and given a average score. The Digital Agency of the Year will be the agency that achieves the highest score across the three cases. | http://daoy.strategyonline.ca/entryinfo/Judging_Process |
I think a big part of my conviction that “I am a writer” came early in my childhood from my love for office supplies. Aren’t all big decisions based on important feelings like that? I see it in my daughter too these days, when she cannot be convinced to leave Staples unless yelled at, even after spending there no less than one hour every single time we visit.
The laptop (why do they insist on calling them “notebooks” these days? Notebooks are the paper thingies, people!) is still my best friend, but a little paper and pen by its side seem to make even more sense. Just like the e-reader keeps good company to the traditionally printed books, right? Exactly.
I remember, as a very young child, before I knew how to write, filling notebook after notebook with squiggles that in my mind looked a lot like real writing. I followed the ink lines left behind by the tip of my pen and it felt like a thing a beauty. And it still does. That’s the weirdest thing. It doesn’t wear off, this childish wonder.
I have never been too fancy about my pens. Anything works. I buy them from the pharmacy, grocery store or Staples, on a whim, and I go for pens that look good and have black ink. I don’t care about the difference between liquid ink and gel, or ball-point and needle point. I am afraid that there are very few pens that I don’t like. But because I don’t like change much, I have settled for now on two favorites: the Pilot Precise V5 and the Uni-Ball Signo Gelstick 0.7. My pen love goes to these two. I’m sure you were dying to know. | https://www.loritironpandit.com/pen-love-or-the-small-pleasures-of-office-supplies/ |
Recently, my wife Christie and I were sitting down at the dinner table with my daughter Mia, who we adopted from Thailand. Before joining our family, Mia spent the early years of her life in an institution. We brought her home in October 2014, shortly after her twelfth birthday, and she’s now 17.
That evening, Mia began asking questions about the adoption of one of our other daughters, Sydney. We adopted Sydney from Korea more than twenty years ago, and she joined our family when she was four months old. Mia had several questions about how old you have to be to adopt a baby and how old a child must be before they can be adopted.
As we talked, Mia mentioned that she could have been adopted as a baby but that nobody wanted her. I immediately stopped her and explained that wasn’t true. Because of issues surrounding Mia’s paperwork in Thailand, she was not legally eligible to be adopted until she was eight years old.
I looked at Mia and told her, “You were able to be adopted for the first time when you were eight. We found out about you when you were 10. After that, it took two years for all the paperwork to get done. Then we were able to get you when you turned 12.”
Christie then chimed in and told Mia we wished we could have adopted her as a baby. We would have loved to have her with us so much earlier. We wish we’d been able to love, support, and raise her in our family and our home during those first 12 years.
This information changed Mia’s story! For her entire life, she believed that nobody wanted her all those years. In reality, it only took two years for her parents to find her and welcome her into a loving family.
No Unwanted Children
Mia’s story is far too common. In many developing countries, children often wait in orphanages because of legal issues or paperwork delays. If those legal issues don’t get resolved, it could be several years before a child is able to be adopted. At that point, the child has been in institutionalized care for years, resulting in damage to their physical, emotional, and psychological development. And tragically, as children get older, their chances of adoption decrease significantly.
These children are not unwanted! Often, they have just fallen through the cracks in a broken system. These children need advocates — and that’s where we come in!
We believe every child deserves a family. That’s why we are working hard to get orphaned and vulnerable children out of institutions and into families while their legal situations get resolved. With your support and partnership, we are currently pioneering and expanding family-style care in several locations in Mexico and Honduras.
But none of this would happen without you — we are so grateful for your partnership! You can learn more about how you are making a difference in the lives of children around the world by visiting the projects page on our website.
Thank you for your generosity. Together, we are changing the way the world cares for orphans! | https://www.hopeeffect.com/author/joe-darago/ |
Ahead, we take a look at who is Melody Prochet dating now, who has she dated, Melody Prochet’s boyfriend, past relationships and dating history. We will also look at Melody’s biography, facts, net worth, and much more.
Who is Melody Prochet dating?
Melody Prochet is currently single, according to our records.
The French Pop Singer was born in France on April 3, 1987. French singer-songwriter who’s known as the lead singer of the band Melody’s Echo Chamber. She released the critically acclaimed EP Melody’s Echo Chamber in 2012.
Relationship status
As of 2023, Melody Prochet’s is not dating anyone. Melody is 35 years old. According to CelebsCouples, Melody Prochet had at least 1 relationship previously. She has not been previously engaged.
Fact: Melody Prochet is turning 36 years old in . Be sure to check out top 10 facts about Melody Prochet at FamousDetails.
About Melody Prochet’s boyfriend
Melody Prochet doesn’t have a boyfriend right now.
All dating histories are fact-checked and confirmed by our users. We use publicly available data and resources to ensure that our dating stats and biographies are accurate.
Who has Melody Prochet dated?
Like most celebrities, Melody Prochet tries to keep her personal and love life private, so check back often as we will continue to update this page with new dating news and rumors.
Melody Prochet boyfriends: She had at least 1 relationship previously. Melody Prochet has not been previously engaged. Melody Prochet has been in a relationship with Kevin Parker (musician) (2012). We are currently in process of looking up information on the previous dates and hookups.
Online rumors of Melody Prochets’s dating past may vary. While it’s relatively simple to find out who’s dating Melody Prochet, it’s harder to keep track of all her flings, hookups and breakups. It’s even harder to keep every celebrity dating page and relationship timeline up to date. If you see any information about Melody Prochet is dated, please let us know.
Relationship Statistics of Melody Prochet
What is Melody Prochet marital status?
Melody Prochet is single.
How many relationships did Melody Prochet have?
Melody Prochet had at least 1 relationship in the past.
How many children does Melody Prochet have?
She has no children.
Is Melody Prochet having any relationship affair?
This information is not available.
Melody Prochet Biography
Melody Prochet was born on a Friday, April 3, 1987 in France. Her birth name is Melody Prochet and she is currently 35 years old. People born on April 3 fall under the zodiac sign of Aries. Her zodiac animal is Rabbit.
She started to play the piano and the viola at an early age. In 2010, she released her debut album, Hunt The Sleeper, with the Dream-Pop band My Bee’s Garden.
Continue to the next page to see Melody Prochet net worth, popularity trend, new videos and more. | https://www.datingcelebs.com/who-is-melody-prochet-dating/ |
PROBLEM TO BE SOLVED: To eliminate generation of chatterings in an angle set condition of two members, make a device superior in operability of angle adjustment and operation stability, so that the device can be manufactured easily by reducing a cost, in the articulated device used in a connection part of the two members which can be flexed to each other.
SOLUTION: First/second metal fittings 1, 2 are pivotally mounted in a first pivotal shaft 4, a ratchet plate 3 and the first metal fitting 1 are pivotally mounted in a second pivotal shaft 5, the ratchet plate 3 has a slot-shaped pivotal support hole 8 for the second pivotal shaft 5 to pass through and a bending hole 9 for the first pivotal shaft 4 to pass through, the first/second pivotal shafts 4, 5 are always positioned in one end side 9a, 8a of the bending hole 9 and the pivotal support hole 8 by energizing of a spring member 6. The second metal fitting 2 can be turned only in the flexing direction by engaging a lock pawl 20 with a retchet 30, when the first pivotal shaft 4 is positioned in one end side 9a of the bending hole 9, and can be turned in the unfolding direction by transfer moving the first pivotal shaft 4 to the other end side 9a of the bending hole 9, when the ratchet plate 3 is displaced by contact of the lock pawl 20 with a disengaging stopper 31, the ratchet plate 3 is reset to the original attitude by the lock pawl 20 abutting to an engaging reset stopper 32.
COPYRIGHT: (C)2000,JPO | |
Plymouth runner-up at Northern Lakes meet
PLYMOUTH — It’s not often you get a lift from one of your opponents.
Nevertheless, that’s exactly what happened to Plymouth at the Northern Lakes Conference Girls Golf Meet Saturday, as the Lady Pilgrims finished second with a 363 behind NLC champ Warsaw’s 354, and Northridge beat out Wawasee to give Plymouth sole possession of second place in the final conference standings.
“I thanked the Northridge coach. I said ‘Man, we appreciate it.’ Our girls needed to pick it up today, and they realized that,” said Plymouth head coach Jack Kinney.
“On Wednesday when we went to practice the six varsity girls actually had a closed door meeting with no one else in the room, and they talked about where they wanted to go and how they wanted to finish this year. Wednesday, Thursday and Friday they really got with it in practice, and you could see a change in the team and the chemistry, and I think it was kind of evident today in the scores that were shot.”
Plymouth was led by Mandy McPherron’s All-NLC turn of 85 at Plymouth Country Club, while Emily Berger earned all-conference honorable mention honors with an 87 in second place for the Pilgrims.
“Emily Berger, she’s gaining confidence every week, and she’s realizing that one or two holes are keeping her from being in the 30s in nine holes and the low 80s for 18,” said Kinney. “Her confidence is growing, and I’m really proud of her for receiving honorable mention.”
“Mandy, I believe that she’s realized that ‘Wow, this is it. I’m a senior, and I don’t want to go out on a note that isn’t something that I would be proud of,’” continued the Plymouth boss. “These past couple weeks Mandy has picked it up to another level. She’s practiced more on her short game with her chipping, her putting, her lob shots. She’s worked hard with Pat Bailey, our assistant coach on all phases of the game. Mandy really deserved all-conference today, and I was happy to see her receive that honor because the last week or two she’s gotten into her mind the way that she wants to go out as a Lady Pilgrim golfer, as someone who’s going to be remembered for doing some great things. We’re really proud of her.”
In the No. 3 slot for Plymouth Saturday, Lauren Rearick shot a 93, while Kelly Henderson turned in a 98, and Emily Denney carded a 108.
Plymouth will play Penn and LaPorte Wednesday before heading into the Warsaw Sectional Saturday.
• NORTHERN LAKES
CONFERENCE MEET
At Plymouth Country Club (par 72)
Team scores: WARSAW 354, PLYMOUTH 363, NORTHRIDGE 369, WAWASEE 374, NORTHWOOD 381, GOSHEN 404, CONCORD 443, MEMORIAL 477
WARSAW (354): Sara Hartle 81, Nikki LaLonde 89, Nikki Grose 91, Elizabeth Meadows 101, Megan Dearlove 93.
PLYMOUTH (363): Emily Berger 87, Lauren Rearick 93, Kelly Henderson 98, Mandy McPherron 85, Emily Denney 108.
NORTHRIDGE (369): Alayna Frauhiger 81, Katie Moniot 93, Tori Roberts 97, Madison Stewart 110, Sidney Reed 98.
WAWASEE (374): Courtney Rassi 83, Kristin Lonn 89, Elizabeth Jackson 90, Courtney McDaniel 112, Alex Goralczyk 129.
NORTHWOOD (381): Heidi Morganthaler 83, Emily Myers 92, Kate Adams 101, Abbey Hersberger 105, Rachel Bebbe 107.
GOSHEN (404): Heather Gladfelter 75, Teage Minier 114, Kaitlyn Karnes 100, Rochelle Rock 115, Katheryn Giddens 122.
CONCORD (443): Kristen Fioritto 93, McKenna Grmiske 113, Abby Roberts 113, Ashlyn Ball 124, Olivia Kaufman 133.
MEMORIAL (477): Brenna Sherwood 77, Kaylin Byers 136, Kharlee Scroy 118, Megan Moore 148, A. Disher 146. | http://www.thepilotnews.com/content/plymouth-runner-northern-lakes-meet?quicktabs_4=1 |
Almost anyone can design a site, but the web design process determines how well it goes from ideation to launch.
As website designers, we tend to think about the overall web design process in terms of putting together visual elements, backend code and wireframes. And to be clear, these are all major parts of web design. But there is more to it than that. What makes a great website stand out from a good one is how well the process and strategy are implemented and aligned with the goals of your customer.
Here are the 10 essential steps that make up a successful web design process for you and your customers.
- Initial Discussion of Goals & Requirements
- Project Timelines & Scope
- The Web Design Proposal
- Summary of Navigation
- Content Creation, Collection, & Visual Elements
- Wireframing & Sitemap
- Prototyping
- Web Development
- Website Testing
- Website Launch
INITIAL DISCUSSION OF GOALS & REQUIREMENTS
It has been said, “A goal without a plan is just a wish.” Such is the case with the web design process.
One of the goals of any company is to have products and services well-known by its target audience. Designers must communicate with all stakeholders to understand these goals and nuances to discover the best way to visually represent a client’s brand and website to the public.
Designers should take time to briefly interview the stakeholders involved to understand each of these points before proceeding in the process. For every minute you spend planning and discussing with the key stakeholders, you will save hours in the actual design and development.
Once this information is collected, document it carefully as you will refer to it throughout the entire web design process.
PROJECT TIMELINES & SCOPE
Understanding the scope of the project (in writing preferably) and setting reasonable timelines to complete the project is a very critical step in the web design process. Having scope and timelines in place helps to set the bar of expectation.
One way to define a web design project scope is to use a Gantt chart. A Gantt chart marks specific activities of the process alongside the start and end dates for each task in the project. It gives a visual representation of what the designer must achieve by certain dates.
Additionally, it helps you to avoid (or at least, be aware of) “scope creep.” This happens when the client’s idea for the project evolves or changes in some way. If this becomes the case, you will easily find your scope of work along with your timelines and budget quickly expanding.
Need a website builder that helps you streamline your web design process? Start a nerDigital free trial today!
THE WEB DESIGN PROPOSAL
Ever heard of “Get it in writing?”
This is where the web design proposal comes in. Getting it in writing means it is a formalized agreement between two parties. At this step, both parties agree to what has been previously discussed for the website.
A solid web design proposal typically includes:
- Project scope
- Project timeline
- Budget and payment method
- Additional work conditions
- Revision rounds
- Specific requirements
A web design proposal tends to reduce misconceptions and misunderstandings during the process. It also helps to keep the project scope in line of the expectations for both parties.
SUMMARY OF NAVIGATION
The site structure or site navigation is a very important part of the web design process. It organizes the main pages and subpages within the website. From a planned site structure, designers can determine how best to display copy and visuals for the end users.
From drop-down menus to hover reveals and side navigation to top navigation, there is no shortage of ways to consider designing the navigation. Think about the needs of the organization and logical ways the end user would expect to see the information he or she is looking for.
CONTENT CREATION, COLLECTION, & VISUAL ELEMENTS
This is the fun and the second most important part. What’s a website without images and content, right?
Ideally, the client will provide the content and visuals they want you to use in the web design process. But you can be instrumental in the process by giving them guidance on what content to add to the website and how to organize it in a way that matters to the audience.
If you are blessed with the gift of design and copy, then here is where you will take the ideas from your client and collate them into compelling website copy and visuals that engage and possibly convert (if that is the goal).
When designing content, consider asking the following:
- Does the content engage the target reader?
- Does the content urge the reader to take the desired action?
- Does the content align with the preferred visual elements?
- Does the content guide the reader or user through the customer journey?
- Will Google care about the content enough to rank it? (Is it SEO-friendly?)
When designing the visual elements, consider asking the following:
- Does the logo match the brand?
- Does the color palette align to the brand’s style guide?
- Are there specific images that reflect the company to be included?
- Will there be videos that will embed in the web design?
- Will the visual elements be interactive?
- Do the font sizes and styles align to the brand’s style guide?
The visual style of the website alongside the content are a large part of what will engage the end user and compel them to take the next step in the customer journey or revisit your website frequently.
Sharp images and videos that don’t limit the bandwidth of a site can help to make it more engaging and communicate a user-friendly and brand-friendly message. This aids in building brand loyalty and trust.
When content and visuals are combined well, users can digest information easily and your customer will see more clicks, conversions, and revenue as a result.
MAPPING OUT & WIREFRAMING THE SITE
This is the step in the process where everything starts coming together. All the ingredients are there and now you are ready to mix them together.
Similar to a GPS that guides you to your destination or a blueprint for constructing a building, a good wireframe is the guide that takes your website from point A to point B in terms of overall design.
PROTOTYPING
Once the web design portion of the overall process has been completed, it is time to send the stakeholders a prototype of what the website looks like.
This is where the designer and stakeholders will ask the question: Did we achieve our goals with this web design project?
At this stage, stakeholders will be tasked with reviewing links, images and proofreading copy. In many cases, the prototyping view is the last view of the project before it goes to launch.
Any bugs, errors, redesigned elements, or revisions are made to improve the website.
ADVANCED WEB DEVELOPMENT
This stage is where the rubber meets the road as they say.
Any advanced features (like custom widgets and integrations) that haven’t been fully developed need to be built out. The web developer (who may be a different person from the designer) should take time to test and optimize for things like pagespeed throughout the development phase of the web design process.
FINAL WEBSITE TESTING & QA
Nothing is worse than a live website with broken links, misspellings or structural issues in copy, fuzzy images, or unoptimized pages.
At this stage, the website is reviewed again. First, by the internal team of people who worked to design and build the site. Second, by the stakeholders who will look at it from an internal and external perspective.
This is the opportunity to check the website in multiple browsers, on multiple mobile devices, and possibly with different IPs as well. You will also want to ensure here that the website is fully mobile responsive.
WEBSITE LAUNCH
This is the exciting part. Your hard work has paid off and you are ready to go live.
The requirements are filled. The content is engaging. The visuals look amazing. Every button is working and every page is linked correctly.
This is the giant leap for mankind. Well, for your stakeholders at least.
While you have taken the site public to the world, the work here isn’t over completely. Web design maintenance and updates will have to be conducted on an ongoing basis if the site is to continue producing successful results for your customer.
ENDING NOTE
Continuous improvement involves adding new content, running user tests frequently, monitoring analytics, and fixing bugs as they arise. An important thing to remember is that a website is never truly done. As outcomes and goals change, tweaks will be made to drive users and the company in those chosen directions.
The key to great web design is having a process that works every time. No matter what type of site you design and launch, your web design process should follow a set path that makes each project a success and every client happy. | https://nerdigital.com/the-10-steps-of-a-successful-web-design-process/ |
GAMBIER — Fiddle and folk melodies performed by Alayne and Gregory Wegner will animate the historic Quarry Chapel on Sunday.
The free concert performed by the daughter-and-father duo is hosted by the Friends of the Quarry Chapel at 10930 Quarry Chapel Road, Gambier, at 3 p.m. Doors open at 2:30 p.m. and parking is free.
In addition to Celtic fiddle tunes, the musicians will tackle classical sonatas on stringed instruments and may pitch in with vocals.
Wegner of Columbus is a recent Kenyon College graduate and a violinist, teacher and arts administrator. She began playing the violin at age 5 and now performs with the Knox County Symphony, the New Albany Symphony Orchestra and the Newark Granville Symphony Orchestra.
Gregory Wegner of Grand Rapids, Michigan, is an amateur musician skilled on the guitar, banjo, concertina, mandolin and violin. Gregory Wegner has been a familiar figure over the years at traditional Irish jam sessions in Grand Rapids. | https://mountvernonnews.com/article/2018/08/16/quarry-chapel-sparks-with-celtic-music-duo/ |
Friday, March 9th, 2018 at 9 pm ET, the spirited and jocular Heidi Hollis of the Heidi Hollis – The Outlander invites dream expert J.M. DeBord to help us understand complex theories from dream psychology with his easy to understand DREAMS 1-2-3 system.
J.M. DeBORD
Author, dream expert, and instructor, J.M. DeBord has appeared on dozens of media programs including Coast to Coast AM with George Noory, teaching his “anyone can do it” system of dream work. Noory praises him as “one heck of a dream interpreter,” adding to a long list of accolades for his skill and knowledge.
DeBord’s DREAMS 1-2-3 system boils down complex theories from dream psychology and combines them with modern interpretation techniques in a way that is easy to understand. His book “Dreams 1-2-3: Remember, Interpret, and Live Your Dreams” is a thorough introduction to dream work that requires no prior knowledge of dream interpretation. It was been widely praised by reviewers. – www.dreams123.net
The Dream Interpretation Dictionary: Symbols, Signs, and Meanings
What do reoccurring dreams reveal? What’s the purpose of nightmares—and can they be stopped? Why do some people show up in dreams? Are some dreams actually warnings? Going beyond superficial explanations, The Dream Interpretation Dictionary: Symbols, Signs and Meanings brings a deep and rich understanding to a variety of images, signs, and symbols. It considers the context to help anyone complete their own personal jigsaw puzzle. It provides the tools to allow anyone to sort through possible connections and to make sense of their dreams.
From entries ranging from “Abandonment” to “Zoo,” this massive tome analyzes sex dreams, money dreams, dreams of falling, running, or paralysis and much, much more. It brings profound insights to thousands of dream messages. It shows what to look for and what to ignore and teaches how to master dream interpretation. Examples of symbols are given. The complexity and context of a dream are explored. Signs and their meanings are illustrated.
Illuminating the intelligence of dreams, decoding clues, explaining symbols, and revealing the universal meanings of each as well as their subtler associations, The Dream Interpretation Dictionary: Symbols, Signs, and Meanings explores the messages delivered by the unconscious mind during sleep. It examines how dreams connect to daily life. It shows how dreams can lead to deeper understanding and self-awareness. Also included are a helpful bibliography and an extensive index, adding to the book’s usefulness. – Get the Book!
Listen to this interview & catch up on all the other shows by joining our IRN Insider program! | https://inceptionradionetwork.com/j-m-debord/ |
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
1. Field of the Invention
The present invention relates to a traffic indicator, especially to a traffic indicator that is capable of displaying different types of traffic lights and symbols with a remaining time indication.
2. Description of the Prior Arts
The traffic signs mainly comprise dynamic type and static type. The static traffic signs, such as road signs, are suitable for indicating traffic rules or speed limitations. The dynamic traffic signs are often composed of lights, for instance, the traffic lights or pedestrian lights.
For crossroads where the amount of traffic is not large, a set of fundamental red, green and yellow lights is enough for controlling the traffic. However, for places with heavy traffic, moving direction lights are often used accompanying the fundamental traffic lights. Therefore, a crossroad may need a lot of lights of different functions to handle the traffic. As a result, drivers may not promptly respond to the changes of these lights.
To overcome the shortcomings, the present invention provides a traffic indicator to mitigate or obviate the aforementioned problems.
The main objective of the present invention is to provide a single traffic indicator capable of simultaneously displaying the traffic lights, traffic symbols and countdown information depends on practical requirements at different crossroads.
The traffic indicator has a base, a circuit board mounted in the base, an LED panel mounted in the base and a protect lid. Based on the control of the circuit board, the LED panel composed of multiple LEDs of different colors is able to display different traffic lights as well as traffic symbols. In other words, a single display screen can show red, green or yellow traffic lights, moving direction symbols and countdown information. Since the traffic indicator has only one display screen, the traffic symbols can be displayed in a larger size so that drivers can easily recognize the traffic symbols. Furthermore, to compare with the conventional LED-based traffic light equipped with multiple display screens, the traffic indicator with only one display screen requires less fabricating and upkeep costs.
Another objective of the present invention is to provide a function of adjusting brightness of the traffic indicator. The traffic indicator uses an optical sensor to detect the intensity of light of surroundings. Based on the detected result, the circuit board can accordingly adjust the brightness of the LEDs on the LED panel.
Other objectives, advantages and novel features of the invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings.
FIG. 1
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With reference to , a traffic indicator () in accordance with the present invention comprises a base (), a circuit board (), an LED panel () and a protect lid ().
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The protect lid () can be made of translucent or transparent material and is mounted on the base () to cover the circuit board () and the LED panel ().
FIG. 2
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With reference to , a control circuit on the circuit board () for the traffic indicator () comprises a control unit (), an output connecting interface (), a timer (), an input device () and an optical sensor ().
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The output connecting interface () is connected between an output terminal of the control unit () and an input terminal of the LED panel (). Through the output connecting interface (), the control unit () can turn on or turn off each of the LEDs, and controls their color and brightness.
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The timer () has an output terminal connecting to the control unit () and an input terminal connecting to the input device ().
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The input device () can be a key set or a switch set, and has output terminals respectively connecting to the control unit () and timer () for setting operation parameters to the control unit () and timer ().
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The optical sensor () connected to the control unit () detects the intensity of the light of the surroundings. The detected result is transmitted to the control unit (). Based on the detected result, the control unit () can adjust the brightness of LED panel ().
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The LED panel () is mounted in the base () and electrically connects to the circuit board (). The LED panel () comprises multiple light emitting diodes (LEDs) of different colors thus, the circuit board () can control the LED panel () to display red, yellow or green light in a large-area to form a traffic light.
FIG. 3
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As shown in , the LED panel () has two display regions. The first display region is an outer ring region (). The second display region is a center circular region () within the outer ring region (). Furthermore, the outer ring region () provides a function of countdown. For example, the entire outer ring region () can be divided into sixty sections corresponding to sixty seconds of a clock. When the center region () displays a traffic symbol or a sign, the originally activated sixty sections of LEDs in the outer ring region () can be either sequentially turned off in a counterclockwise direction to show the remaining seconds or sequentially turned off for one second in a counterclockwise direction to generate a visual effect of a moving dot. Alternatively, the sixty sections of LEDs in the outer ring region () are sequentially turned on for one second in a counterclockwise direction to generate a visual effect of a moving light dot as a moving second hand.
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The displayed color of the outer ring region () is subjected to the color of center region (). The countdown of remained time may be less than 60 seconds or longer than 1 minute subjected to the traffic load at crossroad but the countdown indications above are activated well while the setting of countdown is less than 61 seconds.
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FIG. 4
FIG. 5
FIG. 6
Therefore, the drivers can recognize the remaining seconds from the indications above after which the present displayed traffic symbol will be changed. Furthermore, the center circular region () can display all desired patterns of traffic symbols as long as these patterns are pre-set or pre-programmed in the circuit board (). For example, the center region () shows a right turn symbol on , a right/left turn symbol on and a left turn symbol on . Accordingly, the direction indications above can either illuminate right/left turn symbols alone as a direction indicator or black out the right/left turn symbols when the center circular region () lights up as a traffic light.
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According to the foregoing description, a single traffic indicator in accordance with the present inventions is able to display all desired patterns of traffic symbols based on the pre-set or pre-programmed settings in the circuit board () to meet different requirements of the crossroads. Therefore, the conventional traffic lights can be reduced to one single traffic indicator.
Even though numerous characteristics and advantages of the present invention have been set forth in the foregoing description, together with details of the structure and features of the invention, the disclosure is illustrative only. Changes may be made in the details, especially in matters of shape, size, and arrangement of parts within the principles of the invention to the full extent indicated by the broad general meaning of the terms in which the appended claims are expressed.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is an exploded cross section view of a traffic indicator in accordance with the present invention;
FIG. 2
is a block diagram of a control circuit for the traffic indicator in accordance with the present invention;
FIG. 3
is a plan view of the traffic indicator in accordance with the present invention;
FIG. 4
is an operational view of the traffic indicator in accordance with the present invention showing a red light with a black or other color right turn symbol and a countdown indication of 20 seconds;
FIG. 5
is an operational view of the traffic indicator in accordance with the present invention showing a green light with a black or other color right/left turn symbol and a countdown indication of 35 seconds; and
FIG. 6
is an operational view of the traffic indicator in accordance with the present invention showing a green light with a black or other color left turn symbol and a countdown indication of 46 seconds. | |
What makes a good park? Part 1
I’ve written before that cities are not statistics. In that particular case, I was talking about how we can quantify various aspects of a city or neighborhood, but that those numbers tell us very little about life - the actual experience on the ground, whether people will walk and what kind of economic success it might have. While it frustrates the rational mind, it’s better to start with looking at human behavior, psychology and even sociology. For example, the notion of Walk Appeal. Our collective fascination with numerical analysis extends to park design as well. Standards-making bodies tell us how much park space a city should have, in what configurations, and with what amenities. It’s as if we could simply follow these rules and have successful, well-used parks and public spaces.
Of course, the real world provides no such comfort. Our public spaces vary tremendously in their success – how well they are used, how much they impact adjoining property value, and how much they contribute to people walking or biking.
Parks or plazas of similar sizes show wildly different amounts of usage and success. City officials and residents are often left wondering, why does one park work well when another does not?
Of course, design of the park itself matters. No one has written better about this than William H Whyte, who is discussed in this excellent blog post regarding Brewer Fountain Plaza in Boston. Whyte, like any good researcher, actually studied how people use space, instead of solely relying on design theory. One could say that he excelled at studying humans in their native habitat.
And while Whyte is spot on with those specific criteria for the park/public space, there are a few other bigger-picture criteria from urban design that impact success. For this particular post, I’ll use Savannah, GA as a case study, with its famous Oglethorpe-designed master plan. The primary object of my analysis is Forsyth Park, the largest park in the historic district – not one of the 22 squares that the city is most known for.
For a couple of years now, I’ve not only used Forsyth on a nearly daily basis, but observed how others use it, and how it functions in the community. The park is arguably one of the five or ten best urban parks in America, in my opinion, and a guiding example of how to do it right. While the park certainly nails Whyte’s criteria (water, food, trees, triangulation and much more), it’s how it fits into the larger picture that interests me most. For example:
Location, location, location. So many parks, even ones that have great facilities, are on “left-over” land that was too hard to develop or wasn’t’ in a prime location in the community. In Savannah, Forsyth Park and the squares were integrally-located as part of the neighborhoods, or Wards in this case, as the city developed. So many cities took the opposite approach, as I’ll detail in subsequent posts. This particular land was not an afterthought – it was consciously designed as part of the necessities of living in a city.
Location along key pathways. Again, Forsyth Park is instructive. Located along the axis of Bull Street, Forsyth is on a key spine of the city, extending from City Hall south. The walkway through the middle is a straight shot into the heart of downtown. It’s logical and easy. Residents or visitors can walk from one end to the other without having to worry about sense of direction. Because if its location along this key spine, it encourages casual walking or biking, since the beauty of the park enhances the walking experience.
Integration with the surrounding streets and buildings. While Forsyth is bounded by two streets on the east and west that are one-way, and have traffic that generally moves far too fast, the streets themselves are not wide. This makes them easy to cross for pedestrians, in spite of the high traffic speeds. And, around the park are located small businesses, hotels, bed and breakfasts in addition to the many residences. The park does not feel as though it’s set apart from the neighborhood – it feels as though it’s distinctly part of the neighborhood.
Public space, what we call the “public realm” in planning wonk-speak, is the key element in whether or not people actually walk. The streets, plazas, parks, squares and other public spaces must be thoughtfully designed. Public spaces should be well-located as well, or they simply will not be well-used. Forsyth Park has all the elements Whyte described eloquently, including a vast amount of simple, open land that can be programmed by its users on a daily basis. These things are not easily quantified, but are certainly observable through the study of human behavior. As we consider retrofitting public spaces or building new ones, we are best served by keeping our desire for quantification in check, and looking harder at how design and behavior intersect, whether that’s the scale of a simple playground or an entire neighborhood.
In the next post, I’ll examine Kansas City’s famous parks system designed by George Kessler, and how those parks rate via this criteria.
If you got value from this post, please consider the following: | https://www.kevinklinkenberg.com/blog/good-park-part-1-savannah-ga |
I was born Janice Elaine Davis. Believe it or not, the only part of my name I ever liked was the Davis part. I had a wonderful extended Davis family, and I was and am happy and proud to be a Davis. The Janice Elaine part - bleh!
As soon as I could, about age 7 I adopted my nickname, Jan. Jan, I am.
You may wonder, then, why I named this blog Janice Elaine Sews? Here's the back-story. I've sewn since I was 8,and consider myself a seamstress; but I didn't start quilting until two years ago, when I was 65. Before I started quilting, I had a "beading fling". I approached beading the same way I do quilting, learn as you go. I started gifting some of the necklaces I made with my daughters and some friends. After all, how much jewelry does a person need!
When my youngest daughter, Sharon, would receive compliments on one of my necklaces, she say it was an original "Designed by Janice Elaine". She was soon getting questions about where her co-workers could get a "Design by Janice Elaine". She'd tell them the necklaces were one of a kind, and not for sale. When I meet a new friend of Sharon's they usually call me Janice Elaine.
I've become accustomed to hearing my first and middle names, and when I had to think of a new blog name, I used Janice Elaine Sews, instead of Janice Elaine Quilts. Sews instead of Quilts, because I've sewn longer than I've quilted.
As Sweet as Cinnamon is having a link-up, if you want to share your blog's pedigree, and there are prizes!
May your bobbins always be full, | http://www.jansquilts.com/2011/10/janice-elaine-sews.html |
How to do Steganography in python : You can hide text in a digital image in a way that is completely invisible to the naked eye. This technique is called watermarking. It’s actually steganography pure and simple.
How does it work?
A colour image can be described in computer terms as a triplet matrix. Each triplet gives the colour of a pixel in the RGB system. Today, an image is composed of several million pixels. This allows us to hide a large number of characters in these images.
Integration of the message in the image :
Each character in the text to be hidden will be represented by its extended ASCII code. For example, the ASCII code of “A” is 64, which gives in binary, on one byte (8 bits): 01000001 . The complete text will therefore be a sequence of 0 and 1, each character using 8 bits and implicitly 8 pixels.
In the python code below we will work only on the red component of the pixel (we can of course use all components). The hidden message will be: “AMIRA DATA – All the data you need“. For the first 8 pixels of the image, we will hide the length of the string. It is 34 characters long (each space must be counted so that the original message is not altered), i.e. 00100010 in binary value. The next step is to convert our string to binary. We saw above that the ASCII code of the letter “A” was 64 which gives in binary 01000001.
|pixel (red component)||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16|
|Image originale||231||12||102||202||131||37||45||18||17||154||167||193||7||101||40||38|
|Reduction||230||12||102||202||130||36||44||18||16||154||166||192||6||100||40||38|
|Binary||0||0||1||0||0||0||1||0||0||1||0||0||0||0||0||1|
|Modified image||230||12||103||202||130||36||45||18||16||155||166||192||6||100||40||39|
In the table below, the second row is given the red component of the first 16 pixels. In a first step, in the reduction line, the values are reduced to the largest even number less than or equal to the value . Then, we will add to these even numbers 0 and 1 and it is these 0 and 1, grouped by 8, which will transmit the hidden information.
#-*- coding:Latin-1 -*- from PIL import Image # function to hide the message in the image def encode_message(directory,saved_image,message): im = Image.open(directory) w,h=im.size # we retrieve the dimensions of the image r,g,b=im.split() #Let's split the image in three (red green blue) r=list(r.getdata()) #we turn the image into a list u=len(message) v=bin(len(message))[2:].rjust(8,"0") #we note the length of the string and transform it to binary ascii=[bin(ord(x))[2:].rjust(8,"0") for x in message] #we transform the string into a list of 0 and 1 a=''.join(ascii) #transformation of the list into a chain #the length of the list is encoded in the first 8 red pixels. for j in range(8): r[j]=2*int(r[j]//2)+int(v[j]) #we code the string in the following pixels for i in range(8*u): r[i+8]=2*int(r[i+8]//2)+int(a[i]) # we recreate the red image nr = Image.new("L",(w,h)) nr.putdata(r) # merging the three new images imgnew = Image.merge('RGB',(nr,g,b)) imgnew.save(saved_image)
Retrieving text from the image:
Recovery is a five-step process::
- Retrieve the matrix describing the image.
- Replace an even number by 0, an odd number by 1.
- Group the bits in groups of 8.
- Convert each byte to a decimal number.
- Write the characters corresponding to the ASCII codes obtained. | https://amiradata.com/how-to-do-steganography-in-python/ |
Q:
plot list values against the list index
I have a list of list in which each list contains some numbers, let's say
t = [[5,6,1],[4,6,33],[6,33,5,10],[1,2],[1,22,44,3]]
using python3 I would like to plot each list values against this particular list index, in the above example I should have x-axis from 1 to 3, y-axis from 1 to 50 with a mark on (1,5),(1,6),(1,1),(2,4),(2,6) ...
here is my code
x = list(range(3))
y = [[5,6,1],[4,6,33],[6,33,5,10],[1,2],[1,22,44,3]]
for i in range(len(x)):
purchases = y[i]
for j in range(len(purchases)):
plt.scatter(x,purchases)
it plots the first two indices correctly then i get the error:
x and y must be in the same size
output image
what is the correct way to do this?
A:
I think you need
y = [[5,6,1],[4,6,33],[6,33,5,10],[1,2],[1,22,44,3]]
for v in y:
plt.scatter(range(len(v)),v)
which is basically to calculate the range of each value on the go, because your y list has different-sized lists inside it
If you want to set the same x for each index of y , then do
for i,v in enumerate(y):
plt.scatter([i+1]*(len(v)),v)
Notice that I've done [i+1] because the way you wrote it, seemed you wanted an index starting from 1
| |
About a year after I started baking seriously I wanted to move on to bigger or different shaped pans but really struggled to work out how to scale up recipes, I spent hours googling recipes for certain pan sizes but they always turned out to be different heights.
I had a recipe I knew inside-out, I knew exactly how tall it would be in my usual pan at a certain temperature so I tried this method and it works perfectly for me for any size pan, even novelty pans.
Ive attached a simple spreadsheet but I shall endeavour to explain it here too:
Measure the height of the tin you know works perfectly
Fill it to the brim with water (or if its a lose bottom line it with a bag and fill with flour)
Record the weight or volume of water/flour it takes to fill the pan
Write this number down, for example it takes 1000ml to fill my usual 3 tall pan.
Now say I want to use the same recipe in a much bigger 3 tall pan; it takes 2800ml or water to fill to the brim, so to work out how much I need to scale up my ingredients I divide the new pan size by my usual pan size (2800 divided by 1000) = 2.8
So I need 2.8 times more of each ingredient
The attached spreadsheet will do this for you
Tips;
I find it works better if you scale up this way in single layers rather than a full 3-4 inch deep cake you intend to slice as you need to reduce the oven temperature a lot more with full cake recipes.
Each time you measure the volume of a new pan write it down, that way you dont have to do it again and again!
For novelty pans you clearly cant make them in layers plus they arent a uniform depth so estimate a depth in between the very highest point and the very lowest and keep your estimate on the deeper side as you can always trim but you cant add to a baked cake!
if your pans are different height you may need to take account of this- the spreadsheet attached will do this for you but don't scale up to a 2 inch tall pan if your original recipe makes your cake 3inches tall- divide your original recipe in 3 and make three seperate 1inch layers in your new pan!
This also works for the amount of filling youll need to use (I do find it overestimates filling slightly though)
If I think of any more tips Ill add them to the post
Hope it helps someone!
Thank you for taking the time to do this
Thank you so much for this chart. It's obvious you put quite a bit of time into this. It's nice of you to share it.
ANo I can't see the spread sheet either.
Measuring with water is a great idea, especially for those irregular wilton character pans, and odd shapes like petal pans. For rounds, squares and rectangles, it is also easy to figure out the ratios with some basic math. This also helps if you don't know how many cups of batter your recipe makes. You just need to know what pan size it fills.
(All of the areas given are in square inches)
SQUARE pan (area of a square is one side times one side).
8" 64
9" 81
10 " 100
12" 144
RECTANGLE (short side times long side)
9 x 12 108
11 x 15 165
ROUNDS (Pi times the radius squared - 3.14 times half the diameter squared).
6" 28.25
8" 50.25
9" 63.5
10" 78.5
12" 113
For 2 layers of each size
2 6" 56.5
2 8" 100.5
2 9" 127
2 10" 157
2 12" 226
for example, If your favorite recipe fills 2 9" pans, then it is filling 2 x 63.5 which is 127 square inches. (the true volume is area times height, but if you want the new size cake to be the same height, you can ignore the height and just look at the difference in areas). if you want to fill a 12 " pan for example, the ratio of the 2 9" to the 12" is 127/113. So your 9" recipe will fill one 12" layer very nicely- with the layer being slightly higher, or a little left over.
Or if you double your regular 9" recipe (which would fill 256 inches square), you could do 2 10" (157) and 2 8" inch (100.5). If you want the layers to be taller or shorter, then adjust accordingly
I just keep a chart with the areas handy, and it is easy to add up areas and see how much to increase or decrease my recipe for the sizes needed. HTH!
thanks it does
What you did is so wonderful. I love the cake central community. It is not competitive and everyone is so generous in sharing their knowledge and tips! Thank you for sharing your spreadsheet!
Did any of you guys save a copy of the excel sheet? It doesnt seem to be linked to the forum any more and I've lost my copy :-(
Aww..can't find the excel sheet anymore. Does anyone has a copy please?
Thank you!!! | http://www.cakecentral.com/forum/t/743935/i-hope-this-will-help-if-you-struggle-scaling-up-recipes |
The density depends on what kind of liquid it is, and a smaller amount on the temperature and pressure. Oils are the least dense of the three you mention. … In general, dissolving stuff in water makes it more dense, making vinegar the densest of the three.
Why does oil float above vinegar?
In the case of oil and vinegar, the vinegar is polar and more dense than the oil, so it settles on the bottom of the container. The oil is nonpolar and less dense, so it doesn’t dissolve in the vinegar, and it floats on top.
What’s more dense than oil?
Water molecules are packed more closely together than the long molecules that make up oil. The oxygen atoms in water are also smaller and heavier than the carbon atoms in oil. This contributes to making water more dense than oil.
What floats oil or vinegar?
When two liquids are placed in a container, the denser liquid will fall to the bottom. The less dense liquid will rise to the top. This is why oil float above vinegar.
Is vinegar more dense than olive oil?
You have heard it said that oil floats on water, and cooking oil is no exception — its density is typically 0.92 g/mL. Vinegar is a solution of acetic acid in water. Different manufacturers sell vinegar that is anywhere from 5 to 10% acid, so the density can vary somewhat. However, a typical value is 1.01 g/mL.
How dense is oil?
The density of most oils will range between 700 and 950 kilograms per cubic meter (kg/m3). By definition, water has a density of 1,000 kg/m3. What this means is that most oils will float on water as they are lighter by volume.
How dense is milk?
The density of raw milk depends on its composition and temperature and can usually – literature data vary slightly – be found in the range of 1.026 g/cm3 – 1.034 g/cm3 at 20°C. Since milk is a multi-component system it is not possible to determine the concentration of one component only by density measurement.
Does honey sink in water?
Due to the viscosity of honey, honey is much denser than water. … But as compared to honey it’s density is lower, so it floats.
How dense is honey?
The density of honey typically ranges between 1.38 and 1.45 kg/l at 20 °C.
Is vinegar less dense than oil?
The density depends on what kind of liquid it is, and a smaller amount on the temperature and pressure. Oils are the least dense of the three you mention. … In general, dissolving stuff in water makes it more dense, making vinegar the densest of the three.
What does vinegar float on?
Vinegar is a polar substance, and its molecules are attracted to water molecules (called “hydrophilic”). Therefore, it is able to be mixed with water. It does not technically dissolve, rather, it forms a homogenous solution with water.
What is the density of white vinegar?
Vinegar typically contains between 5 and 20% acetic acid by volume and at least a 2.4Ph. The acetic acid is produced by the fermentation of ethanol or sugars by acetic acid bacteria. Has a density of approximately 0.96 g/mL.
Does white vinegar dissolve in water?
Vinegar is a polar material with molecules that are drawn to water molecules (called “hydrophilic”). As a result, it can be blended with water. It does not dissolve, but rather produces a homogeneous solution with water.
How do you separate oil and vinegar?
For example , you separate oil from vinegar by just letting the mixture sit for a while. you could also cool an oil and vinegar mixture, the oil will solidify before the vinegar does because it has a higher freezing point . You could then just scoop the oil solids away from the liquid.
Is oil more dense than water?
Since the oil is lighter, it is less dense than water and floats on water.
How dense is olive oil?
The density of olive oil is 0.917 kg/l at 20 ∘C.
How dense is peanut oil?
0.91 g/mL at 25 °C (lit.)
Which liquid has highest density?
So in that the liquid which has the highest density is mercury. It is the only metal which exists as liquid in room temperature other metals are hard and rigid and exist in solid phase. It has a density of about 13.546 grams per cubic centimeter.
How dense is maple syrup?
We can have 1 ml or 10 gallons of the syrup in Example 1, but its density will always be 1.15/cm3 at room temperature. The densities of some common pure substances are listed below.
…
Useful Density Values.
|Substance||Density / g cm–3|
|Corn Syrup||1.48|
|Maple syrup||1.33|
|Magnesium||1.74|
|Salt||2.16|
Is water heavier than cream?
If it is full cream milk 1 liter of milk will be lighter than 1 liter of water, because of the fat content. Skimmed milk, from which the fat portion has been removed by churning out the cream will be heavier than water.
Is Honey more dense than water?
Lighter liquids (like water or vegetable oil) are less dense than heavier liquids (like honey or corn syrup) so they float on top of the heavier liquids.
…
How Does It Work.
|Material||Density (g/cm3)|
|Honey||1.42|
|Pancake Syrup||1.37|
|Light Corn Syrup||1.33|
|Dish Soap||1.06|
Does milk float on water?
Physical Science
Cream or milk fat is lighter in density than water and floats on the surface of un-homogenized milk. When you skim off the surface, some of the fat, the denser portions remains and the milk is denser. This explains why skim milk is denser.
Does alcohol float on water?
Alcohol is less dense than water so spirits can float on top of water or juices. They don’t mix because, unless they are stirred up, natural mixing of fluids is actually a very slow process.
Can honey be stored in the fridge?
There are a few ways to go about doing this, but there’s one place you should never store honey: your refrigerator. Keeping honey in the fridge will only increase the speed of crystallization, turning your honey from liquid into a thick, dough-like sludge. … The container you use to store honey is also important.
Is honey bee vomit?
Technically speaking, honey is not bee vomit. … In a bee, the proventriculus and crop are in direct contact with the mouth. The digestion of solid foods in bees begins in the ventriculus and there is no way that a honey bee can bring that food back through the proventriculus, or ‘vomit.
Is honey bee poop or vomit?
No – honey is not bee poop, spit or vomit. Honey is made from nectar by reducing the moisture content after it’s carried back to the hive. While bees store the nectar inside their honey stomachs, the nectar is not vomited or pooped out before it is turned into honey – not technically, at least.
Do bees eat honey?
The majority of honey bee larvae eat honey, but larvae that are chosen to become future queens will be fed with royal jelly. … Only workers forage for food, consuming as much nectar from each flower as they can. After foraging, worker honey bees return to the hive and pass the collected nectar to another worker.
What is vinegar made of?
Introduction. Vinegar is essentially a dilute solution of acetic (ethanoic) acid in water. Acetic acid is produced by the oxidation of ethanol by acetic acid bacteria, and, in most countries, commercial production involves a double fermentation where the ethanol is produced by the fermentation of sugars by yeast.
Does vinegar sink in water?
They don’t mix in water. A small amount of acetic acid is dissolved in household vinegar. vinegar is the densest of the three because it is dissolved in water.
What is lighter vinegar or water?
Originally Answered: Is vinegar more dense than water? Yes it is, slightly. Although vinegar is mostly water it has some acetic acid in it (usually 4–8% by volume) which will make it slightly denser. A quick test is to see which liquid floats in the other.
Why does vinegar and water separate?
This versatile acid can mix with water, oil, alcohol and almost any other kind of liquid — even gasoline — reaching places that other cleaning products can’t. When dissolved in water, acetic acid breaks into two components, the hydrogen and the remainder of the molecule, called the acetate.
What happens when you mix water and vinegar?
This happens because vinegar consists of water and acetic acid. The acetic acid forms strong bonds with water molecules. These bonds slow the movement of the molecules in the solution faster than molecules in pure water, causing the solution to freeze more quickly.
Is white flour more dense than water?
Flour’s bulk density is less than that of water. But its density is more. Putting flour into water is displacing the air around its particles with water. Bulk flour consists of tiny little wheat particles interspersed with air gaps.
What is the density of 5% white vinegar?
Vinegar has a density of approximately 0.96 g/mL.
Is vinegar flammable?
Generally vinegar is not flammable. While there are elements in vinegar that are flammable, the high water content of household vinegar keeps it from being flammable.
How dense is rice vinegar?
2.1 Rice vinegar
used for vinegar production had an alcohol concentration of 6.28 g/100 mL, density 20º/20°C of 0.9890, glucose content of 2.8 g/100 mL and the pH was 3.00.
Is vinegar soluble in oil?
Vinegar dissolves in water but oil does not because vinegar is hydrophilic and oil is hydrophobic.
Does vinegar mix with oil?
Oil and vinegar do not mix or even if they are mixed they will quickly separate when given the opportunity. Some proteins such as eggs are emulsifiers that will cause oil and vinegar to mix.
Can you microwave vinegar?
Among the questions that most people ask is whether vinegar should be microwaved, and the answer is yes, Vinegar can be microwaved for consumption or heated up for use as a microwave cleaner. For consumption purposes, vinegar can be warmed up for several seconds without allowing it to heat up completely.
Can oil and vinegar be separated by decantation?
Separating 2 or More Liquids
A common example is decantation of oil and vinegar. When a mixture of the two liquids is allowed to settle, the oil will float on top of the water so the two components may be separated. Kerosene and water can also be separated using decantation.
Is olive oil and balsamic vinegar good for you?
Both olive oil and balsamic vinegar contain high levels of polyphenols, an antioxidant that is generally believed to reduce inflammation and blood pressure. Olive oil is also high in vitamin E, a nutrient that helps to repair damaged cells.
What kind of mixture is oil and vinegar?
When oil and vinegar are mixed, they form a heterogeneous mixture with two layers, or phases. The oil phase floats on the water, or vinegar, phase. A heterogeneous mixture consists of two or more phases.
Can you float on oil?
Density of oil is 10% less than water. Human body density is closer to water. hence we can not float in oil.
Which liquid is the least dense?
Originally Answered: What is the liquid with lowest density? Probably liquid hydrogen wit a density of 0.07099 g/ml (at 20 K).
Is dense heavy?
Originally Answered: Is dense heavy or light? A lot of dense is heavy. Not much dense is light. And somewhere in between there is a quantity of dense that is Just Right.
What is the thickest edible oil?
Virgin olive oil is slightly higher in acidity than extra-virgin olive oil and has a sharper, stronger taste. Pure olive oil is the thickest oil of the three and has the strongest flavor and aroma.
Is coconut oil denser than olive oil?
As the temperature rises, the oil expands, and therefore its density decreases.
…
Density of some vegetable oils:
|Type of oil||Relative density|
|Olive oil||0.913 – 0.916|
|Palma Oil||0.891 – 0.899|
|Coconut oil||0.908 – 0.921|
|Corn oil||0.917 – 0.925|
What is the heaviest cooking oil?
The density of the oils varies with each type and temperature. The range is from 0.91 to 0.93 g/cm3 between the temperatures of 15 °C and 25 °C. Comparing to water, whose density is 1.00 g/ml, cooking oil is less dense.
…
Density of Cooking Oil. | https://12onions.com/is-vinegar-more-dense-than-oil/ |
As a painter, I tend to focus more on the visual aspect of the nature where I get most of my inspirations for my paintings.
Going though the forest of the Rockies reminded me of how important it is to explore other senses when I'm in nature.
I vividly remember the smell...
February 27, 2018
We've been having some winter rain and cold snap here in Arizona. Although I'm not fond of cold weather, I do appreciate the different senses I can experience which I can interpret to my paintings.
Here is a series of paintings I've been working on based on the Wi...
December 30, 2017
I've been enjoying colors emerging from crisp winter morning.
November 25, 2017
I was watching a spectacular sunset the other day, beautiful colors and a lot of movements. As it was happening, I happened to look across the opposite direction and there it was, calm misty pink and purple blending into a night sky. It was subtle and powerful at the...
November 19, 2017
These two paintings are created based on the theme "Morning".
I have been studying and observing colors and lights in the early mornings. Orange glow created by lights hitting the Catalina mountains, shimmering lights coming through leaves and branches of mesquit...
November 5, 2017
Seeing the landscape of the southwest from a different perspective, in this case from the airplane, is fascinating. You can witness the geological history as it is not covered by a thick forest.
This painting, I had the paper flat on the floor, and le...
October 28, 2017
Autumn, it's a beautiful season to live through but not here in Tucson. It's pretty much a cooler summer.
Sometimes I long for the experiences that inspire me to paint. It's easier when I can visit beautiful places but most of the time, I'm at home thinking about what...
October 21, 2017
The mood created by the moonlight is fascinating and calming at time.
Colors and shapes of objects are muted by the subtle moonlight along with the quietness of night. Just abstract lines and shapes with few colors.
In this sensory overloaded time, I ce...
August 6, 2017
My focus on this painting was textures. Not only in the physical sense of paint textures but also the textures of clouds in the sky, from thick to almost transparent fibrous ones.
There are so many textures in the sky during the monsoon season. Here is my interpr...
August 3, 2017
In this painting, I tried to illustrate the quietness of the sunset after the thunderstorm using defused light and soft brush strokes.
I was imagining the sound of raindrops coming down from the eves and at the distance, seeing the light shimmering through a dark sky. | https://www.takigawastudio.com/blog/tag/landscape |
It’s almost December, typically a time of regular rainstorms and mountain snows in California. But instead millions from the Los Angeles area to San Diego are experiencing a dangerous Santa Ana wind event that is raising fire risks to “critical” levels, the second-highest threat category.
Red Flag warnings along with warnings and advisories for high winds are in effect for downtown L.A., and inland areas of San Diego counties, with wind gusts so far exceeding 80 mph in some mountain areas. Santa Ana winds are expected to continue to whip across the region through Saturday, sending air moving from inland areas to the sea. As the air descends from the mountains it compresses and dries out, leading to extremely low relative humidity levels, in the single digits in some areas.
So far, no major wildfires have ignited during this event, though several small fires have broken out in Southern California.
According to the Weather Service, the cause of this offshore wind event is a low pressure area diving into the Great Basin. The air circulation around this feature is powering winds through mountain passes and up and over high terrain toward heavily populated areas in the state.
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Typically, California’s biggest firestorms have occurred during periods of strong offshore winds such as this one. In an effort to prevent sparks from triggering any fires, Southern California Edison, the area’s biggest utility, is warning more than 100,000 customers that they may lose power as a preventive measure. Most of the customers are in L.A. and San Bernardino counties.
The 2020 wildfire season in California has been unrelenting, due to record warmth, a deepening drought, an abundance of lightning strikes and extreme offshore wind events.
Santa Ana winds resulted in major fires in Orange County in late October, forcing tens of thousands to evacuate on short notice. Now, though, vegetation is even drier, due to the lack of significant rainfall, which makes the strong winds riskier.
The months of August, September and October each ranked as the state’s hottest since records began in 1895.
Without enough rain, high fire danger is continuing into November and December in Southern California, and the driest period is now coinciding with the windiest. Many lower elevation locations, along with the mountains of Los Angeles and Ventura counties, have received less than a quarter inch of rain since Oct. 1, which marks the start of the water year in the state.
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For example, only 0.08 inches of rain has fallen in Camarillo in Ventura County. Typically that area should have received nearly 2 inches of rain by now since Oct. 1.
There are no significant rains in sight for Southern California, either. In fact, computer models project a large area of high pressure to build across the West in early-to-mid December, diverting storm systems to the north, and leaving the region milder and drier than average during the period. The Weather Service forecast office in L.A. is highlighting the potential for two additional Santa Ana wind events to occur during the next week, including what could be a strong one during the middle of the first week of December.
The cool season fire danger this year is reminiscent of December 2017 in Southern California, and is consistent with a delayed arrival of autumn rains over the last several years, an effect that has been predicted to emerge in California due to human-caused climate change.
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The Thomas Fire in Ventura and Santa Barbara counties was ignited by power lines during high winds on Dec. 4, 2017. It burned for more than a month, scorching 281,893 acres and destroying 1,063 structures.
Recent studies have shown that warming and drying fall seasons are amplifying the fire threat, as the number of extreme fire weather days increases and very dry conditions extend later into the year. This trend is the result in part of human-caused climate change and has also been seen in other parts of the world. One study, for example, found that climate change has doubled the days during the fall with extreme wildfire conditions in parts of California since the 1980s.
California is in the midst of its worst wildfire season on record, with about 4.2 million acres burned, more than double the acreage in the previous record-breaking year. At least 10,488 structures have been destroyed and 31 people killed. Five of the top six largest fires on record in the state have occurred this season.
Diana Leonard contributed reporting to this story.
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The most important news stories of the day, curated by Post editors and delivered every morning. | |
Kispiax Village, 1929, by Emily Carr 1000-Piece Pomegranate Artpiece Jigsaw Puzzle
Published with the Art Gallery of Ontario, Toronto.
Canadian artist Emily Carr was initially disappointed by the lackluster public response to her work depicting First Nations villages of the Pacific Northwest and gave up painting for over a decade. Inspired by an encounter with artists of the Group of Seven in 1927, she returned to painting and by 1935 was one of Canada's most celebrated artists. Nevertheless she persisted!
- Puzzle Size: 27" x 20"
- Box Size: 10'' x 13" x 1 7/8"
No reviews found... | https://shop.moxymodern.com/copy-of-pom-puzzle-birducopia-1000.html |
The Number One Question You Must Ask for Physics Collisions
Such a result creates a physicist want to learn more about the system further. The upsilon cells went undetected for such a long time, Litke suggested, mainly since they are just a very small fraction of all of the ganglion cells. Ernest Rutherford, for instance, discovered the essence of the atomic nucleus from such experiments. You also settle on which physics bodies can collide with one another and separately settle on which interactions cause your app to be called.
There’s also a custom preset, which enables you to manually set the method by which the mesh responds to distinct objects in your level. We don’t want to use the entire image, but just a little region of it. At first, the stationary glob doesn’t have any momentum so all of the momentum in the system composed of the 2 globs is maintained by the moving glob. Set to Framerate independent to correct the time step based on the framerate, which may cheap essay lead to the very same simulation to have different results if run twice. In this instance, our mesh is a rather straightforward form. The last velocity of the combined objects is dependent on the masses and velocities of the 2 objects that collided.
My present research is in the discipline of experimental high-energy heavy ion physics. The result is called a ballistic pendulum. The estimating worksheet is intended to direct you get through the estimation practice. These behaviors are completely redundant when using Physics and have zero effect. It was made to direct you get through the estimation practice.
Friction causes a minor decrease in velocity. Otherwise, allow me to explain why it happened. Also be sure that you enable Starter Content. 3 Hit side-on instead of head-on.
It’s essential to note this is an idealized model. We are going to ignore it for now as it’s beyond the range of this tutorial. Pick the one which you think will occur. That grain of sand was not a flash on the monitor. essay outline Overview of the historical evolution of the conventional model.
The directions may change based on the shapes of the bodies and the purpose of impact. But so as to utilize it correctly, ensure you prepare the loopMGR for the scene correctly. Instead, utilize the Immovable property.
You may just have a couple seconds to evaluate the situation and react, but this should be lots of time if you stay calm and remember your driver’s training. Be aware that the camera is centered horizontally and slightly above the vehicle. The classes whom I’ve created allow us to efficiently create both of the above mentioned scenarios (assuming you’ve completed the other, non-music portions of the above).
You can imagine an automobile tire on a road. Within this activity, attempt to predict what’s going to happen in three bumper car collisions. The momentum of the car isn’t conserved. We can think about a couple of cases. We wish to do the exact same thing here. You should comprehend these features and the way in which they work, in the event you should ever drive a car that utilizes them.
The realisation that the excellent diversity of the planet stems from a couple of elementary particles acting under the influence of a couple fundamental forces was among the triumphs of twentieth century physics. Theoretical physicists utilize math to approximate reality and utilize mathematical principles to see whether the approximation they made is accurate in addition to seeing if mathematical operations result in new methods to comprehend the reality they approximated. Air molecules in the atmosphere are continuously bouncing each and every way, so force balance isn’t an ideal description. It seems not one of the primitive shapes quite achieve what we’re looking for.
Here you are at our site. If you click on it, you will observe the options you have to have in order in order to add collision to your mesh. You may want to read about Newton’s laws of motion before starting.
Physicists are searching for many distinct parts of data when studying particle collisions. Collisions play a major part in cue sports. Perfectly elastic collisions are a little ideal.
For each and every action, there’s an equal and opposite reaction. The size of the velocity difference at impact is known as the closing speed. Now you’re ready to fix some momentum and collisions word issues. Now you’re all set to deal with some momentum and collisions word troubles.
Hard collisions may lead to player concussions, but the physics of the way the effect of a helmet hit transfers to the brain aren’t well understood. Say, as an example, that we wish to assist the player feel triumphant as they overcome a particularly tough enemy. Simply speaking, it is a way to make complex procedural animations that mimic physics. Within this undertaking, you’ll experiment with colliding masses, see the method by which they collide, and perhaps learn to use physics to plan the ideal pool shot! | https://www.babystroller.ro/2020/01/09/the-number-one-question-you-must-ask-for-physics-collisions/ |
---
abstract: 'We study analytically the quantum thermalization of two coupled two-level systems (TLSs), which are connected with either two independent heat baths (IHBs) or a common heat bath (CHB). We understand the quantum thermalization in eigenstate and bare-state representations when the coupling between the two TLSs is stronger and weaker than the TLS-bath couplings, respectively. In the IHB case, we find that when the two IHBs have the same temperatures, the two coupled TLSs in eigenstate representation can be thermalized with the same temperature as those of the IHBs. However, in the case of two IHBs at different temperatures, just when the energy detuning between the two TLSs satisfies a special condition, the two coupled TLSs in eigenstate representation can be thermalized with an immediate temperature between those of the two IHBs. In bare-state representation, we find a counterintuitive phenomenon that, under some conditions, the temperature of the TLS connected with the high-temperature bath is lower than that of the other TLS, which is connected with the low-temperature bath. In the CHB case, the coupled TLSs in eigenstate representation can be thermalized with the same temperature as that of the CHB in nonresonant cases. In bare-state representation, the TLS with a larger energy separation can be thermalized to a thermal equilibrium with a lower temperature. In the resonant case, we find a phenomenon of anti-thermalization. We also study the steady-state entanglement between the two TLSs in both the IHB and CHB cases.'
author:
- 'Jie-Qiao Liao'
- 'Jin-Feng Huang'
- 'Le-Man Kuang'
title: 'Quantum thermalization of two coupled two-level systems in eigenstate and bare-state representations'
---
introduction
============
Conventional quantum thermalization [@Breuer; @Gemmerbook] is understood as an irreversibly dynamic process under which a quantum system immersed in a heat bath approaches a thermal equilibrium state with the same temperature as that of the bath. The thermal equilibrium state [@Greiner] of the thermalized system reads as $\rho_{th}(T)=Z_{S}^{-1}\exp[-H_{S}/(k_{B}T)]$, which is merely determined by the Hamiltonian $H_{S}$ of the thermalized system and the temperature $T$ of its environment, where $Z_{S}=\text{Tr}_{S}\{\exp[-H_{S}/(k_{B}T)]\}$ is the partition function of the thermalized system, with $k_{B}$ being the Bolztmann constant. During the course of a quantum thermalization, all of the initial information of the thermalized system is totally erased by its environment. Recently, much attention has been paid to quantum thermalization (e.g., Refs. [@Rigolnature; @Rigol; @Berman; @Reimann; @Linden; @Rajagopal; @Tasaki; @Liao; @Lychkovskiy]). Specifically, a new kind of thermalization, called canonical thermalization (e.g., Refs. [@Popescu; @Goldstein; @Gemmer; @Dong; @Reimann2010]), has been proposed.
The conventional quantum thermalization works for the situations wherein one quantum system is connected with just one environment at thermal equilibrium. When we consider a composite quantum system, which is constructed with many subsystems and connected with many environments, the conventional quantum thermalization is no longer valid. Therefore, the density operator of the composite quantum system at thermal equilibrium can not be written as $\rho_{th}(T)$. In fact, quantum thermalization of a composite quantum system is a very complex problem. On one hand, from the viewpoint of the environments, the composite quantum system could be connected with many independent environments or a common environment. At the same time, the temperatures of these environments could be the same or different. On the other hand, the coupling strengths among these subsystems can affect the physical picture to describe the quantum thermalization of the composite quantum system. When the coupling strengths among these subsystems are stronger than the system-bath couplings, the composite quantum system can be regarded as a single system, while it is regarded as many individual subsystems when the couplings among them are weaker than the system-bath couplings.
The above mentioned situations come true in recent years since quantum systems can be manufactured to be more and more complicated and small, based on the great advances in physics, chemistry, and biology [@Bustamante]. Therefore, the research on *thermodynamics of small systems* becomes very interesting. In particular, quantum thermalization of composite quantum systems becomes an important topic since many important results in this field, such as nonequilibrium work relations and fluctuation theorem (e.g., Refs. [@Jarzynski; @Crooks; @Tasaki2000; @Evans2002; @Hanggi2009]), are based on the thermal equilibrium state of the composite quantum systems. As composite quantum systems are composed of many subsystems, they could be connected either with many independent environments at different temperatures or a common environment. Therefore, it is natural to ask the following questions: (i) How do the couplings among the subsystems affect the quantum thermalization of a composite quantum system? (ii) What are the steady-state properties of a composite quantum system when it is connected with either many independent environments or a common environment?
With these questions, in this paper, we study the quantum thermalization of two coupled two-level systems (TLSs) that are immersed in either two independent heat baths (IHBs) or a common heat bath (CHB). Simple as this model is, it is illustrative. When the coupling between the two TLSs is stronger than the TLS-bath couplings, the two TLSs can be considered as an effective composite system, i.e., a four-level system, and then we understand the quantum thermalization in eigenstate representation of the composite system. In addition, when the coupling between the two TLSs is weaker than the TLS-bath couplings, we understand the quantum thermalization from the viewpoint of each individual TLS. However, due to the TLS-TLS coupling, the effective temperatures of the two TLSs should be different from those for the decoupling case.
As for the environments, there are two kinds of different situations: the IHB case and the CHB case. In the IHB case, we find that, when the two IHBs have the same temperatures, the two coupled TLSs in the eigenstate representation can be thermalized with the same temperature as those of the IHBs. However, in the case where the two IHBs have different temperatures, just when the energy detuning between the two TLSs satisfies a special condition, the two coupled TLSs in eigenstate representation can be thermalized with an immediate temperature between those of the two IHBs. In the bare-state representation, we find a counterintuitive phenomenon that, under some conditions, the temperature of the TLS connected with the high-temperature heat bath is lower than that of the other TLS, which is connected with the low-temperature heat bath.
In the CHB case, we also study the quantum thermalization in eigenstate and bare-state representations. In the eigenstate representation, the present case reduces to the conventional quantum thermalization, \[i.e., one quantum system (an effective four-level system formed by the two coupled TLSs) is thermalized by one environment (the common heat bath) in thermal equilibrium\]. In addition, we also investigate the effective temperatures of the two TLSs in bare-state representation. It is found that the TLS with a larger energy separation can be thermalized with a lower temperature. In particular, in the resonant case, we find a quantum phenomenon of anti-thermalization when the two TLSs are connected with a common heat bath.
This paper is organized as follows: In Sec. \[Sec:2\], we present the physical models and their Hamiltonians. In Sec. \[Sec:3\], we study the quantum thermalization of the two coupled TLSs immersed in two IHBs. In Sec. \[Sec:4\], we consider the case wherein the two TLSs are immersed in a CHB. We conclude this work in Sec. \[Sec:5\]. Finally, we give two appendices, \[appeindependentbath\] and \[appecommontbath\], for detailed derivations of quantum master equations and transition rates for the IHB and CHB cases, respectively.
\[Sec:2\]Physical models and Hamiltonians
=========================================
Let us start with introducing the physical models \[as illustrated in Figs. \[schematic\](a) and \[schematic\](b)\]: two TLSs, denoted by TLS$1$ and TLS$2$ with respective energy separations $\omega_{1}$ and $\omega_{2}$, couple with each other via a dipole-dipole interaction of strength $\xi$. At the same time, the two TLSs couple inevitably with the environments surrounding them. Specifically, in this paper, we consider two kinds of different cases: the IHB case and the CHB case. In the former case, the two TLSs are immersed in two IHBs, while, in the latter case, the two TLSs are immersed in a CHB.
![(Color online) Schematic diagram of the physical models under consideration. Two coupled two-level systems, denoted by TLS$1$ and TLS$2$ with respective energy separations $\protect\omega_{1}$ and $\protect\omega _{2}$, are connected with either (a) two IHBs at temperatures $T_{1}$ and $ T_{2}$ or (b) a CHB at temperature $T$. Between the two TLSs, there exists a dipole-dipole interaction of strength $\protect\xi$. (c) The levels of the four bare states $|\protect\eta_{i}\rangle$ $(i=1,2,3,4)$ of the two free TLSs. (d) The levels of the four eigenstates $|\protect\lambda_{i}\rangle$ $(i=1,2,3,4)$ of Hamiltonian (\[HofTLSs\]) for the two coupled TLSs. In the presence of the baths, there exist bath-induced exciting and damping processes among the four eigenstates. The effective transition rates from states $|\protect\lambda_{i}\rangle$ to $|\protect\lambda_{j}\rangle$ are denoted by $\Gamma_{ij}$. Other cross-dephasing processes are not denoted explicitly.[]{data-label="schematic"}](schematic.eps){width="3.3"}
The Hamiltonian of the total system, including the two TLSs and their environments, is composed of three parts: $$H=H_{\text{TLSs}}+H_{B}+H_{I},$$ where $H_{\text{TLSs}}$ is the Hamiltonian of the two coupled TLSs, $H_{B}$ is the Hamiltonian of the heat baths, and $H_{I}$ describes the interaction Hamiltonian between the two TLSs and their baths.
The Hamiltonian $H_{\text{TLSs}}$ (with $\hbar=1$) reads as $$H_{\text{TLSs}}=\frac{\omega _{1}}{2}\sigma_{1}^{z}+\frac{\omega
_{2}}{2}\sigma_{2}^{z}+\xi(\sigma_{1}^{+}\sigma_{2}^{-}+\sigma_{1}^{-}\sigma
_{2}^{+}). \label{HofTLSs}$$ The first two terms in Eq. (\[HofTLSs\]) are free Hamiltonians of the two TLSs, which are described by the operators $\sigma_{l}^{+}=(
\sigma_{l}^{-})^{\dag}=\vert e\rangle_{ll}\langle g\vert$ and $
\sigma_{l}^{z}=\vert e\rangle_{ll}\langle e\vert-\vert
g\rangle_{ll}\langle g\vert$ $(l=1,2)$, where $|g\rangle_{l}$ and $|e\rangle_{l}$ are, respectively, the ground and excited states of the $l$th TLS (i.e., TLS$l$). The last term in Eq. (\[HofTLSs\]) describes a dipole-dipole interaction of strength $\xi$ between the two TLSs. We note that the Hamiltonian given in Eq. (\[HofTLSs\]) has been widely studied in various physical problems, such as quantum logic gates [@Petrosyan], coherent excitation energy transfer [@Liao2010], decoherence dynamics [@Ferraro2009; @Sinaysky2009; @Ban2009], and nonequilibrium thermal entanglement [@Quiroga2007; @Sinaysky2008].
The Hilbert space of the two TLSs may be spanned by the following four bare states $|\eta_{1}\rangle=|ee\rangle$, $|\eta_{2}\rangle=|eg\rangle$, $|\eta_{3}\rangle=|ge\rangle$, and $|\eta_{4}\rangle=|gg\rangle$ \[as shown in Fig. \[schematic\](c)\], which are eigenstates of the free Hamiltonian $(\omega_{1}\sigma^{z}_{1}+\omega_{2}\sigma^{z}_{2})/2$ of the two TLSs, with the corresponding eigenenergies $E_{\eta_{1}}=-E_{\eta_{4}}=\omega_{m}$ and $E_{\eta_{2}}=-E_{\eta_{3}}=\Delta\omega/2$. Here we have introduced the mean energy separation $\omega_{m}=(\omega_{1}+\omega_{2})/2$ and the energy detuning $\Delta\omega=\omega_{1}-\omega_{2}$.
Due to the dipole-dipole interaction, a stationary single-excitation state should be delocalized and composed of a combination of the single-excitation states in the two TLSs. According to Hamiltonian (\[HofTLSs\]), we can solve the eigenequation $H_{\text{TLSs}}\vert\lambda
_{n}\rangle=E_{\lambda_{n}}\vert\lambda_{n}\rangle$, $(n=1,2,3,4)$ to obtain the following four eigenstates \[as shown in Fig. \[schematic\](d)\]: $$\begin{gathered}
\vert\lambda_{1}\rangle=|\eta_{1}\rangle,\hspace{0.3 cm}
\vert\lambda _{2}\rangle=\cos(\theta/2)|\eta_{2}\rangle
+\sin(\theta/2)|\eta_{3}\rangle,\nonumber\\
\vert\lambda_{3}\rangle=-\sin(\theta/2)|
\eta_{2}\rangle+\cos(\theta/2)|\eta_{3}\rangle,\hspace{0.3 cm}
\vert\lambda_{4}\rangle=|\eta_{4}\rangle,\end{gathered}$$ with the corresponding eigenenergies $E_{\lambda_{1}}=-E_{\lambda_{4}}=\omega_{m}$ and $E_{\lambda_{2}}=-E_{\lambda_{3}}=\sqrt{\Delta\omega^{2}/4+\xi^{2}}$. Here we have introduced a mixing angle $\theta$ $(0<\theta<\pi)$ by $ \tan\theta=2\xi/\Delta\omega$. For a positive $\xi$, when $
\omega_{1}>\omega_{2}$, namely $\Delta\omega>0$, we choose $
\theta=\arctan(2\xi/\Delta\omega)$; When $\omega_{1}<\omega_{2}$, that is $ \Delta\omega<0$, we choose $\theta=\pi+\arctan(2\xi/\Delta\omega)$. The dipole-dipole interaction mixes the two bare states $|\eta_{2}\rangle$ and $
|\eta_{3}\rangle$ with one excitation, and does not change the bare states $ |\eta_{1}\rangle$ with two excitations and $|\eta_{4}\rangle$ with zero excitation.
Aside from the dipole-dipole interaction between the two TLSs, there exist couplings between the TLSs and their environments. In general, when the couplings of a system with its environment are weak, it is universal to model the environment as a harmonic-oscillator heat bath and choose a linear coupling between the system and its environment [@Leggettnp]. In this paper, we consider this situation and model the environments as harmonic-oscillator heat baths. As mentioned above, we will consider two kinds of different cases: one is the IHB case and the other is the CHB case.
In the IHB case, as shown in Fig. \[schematic\](a), the two TLSs are immersed in two IHBs described by the Hamiltonian $$\begin{gathered}
H^{(\text{IHB})}_{B}=H^{(a)}_{B}+H^{(b)}_{B},\nonumber\\
H^{(a)}_{B}=\sum_{j}\omega _{aj}a_{j}^{\dagger }a_{j},\hspace{0.5
cm}H^{(b)}_{B}=\sum_{k}\omega _{bk}b_{k}^{\dagger }b_{k}.\end{gathered}$$ Here $H^{(a)}_{B}$ and $H^{(b)}_{B}$ are, respectively, the Hamiltonians of the baths for TLS$1$ and TLS$2$. The creation and annihilation operators $a^{\dag}_{j}$ ($b^{\dag}_{k}$) and $a_{j}$ ($b_{k}$) describe the $j$th ($k$th) harmonic oscillator with frequency $\omega_{aj}$ ($\omega_{bk}$). The interaction Hamiltonian of the two TLSs with the two IHBs reads $$H^{(\text{IHB})}_{I}=\sum_{j}g_{1j}(a_{j}^{\dagger}\sigma_{1}^{-}+\sigma
_{1}^{+}a_{j})+\sum_{k}g_{2k}(b_{k}^{\dagger}\sigma
_{2}^{-}+\sigma_{2}^{+}b_{k}). \label{nondiacouplingindepH}$$
On the other hand, in the CHB case, as shown in Fig. \[schematic\](b), the two TLSs are immersed in a CHB with the Hamiltonian $$H^{(\text{CHB}
)}_{B}=\sum_{j}\omega _{aj}a_{j}^{\dagger }a_{j},$$ where $a^{\dag}_{j}$ and $%
a_{j}$ are, respectively, the creation and annihilation operators of the $j$th harmonic oscillator with frequency $\omega_{aj}$. The interaction Hamiltonian between the two TLSs and the CHB reads as $$H^{(\text{CHB})}_{I}=\sum_{j}g_{1j}(a_{j}^{\dagger}\sigma_{1}^{-}+\sigma
_{1}^{+}a_{j})+\sum_{j}g_{2j}(a_{j}^{\dagger}\sigma
_{2}^{-}+\sigma_{2}^{+}a_{j}). \label{nondiacouplingH}$$ For simplicity, we have assumed that the TLS-bath coupling strengths $g_{1j}$, $g_{2j}$, and $g_{2k}$ are real numbers.
\[Sec:3\]Quantum thermalization of two coupled TLSs immersed in two IHBs
========================================================================
In this section, we study the quantum thermalization of the two coupled TLSs that are immersed in two IHBs. We depict the evolution of the two TLSs in terms of a quantum master equation. By solving the equations of motion of the density matrix elements to obtain steady-state solution, we study the steady-state properties of the two coupled TLSs.
Equations of motion and steady-state solutions
----------------------------------------------
We consider the situation wherein the environments of the two TLSs are memory-less and the couplings between the TLSs and the environments are weak. Then, we may adopt the usual Born-Markov approximation in derivation of quantum master equation. At the same time, we derive the master equation in eigenstate representation of the two coupled TLSs so that we can safely make the secular approximation (equivalent to the rotating-wave approximation) to obtain a time-independent quantum master equation by neglecting the rapidly oscillating terms [@Breuer]. Therefore, our discussions are under the Born-Markov framework.
In the case of two IHBs, the evolution of the two coupled TLSs is governed by the following Born-Markov master equation in the Schrödinger picture, $$\begin{aligned}
\dot{\rho}_{S}&=&i[\rho_{S},H_{\text{TLSs}}] \notag \\
&&+\sum_{(i,j)}\left[\Gamma_{ji}(2\tau_{ij}\rho
_{S}\tau_{ji}-\tau_{jj}\rho_{S}-\rho_{S}\tau_{jj})\right. \notag \\
&&\left.+\Gamma_{ij}(2\tau_{ji}\rho
_{S}\tau_{ij}-\tau_{ii}\rho_{S}-\rho_{S}\tau_{ii})\right] \notag \\
&&+2\Lambda _{1}(\tau_{42}\rho _{S}\tau_{13} +\tau_{31}\rho _{S}\tau_{24})
\notag \\
&&+2\Lambda _{2}(\tau_{21}\rho _{S}\tau_{34} +\tau_{43}\rho_{S}\tau_{12})
\notag \\
&&+2\Lambda _{3}(\tau_{24}\rho _{S}\tau_{31} +\tau_{13}\rho _{S}\tau_{42})
\notag \\
&&+2\Lambda _{4}(\tau_{12}\rho _{S}\tau_{43} +\tau_{34}\rho _{S}\tau_{21}),
\label{masterequation}\end{aligned}$$ which will be derived in detail in Appendix \[appeindependentbath\]. In Eq. (\[masterequation\]), $\rho_{S}$ is the reduced density operator of the two TLSs. The operators $\tau_{ij}$ are defined by $\tau_{ij}=|\lambda_{i}\rangle\langle\lambda_{j}|$ in the eigenstate representation of the two coupled TLSs. The summation parameters $(i,j)$ in the second line of Eq. (\[masterequation\]) can take $(i,j)=(4,2),(3,1),(2,1),$ and $(4,3)$. In the present model, the effective rates in Eq. (\[masterequation\]) are defined as $\Gamma_{13}=\Gamma_{24}=\Gamma_{1}$, $\Gamma_{31}=\Gamma_{42}=
\Gamma_{2}$, $\Gamma_{12}=\Gamma_{34}=\Gamma_{3}$, and $\Gamma_{21}=
\Gamma_{43}=\Gamma_{4}$, with $$\begin{aligned}
\Gamma_{1}&=&\cos^{2}(\theta/2)A_{1}(\varepsilon_{1})+\sin^{2}(\theta/2)
B_{1}(\varepsilon_{1}), \notag \\
\Gamma _{2} &=&\cos^{2}(\theta/2)A_{2}(\varepsilon_{1})+\sin
^{2}(\theta/2)B_{2}(\varepsilon_{1}), \notag \\
\Gamma _{3} &=&\sin^{2}(\theta/2)A_{1}(\varepsilon_{2})+\cos
^{2}(\theta/2)B_{1}(\varepsilon_{2}), \notag \\
\Gamma _{4} &=&\sin^{2}(\theta/2) A_{2}(\varepsilon_{2})+\cos
^{2}(\theta/2)B_{2}(\varepsilon_{2}), \notag \\
\Lambda _{1} &=&\cos^{2}(\theta/2)A_{1}(\varepsilon_{1})-\sin
^{2}(\theta/2)B_{1}(\varepsilon_{1}), \notag \\
\Lambda _{2} &=&-\sin ^{2}(\theta/2)A_{1}(\varepsilon_{2})+\cos
^{2}(\theta/2)B_{1}(\varepsilon_{2}), \notag \\
\Lambda _{3} &=&\cos^{2}(\theta/2)A_{2}(\varepsilon_{1}) -\sin
^{2}(\theta/2)B_{2}(\varepsilon_{1}), \notag \\
\Lambda _{4} &=&-\sin^{2}(\theta/2)A_{2}(\varepsilon_{2})+\cos
^{2}(\theta/2)B_{2}(\varepsilon_{2}), \label{defofGmmaandLamb}\end{aligned}$$ where we define $A_{1}(\varepsilon_{i})=\gamma_{a}(\varepsilon
_{i})[\bar{n}_{a}(\varepsilon _{i})+1]$, $A_{2}(\varepsilon_{i})=\gamma_{a}(\varepsilon
_{i})\bar{n}_{a}(\varepsilon_{i})$, $B_{1}(\varepsilon_{i})=\gamma_{b}( \varepsilon
_{i})[\bar{n}_{b}(\varepsilon_{i})+1]$, and $B_{2}(
\varepsilon_{i})=\gamma_{b}(\varepsilon
_{i})\bar{n}_{b}(\varepsilon_{i})$ $ (i=1,2)$. The energy separations $\varepsilon_{1}$ and $\varepsilon_{2}$ are introduced as $\varepsilon_{1}=E_{\lambda_{1}}-E_{\lambda_{3}}=E_{
\lambda_{2}}-E_{\lambda_{4}}
=\omega_{m}+\sqrt{\Delta\omega^{2}/4+\xi ^{2}}$ and $\varepsilon_{2}=E_{\lambda_{1}}-E_{\lambda_{2}}=E_{\lambda_{3}}-E_{
\lambda_{4}} =\omega_{m}-\sqrt{\Delta\omega^{2}/4+\xi ^{2}}$. The rates $ \gamma_{a}(\varepsilon _{i})=\pi\varrho_{a}(\varepsilon
_{i})g^{2}_{1}(\varepsilon _{i})$ and $\gamma_{b}(\varepsilon_{i})=\pi \varrho_{b}(\varepsilon
_{i})g^{2}_{2}(\varepsilon _{i})$, where $ \varrho_{a}(\varepsilon
_{i})$ and $\varrho_{b}(\varepsilon _{i})$ are, respectively, the densities of state at energy $\varepsilon _{i}$ of the heat baths for TLS$1$ and TLS$2$. In the following we assume $
\gamma_{a}(\varepsilon _{1})=\gamma_{b}(\varepsilon
_{1})=\gamma_{1}$ and $ \gamma_{a}(\varepsilon
_{2})=\gamma_{b}(\varepsilon _{2})=\gamma_{2}$. In addition, we introduce the average thermal excitation numbers $\bar{n}
_{a}(\varepsilon _{i})=1/[\exp(\varepsilon _{i}/T_{1})-1]$ (with the Boltzmann constant $k_{B}=1$) and $\bar{n} _{b}(\varepsilon
_{i})=1/[\exp(\varepsilon _{i}/T_{2})-1]$ $(i=1,2)$ for the heat baths of TLS$1$ and TLS$2$, at the respective temperatures $T_{1}$ and $T_{2}$ [@Breuer].
Based on quantum master equation (\[masterequation\]), it is straightforward to obtain optical Bloch equations for the density matrix elements of the two TLSs in the eigenstate representation. Denoting $\mathbf{X}(t)=\left(\langle\tau_{11}(t)\rangle,\langle\tau_{22}(t)\rangle,
\langle\tau_{33}(t)\rangle,\langle\tau_{44}(t)\rangle\right)^{T}$ (“$T$" stands for matrix transpose), then the optical Bloch equations for the diagonal density matrix elements in the eigenstate representation can be expressed as $$\dot{\mathbf{X}}(t)=\mathbf{M}^{\text{(IHB)}}\mathbf{X}(t),
\label{OBEforihb}$$ where the coefficient matrix $\mathbf{M}^{\text{(IHB)}}$ is defined by $$\mathbf{M}^{\text{(IHB)}}=-2\left(
\begin{array}{cccc}
\Gamma _{1}+\Gamma _{3} & -\Gamma _{4} & -\Gamma _{2} & 0 \\
-\Gamma _{3} & \Gamma _{1}+\Gamma _{4} & 0 & -\Gamma _{2} \\
-\Gamma _{1} & 0 & \Gamma _{2}+\Gamma _{3} & -\Gamma _{4} \\
0 & -\Gamma _{1} & -\Gamma _{3} & \Gamma _{2}+\Gamma _{4}
\end{array}
\right).$$ From Eq. (\[OBEforihb\]), we can see that the evolution of the diagonal density matrix elements decouples with off-diagonal elements.
The transient solutions of optical Bloch equation (\[OBEforihb\]) can be obtained with the Laplace transform method. To study quantum thermalization, however, it is sufficient to obtain the steady-state solutions, $$\begin{aligned}
\label{steadystatesolution}
\langle\tau_{11}\rangle_{ss}&=&\frac{\Gamma_{2}\Gamma_{4}}{
(\Gamma_{1}+\Gamma_{2}) (\Gamma_{3}+\Gamma_{4})}, \notag \\
\langle\tau_{22}\rangle_{ss}&=&\frac{\Gamma_{2}\Gamma_{3}}{
(\Gamma_{1}+\Gamma_{2}) (\Gamma_{3}+\Gamma_{4})}, \notag \\
\langle\tau_{33}\rangle_{ss}&=&\frac{\Gamma_{1}\Gamma _{4}}{(
\Gamma_{1}+\Gamma_{2})(\Gamma _{3}+\Gamma_{4})}, \notag \\
\langle\tau_{44}\rangle_{ss}&=&\frac{\Gamma_{1}\Gamma _{3}}{
(\Gamma_{1}+\Gamma _{2})(\Gamma_{3}+\Gamma_{4})},\end{aligned}$$ where the subscript “ss" means steady-state solutions.
According to Eq. (\[masterequation\]), we can also obtain the equations of motion for these off-diagonal density matrix elements of the two TLSs as follows: $$\begin{aligned}
\langle \dot{\tau}_{21}(t)\rangle &=&-(2\Gamma _{1}+\Gamma _{3}+\Gamma
_{4}-i\varepsilon _{2})\langle \tau _{21}(t)\rangle +2\Lambda _{3}\langle
\tau _{43}(t)\rangle , \notag \\
\langle \dot{\tau}_{31}(t)\rangle &=&-(\Gamma _{1}+\Gamma _{2}+2\Gamma
_{3}-i\varepsilon _{1})\langle \tau _{31}(t)\rangle +2\Lambda _{4}\langle
\tau _{42}(t)\rangle , \notag \\
\langle \dot{\tau}_{41}(t)\rangle &=&-(\Gamma _{1}+\Gamma _{2}+\Gamma
_{3}+\Gamma _{4}-i\varepsilon _{1}-i\varepsilon _{2})\langle \tau
_{41}(t)\rangle , \notag \\
\langle \dot{\tau}_{32}(t)\rangle &=&-(\Gamma _{1}+\Gamma _{2}+\Gamma
_{3}+\Gamma _{4}-i\varepsilon _{1}+i\varepsilon _{2})\langle \tau
_{32}(t)\rangle , \notag \\
\langle \dot{\tau}_{42}(t)\rangle &=&-(\Gamma _{1}+\Gamma _{2}+2\Gamma
_{4}-i\varepsilon _{1})\langle \tau _{42}(t)\rangle +2\Lambda _{2}\langle
\tau _{31}(t)\rangle , \notag \\
\langle \dot{\tau}_{43}(t)\rangle &=&-(2\Gamma _{2}+\Gamma
_{3}+\Gamma _{4}-i\varepsilon _{2})\langle \tau _{43}(t)\rangle
+2\Lambda _{1}\langle \tau _{21}(t)\rangle.\nonumber\\\end{aligned}$$ Other off-diagonal elements can be obtained via $\langle \tau
_{ij}(t)\rangle ^{\ast }=\langle \tau _{ji}(t)\rangle $. It can be found that the steady-state solutions of the equations of motion for these off-diagonal elements are zero, $$\begin{aligned}
\langle \tau _{ij}\rangle _{ss}=0,\hspace{0.5 cm}i\neq
j.\label{sszero}\end{aligned}$$ Based on the steady-state solutions for these density matrix elements, we can analyze the steady-state properties of the two coupled TLSs.
Quantum thermalization in eigenstate representation
---------------------------------------------------
For the present system with two IHBs, when the coupling between the two TLSs is stronger than the TLS-bath couplings, the two coupled TLSs can be regarded as an effective four-level system connected with two IHBs. Therefore, in the eigenstate representation, the dynamic evolution process of the two-coupled TLSs approaching their steady state of thermal equilibrium can be understood as a non-equilibrium quantum thermalization: thermalization of a quantum system connected with many IHBs at different temperatures [@Jou2003; @Zurcher1990; @Komatsu2008; @Trimper2006; @Eckmann1999; @Huang2009; @Johal2009; @Beer].
As the steady state of the two coupled TLSs is completely mixed in eigenstate representation, we can introduce effective temperatures to characterize the relation between any two eigenstates based on their steady-state populations. From Eq. (\[steadystatesolution\]) we can find the following relations $$\begin{aligned}
\label{ratios}
\frac{\langle\tau_{11}\rangle_{ss}}{\langle\tau_{22}\rangle_{ss}}&=&\frac{
\langle\tau_{33}\rangle_{ss}}{\langle\tau_{44}\rangle_{ss}}=\frac{\Gamma_{4}
}{\Gamma_{3}},\nonumber\\
\frac{\langle\tau_{11}\rangle_{ss}}{
\langle\tau_{33}\rangle_{ss}}&=&\frac{\langle\tau_{22}\rangle_{ss}}{
\langle\tau_{44}\rangle_{ss}}=\frac{\Gamma_{2}}{\Gamma_{1}}.\end{aligned}$$ Generally, it is impossible to define an effective temperature for the effective four-level system at steady state. According to Eq. (\[ratios\]), we can characterize the state of the effective four-level system via introducing the following two effective temperatures [@Quan2005] by $$\begin{aligned}
\frac{\langle\tau_{11}\rangle_{ss}}{\langle\tau_{22}\rangle_{ss}}
&=&\frac{\langle\tau_{33}\rangle_{ss}}{\langle\tau_{44}\rangle_{ss}}
=\exp\left[-\frac{\varepsilon_{2}}{T_{\textrm{eff}}(\varepsilon_{2})}\right], \notag \\
\frac{\langle\tau_{22}\rangle_{ss}}{\langle\tau_{33}\rangle_{ss}}&
=&\exp\left[-\frac{\varepsilon_{1}-\varepsilon_{2}}{T_{\textrm{eff}}
(\varepsilon_{1}-\varepsilon_{2})}\right].\end{aligned}$$ In terms of Eq. (\[ratios\]), we get $$\begin{aligned}
T_{\textrm{eff}}(\varepsilon_{1})&=\frac{\varepsilon_{1}}{\ln\left(\Gamma_{1}/
\Gamma_{2}\right)},\hspace{0.5cm}
T_{\textrm{eff}}(\varepsilon_{2})=\frac{
\varepsilon_{2}}{\ln\left(\Gamma_{3}/\Gamma_{4}\right)}.\end{aligned}$$ When the two IHBs have the same temperatures, namely $T_{1}=T_{2}=T$, we can show that the above two effective temperatures are equal to those of the two IHBs, that is $$\begin{aligned}
T_{\textrm{eff}}(\varepsilon_{1})=T_{\textrm{eff}}(\varepsilon_{2})=T.
\label{equaltemps}\end{aligned}$$ For the nonequilibrium case of $T_{1}\neq T_{2}$, the two effective temperatures are different for general case. We find the following relations $\min(T_{1},T_{2})\leq
T_{\textrm{eff}}(\varepsilon_{1})\leq \max(T_{1},T_{2})$ and $\min(T_{1},T_{2})\leq T_{\textrm{eff}}(\varepsilon_{2})\leq
\max(T_{1},T_{2})$, which mean that the two effective temperatures $T_{\textrm{eff}}(\varepsilon_{1})$ and $T_{\textrm{eff}}(\varepsilon_{2})$ will be within the region from $\min(T_{1},T_{2})$ to $\max(T_{1},T_{2})$ [@Liao]. Under some special conditions, the two effective temperatures could be equal. In this case, we consider that the effective four-level system is thermalized by the non-equilibrium environments: two IHBs at different temperatures.
![(Color online) Plot of the scaled effective temperatures $T_{\textrm{eff}}(\protect\varepsilon_{1})/\gamma$ (solid blue line) and $T_{\textrm{eff}}(\protect \varepsilon_{2})/\gamma$ (dashed red line) vs the mixing angle $\theta$. Other parameters are set as $\gamma_{1}=\gamma_{2}=\gamma$, $\xi/\gamma=10$, $\omega_{m}/\gamma=20$, $T_{1}/\gamma=5$, and $T_{2}/\gamma=10$.[]{data-label="efftemforihb"}](efftemforihb.eps){width="3.3"}
In Fig. \[efftemforihb\], we plot the two effective temperatures $T_{\textrm{eff}}(\varepsilon_{1})$ and $T_{\textrm{eff}}(\varepsilon_{2})$ as a function of the mixing angle $\theta$. It shows that $T_{\textrm{eff}}(\varepsilon_{1})$ and $T_{\textrm{eff}}(\varepsilon_{2})$ can be equal for a special mixing angle $\theta_{0}$, which is determined by $\frac{\varepsilon_{1}}{\ln\left(\Gamma_{1}/\Gamma_{2}\right)}
=\frac{\varepsilon_{2}}{\ln\left(\Gamma_{3}/\Gamma_{4}\right)}$. That is to say the effective four-level system formed by the two coupled TLSs can be thermalized with an effective temperature when $\Delta\omega=\xi/\tan\theta_{0}$. Note that this effective temperature actually is a nonequilibrium temperature [@Hatano2003]. We can also see from Fig. \[efftemforihb\] that the two effective temperatures $T_{\textrm{eff}}(\varepsilon_{1})$ and $T_{\textrm{eff}}(\varepsilon_{2})$ are within the region from $T_{1}$ to $T_{2}$. We point out that the mixing angle $\theta$ should be chosen to make sure that $E_{\lambda_{1}}>E_{\lambda_{2}}$.
Quantum thermalization in bare-state representation
---------------------------------------------------
When the coupling between the two TLSs is weaker than the TLS-bath couplings, we understand the quantum thermalization in bare-state representation. We can express the bare states with the eigenstates as $$\begin{aligned}
|\eta_{1}\rangle&=&\vert\lambda_{1}\rangle, \hspace{0.5
cm}|\eta_{2}\rangle
=\cos(\theta/2)\vert\lambda_{2}\rangle-\sin(\theta /2)\vert
\lambda_{3}\rangle,\nonumber\\
|\eta_{3}\rangle&=&\sin(\theta/2)\vert\lambda_{2} \rangle+\cos(
\theta /2)\vert\lambda_{3}\rangle,\hspace{0.5 cm}|\eta_{4}\rangle=
\vert\lambda_{4}\rangle.\end{aligned}$$ Then we can obtain the relations $$\begin{aligned}
\label{transforma}
\langle\sigma_{l=1,2}^{z}(t)\rangle&=&\langle\tau
_{11}(t)\rangle-\langle\tau _{44}(t)\rangle \notag \\
&&-(-1)^{l}\cos\theta[\langle\tau_{22}(t)\rangle-\langle\tau_{33}(t)\rangle]
\notag \\
&&+(-1)^{l}\sin\theta[\langle\tau _{23}(t)\rangle+\langle\tau
_{32}(t)\rangle ].\end{aligned}$$ According to Eqs. (\[steadystatesolution\]), (\[sszero\]), and (\[transforma\]), the steady-state solution can be obtained as $$\begin{aligned}
\label{sigma12ss}
\langle\sigma_{l=1,2}^{z}\rangle_{ss}&=&\frac{(\Gamma_{2}\Gamma
_{4}-\Gamma_{1}\Gamma _{3})-(-1)^{l}\cos\theta(\Gamma_{2}\Gamma
_{3}-\Gamma_{1}\Gamma _{4})}{(\Gamma_{1}+\Gamma_{2})(\Gamma
_{3}+\Gamma_{4})}
. \notag \\\end{aligned}$$ In addition, the off-diagonal elements of the density matrices of the two TLSs can be expressed as $$\begin{aligned}
\langle \sigma^{+}_{1}(t)\rangle&=&\sin(\theta/2)[\langle
\tau_{14}(t)\rangle-\langle \tau_{34}(t)\rangle] \notag \\
&&+\cos(\theta/2)[\langle \tau_{13}(t)\rangle+\langle \tau_{24}(t)\rangle],
\notag \\
\langle \sigma^{+}_{2}(t)\rangle&=&\sin(\theta/2)[\langle
\tau_{24}(t)\rangle-\langle \tau_{13}(t)\rangle] \notag \\
&&+\cos(\theta/2)[\langle \tau_{12}(t)\rangle+\langle \tau_{34}(t)\rangle].\end{aligned}$$ Because of $\langle \tau_{ij}(t)\rangle_{ss}=0$ $(i\neq j)$, we have $ \langle \sigma^{+}_{1}\rangle_{ss}=\langle
\sigma^{+}_{2}\rangle_{ss}=0$, which implies that the steady states of the two TLSs in bare-state representation are completely mixed. Based on these, it is possible to introduce two effective temperatures of the two TLSs as follows: $$\begin{aligned}
\label{Teff}
T_{\textrm{eff}}(\omega_{l})&=&\frac{\omega_{l}}{\ln\left(\frac{
1-\langle\sigma^{z}_{l}\rangle_{ss}}
{1+\langle\sigma^{z}_{l}\rangle_{ss}} \right)},\hspace{0.5 cm}l=1,2.\end{aligned}$$
![(Color online) Plot of the scaled effective temperatures $T_{\textrm{eff}}(\omega_{1})/\gamma$ (solid blue line) and $T_{\textrm{eff}}(\omega_{2})/\gamma$ (dashed red line) vs the mixing angle $\protect\theta$ for various temperature distributions (shown in figures). Other parameters are set as $\gamma_{1}=\gamma_{2}=\gamma$, $\xi/\gamma=0.1$, and $\omega_{m}/\gamma=20$.[]{data-label="efftemforihbbarerep"}](efftemforihbbarerep.eps){width="3.3"}
In Fig. \[efftemforihbbarerep\], we plot the effective temperatures $T_{\textrm{eff}}(\omega_{1})$ and $T_{\textrm{eff}}(\omega_{2})$ as a function of the mixing angle $\theta$ for various temperature distributions. From Fig. \[efftemforihbbarerep\], we can see the following three interesting results. Firstly, when $\theta\approx\pi/2$, $T_{\textrm{eff}}(\omega_{1})$ and $ T_{\textrm{eff}}(\omega_{2})$ become approximately equal. Specially, at the resonant point $\theta=\pi/2$, the two TLSs have the same temperatures no matter whether the two bath temperatures are the same or not (see the cross point at $\theta=\pi/2$ in figures). This result can be explained as follows: When the two TLSs are nearly in resonance with each other, the dipole-dipole interaction can induce population exchange between the two TLSs such that their temperatures are approximately equal. When $\theta=\pi/2$ ($\omega_{1}=\omega_{2}$), Eq. (\[sigma12ss\]) reduces to $$\begin{aligned}
\langle\sigma_{1}^{z} \rangle_{ss}=\langle\sigma
_{2}^{z}\rangle_{ss}=\frac{\Gamma_{2}\Gamma_{4}-
\Gamma_{1}\Gamma_{3}}
{(\Gamma_{1}+\Gamma_{2})(\Gamma_{3}+\Gamma_{4})},\end{aligned}$$ then we have $T_{\textrm{eff}}(\omega_{1})=T_{\textrm{eff}}(\omega_{2})$.
Secondly, when $\theta$ is near to $0$ and $\pi$, $T_{\textrm{eff}}(\omega_{1})$ and $T_{\textrm{eff}}(\omega_{2})$ approach approximately $T_{1}$ and $T_{2}$, respectively. This result can be understood as follows: Near to $\theta\approx0$ and $\pi$, the energy detuning $|\Delta\omega|=2\xi/|\tan\theta|$ between the two TLSs is very large, and the population exchange between them can be neglected for a weak coupling strength $\xi$. At the same time, the energy separation shifts of the two TLSs are neglectable \[this can be seen from the following effective Hamiltonian (\[effectiveH\])\]. Hence, the temperatures of the two TLSs should be equal to those of their IHBs, respectively.
Thirdly, in Fig. \[efftemforihbbarerep\], there are some regions of $\theta$ where the effective temperature $T_{\textrm{eff}}(\omega_{1})$ of TLS$1$ could be smaller than the temperature $T_{\textrm{eff}}(\omega_{2})$ of TLS$2$ although the bath temperatures $T_{1}>T_{2}$. This is a counterintuitive result. Intuitively, in these cases $T_{\textrm{eff}}(\omega_{2})$ should be smaller than $T_{\textrm{eff}}(\omega_{1})$, because TLS$2$ is connected with a low-temperature bath while TLS$1$ is connected with a high-temperature bath \[In Fig. \[efftemforihbbarerep\](a), when the two bath temperatures are the same, the effective temperatures of the two TLSs should be equal\]. Actually, the counterintuitive result is a net effect of the energy separations of the two TLSs, the bath temperatures, and the coupling between the two TLSs (coupling induces population exchange and energy shifts). From Figs. \[efftemforihbbarerep\](a) to \[efftemforihbbarerep\](d), the curves change gradually with the increase of the temperature difference $T_{1}-T_{2}$ between the two baths. And the counterintuitive region decreases with the increase of the temperature difference $T_{1}-T_{2}$.
In the following, we present a microscopic explanation for Fig. \[efftemforihbbarerep\] as a physical insight of the counterintuitive phenomenon. When the values of $\theta$ are far from the resonant point (not near to $0$ and $\pi$), the two TLSs will have large detuning from each other. Under the large detuning condition $\xi\ll\Delta\omega$, the real population exchange between the two TLSs is compressed, the dipole-dipole interaction induces energy shift to the two TLSs. In this case, we can derive an effective Hamiltonian to describe the two TLSs with the Fröhlich-Nakajima transformation approach [@Frohlich; @Nakajima]. Starting from the Hamiltonian $H_{\textrm{TLSs}}=H'_{0}+H'_{I}$ with $H'_{0}=(\omega_{1}\sigma
_{1}^{z}+\omega_{2}\sigma_{2}^{z})/2$ and $H'_{I}=\xi(\sigma
_{1}^{+}\sigma_{2}^{-}+\sigma_{1}^{-}\sigma_{2}^{+})$, we introduce an operator $S=-\xi(\sigma_{1}^{+}\sigma_{2}^{-}-\sigma_{1}^{-}\sigma_{2}^{+})/\Delta\omega$, which meets the condition $H'_{I}+[H'_{0},S]=0$. Then the effective Hamiltonian reads as $$\begin{aligned}
H_{\textrm{eff}}\equiv H'_{0}+\frac{1}{2}[H'_{I},S]
=\frac{1}{2}\bar{\omega}_{1}\sigma
_{1}^{z}+\frac{1}{2}\bar{\omega}_{2}\sigma
_{2}^{z},\label{effectiveH}\end{aligned}$$ where the shifted energy separations are defined by $\bar{\omega}_{1}=\omega_{1}+\xi^{2}/\Delta\omega$ and $\bar{\omega}_{2}=\omega_{2}-\xi^{2}/\Delta\omega$.
We can see from Eq. (\[effectiveH\]) that, under the large detuning condition, there is no effective coupling between the two TLSs. The dipole-dipole interaction between the two TLSs shifts their energy separations slightly. Hence, when the two TLSs (with the shifted energy separation) are thermalized to thermal equilibrium with their baths, we have the relation $\exp(-\bar{\omega}_{l}/T_{l})=p^{(l)}_{e}/p^{(l)}_{g}$ ($l=1,2$) for the TLS$l$ (with shifted energy separation $\bar{\omega}_{l}$, excited and ground state populations $p^{(l)}_{e}$ and $p^{(l)}_{g}$) in thermal equilibrium at temperature $T_{l}$, and then the effective temperatures defined in Eq. (\[Teff\]) should be $$\begin{aligned}
T_{\textrm{eff}}(\omega_{l})=\frac{\omega_{l}}{\ln(p^{(l)}_{g}/p^{(l)}_{e})}=\frac{\omega_{l}}{\bar{\omega}_{l}}T_{l}.\end{aligned}$$ From Eq. (\[effectiveH\]), we can see that, for a positive $\xi$, when $0<\theta<\pi /2$, we have $\Delta\omega>0$, then $\bar{\omega}_{1}>\omega _{1}$ and $\bar{\omega}_{2}<\omega _{2}$. Hence the effective temperatures $T_{\textrm{eff}}(\omega_{1})<T_{1}$ and $T_{\textrm{eff}}(\omega_{2})>T_{2}$. On the other hand, when $\pi
/2<\theta <\pi$, we have $\Delta \omega<0$, then $\bar{\omega}_{1}<\omega_{1}$ and $\bar{\omega}_{2}>\omega _{2}$. Hence, the effective temperatures $T_{\textrm{eff}}(\omega_{1})>T_{1}$ and $T_{\textrm{eff}}(\omega_{2})<T_{2}$.
According to the above analysis, we can see that, when $T_{1}=T_{2}$ \[Fig. \[efftemforihbbarerep\](a)\], there will exist a counterintuitive region. At the same time, when $\theta$ is near to $0$ and $\pi$, the shifted energy separation $\xi^{2}/\Delta\omega$ approaches zero, then $T_{\textrm{eff}}(\omega_{l})\approx T_{l}$. Hence, with the increase of the bath temperature difference $T_{1}-T_{2}$, the difference between the two effective temperatures also increase, which leads to the counterintuitive region decreases. These results can be seen from Fig. \[efftemforihbbarerep\].
In fact, the above intuitive result is based on the phenomenological master equation $$\begin{aligned}
\dot{\rho}_{S}&=&i[\rho_{S},H_{\text{TLSs}}]
+\mathcal{L}_{1}[\rho_{S}]+\mathcal{L}_{2}[\rho_{S}],
\label{phenomenologicalmeq}\end{aligned}$$ with $$\begin{aligned}
\mathcal{L}_{l=1,2}[\rho_{S}]&=&\frac{\gamma_{l}}{2}(\bar{n} _{l}+1)(
2\sigma_{l}^{-}\rho\sigma_{l}^{+}-\sigma_{l}^{+}\sigma _{l}^{-}\rho -\rho
\sigma_{l}^{+}\sigma_{l}^{-}) \notag \\
&&+\frac{\gamma_{l}}{2}\bar{n}_{l}(2\sigma_{l}^{+}\rho\sigma
_{l}^{-}-\sigma_{l}^{-}\sigma_{l}^{+}\rho-\rho\sigma
_{l}^{-}\sigma_{l}^{+}).\end{aligned}$$ The superoperator $\mathcal{L}_{l}[\rho_{S}]$ describes the dissipation of a TLS$l$ ($l=1,2$) immersed in a heat bath at temperature $T_{l}$ ($\bar{n}_{l}=1/[\exp(\omega_{l}/T_{l})-1]$). Therefore, Eq. (\[phenomenologicalmeq\]) is not valid in the case of two coupled TLSs, especially when the coupling between the two TLSs is stronger than the TLS-bath couplings. In addition, in Eq. (\[phenomenologicalmeq\]), the effects on the TLSs from the two baths are different, one is direct and the other is indirect. For example, the bath of TLS$1$ affects TLS$1$ directly, while the bath of TLS$2$ affects TLS$1 $ indirectly through TLS$2$. On the contrary, our results given in Eq. (\[Teff\]) are based on quantum master equation (\[masterequation\]), which is rigorously derived in the eigen-representation of the two coupled TLSs. Hence, the dissipation is depicted in the eigen-representation of the two coupled TLSs. In other words, the two TLSs play equivalent roles and the two baths directly affect the TLSs. The resonant case is a clear example for the equivalent role of the two TLSs. In the resonant case $\theta=\pi/2$, we obtain $T_{\textrm{eff}}(\omega_{1})=T_{\textrm{eff}}(\omega_{2})$ in terms of quantum master equation (\[masterequation\]), while we get $T_{\textrm{eff}}(\omega_{1})>T_{\textrm{eff}}(\omega_{2})$ from Eq. (\[phenomenologicalmeq\]) when $T_{1}>T_{2}$.
Steady-state entanglement between the two TLSs
----------------------------------------------
In the IHB case, there exists a dipole-dipole interaction between the two TLSs. Therefore, a natural question is: what the quantum entanglement is between the two TLSs after they are thermalized. As we know, during the thermalization processes (not antithermalization), all of the initial information of the two coupled TLSs is totally erased, and the steady state of the two TLSs is determined by the decay rates and bath temperatures. Hence, we need to know the steady-state entanglement in the two TLSs. We note that entanglement dynamics in similar systems has been studied [@Subrahmanyam]. In the following we apply the concurrence to quantify the steady-state entanglement in the two TLSs.
For a $2\times 2$ quantum system (two TLSs) with a density matrix $\rho $ expressed in the bare-state representation, its concurrence [@Wootters] is defined as $$C(\rho )=\max
\{0,\sqrt{s_{1}}-\sqrt{s_{2}}-\sqrt{s_{3}}-\sqrt{s_{4}}\},$$ where $s_{i}$ ($i=1,2,3,4$) are the eigenvalues ($s_{1}$ being the largest one) of the matrix $\rho \tilde{\rho}$. The operator $\tilde{\rho}$ is defined as $$\tilde{\rho}=(\sigma_{1}^{y}\otimes
\sigma_{2}^{y})\rho^{\ast}(\sigma_{1}^{y}\otimes\sigma_{2}^{y}),$$ where $\rho ^{\ast }$ is the complex conjugate of $\rho$ and $\sigma_{l}^{y}$ is the Pauli matrix of TLS$l$. For the $2\times 2$ system, $C=0$ and $C=1$ mean, respectively, the density matrix $\rho$ is an unentangled state and a maximally entangled state. In particular, for the so-called “X"-class state with the density matrix (expressed in the bare-state representation) $$\begin{aligned}
\rho=\left(
\begin{array}{cccc}
\rho_{11} &0 & 0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44} \\
\end{array}
\right),\end{aligned}$$ the concurrence is [@Ikram] $$\begin{aligned}
C(\rho)=2\max\left\{0,|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}},|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}}\right\}.\label{Xstateconcu}\end{aligned}$$
![(Color online) Plot of the steady-state concurrence $C(\rho_{ss})$ vs the scaled bath temperature $T/\gamma$ for various values of $\xi/\gamma$. Other parameters are set as $\gamma_{1}=\gamma_{2}=\gamma$, $\theta=\pi/2$, $T_{1}=T_{2}=T$, and $\omega_{m}/\gamma=20$.[]{data-label="ssentangihb"}](ssentangihb.eps){width="3.3"}
Now, for the present system, its density matrix elements in bare-state representation can be expressed as $\langle\eta _{j}|\rho
|\eta_{i}\rangle=\textrm{Tr}[|\eta_{i}\rangle\langle\eta_{j}|\rho
]=\textrm{Tr}[\mu_{ij}\rho]=\langle\mu_{ij}\rangle$ with the transition operator $\mu_{ij}=|\eta_{i}\rangle \langle \eta _{j}|$. The density matrix elements in the eigenstate representation are expressed by $\langle\lambda_{j}|\rho|\lambda_{i}\rangle
=\textrm{Tr}[|\lambda_{i}\rangle\langle\lambda_{j}|\rho]
=\textrm{Tr}[\tau_{ij}\rho]=\langle\tau_{ij}(t)\rangle$ with $\tau
_{ij}=|\lambda _{i}\rangle \langle \lambda _{j}|$. Since the concurrence is defined in the bare-state representation, and the evolution of the system is expressed in the eigenstate representation. Therefore we need to obtain the transformation between the two representations as follows: $$\begin{aligned}
\langle\mu_{11}(t)\rangle&=&\langle\tau_{11}(t)\rangle,\hspace{0.5cm}\langle\mu
_{44}(t)\rangle=\langle\tau_{44}(t)\rangle, \notag \\
\langle\mu_{22}(t)\rangle&=&\cos^{2}(\theta/2)\langle\tau_{22}(t)\rangle+\sin^{2}(\theta/2)
\langle\tau_{33}(t)\rangle,\nonumber\\
&&-\frac{1}{2}\sin \theta[\langle\tau_{23}(t)\rangle
+\langle\tau_{32}(t)\rangle],\notag \\
\langle\mu_{33}(t)\rangle&=&\sin^{2}(\theta/2)\langle
\tau_{22}(t)\rangle +\cos^{2}(\theta/2)
\langle\tau_{33}(t)\rangle,\notag \\
&&+\frac{1}{2}\sin \theta[\langle\tau_{23}(t)\rangle
+\langle\tau_{32}(t)\rangle],\label{reptransdiag}\end{aligned}$$ and $$\begin{aligned}
\langle\mu_{23}(t)\rangle &=&-\sin^{2}(\theta/2)\langle \tau
_{32}(t)\rangle+\cos^{2}(\theta/2)\langle\tau_{23}(t)\rangle, \notag \\
&&+\frac{1}{2}\sin \theta[\langle\tau_{22}(t)\rangle
-\langle\tau_{33}(t)\rangle],\notag \\
\langle\mu_{12}(t)\rangle&=&\cos(\theta/2)\langle \tau
_{12}(t)\rangle-\sin(\theta /2)\langle\tau _{13}(t)\rangle,\notag \\
\langle\mu_{13}(t)\rangle &=&\sin(\theta/2)\langle \tau
_{12}(t)\rangle +\cos (\theta /2)\langle \tau _{13}(t)\rangle,\notag \\
\langle\mu_{14}(t)\rangle&=&\langle\tau_{14}(t)\rangle,\notag\\
\langle\mu_{24}(t)\rangle &=&\cos(\theta/2)\langle \tau
_{24}(t)\rangle-\sin(\theta/2) \langle \tau
_{34}(t)\rangle,\notag \\
\langle\mu_{34}(t)\rangle &=&\sin(\theta/2) \langle \tau
_{24}(t)\rangle+\cos(\theta/2) \langle \tau
_{34}(t)\rangle.\label{reptransoffdiag}\end{aligned}$$
According to the steady-state solutions given in Eqs. (\[steadystatesolution\]) and (\[sszero\]), the steady-state density matrix of the two TLSs in the bare-state representation can be obtained with the following nonzero elements $$\begin{aligned}
\langle\mu_{11}\rangle_{ss}&=&\langle\tau_{11}\rangle_{ss},
\hspace{0.5 cm}
\langle \mu_{44}\rangle_{ss}=\langle\tau_{44}\rangle_{ss},\notag \\
\langle \mu_{22}\rangle_{ss}
&=&\cos^{2}(\theta/2)\langle\tau_{22}\rangle_{ss}+\sin
^{2}(\theta/2)\langle\tau_{33}\rangle_{ss},\notag\\
\langle\mu_{33}\rangle_{ss}&=&\sin^{2}(\theta/2)\langle\tau_{22}\rangle_{ss}+\cos
^{2}(\theta/2)\langle\tau_{33}\rangle_{ss},\notag\\
\langle \mu _{23}\rangle _{ss} &=&\langle \mu _{32}\rangle _{ss}
=\frac{1}{2}\sin\theta(\langle\tau_{22}\rangle_{ss}-\langle\tau_{33}\rangle_{ss}).\label{nonzeelemens}\end{aligned}$$ The concurrence of this steady state is $$C(\rho_{ss})=2\max \left\{0,|\langle \mu
_{32}\rangle_{ss}|-\sqrt{\langle \mu_{11}\rangle_{ss}\langle \mu
_{44}\rangle_{ss}}\right\}.\label{ssconcu}$$
In the following, we study the steady-state concurrence of the two coupled TLSs in the case of $\theta=\pi/2$ and $T_{1}=T_{2}=T$. In Fig. \[ssentangihb\], we plot the steady-state concurrence $C(\rho_{ss})$ as a function of the bath temperature $T$ for various values of the dipole-dipole interaction strength $\xi$. Figure \[ssentangihb\] shows that a larger steady-state concurrence can be created for a larger $\xi$. We also see the sudden death of the concurrence when the bath temperature increases up to a critical value. Notice that the phenomenon of threshold temperature has also been found in thermal entanglement of spin model [@Quiroga2007; @Sinaysky2008; @Vedral2001; @Wang2002].
\[Sec:4\]Quantum thermalization of two coupled TLSs immersed in a CHB
=====================================================================
In the above section, we have studied the quantum thermalization of two coupled TLSs immersed in two IHBs. However, in some cases, the two coupled TLSs can be considered to be placed in a CHB. In this section, we shall study the quantum thermalization of two coupled TLSs immersed in a CHB.
\[Subsec:4-1\]Equations of motion and steady-state solutions
------------------------------------------------------------
The quantum master equation describing the evolution of the two TLSs immersed in a CHB at temperature $T$ has the same form as Eq. (\[masterequation\]), but the effective rates are not the same as those in the IHB case. In the CHB case, the rates read as $$\begin{aligned}
\Gamma_{12}&=&\left[\sin(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{2})}
+\cos(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{2})}\right]^{2}[\bar{n}
(\varepsilon_{2})+1], \notag \\
\Gamma_{21}&=&\left[\sin(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{2})}
+\cos(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{2})}\right]^{2}\bar{n}
(\varepsilon_{2}), \notag \\
\Gamma_{13}&=&\left[\cos(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{1})}
-\sin(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{1})}\right]^{2}[\bar{n}
(\varepsilon_{1})+1], \notag \\
\Gamma_{31}&=&\left[\cos(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{1})}
-\sin(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{1})}\right]^{2}\bar{n}
(\varepsilon_{1}), \notag \\
\Gamma_{24}&=&\left[\cos(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{1})}
+\sin(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{1})}\right]^{2}[\bar{n}
(\varepsilon_{1})+1], \notag \\
\Gamma_{42}&=&\left[\cos(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{1})}
+\sin(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{1})}\right]^{2}\bar{n}
(\varepsilon_{1}), \notag \\
\Gamma_{34}&=&\left[\sin(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{2})}
-\cos(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{2})}\right]^{2}[\bar{n}
(\varepsilon_{2})+1], \notag \\
\Gamma_{43}&=&\left[\sin(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{2})}
-\cos(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{2})}\right]^{2}\bar{n}
(\varepsilon_{2}), \notag \\
\Lambda_{1}&=&[\cos^{2}(\theta/2)\gamma_{1}(\varepsilon_{1})-\sin^{2}(\theta/2)\gamma_{2}(
\varepsilon_{1})][\bar{n}(\varepsilon_{1})+1], \notag \\
\Lambda_{2}&=&[-\sin^{2}(\theta/2)\gamma_{1}(\varepsilon_{2})+\cos^{2}(\theta/2)\gamma_{2}(
\varepsilon_{2})][\bar{n}(\varepsilon_{2})+1], \notag \\
\Lambda_{3}&=&[\cos ^{2}(\theta/2)\gamma_{1}(\varepsilon_{1})
-\sin^{2}(\theta/2)\gamma_{2}(\varepsilon_{1})]\bar{n}(\varepsilon_{1}),\notag \\
\Lambda_{4}&=&[-\sin^{2}(\theta/2)\gamma_{1}(\varepsilon_{2})
+\cos^{2}(\theta/2)\gamma_{2}(\varepsilon_{2})]\bar{n}(\varepsilon_{2}),
\label{ratesforcommonbath}\end{aligned}$$ where we introduce the rates $\gamma_{1}(\varepsilon
_{i})=\pi\varrho(\varepsilon _{i})g^{2}_{1}(\varepsilon _{i})$, $%
\gamma_{2}(\varepsilon _{i})=\pi\varrho(\varepsilon
_{i})g^{2}_{2}(\varepsilon _{i})$, and the average thermal excitation number $\bar{n}(\varepsilon_{i})=1/[\exp(\varepsilon_{i}/T)-1]$ ($i=1,2$). The detailed derivation of these rates will be given in Appendix [appecommontbath]{}. In comparison with the rates in the IHB case, the rates in the CHB case have correlation terms which are induced by the CHB. In the following discussions, we assume $\gamma_{1}(\varepsilon
_{i})=\gamma_{2}(\varepsilon _{i})\equiv\gamma(\varepsilon _{i})$.
Correspondingly, the optical Bloch equation for the CHB case has the same form as Eq. (\[OBEforihb\]), but the coefficient matrix is replaced by $%
\mathbf{M}^{\text{(CHB)}}$ with the following expression $$\mathbf{M}^{\text{(CHB)}}=-2\left(
\begin{array}{cccc}
\Gamma_{12}+\Gamma_{13} & -\Gamma_{21} & -\Gamma_{31} & 0 \\
-\Gamma_{12} & \Gamma_{21}+\Gamma_{24} & 0 & -\Gamma_{42} \\
-\Gamma_{13} & 0 & \Gamma_{31}+\Gamma_{34} & -\Gamma_{43} \\
0 & -\Gamma_{24} & -\Gamma_{34} & \Gamma_{42}+\Gamma_{43}%
\end{array}
\right).$$ Similar to the IHB case, the equations of motion for diagonal density matrix elements decouple with the off-diagonal elements. The steady-state solutions of the present optical Bloch equation read as $$\begin{aligned}
\label{ssforcommonbath}
\langle \tau _{11}\rangle_{ss} &=&\frac{(\Gamma_{21}+\Gamma_{24}) \Gamma
_{31}\Gamma _{43}+(\Gamma _{31}+\Gamma _{34}) \Gamma _{21}\Gamma _{42}}{A},
\notag \\
\langle \tau_{22}\rangle_{ss} &=&\frac{ (\Gamma _{12}+\Gamma _{13})\Gamma
_{34}\Gamma _{42} +(\Gamma _{42}+\Gamma _{43})\Gamma _{12}\Gamma _{31}}{A},
\notag \\
\langle \tau _{33}\rangle_{ss} &=&\frac{ (\Gamma _{12}+\Gamma _{13})\Gamma
_{43}\Gamma _{24} +(\Gamma _{42}+\Gamma _{43})\Gamma _{21}\Gamma _{13}}{A},
\notag \\
\langle \tau _{44}\rangle_{ss} &=&\frac{(\Gamma _{21}+\Gamma _{24})\Gamma
_{13} \Gamma _{34}+(\Gamma _{31}+\Gamma _{34}) \Gamma _{12}\Gamma _{24}}{A},\end{aligned}$$ with $A=(\Gamma _{12}+\Gamma
_{13})(\Gamma_{34}\Gamma_{42}+\Gamma_{43}\Gamma_{24}) +(\Gamma _{21}+\Gamma
_{24})(\Gamma _{31}\Gamma_{43}+\Gamma _{13}\Gamma_{34}) +(\Gamma
_{31}+\Gamma _{34})(\Gamma _{21}\Gamma_{42}+\Gamma _{12}\Gamma_{24})
+(\Gamma _{42}+\Gamma _{43})(\Gamma _{21}\Gamma_{13}+\Gamma _{12}\Gamma_{31})
$.
We can also obtain the equations of motion for these off-diagonal density matrix elements as follows: $$\begin{aligned}
\langle\dot{\tau}_{21}(t)\rangle&=&-(\Gamma_{21}+\Gamma_{12}+\Gamma
_{13}+\Gamma _{24}-i\varepsilon_{2})\langle \tau _{21}(t)\rangle \notag \\
&&+2\Lambda_{3}\langle\tau_{43}(t)\rangle, \notag \\
\langle\dot{\tau}_{31}(t)\rangle&=&-(\Gamma _{12}+\Gamma
_{31}+\Gamma_{13}+\Gamma_{34}-i\varepsilon_{1})\langle \tau _{31}(t)\rangle
\notag \\
&&+2\Lambda _{4}\langle\tau_{42}(t)\rangle, \notag \\
\langle \dot{\tau}_{41}(t)\rangle&=&-(\Gamma_{12}+\Gamma
_{13}+\Gamma_{42}+\Gamma_{43}-i\varepsilon_{1}-i\varepsilon _{2})
\langle\tau_{41}(t)\rangle, \notag \\
\langle\dot{\tau}_{32}(t)\rangle&=&-(\Gamma_{21}+\Gamma_{31}+\Gamma
_{24}+\Gamma_{34}-i\varepsilon_{1}+i\varepsilon_{2})\langle \tau
_{32}(t)\rangle, \notag \\
\langle\dot{\tau}_{42}(t)\rangle&=&-(\Gamma_{21}+\Gamma _{42}+\Gamma
_{24}+\Gamma _{43}-i\varepsilon _{1})\langle \tau _{42}(t)\rangle \notag \\
&&+2\Lambda _{2}\langle\tau_{31}(t)\rangle, \notag \\
\langle\dot{\tau}_{43}(t)\rangle&=&-(\Gamma_{31}+\Gamma_{42}+\Gamma
_{43}+\Gamma_{34}-i\varepsilon _{2})\langle\tau _{43}(t)\rangle \notag \\
&& +2\Lambda_{1}\langle\tau_{21}(t)\rangle.\end{aligned}$$ The equations of motion for other elements can be obtained by $\langle\tau_{ij}(t)\rangle=\langle\tau^{\ast}_{ji}(t)\rangle$. Obviously, the steady-state solutions of these off-diagonal density matrix elements are zero.
Quantum thermalization in eigenstate representation
---------------------------------------------------
Differently from the IHB case, for the present four-level system, we introduce three effective temperatures to characterize its state. The three effective temperatures $T_{12}$, $T_{13}$, and $T_{34}$ are defined according to the populations of the four levels as follows: $$\begin{aligned}
T_{ij}&=&\frac{E_{\lambda_{i}}-E_{\lambda_{j}}}{\ln\left(\frac{\langle \tau
_{jj}\rangle_{ss}} {\langle \tau _{ii}\rangle_{ss}}\right)}.\end{aligned}$$ We can show that the above introduced temperatures are the same as that of the CHB, $$\begin{aligned}
T_{12}=T_{13}=T_{34}=T, \label{equaltempeforcomba}\end{aligned}$$ which means the two coupled TLSs can approach a thermal equilibrium with the CHB. In other words, the two coupled TLSs in eigenstate representation can be thermalized by the CHB.
According to Eqs. (\[equaltemps\]) and (\[equaltempeforcomba\]), we know that when the temperatures of the heat baths are $T$, irrespective of two IHBs or a CHB, the effective four-level system formed by the two coupled TLSs can be thermalized into a thermal equilibrium state with the same temperature $T$. In other words, based on the thermal equilibrium state at temperature $T$, we can not know whether the two coupled TLSs are connected with two IHBs or a CHB.
Quantum thermalization in bare-state representation
---------------------------------------------------
We also investigate the quantum thermalization of the two TLSs in the bare-state representation. In terms of Eqs. (\[transforma\]) and ([ssforcommonbath]{}), we can obtain the steady-state average values of the two Pauli operators $\sigma^{z}_{1}$ and $\sigma^{z}_{2}$ as follows: $$\begin{aligned}
\langle\sigma_{l=1,2}^{z}\rangle_{ss}&=&\frac{(\Gamma_{21}+\Gamma_{24})(%
\Gamma _{31}\Gamma _{43}-\Gamma _{13} \Gamma _{34})}{A} \notag \\
&&+\frac{(\Gamma _{31}+\Gamma _{34})(\Gamma _{21}\Gamma _{42}-\Gamma
_{12}\Gamma _{24})}{A} \notag \\
&&+(-1)^{l-1}\cos\theta\left[\frac{(\Gamma _{12}+\Gamma _{13})(\Gamma
_{34}\Gamma _{42}-\Gamma _{43}\Gamma _{24})}{A}\right. \notag \\
&&\left.+\frac{(\Gamma _{42}+\Gamma _{43})(\Gamma _{12}\Gamma_{31}-\Gamma
_{21}\Gamma_{13})}{A}\right].\end{aligned}$$ Moreover, we have $\langle \sigma^{+}_{1}\rangle_{ss}=\langle
\sigma^{+}_{2}\rangle_{ss}=0$. Similar to Eq. (\[Teff\]) in the above section, we also introduce two effective temperatures to characterize the state of the two TLSs. In Fig. \[efftemforchb\], we plot the two effective temperatures as a function of the mixing angle $\theta$. We emphasize that the resonant point $\theta=\pi/2$ in Fig. \[efftemforchb\] should be taken out.
![(Color online) Plot of the scaled effective temperatures $T_{\textrm{eff}}(\omega_{1})/\gamma$ (solid blue line) and $T_{\textrm{eff}}(\omega_{2})/\gamma$ (dashed red line) vs the mixing angle $\theta$. Other parameters are set as $\gamma(\varepsilon_{1})=\gamma(\varepsilon_{2})=\gamma$, $\xi/\gamma=0.1$, $\omega_{m}/\gamma=20$, and $T/\gamma=10$. The resonant point $\theta=\pi/2$ is useless.[]{data-label="efftemforchb"}](efftemforchb.eps){width="3.3"}
We can draw a conclusion from Fig. \[efftemforchb\] that the TLS with a larger energy separation can be thermalized to a thermal equilibrium state with a lower temperature. This can be seen as follows: When $ \omega_{1}>\omega_{2}$, we have $0<\theta<\pi/2$, from Fig. \[efftemforchb\] it is clear that the temperature of TLS$1$ is lower than that of TLS$2$. On the other hand, when $\omega_{1}<\omega_{2}$, we have $ \pi/2<\theta<\pi$, Fig. \[efftemforchb\] indicates that the temperature of TLS$2$ is lower than that of TLS$1$ in this region. The physical explanation for Fig. \[efftemforchb\] is the same as that for Fig. \[efftemforihbbarerep\](a).
Quantum anti-thermalization in the resonant case
------------------------------------------------
When $\theta=\pi/2$, the two TLSs are in resonance, and then the decay rates $\Gamma_{13}$, $\Gamma_{31}$, $\Gamma_{34}$, and $\Gamma_{43}$ are zero under the assumption $\gamma_{1}(\varepsilon_{i}) =\gamma_{2}(\varepsilon_{i})$. Hence, the eigenstate $|\lambda_{3}\rangle$ decouples with other eigenstates, resulting in an anti-thermalization phenomenon. The state $|\lambda_{3}\rangle$ is called as a “dark state" [@Gea-Banacloche]. In this case, we need to rewrite new optical Bloch equations for these density matrix elements. We obtain the equations of motion for diagonal density matrix elements $$\begin{aligned}
\langle \dot{\tau}_{11}(t)\rangle &=&-2\Gamma _{12}\langle \tau
_{11}(t)\rangle +2\Gamma _{21}\langle \tau _{22}(t)\rangle, \notag \\
\langle \dot{\tau}_{22}(t)\rangle &= &2\Gamma _{12}\langle \tau
_{11}(t)\rangle-2(\Gamma_{21}+\Gamma_{24}) \langle \tau_{22}(t)\rangle
\notag \\
&&+2\Gamma _{42}\langle\tau_{44}(t)\rangle \notag \\
\langle \dot{\tau}_{33}(t)\rangle &=&0, \notag \\
\langle \dot{\tau}_{44}(t)\rangle &=&2\Gamma _{24}\langle \tau
_{22}(t)\rangle-2\Gamma_{42}\langle\tau_{44}(t)\rangle,\end{aligned}$$ and for off-diagonal density matrix elements $$\begin{aligned}
\langle\dot{\tau}_{21}(t)\rangle&=&-(\Gamma_{21}+\Gamma_{12}+\Gamma
_{24}-i\varepsilon _{2})\langle\tau_{21}(t)\rangle, \notag \\
\langle \dot{\tau}_{31}(t)\rangle &=&-(\Gamma _{12}-i\varepsilon
_{1})\langle \tau _{31}(t)\rangle, \notag \\
\langle \dot{\tau}_{41}(t)\rangle&=&-(\Gamma_{12}+\Gamma _{42}-i\varepsilon
_{1}-i\varepsilon _{2})\langle\tau _{41}(t)\rangle, \notag \\
\langle\dot{\tau}_{32}(t)\rangle&=&-(\Gamma_{21}+\Gamma _{24}-i\varepsilon
_{1}+i\varepsilon _{2}) \langle \tau _{32}(t)\rangle, \notag \\
\langle \dot{\tau}_{42}(t)\rangle&=&-(\Gamma_{21}+\Gamma
_{42}+\Gamma_{24}-i\varepsilon _{1})\langle\tau_{42}(t)\rangle, \notag \\
\langle \dot{\tau}_{43}(t)\rangle&=&-(\Gamma _{42}-i\varepsilon
_{2})\langle\tau_{43}(t)\rangle.\end{aligned}$$ The steady-state solutions for these density matrix elements are $$\begin{aligned}
\label{ssforantitherma}
\langle\tau_{11}\rangle_{ss}&=&\frac{[1-\langle\tau_{33}(0)\rangle]
\Gamma_{21}\Gamma_{42}}{\Gamma_{12}\Gamma_{42}+\Gamma_{12}\Gamma_{42}+
\Gamma_{21}\Gamma_{42}},\notag \\
\langle\tau_{22}\rangle_{ss}&=&\frac{[1-\langle\tau_{33}(0)\rangle]
\Gamma_{12}\Gamma_{42}}{\Gamma_{12}\Gamma_{42}+\Gamma_{12}\Gamma_{42}+
\Gamma_{21}\Gamma_{42}},\notag \\
\langle\tau_{33}\rangle_{ss}&=&\langle\tau_{33}(0)\rangle,\notag \\
\langle\tau_{44}\rangle_{ss}&=&\frac{[1-\langle\tau_{33}(0)\rangle]
\Gamma_{12}\Gamma_{24}}{\Gamma_{12}\Gamma_{42}+\Gamma_{12}\Gamma_{42}+
\Gamma_{21}\Gamma_{42}},\notag \\
\langle\tau_{ij}\rangle_{ss}&=&0,\hspace{0.5 cm}i\neq j,\end{aligned}$$ where $\langle\tau_{33}(0)\rangle=0$ is the initial population of state $ |\lambda_{3}\rangle$. It is obvious that the steady state of the two coupled TLSs depends on its initial state. Therefore, the two coupled TLSs exhibit a phenomenon of anti-thermalization in the sense that the heat bath can not erase totally the initial information of the two TLSs. For example, when initially the two coupled TLSs are prepared in state $|\lambda_{3}\rangle$, they will stay in $|\lambda_{3}\rangle$ forever. However, in the subspace spanned by the three eigenstates $|\lambda_{1}\rangle$, $|\lambda_{2}\rangle$ , and $|\lambda_{4}\rangle$, the initial information of the two coupled TLSs can be totally erased, which can also be seen from Eq. (\[ssforantitherma\] ) when $\langle\tau_{33}(0)\rangle=0$.
Steady-state entanglement between the two TLSs
----------------------------------------------
In the CHB case, in addition to the dipole-dipole interaction between the two TLSs, the common bath can also provide a physical mechanism to entangle the two TLSs. According to Eqs. (\[reptransdiag\]) and (\[reptransoffdiag\]), the steady-state density matrix elements of the two TLSs in the CHB case have the same form as those given in Eq. (\[nonzeelemens\]). However, now the steady-state solutions of the eigenstate populations $\langle\tau_{jj}\rangle_{ss}$ are given by Eq. (\[ssforcommonbath\]). Correspondingly, we can obtain the concurrence between the two TLSs in terms of Eq. (\[ssconcu\]). For a given nonresonant $\theta$, the figure in the CHB case is very similar to Fig. \[ssentangihb\] (so it is not shown here). The phenomenon of threshold temperature also exists in the CHB case.
\[Sec:5\]conclusion and discussions
===================================
In conclusion, we have studied the quantum thermalization of two coupled TLSs which are immersed in either two IHBs or a CHB. We have characterized the temperatures of the two coupled TLSs in eigenstate and bare-state representations when the coupling between the two TLSs is stronger and weaker than the TLS-bath couplings, respectively. In the IHB case, we have found that, when the two IHBs have the same temperatures, the two coupled TLSs could be thermalized in eigenstate representation with the same temperature as those of the heat baths. However, in the case where the two heat baths have different temperatures, just when the energy detuning between the two TLSs satisfies a special condition, the effective four-level system formed by the two coupled TLSs can be thermalized with an immediate temperature between those of the two heat baths. In bare-state representation, we have found a counterintuitive phenomenon that the temperature of the TLS connected with the high-temperature heat bath is lower than that of the other TLS which is connected with the low-temperature heat bath. In the CHB case, the two TLSs in eigenstate representation could be thermalized with the same temperature as that of the heat bath for nonresonant cases. In bare state representation, we have found that the TLS with a larger energy separation can be thermalized to a thermal equilibrium at a lower temperature. We have also found a phenomenon of anti-thermalization of the two TLSs in a common heat bath in the resonant case. In addition, we have studied the steady-state entanglement of the two TLSs in the IHB and CHB cases. It has been found that there exist threshold temperatures for the steady-state entanglement generation.
Finally, we present some discussions on the thermalization time over which the thermalized systems evolve from their initial states to steady states. Mathematically, the thermalization time for a system should be infinite because the long-time limit ($\lim_{t\rightarrow\infty}$) is needed to make sure that these density matrix elements evolve to their steady-state values. From the viewpoint of physics, we might introduce some time scales to describe a thermalization, as the half-life of an exponential decay. However, for present systems, the evolutions of these density matrix elements \[i.e., the transient solution of Eq. (\[OBEforihb\])\] are not purely exponential functions. At the same time, these evolutions depend on the initial conditions. Therefore, it is needed to introduce the time scales under given initial conditions, other than a universal time scale for a quantum thermalization.
Jie-Qiao Liao is grateful to Professor C. P. Sun for many helpful discussions. This work is supported in part by NSFC under Grant No. 11075050, NFRPC under Grant No. 2007CB925204, and PCSIRT under Grant No. IRT0964.
\[appeindependentbath\]Derivation of quantum master equation (\[masterequation\]) for the IHB case
==================================================================================================
In this appendix, we give a detailed derivation of quantum master equation (\[masterequation\]), which describes the evolution of the two TLSs immersed in two IHBs. In the interaction picture with respect to $H_{0}=H_{\textrm{TLSs}}+H^{(\textrm{IHB})}_{B}$, the interacting Hamiltonian (\[nondiacouplingindepH\]) becomes $$\begin{aligned}
H^{(\textrm{IHB})}_{I}(t)=[\tau_{13}B_{13}(t)+\tau_{24}B_{24}(t)]e^{i\varepsilon
_{1}t}+[\tau_{12}B_{12}(t)+\tau_{34}B_{34}(t)]e^{i\varepsilon_{2}t}
+h.c.,\label{HihbIP}\end{aligned}$$ where $\tau_{ij}=|\lambda_{i}\rangle\langle\lambda_{j}|$ and we introduce the noise operators $$\begin{aligned}
B_{12}(t)&=&\sin(\theta/2)A(t)+\cos(\theta/2)B(t),\hspace{0.5 cm}
B_{13}(t)=\cos(\theta/2)A(t)-\sin(\theta/2)B(t),\nonumber\\
B_{24}(t)&=&\cos(\theta/2)A(t)+\sin(\theta/2)B(t), \hspace{0.5 cm}
B_{34}(t)=-\sin(\theta/2)A(t)+\cos(\theta/2)B(t),\end{aligned}$$ with $A(t)=\sum_{j}g_{1j}a_{j}e^{-i\omega _{aj}t}$ and $B(t)=\sum_{k}g_{2k}b_{k}e^{-i\omega_{bk}t}$. Under the Born-Markov approximation [@Breuer], the quantum master equation reads $$\begin{aligned}
\dot{\rho}_{S}=-\int_{0}^{\infty }dt'
\textrm{Tr}_{B}\left[H^{(\textrm{IHB})}_{I}(t),\left[H^{(\textrm{IHB})}_{I}(t-t')
,\rho _{S}(t) \otimes \rho _{B}\right]\right],\end{aligned}$$ where $\textrm{Tr}_{B}$ stands for tracing over the degrees of freedom of the baths. We assume that the two baths are in thermal equilibrium state $\rho_{B}=\rho^{(a)}_{th}\otimes\rho^{(b)}_{th}$ with $\rho^{(a)}_{th}=Z^{-1}_{a}\exp(-\beta_{1}H^{(a)}_{B})$ and $\rho^{(b)}_{th}=Z^{-1}_{b}\exp(-\beta_{2}H^{(b)}_{B})$, where $Z_{a}=\textrm{Tr}_{B_{a}}[\exp(-\beta_{1}H^{(a)}_{B})]$ and $Z_{b}=\textrm{Tr}_{B_{b}}[\exp(-\beta_{2}H^{(b)}_{B})]$ are the partition functions of the two baths, respectively. The parameters $\beta_{1}=1/T_{1}$ and $\beta_{2}=1/T_{2}$ are the inverse temperatures of the baths for TLS$1$ and TLS$2$. Through making the rotating wave approximation, we obtain $$\begin{aligned}
\dot{\rho}_{S}&=&\sum_{(i,j)}\left[\tau _{ij}\rho _{S}\tau
_{ji}\int_{0}^{\infty }dt' e^{i(E_{\lambda_{i}}-E_{\lambda_{j}})t'
}\langle B_{ij}^{\dag }(-t') B_{ij}(0)\rangle-\tau _{jj}\rho
_{S}\int_{0}^{\infty }dt'
e^{-i(E_{\lambda_{i}}-E_{\lambda_{j}})t'}\langle B_{ij}^{\dag }(0)B_{ij}(-t')\rangle\right.\nonumber\\
&&+\left.\tau _{ji}\rho _{S}\tau _{ij}\int_{0}^{\infty }dt'
e^{-i(E_{\lambda_{i}}-E_{\lambda_{j}})t'}\langle
B_{ij}(-t')B^{\dag}_{ij}(0)\rangle-\tau _{ii}\rho
_{S}\int_{0}^{\infty }dt'
e^{i(E_{\lambda_{i}}-E_{\lambda_{j}})t' }\langle B_{ij}(0)B^{\dag }_{ij}(-t')\rangle\right]\nonumber\\
&&+\sum_{(ij,kl)}\left[\tau _{ij}\rho _{S}\tau _{kl}\int_{0}^{\infty
}dt' e^{i(E_{\lambda_{i}}-E_{\lambda_{j}})t' }\langle B_{lk}^{\dag
}(-t') B_{ij}(0)\rangle+\tau _{lk}\rho _{S}\tau
_{ji}\int_{0}^{\infty }dt' e^{i(E_{\lambda_{i}}-E_{\lambda_{j}})t'
}\langle B_{ij}^{\dag }(-t') B_{lk}(0)\rangle\right]
+h.c.,\label{mastereqforoff-diagonalid}\end{aligned}$$ where the summation parameter $(i,j)$ in the first line of Eq. (\[mastereqforoff-diagonalid\]) can take $(i,j)=(1,2),(1,3),(2,3)$, and $(2,4)$, and the summation parameter $(ij,kl)$ in the third line of Eq. (\[mastereqforoff-diagonalid\]) can take $(ij,kl)=(12,43),(13,42),(31,24)$, and $(43,12)$. Here the bath correlation functions are defined by $\langle
X(t)Y(t')\rangle=\textrm{Tr}_{B}[X(t)Y(t')\rho_{B}]$, and we use the property $\langle X(t)Y(t')\rangle=\langle
X(t-t')Y(0)\rangle=\langle X(0)Y(t'-t)\rangle$ of the correlation functions. To derive the quantum master equation, we need to calculate the one-side Fourier transform of the correlation functions in Eq. (\[mastereqforoff-diagonalid\]). For simplicity, in the following we only keep the real parts of the one-side Fourier transforms of the correlation functions and neglect their imaginary parts since the imaginary parts only contribute to the Lamb shifts, which are neglected in this work. The real parts of the one-side Fourier transform of the correlation functions can be obtained as follows: $$\begin{aligned}
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{1}t'
}\langle B_{24}(0)B^{\dag}_{24}(-t')\rangle\right]=
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{1}t'
}\langle
B_{13}(0)B^{\dag}_{13}(-t')\rangle\right]=\Gamma_{1},\nonumber\\
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{1}t'
}\langle
B_{24}^{\dag}(0)B_{24}(-t')\rangle\right]=\textrm{Re}\left[\int_{0}^{\infty
}dt' e^{-i\varepsilon _{1}t'
}\langle B_{13}^{\dag}(0)B_{13}(-t')\rangle\right]=\Gamma_{2},\nonumber\\
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{2}t'
}\langle B_{12}(0)B^{\dag}_{12}(-t')\rangle\right]=
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{2}t'
}\langle
B_{34}(0)B^{\dag}_{34}(-t')\rangle\right]=\Gamma_{3},\nonumber\\
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{2}t'
}\langle B_{12}^{\dag}(0)B_{12}(-t')\rangle\right]=
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{2}t'
}\langle B_{34}^{\dag }(0)B_{34}(-t')\rangle\right]=\Gamma_{4},\end{aligned}$$ and $$\begin{aligned}
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon_{1}t'
}\langle B_{24}(-t')B^{\dag}_{13}(0)\rangle\right]
=\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{1}t'
}\langle B_{13}(-t')B^{\dag}_{24}(0)\rangle\right]=\Lambda_{1},\nonumber\\
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{2}t'
}\langle B_{12}(-t')B^{\dag}_{34}(0)\rangle\right]
=\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{2}t'
}\langle B_{34}(-t')B^{\dag}_{12}(0)\rangle\right]=\Lambda_{2},\nonumber\\
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{1}t'
}\langle B^{\dag}_{24}(-t')B_{13}(0)\rangle\right]
=\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{1}t'
}\langle
B^{\dag}_{13}(-t')B_{24}(0)\rangle\right]=\Lambda_{3},\nonumber\\
&&\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon
_{2}t'}\langle B^{\dag}_{12}(-t')B_{34}(0)\rangle\right]
=\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon _{2}t'
}\langle B^{\dag}_{34}(-t')B_{12}(0)\rangle\right]=\Lambda_{4},\end{aligned}$$ where the parameters $\Gamma_{i}$ and $\Lambda_{i}$ have been defined in Eq. (\[defofGmmaandLamb\]). Based on the above one-side Fourier transforms of these correlation functions, we can obtain those for other correlation functions. By substituting them into quantum master equation (\[mastereqforoff-diagonalid\]) and returning to the Schrödinger picture, we can obtain quantum master equation (\[masterequation\]).
In the following we give an example for calculation of the one-side Fourier transform of correlation function, $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{1}t'
}\langle B_{24}^{\dag}(0)B_{24}(-t')\rangle\right]
&=&\cos^{2}(\theta/2)\textrm{Re}\left[\int_{0}^{\infty }dt'
e^{-i\varepsilon _{1}t' }\langle
A^{\dag}(0)A(-t')\rangle\right]\nonumber\\
&&+\sin^{2}(\theta/2)\textrm{Re}\left[\int_{0}^{\infty }dt'
e^{-i\varepsilon _{1}t' }\langle B^{\dag}(0)B(-t')\rangle\right],\end{aligned}$$ which is based on the fact that there is no correlation between the two IHBs. We can calculate $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{1}t'
}\langle A^{\dag}(0)A(-t')\rangle\right]
=\sum_{j}g^{2}_{1j}\bar{n}_{a}(\omega_{aj})\pi\delta(\omega_{aj}-\varepsilon_{1})
=\pi\varrho(\varepsilon_{a})g^{2}_{1}(\varepsilon_{1})\bar{n}_{a}(\varepsilon_{1})
=\gamma_{a}(\varepsilon_{1})\bar{n}_{a}(\varepsilon_{1}),\end{aligned}$$ where we introduce the rate $\gamma_{a}(\varepsilon_{1})=\pi\varrho_{a}(\varepsilon_{1})g^{2}_{1}(\varepsilon_{1})$ and the average thermal excitation $\bar{n}_{a}(\varepsilon_{1})=1/[\exp(\varepsilon_{1}/T_{1})-1]$. Note that here we have also used the formula $$\begin{aligned}
\int_{0}^{\infty }dt' e^{\pm i\omega t' }=\pi\delta(\omega)\pm
i\mathbf{P}\frac{1}{\omega },\end{aligned}$$ where the sign “$\mathbf{P}$" stands for the principal value integral. Similarly, we can obtain $\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{1}t'
}\langle B^{\dag}(0)B(-t')\rangle\right]
=\gamma_{b}(\varepsilon_{1})\bar{n}_{b}(\varepsilon_{1})$ with $\gamma_{b}(\varepsilon_{1})=\pi\varrho_{b}(\varepsilon_{1})g^{2}_{2}(\varepsilon_{1})$. Therefore, we have $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{-i\varepsilon _{1}t'
}\langle B_{24}^{\dag}(0)B_{24}(-t')\rangle\right]=\cos^{2}( \theta
/2)\gamma_{a}(\varepsilon_{1})\bar{n}_{a}(\varepsilon _{1})+\sin
^{2}(\theta/2)\gamma_{b}(\varepsilon
_{1})\bar{n}_{b}(\varepsilon_{1})=\Gamma_{2}.\end{aligned}$$ With the same method, the one-side Fourier transform for other correlation functions can also be obtained.
\[appecommontbath\]Derivation of the rates in Eq. (\[ratesforcommonbath\]) for the CHB case
===========================================================================================
In the CHB case, the interaction Hamiltonian has the same form as Eq. (\[HihbIP\]). But now the noise operators become $$\begin{aligned}
B_{12}(t)&=&\sin(\theta/2) A_{1}(t)
+\cos(\theta/2)A_{2}(t),\hspace{0.5 cm}
B_{13}(t)=\cos(\theta/2) A_{1}(t)-\sin(\theta/2)A_{2}(t),\nonumber\\
B_{24}(t)&=&\cos(\theta/2) A_{1}(t)+\sin(\theta/2)
A_{2}(t),\hspace{0.5 cm}
B_{34}(t)=-\sin(\theta/2)A_{1}(t)+\cos(\theta/2) A_{2}(t),\end{aligned}$$ with $A_{1}(t)=\sum_{j}g_{1j}a_{j}e^{-i\omega_{j}t}$ and $
A_{2}(t)=\sum_{j}g_{2j}a_{j}e^{-i\omega_{j}t}$. The quantum master equation for the CHB case has the same form as Eq. (\[mastereqforoff-diagonalid\]), but now the real parts of the one-side Fourier transform for the correlation functions become $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty}e^{-i\varepsilon _{2}t'}\langle
B_{12}(-t')B_{12}^{\dag }(0)\rangle
dt'\right]&=&\Gamma_{12},\hspace{0.5 cm}
\textrm{Re}\left[\int_{0}^{\infty }e^{-i\varepsilon _{1}t' }\langle
B_{13}(-t')B_{13}^{\dag }(0)\rangle dt'
\right]=\Gamma_{13},\nonumber\\
\textrm{Re}\left[ \int_{0}^{\infty }e^{-i\varepsilon _{1}t' }\langle
B_{24}(-t')B_{24}^{\dag }(0)\rangle dt' \right]
&=&\Gamma_{24},\hspace{0.5 cm} \textrm{Re}\left[ \int_{0}^{\infty
}e^{-i\varepsilon _{2}t' }\langle B_{34}( -t')B_{34}^{\dag
}(0)\rangle dt' \right] =\Gamma_{34},\nonumber\\
\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t' }\langle
B_{12}^{\dag}(-t')B_{12}(0)\rangle dt'
\right]&=&\Gamma_{21},\hspace{0.5 cm}
\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{1}t' }\langle
B_{13}^{\dag}(-t')B_{13}(0)\rangle dt'
\right]=\Gamma_{31},\nonumber\\
\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{1}t' }\langle
B_{24}^{\dag}(-t')B_{24}(0)\rangle dt'
\right]&=&\Gamma_{42},\hspace{0.5 cm}
\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t' }\langle
B_{34}^{\dag}(-t')B_{34}(0)\rangle dt' \right]=\Gamma_{43},\end{aligned}$$ and $$\begin{aligned}
\textrm{Re}\left[ \int_{0}^{\infty }e^{-i\varepsilon _{1}t' }\langle
B_{24}\left(-t' \right)B_{13}^{\dag }\left(0\right)\rangle dt'
\right]&=&\textrm{Re}\left[ \int_{0}^{\infty }e^{-i\varepsilon
_{1}t' }\langle B_{13}\left( -t' \right)B_{24}^{\dag
}\left(0\right)\rangle dt' \right]
=\Lambda_{1},\nonumber\\
\textrm{Re}\left[ \int_{0}^{\infty }e^{-i\varepsilon _{2}t' }\langle
B_{12}\left(-t' \right)B_{34}^{\dag }\left(0\right)\rangle dt'
\right]&=&\textrm{Re}\left[ \int_{0}^{\infty }e^{-i\varepsilon
_{2}t' }\langle B_{34}\left(-t' \right)B_{12}^{\dag
}\left(0\right)\rangle dt'
\right]=\Lambda_{2},\nonumber\\
\textrm{Re}\left[ \int_{0}^{\infty }e^{i\varepsilon _{1}t' }\langle
B_{24}^{\dag }(-t')B_{13}(0)\rangle dt' \right]&=&\textrm{Re}\left[
\int_{0}^{\infty }e^{i\varepsilon _{1}t' }\langle B_{13}^{\dag
}(-t')B_{24}(0)\rangle dt' \right]
=\Lambda_{3},\nonumber\\
\textrm{Re}\left[ \int_{0}^{\infty }e^{i\varepsilon _{2}t' }\langle
B_{34}^{\dag }(-t') B_{12}(t)\rangle dt' \right]
&=&\textrm{Re}\left[ \int_{0}^{\infty }e^{i\varepsilon _{2}t'
}\langle B_{12}^{\dag}(-t') B_{34}(0) \rangle dt' \right]
=\Lambda_{4},\end{aligned}$$ where the parameters $\Gamma_{ij}$ and $\Lambda_{i}$ have been defined in Eq. (\[ratesforcommonbath\]). The one-side Fourier transform of other correlation functions can be obtained in terms of the above results. Below, we give an example for calculation of the one-side Fourier transform of correlation functions, $$\begin{aligned}
&&\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t' }\langle
B_{12}^{\dag}(-t')B_{12}(0)\rangle dt' \right]\nonumber\\
&=&\sin
^{2}\left( \theta /2\right)\textrm{Re}\left[\int_{0}^{\infty
}e^{i\varepsilon _{2}t' }\langle A_{1}^{\dag}(-t')A_{1}(0)\rangle
dt' \right]+\cos ^{2}\left( \theta
/2\right)\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t'
}\langle A_{2}^{\dag}(-t')A_{2}(0)\rangle dt'
\right]\nonumber\\&&+\frac{1}{2}\sin \theta
\left\{\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t'
}\langle A_{1}^{\dag}(-t')A_{2}(0)\rangle dt'
\right]+\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t'
}\langle A_{2}^{\dag}(-t')A_{1}(0)\rangle dt' \right]\right\}.\end{aligned}$$ We calculate $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty }dt'
e^{i\varepsilon_{2}t' }\langle
A_{1}^{\dag}(-t')A_{1}(0)\rangle\right]
=\sum_{j}g^{2}_{1j}\bar{n}(\omega_{j})\pi\delta(\omega_{j}-\varepsilon_{2})
=\pi\varrho(\varepsilon_{2})g^{2}_{1}(\varepsilon_{2})\bar{n}(\varepsilon_{2})
=\gamma_{1}(\varepsilon_{2})\bar{n}(\varepsilon_{2}),\end{aligned}$$ where we introduce the rate $\gamma_{1}(\varepsilon_{2})=\pi\varrho(\varepsilon_{2})g^{2}_{1}(\varepsilon_{2})$ and the average thermal excitation $\bar{n}(\varepsilon_{2})=1/[\exp(\varepsilon_{2}/T)-1]$. Using the same method, we can obtain $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon_{2}t'
}\langle A_{1}^{\dag}(-t')A_{2}(0)\rangle\right]
=\sum_{j}g_{1j}g_{2j}\bar{n}(\omega_{j})\pi\delta(\omega_{j}-\varepsilon_{2})
=\pi\varrho(\varepsilon_{2})g_{1}(\varepsilon_{2})g_{2}(\varepsilon_{2})\bar{n}(\varepsilon_{2})
=\gamma_{12}(\varepsilon_{2})\bar{n}(\varepsilon_{2}),\end{aligned}$$ where the rate $\gamma_{12}(\varepsilon_{2})=\pi\varrho(\varepsilon_{2})g_{1}(\varepsilon_{2})g_{2}(\varepsilon_{2})=\sqrt{\gamma_{1}(\varepsilon_{2})\gamma_{2}(\varepsilon_{2})}$. Similarly, we have $\textrm{Re}\left[\int_{0}^{\infty }dt'
e^{i\varepsilon_{2}t'}\langle
A_{2}^{\dag}(-t')A_{2}(0)\rangle\right]=\gamma_{2}(\varepsilon_{2})\bar{n}(\varepsilon_{2})$ and $\textrm{Re}\left[\int_{0}^{\infty }dt' e^{i\varepsilon_{2}t'
}\langle
A_{2}^{\dag}(-t')A_{1}(0)\rangle\right]=\gamma_{12}(\varepsilon_{2})\bar{n}(\varepsilon_{2})$. Therefore, we obtain $$\begin{aligned}
\textrm{Re}\left[\int_{0}^{\infty }e^{i\varepsilon _{2}t' }\langle
B_{12}^{\dag}(-t')B_{12}(0)\rangle dt'
\right]=\left[\sin(\theta/2)\sqrt{\gamma_{1}(\varepsilon_{2})}
+\cos(\theta/2)\sqrt{\gamma_{2}(\varepsilon_{2})}
\right]^{2}\bar{n}(\varepsilon_{2})=\Gamma_{21}.\end{aligned}$$ The one-side Fourier transform for other correlation functions can also be obtained with the same method.
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