text:
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# Characterization of dragon fruit (Hylocereus spp.) components with valorization potential

(2013)
Author
Promoter
(UGent)
Organization
Abstract
Dragon fruit (Hylocereus spp.), also known as pitaya or pitahaya, is increasingly gaining interest in many countries, including Thailand which is a country with a climate ideal for breeding different varieties of tropical and subtropical fruits in general, and dragon fruit more specifically. The benefits of dragon fruit for human health can be explained by its essential nutrients such as vitamins, minerals, complex carbohydrates, dietary fibres and antioxidants. Dragon fruit is also an essential source of betacyanin which serves as a red/purple pigment with antioxidative properties. In Thailand, most of the fresh dragon fruits are consumed domestically, while some are traded globally as fresh fruit and processed products as juice or puree. As it is, dragon fruit peel and seeds are often considered as waste or by-products from fruit processing and have been less successful at adding value to the fruit. Currently, there is limited literature on dragon fruit properties, its processed products and constituents as well as potential utilization. Therefore, the main objective of this doctoral research was to gain deeper insight in the characteristics of dragon fruit and its derived products and components, specifically of two species of dragon fruit, i.e. white-flesh dragon fruit (H. undatus) and red-flesh dragon fruit (H. polyrhizus). The characteristics of these two species of dragon fruit and their possible application are extensively discussed in the doctoral thesis/research. Chapter 1 provides a state-of-the-art of current and relevant research on dragon fruit with respect to botanical classification, cultivation, economic aspects and chemical composition (particularly nutrients, pigment and antioxidative components) as well as an overview of techniques to process fruits (processing), i.e. freeze-drying and thermal processing. The impact of these processes on quality attributes (e.g. physicochemical and rheological properties) of different fruits is critically reviewed. The review also aims at giving a summarized overview of the most important aspects of seed oil (e.g. chemical composition and lipid oxidation) and cell wall polysaccharides (e.g. structure of pectic and hemicellulosic substances). In Chapter 2, the properties/characteristics of the whole fruit, peel, pulp (seed-free flesh) and puree (flesh containing seeds) of the two species of dragon fruit were performed. Results demonstrated that the peel of both dragon fruit species as well as the pulp and puree of the red-flesh species contained large amounts of betacyanin. As it is, they have the potential to be utilized as a natural colouring agent. Dragon fruit pulp and puree also exhibited significant antioxidative activities. This was even more pronounced for the red-flesh species due to the presence of betacyanin. The pulp and puree showed shear-thinning behaviour due to the presence of mucilaginous components. Throughout this doctoral research, these characteristics of dragon fruit, for example, pigment, antioxidative activities and rheological properties, provide a deeper insight with regard to further valorization of the fruit components and optimization of the fruit processing. Chapter 3 describes the characterization of the freeze-dried pulp and peel of the two species of dragon fruit to be further utilized of dragon fruit’s pigment (betacyanin) as a food additive. It was found that freeze-drying could satisfactorily preserve colour and pigment concentration. The freeze-dried red-flesh pulp and the freeze-dried dragon fruit peel from both species contained high contents of betacyanin. The freeze-dried dragon fruit pulp was well-soluble in water, whereas this was not the case for the freeze-dried dragon fruit peel. Additionally, due to the pH-sensitivity of betacyanin, the influence of pH (1-11) on the colour shift of the freeze-dried dragon fruit peel and pulp was monitored. The colour of the freeze-dried peel was stable within a pH range of 3-7, whereas the freeze-dried red-flesh pulp had high colour stability over a pH range of 1-11 and is available in a convenient form making it suitable for the use as a food colourant. In order to gain insight into the effects of thermal treatment on the characteristics of the white-flesh and red-flesh dragon fruit purees, the antioxidative, rheological, physicochemical and microbiological properties of the purees were investigated at different process conditions (between 50 and 90 °C for 60 min). The results are presented and discussed in Chapter 4. It is interesting to see that the antioxidative properties of the heated dragon fruit puree increased during heating treatment probably because of the superior antioxidative attributes of the dragon fruit seeds present in the puree and the formation of Maillard reaction products. During thermal processing, the L* value (lightness) and b* value (yellowness) of the processed dragon fruit puree can be used to control the quality online. Total colour change (TCC), which is an indicator of different colour between the unheated and heated puree samples, of the red-flesh dragon fruit puree showed a strong negative correlation with betacyanin content. The kinetics of colour changes and betacyanin degradation in the heated dragon fruit puree followed a second-order models. The apparent viscosity of the heated dragon fruit puree increased with heating time and temperature. The rheological data fitted very well with the power-law model, showing shear-thinning behaviour. Thus, the heated dragon fruit puree, particularly the red-flesh puree, offer possibilities to be applied in foodstuffs due to their interesting attributes after thermal treatment. The dragon fruit seeds are considered to have a high antioxidative potential. As it is, oil was extracted from the white-flesh and red-flesh dragon fruit seeds with petroleum ether as cold extraction. The characterization of the seed oil is described in Chapter 5. Dragon fruit seeds contained significant amounts of oil, which accounted for about 32-34% of the dried seed weight. The predominant fatty acids of both dragon fruit seed oils were linoleic acid (C18:2, 45-55%), oleic acid (C18:1, 19-24%), palmitic acid (C16:0, 15-18%) and stearic acid (C18:0, 7-8%), respectively. Dragon fruit seed oil is interesting from a nutritional point of view as it contains a high amount of essential fatty acids, amounting to 56%. The triacylglycerol (TAG) composition in the seed oil was also analyzed. It was shown to contain mainly triunsaturated and diunsaturated TAGs. A significant amount of tocopherols in the dragon fruit seed oil was clearly observed in which α-tocopherol was the most abundant tocopherol in the oil (72% of total tocopherol content). In addition, a storage assessment of 3 months was performed, monitoring the oxidative and tocopherol stabilities in the dragon fruit seed oil. A low oxidation rate of both dragon fruit seed oils was obtained, while tocopherol content decreased on storage. However, the concentration of tocopherols in the dragon fruit seed oil after 12 weeks remained high content compared to the initial state. Thus, the dragon fruit seed oil could be considered as a good source of essential fatty acids and tocopherols, with a high oxidative stability. In Chapter 6, the activity of pectic enzymes, i.e. pectin methylesterase (PME) and polygalacturonase (PG), of the pulp and peel of white-flesh and red-flesh dragon fruits as well as inactivation of pectic enzymes by thermal treatments (30-90 °C for 10 min) were examined. The untreated white-flesh dragon fruit had a higher PME activity compared to the red-flesh dragon fruit, whereas PG activity was absent in all dragon fruit samples. Results also demonstrated that no significant effects of PME activity at temperature below 70 °C for 10 min in either the pulp or the peel of dragon fruit were observed, while a moderate heat treatment (80 and 90 °C for 10 min) could efficiently inactivate PME. In the last part of this doctoral study, the polysaccharides (pectic and hemicellulosic substances) from cell wall materials of the pulp and peel of white-flesh and red-flesh dragon fruits were isolated and structurally compared, as described in Chapter 7. Prior to cell wall polysaccharide isolation, blanching at 80 °C for 10 min allowed PME inactivation. Subsequently, cell wall material was extracted and sequentially fractionated using various solutions to obtain three different pectic fractions, a hemicellulosic fraction and a remaining residue fraction which could not be solubilized by the procedure used. These polysaccharide fractions were chemically characterized in terms of galacturonic acid (GalA) content, degree of methoxylation (DM), neutral sugar composition, molar mass distribution and affinity towards some specific anti-pectin antibodies. Results showed that the cell wall polysaccharides in the pulp and, to an even greater extent, in the peel of both white-flesh and red-flesh species contained significant amounts of pectic substances. The pectic substances in the dragon fruit peel as well as in the dragon fruit pulp were shown to be lowly methyl-esterified pectin (DM < 39%) and highly water-soluble pectin (38-47%). A higher contribution of pectic homogalacturonan (HG) region in the peel samples was observed compared to the pulp samples, whereas there were no large differences between the pectic substances of both dragon fruit species. Despite the low average DM value of the pulp and peel pectins, pectic substances in both pulp and peel samples showed an affinity for anti-HG antibodies with different specificities, indicating that a wide range of epitopes (including long blocks of unesterified GalA residues as well as a few consecutive esterified GalA residues) was present. In this doctoral research, the key findings of the experimental work are clearly shown and summarized in Chapter 8. Dragon fruit is an excellent source of bioactive compounds such as betacyanin and tocopherol which both have high antioxidative properties and health-promoting functions. Dragon fruit also provides many valuable products and components that could have high potential of being applied in food industry. For example, the freeze-dried dragon fruit pulp could be used as a red/purple colouring agent (betacyanin) as well as an instant juice powder. The heated dragon fruit puree with significant antioxidative activity could serve as a semi-processed product, the dragon fruit seed oil, on the other hand, was defined as a high-value oil. The extracted cell wall polysaccharide from dragon fruit peel demonstrated to be a low methyl-esterified pectin. Thus, the dissemination of the research findings may significantly increase the value of dragon fruit, resulting in an increased revenue for growers and producers.
Keywords
low-esterified pectic substances, hemicellulose, oxidative stability, pectic enzymes, pectin, physicochemical properties, power law model, storage conditions, neutral sugar composition, thermal properties, triacylglycerol, colour, dragon fruit puree, essential fatty acids, rheological parameters, dragon fruit seed oil, freeze-dried, thermal treatment, tocopherols, pulp, peel, vitamin c, betacyanin, Hylocereus spp., dragon fruit, pitaya, pectin structure, antioxidative properties, food colouring, anti-pectin antibodies

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## Citation

Chicago
Liaotrakoon, Wijitra. 2013. “Characterization of Dragon Fruit (Hylocereus Spp.) Components with Valorization Potential”. Ghent, Belgium: Ghent University. Faculty of Bioscience Engineering.
APA
Liaotrakoon, W. (2013). Characterization of dragon fruit (Hylocereus spp.) components with valorization potential. Ghent University. Faculty of Bioscience Engineering, Ghent, Belgium.
Vancouver
1.
Liaotrakoon W. Characterization of dragon fruit (Hylocereus spp.) components with valorization potential. [Ghent, Belgium]: Ghent University. Faculty of Bioscience Engineering; 2013.
MLA
Liaotrakoon, Wijitra. “Characterization of Dragon Fruit (Hylocereus Spp.) Components with Valorization Potential.” 2013 : n. pag. Print.
@phdthesis{4093845,
abstract     = {Dragon fruit (Hylocereus spp.), also known as pitaya or pitahaya, is increasingly gaining interest in many countries, including Thailand which is a country with a climate ideal for breeding different varieties of tropical and subtropical fruits in general, and dragon fruit more specifically. The benefits of dragon fruit for human health can be explained by its essential nutrients such as vitamins, minerals, complex carbohydrates, dietary fibres and antioxidants. Dragon fruit is also an essential source of betacyanin which serves as a red/purple pigment with antioxidative properties. In Thailand, most of the fresh dragon fruits are consumed domestically, while some are traded globally as fresh fruit and processed products as juice or puree. As it is, dragon fruit peel and seeds are often considered as waste or by-products from fruit processing and have been less successful at adding value to the fruit. Currently, there is limited literature on dragon fruit properties, its processed products and constituents as well as potential utilization. Therefore, the main objective of this doctoral research was to gain deeper insight in the characteristics of dragon fruit and its derived products and components, specifically of two species of dragon fruit, i.e. white-flesh dragon fruit (H. undatus) and red-flesh dragon fruit (H. polyrhizus). The characteristics of these two species of dragon fruit and their possible application are extensively discussed in the doctoral thesis/research.
Chapter 1 provides a state-of-the-art of current and relevant research on dragon fruit with respect to botanical classification, cultivation, economic aspects and chemical composition (particularly nutrients, pigment and antioxidative components) as well as an overview of techniques to process fruits (processing), i.e. freeze-drying and thermal processing. The impact of these processes on quality attributes (e.g. physicochemical and rheological properties) of different fruits is critically reviewed. The review also aims at giving a summarized overview of the most important aspects of seed oil (e.g. chemical composition and lipid oxidation) and cell wall polysaccharides (e.g. structure of pectic and hemicellulosic substances).
In Chapter 2, the properties/characteristics of the whole fruit, peel, pulp (seed-free flesh) and puree (flesh containing seeds) of the two species of dragon fruit were performed. Results demonstrated that the peel of both dragon fruit species as well as the pulp and puree of the red-flesh species contained large amounts of betacyanin. As it is, they have the potential to be utilized as a natural colouring agent. Dragon fruit pulp and puree also exhibited significant antioxidative activities. This was even more pronounced for the red-flesh species due to the presence of betacyanin. The pulp and puree showed shear-thinning behaviour due to the presence of mucilaginous components. Throughout this doctoral research, these characteristics of dragon fruit, for example, pigment, antioxidative activities and rheological properties, provide a deeper insight with regard to further valorization of the fruit components and optimization of the fruit processing.
Chapter 3 describes the characterization of the freeze-dried pulp and peel of the two species of dragon fruit to be further utilized of dragon fruit{\textquoteright}s pigment (betacyanin) as a food additive. It was found that freeze-drying could satisfactorily preserve colour and pigment concentration. The freeze-dried red-flesh pulp and the freeze-dried dragon fruit peel from both species contained high contents of betacyanin. The freeze-dried dragon fruit pulp was well-soluble in water, whereas this was not the case for the freeze-dried dragon fruit peel. Additionally, due to the pH-sensitivity of betacyanin, the influence of pH (1-11) on the colour shift of the freeze-dried dragon fruit peel and pulp was monitored. The colour of the freeze-dried peel was stable within a pH range of 3-7, whereas the freeze-dried red-flesh pulp had high colour stability over a pH range of 1-11 and is available in a convenient form making it suitable for the use as a food colourant.
In order to gain insight into the effects of thermal treatment on the characteristics of the white-flesh and red-flesh dragon fruit purees, the antioxidative, rheological, physicochemical and microbiological properties of the purees were investigated at different process conditions (between 50 and 90 {\textdegree}C for 60 min). The results are presented and discussed in Chapter 4. It is interesting to see that the antioxidative properties of the heated dragon fruit puree increased during heating treatment probably because of the superior antioxidative attributes of the dragon fruit seeds present in the puree and the formation of Maillard reaction products. During thermal processing, the L* value (lightness) and b* value (yellowness) of the processed dragon fruit puree can be used to control the quality online. Total colour change (TCC), which is an indicator of different colour between the unheated and heated puree samples, of the red-flesh dragon fruit puree showed a strong negative correlation with betacyanin content. The kinetics of colour changes and betacyanin degradation in the heated dragon fruit puree followed a second-order models. The apparent viscosity of the heated dragon fruit puree increased with heating time and temperature. The rheological data fitted very well with the power-law model, showing shear-thinning behaviour. Thus, the heated dragon fruit puree, particularly the red-flesh puree, offer possibilities to be applied in foodstuffs due to their interesting attributes after thermal treatment.
The dragon fruit seeds are considered to have a high antioxidative potential. As it is, oil was extracted from the white-flesh and red-flesh dragon fruit seeds with petroleum ether as cold extraction. The characterization of the seed oil is described in Chapter 5. Dragon fruit seeds contained significant amounts of oil, which accounted for about 32-34\% of the dried seed weight. The predominant fatty acids of both dragon fruit seed oils were linoleic acid (C18:2, 45-55\%), oleic acid (C18:1, 19-24\%), palmitic acid (C16:0, 15-18\%) and stearic acid (C18:0, 7-8\%), respectively. Dragon fruit seed oil is interesting from a nutritional point of view as it contains a high amount of essential fatty acids, amounting to 56\%. The triacylglycerol (TAG) composition in the seed oil was also analyzed. It was shown to contain mainly triunsaturated and diunsaturated TAGs. A significant amount of tocopherols in the dragon fruit seed oil was clearly observed in which \ensuremath{\alpha}-tocopherol was the most abundant tocopherol in the oil (72\% of total tocopherol content). In addition, a storage assessment of 3 months was performed, monitoring the oxidative and tocopherol stabilities in the dragon fruit seed oil. A low oxidation rate of both dragon fruit seed oils was obtained, while tocopherol content decreased on storage. However, the concentration of tocopherols in the dragon fruit seed oil after 12 weeks remained high content compared to the initial state. Thus, the dragon fruit seed oil could be considered as a good source of essential fatty acids and tocopherols, with a high oxidative stability.
In Chapter 6, the activity of pectic enzymes, i.e. pectin methylesterase (PME) and polygalacturonase (PG), of the pulp and peel of white-flesh and red-flesh dragon fruits as well as inactivation of pectic enzymes by thermal treatments (30-90 {\textdegree}C for 10 min) were examined. The untreated white-flesh dragon fruit had a higher PME activity compared to the red-flesh dragon fruit, whereas PG activity was absent in all dragon fruit samples. Results also demonstrated that no significant effects of PME activity at temperature below 70 {\textdegree}C for 10 min in either the pulp or the peel of dragon fruit were observed, while a moderate heat treatment (80 and 90 {\textdegree}C for 10 min) could efficiently inactivate PME.
In the last part of this doctoral study, the polysaccharides (pectic and hemicellulosic substances) from cell wall materials of the pulp and peel of white-flesh and red-flesh dragon fruits were isolated and structurally compared, as described in Chapter 7. Prior to cell wall polysaccharide isolation, blanching at 80 {\textdegree}C for 10 min allowed PME inactivation. Subsequently, cell wall material was extracted and sequentially fractionated using various solutions to obtain three different pectic fractions, a hemicellulosic fraction and a remaining residue fraction which could not be solubilized by the procedure used. These polysaccharide fractions were chemically characterized in terms of galacturonic acid (GalA) content, degree of methoxylation (DM), neutral sugar composition, molar mass distribution and affinity towards some specific anti-pectin antibodies. Results showed that the cell wall polysaccharides in the pulp and, to an even greater extent, in the peel of both white-flesh and red-flesh species contained significant amounts of pectic substances. The pectic substances in the dragon fruit peel as well as in the dragon fruit pulp were shown to be lowly methyl-esterified pectin (DM {\textlangle} 39\%) and highly water-soluble pectin (38-47\%). A higher contribution of pectic homogalacturonan (HG) region in the peel samples was observed compared to the pulp samples, whereas there were no large differences between the pectic substances of both dragon fruit species. Despite the low average DM value of the pulp and peel pectins, pectic substances in both pulp and peel samples showed an affinity for anti-HG antibodies with different specificities, indicating that a wide range of epitopes (including long blocks of unesterified GalA residues as well as a few consecutive esterified GalA residues) was present.
In this doctoral research, the key findings of the experimental work are clearly shown and summarized in Chapter 8. Dragon fruit is an excellent source of bioactive compounds such as betacyanin and tocopherol which both have high antioxidative properties and health-promoting functions. Dragon fruit also provides many valuable products and components that could have high potential of being applied in food industry. For example, the freeze-dried dragon fruit pulp could be used as a red/purple colouring agent (betacyanin) as well as an instant juice powder. The heated dragon fruit puree with significant antioxidative activity could serve as a semi-processed product, the dragon fruit seed oil, on the other hand, was defined as a high-value oil. The extracted cell wall polysaccharide from dragon fruit peel demonstrated to be a low methyl-esterified pectin. Thus, the dissemination of the research findings may significantly increase the value of dragon fruit, resulting in an increased revenue for growers and producers.},
author       = {Liaotrakoon, Wijitra},
isbn         = {9789059896277},
language     = {eng},
pages        = {XVIII, 217},
publisher    = {Ghent University. Faculty of Bioscience Engineering},
school       = {Ghent University},
title        = {Characterization of dragon fruit (Hylocereus spp.) components with valorization potential},
year         = {2013},
}


text:
# hessian¶

Graph.hessian(tensor, variables, *, name=None)

Calculate a single Hessian matrix for all the variables.

The Hessian is a matrix containing all the second partial derivatives of the tensor with respect to the variables.

Parameters
• tensor (Tensor(scalar, real)) – The real scalar tensor $$T$$ whose Hessian matrix you want to calculate.

• variables (list[Tensor(real)]) – The list of real variables $$\{\theta_i\}$$ with respect to which you want to take the second partial derivatives of the tensor. If any of the tensors of the list is not scalar, this function treats each of the elements of the tensor as a different variable. It does this by flattening all tensors and concatenating them in the same sequence that you provided in this list.

• name (str, optional) – The name of the node.

Returns

The real Hessian matrix $$H$$ containing the second partial derivatives of the tensor $$T$$ with respect to the variables $$\{\theta_i\}$$.

Return type

Tensor(2D, real)

Warning

This function currently doesn’t support calculating a Hessian matrix for a graph which includes an infidelity_pwc node if it involves a Hamiltonian with degenerate eigenvalues at any segment. In that case, the function returns a nan Hessian matrix.

Notes

The $$(i,j)$$ element of the Hessian contains the partial derivative of the tensor with respect to the ith and the jth variables:

$H_{i,j} = \frac{\partial^2 T}{\partial \theta_i \partial \theta_j}.$

The variables $$\{\theta_i\}$$ follow the same sequence as the input list of variables. If some of the variables are not scalars, this function flattens them and concatenates them in the same order of the list of variables that you provided to create the sequence of scalar variables $$\{\theta_i\}$$.

If you choose a negative log-likelihood as the tensor $$T$$, you can use this Hessian as an estimate of the Fisher information matrix.

format: plain


text:
Required field
Required field
Required field

# Advanced Tutorial: Natural Convection of a LED Spotlight

This tutorial shows how to perform a natural convection simulation on a LED spotlight, aiming to satisfy maximum temperature requirements.

This tutorial teaches how to:

• Set up and run a natural convection simulation using the Conjugate heat transfer v2.0 solver;
• Assign boundary conditions, material, and other properties to the simulation;
• Mesh with the SimScale standard meshing algorithm.

We are following the typical SimScale workflow:

1. Prepare the CAD model for the simulation.
2. Set up the simulation.
3. Create the mesh.
4. Run the simulation and analyze the results.

Did you know?

For applications with LEDs, it’s important to control the temperatures developing on the chips. A poor design, with high temperatures, is harmful to the lifetime of a LED package.

In this tutorial, we will simulate how effective natural convection is in cooling our geometry:

## 1. Prepare the CAD Model and Select the Analysis Type

As a first step, please click on the button below. It will import the tutorial project directly into your Workbench.

The following picture demonstrates what should be visible after importing the tutorial project:

Note that the project contains two geometries. The first one consists of the complete LED spotlight geometry, and the second one consists of a 90 degrees slice.

Did you know?

It’s possible to use the symmetrical nature of the LED spotlight geometry in our favor. Since we expect the flow to be mirrored along the symmetry planes, we can use just a quarter of the geometry. This brings a series of benefits, such as:

– Allows faster meshing operations and simulation runs
– Possibility to use finer cells, thus improving the domain discretization

### 1.1. Create an Enclosure

The initial CAD geometry only contains the LED spotlight parts. Therefore, it’s necessary to create an external air domain. In SimScale, this is possible with an enclosure operation.

To do that, please right-click on the LED Spotlight – Quarter Geometry and select an ‘Enclosure‘ operation.

The enclosure should be large enough to prevent the boundary conditions from interfering with the results. In Figure 6, you will find a good rule of thumb for the enclosure dimensions:

• Top: 6 times the height of the LED spotlight
• Down: 4 times the height of the geometry
• Sides: 8 times the width of the geometry

Please set up the enclosure dimensions as in Figure 7:

1. Enable Keep existing parts. This step is important for conjugate heat transfer simulations. This way, the solid parts won’t get deleted during the enclosure operation.
2. Set the enclosure dimensions to the following:
• Minimum (x) value: 0 meters
• Minimum (y) value: -0.3 meters
• Minimum (z) value: 0 meters
• Maximum (x) value: 0.25 meters
• Maximum (y) value: 0.4 meters
• Maximum (z) value: 0.25 meters
3. ‘Start’ the operation.

### 1.2. Imprint Operation

Another important step is to run an Imprint operation. In our LED geometry, a series of interfaces between solid/solid and solid/flow regions are present. The imprint operation improves the automatic detection of such interfaces.

Using Figure 5 as a reference, create an Imprint operation and click ‘Start’.

### 1.3. Create the Simulation

Figure 8 shows the Workbench once the Imprint operation finishes running. Click on the ‘Create Simulation’ button to proceed.

The analysis type choice widget opens up. From the options, pick ‘Conjugate heat transfer v2.0’ and ‘Create the Simulation’:

As a result of a strongly coupled energy equation, the Conjugate heat transfer v2.0 analysis type provides much faster convergence rates, in comparison to the conventional Conjugate heat transfer (CHT).

Did you know?

The analysis type you choose within the simulation library depends on what results you are interested in, and what given parameters you have.

• With a Conjugate Heat Transfer (CHT or CHT v2.0) analysis type, it’s possible to simulate heat transfer between solids and also between the solid and fluid domains. This is the common choice for electronics cooling.
• If you are only interested in the heat transfer within solids, you can simplify the simulation process by performing a Heat Transfer analysis.
• Lastly, for simulating heat transfer only in the fluid domain, the appropriate analysis type is a Convective Heat Transfer.

After creating a simulation, the simulation tree will be visible in the left-hand side panel. It’s necessary to set up all entries to be able to run the simulation. At this point, SimScale will also automatically detect all Contacts within the geometry. In our case, there are a total of 64 interfaces.

The global simulation settings remain as default, as in Figure 10. Since it’s a natural convection simulation, the velocities in the domain are small, and a ‘Laminar’ turbulence model is appropriate.

Did you know?

In natural convection simulations, where the temperature gradients in the air domain are small, it’s common to use the Boussinesq approximation to account for buoyancy.

In Figure 10, one can use the Boussinesq approximation by keeping Compressible toggled off. For further information, please check this documentation page about buoyancy.

## 2. Assigning the Material and Boundary Conditions

In the following sections, we will set up the physics of the simulation.

### 2.1. Model

In the Model tab, it’s possible to define the direction and magnitude of gravity:

Please define gravity as ‘-9.81’ $$m/s^2$$ in the y-direction.

### 2.2. Defining the Materials

In conjugate heat transfer simulations, it’s necessary to define both solid and fluid materials. Let’s first go through the fluid domain.

Important

Each volume in the domain needs to receive one material assignment.

Assigning more than one material to a volume or leaving parts without any assignments will result in an error.

2.2.1 Fluid Materials

To add a fluid material, please click on the ‘+ button’ next to Fluid:

The fluid material library pops up. From the list, choose ‘Air‘:

The geometry contains two air regions: one internal to the LED, and the external flow region. You can use the right-hand side panel to quickly select the volumes.

Did you know?

For this tutorial, we will use the default values for the air properties. It’s possible, however, to edit each one of the values from Figure 14 by clicking on them. Find more details about using custom materials in this article.

2.2.2 Solid Materials

Let’s now add the solid materials. First, click on the ‘+ button’ next to Solids, as in figure 12. This time, a library of solid materials appear.

From the list, please choose ‘PVC’, which is a common material for circuit boards:

The materials library contains common properties for PVC. In this tutorial, PVC is used in a circuit board, so we will adjust the material properties to account for the circuits. Figure 16 shows the changes:

• Set $$(\kappa)$$ Thermal conductivity to 2.5 $$\frac {W}{m.K}$$.
• The new Specific heat is 1004 $$\frac {J}{kg.K}$$.
• Set $$(\rho)$$ Density to 1900 $$\frac {kg}/{m^3}$$.
• Lastly, assign the Board volume.

All the remaining solid materials maintain the default settings. Find below a list of materials and the respective assignments.

• Aluminium: Core
• Brass: Connector Base
• Copper: Connector Pin
• Glass: Glass
• Silicon: 0.25 W Chip, 0.5 W Chip, and 1 W Chip

### 2.3. Initial Conditions

In CFD simulations, it’s a good practice to initialize the parameters close to the expected solution. With this approach, it’s possible to improve the convergence rate of a simulation, achieving the final result faster.

In this tutorial, we will change the initialization of temperature and velocity.

Did you know?

In SimScale, there are two ways to define the initialization of a parameter:

• Uniform initialization, which initializes a parameter in the entire domain with the same value
• Subdomain initialization, which initializes a parameter in specific volumes within the domain

2.3.1 Temperature

The LED geometry contains three chips, which are dissipating heat. From a design perspective, it’s important to keep the chip temperatures under control, otherwise, the lifetime of the LED gets shorter.

Usual temperatures on the chips are around 100°C, thus, initializing the LED parts at 75°C is a good initial guess. Figure 17 shows the initial steps for initialization with subdomains.

1. Add a new subdomain by clicking on the ‘+ button’ next to Subdomains.
2. Input the initialization value, in this case, 75°C.

We want to assign all the LED parts to this subdomain. The quickest way to do that is by using an Invert visible assignment function. Please proceed as below:

1. Select the Flow region volume.
2. By right-clicking in the viewer, a window with options appears.
3. Click on Invert visible assignment. This way, the 9 LED volumes are assigned.

2.3.2 Velocity

As a result of the high temperatures on the LED geometry, a natural convection plume develops. Therefore, we strongly recommend initializing the velocity field around the LED, which enhances the convergence rate of the simulation.

In this tutorial, we will initialize the velocity field using a cylinder geometry primitive, which is a good option for the shape of the LED geometry. Figure 20 shows the initial steps:

1. Add a new subdomain by clicking on the ‘+ button’ next to Subdomains.
2. Using the orientation cube as a reference, initialize the velocity in the y-direction with 0.05 $$m/s$$.
3. Click on the ‘+ button’ next to Geometry primitives and select a Cylinder.

Now, a window opens up, where the user can define the cylinder orientation and dimensions. Using a cylinder slightly wider than the LED geometry yields good results. Therefore, please input the values from Figure 21:

After saving the cylinder primitive, make sure to assign it to the subdomain initialization.

Important

The optimal value for the velocity initialization may vary from case to case. For natural convection applications, initializing velocity with velocities between 0.05 and 0.2 m/s provides a very good convergence in most cases.

### 2.4. Boundary Conditions

For boundary condition definition, we will use Figure 22 as reference:

2.4.1 Natural Convection Inlet/Outlet

The natural convection inlet/outlet boundary condition is exclusive to SimScale. This boundary condition allows fluid to go in and out of the domain, therefore it’s a good option when it’s not clear what the flow behavior will be. For more details on the mathematical implementation, see this dedicated documentation page.

To create a new boundary condition, click on the ‘+ button’ next to Boundary conditions, and select the desired type from the drop-down menu.

Using figure 22 as a reference, the top, bottom, and both side faces will receive a natural convection inlet/outlet boundary condition.

2.4.2 Symmetry

Please create a second boundary condition, as in figure 21. This time, however, select a Symmetry condition:

Note

Please note that all 16 faces in the symmetry plane receive a symmetry boundary condition. This includes all faces shown in Figure 26.

Additionally, the geometry contains three chips, which will be defined as power sources. Figure 27 contains further details:

In SimScale, there are two types of Power sources:

• Absolute power source: the user defines the heat flux in $$W$$ or $$\frac {btu}{s}$$
• Specific power source: the definition of the flux is in $$W/m^3$$ or $$\frac {btu}{s.in^3}$$

Hence, the Absolute power source works best in our case, as we already know the exact heat flux in each chip.

Please proceed to set up the first power source, as in Figure 28:

1. Create a new power source by clicking on the ‘+ button’ next to Power sources.
2. Select an Absolute power source and set the heat flux to 1 watt.
3. Assign the 1 W Chip volume to this power source, which can be done using the right-hand side geometry panel in the Workbench.

Please create two new power sources, for the 0.5 W Chip and the 0.25 W Chip volumes. Follow the same steps outlined above, remembering to adjust the Heat flux accordingly.

### 2.6. Numerics & Simulation Control

The default settings in the Numerics tab work well for most simulations. In this tutorial, no changes are required.

In the Simulation control tab, it’s possible to define a series of parameters that control the simulation process, including the number of iterations to perform, and the maximum runtime for the simulation run. Figure 29 highlights the changes in the simulation control tab:

• Define an End time of 750 iterations and a Write interval of 750 iterations. With these settings, only the result set from the last iteration will be written, which is a common practice for steady-state simulations.
• Change the Maximum runtime of the simulation to 30000 seconds.

Did you know?

In a steady-state simulation, the End time and Delta t parameters control the number of iterations to be performed.

For more notes on the simulation control settings, please visit the following page: How to set up simulation control parameters in a steady-state CFD analysis.

### 2.7. Results Control

Setting up result controls is a crucial step in a simulation. Since they help us to assess convergence, it’s important to set meaningful result controls for our parameters of interest.

For a LED simulation, some parameters of interest are, for example, the temperature on the chips. Therefore, we will set Area average result controls on them. To do that, please proceed as in Figure 30:

1. Create a new Area average control by clicking on the ‘+ button’ next to Surface data.
2. Define a Write interval of 5 iterations. This way, the algorithm will plot the area averages every 5 iterations, which is good enough for steady-state.
3. To make the face selection easier, please hide the Board volume, by clicking on the ‘eye icon’ next to it.
4. For the first result control, please select the top face of the 1-watt chip.

Afterward, following the same procedure, please create two additional area average controls, for the top faces of the 0.5 W Chip and the 0.25 W Chip.

## 3. Mesh

For the meshing operation, we recommend using the standard algorithm, which is a good choice in general, as it is quite automated and delivers good results for most geometries.

After clicking on Mesh in the simulation tree, a setup window opens. The main default settings are good for an initial study. Under Advanced settings, please set the Number of gap elements to 3.

Did you know?

In CHT simulations, oftentimes the solid parts will have thin sections. It’s important to add at least 2 or 3 cells to correctly resolve the temperature gradients within these small parts.

By defining the Number of gap elements to 3, the meshing algorithm will ensure at least 3 elements capturing the small sections.

With these mesh settings, we already have a good mesh that allows us to run a successful simulation.

In the box below, we will show optional steps to configure a region refinement for the mesh, which allows us to capture the convection plume more accurately. If you would like to proceed without region refinements, please skip to section 4.

(Optional) Region refinement for the convection plume

In natural convection simulations, a convection plume develops above the hot parts. To capture the plume more accurately, we can use finer mesh cells around that region.

With the standard mesher, a region refinement is appropriate to capture the developing plume. Therefore, please click on the ‘+ button’ next to Refinements:

In the configuration window that opens, you can define the Maximum edge length for the cells inside the refinement region. In Figure 35, you will find the necessary steps:

1. Define a Maximum edge length of 0.0025 meters
2. Click on the ‘+ button’ next to Geometry primitives and choose a ‘Cartesian box’

Next, it’s necessary to define the dimensions of the cartesian box. Keeping the convection plume in mind, please input the following dimensions:

• Minimum (x) value: 0 m
• Minimum (y) value: -0.1 m
• Minimum (z) value: 0 m
• Maximum (x) value: 0.05 m
• Maximum (y) value: 0.25 m
• Maximum (z) value: 0.05 m

As a final step, return to the Region refinement previously created, and toggle on the cartesian box assignment.

This concludes the configuration of the region refinement.

Did you know?

The CHTv2.0 solver uses a special kind of meshes, named conformal meshes. In a conformal mesh, the cells at the interface between two parts will match perfectly, which allows for a more robust setup and faster convergence.

For this reason, meshes created for other analysis types can’t be used in a CHTv2.0 simulation.

## 4. Start the Simulation

To start a new simulation, please click on the ‘+ button’ next to Simulation Runs.

This way, the mesh will be generated, and afterward the simulation run will start automatically. While the simulation results are being calculated, you can already have a look at the intermediate results in the post-processor. They are being updated in real-time! You will also find a link to the finished project at the end of the tutorial.

The simulation run takes from 110 to 150 minutes to finish. At this point, you can access the post-processing environment by clicking on ‘Solution Fields’ or ‘Post-process results’:

## 5. Post-Processing

### 5.1 Result Controls

The average chip temperatures are available under the ‘Area averages’ tab:

The plot shows how the chip temperature is evolving as the algorithm performs iterations. In a converged simulation, the temperatures will stabilize and no longer change between iterations. As a result of the strongly coupled formulation, the CHT v2.0 algorithm the temperature converges quickly, in just over 400 iterations.

Did you know?

With SimScale, it’s possible to run multiple simulations in parallel for different LED designs. The aim is to obtain an optimized geometry, with lower temperatures on the chips.

This way, you can evaluate various geometries faster, speeding up the design process.

### 5.2 Post-Processing Pictures

After accessing the post-processor by clicking on ‘Solution Fields‘, you can use several post-processing filters to further analyze the results.

In the animation below, we can see the flow field that develops around the LED spotlight. Air is drawn from the bottom and the sides of the spotlight, due to the temperatures in the solid parts.

Have a look at our post-processing guide to learn how to use the post-processor for this simulation.

Congratulations! You finished the tutorial!

Note

If you have questions or suggestions, please reach out either via the forum or contact us directly.

Last updated: September 20th, 2020


text:
# eledpar doesn't respect changes of line spacing

I try to typeset a bilingual text on facing pages. For this I use the eledmac/eledpar packages. The latin text on the even pages should get displayed larger than the translation on the odd pages. However, within the environments of the eledpar package, the linespread does not get sufficiently increased.

The difference to a normal paragraph outside the environments of eledpar with larger font size is small but noticeable and leads to a suboptimal outcome.

MWE:

\documentclass{scrbook}
\usepackage[latin,english]{babel}
\usepackage{blindtext}
\usepackage{eledmac,eledpar}
\begin{document}
\begin{pages}\numberlinefalse{}
\begin{Leftside}\large \selectlanguage{latin}
\beginnumbering
\pstart \blindtext \blindtext \pend
\endnumbering
\par
\end{Leftside}

\begin{Rightside}\selectlanguage{english}
\beginnumbering
\pstart \blindtext \blindtext \pend
\endnumbering
\end{Rightside}
\Pages
\end{pages}

The following has correct line spacing
\selectlanguage{latin}
{\large
\blindtext \blindtext\par
}
\end{document}

Am I doing something wrong?

(The parallel package seems to work correctly, but it is not an option for me as it can't set even basic footnotes on the pages which are typeset in parallel.)

EDIT: I get the impression that eledpar doesn't respect alterings of the line spacing at all -- see this MWE:

\documentclass{scrbook}
\usepackage[latin,english]{babel}
\usepackage{blindtext}
\usepackage{eledmac,eledpar}
\begin{document}
\begin{pages}\numberlinefalse{}
\begin{Leftside}\selectlanguage{latin}
\beginnumbering
\pstart \linespread{1.5}\selectfont \blindtext \blindtext \par \pend
\endnumbering
\end{Leftside}

\begin{Rightside}\selectlanguage{english}
\beginnumbering
\pstart \blindtext \blindtext \pend
\endnumbering
\end{Rightside}
\Pages
\end{pages}


text:
# Measure distance for spatialLine in R returns bad value

I have this spatialLine in R:

library(sp)
library(maptools)
library(SDraw)
library(leaflet)

x <- c(153.9, 152.67072,147.9879,144.25364,135.45557,116.03998,76.92708,45.10622,37.02522,36.76888,36.41,37.19583,41.55226,50.99487,53.18)

y <- c(73.52,75.80461,78.28467,80.82467,83.32397,85.49637,86.39683,85.16955,82.60437,78.01859,73.35001,72.87417,71.23226,71.55405,68.73)

d <- SpatialLines(list(Lines(Line(cbind(x,y)), ID='a')))
crs(d) <- "+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs"

leaflet(data = d) %>%
addPolylines() # just plot the line

I want to measure its length. I tried:

> lineLength(d, byid = T)
[1] 148.407

However, this is not good (the actual size of this would be aroung ~4000 km as you can see). How can I actually measure the path length?

• The measurement is correct, if you want the output in Cartesian degrees (which are meaningless). You probably want to sum on geodetic distance. – Vince Jan 12 at 23:37
• try sp::LineLength(d, longlat = TRUE) – mdsumner Jan 14 at 5:24

Edit: use geosphere...

There's a few packages that can compute true distance given lat-long coordinates, for example fields has the rdist.earth and rdist.earth.vec functions.

rdist.earth.vec takes two lat-long matrix arguments with the same number of rows and computes the distance from the points in matching rows. So for your simple single line feature we can get all the coordinates with:

> xy = coordinates(d)[[1]][[1]]

By dropping off the last row and the first row we can create two matrices where the rows are points 1 and 2, and then 2 and 3, and then 3 and 4 and so on, and then sum the total distance between those pairs:

> sum(fields::rdist.earth.vec(xy[-15,],xy[-1,], miles=FALSE))
[1] 4367.507

I think the package authors are from the USA, so don't forget to use miles=FALSE to get kilometres!

Or even easier, use geosphere package:

> library(geosphere)
> lengthLine(d)
[1] 4380327

This is in metres, and is slightly different to the rdist.earth measurement. It uses an ellipsoid model for the earth rather than a spherical model that rdist.earth uses. It also works directly on Spatial data objects.

Another solution is to convert lat-long to a cartesian coordinate system and then use pythagoras theorem, but that is tricky for data over a large area and is likely to be less accurate.

• thank you very much! I have multiple features in the spatialLine in fact but that should be easy to do with a loop. thanks for the help – Andrei Niță Jan 13 at 12:18
• I think geosphere is more accurate and convenient - see edit. – Spacedman Jan 13 at 13:13


text:
# Features for time series classification

I consider the problem of (multiclass) classification based on time series of variable length $T$, that is, to find a function $$f(X_T) = y \in [1..K]\\ \text{for } X_T = (x_1, \dots, x_T)\\ \text{with } x_t \in \mathbb{R}^d ~,$$ via a global representation of the time serie by a set of selected features $v_i$ of fixed size $D$ independent of $T$, $$\phi(X_T) = v_1, \dots, v_D \in \mathbb{R}~,$$ and then use standard classification methods on this feature set. I'm not interested in forecasting, i.e. predicting $x_{T+1}$. For example, we may analyse the way a person walks to predict the gender of the person.

What are the standard features that I may take into account ? In example, we can obviously use the mean and variance of the serie (or higher order moments) and also look into the frequency domain, like the energy contained in some interval of the Discrete Fourier Transform of the serie (or Discrete Wavelet Transform).

## Simple statistical features

• Means in each of the $d$ dimensions
• Standard deviations of the $d$ dimensions
• Skewness, Kurtosis and Higher order moments of the $d$ dimensions
• Maximum and Minimum values

## Time serie analysis related features

• The $d \times d-1$ Cross-Correlations between each dimension and the $d$ Auto-Correlations
• Orders of the autoregressive (AR), integrated (I) and moving average (MA) part of an estimated ARIMA model
• Parameters of the AR part
• Parameters of the MA part

## Frequency domain related features

See Morchen03 for a study of energy preserving features on DFT and DWT

• frequencies of the $k$ peaks in amplitude in the DFTs for the detrended $d$ dimensions
• $k$-quantiles of these DFTs
• Emile, This question is similar to one I just posed (stats.stackexchange.com/questions/51475/…). Would you be able to post any R code for the DFT features? – B_Miner Mar 6 '13 at 15:43
• Is there any method based on shapelets for variable length time series? – Simone Jun 9 '15 at 4:18

Emile, I think the features listed in your answer are pretty good starting points, though as always, I think some domain expertise (or at least a good long think) about your problem is equally important.

You may want to consider including features calculated from the derivatives (or integrals) of your signal. For example, I would wager that rapid acceleration/deceleration is a reasonably good predictors of accident-prone driving. That information is obviously still present in the position signal, but it's not nearly as explicit.

You may also want to consider replacing the Fourier coefficients with a wavelet or wavelet packet representation. The major advantage of wavelets is that they allow you to localize a feature in both frequency and time, while the traditional Fourier coefficients are restricted to only time. This might be particularly useful if your data contains components that switch on/off irregularly or has square wave-like pulses that can be problematic for Fourier methods.

As the other answers suggested, there is a huge number of time series characteristics that can be used as potential features. There are simple features such as the mean, time series related features such as the coefficients of an AR model or highly sophisticated features such as the test statistic of the augmented dickey fuller hypothesis test.

# Comprehensive Overview over possible time series features

The python package tsfresh automates the extraction of those features. Its documentation describes the different calculated features. You can find the page with the calculated features here.

Disclaimer: I am one of the authors of tsfresh.

I suggest you, instead of using classic approaches for extracting hand-engineered features, utilise autoencoders. Autoencoders plays an important role in the feature extraction of deep learning architecture.

The autoencoder tries to learn a function $f(X_T)≈X_T$. In other words, it is trying to learn an approximation to the identity function, so as to output $\hat X_T$ that is similar to $X_T$.

The identity function seems a particularly trivial function to be trying to learn; but by placing constraints on the network, such as by limiting the number of hidden units, we can discover interesting structure about the data.

In this way, your desired $ϕ(X_T)=v_1,…,v_D \in R$ will be equivalent to the output values of the middlemost layer in a deep autoencoder, If you limit the number of hidden units in the middlemost to $D$.

Additionally, you can use many flavors of autoencoder for finding the best solution to your problem.

The linked paper will be somewhat enlightening, since it is interested in the more or less the same issue in another context.

Paper abstract (in the Internet Archive)

Paper PDF

Depending on the length of your time series, the usual approach is to epoch the data into segments, e.g. 10 secs.

However, often prior to breaking the time-series into segments it is necessary to perform some preprocessing such as filtering and artifact rejection. You can then compute a variety of features such as those based on frequency (i.e. take an FFT for each epoch), time (e.g. mean, variance etc of the time-series in that epoch) or morphology, (i.e. the shape of the signal/time-series in each epoch).

Usually the features used to classify segments (epochs) of a time-series/signal are domain-specific but Wavelet/Fourier analysis are simply tools to allow you examine your signal in the frequency/time-frequency domains rather than being features in themselves.

In a classification problem each epoch will have a class label e.g. 'happy' or 'sad', you would then train a classifier to distinguish between 'happy' and 'sad' epochs using the 6 features calculated for each epoch.

In the event that each time series represents a single case for classification, you need to calculate each feature across all samples of the time series. The FFT is only relevant here if the signal is linear time invariant (LTI), i.e. if the signal can be considered to be stationary over the whole time series, if the signal is not stationary over the period of interest, a wavelet analysis may be more appropriate. This approach will mean that each time series will produce one feature vector and will constitute one case for classification.

• I've always felt that breaking time series, a naturally continuous process, into discrete periods (or epochs), results in a loss of information. Unless there is natural epochs in the series, how does one choose epochs? Likely epochs can be found that fits any desired outcome. – Cam.Davidson.Pilon Feb 25 '13 at 16:02
• Not sure that an epoch can be found to suit every outcome but for any practical problem With a non-stationary signal you need to find someway to take into account temporal variations (if a signal/time-series adequately described analytically or is linear time invariant this not necessary). The epoch length is again domain specific but usually chosen to be short enough that the signal is stationary under the time period (epoch) of interest. – BGreene Feb 25 '13 at 16:10
• BGreene, could you explain what is the benefit of partitioning the serie into epochs for selecting features used in classification ? I see this partitioning as a helper for later computations (i.e. FFT), but not as something related to the selection of features themselves. Maybe it is related to the "morphology" you mentioned. – Emile Feb 25 '13 at 16:20
• The partitioning into epochs has nothing to do with feature selection. If you have a long signal (e.g. 10 hours of recording sampled @100Hz) then in order to examine how the signal changes over time you need to break it into epochs. In a classification problem each epoch will have a class label e.g. 'happy' or 'sad', you would then train a classifier to distinguish between 'happy' and 'sad' epochs using the 6 features calculated for each epoch. – BGreene Feb 25 '13 at 16:39
• Ok:) It is definitely unrelated to my original question. I consider the case where the class labels the whole serie. I'm going to edit my question to add a clarifying example. – Emile Feb 25 '13 at 16:52


text:
# What are the possible rational roots of 6x^4+35x^3-x^2-7x-1=0 and then determine the rational roots?

Jan 9, 2017

The possible rational roots are +-1;+-1/2;+-1/3;+-1/6;
The rational roots are 1/2;-1/3

#### Explanation:

The possible rational roots are obtained by dividing each of the factors of the known term (-1) by each of the factors of the maximum degree term's coefficient (6).

Then they are:

+-1;+-1/2;+-1/3;+-1/6

To determine the rational roots, let's use the remainder rule:

$\textcolor{red}{x = 1} \to 6 + 35 - 1 - 7 - 1 = 32 \ne 0$
$\textcolor{red}{x = - 1} \to 6 - 35 - 1 + 7 - 1 = - 24 \ne 0$
$\textcolor{red}{x = \frac{1}{2}} : 6 {\left(\frac{1}{2}\right)}^{4} + 35 {\left(\frac{1}{2}\right)}^{3} - {\left(\frac{1}{2}\right)}^{2} - 7 \left(\frac{1}{2}\right) - 1 = 0$

Then $x = \frac{1}{2}$ is a rational root.

You also can use a spreadsheet to find zeros faster.

Then you would find only another rational root $x = - \frac{1}{3}$


text:
# Can two perfect squares average to a third perfect square? [duplicate]

My question is does there exist a triple of integers, $a<b<c$ such that $b^2 = \frac{a^2+c^2}{2}$

I suspect that the answer to this is no but I have not been able to prove it yet. I realize this is very similar to the idea of Pythagorean triples but I am not versed enough in this subject to try and modify the theory for this case. One simple observation is that in order to have any hope of this working is that $a$ and $c$ must be of the same parity. Furthermore if such a triple exists we can build an infinite sequence since $(2b)^2 = \frac{(2a)^2+(2c)^2}{2}$ if and only if $b^2 = \frac{a^2+c^2}{2}$

Any help on determining this to be either true or false would be much appreciated. I am hoping it does end up being impossible, so if someone does find a desired triple I would next move up to cubes instead of squares

Edit: Thank you for the comments, I have foolishly overlooked a simple example and see that there are many solutions to this based on the theory of diophantine equations. However this is unfortunate for me because i was hoping NOT to be able to solve this. This question arose while studying a certain type of graph labeling. What I desire is to be able to create a sequence, $S$, of arbitrary length (since each member of the sequence is to be a label of a vertex in the graph) such that for every $x \in S$, $|x-s_i| \neq |x-s_j|$ for $i \neq j$. I was naively hoping that the sequence of squares was sufficient to satisfy this condition.

Further edit, I have found that the sequence $2^n$ works but it would be nice if I could find a polynomial sequence.

## marked as duplicate by mathlove, Joel Reyes Noche, Najib Idrissi, Travis, Jonas MeyerMay 30 '15 at 17:38

• (1 + 49) / 2 = 25 – Will Jagy May 28 '15 at 22:18
• @WillJagy I'd say that qualifies as an answer. – AlexR May 28 '15 at 22:22
• Now the next step is to determine if there is an infinite number of such triplets, or just a finite number – Tryss May 28 '15 at 22:24
• There are plenty of 3AP (arithmetic progressions of three elements) made of squares. However, there is no 4AP of squares. This is more or less equivalent to the case $n=4$ of the Fermat Last Theorem. – Jack D'Aurizio May 28 '15 at 22:35
• I think it gets even better, since you can have an actually arbitrary dense sequence of such integers, as demonstrated here : ncbi.nlm.nih.gov/pmc/articles/PMC1078964 – C4stor May 29 '15 at 14:38

$$\frac{1 + 49}{2} = 25$$

There are infinitely many solutions to $x^2 + y^2 = 2 z^2.$ I am pretty sure that version has been asked on MSE before, and formulas giving all integer answers were given. Give me a few minutes, I will find that or do it over.

here is a recent discussion Diophantine Equations : Solving $a^2+ b^2=2c^2$

• could you please add the link from MSE? could not find it. – jermenkoo May 29 '15 at 13:30

Parametric solution: $$a = |x^2 - 2 x y - y^2|, \ b = x^2 + y^2,\ c = x^2 + 2 x y - y^2$$

• $a\lt b\lt c{}$? – Barry Cipra May 28 '15 at 22:59
• $a < b < c$ if $0 < y < x$. – Robert Israel May 28 '15 at 23:37

If $b^2=|x+iy|^2$, then $2b^2=|(1+i)(x+iy)|^2$, so all you need to get rolling are Pythagorean triples (written here as $b^2=x^2+y^2$).

• Barry, I did not initially put it together, but we get infinitely many solutions to any $x^2 + y^2 = n z^2$ as long as $n = u^2 + v^2$ and $a^2 + b^2 = c^2$ is Pythagorean, take $(ua \pm vb)^2 + (ub \mp va)^2 = n c^2.$ – Will Jagy May 29 '15 at 0:10

This is an off-the-wall answer/consequence but I think it's interesting.

Consider the equation $$x^2 + y^2 = 2z^2$$ this can be rewritten as $$(x + z)(x - z) = (z + y)(z - y).$$

Let $A = x + z, \quad B = x - z, \quad D = z + y, \quad C = z - y.$

Also let $P = AB = CD.$

Note that

\begin{align} (\xi + A)(\xi - B) &= \xi^2 + 2z\xi - P\tag{1}\\ (\xi + C)(\xi + D) &= \xi^2 + 2z\xi + P\tag{2}\\ \end{align}

This implies that each of the four polynomials $\xi^2 \pm 2z\xi \pm P$ is factorable over the set of integers.

Further, if $A, B, C,$ and $D$ that solve $(1)$ and $(2)$ can be found, then

$$x = \frac 1 2 (A + B), \quad y = \frac 1 2(D - C), \quad z = \frac 1 2(A - B) = \frac 1 2(C + D)$$

Actually, $x, y,$ and $z$ as show above will usually come out to be multiples of $\frac 1 2.$ If that is the case, then you have to use $2x, 2y,$ and $2z$.

I have a heuristic for remembering how to find values of $A, B, C,$ and $D.$ Choose any two relatively prime, opposite-parity numbers $n > m$ and fill in the following table.

\begin{array}{|c|c|c} \hline n & n+m & A = n(n+m) \\ \hline n-m & m & B = m(n-m) \\ \hline C = n(n-m) & D = m(n+m) & \end{array}

It follows that $$x = \dfrac{n^2 - m^2}{2} + mn, \quad y = \dfrac{n^2 - m^2}{2} - mn, \quad z = \dfrac{n^2 + m^2}{2}$$

To show what this is like with some numbers, let n = 3 and m = 2. We get

\begin{array}{|c|c|c} \hline 3 & 5 & A = 15 \\ \hline 1 & 2 & B = 2 \\ \hline C = 3 & D = 10 & \end{array}

We see that \begin{align} (\xi + 15)(\xi - 2) &= \xi^2 + 13\xi - 30\\ (\xi + 3)(\xi + 10) &= \xi^2 + 13\xi + 30\\ \end{align}

And, doubling our answers, $2x = 17, \quad 2y = 7, \quad 2z = 13$

And we see that $x^2 + y^2 = z^2$

Suppose $0<a<b<c$ is a pythagorean triple, i.e., $a^2+b^2=c^2$. Then, $2c^2$ is also a sum of two squares: $$2c^2 = 2(a^2+b^2) = (1^2+1^2)(a^2+b^2) = (b+a)^2+(b-a)^2.$$ Thus, the pythagorean triple $(a,b,c)$ produces a solution $(b-a,b+a,c)$ of $x^2+y^2=2z^2$. For instance, $(3,4,5)$ produces $(1,7,5)$, or $1+7^2=2\cdot 5^2$.
Conversely, if $(e,f,g)$ is a solution of $x^2+y^2=2z^2$ with $e<f$, then $g^2$ is also a sum of two squares, namely $$g^2 = \left(\frac{f+e}{2}\right)^2 + \left(\frac{f-e}{2}\right)^2.$$ Hence $(e,f,g)$ produces a pythagorean triple $((f-e)/2,(f+e)/2,g)$. (Note that $f^2+e^2$ is even, and so $f\equiv e \bmod 2$, and therefore $f+e$ and $f-e$ are even too.)
It follows that there is a bijection between pythagorean triples and solutions of $x^2+y^2=2z^2$: $$(a,b,c) \mapsto (b-a,b+a,c).$$
Since every pythagorean triple can be written as: $$(\lambda(n^2-m^2),\lambda 2mn,\lambda(n^2+m^2))$$ for some $n>m>0$ and some $\lambda\geq 1$, we have that every solution of $x^2+y^2=2z^2$ is of the form $$(\lambda(2mn-(n^2-m^2)), \lambda(n^2-m^2+2mn),\lambda(n^2+m^2))$$ for some $\lambda\geq 1$ and some $0<m<n$.


text:
### 机器视觉代写|图像处理作业代写Image Processing代考|CHAIN CODE

statistics-lab™ 为您的留学生涯保驾护航 在代写图像处理Image Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图像处理Image Processing代写方面经验极为丰富，各种代写图像处理Image Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器视觉代写|图像处理作业代写Image Processing代考|CHAIN CODE

Chain codes are a notation for recording the list of boundary pixels of an object. The chain code uses a logically connected sequence of straight-line segments with specified length and direction to represent the boundary [45]. A chain code can be created by tracking a boundary in some direction, say clockwise, and assigning a direction to the segments connecting every pair of pixels. The direction of each segment is coded by using a 4- or 8 -connected numbering scheme, as shown in Figure 2.18. An example of the representations of an object boundary by using 4 – and 8 directional chain codes are shown in Figure 2.19.

Figure $2.18$ Numbering scheme of the chain code.
Taking an 8 -connected numbering scheme, for example, each code indicates the change of angular direction (in multiples of $45^{\circ}$ ) from one boundary pixel to the next. The even codes $0,2,4$, and 6 correspond to horizontal and vertical directions, while the odd codes $1,3,5$, and 7 correspond to the diagonal directions. The boundary has changed direction when a change occurs between two consecutive chain codes, and the change in the code direction usually indicates a corner on the boundary. By using the chain code, a complete description of an object boundary can be represented by the coordinates of the starting point together with the list of chain codes leading to subsequent boundary pixels, as shown in Figure $2.20$. This representation of a list of boundary pixels becomes more succinct than using all boundary pixels’ coordinates.
However, the chain code depends on the starting point, and different starting points result in different chain codes for the same boundary. To address this, the chain code for a boundary can be normalized with respect to the starting point by treating it as a circular or periodic sequence of direction numbers, and redefining the starting point such that the resulting sequence of numbers is of minimum magnitude.

## 机器视觉代写|图像处理作业代写Image Processing代考|IMAGE INTERPOLATION

An image gives the intensity values at the integral lattice locations, that is, the coordinates of each pixel are both integers. Image interpolation is the process of using known pixel intensity values to estimate the values at arbitrary locations other than those defined exactly by the integral lattice locations.

Image interpolation is a fundamental operation in image processing and has been widely used in image zooming, rotating, geometric calibration, etc. For example, as seen in Figure 2.22, suppose the input image coordinates $(x, y)$ are assigned to another pair of image coordinates $(\eta, \xi)$ by some coordinate transformation T:
$$(\eta, \xi)=\mathrm{T}{(x, y)}$$
Then the intensity values of the input image also have to be assigned to the corresponding locations of the transformed image. However, with the coordinate transform $\mathrm{T}$, some output pixels with coordinates calculated by Equation $2.32$ may be located between the integer-valued grid points in the $x y$-plane. Thus, the image interpolation techniques are applied to determine the intensity values at those in-between locations. Note also that two or more pixels in the input image can be mapped into the same pixel in the output image by the coordinate transform, in which case the image interpolation techniques can be used to combine multiple input pixel values into a common output pixel value.

## 机器视觉代写|图像处理作业代写Image Processing代考|NEAREST NEIGHBOR INTERPOLATION

The nearest neighbor interpolation, also called zero-order interpolation, assigns to each output pixel the intensity value of its nearest neighbor in the input image. To perform nearest neighbor interpolation method, the coordinates of every pixel in the output image, denoted as $(m, n)$, are first mapped into the input image by:
$$(u, v)=\mathrm{T}^{-1}{(m, n)}$$
where $(u, v)$ becomes the corresponding coordinates in the input image. Then the intensity value of the pixel located at $(m, n)$ in the output image is set as the value of the pixel that has the shortest distance to $(u, v)$ in the input image. This process is illustrated by Figure $2.23$.The nearest neighbor interpolation method is computationally very simple and fast. However, this method only uses the value of the pixel that is closest to the interpolated location, without taking account of the influence of other neighboring pixels. As a result, this method may produce severe mosaic and saw-tooth effect.

(这,X)=吨(X,是)

(在,在)=吨−1(米,n)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


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# Define

Question:

Define

(i) rational numbers

(ii) irrational numbers

(iii) real numbers.

Solution:

Rational numbers: The numbers of the form $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$ are called rational numbers.

Example: $\frac{2}{3}$

Irrational numbers: The numbers which when expressed in decimal form are expressible as non-terminating and non-repeating decimals are called irrational numbers.

Example: $\sqrt{2}$

Real numbers: The numbers which are positive or negative, whole numbers or decimal numbers and rational number or irrational number are called real numbers.

Example: $2, \frac{1}{3}, \sqrt{2},-3$ etc.


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# Construct Elliptic Curves as Jacobians¶

An elliptic curve is a genus one curve with a designated point. The Jacobian of a genus-one curve can be defined as the set of line bundles on the curve, and is isomorphic to the original genus-one curve. It is also an elliptic curve with the trivial line bundle as designated point. The utility of this construction is that we can construct elliptic curves without having to specify which point we take as the origin.

EXAMPLES:

sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(u^4+v^4+w^2)
Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field

sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

One can also define Jacobians of varieties that are not genus-one curves. These are not implemented in this module, but we call the relevant functionality:

sage: R.<x> = PolynomialRing(QQ)
sage: f = x**5 + 1184*x**3 + 1846*x**2 + 956*x + 560
sage: C = HyperellipticCurve(f)
sage: Jacobian(C)
Jacobian of Hyperelliptic Curve over Rational Field defined
by y^2 = x^5 + 1184*x^3 + 1846*x^2 + 956*x + 560

REFERENCES:

 [WpJacobianVariety] http://en.wikipedia.org/wiki/Jacobian_variety
sage.schemes.elliptic_curves.jacobian.Jacobian(X, **kwds)

Return the Jacobian.

INPUT:

• X – polynomial, algebraic variety, or anything else that has a Jacobian elliptic curve.
• kwds – optional keyword arguments.

The input X can be one of the following:

EXAMPLES:

sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: Jacobian(C, morphism=True)
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
u^3 + v^3 + w^3
To:   Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
Defn: Defined on coordinates by sending (u : v : w) to
(u*v^7*w + u*v^4*w^4 + u*v*w^7 :
v^9 + 3/2*v^6*w^3 - 3/2*v^3*w^6 - w^9 :
-v^6*w^3 - v^3*w^6)

sage.schemes.elliptic_curves.jacobian.Jacobian_of_curve(curve, morphism=False)

Return the Jacobian of a genus-one curve

INPUT:

• curve – a one-dimensional algebraic variety of genus one.

OUTPUT:

Its Jacobian elliptic curve.

EXAMPLES:

sage: R.<u,v,w> = QQ[]
sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage.schemes.elliptic_curves.jacobian.Jacobian_of_equation(polynomial, variables=None, curve=None)

Construct the Jacobian of a genus-one curve given by a polynomial.

INPUT:

• F – a polynomial defining a plane curve of genus one. May be homogeneous or inhomogeneous.
• variables – list of two or three variables or None (default). The inhomogeneous or homogeneous coordinates. By default, all variables in the polynomial are used.
• curve – the genus-one curve defined by polynomial or # None (default). If specified, suitable morphism from the jacobian elliptic curve to the curve is returned.

OUTPUT:

An elliptic curve in short Weierstrass form isomorphic to the curve polynomial=0. If the optional argument curve is specified, a rational multicover from the Jacobian elliptic curve to the genus-one curve is returned.

EXAMPLES:

sage: R.<a,b,c> = QQ[]
sage: f = a^3+b^3+60*c^3
sage: Jacobian(f)
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
sage: Jacobian(f.subs(c=1))
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field

If we specify the domain curve the birational covering is returned:

sage: h = Jacobian(f, curve=Curve(f));  h
Scheme morphism:
From: Projective Curve over Rational Field defined by a^3 + b^3 + 60*c^3
To:   Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
Defn: Defined on coordinates by sending (a : b : c) to
(216000*a*b^7*c + 12960000*a*b^4*c^4 + 777600000*a*b*c^7 :
216000*b^9 + 19440000*b^6*c^3 - 1166400000*b^3*c^6 - 46656000000*c^9 :
-216000*b^6*c^3 - 12960000*b^3*c^6)
sage: h([1,-1,0])
(0 : 1 : 0)

Plugging in the polynomials defining $$h$$ allows us to verify that it is indeed a rational morphism to the elliptic curve:

sage: E = h.codomain()
sage: E.defining_polynomial()(h.defining_polynomials()).factor()
(-10077696000000000) * c^3 * b^3 * (a^3 + b^3 + 60*c^3) * (b^6 + 60*b^3*c^3 + 3600*c^6)^3

By specifying the variables, we can also construct an elliptic curve over a polynomial ring:

sage: R.<u,v,t> = QQ[]
sage: Jacobian(u^3+v^3+t, variables=[u,v])
Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over
Multivariate Polynomial Ring in u, v, t over Rational Field

TESTS:

sage: from sage.schemes.elliptic_curves.jacobian import Jacobian_of_equation
sage: Jacobian_of_equation(f, variables=[a,b,c])
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field

#### Previous topic

Elliptic curve constructor

#### Next topic

Elliptic curves over a general ring


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homechevron_rightStudy

# Area expression with different units

Calculator expresses the given area through the different units of area measurement.

Calculator expresses the given area through the different units of area measurement.
For example 6 hectares 56 ares it's 656 ares or 65600 square metres or 656000000 square centimetres. This kind of task - to express the area through different units, such as hectares through square meters, can often be seen in math textbooks for 2-4 grades, so this calculator can be helpful to check the task.

Set value


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# Precalculus

Considering the line segment joining A(-5, 2) and B(7, 8), find an equation, in the form of Ax+By=C, that expressed the fact that a point P(x, y) is equidistant from A and from B. How would I go about even attempting this problem?

1. any point on the perpendicular bisector of AB satisfies your requirement.

what is the slope of AB ?
slope = (8-2)/(7+5) = 6/12 = 1/2
so the perpendicular has slope
m = -2
It must go through the midpoint of AB which is at:
x =(7-5)/2 = 1
y = (8+2)/5 = 2
or (1,2)
so
2 = -2(1) + b
b = 4
so
y = -2 x + 4
2 x + y = 4

posted by Damon

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# Prime Generating Sequences

A couple of months ago (really, a long, long time ago) I found an interesting question on Mathematics Stack Exchange (another site to effectively waste away hours of your life). It reminded me of my Bachelor's thesis (which I wrote a really, really long time ago) about the sequence

where $a_1=7$ and

Here, $\mathrm{gcd}(a,b)=(a,b)$ stands for the greatest common divisor,1 i.e., the largest integer $d$ that divides both $a$ and $b$. This may not seem terribly interesting at first sight, but if you look at the first few values for $g_n$ you'll notice something curious:

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101...

There are for sure a lot of ones in there, but other than that, all the numbers are primes. This is not a bias in the first few example -- Eric Rowland proved that all values of $g_n$ are either $1$ or a prime in a beautiful little paper back in 2008. Continue reading Prime Generating Sequences

1. It's common to abbreviate gcd(a,b)=(a,b) in number theory, and I shall do so in the remainder of the article. Similarly, it's convention to write lcm(a,b)=[a,b].


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# Heat is transferred to a heat engine from a furnace at a rate of 80 MW. If the rate of waste heat rejection to a nearby river is 50 MW, what is the net power output?

This question was previously asked in
MPSC AMVI Official Paper 2: Set A/2013
View all MPSC Motor Vehicle Inspector Papers >
1. 130 MW
2. 1.6 MW
3. 16 MW
4. 30 MW

Option 4 : 30 MW
Free
EKT ME Full Test 1
1994
50 Questions 150 Marks 45 Mins

## Detailed Solution

Concept:

Net power output of heat engine (Wnet) = QH - QL

Efficiency of heat engine $$(η_{HE}) = \frac{W_{net}}{Q_H}$$

where QH = Heat supplied to heat engine and QL = Heat rejected from heat engine.

Calculation:

Given:

QH = 80 MW and QL = 50 MW

Wnet = QH - QL

∴ Wnet = 80 - 50 ⇒ 30 MW.


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# lps2lts¶

The tool lps2lts generates a labelled transition system from a linear process (with extension .lps). While doing so, various checks can be performed, such as checks for deadlock, livelock, and the presence of certain actions.

The generated transition system can be saved in various formats (.aut, .lts, .fsm and .dot). Generally not saving, or saving in .aut format is the fastest, but it only contains information about transitions. The .lts format has the advantage that all information about the transition system is included, including data types, action declarations and state information. It is however not human readable. The .fsm format contains transition and state information, and is human readable.

A typical use of lps2lts is as follows:

lps2lts --verbose infile.lps outfile.aut

This automatically generates the file in .aut format. If no outfile is given, the transition system will be generated but not saved. The flag –verbose is the verbose flag, such that lps2lts gives some information about progress.

The labelled transition systems can be inspected using graphical tools like ltsgraph, ltsview and diagraphica. Using the tool ltsconvert the transition systems can be reduced modulo behavioural equivalences. Other tools allow to compare labelled transition systems ltsconvert, verify modal formulas lts2pbes, etc.

There are options to search for particular action labels, multi-actions with data, deadlocks and livelocks. The –trace option instructs the tool to save (a limited number) of traces to such states. Only one trace per state is saved. If the trace count is reached lps2lts stops. As a typical example, searching for an error action, saving one state and stop, can be done as follows:

lps2lts --verbose --action=error --trace=1 infile.lps

Generating a labelled transition system can be quite time consuming. Choosing the compiling rewriter using the –rewriter=jittyc can speed up the generation with a factor 10. The compiling rewriter is not available on all platforms. The use of the flag –cached may also have a dramatic influence on the generation speed, at the expense of using more memory. It caches the results of evaluating conditions in each summand in the linear process.

There are several options to traverse the state space. Default is breadth-first. But depth-first, random, and prioritised are also possible. Of special note is highway search [EGWW09]. When exploring the state space there is a stack of encountered states not yet explored. Using:

lps2lts --verbose --todo-max=100000 --action=error infile.lps

the size of this stack is limited to in this case 100000 states. If the stack exceeds this size additional states are randomly ignored. High way search does not guarantee that all states are seen, but it an effective way to explore a state space, far more effective than random simulation.

When the linear process has confluent tau actions (which can be proven using the tool lpsconfcheck) then the flag –confluent generates a state space giving priority to confluent tau’s [GM14]. In certain cases this can give an exponential reduction. The confluent tau is by default called ctau.

When generating the transition system is taking too much time, the generation can be aborted. lps2lts will attempt to save the transition system before terminating. Using the flag –max the size of the state space can also be limited a priori.

Warning

Generation of an labelled transition system typically fails when

• The rewrite system is not terminating. A typical non terminating rewrite rule is eqn f(x) = if(x==0, 0, f(x-1)); as f can infinitely be rewritten. Use conditions in rewrite rules to avoid this. Non terminating rewrite systems lead to an overflow of the stack and a subsequent crash of lps2lts.
• There are sums over infinite domains in the linear process. E.g. having a summand sum x:Nat. a(x).X will cause the tool to enumerate all natural numbers first, filling up all the memory, not generating any transition.
• The size of the state space is large as some process process parameters can attain an infinite or large amount of different values. An effective way to figure this out is to generate part of the transition system in a .fsm file. At the beginning of this file all process parameters and all the values that they got in some state are listed.

[EGWW09] T.A.N. Engels, J.F. Groote, M.J. van Weerdenburg and T.A.C. Willemse. Search algorithms for automated validation. Journal of Logic and Algebraic Programming 78(4), 274-287, 2009.

[GM14] J.F. Groote and M.R. Mousavi. Modeling and analysis of communicating systems. The MIT Press 2014.

## Manual page for lps2lts¶

### Usage¶

lps2lts   [OPTION]... [INFILE [OUTFILE]]

### Description¶

Generate an LTS from the LPS in INFILE and save the result to OUTFILE. If INFILE is not supplied, stdin is used. If OUTFILE is not supplied, the LTS is not stored.

If the ‘jittyc’ rewriter is used, then the MCRL2_COMPILEREWRITER environment variable (default value: ‘mcrl2compilerewriter’) determines the script that compiles the rewriter, and MCRL2_COMPILEDIR (default value: ‘.’) determines where temporary files are stored.

Note that lps2lts can deliver multiple transitions with the same label betweenany pair of states. If this is not desired, such transitions can be removed byapplying a strong bisimulation reducton using for instance the tool ltsconvert.

The format of OUTFILE is determined by its extension (unless it is specified by an option). The supported formats are:

‘aut’ for the Aldebaran format (CADP), ‘dot’ for the GraphViz format (no longer supported as input format), ‘fsm’ for the Finite State Machine format, or ‘lts’ for the mCRL2 LTS format

If the jittyc rewriter is used, then the MCRL2_COMPILEREWRITER environment variable (default value: mcrl2compilerewriter) determines the script that compiles the rewriter, and MCRL2_COMPILEDIR (default value: ‘.’) determines where temporary files are stored. Note that lps2lts can deliver multiple transitions with the same label between any pair of states. If this is not desired, such transitions can be removed by applying a strong bisimulation reducton using for instance the tool ltsconvert.

### Command line options¶

-aNAMES , --actionNAMES

report whether an action from NAMES occurs in the transitions system, where NAMES is a comma-separated list. A message is printed for every occurrence of one of these action names. With the -t flag traces towards these actions are generated. When using -tN only N traces are generated after which the generation of the state space stops.

-b[NUM] , --bit-hash[NUM]

use bit hashing to store states and store at most NUM states. This means that instead of keeping a full record of all states that have been visited, a bit array is used that indicate whether or not a hash of a state has been seen before. Although this means that this option may cause states to be mistaken for others (because they are mapped to the same hash), it can be useful to explore very large LTSs that are otherwise not explorable. The default value for NUM is 2*10^8 (this corresponds to 25MB of memory).

--cached

use enumeration caching techniques to speed up state space generation.

-c[NAME] , --confluence[NAME]

apply prioritization of transitions with the action label NAME. (when no NAME is supplied (i.e., ‘-c’) priority is given to the action ‘ctau’. To give priority to to tau use the flag -ctau. Note that if the linear process is not tau-confluent, the generated state space is necessarily branching bisimilar to the state space of the lps. The generation algorithm that is used does not require the linear process to be tau convergent.

-D , --deadlock

-F , --divergence

detect divergences (i.e. for every state with a divergence (=tau loop) a message is printed). The algorithm to detect the divergences is linear for every state, so state space exploration becomes quadratic with this option on, causing a state space exploration to become slow when this option is enabled.

-yBOOL , --dummyBOOL

replace free variables in the LPS with dummy values based on the value of BOOL: ‘yes’ (default) or ‘no’.

--error-trace

if an error occurs during exploration, save a trace to the state that could not be explored.

--init-tsizeNUM

set the initial size of the internally used hash tables (default is 10000).

-lNUM , --maxNUM

explore at most NUM states

-mNAMES , --multiactionNAMES

detect and report multiactions in the transitions system from NAMES, a comma-separated list. Works like -a, except that multi-actions are matched exactly, including data parameters.

--no-info

do not add state information to OUTFILE. Without this option lps2lts adds state vector to the LTS. This option causes this information to be discarded and states are only indicated by a sequence number. Explicit state information is useful for visualisation purposes, for instance, but can cause the OUTFILE to grow considerably. Note that this option is implicit when writing in the AUT format.

-n , --nondeterminism

detect nondeterministic states, i.e. states with outgoing transitions with the same label to different states.

-oFORMAT , --outFORMAT

save the output in the specified FORMAT.

--prune

use summand pruning to speed up state space generation.

-QNUM , --qlimitNUM

limit enumeration of quantifiers to NUM variables. (Default NUM=1000, NUM=0 for unlimited).

-rNAME , --rewriterNAME

use rewrite strategy NAME:

jitty

jitty rewriting

jittyc

compiled jitty rewriting

jittyp

jitty rewriting with prover

-sNAME , --strategyNAME

explore the state space using strategy NAME:

b, breadth

d, depth

depth-first search

p, prioritized

prioritize single actions on its first argument being of sort Nat where only those actions with the lowest value for this parameter are selected. E.g. if there are actions a(3) and b(4), a(3) remains and b(4) is skipped. Actions without a first parameter of sort Nat and multactions with more than one action are always chosen (option is experimental)

q, rprioritized

prioritize actions on its first argument being of sort Nat (see option –prioritized), and randomly select one of these to obtain a prioritized random simulation (option is experimental)

r, random

random simulation. Out of all next states one is chosen at random independently of whether this state has already been observed. Consequently, random simultation only terminates when a deadlocked state is encountered.

--suppress

in verbose mode, do not print progress messages indicating the number of visited states and transitions. For large state spaces the number of progress messages can be quite horrendous. This feature helps to suppress those. Other verbose messages, such as the total number of states explored, just remain visible.

--tauACTNAMES

consider actions with a name in the comma separated list ACTNAMES to be internal. This list is only used and allowed when searching for divergencies.

--timings[FILE]

append timing measurements to FILE. Measurements are written to standard error if no FILE is provided

--todo-maxNUM

keep at most NUM states in todo lists; this option is only relevant for breadth-first search, where NUM is the maximum number of states per level, and for depth first search, where NUM is the maximum depth.

-t[NUM] , --trace[NUM]

Write a shortest trace to each state that is reached with an action from NAMES with the option –action, is a deadlock with the option –deadlock, is nondeterministic with the option –nondeterminism, or is a divergence with the option –divergence to a file. No more than NUM traces will be written. If NUM is not supplied the number of traces is unbounded. For each trace that is to be written a unique file with extension .trc (trace) will be created containing a shortest trace from the initial state to the deadlock state. The traces can be pretty printed and converted to other formats using tracepp.

-u , --unused-data

do not remove unused parts of the data specification.

#### Standard options¶

-q , --quiet

do not display warning messages

-v , --verbose

display short intermediate messages

-d , --debug

display detailed intermediate messages

--log-levelLEVEL

display intermediate messages up to and including level

-h , --help

display help information

--version

display version information

### Author¶

Muck van Weerdenburg


text:
# Terminology for matrix whose rows are permutations of a given multiset.

Let $X=\{a_{1},a_{2},\ldots,a_{m}\}$ be a multiset. Is there a name for an $n\times m$ matrix $A$ such that the entries of each row of $A$ are equal to the set $X$.

For example, if $X=\{1,1,2,3,3\}$ and $n=5$, then we could take

$A=\left( \begin{array}{ccccc} 1 & 1 & 2 & 3 & 3 \\ 1 & 3 & 1 & 2 & 3 \\ 3 & 3 & 1 & 1 & 2 \\ 2 & 1 & 3 & 1 & 3 \\ 1 & 1 & 3 & 3 & 2 \\ \end{array} \right).$

-


text:
• # question_answer A photon, an electron and a uranium nucleus all have the same wavelength. The one with the most energy        A) Is the photon B) Is the electron C) Is the uranium nucleus D) Depends upon the wavelength and the properties of the particle.

[a] $\lambda =\frac{h}{mv}=\frac{h}{\sqrt{2mE}}:$ $\therefore \ E=\frac{{{h}^{2}}}{2m{{\lambda }^{2}}}$ $\lambda$ is same for all, so $E\propto \frac{1}{m}$. Hence energy will be maximum for particle with lesser mass.


text:
Titan Football Manufacturing had the following operating results for 2016: sales = $19,910; cost... Question: Titan Football Manufacturing had the following operating results for 2016: sales =$19,910; cost of goods sold = $13,850; depreciation expense =$2,240; interest expense = $280; dividends paid =$680. At the beginning of the year, net fixed assets were $20,300, current assets were$3,070, and current liabilities were $1,940. At the end of the year, net fixed assets were$24,140, current assets were $3,540, and current liabilities were$2,030. The tax rate for 2016 was 40%.

a. What is net income for 2016? (Do not round intermediate calculations and round your answer to the nearest whole number, e.g., 32.)

Net income $_____ b. What is the operating cash flow for 2016? (Do not round intermediate calculations and round your answer to the nearest whole number, e.g., 32.) Operating cash flow$ _____

c. What is the cash flow from assets for 2016? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and round your answer to the nearest whole number, e.g., 32.)

Cash flow from assets $_____ Assume no new debt was issued during the year. d. What is the cash flow to creditors for 2016? (Do not round intermediate calculations and round your answer to the nearest whole number, e.g., 32.) Cash flow to creditors$ _____

e. What is the cash flow to stockholders for 2016? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and round your answer to the nearest whole number, e.g., 32.)

Cash flow to stockholders $_____ Income Statement: An income statement is a primary financial statement that indicates the financial performance of a company during a fiscal year by determining the net income or loss generated by the business. The net income is the excess of revenues over the expenses incurred to generate the revenues. By making adjustments to net income, the operating cash flow can be determined and free cash flow can be determined from the operating cash flow. Answer and Explanation: Answer: The computations for Titan Football Manufacturing are made below: a) The net income for 2016:  Net income$2,124

Computation:

 Amount Sales $19,910 Cost of goods sold$13,850 Depreciation expense $2,240 EBIT$3,820 Interest expense $280 EBT$3,540 Taxes @ 40% $1,416 Net income$2,124

b) The operating cash flow for 2016:

 Operating cash flow $4,644 Computation: • Operating cash flow = EBIT + Depreciation expense - Taxes • Operating cash flow =$3,820 + $2,240 -$1,416
• Operating cash flow = $4,644 c) The cash flow from assets for 2016:  Cash flow from assets -$1,816

Explanation:

The first step is to determine the changes in net working capital:

Computation:

• Change in net working capital = (Ending current assets - Ending current liabilities) - (Beginning current assets - Beginning current liabilities)
• Change in net working capital = ($3,540 -$2,030) - ($3,070 -$1,940)
• Change in net working capital = $1,510 -$1,130
• Change in net working capital = $380 The second step is to determine the net capital spending: Computation: • Net capital spending = (Ending fixed assets - Beginning fixed assets) + Depreciation expense • Net capital spending = ($24,140 - $20,300 +$2,240)
• Net capital spending = $6,080 The last step is to determine the cash flow from assets: Computation: • Cash flow from assets = Operating cash flow - Change in net working capital - Net capital spending • Cash flow from assets =$4,644 - $380 -$6,080
• Cash flow from assets = -$1,816 d) The cash flow to creditors for 2016:  Cash flow to creditors$280

Computation:

• Cash flow to creditors = Interest expense - Net new borrowings
• Cash flow to creditors = $280 -$0
• Cash flow to creditors = $280 e) The cash flow to stockholders for 2016:  Cash flow to stockholders -$2,096

Computation:

• Cash flow from assets = Cash flow to creditors + Cash flow to stockholders
• Cash flow to stockholders = -$1,816 -$280
• Cash flow to stockholders = -\$2,096