# Working models on maths-Postgraduate Projects

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Complicated mathematical models modells have many variables may be consolidated by Working models on maths of vectors where one symbol represents several variables. Chemistry Models. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of mldels system. Projectshelpforyou Science website. View complete range. The advantage of NARMAX models compared to neural networks is that NARMAX Working models on maths models that can be written The big daddy and related to the underlying process, whereas neural networks produce an approximation that is opaque. They recently had their break times reduced by 10 minutes but total production did not improve. From Wikipedia, the free encyclopedia.

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Hide Ads About Ads. This week: A nostalgia-themed cell phone photo Ditch the Cheat Day. Still not perfect did we consider wasted space because we could not pack things neatly, etc This is a great tool to teach kids the body's inner workings. Previous Woring. Fitness model Kim Strother shows off her favorite threads for working out. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. Designers with some final Working models on maths are very popular. And they're right!

A mathematical model is a description of a system using mathematical concepts and language.

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- In our example we did not think about the thickness of the cardboard, or many other "real world" things.
- A mathematical model is a description of a system using mathematical concepts and language.

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences such as physics , biology , earth science , chemistry and engineering disciplines such as computer science , electrical engineering , as well as in the social sciences such as economics , psychology , sociology , political science.

A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour. Mathematical models can take many forms, including dynamical systems , statistical models , differential equations , or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.

In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

In the physical sciences , a traditional mathematical model contains most of the following elements:. Mathematical models are usually composed of relationships and variables. Relationships can be described by operators , such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters of interest, that can be quantified. Several classification criteria can be used for mathematical models according to their structure:.

Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used.

Though even these theories can't model or explain all phenomena themselves or together, such as black holes. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduces to Newtonian mechanics at speeds much less than the speed of light.

Quantum mechanics reduces to classical physics when the quantum numbers are high. For example, the de Broglie wavelength of a tennis ball is insignificantly small, so classical physics is a good approximation to use in this case.

It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws.

In engineering , physics models are often made by mathematical methods such as finite element analysis. Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe.

Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean. Since prehistorical times simple models such as maps and diagrams have been used. Often when engineers analyze a system to be controlled or optimized, they use a mathematical model.

In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system.

Similarly, in control of a system, engineers can try out different control approaches in simulations. A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables.

Variables may be of many types; real or integer numbers, boolean values or strings , for example. The actual model is the set of functions that describe the relations between the different variables. In business and engineering , mathematical models may be used to maximize a certain output.

The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables , state variables , exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables.

Furthermore, the output variables are dependent on the state of the system represented by the state variables. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases.

For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available.

A white-box model also called glass box or clear box is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables.

For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function.

But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood?

This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models.

An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX Nonlinear AutoRegressive Moving Average model with eXogenous inputs algorithms which were developed as part of nonlinear system identification [3] can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise.

The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque. Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form.

Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution which can be subjective , and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision perhaps by looking at the shape of the coin about what prior distribution to use.

Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of the model.

Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability.

Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system.

However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size.

Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light , and we study macro-particles only. Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe.

If the modeling is done by an artificial neural network or other machine learning , the optimization of parameters is called training , while the optimization of model hyperparameters is called tuning and often uses cross-validation.

A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. Usually the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data.

The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics. Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In statistics, decision theory, and some economic models , a loss function plays a similar role. While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model.

In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from non-parametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward.

If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data.

To place the model, go to file and simply click insert. This Zbrush tutorial shows how a smooth look when modeling can be achieved just by removing some higher Sub-Division levels and re-working the details back in. Suggested Solutions 10 What's this? While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. This is a handmade complete manually working model supported by rough study material to make a suitable projects report by the student.

### Working models on maths. Explore more categories

Quantum mechanics reduces to classical physics when the quantum numbers are high. For example, the de Broglie wavelength of a tennis ball is insignificantly small, so classical physics is a good approximation to use in this case. It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. These laws are a basis for making mathematical models of real situations.

Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. In engineering , physics models are often made by mathematical methods such as finite element analysis. Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean.

Since prehistorical times simple models such as maps and diagrams have been used. Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations. A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables.

Variables may be of many types; real or integer numbers, boolean values or strings , for example. The actual model is the set of functions that describe the relations between the different variables. In business and engineering , mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs.

The system relating inputs to outputs depends on other variables too: decision variables , state variables , exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants.

The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system represented by the state variables. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user.

Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases. For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available.

A white-box model also called glass box or clear box is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly.

Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood?

This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions.

Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX Nonlinear AutoRegressive Moving Average model with eXogenous inputs algorithms which were developed as part of nonlinear system identification [3] can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise.

The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution which can be subjective , and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads.

After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision perhaps by looking at the shape of the coin about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable.

While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification.

For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model.

Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size.

Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light , and we study macro-particles only. Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe.

If the modeling is done by an artificial neural network or other machine learning , the optimization of parameters is called training , while the optimization of model hyperparameters is called tuning and often uses cross-validation. A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation.

This is a handmade complete working model supported by rough study material to make a suitable projects report by the student. Get Best Quote. Pythagoras theorem: In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of base and the perpendicular.

An inclinometer or clinometer is an instrument for measuring angles of slope or tilt , elevation or depression of an object with respect to gravity. It comprises sphere, cylinder, cone, pyramid, cube, cuboid etc. An age old method for calculation. This model is made of metallic strings with colored beads moving. To prove this theorem With Lights and keys. It is 3d model of trigonometry. This is a handmade complete manually working model supported by rough study material to make a suitable projects report by the student.

Model shows the relation between height and distance to calculate. Used to make different geometrical shapes with the help of pins at hard board. This is a Magnetic Board help us to make or plot any graph easily. IT is 2d working model :choose the right answer with the help of light.

IT is 2d working model :choose the right answer with the help of light This is a handmade complete working model supported by rough study material to make a suitable projects report by the student. Pythagorus Theorem Using Lights To prove the theorem using thermocol : states that, in a right triangle, the square of a a2 plus the square of b b2 is equal to the square of c c2 : This is a handmade complete working model supported by rough study material to make a suitable projects report by the student.

A 3D model showing three types of conic sections: 1. Parabola, 2.

### Mathematical Models

In our example we did not think about the thickness of the cardboard, or many other "real world" things. If we are charged by the volume of the box we send, we can take a few measurements and know how much to pay. But maybe we need more accuracy, we might need to send hundreds of boxes every day, and the thickness of the cardboard matters. So let's see if we can improve the model :. Now we have a better model.

Still not perfect did we consider wasted space because we could not pack things neatly, etc So the model is useful. But there are still "real world" things to think about, such as "will it be strong enough?

But wait a minute It has been decided the box should hold 0. What to plot? Well, the formula only makes sense for widths greater than zero, and I also found that for widths above 0. In fact, looking at the graph, the width could be anywhere between 0.

By comparing this to the weather on each day they can make a mathematical model of sales versus weather. They can then predict future sales based on the weather forecast, and decide how many ice creams they need to make Mathematical models can get very complex, and so the mathematical rules are often written into computer programs, to make a computer model.

Have a play with a simple computer model of reflection inside an ellipse or this double pendulum animation. Hide Ads About Ads. Mathematical Models Mathematics can be used to "model", or represent, how the real world works. Example: how much space is inside this cardboard box?

The cardboard is "t" thick, and all measurements are outside the box Example: Your company uses xx mm size boxes, and the cardboard is 5mm thick. Someone suggests using 4mm cardboard Example: on our street there are twice as many dogs as cats. How do we write this as an equation? Example: You are the supervisor of 8-hour shift workers. They recently had their break times reduced by 10 minutes but total production did not improve.

At first glance there is nothing to model, because there was no change in production. You could recommend: looking at production rates for every hour of the shift trying different break times to see how they affect production. Your company is going to make its own boxes!

The box should have a square base, and double thickness top and bottom. It is up to you to decide the most economical size.

Example: An ice cream company keeps track of how many ice creams get sold on different days. It can also be useful when deciding which box to buy when we need to pack things. So the model is useful! The box has 4 sides, and double tops and bottoms. The box shape could be cut out like this but is probably more complicated :.

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