Exploring quantum physics through hands-on projects pdf download

Ridge Regression Equation In other words, Lasso Regression automatically performs feature selection and outputs a sparse model i. Lasso versus Ridge regularization On the Lasso cost function, the BGD path tends to bounce across the gutter toward the end. You need to gradually reduce the learning rate in order to actually converge to the global minimum. It is almost always preferable to have at least a little bit of regularization, so generally you should avoid plain Linear Regression.

In general, Elastic Net is preferred over Lasso since Lasso may behave erratically when the number of features is greater than the number of training instances or when several features are strongly correlated. This is called early stopping. Figure shows a complex model in this case a high-degree Polynomial Regression model being trained using Batch Gradient Descent. As the epochs go by, the algorithm learns and its prediction error RMSE on the training set naturally goes down, and so does its prediction error on the validation set.

However, after a while the validation error stops decreasing and actually starts to go back up. This indicates that the model has started to overfit the training data. Early stopping regularization With Stochastic and Mini-batch Gradient Descent, the curves are not so smooth, and it may be hard to know whether you have reached the minimum or not.

One solution is to stop only after the validation error has been above the minimum for some time when you are confident that the model will not do any better , then roll back the model parameters to the point where the validation error was at a minimum.

This makes it a binary classifier. Estimating Probabilities So how does it work? It is defined as shown in Equation and Figure Indeed, if you compute the logit of the estimated probability p, you will find that the result is t. The logit is also called the log-odds, since it is the log of the ratio between the estimated probability for the positive class and the estimated probability for the negative class. Training and Cost Function Good, now you know how a Logistic Regression model estimates probabilities and makes predictions.

But how is it trained? This idea is captured by the cost function shown in Equation for a single training instance x. On the other hand, — log t is close to 0 when t is close to 1, so the cost will be close to 0 if the estimated probability is close to 0 for a negative instance or close to 1 for a positive instance, which is precisely what we want. It can be written in a single expression as you can verify easily , called the log loss, shown in Equation Once you have the gradient vector containing all the partial derivatives you can use it in the Batch Gradient Descent algorithm.

For Stochastic GD you would of course just take one instance at a time, and for Mini-batch GD you would use a mini-batch at a time. This is a famous dataset that contains the sepal and petal length and width of iris flowers of three different species: Iris-Setosa, Iris-Versicolor, and Iris-Virginica see Figure Gordon E.

Estimated probabilities and decision boundary The petal width of Iris-Virginica flowers represented by triangles ranges from 1. In between these extremes, the classifier is unsure.

Therefore, there is a decision boundary at around 1. Note that it is a linear boundary. The hyperparameter controlling the regularization strength of a Scikit-Learn LogisticRegression model is not alpha as in other linear models , but its inverse: C.

The higher the value of C, the less the model is regularized. Softmax Regression The Logistic Regression model can be generalized to support multiple classes directly, without having to train and combine multiple binary classifiers as discussed in Chapter 3. The idea is quite simple: when given an instance x, the Softmax Regression model first computes a score sk x for each class k, then estimates the probability of each class by applying the softmax function also called the normalized exponential to the scores.

The scores are generally called logits or log-odds although they are actually unnormalized log- odds. Just like the Logistic Regression classifier, the Softmax Regression classifier predicts the class with the highest estimated probability which is simply the class with the highest score , as shown in Equation The Softmax Regression classifier predicts only one class at a time i.

You cannot use it to recognize multiple people in one picture. The objective is to have a model that estimates a high probability for the target class and consequently a low probability for the other classes. Minimizing the cost function shown in Equation , called the cross entropy, should lead to this objective because it penalizes the model when it estimates a low probability for a target class.

In general, it is either equal to 1 or 0, depending on whether the instance belongs to the class or not. Cross Entropy Cross entropy originated from information theory.

Suppose you want to efficiently transmit information about the weather every day. If there are eight options sunny, rainy, etc. Cross entropy measures the average number of bits you actually send per option.

If your assumption about the weather is perfect, cross entropy will just be equal to the entropy of the weather itself i. For more details, check out this video. Notice that the decision boundaries between any two classes are linear. The figure also shows the probabilities for the Iris-Versicolor class, represented by the curved lines e.

What Linear Regression training algorithm can you use if you have a training set with millions of features? Suppose the features in your training set have very different scales. What can you do about it?

Can Gradient Descent get stuck in a local minimum when training a Logistic Regression model? Do all Gradient Descent algorithms lead to the same model provided you let them run long enough? Suppose you use Batch Gradient Descent and you plot the validation error at every epoch. If you notice that the validation error consistently goes up, what is likely going on? How can you fix this? Which Gradient Descent algorithm among those we discussed will reach the vicinity of the optimal solution the fastest?

Which will actually converge? How can you make the others converge as well? Suppose you are using Polynomial Regression. You plot the learning curves and you notice that there is a large gap between the training error and the validation error. What is happening? What are three ways to solve this? Suppose you are using Ridge Regression and you notice that the training error and the validation error are almost equal and fairly high.

Would you say that the model suffers from high bias or high variance? This chapter will explain the core concepts of SVMs, how to use them, and how they work. Figure shows part of the iris dataset that was introduced at the end of Chapter 4.

The two classes can clearly be separated easily with a straight line they are linearly separable. The left plot shows the decision boundaries of three possible linear classifiers. The model whose decision boundary is represented by the dashed line is so bad that it does not even separate the classes properly.

The other two models work perfectly on this training set, but their decision boundaries come so close to the instances that these models will probably not perform as well on new instances.

You can think of an SVM classifier as fitting the widest possible street represented by the parallel dashed lines between the classes. This is called large margin classification. These instances are called the support vectors they are circled in Figure SVMs are sensitive to the feature scales, as you can see in Figure on the left plot, the vertical scale is much larger than the horizontal scale, so the widest possible street is close to horizontal. After feature scaling e.

Sensitivity to feature scales Soft Margin Classification If we strictly impose that all instances be off the street and on the right side, this is called hard margin classification. Figure shows the iris dataset with just one additional outlier: on the left, it is impossible to find a hard margin, and on the right the decision boundary ends up very different from the one we saw in Figure without the outlier, and it will probably not generalize as well.

Hard margin sensitivity to outliers To avoid these issues it is preferable to use a more flexible model. The objective is to find a good balance between keeping the street as large as possible and limiting the margin violations i.

This is called soft margin classification. Figure shows the decision boundaries and margins of two soft margin SVM classifiers on a nonlinearly separable dataset. On the right, using a low C value the margin is quite large, but many instances end up on the street. On the left, using a high C value the classifier makes fewer margin violations but ends up with a smaller margin. Large margin left versus fewer margin violations right If your SVM model is overfitting, you can try regularizing it by reducing C.

The resulting model is represented on the left of Figure The LinearSVC class regularizes the bias term, so you should center the training set first by subtracting its mean. This is automatic if you scale the data using the StandardScaler. Moreover, make sure you set the loss hyperparameter to "hinge", as it is not the default value. Finally, for better performance you should set the dual hyperparameter to False, unless there are more features than training instances we will discuss duality later in the chapter.

One approach to handling nonlinear datasets is to add more features, such as polynomial features as you did in Chapter 4 ; in some cases this can result in a linearly separable dataset. Consider the left plot in Figure it represents a simple dataset with just one feature x1. This dataset is not linearly separable, as you can see. Linear SVM classifier using polynomial features Polynomial Kernel Adding polynomial features is simple to implement and can work great with all sorts of Machine Learning algorithms not just SVMs , but at a low polynomial degree it cannot deal with very complex datasets, and with a high polynomial degree it creates a huge number of features, making the model too slow.

Fortunately, when using SVMs you can apply an almost miraculous mathematical technique called the kernel trick it is explained in a moment. It makes it possible to get the same result as if you added many polynomial features, even with very high- degree polynomials, without actually having to add them. This trick is implemented by the SVC class. On the right is another SVM classifier using a 10th- degree polynomial kernel.

Conversely, if it is underfitting, you can try increasing it. The hyperparameter coef0 controls how much the model is influenced by high- degree polynomials versus low-degree polynomials.

SVM classifiers with a polynomial kernel A common approach to find the right hyperparameter values is to use grid search see Chapter 2.

It is often faster to first do a very coarse grid search, then a finer grid search around the best values found. Adding Similarity Features Another technique to tackle nonlinear problems is to add features computed using a similarity function that measures how much each instance resembles a particular landmark.

Now we are ready to compute the new features. As you can see, it is now linearly separable. The simplest approach is to create a landmark at the location of each and every instance in the dataset.

This creates many dimensions and thus increases the chances that the transformed training set will be linearly separable.

The downside is that a training set with m instances and n features gets transformed into a training set with m instances and m features assuming you drop the original features. If your training set is very large, you end up with an equally large number of features. Gaussian RBF Kernel Just like the polynomial features method, the similarity features method can be useful with any Machine Learning algorithm, but it may be computationally expensive to compute all the additional features, especially on large training sets.

However, once again the kernel trick does its SVM magic: it makes it possible to obtain a similar result as if you had added many similarity features, without actually having to add them. For example, some kernels are specialized for specific data structures. With so many kernels to choose from, how can you decide which one to use? If the training set is not too large, you should try the Gaussian RBF kernel as well; it works well in most cases. The algorithm takes longer if you require a very high precision.

In most classification tasks, the default tolerance is fine. This algorithm is perfect for complex but small or medium training sets.

However, it scales well with the number of features, especially with sparse features i. In this case, the algorithm scales roughly with the average number of nonzero features per instance. Platt There is little regularization on the left plot i. In this chapter, we will use a different convention, which is more convenient and more common when you are dealing with SVMs: the bias term will be called b and the feature weights vector will be called w.

No bias feature will be added to the input feature vectors. Decision function for the iris dataset The dashed lines represent the points where the decision function is equal to 1 or —1: they are parallel and at equal distance to the decision boundary, forming a margin around it. Training a linear SVM classifier means finding the value of w and b that make this margin as wide as possible while avoiding margin violations hard margin or limiting them soft margin.

In other words, dividing the slope by 2 will multiply the margin by 2. The smaller the weight vector w, the larger the margin. However, if we also want to avoid any margin violation hard margin , then we need the decision function to be greater than 1 for all positive training instances, and lower than —1 for negative training instances.

Optimization algorithms work much better on differentiable functions. We now have two conflicting objectives: making the slack variables as small as possible to 1 reduce the margin violations, and making 2 wT w as small as possible to increase the margin. This gives us the constrained optimization problem in Equation Many off-the-shelf solvers are available to solve QP problems using a variety of techniques that are outside the scope of this book.

So one way to train a hard margin linear SVM classifier is just to use an off-the-shelf QP solver by passing it the preceding parameters. Similarly, you can use a QP solver to solve the soft margin problem see the exercises at the end of the chapter.

Luckily, the SVM problem happens to meet these conditions,6 so you can choose to solve the primal problem or the dual problem; both will have the same solution.

Equation shows the dual form of the linear SVM objective if you are interested in knowing how to derive the dual problem from the primal problem, see Appendix C. So what is this kernel trick anyway? Kernelized SVM Suppose you want to apply a 2nd-degree polynomial transformation to a two- dimensional training set such as the moons training set , then train a linear SVM classifier on the transformed training set.

However, in Machine Learning, vectors are frequently represented as column vectors i. To remain consistent with the rest of the book, we will use this notation here, ignoring the fact that this technically results in a single-cell matrix rather than a scalar value. The result will be strictly the same as if you went through the trouble of actually transforming the training set then fitting a linear SVM algorithm, but this trick makes the whole process much more computationally efficient.

This is the essence of the kernel trick. Equation lists some of the most commonly used kernels. There is still one loose end we must tie. But how can you make predictions without knowing w? This makes it possible to use the kernel trick, once again Equation Of course, you also need to compute the bias term b , using the same trick Equation Unfortunately it converges much more slowly than the methods based on QP. Minimizing this term ensures that the model makes the margin violations as small and as few as possible Hinge Loss The function max 0, 1 — t is called the hinge loss function represented below.

For large- scale nonlinear problems, you may want to consider using neural networks instead see Part II. What is the fundamental idea behind Support Vector Machines? What is a support vector? Why is it important to scale the inputs when using SVMs?

Can an SVM classifier output a confidence score when it classifies an instance? What about a probability? Should you use the primal or the dual form of the SVM problem to train a model on a training set with millions of instances and hundreds of features?

What about C? Train a LinearSVC on a linearly separable dataset. See if you can get them to produce roughly the same model. Since SVM classifiers are binary classifiers, you will need to use one-versus-all to classify all 10 digits. Cauwenberghs, T. Poggio Bordes, S. Ertekin, J. Weston, L. Bottou What accuracy can you reach? Train an SVM regressor on the California housing dataset. They are very powerful algorithms, capable of fitting complex datasets.

Then we will go through the CART training algorithm used by Scikit-Learn, and we will discuss how to regularize trees and use them for regression tasks. Finally, we will discuss some of the limitations of Decision Trees. The following code trains a DecisionTreeClassifier on the iris dataset see Chapter 4 : from sklearn.

Suppose you find an iris flower and you want to classify it. In this case, it is a leaf node i. Now suppose you find another flower, but this time the petal length is greater than 2. If it is, then your flower is most likely an Iris-Versicolor depth 2, left. If not, it is likely an Iris-Virginica depth 2, right. One of the many qualities of Decision Trees is that they require very little data preparation. For example, training instances have a petal length greater than 2.

For example, since the depth-1 left node applies only to Iris-Setosa training instances, it is pure and its gini score is 0. Another impurity measure is discussed shortly. Since the left area is pure only Iris-Setosa , it cannot be split any further. Such models are often called white box models.

They make great predictions, and you can easily check the calculations that they performed to make these predictions; nevertheless, it is usually hard to explain in simple terms why the predictions were made. Her mouth? Her nose? Her shoes? Or even the couch that she was sitting on?

Conversely, Decision Trees provide nice and simple classification rules that can even be applied manually if need be e. For example, suppose you have found a flower whose petals are 5 cm long and 1. And of course if you ask it to predict the class, it should output Iris-Versicolor class 1 since it has the highest probability.

Notice that the estimated probabilities would be identical anywhere else in the bottom-right rectangle of Figure —for example, if the petals were 6 cm long and 1. How does it choose k and tk? It searches for the pair k, tk that produces the purest subsets weighted by their size. The cost function that the algorithm tries to minimize is given by Equation Once it has successfully split the training set in two, it splits the subsets using the same logic, then the sub-subsets and so on, recursively.

It does not check whether or not the split will lead to the lowest possible impurity several levels down. A greedy algorithm often produces a reasonably good solution, but it is not guaranteed to be the optimal solution. Computational Complexity Making predictions requires traversing the Decision Tree from the root to a leaf. Decision Trees are generally approximately balanced, so traversing the Decision Tree requires going through roughly O log2 m nodes.

Gini Impurity or Entropy? By default, the Gini impurity measure is used, but you can select the entropy impurity measure instead by setting the criterion hyperparameter to "entropy". The concept of entropy originated in thermodynamics as a measure of molecular disorder: entropy approaches zero when molecules are still and well ordered. NP is the set of problems whose solutions can be verified in polynomial time. Equation shows the definition of the entropy of the ith node. The truth is, most of the time it does not make a big difference: they lead to similar trees.

Gini impurity is slightly faster to compute, so it is a good default. However, when they differ, Gini impurity tends to isolate the most frequent class in its own branch of the tree, while entropy tends to produce slightly more balanced trees.

If left unconstrained, the tree structure will adapt itself to the training data, fitting it very closely, and most likely overfitting it. Such a model is often called a nonparametric model, not because it does not have any parameters it often has a lot but because the number of parameters is not determined prior to training, so the model structure is free to stick closely to the data. In contrast, a parametric model such as a linear model has a predetermined number of parameters, so its degree of freedom is limited, reducing the risk of overfitting but increasing the risk of underfitting.

As you know by now, this is called regularization. The regularization hyperparameters depend on the algorithm used, but generally you can at least restrict the maximum depth of the Decision Tree. Other algorithms work by first training the Decision Tree without restrictions, then pruning deleting unnecessary nodes.

A node whose children are all leaf nodes is considered unnecessary if the purity improvement it provides is not statistically significant. The pruning continues until all unnecessary nodes have been pruned. Figure shows two Decision Trees trained on the moons dataset introduced in Chapter 5.

On the left, the Decision Tree is trained with the default hyperparameters i. It is quite obvious that the model on the left is overfitting, and the model on the right will probably generalize better. A Decision Tree for regression This tree looks very similar to the classification tree you built earlier. This prediction is simply the average target value of the training instances associated to this leaf node. The algorithm splits each region in a way that makes most training instances as close as possible to that predicted value.

Equation shows the cost function that the algorithm tries to minimize. Without any regularization i. It is obviously overfitting the training set very badly. However they do have a few limitations. First, as you may have noticed, Decision Trees love orthogonal decision boundaries all splits are perpendicular to an axis , which makes them sensitive to training set rotation.

Although both Decision Trees fit the training set perfectly, it is very likely that the model on the right will not generalize well. Sensitivity to training set rotation More generally, the main issue with Decision Trees is that they are very sensitive to small variations in the training data.

For example, if you just remove the widest Iris- Versicolor from the iris training set the one with petals 4. As you can see, it looks very different from the previous Decision Tree Figure Sensitivity to training set details Random Forests can limit this instability by averaging predictions over many trees, as we will see in the next chapter.

What is the approximate depth of a Decision Tree trained without restrictions on a training set with 1 million instances? If a Decision Tree is underfitting the training set, is it a good idea to try scaling the input features?

If it takes one hour to train a Decision Tree on a training set containing 1 million instances, roughly how much time will it take to train another Decision Tree on a training set containing 10 million instances? Train and fine-tune a Decision Tree for the moons dataset. Use grid search with cross-validation with the help of the GridSearchCV class to find good hyperparameter values for a DecisionTreeClassifier. Grow a forest. Continuing the previous exercise, generate 1, subsets of the training set, each containing instances selected randomly.

Train one Decision Tree on each subset, using the best hyperparameter values found above. Evaluate these 1, Decision Trees on the test set. Now comes the magic. There will be a few projects throughout semester that will build on the course material and utilize open source software and open data in physics and related fields.

The list of topics will evolve, according to the interests of the class and instructors. Material will be clustered into units of varying duration, as indicated below. The lists of suggested readings and references are advisory; a large amount of material of excellent quality is now available on the worldwide web, particularly on the sites of university courses addressing the topics of each unit.

A distinguishing feature of this course is its sharp focus on endeavors in the data-rich physical sciences as the arenas in which modern machine learning techniques are taught. The course uses open scientific data, open source software from data science and physics-related fields, and publicly-available information as enabling elements.

Research-inspired projects are an important part of the course and students will not only execute them but will play an active role in helping define and shape them. Example projects might include machine learning approaches to searches for new particles or interactions at high-energy colliders; methods of particle tracking and reconstruction; identification, classification and measurement of astrophysical phenomena; novel approaches to medical imaging and simulation using techniques from physics and machine learning; machine learning in quantum information science.

Through these projects and the course material, students will learn how large datasets in physics are generated, curated, and analyzed, using machine learning as a tool to generate key insights in both experimental and theoretical science. No professional credit. Single-particle distribution functions; classical and quantum mechanical systems, Boltzmann equation, virial theorem, and equations of state for gases; formal theory: ensembles, identical particles, thermodynamics of simple systems, and distribution functions; nonequilibrium problems; conservation laws and hydrodynamic equations, sound waves, and transport coefficients; plasmas, normal Fermi fluid, superfluids, and systems with internal degrees of freedom.

Review of Maxwell's equations; relativistic formulation of the electromagnetic field and the motion of charged particles; plane and guided waves; retarded potentials; radiation from simple antennas; radiation from accelerated charged particles; scattering and further topics.

Core techniques of mathematical physics widely used in the physical sciences. Calculus of variations and its applications; partial differential equations of mathematical physics including classification and boundary conditions ; separation of variables, series solutions of ordinary differential equations and Sturm-Liouville eigenproblems; Legendre polynomials, spherical harmonics, Bessel functions and their applications; normal mode eigenproblems including the wave and diffusion equations ; inhomogeneous ordinary differential equations including variation of parameters ; inhomogeneous partial differential equations and Green functions; potential theory; integral equations including Fredholm theory.

Prerequisite: MATH Further core techniques of mathematical physics widely used in the physical sciences. Complex variables; group theory in classical and quantum systems; tensors in physics; differential forms and their applications in mechanics; electromagnetism.

Broad introduction to nonlinear dynamics of physical systems with varying degrees of complexity; survey of a variety of concepts associated with bifurcation phenomena, mappings, nonlinear oscillations, chaotic behavior, strange attractors, and solitons.

Topics of current interest. Experimental and theoretical fundamentals of quantum information, using nonclassical features of quantum physics wave-particle duality, superposition, and entanglement to surpass the information-processing capabilities of classical systems.

Underlying fundamental quantum phenomena, including tests of nonlocality, quantum erasers, the quantum Zeno effect, squeezed light, multi-particle interference, state transformations of the Bloch sphere, and decoherence; quantum cryptography and teleportation; quantum information theory; quantum computation algorithms and techniques for error correction; experimental "qubit" systems.

Rigorous survey of modern atomic, molecular, and optical physics, including a functional approach to theory and an overview of experimental techniques. Atomic structure, including fine and hyperfine structure, multi-electron atoms, and relativistic effects; interaction of single atoms with dynamic and static electromagnetic fields, ultra-cold collisions between atoms; laser cooling, evaporative cooling, and magnetic trapping; Paul and Penning traps; quantum degenerate gases; atom interferometry.

Systematic introduction to Einstein's theory, with emphasis on modern coordinate-free methods of computation. Review of special relativity, modern differential geometry, foundations of general relativity, laws of physics in the presence of a gravitational field, linearized theory, and experimental tests of gravitation theories.

Same as ASTR Continuation of PHYS with emphasis on applications to astrophysics and cosmology. Relativistic stars, gravitational collapse, black holes, gravitational waves, numerical relativity, and cosmology. In this two-semester course students will engage in the collaborative design and execution of a year-long Instrumentation and measurement-intensive technical project.

Required activities will include a written project proposal of work to be undertaken, informal group-generated oral presentations on technical issues, periodic formal written progress reports, a final project oral presentation, and a final project paper.

The set of projects might include investigations suggested by industry partners. There will be two class meetings per week, each of three hours duration. In addition to the project work, we will bring in local experts to discuss a number of relevant topics with the class; these are shown in the syllabus, below. Note that readings will consist primarily of technical materials and documentation by the producers of components used by individual projects. As a result, readings and external materials will vary from group to group.

May be repeated in separate terms to a total of 8 hours. Prerequisite: Students must enroll in consecutive fall and spring semesters to meet the requirements of the Concentration in Instrumentation and Applied Physics under the Master of Engineering in Engineering degree.

Introduce students to a broader spectrum of devices than they can be expected to encounter in their Physics projects. The goal of the course is to familiarize students with some of the techniques available to them when defining and proposing a technical project in an unfamiliar domain.

There will be two 50 minute classes each week, split into a discussion of basic principles and a simple hands-on laboratory exercise. The list of topics—which is not intended to be exhaustive—will evolve, according to the interests of the class and instructors. There are no formal prerequisites other than prior completion of a rigorous undergraduate major or minor in physics, astronomy, or a related field.

Introduces students to the underlying physical principles employed by various devices. As in Physics , we will introduce students to a broader spectrum of device principles than they will encounter in their Physics projects.

There will be two 50 minute classes each week, split into discussion and laboratory exercises. Fundamental aspect of astrophysics and cosmology and new developments in these fields. Basic physical concepts and principles, the key observational evidence, and illustrative calculations. Relativistic cosmological models, inflation, Big-Bang nucleosynthesis, and the cosmic microwave background; formation and evolution of galaxy clusters, galaxies, and stars; formation, structure, and evolution of white dwarfs, neutron stars, and black holes; rotation- and accretion-powered pulsars, X-ray and y-ray stars, and gravitational radiation.

Rigorous survey of the physical properties of black holes, white dwarfs, and neutron stars. Formation of compact objects. The subject matter level and scope of the course are such that it is acceptable as the required elective course in the Physics major.

See Department of Physics course announcements. Terms offered: Fall , Fall , Fall This course is designed to assist physics and other physical sciences transfer students in their transition to UC Berkeley. Over the course of a semester, students will learn about campus resources, how to navigate the campus, establish connections with other students in their cohorts, receive physics transfer peer mentorship and advising. Students will work in small-groups to solve challenging mathematical and physics concepts to assist with academic success.

Prerequisites: Open only to physics and other physical sciences transfer students. Terms offered: Spring , Spring , Spring Elements of general relativity. Physics of pulsars, cosmic rays, black holes. The cosmological distance scale, elementary cosmological models, properties of galaxies and quasars. The mass density and age of the universe. Evidence for dark matter and dark energy and concepts of the early universe and of galaxy formation. Reflections on astrophysics as a probe of the extrema of physics.

Relativistic Astrophysics and Cosmology: Read Less [-]. Terms offered: Spring , Spring , Spring We will review the structure of proteins, nucleic acids, carbohydrates, lipids, and the forces and interactions maintaining their structure in solution. We will describe the thermodynamics and kinetics of protein folding. The principles of polymer chain statistics and of helix-coil transitions in biopolymers will be reviewed next, together with biopolymer dynamics. We will then cover the main structural methods in biology: X-ray crystallography , MNR and fluorescence spectroscopy, electron and probe microscopy, and single molecular methods.

Principles of Molecular Biophysics: Read Less [-]. Terms offered: Fall , Fall , Fall The course design covers data analysis and machine learning, highlighting their importance to the physical sciences. It covers data analysis with linear and nonlinear regression, logistic regression, and gaussian processes.

It covers concepts in machine learning such as unsupervised and supervised regression and classification learning. It develops Bayesian statistics and information theory, covering concepts such as information, entropy, posteriors , MCMC, latent variables, graphical models and hierarchical Bayesian modeling.

It covers numerical analysis topics such as integration and ODE, linear algebra, multi-dimensional optimization, and Fourier transforms. Terms offered: Spring , Spring , Spring A seminar which includes study and reports on current theoretical and experimental problems. Open only to students officially in the physics honors program or with consent of instructor. Terms offered: Fall , Fall , Spring This multidisciplinary course provides an introduction to fundamental conceptual aspects of quantum mechanics from a computational and informational theoretic perspective, as well as physical implementations and technological applications of quantum information science.

Basic sections of quantum algorithms, complexity, and cryptography, will be touched upon, as well as pertinent physical realizations from nanoscale science and engineering. Terms offered: Fall , Spring , Fall Thesis work under the supervision of a faculty member. To obtain credit the student must, at the end of two semesters, submit a satisfactory thesis. A total of four units must be taken. The units may be distributed between one or two semesters in any way.

This is part one of a year long series course. A provisional grade of IP in progress will be applied and later replaced with the final grade after completing part two of the series. Terms offered: Spring , Fall , Spring Thesis work under the supervision of a faculty member. This is part two of a year long series course. Upon completion, the final grade will be applied to both parts of the series. Terms offered: Spring , Fall , Spring Enrollment restrictions apply; see the Introduction to Courses and Curricula section in this catalog.

Summer: 6 weeks - 2. Terms offered: Prior to Discussion-based introduction to contemporary research in physics for advanced undergraduates. Presentation of different weekly topics in physics research led by graduate students, postdocs, or professors in a particular field to connect upper division physics majors with contemporary research and to increase dialogue between upper division undergraduates and researchers in the department. Course Objectives: -- To connect upper division physics majors with contemporary research in a way that traditional coursework does not.

This has resulted in research projects for several students -- Students received mentoring from the graduate student on many career path issues. Alternative to final exam. Terms offered: Fall , Spring , Fall Enrollment restrictions apply; see the Introduction to Courses and Curricula section in this catalog.

Terms offered: Spring , Spring , Spring A three-module introduction to the fundamental topics of Nano-Science and Engineering NSE theory and research within chemistry, physics, biology, and engineering. This course includes quantum and solid-state physics; chemical synthesis, growth fabrication, and characterization techniques; structures and properties of semiconductors, polymer, and biomedical materials on nanoscales; and devices based on nanostructures.

Prerequisites: Major in physical science such as chemistry, physics, etc. Terms offered: Spring , Spring , Spring Principles of gas dynamics, self-gravitating fluids, magnetohydrodynamics and elementary kinetic theory. Aspects of convection, fluid oscillations, linear instabilities, spiral density waves, shock waves, turbulence, accretion disks, stellar winds, and jets. Terms offered: Spring , Spring , Spring A multidisciplinary overview of computational nanoscience for both theorists and experimentalists.

This course teaches the main ideas behind different simulation methods; how to decompose a problem into "simulatable" constituents; how to simulate the same thing two different ways; knowing what you are doing and why thinking is still important; the importance of talking to experimentalists; what to do with your data and how to judge its validity; why multiscale modeling is both important and nonsense. Terms offered: Spring , Spring , Spring Lagrange and Hamiltonian dynamics, variational methods, symmetry, kinematics and dynamics of rotation, canonical variables and transformations, perturbation theory, nonlinear dynamics, KAM theory, solitons and integrable pdes.

Terms offered: Spring , Spring , Fall Nonlinear dynamics of dissipative systems, attractors, perturbation theory, bifurcation theory, pattern formation. Emphasis on recent developments, including turbulence. Terms offered: Fall , Fall , Fall An introduction to the basic physics of astronomy and astrophysics at the graduate level. Principles of energy transfer by radiation. Elements of classical and quantum theory of photon emission; bremsstrahlung, cyclotron and synchrotron radiation.

Compton scattering, atomic, molecular and nuclear electromagnetic transitions. Collisional excitation of atoms, molecules and nuclei. Radiation Processes in Astronomy: Read Less [-]. Terms offered: Fall , Fall , Fall Maxwell's equations, gauge transformations and tensors.

Complete development of special relativity, with applications. Plane waves in material media, polarization, Fresnel equations, attenuation, and dispersion.

Wave equation with sources, retarded solution for potentials, and fields. Cartesian and spherical multipole expansions, vector spherical harmonics, examples of radiating systems, diffraction, and optical theorem.

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Learn more about the News Feed. Includes: Solid fuel tablets 2. These are sometimes alternatively called fire starters or fire starting tinder.

Important : Be sure to keep them away from children and pets. Slideshow with step-by-step instructions viewable online. Click through the slideshow to see how the chemical skewers are prepared.

Click through the slideshow to see how to conduct the flame tests. Log in to add favorite More Menu Read More. Physics Teacher. Chemistry Teacher. Variations You could try this science project using other metal compounds known to produce certain flame colors when burned. Check out the resources in the Bibliography section for some ideas on how to do this. Be sure to always look into and follow all proper safety precautions when handling and burning different chemicals.

What colors do other metal compounds make when burned? View feedback on this project from other users. Hide feedback on this project from other users Recent Feedback Submissions Sort by Date Sort by User Name Rodrox said: What was the most important thing you learned? Can you suggest any improvements or ideas? Very Good What is your enthusiasm for science after doing your project?

High Compared to a typical science class, please tell us how much you learned doing this project. Much more Do you agree? Excellent What is your enthusiasm for science after doing your project? Very high Compared to a typical science class, please tell us how much you learned doing this project. Ask an Expert The Ask an Expert Forum is intended to be a place where students can go to find answers to science questions that they have been unable to find using other resources.

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