The aim of this theses is to create application, which is able to calculate buckling load of structure made from 1D bar elements, using finite element method. introduction is devoted to basic principles of buckling and derivation of necessary formulas. Then are described all operations and numerical methods needed for the application. At the and is in detail analyzed few structures and results are compared with known solutions or with other applications.

A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.

This thesis describes improvements of the periodic QZ algorithm and several variants of the Schur algorithm for block Toeplitz matrices. Documentation of the available software is included.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

It is an important question in machine learning to understand how neural networks learn. This thesis sheds further light onto this by studying autoencoder neural networks which can memorize data by storing it as attractors.What this means is that an autoencoder can learn a training set and later produce parts or all of this training set even when using other inputs not belonging to this set. We seek out to illuminate the effect on how ReLU networks handle memorization when trained with different setups: with and without bias, for different widths and depths, and using two different types of training images -- from the CIFAR10 dataset and randomly generated. For this, we created controlled experiments in which we train autoencoders and compute the eigenvalues of their Jacobian matrices to discern the number of data points stored as attractors.We also manually verify and analyze these results for patterns and behavior. With this thesis we broaden the understanding of ReLU autoencoders: We find that the structure of the data has an impact on the number of attractors. For instance, we produced autoencoders where every training image became an attractor when we trained with random pictures but not with CIFAR10. Changes to depth and width on these two types of data also show different behaviour.Moreover, we observe that loss has less of an impact than expected on attractors of trained autoencoders.

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.

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