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Fixed point theorems in the theory of differential equations
Zelina, Michael ; Pražák, Dalibor (advisor) ; Bárta, Tomáš (referee)
This thesis is devoted to show various applications of fixed point theorems on dif- ferential equations. In the beginning we use a notion of topological degree to derive several fixed points theorems, primarily Brouwer, Schauder and Kakutani-Ky Fan the- orem. Then we apply them on a wide range of relatively simple problems from ordinary and partial differential equations (ode and pde). Finally, we take a look on a few more complex problems. First is an existence of a solution to the model of mechanical os- cillator with non-monotone dependence of both displacement and velocity. Second is a solution to so called Gause predator-prey model with a refuge. The last one is cer- tain partial differential equation with a constraint which determines maximal monotone graph. 1
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Linear version of Holub's algorithm
Tvrdý, David ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
This work studies a linear agorithm which decides if a given word is a fixed point of any nontrivial morphism. This work also contains a description of auxiliary data structures which are crucial for linear time complexity of the algorithm. A Java implementation of the algorithm is provided along with a step-by-step visualization for particular input words. 1
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SSI Dividing Numerical Integrator
Suntcov, Roman ; Veigend, Petr (referee) ; Šátek, Václav (advisor)
The thesis deals with numerical integration and hardware division operations. The reader is familiar with the numerical solution of differential equations through several different methods, for example Taylor's series. Furthermore, it is discussed the operation of division in the hardware and the method of its implementation in the FPGA. Subsequently, a parallel-parallel and serial-parallel integrator is designed. The practical aim of the thesis is to design and implement a serial-serial dividing integrator and create a simulator for it.
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A Library for Convolutional Neural Network Design
Rek, Petr ; Mrázek, Vojtěch (referee) ; Sekanina, Lukáš (advisor)
In this diploma thesis, the reader is introduced to artificial neural networks and convolutional neural networks. Based on that, the design and implementation of a new library for convolutional neural networks is described. The library is then evaluated on widely used datasets and compared to other publicly available libraries. The added benefit of the library, that makes it unique, is its independence on data types. Each layer may contain up to three independent data types - for weights, for inference and for training. For the purpose of evaluating this feature, a data type with fixed point representation is also part of the library. The effects of this representation on trained net accuracy are put to a test.
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Hardware Realization of Higher Order Numerical Integrator
Matečný, František ; Veigend, Petr (referee) ; Šátek, Václav (advisor)
This work describes numerical integration and solution for ordinary differential equations by the Taylor series by different types of integrators. The next part is a description of floating point and fixed point arithmetic. Subsequently, we are presenting designs and calculation methods for parallels multiplication and division integrators in floating point and fixed point arithmetic. The designs were realized in VHDL and implemented on FPGA. Finally we summarizes the proposed solution and compare time complexity with another numerical methods.
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Specialized Computer System Automatic Control
Opálka, Jan ; Šátek, Václav (referee) ; Kunovský, Jiří (advisor)
This work deals with the automatic control of calculations of specialized system. The reader is acquainted with the numerical solution of differential equations by Taylor series method and numerical integrators. The practical aim of this work is to analyze parallel characteristics of Taylor series method, specification of parallel math operations and design of control of this operations.
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Arithmetical completeness of the logic R
Holík, Lukáš ; Švejdar, Vítězslav (advisor) ; Bílková, Marta (referee)
The aim of this work is to use contemporary notation to build theory of Rosser logic, explain in detail its relation to Peano arithmetic, show its Kripke semantics and finally using plural self-reference show the proof of arithmetical completeness. In the last chapter we show some of the properties of Rosser sentences. Powered by TCPDF (www.tcpdf.org)
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Analysis of the CubeHash proposal
Stankovianska, Veronika ; Tůma, Jiří (advisor) ; Hojsík, Michal (referee)
The present thesis analyses the proposal of CubeHash with spe- cial emphasis on the following papers: "Inside the Hypercube" [1], "Sym- metric States and Their Improved Structure" [7] and "Linearisation Frame- work for Collision Attacks" [6]. The CubeHash algorithm is presented in a concise manner together with a proof that the CubeHash round function R : ({0, 1}32 )32 → ({0, 1}32 )32 is a permutation. The results of [1] and [7] con- cerning the CubeHash symmetric states are reviewed, corrected and substan- tiated by proofs. More precisely, working with a definition of D-symmetric state, based on [7], the thesis proves both that for V = Z4 2 and its linear subspace D, there are 22 |V | |D| D-symmetric states and an internal state x is D-symmetric if and only if the state R(x) is D-symmetric. In response to [1], the thesis presents a step-by-step computation of a lower bound for the num- ber of distinct symmetric states, explains why the improved preimage attack does not work as stated and gives a mathematical background for a search for fixed points in R. The thesis further points out that the linearisation method from [6] fails to consider the equation (A ⊕ α) + β = (A + β) ⊕ α (∗), present during the CubeHash iteration phase. Necessary and sufficient conditions for A being a solution to (∗) are...
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Algorithm for word morphisms fixed points
Matocha, Vojtěch ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
In the present work we study the first polynomial algorithm, which tests if the given word is a fixed point of a nontrivial morphism. This work contains an improved worst-case complexity estimate O(m · n) where n denotes the word length and m denotes the size of the alphabet. In the second part of this work we study the union-find problem, which is the crucial part of the described algorithm, and the Ackermann function, which is closely linked to the union-find complexity. We summarize several common methods and their time complexity proofs. We also present a solution for a special case of the union-find problem which appears in the studied algorithm. The rest of the work focuses on a Java implementation, whose time tests correspond to improved upper bound, and a visualization useful for particular entries.
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Division Operation Simulation
Matečný, František ; Šátek, Václav (referee) ; Kunovský, Jiří (advisor)
This work deals with numerical integration and division operation. The reader is acquainted with the numerical solution of differential equations using division by the Taylor series. Next is explained the principle of SRT division in hardware and introduction of draft of design series-parallel and parallel division integrator in fixed point arithmetic. The practical aim of this work is implementation parallel division integrator and development of a software simulation of this integrator.
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