x - UCSB ECE
... Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn ! R such that Then xeq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, x(t) converges to the largest invariant set M contained in E { z 2 Rn : W(z) = 0 } Note ...
... Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn ! R such that Then xeq is a Lyapunov stable equilibrium and the solution always exists globally. Moreover, x(t) converges to the largest invariant set M contained in E { z 2 Rn : W(z) = 0 } Note ...
Lecture #7 Stability and convergence of ODEs Hybrid Control and
... The equilibrium point xeq ∈ Rn is (Lyapunov) stable if ∃ α ∈ 2: ||x(t) – xeq|| · α(||x(t0) – xeq||) ∀ t≥ t0≥ 0, ||x(t0) – xeq||· c Suppose we could show that ||x(t) – xeq|| always decreases along solutions to the ODE. Then ||x(t) – xeq|| · ||x(t0) – xeq|| ∀ t≥ t0≥ 0 we could pick α(s) = s ⇒ Lyapunov ...
... The equilibrium point xeq ∈ Rn is (Lyapunov) stable if ∃ α ∈ 2: ||x(t) – xeq|| · α(||x(t0) – xeq||) ∀ t≥ t0≥ 0, ||x(t0) – xeq||· c Suppose we could show that ||x(t) – xeq|| always decreases along solutions to the ODE. Then ||x(t) – xeq|| · ||x(t0) – xeq|| ∀ t≥ t0≥ 0 we could pick α(s) = s ⇒ Lyapunov ...
Necessary conditions on minimal system configuration for general
... What are the necessary conditions under which fuzzy systems can possibly be universal approximators but with as minimal system configuration as possible? We have established such necessary conditions for the general single-input single-output (SISO) Mamdani fuzzy systems and a MISO Mamdani fuzzy sys ...
... What are the necessary conditions under which fuzzy systems can possibly be universal approximators but with as minimal system configuration as possible? We have established such necessary conditions for the general single-input single-output (SISO) Mamdani fuzzy systems and a MISO Mamdani fuzzy sys ...
OPVK_Computer viruses and security
... Ivan Zelinka Advanced Computer Security for joint teaching programme of BUT and VSB-TUO Vysoká škola báňská-Technická univerzita Ostrava 17. listopadu 15 Ostrava - Poruba first ...
... Ivan Zelinka Advanced Computer Security for joint teaching programme of BUT and VSB-TUO Vysoká škola báňská-Technická univerzita Ostrava 17. listopadu 15 Ostrava - Poruba first ...
Assignment • Hat Curve Fractal Handout
... Mathematical Fractals "I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world.“ --Al Gore ...
... Mathematical Fractals "I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world.“ --Al Gore ...
Activation process in excitable systems with multiple noise sources
... (iii) the oscillatory state at c > cH,L . The case (ii) involves coexistence between the fixed point and the large limit cycle, whose basins of attraction are separated by the stable manifold of the saddle cycle. In strict terms, such a scenario does not conform to Izhikevich’s definition of excitab ...
... (iii) the oscillatory state at c > cH,L . The case (ii) involves coexistence between the fixed point and the large limit cycle, whose basins of attraction are separated by the stable manifold of the saddle cycle. In strict terms, such a scenario does not conform to Izhikevich’s definition of excitab ...
Reaction rate theory: What it was, where is it today, and where is it
... half of a century until the transition state was precisely defined by Pechukas 关Dynamics of Molecular Collisions B, edited by W. H. Miller 共Plenum, New York, 1976兲兴, but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the questio ...
... half of a century until the transition state was precisely defined by Pechukas 关Dynamics of Molecular Collisions B, edited by W. H. Miller 共Plenum, New York, 1976兲兴, but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the questio ...
the nekhoroshev theorem and long–term stabilities in the solar system
... required. After the first formulation of Nekhoroshev’s theorem in 1977, many theoretical improvements have been achieved. On the one hand, alternative proofs of the theorem itself led to consistent improvements of the stability estimates; on the other hand, the extensions which were necessary to app ...
... required. After the first formulation of Nekhoroshev’s theorem in 1977, many theoretical improvements have been achieved. On the one hand, alternative proofs of the theorem itself led to consistent improvements of the stability estimates; on the other hand, the extensions which were necessary to app ...
101 uses of a quadratic equation: Part II
... Image courtesy of NASA With the coming of the Renaissance, deep thinkers started to look at the world in a different way. One of these was Copernicus, who made history by proposing that the Earth went round the Sun rather than the other way round. Copernicus thought that the orbit of the Earth was a ...
... Image courtesy of NASA With the coming of the Renaissance, deep thinkers started to look at the world in a different way. One of these was Copernicus, who made history by proposing that the Earth went round the Sun rather than the other way round. Copernicus thought that the orbit of the Earth was a ...
On Cantorian spacetime over number systems with division by zero
... separate additive and multiplicative groups with distinct sets of numbers, but all operations are performed on single operational structure, as if nature operates on bigroups or multigroups, in general. Physics should deploy models that match the nature’s mode of operations, rather than some pure-ma ...
... separate additive and multiplicative groups with distinct sets of numbers, but all operations are performed on single operational structure, as if nature operates on bigroups or multigroups, in general. Physics should deploy models that match the nature’s mode of operations, rather than some pure-ma ...
数学专著目录
... Integral Equations - H. Hochstadt.djv Introduction to Complex Analysis - R. Nevanlinna, V. Paatero Introduction to Complex Analysis Lecture notes - W. Chen *Introduction to Numerical Analysis 2 ed - J.Stoer,R.Bulirsch Introduction To p-adic Numbers and p-adic Analysis - A. Baker Introduction to the ...
... Integral Equations - H. Hochstadt.djv Introduction to Complex Analysis - R. Nevanlinna, V. Paatero Introduction to Complex Analysis Lecture notes - W. Chen *Introduction to Numerical Analysis 2 ed - J.Stoer,R.Bulirsch Introduction To p-adic Numbers and p-adic Analysis - A. Baker Introduction to the ...
forensics repository
... Advanced Calculus With Applications In Statistics - A Khuri An introduction to the fractional calculus and fractional differential equations - Miller K.S., Ross B. Calculus 5th Edition - James Stewart solution Calculus 5th Edition - James Stewart Calculus Bible Calculus Concepts and Contexts 2nd Ed ...
... Advanced Calculus With Applications In Statistics - A Khuri An introduction to the fractional calculus and fractional differential equations - Miller K.S., Ross B. Calculus 5th Edition - James Stewart solution Calculus 5th Edition - James Stewart Calculus Bible Calculus Concepts and Contexts 2nd Ed ...
Joseph L. Doob - National Academy of Sciences
... called it the “separable” version. Although Kolmogorov’s construction is no longer the method of choice, and therefore Doob’s result is seldom used today, at the time even Kolmogorov acknowledged it as a substantive contribution. Doob’s renown did not rest on his separability theorem. Instead, he wa ...
... called it the “separable” version. Although Kolmogorov’s construction is no longer the method of choice, and therefore Doob’s result is seldom used today, at the time even Kolmogorov acknowledged it as a substantive contribution. Doob’s renown did not rest on his separability theorem. Instead, he wa ...
t - KTH
... • For Poisson arrival process: the time until the next arrival does not depend on the time spent after the previous arrival Poisson arrival () ...
... • For Poisson arrival process: the time until the next arrival does not depend on the time spent after the previous arrival Poisson arrival () ...
Comparing mathematical provers - Institute for Computing and
... For most systems it is clear why they are in this list, but a few need explanation. Otter is not designed for the development of a structured body of mathematics in the QED style, but instead is used in a ‘one shot’ way to solve logical puzzles. Also it is only one of the members (although the best ...
... For most systems it is clear why they are in this list, but a few need explanation. Otter is not designed for the development of a structured body of mathematics in the QED style, but instead is used in a ‘one shot’ way to solve logical puzzles. Also it is only one of the members (although the best ...
Periodic Orbits in Generalized Mushroom Billiards SURF 2006
... confined point particle moving with constant speed and colliding perfectly elastically against its boundary. Studying the periodic orbits of a billiard provides insight into its dynamics, which can be chaotic, regular (non-chaotic), or a mixture of the two. A chaotic billiard is one in which initiall ...
... confined point particle moving with constant speed and colliding perfectly elastically against its boundary. Studying the periodic orbits of a billiard provides insight into its dynamics, which can be chaotic, regular (non-chaotic), or a mixture of the two. A chaotic billiard is one in which initiall ...
Fractals and Dimension
... at x = 1). But if we iterate both together, choosing f1 or f2 at random at each iteration, with probability 1/2, we obtain the whole of C as an attractor for the system. The orbit of any initial point x0 approaches closer and closer to C and moreover with probability 1 it approaches arbitrarily clos ...
... at x = 1). But if we iterate both together, choosing f1 or f2 at random at each iteration, with probability 1/2, we obtain the whole of C as an attractor for the system. The orbit of any initial point x0 approaches closer and closer to C and moreover with probability 1 it approaches arbitrarily clos ...
1.7. Stability and attractors. Consider the autonomous differential
... (i) The origin is stable if and only if Re σ(A) ≤ 0 and the eigenvalues with zero real parts have simple elementary divisors; that is, each Jordan block has dimension one; (ii) The origin is a global attractor for (5.1) if an only if Re σ(A) < 0. Proof. If the origin is stable, then we must have Re ...
... (i) The origin is stable if and only if Re σ(A) ≤ 0 and the eigenvalues with zero real parts have simple elementary divisors; that is, each Jordan block has dimension one; (ii) The origin is a global attractor for (5.1) if an only if Re σ(A) < 0. Proof. If the origin is stable, then we must have Re ...
Americana, American History Mathematics
... Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. Appropriate for advanced undergraduates and graduate students, it defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, fo ...
... Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. Appropriate for advanced undergraduates and graduate students, it defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, fo ...
Numerical Techniques for Approximating Lyapunov Exponents and
... exponents. In particular, in this paper we describe the algorithms that are implemented in these robust, reliable computational procedures. The need for reliable codes to approximate Lyapunov exponents and related quantities is apparent in many application areas. In [42] Lyapunov exponents are comp ...
... exponents. In particular, in this paper we describe the algorithms that are implemented in these robust, reliable computational procedures. The need for reliable codes to approximate Lyapunov exponents and related quantities is apparent in many application areas. In [42] Lyapunov exponents are comp ...
$doc.title
... This book aims to construct a general framework for the analysis of a large class of random fields, also known as multiparameter processes. A great part of one-parameter theory is also included, with the goal to keep the book self-contained. The book is divided into two parts, devoted to discrete-pa ...
... This book aims to construct a general framework for the analysis of a large class of random fields, also known as multiparameter processes. A great part of one-parameter theory is also included, with the goal to keep the book self-contained. The book is divided into two parts, devoted to discrete-pa ...
Mathematical Aspects of Artificial Intelligence
... Artificial Intelligence (AI) is an important and exciting field. It is an active research area and is considered to have enormous research opportunities and great potential for applications. At the same time, AI is highly controversial. There is a history of great expectations, and large investments ...
... Artificial Intelligence (AI) is an important and exciting field. It is an active research area and is considered to have enormous research opportunities and great potential for applications. At the same time, AI is highly controversial. There is a history of great expectations, and large investments ...
Differential Equations
... state variables, since their values at any time describe the state of the system. Similarly, the vector u = u1i + u2j is called the state vector of the system. The u1u2plane itself is called the state space. If there are only two state variables, the u1u2plane may be called the state plane or, mor ...
... state variables, since their values at any time describe the state of the system. Similarly, the vector u = u1i + u2j is called the state vector of the system. The u1u2plane itself is called the state space. If there are only two state variables, the u1u2plane may be called the state plane or, mor ...
satellite frequency assignments using transiently chaotic neural
... FAP, we focus on minimization of system interference for fixed frequency assignments in satellite communications. Interference in satellite communications depends on transmitter power, channel loss, receiver sensitivity, and antenna gains. Frequency rearrangements are an effective complement alongs ...
... FAP, we focus on minimization of system interference for fixed frequency assignments in satellite communications. Interference in satellite communications depends on transmitter power, channel loss, receiver sensitivity, and antenna gains. Frequency rearrangements are an effective complement alongs ...
Random Matrix Approach to Linear Control Systems
... pattern. Thus, we must show why this pattern arises naturally in these systems. ...
... pattern. Thus, we must show why this pattern arises naturally in these systems. ...
Chaos theory
Chaos theory is the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:Chaos: When the present determines the future, but the approximate present does not approximately determine the future.Chaotic behavior exists in many natural systems, such as weather and climate. This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in several disciplines, including meteorology, sociology, physics, engineering, economics, biology, and philosophy.