Research on Evolution Equation Compendium, Volume 1

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Sports Physiol. Perform 8, — Impact of a soccer match on the cardiac autonomic control of referees. Relationship between aerobic capacity and Yo-Yo IR1 performance in Brazilian professional futsal players. Asian J. Bradley, P. The application of the Yo-Yo intermittent endurance level 2 test to elite female soccer populations. Sports 24, 43— Match performance and physical capacity of players in the top three competitive standards of English professional soccer. Sub-maximal and maximal Yo-Yo intermittent endurance test level 2: heart rate response, reproducibility and application to elite soccer.

Brink, Y. Clinical instruments: reliability and validity critical appraisal. Brito, J. Factors influencing the performance of young football players in the yo-yo intermittent endurance test Level 2. Brocherie, F. Association of hematological variables with team-sport specific fitness performance. Bruce, L. Validity of the Intermittent Fitness Test in sub-elite female athletes. CrossRef Full Text. Buchheit, M. Adding heat to the live-high train-low altitude model: a practical insight from professional football.

Physiological and performance adaptations to an in-season soccer camp in the heat: associations with heart rate and heart rate variability. Sports 21, e—e Medium-sided games in soccer: physical and heart rate demands throughout successive working periods. Sport Exerc. Campos-Vazquez, M. Relationships between rating-of-perceived-exertion- and heart-rate-derived internal training load in professional soccer players: a comparison of on-field integrated training sessions.

Comparison of the effect of repeated-sprint training combined with two different methods of strength training on young soccer players. Casamichana, D. Influence of the type of marking and the number of players on physiological and physical demands during sided games in soccer. Effect of number of touches and exercise duration on the kinematic profile and heart rate response during small-sided games in soccer. Castagna, C. Competitive-level differences in Yo-Yo intermittent recovery and twelve minute run test performance in soccer referees. Effects of intermittent-endurance fitness on match performance in young male soccer players.

Cardiorespiratory responses to Yo-yo Intermittent Endurance Test in nonelite youth soccer players. Aerobic fitness and yo-yo continuous and intermittent tests performances in soccer players: a correlation study. The Yo-Yo intermittent recovery test in basketball players. Sport 11, — Castillo, D. Physical fitness and physiological characteristics of soccer referees. Sports 31, 27— Chan, H. Power and endurance in Hong Kong professional football players. Asia Pac. Chaouachi, A. Intermittent endurance and repeated sprint ability in soccer players. Cholewa, J.

The effects of sodium bicarbonate supplementation on asoccer specific conditioning test in division III soccer players. Christensen, P. VO2 kinetics and performance in soccer players after intense training and inactivity. Chtourou, H. The effect of ramadan fasting on physical performances, mood state and perceived exertion in young footballers. Chuman, K. Relationships between Yo-Yo intermittent recovery tests and development of aerobic and anaerobic fitness in U and U soccer players.

Cihan, H. Comparison of recovering times and aerobic capacity according to playing positions of elite football players. Clarke, A. Critical velocity as a measure of aerobic fitness in women's rugby sevens. Sport 17, — Cobley, J. N-Acetylcysteine's attenuation of fatigue after repeated bouts of intermittent exercise: practical implications for tournament situations. Sport Nutr. Coelho, D. Analysis of chronic physiological demand of an annual soccer season.

Kineanthropometry Hum. Cone, J. Effects of an individualized soccer match simulation on vertical stiffness and impedance. Cooper, K. A means of assessing maximal oxygen intake. Correlation between field and treadmill testing. JAMA , — Coratella, G. The specificity of the Loughborough Intermittent Shuttle Test for recreational soccer players is independent of their intermittent running ability. Cullen, B. Fitness profiling of elite level adolescent Gaelic football players.

Darrall-Jones, J. Anthropometric and physical profiles of english academy rugby union players. Anthropometric, sprint, and high-intensity running profiles of english academy rugby union players by position. Da Silva, J. Validity and reliability of a new field test Carminatti's test for soccer players compared with laboratory-based measures.

Deliceoglu, G. The investigation of heart rate variation on endurance of professional soccer players. Middle-East J. Deprez, D. Reliability and validity of the Yo-Yo intermittent recovery test level 1 in young soccer players. Characteristics of high-level youth soccer players: variation by playing position. The Yo-Yo intermittent recovery test level 1 is reliable in young high-level soccer players. Sport 32, 65— Relative age effect and Yo-Yo IR1 in youth soccer.

De Souza, J. Changes in metabolic and motor performance variables induced by training in handball players. Dinardi, R. Online 20, 92— Dixon, D. A retrospective study of the Yo-Yo IE2 Test: can it be used to differentiate between different levels of futsal referees? Dixon, H. Sodium bicarbonate ingestion improves Yo-Yo intermittent recovery test 1 performance: a randomized crossover trial. Dupont, G. Yo-Yo intermittent recovery test versus the Universite de Montreal Track Test: relation with a high-intensity intermittent exercise.

Sport 13, — Eaton, T. A combination of amino acids and caffeine enhances sprint running capacity in a hot, hypoxic environment. Fabregat-Andres, O. Evaluation of a new shirt-based electrocardiogram device for cardiac screening in soccer players: comparative study with treadmill ergospirometry. Fanchini, M. Are the Yo-Yo intermittent recovery test levels 1 and 2 both useful?

Reliability, responsiveness and interchangeability in young soccer players. Sports Sci 32, — Effect of training-session intensity distribution on session rating of perceived exertion in soccer players. Faude, O. Combined strength and power training in high-level amateur football during the competitive season: a randomised-controlled trial. Ferioli, D. Different training loads partially influence physiological responses to preparation period in basketball.

Flatt, A. Evaluating individual training adaptation with smartphone-derived heart rate variability in a collegiate female soccer team. Evaluating a nationwide recreational football intervention: recruitment, attendance, adherence, exercise intensity, and health effects. Biomed Res. Furlan, N. Gatterer, H. Effects of a day maximal shuttle-run shock microcycle in hypoxia on soccer specific performance and oxidative stress.

Gibson, N. Relationship between measures of aerobic fitness, speed and repeated sprint ability in full and part time youth soccer players. Fitness 53, 9— Gunnarsson, T. Effect of additional speed endurance training on performance and muscle adaptations. Hamlin, M.

Hypoxic repeat sprint training improves rugby player's repeated sprint but not endurance performance. Hammouda, O. Does Ramadan fasting affect the diurnal variations in metabolic responses and total antioxidant capacity during exercise in young soccer players? Hasegawa, N. Physical characteristics of collegiate women's football players. Football Sci. Heaney, N. The Effect of a 4 week aerobic interval training block using maximal aerobic speed as the intensity measure with elite female hockey players.

Henrique Borges, P. Impact of aerobic power, strength of lower limbs and speed on technical skills in young soccer players. Online 20, — Hermassi, S. Relationships between the yo-yo intermittent recovery test and anaerobic performance tests in adolescent handball players. Relationship between explosive performance measurements of the lower limb and repeated shuttle-sprint ability in elite adolescent handball players. Effects of in-season short-term aerobic and high-intensity interval training program on repeated sprint ability and jump performance in handball players.

Fitness 58, 50— Higgins, J. Quantifying heterogeneity in a meta-analysis. Higham, D. Physiological, anthropometric, and performance characteristics of rugby sevens players. Hogarth, L. Activity profiles and physiological responses of representative tag football players in relation to playing position and physical fitness. Hogarth, W. The relationship between physical capacity and match running performance in men's tag football. Sport 15, — Iacono, A. Improving fitness of elite handball players: small-sided games vs. Iaia, F. Short- or long-rest intervals during repeated-sprint training in soccer?

The effect of two speed endurance training regimes on performance of soccer players. Idrizovic, K. Physical and anthropometric profiles of elite female soccer players. Dello Sport 67, — The Correlation between aerobic power, acceleration, repeated-sprint and speed endurance in elite female football. Sport Health 2, 51— Ingebrigtsen, J.

Yo-Yo IR2 testing of elite and sub-elite soccer players: performance, heart rate response and correlations to other interval tests. Relationships between field performance tests in high-level soccer players. Performance effects of 6 weeks of aerobic production training in junior elite soccer players. Inness, M. Team-sport athletes' improvement of performance on the yo-yo intermittent recovery test level 2, but not of time-trial performance, with intermittent hypoxic training.

Jamurtas, A. Iron status markers are only transiently affected by a football game. Johnston, R. Influence of playing standard and physical fitness on activity profiles and post-match fatigue during intensified junior rugby league competition. Open Jones, B. Physical qualities of international female rugby league players by playing position. Joo, C.

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The effects of short-term detraining on exercise performance in soccer players. Julian, R. The effects of menstrual cycle phase on physical performance in female soccer players. Karavelioglu, M. Anthropologist 18, — Pulse rate before and after Yo-Yo 2 test at young footbalers and deciding changes at some blood parameters. Karsten, B. The effects of a 6-week strength training on critical velocity, anaerobic running distance, M sprint and Yo-Yo intermittent running test performances in male soccer players.

Kavaliauskas, M. Effects of in-season uphill sprinting on physical characteristics in semi-professional soccer players. Fitness 57, — Kelly, R. The seasonal variations in anthropometric and performance characteristics of elite inter county Gaelic football players. Kilding, A. Effects of acutely intermittent hypoxic exposure on running economy and physical performance in basketball players. Kilit, B. Laboratory and field-based assessment of maximal aerobic and anaerobic power in professional tennis players. Comparison of the physiological responses and time-motion characteristics of young soccer players in small-sided games: the effect of goalkeeper.

Krustrup, P. Physiological demands of top-class soccer refereeing in relation to physical capacity: effect of intense intermittent exercise training. The Yo-Yo IE2 test: physiological response for untrained men versus trained soccer players. Muscle adaptations and performance enhancements of soccer training for untrained men. Long-term musculoskeletal and cardiac health effects of recreational football and running for premenopausal women.

Sports 20, 58— Physical demands in competitive ultimate frisbee. The yo-yo intermittent recovery test: physiological response, reliability, and validity. The Yo-Yo IR2 test: physiological response, reliability, and application to elite soccer. Broad-spectrum health improvements with one year of soccer training in inactive mildly hypertensive middle-aged women. Sports 27, — Game-induced fatigue patterns in elite female soccer.

Kvorning, T. Strength and conditioning training by the danish national handball team before an olympic tournament. Lategan, L. Physiological profiles of South African soccer referees and assistant referees. Health Sci. A maximal multistage m shuttle run test to predict VO2 max. An indirect continuous running multistage field test: the Universite de Montreal track test. Sport Sci. Leme, L. For the student who has already reached calculus I suggest Gullberg as a reference. With the preceding in mind I prefer books in the workbook format. An excellent textbook series is the series by Bittinger published by Aison-Wesley.

Trig like pre-calculus algebra and calculus itself tends to be remarkably similar from one text to another. A good example of the genre is: Keedy, Mervin L. Trigonometry: Triangles and Functions. There is a recent book about trig for the serious student. This is a much needed book and has my highest recommendation: Maor, Eli.

Trigonometric Delights. Princeton University. Selected Papers on Precalculus. Stillwell, John. Numbers and Geometry. First, see Principle of Learning Calculus. Kline, Morris. There are a great many competent texts in this area. The best is Strang, Gilbert. Linear Algebra and Its Applications. It undoubtedly the most influential book in its area since Halmos's Finite Dimensional Vector Spaces. This is a more appropriate text for the classroom, especially at the sophomore level: Strang, Gilbert.

Introduction to Linear Algebra. A more recent book along similar lines is: Curtis, Morton L. Abstract Linear Algebra. Linear Algebra Done Right. An advanced applied text is: Lax, Peter D. Linear Algebra. A book emphasizing that is: Banchoff, Thomas, and John Wermer. Linear Algebra Through Geometry. It is certainly an interesting text after the first course.

Linear Algebra and Its Applications , 2 nd ed. He does a nice job of introducing a surprising number of the key ideas in the first chapter. I think somehow that this has a great pedagogical payoff. Although it is very similar to many other texts, I like this particular text a great deal. Personally though I prefer the introductory text by Strang If choosing a text for a sophomore level course, I myself would choose the book by Lay or the one by Strang Wellesley-Cambridge Press. The following book has merit and might work well as an adjunct book in the basic linear algebra course.

It is the book for the student just learning mathematics who wants to get into computer graphics. Farin, Gerald and Dianne Hansford. Basic Matrix Algebra with Algorithms and Applications. Chapman and Hall. Also strong on applications. An excellent choice for a second book: Robert, Alain M. World Scientific. It is well written and is abstract but will throw in a section for physicists. I like this book quite a bit. Further Linear Algebra. Also see Courant and John. Most standard calculus texts have a section on multivariable calculus and many sell these sections as separate texts as an option.

For example the Harvard Calculus Consortium mentioned in Calculus sell their multivariable volume separately. The most informal treatment is the second half of a series. This is a great book for the student in third semester calculus to have on the side. Multivariate Calculus : A Geometric Approach. A particularly good example is: Kaplan, Wilfred.

Advanced Calculus , 3 rd ed. Functions of Two Variables. Multivariate Calculus and Geometry. Functions of Several Variables. Advanced Calculus of Several Variables. It uses a fair bit of linear algebra which is presented in the text, but I suggest linear algebra as a prerequisite. Its orientation is economics, so there is no Divergence Theorem or Stokes Theorem.

Binmore, Ken and Joan Davies. Calculus: Concepts and Methods. Bachman, David.

McGraw Hill. Like in some other areas, many books on differential equations are clones. The standard text is often little more than a cookbook containing a large variety of tools for solving d. Most people use only a few of these tools. Moreover, after the course, math majors usually forget all the techniques. Engineering students on the other hand can remember a great deal more since they often use these techniques.

A good example of the standard text is: Ross, Shepley L. Introduction to Ordinary Differential Equations , 4 th ed. Bronson, Richard. Theory and Problems of Differential Equations , 2 nd ed. Schaum McGraw-Hill. Differential Equations and Their Applications , 3 rd ed. Differential Equations with Applications and Historical Notes , 2 nd ed. It covers the main topics very succinctly and is well written. Given its very modest price and clarity I recommend it as a study aid to all students in the basic d.

Many others would appreciate it as well. Bear, H. Differential Equations: A Concise Course. For a personal library or reference I would prefer the Braun and Simmons. An introductory volume that emphasizes ideas and the graphical underpinnings of d. Another book that may be the best textbook here which is strong on modeling is Borrelli and Coleman. Differential Equations: A Modeling Perspective. Both are quite good. The following book can be considered a supplementary text for either the student or the teacher in d. Braun, Martin, Courtney S. Coleman, Donald A. Differential Equation Models.

Similar to the Kostelich and Armruster volume above these emphasize geometry. These volumes rely on the geometrical view all the way through.


Note that the second volume can be read independently of the first. Hubbard, J. Part 1. What it does exceptionally well is to use complex arithmetic to simplify complex problems. Redheffer, Raymond M. Introduction to Differential Equations. Jones and Bartlett. Nonetheless, it is aimed at roughly the junior level. O'Malley, Robert E. Thinking About Ordinary Differential Equations. An undergraduate text that emphasizes theory and moves along at a fair clip is: Birkhoff, Garrett. Gian-Carlo Rota. Ordinary Differential Equations.

See Dynamical Systems and Calculus. Schiff, Joel L. The Laplace Transform. The standard text in this area has been: Ward, James Brown. Ruel V. Fourier Series and Boundary Value Problems. Partial Differential Equations for Scientists and Engineers. Lots of pictures. A new book that is also very attractive: O'Neil, Peter V. Beginning Partial Differential Equations. Applied Partial Differential Equations. A Classic introduction. Elementary and a quick read. Goldberg, Samuel.

Introduction to Difference Equations. Both are considerably more in depth than Goldberg's. Read his first. Elaydi, Saber, N. An Introduction to Difference Equations, 2nd ed. Difference Equations: An Introduction with Applications. Two classics that precede the current era of hyper-interest in this area are both are linear algebra intensive Luenberger, David G.

Hirsch, Morris W. Three elementary books follow. The second and third seem to be particularly suited as texts at the sophomore-junior level. They emphasize linear algebra whereas Acheson is more differential equations and physics. Acheson, David. This section is not for beginners! If you are just learning calculus go to the section Calculus. The genesis, by the creator, is tough reading: Robinson, Abraham. Non-Standard Analysis. Infinitesimal Calculus. A book that is supposed to be easy but is very abstract is: Robert, Alain.

Nonstandard Analysis. A Primer of Infinitesimal Analysis. An Introduction to Nonstandard Real Analysis. The following book is a primer on complex numbers that ends with a short introduction to Complex Analysis. It is a perfect book for the sophomore in math or engineering.

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Great book:. Nahin, Paul J. Perhaps the most remarkable book in this area; truly great book is: Needham, Tristan. Visual Complex Analysis. Also, it is a great reference during the first course. A wonderful book that is concise, elegant, clear: a must have: Bak, Joseph and Donald J. Complex Analysis , 2 nd ed. Complex Analysis. Calculus with Complex Numbers. Taylor and Francis. A thorough well written text I like is: Ablowitz, Mark J.

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Complex Variables: Introduction and Applications. Complex Variables and Applications 6 th ed. Complex Variables with Applications , 2 nd ed. Still another superb first text is formatted exactly as elementary calculus texts usually are: Saff, E. An Introduction to Complex Function Theory. Introduction to Complex Analysis. Basic Complex Analysis , 2 nd ed. Invitation to Complex Analysis. Birkhauser Boston. Titmarsh, E.

The Theory of Functions , 2 nd ed. A reference that I expect to sell very well to a wide audience: Krantz, Steven G. Handbook of Complex Analysis. The author says it should get you ready for Ph. Definitely a superior work. Gamelin, Theodore W. A great pedagogical work most highly recommended especially to electrical engineers Schey, H. Vector Calculus , 4 rd ed. This may be the best book to have. It is very good. Vector and Tensor Analysis.

Vector Calculus. Vector Calculus , 2 nd ed. Books on differential forms and tensors can often merely enhance the reputations of those areas for being difficult. However, there are exceptions. On tensors I like two books which complement each other well. The book by Danielson is more application oriented. If you are serious about this area get both books. Also, the Schaum outline series volume on tensors has merit.

Simmonds, James G. A Brief on Tensor Analysis , 2 nd ed. Vectors and Tensors in Engineering and Physics , 2 nd ed. A Geometric Approach to Differential Forms. Second Year Calculus. There are roughly 37 zillion books on applied math with titles like Mathematics for Left-Handed Quantum Engineers Check out Gullberg , it was specifically written for engineering students though it is appropriate for all students of math A great book which, appropriate for its author, emphasizes linearity is: Strang, Gilbert.

Computational Science and Engineering. Wellesley-Cambridge Press. A recent book that is pedagogically very nice and goes though junior level material with wide coverage extending to group theory is Riley et al. A great tool for applied mathematicians: Andrews, Larry C. Special Functions of Mathematics for Engineers , 2 nd ed. A Course in Mathematics for Students of Physics. Mathematical Methods in the Physical Sciences , 3 rd ed. Also, this demonstrates how completely impartial I am, since Professor Boas detests me. A tour de force at the graduate level; a book for the serious student: Gershenfeld, Neil.

The Nature of Mathematical Modeling. Metric Spaces: Iteration and Application. When Least is Best. Shiflet, Angela B. Princeton University Press. Courant and John A great reference is the last edition of Courant's great classic work on calculus. Nonetheless they are relatively not expensive and they are great references. Volume I is a superb work on analysis.

Courant, Richard and Fritz John. Introduction to Calculus and Analysis. Vol I. Check out Gullberg. A classic originally published more than fifty years ago : Hogben, Lancelot. Although ostensibly written for the layman, it is not a light work. Its treatment of geometry is particularly good Courant, Richard, Herbert Robins. Revised by Ian Stewart. What is Mathematics. A sweet book that is similar in spirit to Stillwell's and that should be of interest to students of analysis is Pontrjagin, Lev S. Learning Higher Mathematics. The Mathematical Experience. The late Morris Kline wrote several good books for the layman as well as for the professional.

My personal favorite is strong on history and art and I think deserves more attention than it has ever had. I think it is more important now then when it was first published in the 's : Kline, Morris. Mathematics in Western Culture. The following is a book I think every undergraduate math major who is at all serious should have: Hewson, Stephen Fletcher. The books here tend cover algorithms and computability but don't forget to go the sections Algorithms and Logic and Computability.

Dewdney wrote a book of 66 chapters to briefly and succinctly cover the interesting topics of computer science. The emphasis here is theory. This is a book every computer science major should have, and probably every math major and certainly anyone with a serious interest in computer science.

Dewdney, A. The New Turing Omnibus. Algorithmics: The Spirit of Computing , 2 nd ed. Algorithms and Complexity. Combinatorics Including Graph Theory. Posamentier, Alfred S. The Fabulous Fibonacci Numbers. Numerical Analysis. Most books on numerical analysis are written to turn off the reader and to encourage him or her to go into a different, preferably unrelated, field. Secondly, almost all of the books in the area are written by academics or researchers at national labs, i.

The kind of industry I use to work in was a little different than that. The problem is partly textbook evolution. I've seen books long out of print that would work nicely in the classroom. However, textbook competition requires that newer books contain more and more material until the book can become rather unwieldy in several senses for the classroom.

The truth is that the average book has far too much material for a course. Numerical analysis touches upon so many other topics this makes it a more demanding course than others. A marvelous exception to the above is the book by G. It avoids the problem just mentioned because it is based upon notes from a course. It is concise and superbly written. It is the one I am now teaching out of.

Stewart, G. Afternotes on Numerical Analysis. The first is absolutely superb. Both books are great to read, but I don't like either as a text. Acton, Forman. Numerical Methods That Work. An interesting book that seems in the spirit of the first book by Acton above is: Breuer, Shlomo, Gideon Zwas. Numerical Mathematics: A Laboratory Approach. I would like to know however how it has done as a text. A book by a great applied mathematician that is worth having is: Hamming, R.

Numerical Methods for Scientists and Engineers , 2 nd ed.. Numerical Analysis: Theory and Practice. Douglas, Richard Burden. Numerical Methods, 2 nd ed. This is the problem with teaching the course. On the flip side of course, it covers less material e. Also, it does not give pseudo-code for algorithms.

This is okay with me for the following reasons. Given a textbook with good pseudo-code, no matter how much I lecture the students on its points and various alternatives, they usually copy the pseudocode as if it the word of God rather than regarding my word as the word of God. It is useful to make them take the central idea of the algorithm and work out the details their selves. This text also has an associated instructors guide and student guides.

See the book by Cooper. Fourier Analysis. The best book on Fourier analysis is the one by Korner. However, it is roughly at a first year graduate level and is academic rather than say engineering oriented. Any graduate student in analysis should have this book. Korner, T. The Fourier Transform and Its Applications , 2 nd ed. Another book in a similar vein has been reprinted recently I think : Papoulis, Athanasios.

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The Fourier Integral and Its Applications. A book with many applications to engineering is Folland, Gerald B. Fourier Analysis and its Applications. Introduction to Fourier Analysis. A fairly short book pp that is worthwhile is: Solymar, L. Lectures on Fourier Series. Fourier Series and Integral Transforms. Another short concise work: Bhatia, Rajendra. Fourier Series. Number theory is one of the oldest and most loved mathematical disciplines and as a result there have been many great books on it.

The serious student will also need to study abstract algebra and in particular group theory. Let me list four superb introductions. These should be accessible to just about anyone. The book by Davenport appears to be out of print, but not long ago it was being published by two publishers.

It might return soon. The second book by Ore gives history without it getting in the way of learning the subject. Ore, Oystein. Invitation to Number Theory. Number Theory and its History. An Adventurer's Guide to Number Theory. Here are five excellent elementary texts that last I knew are still in print. Silverman, Joseph H.

Dudley, Underwood. Elementary Number Theory , 2 nd Ed. Elementary Number Theory and its Applications , 5 th ed. Maybe the text to have. Burton, David M. Elementary Number Theory , 4 th Ed.


He gets into arithmetic functions before he does Euler's generalization of Fermat's Little Theorem. However, many of the proofs are very nice. I like this one quite bit. Like Rosen, the later editions are indeed better. An Introductory Text that has a lot going for it is the one by Stillwell. It has great material but is too fast for most beginners. Should require a course in abstract algebra. Maybe the best second book around on number theory.

Elements of Number Theory. Zuckerman, Hugh L. An Introduction to the Theory of Numbers , 5 th ed. A Concise Introduction to the Theory of Numbers. Number Theory. An Introduction to Number Theory. Lectures on Elementary Number Theory. The Theory of Numbers. Schroeder, M. Second, an improper selection of the excitation strategy may lead the problem unsolvable.

As a simple example to this situation, for a problem involving only the suppression of specific harmonic levels, the selection of a VAS or PS switching scheme makes the problem unsolvable, since the harmonic levels are bounded to each other in these schemes. Hence, the independent suppression of different harmonics is mathematically impossible for this problem using the mentioned time schemes.

Since many problems may be solved via only correct switching strategy, this selection is not crucial and the traditional parameters may be excluded from the problem. However, adding some extra degrees of freedom will relax the solution of the problem and in some specific cases such as main beam steering, inclusion of the excitation phase to the parameter vector is inevitable.

From this point on, the rest of the procedure is the application of the optimizer to the problem.

The key points in the TMA design may be summarized as Defining the problem correctly,. The first study dealing with harmonic suppression via an optimizer has been appeared by Yang et al. In these researches, the pattern of the sidebands has been sampled in a certain precision, and the maximum of these sample set has been taken as the maximum sideband level SBL which has been used as the parameter to be minimized.

The basic idea behind this approach is to lower the maximum level of the infinite harmonics below a certain level usually a certain communication threshold. However, her the main problem is: there exist infinite harmonics in number and the question is how many of them should be considered? For this issue, usually the first M harmonics are taken into account in practice and M depends on the problem at hand.

This idea is not entirely true since there exist situations that the first harmonic is not the highest especially in forced cases e. SBL suppression is a generally applicable generic way to gather information about sidebands; however, calculating sidebands from array factor definition with a certain accuracy is a time- and system resource-consuming process even conducting operation in the first harmonic. Over and above, in azimuthal asymmetric cases such as planar and volumetric situations, the calculation time is enormously increasing exponentially in proportion to the azimuth resolution.

Hence, there exists a solution time-accuracy trade-off in this technique, which always needed to be considered. An illustrative block diagram of this technique may be found in [ 31 ]. In the cost function of an optimizer, this SBL information may be directly added. As an example, for a linear array and a problem involving only sideband suppression, the cost function of the metaheuristic algorithm may be written as.

For this kind cost function, the expected value of the cost of the solution is zero. Extracting harmonic information to use in an optimizer from harmonic pattern samples is a generic way, but it is not the only way to gather information about sidebands. Calculating the power in sidebands and forming a bound function may also be used in sideband suppression problems. These two methods will be discussed in separate sections. In , Bregains et al. Since the infinite series of absolute squared complex Fourier coefficients is convergent which appears in the raw equation, the infinite summation reduces to a specific function involving switch-on durations.

After taking the integration over elevation, the result becomes an elegant closed form equation which gives an asymptotic approximation of the total power in infinite harmonics. After this first paper, Poli et al. The main idea behind these papers is the integration of the array factor in planar case, which may be expressed as ordinary Bessel functions.

One handicap of this equation is that it only holds for the VAS scheme. For a shifted case i. In , this situation is added and combined with original equation by Aksoy and Afacan [ 28 ]. But the equation still did not include volumetric cases, and in , Aksoy published a paper to close this gap [ 29 ]. The key point in volumetric calculations is solving a definite integral over elevation involving an ordinary Bessel function times a complex exponential function.

Hence, by the publication of [ 29 ], the equation has been taken its final and more general form for VAS and PS switched arrays. Here, it must be noted that the studies mentioned so far have been conducted using ideal pulses i. In addition to the studies given above, a pulse model having transition region has been studied by Bekele et al. The summary of the derivation of the general form of power equation from this point on, it will be referred as SR equation is given in the next subsection, and for more information, references given above may be followed.

In this section, the general form of the harmonic power equation will be briefly derived. Before beginning derivation, some remarks should be noted. Since the wave front of a spherically propagating wave may be assumed as planar in Fraunhofer region, this assumption helps to write the array factor in a simple summation consisting of planar wave front oriented phase shifts.

Hence, in the region where the maximum phase error is below a certain tolerable level, the wave may be assumed as planar wave. Under these approximations, assume that N isotropic radiators are randomly oriented in a three-dimensional space. In this general case, the time average power at harmonics may be represented as.

In Eq. Here, the Fourier coefficients depend on the switching strategy. If an ideal PS time scheme consisting of rectangular pulses is considered, the switching function may be modeled as. For this time scheme, the Fourier coefficients may be written as. By using relation for summable series, see [25, Appendix], Eq. For above equation, there exist 53 different situations in terms of equality and inequality conditions of switching instants. If these six equality and inequality possibility of starting and finishing instants of the pulses are considered, these six conditions may be written in one form given by.

For more detailed calculations, [ 28 ] may be followed. By substituting this result into Eq. Solving this integral is not an easy task, and some manipulations should be conducted. It can be started from to write some exponential terms in terms of Bessel functions. To do that, the following fact may be used. By using this conversion, Eq. Multiplying both sides of Eq. If this result is substituted to Eq. Once more, in Eq. For a more detailed proof, [ 29 ] may be followed, and for linear and planar case, see [ 25 ] and [ 27 ], respectively.

The first usage of total power in harmonic suppression problems was conducted by Poli et al. The main idea behind this technique is based on the idea of the total power reduction in harmonics will concentrate the power to main radiation, hence the harmonics become suppressed. Since the result of the equation does not meet the actual power in harmonics, the ratio of the harmonic power to the actual power is more meaningful in practice. Hence, usually the direct result of the Eq. Instead of using the direct result, the ratio of the harmonic power to the actual power is more meaningful which may be written by.

Without loss of generality for a problem concerning only harmonic reduction, the usage of Eq. As shown in [ 30 ] that neither harmonic level reduction nor power reduction ensures the total suppression in terms of both power and communication level. Hence, a combined way may be a more suitable approach if the both level and power reduction is necessary. As mentioned earlier, the SBL method extracts the harmonic level information in an algorithmic way similar to sidelobe calculations.

Since there exists no analytical solution to find a maximum point in an unknown sidelobe region, the sidelobe calculations are being conducted such an operation involving sampling the pattern in sidelobe region and finding its maximum. In contrast, the sideband calculations are being conducted in whole visible region, which makes a difference in both SLL and SBL calculations.

In other words, since the SBL calculations are being operated over a complete elevation and azimuth space, in some cases, the maximum of a pattern may be extracted analytically. Since the maximum points of all individual harmonic patterns can be calculated, they form a maximum sideband level set, and the maximum point of this set bounds the whole harmonic maxima. Hence, this maximum appears as a bound function covering all individual maximum harmonic levels which are infinite in number.