Showing papers in "American Mathematical Monthly in 1974"
TL;DR: In this article, the authors present an approach for ODE's Phase Plane, Qualitative Methods, and Partial Differential Equations (PDE's) to solve ODE problems.
Abstract: PART A: ORDINARY DIFFERENTIAL EQUATIONS (ODE'S). Chapter 1. First-Order ODE's. Chapter 2. Second Order Linear ODE's. Chapter 3. Higher Order Linear ODE's. Chapter 4. Systems of ODE's Phase Plane, Qualitative Methods. Chapter 5. Series Solutions of ODE's Special Functions. Chapter 6. Laplace Transforms. PART B: LINEAR ALGEBRA, VECTOR CALCULUS. Chapter 7. Linear Algebra: Matrices, Vectors, Determinants: Linear Systems. Chapter 8. Linear Algebra: Matrix Eigenvalue Problems. Chapter 9. Vector Differential Calculus: Grad, Div, Curl. Chapter 10. Vector Integral Calculus: Integral Theorems. PART C: FOURIER ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS. Chapter 11. Fourier Series, Integrals, and Transforms. Chapter 12. Partial Differential Equations (PDE's). Chapter 13. Complex Numbers and Functions. Chapter 14. Complex Integration. Chapter 15. Power Series, Taylor Series. Chapter 16. Laurent Series: Residue Integration. Chapter 17. Conformal Mapping. Chapter 18. Complex Analysis and Potential Theory. PART E: NUMERICAL ANALYSIS SOFTWARE. Chapter 19. Numerics in General. Chapter 20. Numerical Linear Algebra. Chapter 21. Numerics for ODE's and PDE's. PART F: OPTIMIZATION, GRAPHS. Chapter 22. Unconstrained Optimization: Linear Programming. Chapter 23. Graphs, Combinatorial Optimization. PART G: PROBABILITY STATISTICS. Chapter 24. Data Analysis: Probability Theory. Chapter 25. Mathematical Statistics. Appendix 1: References. Appendix 2: Answers to Odd-Numbered Problems. Appendix 3: Auxiliary Material. Appendix 4: Additional Proofs. Appendix 5: Tables. Index.
2,257 citations
271 citations
TL;DR: In this paper, Euler and the Zeta Function were discussed. But they did not consider the Euler-Zeta function. And the Zetal Function was not considered.
Abstract: (1974) Euler and the Zeta Function The American Mathematical Monthly: Vol 81, No 10, pp 1067-1086
201 citations
TL;DR: In this paper, the power mean and logarithmic mean have been compared in the context of the Logarithm of the Power Mean and Power Mean, and the results show that the latter is more accurate than the latter.
Abstract: (1974). The Power Mean and the Logarithmic Mean. The American Mathematical Monthly: Vol. 81, No. 8, pp. 879-883.
171 citations
TL;DR: A personal view of how this subject interacts with Mathematics is given, by discussing the similarities and differences between the two fields, and by examining some of the ways in which they help each other.
Abstract: (1974). Computer Science and its Relation to Mathematics. The American Mathematical Monthly: Vol. 81, No. 4, pp. 323-343.
155 citations
TL;DR: In this article, the optimal velocity in a race is defined as the ratio of the maximum velocity to the minimum velocities of all the runners in the race, with respect to the number of competitors.
Abstract: (1974). Optimal Velocity in a Race. The American Mathematical Monthly: Vol. 81, No. 5, pp. 474-480.
98 citations
TL;DR: In this article, the authors consider the question "Is Mathematical Truth Time-Dependent? The American Mathematical Monthly: Vol. 81, No. 4, pp. 354-365.
Abstract: (1974). Is Mathematical Truth Time-Dependent? The American Mathematical Monthly: Vol. 81, No. 4, pp. 354-365.
98 citations
TL;DR: In this paper, the recurrence relation of (1.2) has been rediscovered, and it has been used to derive formulas in the form of (0.314) for rational real p.
Abstract: giving a way to calculate as many of the B's as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburg's student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a
87 citations
84 citations
TL;DR: In this article, Strong Derivatives and Inverse Mappings are discussed. But they focus on strong derivatives and inverse mappings, and do not consider strong inverse mappings.
Abstract: (1974). Strong Derivatives and Inverse Mappings. The American Mathematical Monthly: Vol. 81, No. 9, pp. 969-980.
68 citations
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TL;DR: In this article, the inner product spaces were studied in the context of inner product space theory and inner productspaces were used to describe the inner products of a set of inner products.
Abstract: (1974). Inner Product Spaces. The American Mathematical Monthly: Vol. 81, No. 1, pp. 29-36.
TL;DR: A Curious Nim-Type Game (CNG) as discussed by the authors is a Nim-type game with a game-theoretic complexity of 0.1. The American Mathematical Monthly: Vol. 81, No. 8, pp. 876-879.
Abstract: (1974). A Curious Nim-Type Game. The American Mathematical Monthly: Vol. 81, No. 8, pp. 876-879.
TL;DR: The authors present programming as a mathematical activity without undertaking the arduous task of supplying a definition of "mathematics" that will please all mathematicians, nor of defining "programming" in a way that is palatable to all programmers.
Abstract: In this article I intend to present programming as a mathematical activity without undertaking the arduous task of supplying a definition of "mathematics" that will please all mathematicians, nor of defining "programming" in a way that is palatable to all programmers. With respect to mathematics I believe, however, that most of us can agree upon the following characteristics of most mathematical work: (1) Compared with other fields of intellectual activity, mathematical assertions tend to be unusually precise. (2) Mathematical assertions tend to be general in the sense that they are applicable to a large (often infinite) class of instances. (3) Mathematics embodies a discipline of reasoning allowing such assertions to be made with an unusually high confidence level.
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TL;DR: The American Mathematical Monthly: Vol. 81, No. 4, No 4, pp. 343-349 as mentioned in this paper, was the first publication of this paper. But it was published in 1974.
Abstract: (1974). Maxwell's Equations. The American Mathematical Monthly: Vol. 81, No. 4, pp. 343-349.
TL;DR: In this paper, another proof that Convex Functions are Locally Lipschitz is given, which is the only known proof that convex functions are locally Lipschiitz functions.
Abstract: (1974). Another Proof that Convex Functions are Locally Lipschitz. The American Mathematical Monthly: Vol. 81, No. 9, pp. 1014-1016.
TL;DR: A bit of experimenting will show that there are other possibilities; point d in Figure 1 is a new type, for instance, and point d disappears as soon as the bottom fold is pulled to the right by any amount whatsoever as mentioned in this paper.
Abstract: A bit of experimenting will show that there are other possibilities; point d in Figure 1 is a new type, for instance. Further experimenting, however, leads to the conclusion that arbitrarily small alterations in the position of the fabric can eliminate any points not of the first three types. The new shape at d disappears, for example, as soon as the bottom fold is pulled to the right by any amount whatsoever (Figure 2).
TL;DR: In this article, the authors present a method for the calculation of functions and roots of polynomials, and the care and treatment of singularities in the context of network problems.
Abstract: Part I. Fundamental Methods: 1. The calculation of functions 2. Roots of transcendental equations 3. Interpolation - and all that 4. Quadrature 5. Ordinary differential equations - initial conditions 6. Ordinary differential equations - boundary conditions 7. Strategy versus tactics - roots of polynomials 8. Eigenvalues I 9. Fourier series Part II. Double Trouble: 10. Evaluation of integrals 11. Power series, continued fractions, and rational approximations 12. Economization of approximations 13. Eigenvalues II - rotational methods 14. Roots of equations - again 15. The care and treatment of singularities 16. Instability in extrapolation 17. Minimum methods 18. Laplace's equation - an overview 19. Network problems.
TL;DR: The American Mathematical Monthly (AMM) Vol. 81, No. 4, pp. 349-354 as mentioned in this paper, was the first publication of this paper. But it was published in 1974.
Abstract: (1974). Everywhere Differentiable, Nowhere Monotone, Functions. The American Mathematical Monthly: Vol. 81, No. 4, pp. 349-354.
TL;DR: An elementary proof of Kolmogorov's theorem is given in this paper, where it is shown that an elementary proof can be found in the standard proof of the theorem.
Abstract: (1974). An Elementary Proof of Kolmogorov's Theorem. The American Mathematical Monthly: Vol. 81, No. 5, pp. 480-486.
TL;DR: In this paper, the authors consider the case where items whose lengths are between l 1 and l 2 are inserted into and withdrawn from a linear store, subject to the restrictions that at no moment...
Abstract: Suppose l ≧ l2 ≧ l1 > 0. Suppose that, at random times, items whose lengths are between l1 and l2 are inserted into and withdrawn from a linear store, subject to the restrictions that at no moment ...
TL;DR: An elementary proof of the Kronecker-Weber Theorem is given in this article, which is the basis for this paper. But it is not a proof of its correctness.
Abstract: (1974). An Elementary Proof of the Kronecker-Weber Theorem. The American Mathematical Monthly: Vol. 81, No. 6, pp. 601-607.
TL;DR: In this paper, the topological properties of manifolds are investigated and the authors propose a topological model for the topology properties of manifolds, which they call Topological Properties of Manifolds.
Abstract: (1974). Topological Properties of Manifolds. The American Mathematical Monthly: Vol. 81, No. 6, pp. 633-636.
TL;DR: In this paper, it was shown that n + 2 points in En with Odd Integral Distances can be seen as a convex combination of n + 1 points and 2 points.
Abstract: (1974). Are there n + 2 Points in En with Odd Integral Distances? The American Mathematical Monthly: Vol. 81, No. 1, pp. 21-25.
TL;DR: In this article, the generalized Pythagorean theorem has been studied in the context of algebraic geometry, and the results show that it is NP-hard to prove. But it is possible.
Abstract: (1974). Generalized Pythagorean Theorem. The American Mathematical Monthly: Vol. 81, No. 3, pp. 262-265.
TL;DR: The American Mathematical Monthly (AMM) Vol. 81, No. 7, pp. 724-738 as discussed by the authors has been used for recursive undecidability.
Abstract: (1974). Recursive Undecidability — An Exposition. The American Mathematical Monthly: Vol. 81, No. 7, pp. 724-738.