I.C.F. Ipsen and C.D. Meyer, The idea behind krylov subspaces, American Mathematical Monthly 105 (1998), no. 10, 889--899.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....these connections, at demonstrating patterns by generalizable example, at utilizing his summation formula only until it begins to diverge , and at determining the relevant Euler Maclaurin constant in each application of the summation formula. His work also inaugurated study of the zeta function [2, 24]. Euler s accomplishments throughout this entire arena are discussed from di#erent points of view in many modern books [5] 11, pp. 119 136] 12, II.10] 13, ch. XIII] 15, p. 197f] 19, ch. XIV] 26, p. 184, 257 285] 28, p. 338f] Euler included all of these discoveries and others in beautifully ....

R. Ayoub, Euler and the zeta function, American Mathematical Monthly 81 (1974), 1067--1086.

....of the theorems discussed in this paper can be found throughout the literature. Some applications in signal enhancement, restoration and more are addressed in [1] 12] 17 23] The books [24] 25] might also be of interest. A review of several important fixed point theorems can be found in [26]. ....

R. H. Bing. The elusive fixed-point property. The American Mathematical Monthly, 76:119--132, February 1969.

....The basic fixed point theorems are Banach s theorem [1, 2] for contractive mappings, Brouwer s theorem [3 6] for continuous mappings in a finite dimensional space, and Schauder s generalization [4] of Brouwer s theorem to infinite dimensional Banach spaces. Many other results are discussed in [7, 8]. These fixed point theorems are tools of great importance in signal and image reconstruction, tomography, telecommunications, interpolation, extrapolation, signal enhancement, filter design, among many others [9 16] A quick glance through [9] for example, should convince any reader of the ....

R. H. Bing. The elusive fixed-point property. The American Mathematical Monthly, 76:119--132, February 1969.

....2] 2, 2] is a subconstraint of the bounded constraint (#y x = 1, 2, 2] 3 The main tool that we use in the proofs is the following theorem, which is a reformulation of Theorem 5.3.7 of A. Neumaier s book on Interval Methods [25] a parametrized generalization of Miranda s Theorem [19, 22] that ensures the continuity of the parametrized solution space. Miranda s theorem is a generalization of the Boltzmann intermediate value theorem to systems of equations, and it is one of the main tools used in interval analysis [25, 16] Readers who are not interested in proofs can savely skip ....

W. Kulpa. The Poincare-Miranda theorem. The American Mathematical Monthly, 104(6):545--550, 1997.

....conjecture according to which, for any n 0, there is a finite number k of iterations such that f k (n) 1, is obscure. It is traditionally credited to Lothar Collatz in 1932 but it is also known as Gelfond serie. The proof of this conjecture seems to be a really intractable open problem (see [Lag85]) We do not have here any pretense to give a solution to this problem, we simply want to see it as a pathologic case in computer program verification: a very simple program for a very complicated problem. In terms of graphs, the Collatz problem can be seen as a reachability problem on the graph G ....

J. C. Lagarias. The 3x+1 problem and its generalizations. The American Mathematical Monthly, 92(1):3--23, 1985.

....by Parker Brothers (but ArthurMerlin games are something else again) Quite a bit is known about such solitaire games. Anyway, a huge literature has been accumulating on cellular automata. A small example, intersecting with solitaire games, is [Gol91] Pel87] Sto89] Sut88] Sut89] [Sut90], Sut95] Incidentally, related but di erent solitaires are chip ring games , see e.g. BL92] L op97] Big99] What seems more attractive and new is to transform these solitaire games into two player games, where the player rst achieving 0s on all the non leaf vertices wins and the ....

K. Sutner. The -game and cellular automata. American Mathematical Monthly, 97(1):24-34, 1990.

....and the proof is really long. Y. Colin de Verdi ere [13] shows the result, only for triangulated graphs, on arbitrary surfaces of non positive curvature using the Gauss Bonnet formula; the same technique does not seem to extend to 3 connected graphs [12] In the same spirit as Thomassen [34, 35, 36], who gave simple proofs of basic theorems in graph theory, we provide a short, self contained proof of Tutte s theorem. Its attractive points are its shortness and its simplicity (very few graph theory is required, the geometric aspects are emphasized) The proof is transparent and progressive; ....

....vertex. Proof. Suppose that there is a inactive vertex v. Figure 3 summarizes the proof: we show that the planar graph G contains a subdivision of the bipartite graph K 3;3 , which is impossible 6 PSfrag replacements v v Figure 3: A summary of the proof of Lemma 7. see for example [36]) Using the fact that G is 3 connected, we can see the existence, in G( of three distinct active vertices v i ; i = 1; 2; 3, and three paths P i joining v to v i , so that, for any i, the path P i does not contain any vertex v j for j 6= i. Indeed, let w be a vertex of G so that (w) 6= 0. ....

C. Thomassen. The Jordan-Schon ies theorem and the classication of surfaces. American Mathematical Monthly, 99(2):116-130, 1992.

....we confine our attention to this. Figure 1 illustrates the input to the problem. In this paper we look at achieving a faster running time by approximating the shortest path. 1.1 Previous Results The first (mathematical) result on exact shortest paths in a line arrangement is due to C. Davis [3] in 1948. He proves that a shortest path would not travel on certain segments of the arrangement. As mentioned, the best known algorithm for the exact shortest path required (n 2 ) space and time in the worst case. In 1998, Chen et al. 2] found a way to report the shortest path using O(n ....

....contains an exact shortest path of the original arrangement. The simple structure of the quadrilateral grid is the first step to making our approximation algorithm work. The proof uses a theorem, which we call the Sandwich Theorem, proven in part by Eppstein and Hart [5] and in part by Davis [3]. 2 The Algorithm 2.1 Intuition The key ideas of this algorithm are unintuitive, and some of what it does is quite difficult to picture (or to illustrate) despite the fact that we are working with lines in the plane. The algorithm would be quite simple if not for a worst case scenario, where ....

C. Davis. The short-cut problem. American Mathematical Monthly, pp. 147-150, March, 1948.

....an order, is constructive, and as far as I know, every constructive definition is recursive. 8 7 G odel s brief and enigmatic remark (as to a proof of the equivalence) is elucidated in [60] and [61] 8 The quotation continues directly the above quotation from this letter. Church s paper [6] was given on December 30, 1933 to a meeting of the Mathematical Association; incidentally, G odel presented his [27] in the very same session of that meeting. cf. also [31] reviewing [7] STEP BY RECURSIVE STEP: CHURCH S ANALYSIS OF EFFECTIVE CALCULABILITY 159 The last remark is actually ....

....a meeting of the Mathematical Association; incidentally, G odel presented his [27] in the very same session of that meeting. cf. also [31] reviewing [7] STEP BY RECURSIVE STEP: CHURCH S ANALYSIS OF EFFECTIVE CALCULABILITY 159 The last remark is actually reminiscent of part of the discussion in [6], where Church claims that . it appears to be possible that there should be a system of symbolic logic containing a formula to stand for every definable function of positive integers, and I fully believe that such systems exist . p. 358) From the context it is clear that constructive ....

[Article contains additional citation context not shown here]

, The Richard paradox, American Mathematical Monthly, vol. 41 (1934), pp. 356--61.

....of the theorems discussed in this paper can be found throughout the literature. Some applications in signal enhancement, restoration and more are addressed in [1] 12] 17 23] The books [24] 25] might also be of interest. A review of several important fixed point theorems can be found in [26]. ....

R. H. Bing. The elusive fixed-point property. The American Mathematical Monthly, 76:119--132, February 1969.

....The basic fixed point theorems are Banach s theorem [1, 2] for contractive mappings, Brouwer s theorem [3 6] for continuous mappings in a finite dimensional space, and Schauder s generalization [4] of Brouwer s theorem to infinite dimensional Banach spaces. Many other results are discussed in [7, 8]. These fixed point theorems are tools of great importance in signal and image reconstruction, tomography, telecommunications, interpolation, extrapolation, signal enhancement, filter design, among many others [9 16] A quick glance through [9] for example, should convince any reader of the ....

R. H. Bing. The elusive fixed-point property. The American Mathematical Monthly, 76:119--132, February 1969.

....much reduces the usefulness or significance of such a proof. 15 [M9] It seems that Hilbert s intuition had, for once, led him to slightly exaggerated hopes, and there are to day good reasons for doubting the possibility of such [consistency] proofs . 16 See [M4] and [M2] or its translation, [M3]. 17 Is this the first occurrence in history of this odious phrase August 22, 1990 : The Ignorance of Bourbaki : Page 4 As you might by now expect, there is no mention, or even hint in ....

N. Bourbaki, The Architecture of Mathematics. American Mathematical Monthly 57 (1950, 221--232

....style is given by Backus (1978) the main inventor of FORTRAN. A good elementary introduction to CAML Light and functional programming is Mauny (1995) Paulson (1991) is another good textbook, though based on Standard ML. 8 A good survey of this problem, and attempts to solve it, is given by Lagarias (1985). Strictly, we should use unlimited precision integers rather than machine arithmetic. We will see later how to do this. Chapter 3 Further CAML In this chapter, we consolidate the previous examples by specifying the basic facilities of CAML and the syntax of phrases more precisely, and then go ....

Lagarias, J. (1985) The 3x + 1 problem and its generalizations. The American Mathematical Monthly , 92, 3-23. Available on the Web as http://www.cecm.sfu.ca/organics/papers/lagarias/index.html.

....a rewriting to a normal form. In the table, 3 2 and 1 represent the same value, and that can be checked after evaluating the arithmetic expression. The third column is an example of a semantic domain, natural functions, where the equivalence of two elements may not be proved. There is a conjecture [Lag85] that the two functions represented in this column are equivalent and although empirically checked it was not proven and it is possible that no formal proof exists. For comparable languages it can be checked if the semantics of any two sentences are equivalent. Some languages used in CS, whose ....

Je Lagarias. The 3x+1 problem and its generalizations. American Mathematical Monthly, 92:3 - 23, 1985.

No context found.

Gilbert Strang The Fundamental Theorem of Linear Algebra, American Mathematical Monthly, Nov. 1993, p. 848 -- 855.

No context found.

Albert Wilansky, The Row-Sums of the Inverse Matrix, American Mathematical Monthly, volume 58 number 9 (Nov. 1951), p. 614.

No context found.

Hill, T. (1995b) The significant-digit phenomenon. American Mathematical Monthly, 102, 322--327.

....from the search. Appendix B gives a brief history of the search. 1. 1 The 3x 1 Problem The 3x 1 problem concerns iterates of the following function: T (n) 3n 1) 2; if n j 1 (mod 2) n=2; if n j 0 (mod 2) 1:1) which takes odd integers n to (3n 1) 2 and even integers n to n=2 [Lag85, Page 4] The 3x 1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1 [Lag85, Page 3] This conjecture, as Lagarias states, is apparently intractable. The program discussed in this report is not concerned with ....

.... = 3n 1) 2; if n j 1 (mod 2) n=2; if n j 0 (mod 2) 1:1) which takes odd integers n to (3n 1) 2 and even integers n to n=2 [Lag85, Page 4] The 3x 1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1 [Lag85, Page 3] This conjecture, as Lagarias states, is apparently intractable. The program discussed in this report is not concerned with validating or disproving the 3x 1 conjecture. Instead, the program is designed to investigate certain measures related to the iterates of T (and the function H ....

[Article contains additional citation context not shown here]

J. C. Lagarias. The 3x+1 Problem and its Generalizations. The American Mathematical Monthly, 92(1):3--23, January 1985.

No context found.

I.C.F. Ipsen and C.D. Meyer, The idea behind krylov subspaces, American Mathematical Monthly 105 (1998), no. 10, 889--899.

No context found.

, The logic of provability, American Mathematical Monthly, vol. 91 (1984), pp. 470--480.

No context found.

G.-C. Rota. The Number of Partitions of a Set. American Mathematical Monthly, 71(5):498--504, May 1964.

No context found.

Jeffrey Lagarias, The 3x + 1 problem and its generalizations, American Mathematical Monthly, vol 92, no. 1, January 1985, pages 3--23.

No context found.

.1oaclxim Landek. The nathemaicsofsentencc structure. American Mathematical Monthly, 65:15,1-170, 1958.

No context found.

I.F. Ipsen, C. Meyer, The Idea Behind Krylov Methods, American Mathematical Monthly, 105 (1998) pp. 889--899.

No context found.

R. W. Hamming. The unreasonable eectiveness of mathematics. The American Mathematical Monthly, 87(2), February 1980.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC