Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Size: px
Start display at page:

Download "Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission."

Transcription

1 On the Probability of Covering the Circle by Rom Arcs Author(s): F. W. Huffer L. A. Shepp Source: Journal of Applied Probability, Vol. 24, No. 2 (Jun., 1987), pp Published by: Applied Probability Trust Stable URL: Accessed: 22/11/ :31 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms Conditions of Use, available at JSTOR's Terms Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, students discover, use, build upon a wide range of content in a trusted digital archive. We use information technology tools to increase productivity facilitate new forms of scholarship. For more information about JSTOR, please contact Applied Probability Trust is collaborating with JSTOR to digitize, preserve extend access to Journal of Applied Probability.

2 J. Appl. Prob. 24, (1987) Printed in Israel C Applied Probability Trust 1987 ON THE PROBABILITY OF COVERING THE CIRCLE BY RANDOM ARCS F. W. HUFFER,* Florida State University L. A. SHEPP,** AT& T Bell Laboratories Abstract Arcs of length lk, 0 < lk < 1, k = 1, 2, *, n, are thrown independently uniformly on a circumference W having unit length. Let P(11, 12, * * *,) be the probability that W is completely covered by the n rom arcs. We show that P(I?, 12,*., 14) is a Schur-convex function that it is convex in each argument when the others are held fixed. COVERAGE PROBABILITIES; SCHUR-CONVEX; GEOMETRICAL PROBABILITY 1. Introduction Suppose arcs of length lk, 0 < k <, 1 _k _ n, are thrown independently uniformly on a circumference W having unit length. Let P(1l, 12, ***, n) be the probability that W is completely covered by the n rom arcs. Stevens (1939) has explicitly evaluated P(11, 12, **, I) in the case 11 = 12 =. = I4. It seems hopeless to give a simple formula for the case of general arc lengths. In this paper we examine qualitative properties of the function P(l, 12, *, In). In particular, we show that (1.1) P(1l, 12,.., ) is Schur-convex, (1.2) P(ll, 12, *, In) is convex in each argument (keeping the others fixed). These results are proved in Sections 2 3. There is an extensive literature concerned with rom arcs coverage problems. Much of this literature deals exclusively with the case of equal arc Received 16 July 1985; revision received 18 February * Postal address: Department of Statistics, Florida State University, Tallahassee, FL , USA. Research supported by Office of Naval Research Contract N C ** Postal address: AT & T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA. 422

3 On the probability of covering the circle by rom arcs 423 lengths. A few of the papers concerned with coverage of the circle by arcs of different lengths are Shepp (1972), Siegel Hoist (1982), Janson (1983) Huffer (1986). Shepp works with infinitely many arcs having fixed lengths given by some sequence {l,}. In the last three papers, the arc lengths are taken to be independent rom variables. A survey of some of the coverage literature may be found in Solomon (1978). Our main result is the Schur-convexity of P which was proposed as a conjecture by Frank Proschan. Let {xi, }) {y i},.. be collections of arc lengths with xl > x2 *** > xn Yl > Y2 * * *? y Yn. Schur-convexity says the following: If then i-i k k x _-< Yi for 1l k_ n-1 i-i i-i n y i = P(x,, x2,.., Xn) n i-i Y P(yl, Y2,' *', Yn)- Loosely speaking, if xl, x2, * *, xn are 'less spread out' or 'more nearly equal' than are Y1, Y2, *, Yn, then P(x1, x2,..*, X) ' P(1, y2,., Yn). Schur-convexity allows us to compute explicit bounds in some cases. For instance, P(l, 12,, In) > P(l,, *,) where I = n- ~ 1i Stevens' formula gives P(I, 1,, )= z.. (-l) k(n)(1-kl) kn See Marshall Olkin (1979) for further information on Schur-convexity. One can also consider covering the circle by n romly placed sets (not necessarily arcs). A result due to Wschebor (1973) states roughly that among sets with measures 11, 12, ***,,, the coverage probability is largest when all the sets are arcs (intervals). Our result (1.1) shows that among arcs with ?? + In = c, greater inequality in the lengths leads to a greater probability of coverage. 2. Proof of Schur-convexity Assume without loss of generality that To demonstrate Schurconvexity it suffices to show that

4 424 F. W. HUFFER AND L. A. SHEPP (2.1) P(,l, 1213,, l,n) P(I, + e, 12 - e, 13,., In) for all sufficiently small e > 0 (see A.2.b on p. 55 of Marshall Olkin (1979)). An outline of the proof follows. To prove (2.1) we must compare two situations: one in which the first two arcs have lengths 11, 12 the other in which these arcs have lengths 1, + e, 12 - e. To do this we construct these two situations on the same probability space then condition on the positions of the first two arcs. Having fixed the first two arcs, we examine the conditional probability of covering the remaining part of the circle by the remaining n - 2 arcs with lengths 13,?., In. It is easy to hle the case when the first two arcs overlap. When the first two arcs are disjoint, a more complicated proof is needed. For this case we use a reflection argumento express the coverage probability inequality in the form given in (2.5). A further conditional argumenthen reduces this to a consequence of the lemma given at the close of this section. For notational convenience, we parameterize the circle by using the real line modulo 1. A real number x corresponds to a unique point on ' which we denote by [x]. [x] = [y] if only if x - y is an integer. An interval (a, b) on the line corresponds to a unique arc on W which we denote by [(a, b)]. Let Xi, X2,... X, be independent uniformly distributed on (0, 1). Define l = [(X1, XI + 14)], -2 = [(X2, X2 + 12)], S; = [(XI, Xl + 1l + E)], 2 = [(X2 + E, X2 + 12)] n e = U i-3 [(Xi,x,+,)]. Formula (2.1) is equivalento (2.2) P{ C Sl U d2 U J} <P{ 6" C ' U a2 JUf}. To prove (2.2) we show that (2.3) P{ c,1 u a2 U I X,i, X,) } P{' c.\ U 2 U 1 Xl, x2} almost surely. There are three cases to consider: (i) [X2] E 1,, (ii) [X2]d1l [X2 + 12] E, (iii) si, r n 2 = 0 In case (i) we have Si U ds2 C d U 2. In case (ii) both il U d2 d; U 2 are arcs on ' the length of 1, U d2 is less than or equal to the

5 On the probability of covering the circle by rom arcs 425 length of d/[ U.2/. The reader can easily verify these facts. Thus (2.3) holds in both cases (i) (ii). The remainder of the proof deals with case (iii) so that in the following we take XI X2 to be fixed (non-rom) values such that /i, n r42 0. This implies./1 n 2/ = 0. Let V V' denote the complements of li U s2./4 U.2 respectively. We can now rewrite (2.3) as (2.4) Pt V c X} _ P{ V' C X}. V consists of two disjoint arcs separated by a distance of 12. V' consists of disjoint arcs having the same lengths but situated closer together (separated by 12 - e). Let A, B, C C*be the disjoint arcs given in Figure 1. Both C C* have length e. With this notation V =A U B U C V' =-A U B U C*. Let A* be the reflection of A about the diagonal through the midpoint of B. Noting that the reflection of C is C* the reflection of B is B, symmetry yields so that (2.4) is equivalento Since P{A u B U C c } = P{A* u B U Cc X} P{A U B U Cc Y}) P{A* U B U Cc Y}. P{A u B u C c =P{A UB u B C c c-pa U,,C t} by symmetry P{A U B C XW} = P{A* U B C Jf}, this may be rewritten as P{A U B c Jr, cg r} > P{A* u B c r, C4 0} B Figure 1

6 426 F. W. HUFFER AND L. A. SHEPP or equivalently (2.5) P{A c li B /c, Ct } > P{A* CY lb c, C e1 }. In proving (2.5), in addition to conditioning on the event {B c ar, C?t Xj}, we shall assume we know which of the rom arcs intersect B the exact position of these arcs. More precisely, we are given (B C XW, C? J') the value Xi for all i such that [(Xi, Xi + 1/)] intersects B. For convenience we use y. to denote this conditioning. To prove (2.5) complete the proof of our tlt/orem it suffices to show that (2.6) P{A c JF} >- P{A* c }. Let D be the union of all the rom arcs [(X,, Xi + 1,)] which intersect B. Given F, we know that D is an arc containing B that one of its endpoints (call it Y) must lie in C. We can ignore the case where D is very large has both endpoints in Cbecause in that case both conditional probabilities in (2.6) are equal to 1. At this point we shall assume that e (which is the length of C) is less than the minimum of 13, 14,, **,. Let ( be the collection of those rom arcs which do not intersect B, that is, = {i: i 3 B n [(,, Xi + l)] = 0}. Given 3F, the arcs in i cannot intersect D n C, for this would lead to complete coverage of C. Thus, given 3F, the arcs in r are uniformly independently distributed inside the arc extending from Y to Z (where Z is the endpoint of B on the side not adjoining C) which we denote by E. Various items introduced in the previous paragraphs are pictured in Figure 2. The diagram depicts A A* as being disjoint, but this will not always be the case. B Figure 2

7 On the probability of covering the circle by rom arcs 427 Define A,=A -D, A*=A*-D Then P{A C O = U [(Xi, + )]. iec J.y} = P{Ar C,r} P{A* C t} j } = P{A*C Jr}. If D intersects A*, then A* c A* so that (2.6) holds. Now consider the case when D A* are disjoint so that A* = A*. Because A A* have the same length A is closer than A* to the midpoint of E, the following lemma shows that P{A c Y} ) P{A* C Y,r Since Ar A ra we w} have P C P(A C atr} so that again (2.6) holds. This completes the proof. It remains to present prove the lemma needed above. This will be done independently of the previous material. Lemma. Romly place k intervals of length 01, 02,' *, Ok inside the interval (0, L). The left endpoint of the interval of length Oi is uniformly distributed on (0, L - 60). Fix a value b in (0, L). For 0 _ 2 L - b define Q(A) to be the probability that (A, i + b) is completely covered by the k rom intervals. Q(A) is non-decreasing for 0 _ < (L - b)/2. That is, for an interval of fixed length, the probability of complete coverage increases as the interval is moved toward the center of (0, L). Proof. The k rom intervals will be denoted I, I2, *., Ik. Let B= (A, i + b) with A <(L-b)/2. Choose e small enough so that ; + e (L - b)/2. Instead of comparing the coverage probabilities of the intervals B = (-A, + b) (A + e, A + e + b), we shall consider only B transform the intervals II, I2,., Ik uniformly distributed in (0, L) into intervals I*, I2*,. *, Ik uniformly distributed in (- e, L - e). The intervals I* will be constructed so that (2.7) if Bc U Ii, thenb C U I*. i-= i=l k k

8 428 F. W. HUFFER AND L. A. SHEPP The lemma follows immediately from (2.7). Before describing the construction we need some preliminaries. Let C= U {Ii: Ii?(0, b + 2A))}. Let 9 be the event that B n Cis non-empty there exists j such that B n C c Ij Ij n (L -, L) # 0. Note that (O, b + 2l) (L - e, L) are disjoint. We use R to denote the operation of reflection about the midpoint of B. The interval B is centered inside (0, b + 2A) so that if Ii C (0, b + 2A) then also R(I,) c (0, b + 2A). An interval Ii = (c, d) is transformed into J* in (- e, L - e) according to this rule: if d >L -e, then = (d - L, d - L +(d - c)); if Ii c (0, b + 2A)! occurs, then i = R(Ii); otherwise, take It = Ii. To verify that this construction works, condition on the exact positions of those intervals Ii for which I,i(0, b + 2)). Conditionally, the remaining rom intervals will be independently uniformly distributed inside (0, b + 22). The reader is left to complete the argument. A crucial observation is that p(b n I.) - (B n IP) for all i where # denotes Lebesgue measure. 3. Convexity in each argument We now prove (1.2). Let Xl, X2,, X, be independent uniformly distributed on (0, 1). Choose y z satisfying 0 <y < z <y + z < 1. Define, = [(X1, X X2 +)], [(X + Y, X1 + Y + Z)], 3= [(X, + y, x, + )], 4 = [(X, X1 + y + Z)], n W= U i-2 [(, x, + )]. Here we use the bracket notation described early in Section 2. For 1 _ i _ 4 define J, to be the indicator of the event { C Ri U C). Because X, n 92 = a3 Xl U - S2= 4, it is easily seen that Taking expected values leads to J1 + J2 ' J3 + J4. P(z, 12,'", In) _-< :?[P(z - y, 12,, * n) + P(z + y, 12, * *, In)] This suffices to show that P(z, 12, * *, In) is convex in z when 12, ***, 1, are held fixed. The argument above is essentially that used by Huffer (1987).

9 On the probability of covering the circle by rom arcs 429 Acknowledgements The authors wish to thank Frank Proschan for conjecturing (1.1), Andrew Odlyzko for remarks which were helpful in proving (1.1), Doug Critchlow for a suggestion which improved the proof of the lemma. References HUFFER, F. W. (1987) Inequalities for the M/G/oo queue related shot noise processes. J. Appl. Prob. 24(4). HUFFER, F. W. (1986) Variability orderings related to coverage problems on the circle. J. Appl. Prob. 23, JANSON, S. (1983) Rom coverings of the circle with arcs of rom lengths. In Probability Mathematical Statistics: Essays in Honour of Carl-Gustav Esseen, ed. A. Gut L. Holst. Department of Mathematics, Uppsala University, Sweden. MARSHALL, A. W. AND OLKIN, I. (1979) Inequalities: Theory of Majorization Its Applications. Academic Press, New York. SHEPP, L. A. (1972) Covering the circle with rom arcs. Israel J. Math. 11, SIEGEL, A. F. AND HOLST, L. (1982) Covering the circle with rom arcs of rom sizes. J. Appl. Prob. 19, SOLOMON, H. (1978) Geometric Probability. SIAM, Philadelphia, PA. STEVENS, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics 9, WSCHEBOR, M. (1973) Sur le recouvrement du cercle par des ensembles places au hasard. Israel J. Math. 15, 1-11.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. On the Bound for a Pair of Consecutive Quartic Residues of a Prime Author(s): R. G. Bierstedt and W. H. Mills Source: Proceedings of the American Mathematical Society, Vol. 14, No. 4 (Aug., 1963), pp.

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Uncountably Many Inequivalent Analytic Actions of a Compact Group on Rn Author(s): R. S. Palais and R. W. Richardson, Jr. Source: Proceedings of the American Mathematical Society, Vol. 14, No. 3 (Jun.,

More information

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Biometrika Trust An Improved Bonferroni Procedure for Multiple Tests of Significance Author(s): R. J. Simes Source: Biometrika, Vol. 73, No. 3 (Dec., 1986), pp. 751-754 Published by: Biometrika Trust Stable

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Merging of Opinions with Increasing Information Author(s): David Blackwell and Lester Dubins Source: The Annals of Mathematical Statistics, Vol. 33, No. 3 (Sep., 1962), pp. 882-886 Published by: Institute

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. On Runs of Residues Author(s): D. H. Lehmer and Emma Lehmer Source: Proceedings of the American Mathematical Society, Vol. 13, No. 1 (Feb., 1962), pp. 102-106 Published by: American Mathematical Society

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Indices of Torsion-Free Subgroups of Fuchsian Groups Author(s): R. G. Burns and Donald Solitar Source: Proceedings of the American Mathematical Society, Vol. 89, No. 3 (Nov., 1983), pp. 414-418 Published

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Modalities in Ackermann's "Rigorous Implication" Author(s): Alan Ross Anderson and Nuel D. Belnap, Jr. Source: The Journal of Symbolic Logic, Vol. 24, No. 2 (Jun., 1959), pp. 107-111 Published by: Association

More information

Journal of Applied Probability, Vol. 13, No. 3. (Sep., 1976), pp

Journal of Applied Probability, Vol. 13, No. 3. (Sep., 1976), pp Buffon's Problem with a Long Needle Persi Diaconis Journal of Applied Probability, Vol. 13, No. 3. (Sep., 1976), pp. 614-618. Stable URL: http://links.jstor.org/sici?sici=0021-9002%28197609%2913%3a3%3c614%3abpwaln%3e2.0.co%3b2-p

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. 6625 Author(s): Nicholas Strauss, Jeffrey Shallit, Don Zagier Source: The American Mathematical Monthly, Vol. 99, No. 1 (Jan., 1992), pp. 66-69 Published by: Mathematical Association of America Stable

More information

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Biometrika Trust A Stagewise Rejective Multiple Test Procedure Based on a Modified Bonferroni Test Author(s): G. Hommel Source: Biometrika, Vol. 75, No. 2 (Jun., 1988), pp. 383-386 Published by: Biometrika

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Variance of the Product of Random Variables Author(s): Leo A. Goodman Source: Journal of the American Statistical Association, Vol. 57, No. 297 (Mar., 1962), pp. 54-60 Published by: American Statistical

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. A Generic Property of the Bounded Syzygy Solutions Author(s): Florin N. Diacu Source: Proceedings of the American Mathematical Society, Vol. 116, No. 3 (Nov., 1992), pp. 809-812 Published by: American

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Selecting the Better Bernoulli Treatment Using a Matched Samples Design Author(s): Ajit C. Tamhane Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 42, No. 1 (1980), pp.

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. There are no $Q$-Points in Laver's Model for the Borel Conjecture Author(s): Arnold W. Miller Source: Proceedings of the American Mathematical Society, Vol. 78, No. 1 (Jan., 1980), pp. 103-106 Published

More information

The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp

The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp A Visual Explanation of Jensen's Inequality Tristan Needham The American Mathematical Monthly, Vol. 100, No. 8. (Oct., 1993), pp. 768-771. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28199310%29100%3a8%3c768%3aaveoji%3e2.0.co%3b2-8

More information

ON THE EVOLUTION OF ISLANDS

ON THE EVOLUTION OF ISLANDS ISRAEL JOURNAL OF MATHEMATICS, Vol. 67, No. 1, 1989 ON THE EVOLUTION OF ISLANDS BY PETER G. DOYLE, Lb COLIN MALLOWS,* ALON ORLITSKY* AND LARRY SHEPP t MT&T Bell Laboratories, Murray Hill, NJ 07974, USA;

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Fermat's Little Theorem: Proofs That Fermat Might Have Used Author(s): Bob Burn Source: The Mathematical Gazette, Vol. 86, No. 507 (Nov., 2002), pp. 415-422 Published by: The Mathematical Association Stable

More information

The College Mathematics Journal, Vol. 16, No. 2. (Mar., 1985), pp

The College Mathematics Journal, Vol. 16, No. 2. (Mar., 1985), pp On Rearrangements of the Alternating Harmonic Series Fon Brown; L. O. Cannon; Joe Elich; David G. Wright The College Mathematics Journal, Vol. 16, No. 2. (Mar., 1985), pp. 135-138. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28198503%2916%3a2%3c135%3aorotah%3e2.0.co%3b2-q

More information

The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.

The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. On the Optimal Character of the (s, S) Policy in Inventory Theory Author(s): A. Dvoretzky, J. Kiefer, J. Wolfowitz Reviewed work(s): Source: Econometrica, Vol. 21, No. 4 (Oct., 1953), pp. 586-596 Published

More information

Variance of Lipschitz Functions and an Isoperimetric Problem for a Class of Product Measures

Variance of Lipschitz Functions and an Isoperimetric Problem for a Class of Product Measures Variance of Lipschitz Functions and an Isoperimetric Problem for a Class of Product Measures Sergei G. Bobkov; Christian Houdré Bernoulli, Vol. 2, No. 3. (Sep., 1996), pp. 249-255. Stable URL: http://links.jstor.org/sici?sici=1350-7265%28199609%292%3a3%3c249%3avolfaa%3e2.0.co%3b2-i

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Measuring the Speed and Altitude of an Aircraft Using Similar Triangles Author(s): Hassan Sedaghat Source: SIAM Review, Vol. 33, No. 4 (Dec., 1991), pp. 650-654 Published by: Society for Industrial and

More information

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Management Science.

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Management Science. On the Translocation of Masses Author(s): L. Kantorovitch Source: Management Science, Vol. 5, No. 1 (Oct., 1958), pp. 1-4 Published by: INFORMS Stable URL: http://www.jstor.org/stable/2626967. Accessed:

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at American Society for Quality A Note on the Graphical Analysis of Multidimensional Contingency Tables Author(s): D. R. Cox and Elizabeth Lauh Source: Technometrics, Vol. 9, No. 3 (Aug., 1967), pp. 481-488

More information

Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp

Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp Abscissas and Weight Coefficients for Lobatto Quadrature H. H. Michels Mathematics of Computation, Vol. 17, No. 83. (Jul., 1963), pp. 237-244. Stable URL: http://links.jstor.org/sici?sici=0025-5718%28196307%2917%3a83%3c237%3aaawcfl%3e2.0.co%3b2-7

More information

The College Mathematics Journal, Vol. 21, No. 4. (Sep., 1990), pp

The College Mathematics Journal, Vol. 21, No. 4. (Sep., 1990), pp Tabular Integration by Parts David Horowitz The College Mathematics Journal, Vol. 21, No. 4. (Sep., 1990), pp. 307-311. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28199009%2921%3a4%3c307%3atibp%3e2.0.co%3b2-o

More information

The American Mathematical Monthly, Vol. 81, No. 4. (Apr., 1974), pp

The American Mathematical Monthly, Vol. 81, No. 4. (Apr., 1974), pp Everywhere Differentiable, Nowhere Monotone, Functions Y. Katznelson; Karl Stromberg The American Mathematical Monthly, Vol. 81, No. 4. (Apr., 1974), pp. 349-354. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28197404%2981%3a4%3c349%3aednmf%3e2.0.co%3b2-j

More information

Journal of the American Mathematical Society, Vol. 2, No. 2. (Apr., 1989), pp

Journal of the American Mathematical Society, Vol. 2, No. 2. (Apr., 1989), pp An Estimate for Character Sums Nicholas M. Katz Journal of the American Mathematical Society, Vol. 2, No. 2. (Apr., 1989), pp. 197-200. Stable URL: http://links.jstor.org/sici?sici=0894-0347%28198904%292%3a2%3c197%3aaefcs%3e2.0.co%3b2-l

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust Robust Regression via Discriminant Analysis Author(s): A. C. Atkinson and D. R. Cox Source: Biometrika, Vol. 64, No. 1 (Apr., 1977), pp. 15-19 Published by: Oxford University Press on

More information

The College Mathematics Journal, Vol. 24, No. 4. (Sep., 1993), pp

The College Mathematics Journal, Vol. 24, No. 4. (Sep., 1993), pp Taylor Polynomial Approximations in Polar Coordinates Sheldon P. Gordon The College Mathematics Journal, Vol. 24, No. 4. (Sep., 1993), pp. 325-330. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28199309%2924%3a4%3c325%3atpaipc%3e2.0.co%3b2-m

More information

Mind Association. Oxford University Press and Mind Association are collaborating with JSTOR to digitize, preserve and extend access to Mind.

Mind Association. Oxford University Press and Mind Association are collaborating with JSTOR to digitize, preserve and extend access to Mind. Mind Association Response to Colyvan Author(s): Joseph Melia Source: Mind, New Series, Vol. 111, No. 441 (Jan., 2002), pp. 75-79 Published by: Oxford University Press on behalf of the Mind Association

More information

Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds

Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds Guozhen Lu Proceedings of the American Mathematical Society, Vol. 117, No. 4. (Apr., 1993), pp. 971-978. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28199304%29117%3a4%3c971%3aclaaat%3e2.0.co%3b2-t

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Regression Analysis when there is Prior Information about Supplementary Variables Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 22, No. 1 (1960),

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Borsuk-Ulam Theorem and Bisection of Necklaces Author(s): Noga Alon and Douglas B. West Source: Proceedings of the American Mathematical Society, Vol. 98, No. 4 (Dec., 1986), pp. 623-628 Published

More information

E. DROR, W. G. DWYER AND D. M. KAN

E. DROR, W. G. DWYER AND D. M. KAN Self Homotopy Equivalences of Postnikov Conjugates Author(s): E. Dror, W. G. Dwyer, D. M. Kan Reviewed work(s): Source: Proceedings of the American Mathematical Society, Vol. 74, No. 1 (Apr., 1979), pp.

More information

Annals of Mathematics

Annals of Mathematics Annals of Mathematics The Clifford-Klein Space Forms of Indefinite Metric Author(s): Joseph A. Wolf Reviewed work(s): Source: The Annals of Mathematics, Second Series, Vol. 75, No. 1 (Jan., 1962), pp.

More information

American Journal of Mathematics, Vol. 109, No. 1. (Feb., 1987), pp

American Journal of Mathematics, Vol. 109, No. 1. (Feb., 1987), pp A Simple Algorithm for Cyclic Vectors Nicholas M. Katz American Journal of Mathematics, Vol. 109, No. 1. (Feb., 1987), pp. 65-70. Stable URL: http://links.jstor.org/sici?sici=0002-9327%28198702%29109%3a1%3c65%3aasafcv%3e2.0.co%3b2-a

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at A Note on the Efficiency of Least-Squares Estimates Author(s): D. R. Cox and D. V. Hinkley Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 30, No. 2 (1968), pp. 284-289

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust Some Simple Approximate Tests for Poisson Variates Author(s): D. R. Cox Source: Biometrika, Vol. 40, No. 3/4 (Dec., 1953), pp. 354-360 Published by: Oxford University Press on behalf of

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Early History of the Ham Sandwich Theorem Author(s): W. A. Beyer and Andrew Zardecki Source: The American Mathematical Monthly, Vol. 111, No. 1 (Jan., 2004), pp. 58-61 Published by: Mathematical Association

More information

A Note on a Method for the Analysis of Significances en masse. Paul Seeger. Technometrics, Vol. 10, No. 3. (Aug., 1968), pp

A Note on a Method for the Analysis of Significances en masse. Paul Seeger. Technometrics, Vol. 10, No. 3. (Aug., 1968), pp A Note on a Method for the Analysis of Significances en masse Paul Seeger Technometrics, Vol. 10, No. 3. (Aug., 1968), pp. 586-593. Stable URL: http://links.jstor.org/sici?sici=0040-1706%28196808%2910%3a3%3c586%3aanoamf%3e2.0.co%3b2-j

More information

Some Results Concerning Uniqueness of Triangle Sequences

Some Results Concerning Uniqueness of Triangle Sequences Some Results Concerning Uniqueness of Triangle Sequences T. Cheslack-Postava A. Diesl M. Lepinski A. Schuyler August 12 1999 Abstract In this paper we will begin by reviewing the triangle iteration. We

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Problem set 1, Real Analysis I, Spring, 2015.

Problem set 1, Real Analysis I, Spring, 2015. Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n

More information

Proceedings of the American Mathematical Society, Vol. 17, No. 5. (Oct., 1966), pp

Proceedings of the American Mathematical Society, Vol. 17, No. 5. (Oct., 1966), pp On the Existence and Uniqueness of the Real Logarithm of a Matrix Walter J. Culver Proceedings of the American Mathematical Society, Vol. 17, No. 5. (Oct., 1966), pp. 1146-1151. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28196610%2917%3a5%3c1146%3aoteauo%3e2.0.co%3b2-b

More information

Mathematics of Operations Research, Vol. 2, No. 2. (May, 1977), pp

Mathematics of Operations Research, Vol. 2, No. 2. (May, 1977), pp New Finite Pivoting Rules for the Simplex Method Robert G. Bland Mathematics of Operations Research, Vol. 2, No. 2. (May, 1977), pp. 103-107. Stable URL: http://links.jstor.org/sici?sici=0364-765x%28197705%292%3a2%3c103%3anfprft%3e2.0.co%3b2-t

More information

COMPLETION OF PARTIAL LATIN SQUARES

COMPLETION OF PARTIAL LATIN SQUARES COMPLETION OF PARTIAL LATIN SQUARES Benjamin Andrew Burton Honours Thesis Department of Mathematics The University of Queensland Supervisor: Dr Diane Donovan Submitted in 1996 Author s archive version

More information

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Biometrika Trust Discrete Sequential Boundaries for Clinical Trials Author(s): K. K. Gordon Lan and David L. DeMets Reviewed work(s): Source: Biometrika, Vol. 70, No. 3 (Dec., 1983), pp. 659-663 Published

More information

CLASSES OF DIOPHANTINE EQUATIONS WHOSE POSITIVE INTEGRAL SOLUTIONS ARE BOUNDED*

CLASSES OF DIOPHANTINE EQUATIONS WHOSE POSITIVE INTEGRAL SOLUTIONS ARE BOUNDED* 1929.1 DIOPHANTINE EQUATIONS 859 CLASSES OF DIOPHANTINE EQUATIONS WHOSE POSITIVE INTEGRAL SOLUTIONS ARE BOUNDED* BY D. R. CURTISS 1. Introduction. If upper bounds have been found for the positive integral

More information

Philosophy of Science Association

Philosophy of Science Association Philosophy of Science Association Why Bohm's Theory Solves the Measurement Problem Author(s): Tim Maudlin Source: Philosophy of Science, Vol. 62, No. 3 (Sep., 1995), pp. 479-483 Published by: The University

More information

Proceedings of the American Mathematical Society, Vol. 57, No. 2. (Jun., 1976), pp

Proceedings of the American Mathematical Society, Vol. 57, No. 2. (Jun., 1976), pp Elliptic Curves and Dedekind Domains Michael Rosen Proceedings of the American Mathematical Society, Vol. 57, No. 2. (Jun., 1976), pp. 197-201. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28197606%2957%3a2%3c197%3aecadd%3e2.0.co%3b2-7

More information

Some inequalities for sum and product of positive semide nite matrices

Some inequalities for sum and product of positive semide nite matrices Linear Algebra and its Applications 293 (1999) 39±49 www.elsevier.com/locate/laa Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b,

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. On the Hahn-Banach Extension Property Author(s): Jonathan M. Borwein Source: Proceedings of the American Mathematical Society, Vol. 86, No. 1, (Sep., 1982), pp. 42-46 Published by: American Mathematical

More information

Bichain graphs: geometric model and universal graphs

Bichain graphs: geometric model and universal graphs Bichain graphs: geometric model and universal graphs Robert Brignall a,1, Vadim V. Lozin b,, Juraj Stacho b, a Department of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, United

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at On the Estimation of the Intensity Function of a Stationary Point Process Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 2 (1965), pp. 332-337

More information

On completing partial Latin squares with two filled rows and at least two filled columns

On completing partial Latin squares with two filled rows and at least two filled columns AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 68(2) (2017), Pages 186 201 On completing partial Latin squares with two filled rows and at least two filled columns Jaromy Kuhl Donald McGinn Department of

More information

Annals of Mathematics

Annals of Mathematics Annals of Mathematics Asymptotically Neutral Distributions of Electrons and Polynomial Approximation Author(s): Jacob Korevaar Source: The Annals of Mathematics, Second Series, Vol. 80, No. 3 (Nov., 1964),

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. First Passage Time for a Particular Gaussian Process Author(s): L. A. Shepp Source: The Annals of Mathematical Statistics, Vol. 42, No. 3 (Jun., 1971), pp. 946-951 Published by: Institute of Mathematical

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Classification of 1-Manifolds: A Take-Home Exam Author(s): David Gale Source: The American Mathematical Monthly, Vol. 94, No. 2 (Feb., 1987), pp. 170-175 Published by: Mathematical Association of America

More information

An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions

An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions An Improved Approximate Formula for Calculating Sample Sizes for Comparing Two Binomial Distributions J. T. Casagrande; M. C. Pike; P. G. Smith Biometrics, Vol. 34, No. 3. (Sep., 1978), pp. 483-486. Stable

More information

1. To be a grandfather. Objects of our consideration are people; a person a is associated with a person b if a is a grandfather of b.

1. To be a grandfather. Objects of our consideration are people; a person a is associated with a person b if a is a grandfather of b. 20 [161016-1020 ] 3.3 Binary relations In mathematics, as in everyday situations, we often speak about a relationship between objects, which means an idea of two objects being related or associated one

More information

A COMPOUND POISSON APPROXIMATION INEQUALITY

A COMPOUND POISSON APPROXIMATION INEQUALITY J. Appl. Prob. 43, 282 288 (2006) Printed in Israel Applied Probability Trust 2006 A COMPOUND POISSON APPROXIMATION INEQUALITY EROL A. PEKÖZ, Boston University Abstract We give conditions under which the

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Queues with Time-Dependent Arrival Rates: II. The Maximum Queue and the Return to Equilibrium Author(s): G. F. Newell Source: Journal of Applied Probability, Vol. 5, No. 3 (Dec., 1968), pp. 579-590 Published

More information

TRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS

TRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS Math. J. Okayama Univ. 53 (2011), 185 216 TRIANGLE CENTERS DEFINED BY QUADRATIC POLYNOMIALS Yoshio Agaoka Abstract. We consider a family of triangle centers whose barycentric coordinates are given by quadratic

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. A Consistency Result on Thin-Tall Superatomic Boolean Algebras Author(s): Juan Carlos Martínez Source: Proceedings of the American Mathematical Society, Vol. 115, No. 2 (Jun., 1992), pp. 473-477 Published

More information

Detection of Influential Observation in Linear Regression. R. Dennis Cook. Technometrics, Vol. 19, No. 1. (Feb., 1977), pp

Detection of Influential Observation in Linear Regression. R. Dennis Cook. Technometrics, Vol. 19, No. 1. (Feb., 1977), pp Detection of Influential Observation in Linear Regression R. Dennis Cook Technometrics, Vol. 19, No. 1. (Feb., 1977), pp. 15-18. Stable URL: http://links.jstor.org/sici?sici=0040-1706%28197702%2919%3a1%3c15%3adoioil%3e2.0.co%3b2-8

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Equilibrium Points of Bimatrix Games Author(s): C. E. Lemke and J. T. Howson, Jr. Source: Journal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 2 (Jun., 1964), pp. 413-423 Published

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at A Simple Non-Desarguesian Plane Geometry Author(s): Forest Ray Moulton Source: Transactions of the American Mathematical Society, Vol. 3, No. 2 (Apr., 1902), pp. 192-195 Published by: American Mathematical

More information

Topological properties of Z p and Q p and Euclidean models

Topological properties of Z p and Q p and Euclidean models Topological properties of Z p and Q p and Euclidean models Samuel Trautwein, Esther Röder, Giorgio Barozzi November 3, 20 Topology of Q p vs Topology of R Both R and Q p are normed fields and complete

More information

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. A Proof of Weierstrass's Theorem Author(s): Dunham Jackson Reviewed work(s): Source: The American Mathematical Monthly, Vol. 41, No. 5 (May, 1934), pp. 309-312 Published by: Mathematical Association of

More information

Philosophy of Science, Vol. 43, No. 4. (Dec., 1976), pp

Philosophy of Science, Vol. 43, No. 4. (Dec., 1976), pp Many Worlds Are Better than None Stanley Kerr Philosophy of Science, Vol. 43, No. 4. (Dec., 1976), pp. 578-582. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28197612%2943%3a4%3c578%3amwabtn%3e2.0.co%3b2-c

More information

The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica.

The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision Author(s): Kenneth O. May Source: Econometrica, Vol. 20, No. 4 (Oct., 1952), pp. 680-684 Published by: The Econometric

More information

Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp

Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp Approximate Identities and H1(R) Akihito Uchiyama; J. Michael Wilson Proceedings of the American Mathematical Society, Vol. 88, No. 1. (May, 1983), pp. 53-58. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28198305%2988%3a1%3c53%3aaia%3e2.0.co%3b2-4

More information

On Kusuoka Representation of Law Invariant Risk Measures

On Kusuoka Representation of Law Invariant Risk Measures MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of

More information

A CONVEXITY CONDITION IN BANACH SPACES AND THE STRONG LAW OF LARGE NUMBERS1

A CONVEXITY CONDITION IN BANACH SPACES AND THE STRONG LAW OF LARGE NUMBERS1 A CONVEXITY CONDITION IN BANACH SPACES AND THE STRONG LAW OF LARGE NUMBERS1 ANATOLE BECK Introduction. The strong law of large numbers can be shown under certain hypotheses for random variables which take

More information

International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics.

International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics. 400: A Method for Combining Non-Independent, One-Sided Tests of Significance Author(s): Morton B. Brown Reviewed work(s): Source: Biometrics, Vol. 31, No. 4 (Dec., 1975), pp. 987-992 Published by: International

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Relationship Between the Zeros of Best Approximations and Differentiability Author(s): Peter B. Borwein Source: Proceedings of the American Mathematical Society, Vol. 92, No. 4, (Dec., 1984), pp. 528-532

More information

Introduction and Preliminaries

Introduction and Preliminaries Chapter 1 Introduction and Preliminaries This chapter serves two purposes. The first purpose is to prepare the readers for the more systematic development in later chapters of methods of real analysis

More information

NOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES

NOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES NOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES G. GRÄTZER AND E. KNAPP Abstract. Let D be a finite distributive lattice with

More information

MATH 532, 736I: MODERN GEOMETRY

MATH 532, 736I: MODERN GEOMETRY MATH 532, 736I: MODERN GEOMETRY Test 2, Spring 2013 Show All Work Name Instructions: This test consists of 5 pages (one is an information page). Put your name at the top of this page and at the top of

More information

ERDŐS AND CHEN For each step the random walk on Z" corresponding to µn does not move with probability p otherwise it changes exactly one coor dinate w

ERDŐS AND CHEN For each step the random walk on Z corresponding to µn does not move with probability p otherwise it changes exactly one coor dinate w JOURNAL OF MULTIVARIATE ANALYSIS 5 8 988 Random Walks on Z PAUL ERDŐS Hungarian Academy of Sciences Budapest Hungary AND ROBERT W CHEN Department of Mathematics and Computer Science University of Miami

More information

THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter

THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,

More information

An Addition Theorem Modulo p

An Addition Theorem Modulo p JOURNAL OF COMBINATORIAL THEORY 5, 45-52 (1968) An Addition Theorem Modulo p JOHN E. OLSON The University of Wisconsin, Madison, Wisconsin 53706 Communicated by Gian-Carlo Rota ABSTRACT Let al,..., a=

More information

Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming

Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Yuval Filmus April 4, 2017 Abstract The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of

More information

A UNIVERSAL SEQUENCE OF CONTINUOUS FUNCTIONS

A UNIVERSAL SEQUENCE OF CONTINUOUS FUNCTIONS A UNIVERSAL SEQUENCE OF CONTINUOUS FUNCTIONS STEVO TODORCEVIC Abstract. We show that for each positive integer k there is a sequence F n : R k R of continuous functions which represents via point-wise

More information

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3

More information

Uniformly discrete forests with poor visibility

Uniformly discrete forests with poor visibility Uniformly discrete forests with poor visibility Noga Alon August 19, 2017 Abstract We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any

More information

Decomposing oriented graphs into transitive tournaments

Decomposing oriented graphs into transitive tournaments Decomposing oriented graphs into transitive tournaments Raphael Yuster Department of Mathematics University of Haifa Haifa 39105, Israel Abstract For an oriented graph G with n vertices, let f(g) denote

More information

The American Mathematical Monthly, Vol. 104, No. 8. (Oct., 1997), pp

The American Mathematical Monthly, Vol. 104, No. 8. (Oct., 1997), pp Newman's Short Proof of the Prime Number Theorem D. Zagier The American Mathematical Monthly, Vol. 14, No. 8. (Oct., 1997), pp. 75-78. Stable URL: http://links.jstor.org/sici?sici=2-989%2819971%2914%3a8%3c75%3anspotp%3e2..co%3b2-c

More information

4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu**

4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu** 4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN Robin Thomas* Xingxing Yu** School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA May 1991, revised 23 October 1993. Published

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Convex Real Projective Structures on Closed Surfaces are Closed Author(s): Suhyoung Choi and William M. Goldman Source: Proceedings of the American Mathematical Society, Vol. 118, No. 2 (Jun., 1993), pp.

More information

Convergence in shape of Steiner symmetrized line segments. Arthur Korneychuk

Convergence in shape of Steiner symmetrized line segments. Arthur Korneychuk Convergence in shape of Steiner symmetrized line segments by Arthur Korneychuk A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Mathematics

More information

The Periodogram and its Optical Analogy.

The Periodogram and its Optical Analogy. The Periodogram and Its Optical Analogy Author(s): Arthur Schuster Reviewed work(s): Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character,

More information

Abstract. 2. We construct several transcendental numbers.

Abstract. 2. We construct several transcendental numbers. Abstract. We prove Liouville s Theorem for the order of approximation by rationals of real algebraic numbers. 2. We construct several transcendental numbers. 3. We define Poissonian Behaviour, and study

More information

Efficient packing of unit squares in a square

Efficient packing of unit squares in a square Loughborough University Institutional Repository Efficient packing of unit squares in a square This item was submitted to Loughborough University's Institutional Repository by the/an author. Additional

More information

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. A Rearrangement Inequality and the Permutahedron Author(s): A. Vince Reviewed work(s): Source: The American Mathematical Monthly, Vol. 97, No. 4 (Apr., 1990), pp. 319-323 Published by: Mathematical Association

More information

2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016

2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016 6 EF Exam Texas A&M High School Students Contest Solutions October, 6. Assume that p and q are real numbers such that the polynomial x + is divisible by x + px + q. Find q. p Answer Solution (without knowledge

More information

The Generalized Coupon Collector Problem

The Generalized Coupon Collector Problem The Generalized Coupon Collector Problem John Moriarty & Peter Neal First version: 30 April 2008 Research Report No. 9, 2008, Probability and Statistics Group School of Mathematics, The University of Manchester

More information

LAYERED NETWORKS, THE DISCRETE LAPLACIAN, AND A CONTINUED FRACTION IDENTITY

LAYERED NETWORKS, THE DISCRETE LAPLACIAN, AND A CONTINUED FRACTION IDENTITY LAYERE NETWORKS, THE ISCRETE LAPLACIAN, AN A CONTINUE FRACTION IENTITY OWEN. BIESEL, AVI V. INGERMAN, JAMES A. MORROW, AN WILLIAM T. SHORE ABSTRACT. We find explicit formulas for the conductivities of

More information

Spheres with maximum inner diameter

Spheres with maximum inner diameter Carnegie Mellon University Research Showcase @ CMU Department of Mathematical Sciences Mellon College of Science 1970 Spheres with maximum inner diameter Juan Jorge Schäffer Carnegie Mellon University,

More information