Journal of Inequalities in Pure and Applied Mathematics
|
|
- Laurence Nash
- 6 years ago
- Views:
Transcription
1 Journal of Inequalities in Pure and Applied Mathematis A NEW ARRANGEMENT INEQUALITY MOHAMMAD JAVAHERI University of Oregon Department of Mathematis Fenton Hall, Eugene, OR javaheri@uoregon.edu URL: javaheri volume 7, issue 5, artile 162, Reeived 29 April, 2006; aepted 17 November, Communiated by: G. Bennett Abstrat Home Page 2000 Vitoria University ISSN eletroni):
2 Abstrat In this paper, we disuss the validity of the inequality x i x a i x b i+1 ) 2 x 1+a+b)/2 i, where 1, a, b are the sides of a triangle and the indies are understood modulo n. We show that, although this inequality does not hold in general, it is true when n 4. For general n, we show that any given set of nonnegative real numbers an be arranged as x 1, x 2,..., x n suh that the inequality above is valid Mathematis Subjet Classifiation: 26Dxx. Key words: Inequality, Arrangement. I would like to thank Omar Hosseini for bringing to my attention the ase a = b = 1 of the inequality 1.2). I would also like to thank Professor G. Bennett for reviewing the artile and making useful omments. 1 Main Statements Proofs Referenes Page 2 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
3 1. Main Statements Let a, b, x 1, x 2,..., x n be nonnegative real numbers. If a + b = 1 then, by the Rearrangement inequality [1], we have 1.1) x a i x b i+1 x i, where throughout this paper, the indies are understood to be modulo n. In an attempt to generalize this inequality, we onsider the following 2 1.2) x a i x b i+1 xi), x i where = a + b + 1)/2. It turns out that if a + b 1 then the inequality 1.2) is false for n large enough f. Prop. 2.2). However, we show that if 1.3) b a + 1, a b + 1, 1 a + b, then the inequality 1.2) is true in the ase of n = 4 f. Prop. 2.1). Moreover, under the same onditions on a, b as in 1.3), we show that one an always find a permutation µ of {1, 2,..., n} suh that f. Prop. 2.4) 2 1.4) x a µi)x b µi+1) xi). x i The onditions in 1.3) annot be ompromised in the sense that if for all nonnegative x 1, x 2,..., x n there exists a permutation µ suh that the onlusion Page 3 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
4 1.4) holds, then a, b must satisfy 1.3). To see this, let x 1 = x > 0 be arbitrary and x i = 1, i = 2,..., n. Then, for any permutation µ, the inequality 1.4) reads the same as: 1.5) x + n 1)x a + x b + n 2) x + n 1) 2. If the above inequality is true for all x and n, by omparing the oeffiients of n on both sides of the inequality 1.5), we should have x a + x b + x 3 2x 2. Sine x > 0 is arbitrary, 1, a, b and onditions 1.3) follow. The ase of a = b = 1 of 1.2) is partiularly interesting: 1.6) x i x i x i+1 x 3/2 i ) 2. There is a ounterexample to 1.6) when n = 9, e.g. take 1.7) x 1 = x 9 = 8.5, x 2 = x 8 = 9, x 3 = x 7 = 10, x 4 = x 6 = 11.5, x 5 = 12, and subsequently the inequality 1.6) is false for all n 9 f. prop. 2.2)). Proposition 2.1 shows that the inequality 1.6) is true for n 4, and there seems to be a omputer-based proof [2] for the ases n = 5, 6, 7 whih, if true, leaves us with the only remaining ase n = 8. Page 4 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
5 2. Proofs Applying Jensen s inequality [1, 3.14] to the onave funtion log x gives 2.1) u r v s w t ru + sv + tw, where u, v, w, r, s, t are nonnegative real numbers and r + s + t = 1. If, in addition, we have r, s, t > 0 then the equality ours iff u = v = w. However, if t = 0 and r, s, w > 0 then the equality ours iff u = v. We use this inequality in the proof of the proposition below. Proposition 2.1. Let a, b 0 suh that a + 1 b, b + 1 a and a + b 1. Then for all nonnegative real numbers x, y, z, t, 2.2) x + y + z + t)x a y b + y a z b + z a t b + t a x b ) x + y + z + t ) 2, where = a + b + 1)/2. The equality ours if and only if {a, b} = {0, 1} or x = y = z = t. Proof. We apply the inequality 2.1) to 2.3) and obtain: 2.4) x a y b z u = yz), v = xz), w = xy), r = 1 a, s = 1 b, t = 1 1, 1 a ) yz) + 1 b ) xz) ) xy). Page 5 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
6 Notie that the assumptions on a, b in the lemma are made exatly so that r, s, t are nonnegative. Similarly, by replaing z with t in 2.4), we have: 2.5) x a y b t 1 a ) yt) + 1 b ) tx) ) xy). Next, apply 2.1) to 2.6) u = x 2, v = xy), w = 1, r = 1 b, s = b, t = 0, and get 2.7) x a+1 y b 1 b ) x 2 + b xy). Similarly, by interhanging a and b, one has 2.8) x a y b+1 1 a ) x 2 + a xy). Adding the inequalities 2.4), 2.5), 2.7) and 2.8) gives: 2.9) Sx a y b 1 x ) xy) + 1 a ) yz) a ) yt) + 1 b ) tx) 1 b ) xz), Page 6 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
7 where S = x+y+z+t. There are three more inequalities of the form above that are obtained by replaing the pair x, y) by y, z), z, t) and t, x). By adding all four inequalities or by taking the yli sum of 2.9)) we have 2.10) ST 1 x ) x + z )y + t ) + 2 { xz) + yt) }, where ST stands for the left hand side of the inequality 2.2). The right hand side of the above inequality is equal to 2.11) ) ) 2 x 1 {x z ) 2 + y + t ) 2 2x + z )y + t ) }, whih is less than or equal to x ) 2, sine 1. This onludes the proof of the inequality 2.2). Next, suppose the equality ours in 2.2) and so the inequalities 2.4) 2.8) are all equalities. If a = 0 then we have x x b = x ) 2 and so, by the equality ase of Cauhy-Shwarz, the two vetors x, y, z, t) and x b, y b, z b, t b ) have to be proportional. Then either b = = 1 or x = y = z = t. Thus suppose a, b 0. Sine = a = b is impossible, without loss of generality suppose that b. Sine the inequality 2.7) must be an equality, x 2 = x y f. the disussion on the equality ase of 2.1)). Similarly y 2 = y z, z 2 = z t and t 2 = t x. It is then not diffiult to see that x = y = z = t. Let Na, b) denote the largest integer n for whih the inequality 1.2) holds for all nonnegative x 1, x 2,..., x n. By the above proposition, we have Na, b) 4. Page 7 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
8 Proposition 2.2. Let a, b 0 suh that a + b 1. Then Na, b) <. Moreover, if n Na, b) then the inequality 1.2) is valid for all nonnegative x 1,..., x n. Proof. The proof is divided into two parts. First we show that the inequality 1.2) annot be true for all n. Proof is by ontradition. If a = b = 0 then 1.2) is false for n = 2 e.g. take x 1 = 1, x 2 = 2). Thus, suppose a + b > 0 and that the inequality 1.2) is true for all n. Let f be a non-onstant positive ontinuous funtion on the interval I = [0, 1] suh that f0) = f1). Let ) i ) x i = f, y i = x a i x b n i+1) 1/a+b), i = 1,..., n. Sine y i is a number between x i and x i+1 possibly equal to one of them), by the Intermediate-value theorem [3, Th 3.3], there exists t i I i suh that ft i ) = y i. By the definition of integral we have: 2.13) I fx)dx f a+b 1 x)dx = lim I n n 2 1 = lim n n 2 1 lim n n 2 x i x i y a+b i x a i x b i+1 2 xi) = I f x)dx) 2, where we have applied the inequality 1.2) to the x i s. On the other hand, by Page 8 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
9 the Cauhy-Shwarz inequality for integrals, we have ) 2 fx)dx f a+b x)dx f 1 a+b 2.14) 2 x)f 2 x)dx I I = I I f x)dx) 2, with equality iff f and f a+b are proportional. The statements 2.13) and 2.14) imply that the equality indeed ours. Sine a + b 1 and f is not a onstant funtion, the two funtions f and f a+b annot be proportional. This ontradition implies that 1.2) ould not be true for all n i.e. Na, b) <. Next, we show that 1.2) is valid for all n N. It is suffiient to show that if the inequality 1.2) is true for all ordered sets of k+1 nonnegative real numbers, then it is true for all ordered sets of k nonnegative real numbers. Let y 1,..., y k be nonnegative real numbers and set 2.15) S = k y i, A = k yi a yi+1, b P = k yi. Without loss of generality we an assume P = 1. For eah 1 i k, define an ordered set of k + 1 nonnegative real numbers by setting: y j 1 j i + 1 x j = y j 1 i + 2 j k + 1 Applying the inequality 1.2) to x 1,..., x k+1 gives 2.16) S + y i )A + y a+b i ) P + yi ) 2 = 1 + yi 2 + 2yi. Page 9 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
10 Adding these inequalities for i = 1,..., k, yields: 2.17) ksa + S i y a+b i + AS k + 2. On the other hand, by the Rearrangement inequality [1] we have 2.18) k yi a yi+1 b k y a+b i, and the lemma follows by putting together the inequalities 2.17) and 2.18). The inequality 1.1) translates to Na, b) = when a + b = 1. We expet that Na, b) as a + b 1. The following proposition supports this onjeture. We define 2.19) A n a, b) = sup x i 2 x a i x b i+1 xi) max x i = 1 1 i n. This number roughly measures the validity of the inequality 1.2). Also let 2.20) σ t = 1 x t n i. By the Hölder inequality [1], if α, β > 0 and α + β = 1 then for any s, t > 0 we have: 2.21) σ α s σ β t σ αs+βt. Page 10 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
11 Proposition 2.3. Nu, u) is a non-inreasing funtion of u 1/2. Moreover, for all n and a, b ) lim a+b 1 A na, b) = 0. Proof. Suppose u > v > 1/2. We show that Nu, u) Nv, v). Without loss of generality we an assume: 2.23) u v < 1 4. By the definition of N = Nv, v), there must exist N + 1 nonnegative integers x 1,..., x N+1 suh that the inequality 1.2) is false and so N+1 N+1 N+1 ) ) x i x v i x v i+1 >. x v+1/2 i We show that the nonnegative numbers y i = x u/v i, i = 1,..., N + 1 give a ounterexample to 1.2) when a = b = u. In light of 2.24), one just needs to show N+1 ) 2 / N+1 N+1 ) 2 / N ) x i. To prove this, first let 2.26) α = x u+1/2v i x u/v i x u+1/2 i u + 1/2v) u/v u + 1/2v) 1, β = u/v 1 u + 1/2v) 1, s = 1, t = u + 1 2v. Page 11 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
12 The numbers above are simply hosen suh that α + β = 1 and αs + βt = u/v. We briefly hek that α, β > 0. The denominator of frations above is positive, sine u + 1/2v) v + 1/v)/2 1. This implies β > 0. Now the positivity of α > 0 is equivalent to u1 v) < 1/2. If v 1 then u1 v) 0 < 1/2. So suppose v 1. By using 2.23), we have: 2.27) u1 v) v + 1 ) 1 v) = v v < 1 2, for all v 0. Now we an safely plug α, β, s, t in 2.21) and get 2.28) σ α 1 σ β u+1/2v σ u/v. Next, let α = 1 α)/2 and β = 1 β/2. Sine α + β = 1 and α, β > 0, we an use Hölder s inequality 2.21) with α, β instead of α and β and the same s, t as before) and get this time α s + β t = u + 1/2): 2.29) σ 1 α)/2 1 σ 1 β/2 1+1/2v σ u+1/2. Now we square the above inequality and multiply it with 2.28) to obtain: 2.30) σ 1 σ 2 1+1/2v σ u/v σ 2 u+1/2, whih is equivalent to the inequality 2.25). So far we have shown the existene of a ounterexample to 1.2) for a = b = u when n = N + 1. Then Prop. 2.2 gives Nu, u) N = Nv, v) and this onludes the proof of the monotoniity of N. Page 12 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
13 It remains to prove that A n a, b) onverges to 0 as a+b 1. To the ontrary, assume there exists ɛ > 0 and a sequene a j, b j ) suh that A n a j, b j ) > ɛ and a j + b j 1. Then by definition, for eah j, there exists an n-tuple X j = x 1j,..., x nj ) suh that max x ij = 1 and 2.31) x ij x a j ij xb j i+1j x j ij ) 2 ɛ 2, where j = a j + b j + 1)/2. Sine X j is a bounded sequene, it follows that, along a subsequene j k, the X jk s onverge to some X = x 1,..., x n ). On the other hand, along a subsequene of j k denoted again by j k ), a jk a and b jk b for some a, b 0. Sine a j + b j 1, we have a + b = 1. By taking the limits of the inequality 2.31) along this subsequene, we should have ) ) x a i x b i+1 x i ɛ 2 > 0, x i whih ontradits the inequality 1.1). This ontradition establishes the equation 2.22). The next proposition shows that the inequality 1.2) holds if one mixes up the order of the x i s. The proof is simple and makes use of the monotoniity of σ t ) 1/t where σ t is defined by the equation 2.20). It is well-known that σ t ) 1/t is a non-dereasing funtion of t [1, Th. 16]. Proposition 2.4. Let a, b, be as in Proposition 2.1. Then for any given set of n nonnegative real numbers there exists an arrangement of them as x 1,..., x n suh that the inequality 1.2) holds. Page 13 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
14 Proof. Equivalently, we show that if x 1, x 2,..., x n are nonnegative then there exists a permutation µ of the set {1, 2,..., n} suh that the inequality 1.4) holds. Let 2.33) S = x i, T = x a i x b j. j i Then ST = nσ 1 n 2 σ a σ b nσ a+b ) = n 3 σ 1 σ a σ b n 2 σ 1 σ a+b. Now by the Cauhy- Shwarz inequality [4], σ 2 σ 1 σ a+b. On the other hand by the monotoniity of σ 1/t t, we have σ 1 σ 1/, σ a σ a/, σ b σ b/, and so σ 1 σ a σ b σ 2. It follows from these inequalities that 2.34) ST n 2 n 1)σ 2. Now for a permutation µ of 1, 2,..., n, let: 2.35) A µ = x a µi)x b µi+1). We would like to show that SA µ nσ ) 2 for some permutation µ. It is suffiient to show that the average of SA µ over all permutations µ is less than or equal to nσ ) 2. To show this, observe that the average of SA µ is equal to ST/n 1) and so the laim follows from the inequality 2.34). The symmetri group S n ats on R n in the usual way, namely for µ S n and x 1,..., x n ) R n let 2.36) µ x 1,..., x n ) = x µ1),..., x µn) ). Page 14 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
15 Let R be a region in R n that is invariant under the ation of permutations i.e. µ R R for all µ). define: 2.37) λr) = x ) 2 1,..., x n ) R x i x a i x b i+1 x i. By Proposition 2.4: 2.38) R µ S n µ λr). In partiular, by taking the Lebesgue measure of the sides of the inlusion above, we get 2.39) vol λr) vol R. n! We prove a better lower bound for vol λr) when n is a prime number similar but weaker results an be proved in general). Proposition 2.5. Let a, b be as in Proposition 2.1 and n be a prime number. Let R R n + be a Lebesgue-measurable bounded set that is invariant under the ation of permutations. Let λr) denote the set of all x 1,..., x n ) R for whih the inequality 1.2) holds. Then 2.40) vol λr) vol R n 1. Page 15 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
16 Proof. For m {1, 2,..., n 1} let µ m S n and denote the permutation 2.41) µ m i) = mi, where all the numbers are understood to be modulo n in partiular µ m n) = n for all m). Now reall the definition of A µ from the equation 2.35) and observe that: 2.42) n 1 m=1 A µm = = n 1 m=1 x a mix b mi+m = n 1 x a j j=1 m=1 x b j+m = n 1 m=1 j=1 j=1 x a j x a j x b j+m x b i. i j Then, the same argument in the proof of Prop. 2.4 implies that, for some m {1,..., n 1}, we have A µm nσ ) 2. We onlude that 2.43) R n 1 m=1 whih in turn implies the inequality 2.40). µ m λr), Page 16 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
17 Referenes [1] G.H. HARDY, J.E. LITTLEWOOD AND G. PÓLYA, Inequalities, Cambridge University Press, 2nd ed. 1988). [2] O. HOSSEINI, Private Communiation, Fall [3] M.H. PROTTER AND C.B. Jr MORREY, A First Course in Real Analysis, Springer 1997). [4] M.J. STEELE, The Cauhy-Shwarz Master Class: An Introdution to the Art of Mathematial Inequalities, Cambridge University Press 2004). Page 17 of 17 J. Ineq. Pure and Appl. Math. 75) Art. 162, 2006
Convergence of the Logarithmic Means of Two-Dimensional Trigonometric Fourier Series
Bulletin of TICMI Vol. 0, No., 06, 48 56 Convergene of the Logarithmi Means of Two-Dimensional Trigonometri Fourier Series Davit Ishhnelidze Batumi Shota Rustaveli State University Reeived April 8, 06;
More informationOrdered fields and the ultrafilter theorem
F U N D A M E N T A MATHEMATICAE 59 (999) Ordered fields and the ultrafilter theorem by R. B e r r (Dortmund), F. D e l o n (Paris) and J. S h m i d (Dortmund) Abstrat. We prove that on the basis of ZF
More informationA Characterization of Wavelet Convergence in Sobolev Spaces
A Charaterization of Wavelet Convergene in Sobolev Spaes Mark A. Kon 1 oston University Louise Arakelian Raphael Howard University Dediated to Prof. Robert Carroll on the oasion of his 70th birthday. Abstrat
More informationmax min z i i=1 x j k s.t. j=1 x j j:i T j
AM 221: Advaned Optimization Spring 2016 Prof. Yaron Singer Leture 22 April 18th 1 Overview In this leture, we will study the pipage rounding tehnique whih is a deterministi rounding proedure that an be
More informationMethods of evaluating tests
Methods of evaluating tests Let X,, 1 Xn be i.i.d. Bernoulli( p ). Then 5 j= 1 j ( 5, ) T = X Binomial p. We test 1 H : p vs. 1 1 H : p>. We saw that a LRT is 1 if t k* φ ( x ) =. otherwise (t is the observed
More informationThe Hanging Chain. John McCuan. January 19, 2006
The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a
More informationProduct Policy in Markets with Word-of-Mouth Communication. Technical Appendix
rodut oliy in Markets with Word-of-Mouth Communiation Tehnial Appendix August 05 Miro-Model for Inreasing Awareness In the paper, we make the assumption that awareness is inreasing in ustomer type. I.e.,
More informationA Functional Representation of Fuzzy Preferences
Theoretial Eonomis Letters, 017, 7, 13- http://wwwsirporg/journal/tel ISSN Online: 16-086 ISSN Print: 16-078 A Funtional Representation of Fuzzy Preferenes Susheng Wang Department of Eonomis, Hong Kong
More information(q) -convergence. Comenius University, Bratislava, Slovakia
Annales Mathematiae et Informatiae 38 (2011) pp. 27 36 http://ami.ektf.hu On I (q) -onvergene J. Gogola a, M. Mačaj b, T. Visnyai b a University of Eonomis, Bratislava, Slovakia e-mail: gogola@euba.sk
More informationAlienation of the logarithmic and exponential functional equations
Aequat. Math. 90 (2016), 107 121 The Author(s) 2015. This artile is published with open aess at Springerlink.om 0001-9054/16/010107-15 published online July 11, 2015 DOI 10.1007/s00010-015-0362-2 Aequationes
More informationOn maximal inequalities via comparison principle
Makasu Journal of Inequalities and Appliations (2015 2015:348 DOI 10.1186/s13660-015-0873-3 R E S E A R C H Open Aess On maximal inequalities via omparison priniple Cloud Makasu * * Correspondene: makasu@uw.a.za
More informationAsymptotic non-degeneracy of the solution to the Liouville Gel fand problem in two dimensions
Comment. Math. Helv. 2 2007), 353 369 Commentarii Mathematii Helvetii Swiss Mathematial Soiety Asymptoti non-degeneray of the solution to the Liouville Gel fand problem in two dimensions Tomohio Sato and
More informationMEASURE AND INTEGRATION: LECTURE 15. f p X. < }. Observe that f p
L saes. Let 0 < < and let f : funtion. We define the L norm to be ( ) / f = f dµ, and the sae L to be C be a measurable L (µ) = {f : C f is measurable and f < }. Observe that f = 0 if and only if f = 0
More informationDiscrete Bessel functions and partial difference equations
Disrete Bessel funtions and partial differene equations Antonín Slavík Charles University, Faulty of Mathematis and Physis, Sokolovská 83, 186 75 Praha 8, Czeh Republi E-mail: slavik@karlin.mff.uni.z Abstrat
More informationSolutions Manual. Selected odd-numbered problems in. Chapter 2. for. Proof: Introduction to Higher Mathematics. Seventh Edition
Solutions Manual Seleted odd-numbered problems in Chapter for Proof: Introdution to Higher Mathematis Seventh Edition Warren W. Esty and Norah C. Esty 5 4 3 1 Setion.1. Sentenes with One Variable Chapter
More informationSECOND HANKEL DETERMINANT PROBLEM FOR SOME ANALYTIC FUNCTION CLASSES WITH CONNECTED K-FIBONACCI NUMBERS
Ata Universitatis Apulensis ISSN: 15-539 http://www.uab.ro/auajournal/ No. 5/01 pp. 161-17 doi: 10.1711/j.aua.01.5.11 SECOND HANKEL DETERMINANT PROBLEM FOR SOME ANALYTIC FUNCTION CLASSES WITH CONNECTED
More informationThe law of the iterated logarithm for c k f(n k x)
The law of the iterated logarithm for k fn k x) Christoph Aistleitner Abstrat By a lassial heuristis, systems of the form osπn k x) k 1 and fn k x)) k 1, where n k ) k 1 is a rapidly growing sequene of
More informationKAMILLA OLIVER AND HELMUT PRODINGER
THE CONTINUED RACTION EXPANSION O GAUSS HYPERGEOMETRIC UNCTION AND A NEW APPLICATION TO THE TANGENT UNCTION KAMILLA OLIVER AND HELMUT PRODINGER Abstrat Starting from a formula for tan(nx in the elebrated
More informationComplexity of Regularization RBF Networks
Complexity of Regularization RBF Networks Mark A Kon Department of Mathematis and Statistis Boston University Boston, MA 02215 mkon@buedu Leszek Plaskota Institute of Applied Mathematis University of Warsaw
More informationMaximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More informationMost results in this section are stated without proof.
Leture 8 Level 4 v2 he Expliit formula. Most results in this setion are stated without proof. Reall that we have shown that ζ (s has only one pole, a simple one at s =. It has trivial zeros at the negative
More informationSQUARE ROOTS AND AND DIRECTIONS
SQUARE ROOS AND AND DIRECIONS We onstrut a lattie-like point set in the Eulidean plane that eluidates the relationship between the loal statistis of the frational parts of n and diretions in a shifted
More informationHOW TO FACTOR. Next you reason that if it factors, then the factorization will look something like,
HOW TO FACTOR ax bx I now want to talk a bit about how to fator ax bx where all the oeffiients a, b, and are integers. The method that most people are taught these days in high shool (assuming you go to
More informationPacking Plane Spanning Trees into a Point Set
Paking Plane Spanning Trees into a Point Set Ahmad Biniaz Alfredo Garía Abstrat Let P be a set of n points in the plane in general position. We show that at least n/3 plane spanning trees an be paked into
More informationREFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS. 1. Introduction
Version of 5/2/2003 To appear in Advanes in Applied Mathematis REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS MATTHIAS BECK AND SHELEMYAHU ZACKS Abstrat We study the Frobenius problem:
More informationOn Component Order Edge Reliability and the Existence of Uniformly Most Reliable Unicycles
Daniel Gross, Lakshmi Iswara, L. William Kazmierzak, Kristi Luttrell, John T. Saoman, Charles Suffel On Component Order Edge Reliability and the Existene of Uniformly Most Reliable Uniyles DANIEL GROSS
More informationStability of alternate dual frames
Stability of alternate dual frames Ali Akbar Arefijamaal Abstrat. The stability of frames under perturbations, whih is important in appliations, is studied by many authors. It is worthwhile to onsider
More informationON THE GENERAL QUADRATIC FUNCTIONAL EQUATION
Bol. So. Mat. Mexiana (3) Vol. 11, 2005 ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION JOHN MICHAEL RASSIAS Abstrat. In 1940 and in 1964 S. M. Ulam proposed the general problem: When is it true that by hanging
More informationEDGE-DISJOINT CLIQUES IN GRAPHS WITH HIGH MINIMUM DEGREE
EDGE-DISJOINT CLIQUES IN GRAPHS WITH HIGH MINIMUM DEGREE RAPHAEL YUSTER Abstrat For a graph G and a fixed integer k 3, let ν k G) denote the maximum number of pairwise edge-disjoint opies of K k in G For
More informationCan Learning Cause Shorter Delays in Reaching Agreements?
Can Learning Cause Shorter Delays in Reahing Agreements? Xi Weng 1 Room 34, Bldg 2, Guanghua Shool of Management, Peking University, Beijing 1871, China, 86-162767267 Abstrat This paper uses a ontinuous-time
More informationHILLE-KNESER TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES
HILLE-KNESER TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES L ERBE, A PETERSON AND S H SAKER Abstrat In this paper, we onsider the pair of seond-order dynami equations rt)x ) ) + pt)x
More informationHYPERSTABILITY OF THE GENERAL LINEAR FUNCTIONAL EQUATION
Bull. Korean Math. So. 52 (2015, No. 6, pp. 1827 1838 http://dx.doi.org/10.4134/bkms.2015.52.6.1827 HYPERSTABILITY OF THE GENERAL LINEAR FUNCTIONAL EQUATION Magdalena Piszzek Abstrat. We give some results
More informationarxiv: v2 [math.pr] 9 Dec 2016
Omnithermal Perfet Simulation for Multi-server Queues Stephen B. Connor 3th Deember 206 arxiv:60.0602v2 [math.pr] 9 De 206 Abstrat A number of perfet simulation algorithms for multi-server First Come First
More informationOn the density of languages representing finite set partitions
On the density of languages representing finite set partitions Nelma Moreira Rogério Reis Tehnial Report Series: DCC-04-07 Departamento de Ciênia de Computadores Fauldade de Ciênias & Laboratório de Inteligênia
More informationON THE LEAST PRIMITIVE ROOT EXPRESSIBLE AS A SUM OF TWO SQUARES
#A55 INTEGERS 3 (203) ON THE LEAST PRIMITIVE ROOT EPRESSIBLE AS A SUM OF TWO SQUARES Christopher Ambrose Mathematishes Institut, Georg-August Universität Göttingen, Göttingen, Deutshland ambrose@uni-math.gwdg.de
More informationA xed point approach to the stability of a nonlinear volterra integrodierential equation with delay
Haettepe Journal of Mathematis and Statistis Volume 47 (3) (218), 615 623 A xed point approah to the stability of a nonlinear volterra integrodierential equation with delay Rahim Shah and Akbar Zada Abstrat
More informationMath 151 Introduction to Eigenvectors
Math 151 Introdution to Eigenvetors The motivating example we used to desrie matrixes was landsape hange and vegetation suession. We hose the simple example of Bare Soil (B), eing replaed y Grasses (G)
More informationThe Second Postulate of Euclid and the Hyperbolic Geometry
1 The Seond Postulate of Eulid and the Hyperboli Geometry Yuriy N. Zayko Department of Applied Informatis, Faulty of Publi Administration, Russian Presidential Aademy of National Eonomy and Publi Administration,
More informationThe Effectiveness of the Linear Hull Effect
The Effetiveness of the Linear Hull Effet S. Murphy Tehnial Report RHUL MA 009 9 6 Otober 009 Department of Mathematis Royal Holloway, University of London Egham, Surrey TW0 0EX, England http://www.rhul.a.uk/mathematis/tehreports
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationHankel Optimal Model Order Reduction 1
Massahusetts Institute of Tehnology Department of Eletrial Engineering and Computer Siene 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Hankel Optimal Model Order Redution 1 This leture overs both
More informationNonreversibility of Multiple Unicast Networks
Nonreversibility of Multiple Uniast Networks Randall Dougherty and Kenneth Zeger September 27, 2005 Abstrat We prove that for any finite direted ayli network, there exists a orresponding multiple uniast
More informationKRANNERT GRADUATE SCHOOL OF MANAGEMENT
KRANNERT GRADUATE SCHOOL OF MANAGEMENT Purdue University West Lafayette, Indiana A Comment on David and Goliath: An Analysis on Asymmetri Mixed-Strategy Games and Experimental Evidene by Emmanuel Dehenaux
More informationFuzzy inner product space and its properties 1
International Journal of Fuzzy Mathematis and Systems IJFMS). ISSN 48-9940 Volume 5, Number 1 015), pp. 57-69 Researh India Publiations http://www.ripubliation.om Fuzzy inner produt spae and its properties
More informationRATIONALITY OF SECANT ZETA VALUES
RATIONALITY OF SECANT ZETA VALUES PIERRE CHAROLLOIS AND MATTHEW GREENBERG Abstrat We use the Arakawa-Berndt theory of generalized η-funtions to prove a onjeture of Lalìn, Rodrigue and Rogers onerning the
More informationLECTURE NOTES FOR , FALL 2004
LECTURE NOTES FOR 18.155, FALL 2004 83 12. Cone support and wavefront set In disussing the singular support of a tempered distibution above, notie that singsupp(u) = only implies that u C (R n ), not as
More informationChapter 8 Hypothesis Testing
Leture 5 for BST 63: Statistial Theory II Kui Zhang, Spring Chapter 8 Hypothesis Testing Setion 8 Introdution Definition 8 A hypothesis is a statement about a population parameter Definition 8 The two
More informationAn Integer Solution of Fractional Programming Problem
Gen. Math. Notes, Vol. 4, No., June 0, pp. -9 ISSN 9-784; Copyright ICSRS Publiation, 0 www.i-srs.org Available free online at http://www.geman.in An Integer Solution of Frational Programming Problem S.C.
More informationResearch Article Approximation of Analytic Functions by Solutions of Cauchy-Euler Equation
Funtion Spaes Volume 2016, Artile ID 7874061, 5 pages http://d.doi.org/10.1155/2016/7874061 Researh Artile Approimation of Analyti Funtions by Solutions of Cauhy-Euler Equation Soon-Mo Jung Mathematis
More informationRemark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the function V ( x ) to be positive definite.
Leture Remark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the funtion V ( x ) to be positive definite. ost often, our interest will be to show that x( t) as t. For that we will need
More informationEXACT TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED KURAMOTO-SIVASHINSKY EQUATION
Journal of Mathematial Sienes: Advanes and Appliations Volume 3, 05, Pages -3 EXACT TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED KURAMOTO-SIVASHINSKY EQUATION JIAN YANG, XIAOJUAN LU and SHENGQIANG TANG
More informationON A PROCESS DERIVED FROM A FILTERED POISSON PROCESS
ON A PROCESS DERIVED FROM A FILTERED POISSON PROCESS MARIO LEFEBVRE and JEAN-LUC GUILBAULT A ontinuous-time and ontinuous-state stohasti proess, denoted by {Xt), t }, is defined from a proess known as
More informationA Variational Definition for Limit and Derivative
Amerian Journal of Applied Mathematis 2016; 4(3): 137-141 http://www.sienepublishinggroup.om/j/ajam doi: 10.11648/j.ajam.20160403.14 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) A Variational Definition
More informationON THE MOVING BOUNDARY HITTING PROBABILITY FOR THE BROWNIAN MOTION. Dobromir P. Kralchev
Pliska Stud. Math. Bulgar. 8 2007, 83 94 STUDIA MATHEMATICA BULGARICA ON THE MOVING BOUNDARY HITTING PROBABILITY FOR THE BROWNIAN MOTION Dobromir P. Kralhev Consider the probability that the Brownian motion
More informationa n z n, (1.1) As usual, we denote by S the subclass of A consisting of functions which are also univalent in U.
MATEMATIQKI VESNIK 65, 3 (2013), 373 382 September 2013 originalni nauqni rad researh paper CERTAIN SUFFICIENT CONDITIONS FOR A SUBCLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR S. Sivasubramanian,
More informationNUMERICALLY SATISFACTORY SOLUTIONS OF HYPERGEOMETRIC RECURSIONS
MATHEMATICS OF COMPUTATION Volume 76, Number 259, July 2007, Pages 1449 1468 S 0025-5718(07)01918-7 Artile eletronially published on January 31, 2007 NUMERICALLY SATISFACTORY SOLUTIONS OF HYPERGEOMETRIC
More informationInternet Appendix for Proxy Advisory Firms: The Economics of Selling Information to Voters
Internet Appendix for Proxy Advisory Firms: The Eonomis of Selling Information to Voters Andrey Malenko and Nadya Malenko The first part of the Internet Appendix presents the supplementary analysis for
More informationA Queueing Model for Call Blending in Call Centers
A Queueing Model for Call Blending in Call Centers Sandjai Bhulai and Ger Koole Vrije Universiteit Amsterdam Faulty of Sienes De Boelelaan 1081a 1081 HV Amsterdam The Netherlands E-mail: {sbhulai, koole}@s.vu.nl
More informationVolume 29, Issue 3. On the definition of nonessentiality. Udo Ebert University of Oldenburg
Volume 9, Issue 3 On the definition of nonessentiality Udo Ebert University of Oldenburg Abstrat Nonessentiality of a good is often used in welfare eonomis, ost-benefit analysis and applied work. Various
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 371 (010) 759 763 Contents lists available at SieneDiret Journal of Mathematial Analysis an Appliations www.elsevier.om/loate/jmaa Singular Sturm omparison theorems Dov Aharonov, Uri
More informationChapter 2: Solution of First order ODE
0 Chapter : Solution of irst order ODE Se. Separable Equations The differential equation of the form that is is alled separable if f = h g; In order to solve it perform the following steps: Rewrite the
More informationarxiv:math/ v4 [math.ca] 29 Jul 2006
arxiv:math/0109v4 [math.ca] 9 Jul 006 Contiguous relations of hypergeometri series Raimundas Vidūnas University of Amsterdam Abstrat The 15 Gauss ontiguous relations for F 1 hypergeometri series imply
More informationSuper edge-magic total labeling of a tree
Super edge-magi total labeling of a tree K. Ali, M. Hussain, A. Razzaq Department of Mathematis, COMSATS Institute of Information Tehnology, Lahore, Pakistan. {akashifali, mhmaths, asimrzq}@gmail.om Abstrat.
More informationarxiv: v2 [math.gm] 29 Jan 2009
ON A CLASS OF ENTIRE FUNCTIONS, ALL THE ZEROS OF WHICH ARE REAL HIDAYAT M. HUSEYNOV arxiv:9.57v [ath.gm] 9 Jan 9 Abstrat. For a lass of funtions the reality of all the zeros of the Fourier transfor is
More information(, ) Anti Fuzzy Subgroups
International Journal of Fuzzy Mathematis and Systems. ISSN 2248-9940 Volume 3, Number (203), pp. 6-74 Researh India Publiations http://www.ripubliation.om (, ) Anti Fuzzy Subgroups P.K. Sharma Department
More informationAxioms for the Real Number System
Axioms for the Real Number System Math 361 Fall 2003 Page 1 of 9 The Real Number System The real number system consists of four parts: 1. A set (R). We will call the elements of this set real numbers,
More informationFundamental Theorem of Calculus
Chater 6 Fundamental Theorem of Calulus 6. Definition (Nie funtions.) I will say that a real valued funtion f defined on an interval [a, b] is a nie funtion on [a, b], if f is ontinuous on [a, b] and integrable
More informationSoft BCL-Algebras of the Power Sets
International Journal of lgebra, Vol. 11, 017, no. 7, 39-341 HIKRI Ltd, www.m-hikari.om https://doi.org/10.1988/ija.017.7735 Soft BCL-lgebras of the Power Sets Shuker Mahmood Khalil 1 and bu Firas Muhamad
More informationModal Horn Logics Have Interpolation
Modal Horn Logis Have Interpolation Marus Kraht Department of Linguistis, UCLA PO Box 951543 405 Hilgard Avenue Los Angeles, CA 90095-1543 USA kraht@humnet.ula.de Abstrat We shall show that the polymodal
More informationSome facts you should know that would be convenient when evaluating a limit:
Some fats you should know that would be onvenient when evaluating a it: When evaluating a it of fration of two funtions, f(x) x a g(x) If f and g are both ontinuous inside an open interval that ontains
More informationCONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION. Smail Mahdi
Serdia Math. J. 30 (2004), 55 70 CONDITIONAL CONFIDENCE INTERVAL FOR THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION Smail Mahdi Communiated by N. M. Yanev Abstrat. A two-sided onditional onfidene interval
More informationUniversity of Groningen
University of Groningen Port Hamiltonian Formulation of Infinite Dimensional Systems II. Boundary Control by Interonnetion Mahelli, Alessandro; van der Shaft, Abraham; Melhiorri, Claudio Published in:
More informationStudy of EM waves in Periodic Structures (mathematical details)
Study of EM waves in Periodi Strutures (mathematial details) Massahusetts Institute of Tehnology 6.635 partial leture notes 1 Introdution: periodi media nomenlature 1. The spae domain is defined by a basis,(a
More informationCOMPARISON OF GEOMETRIC FIGURES
COMPARISON OF GEOMETRIC FIGURES Spyros Glenis M.Ed University of Athens, Department of Mathematis, e-mail spyros_glenis@sh.gr Introdution the figures: In Eulid, the geometri equality is based on the apability
More informationCase I: 2 users In case of 2 users, the probability of error for user 1 was earlier derived to be 2 A1
MUTLIUSER DETECTION (Letures 9 and 0) 6:33:546 Wireless Communiations Tehnologies Instrutor: Dr. Narayan Mandayam Summary By Shweta Shrivastava (shwetash@winlab.rutgers.edu) bstrat This artile ontinues
More informationSURFACE WAVES OF NON-RAYLEIGH TYPE
SURFACE WAVES OF NON-RAYLEIGH TYPE by SERGEY V. KUZNETSOV Institute for Problems in Mehanis Prosp. Vernadskogo, 0, Mosow, 75 Russia e-mail: sv@kuznetsov.msk.ru Abstrat. Existene of surfae waves of non-rayleigh
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics NOTES ON AN INTEGRAL INEQUALITY QUÔ C ANH NGÔ, DU DUC THANG, TRAN TAT DAT, AND DANG ANH TUAN Department of Mathematics, Mechanics and Informatics,
More informationCommon Value Auctions with Costly Entry
Common Value Autions with Costly Entry Pauli Murto Juuso Välimäki June, 205 preliminary and inomplete Abstrat We onsider a model where potential bidders onsider paying an entry ost to partiipate in an
More informationSome examples of generated fuzzy implicators
Some examples of generated fuzzy impliators Dana Hliněná and Vladislav Biba Department of Mathematis, FEEC, Brno University of Tehnology Tehniká 8, 616 00 Brno, Czeh Rep. e-mail: hlinena@fee.vutbr.z, v
More informationMonomial strategies for concurrent reachability games and other stochastic games
Monomial strategies for onurrent reahability games and other stohasti games Søren Kristoffer Stiil Frederisen and Peter Bro Miltersen Aarhus University Abstrat. We onsider two-player zero-sum finite but
More informationarxiv: v1 [math.co] 16 May 2016
arxiv:165.4694v1 [math.co] 16 May 216 EHRHART POLYNOMIALS OF 3-DIMENSIONAL SIMPLE INTEGRAL CONVEX POLYTOPES YUSUKE SUYAMA Abstrat. We give an expliit formula on the Ehrhart polynomial of a 3- dimensional
More informationSolving a system of linear equations Let A be a matrix, X a column vector, B a column vector then the system of linear equations is denoted by AX=B.
Matries and Vetors: Leture Solving a sstem of linear equations Let be a matri, X a olumn vetor, B a olumn vetor then the sstem of linear equations is denoted b XB. The augmented matri The solution to a
More informationThe Laws of Acceleration
The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the
More informationCommutators Involving Matrix Functions
Eletroni Journal of Linear Algera Volume 33 Volume 33: Speial Issue for the International Conferene on Matrix Analysis and its Appliations MAT TRIAD 017 Artile 11 018 Commutators Involving Matrix untions
More informationAssessing the Performance of a BCI: A Task-Oriented Approach
Assessing the Performane of a BCI: A Task-Oriented Approah B. Dal Seno, L. Mainardi 2, M. Matteui Department of Eletronis and Information, IIT-Unit, Politenio di Milano, Italy 2 Department of Bioengineering,
More informationEstimating the probability law of the codelength as a function of the approximation error in image compression
Estimating the probability law of the odelength as a funtion of the approximation error in image ompression François Malgouyres Marh 7, 2007 Abstrat After some reolletions on ompression of images using
More informationRIEMANN S FIRST PROOF OF THE ANALYTIC CONTINUATION OF ζ(s) AND L(s, χ)
RIEMANN S FIRST PROOF OF THE ANALYTIC CONTINUATION OF ζ(s AND L(s, χ FELIX RUBIN SEMINAR ON MODULAR FORMS, WINTER TERM 6 Abstrat. In this hapter, we will see a proof of the analyti ontinuation of the Riemann
More informationarxiv:gr-qc/ v2 6 Feb 2004
Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this
More informationThe First Integral Method for Solving a System of Nonlinear Partial Differential Equations
ISSN 179-889 (print), 179-897 (online) International Journal of Nonlinear Siene Vol.5(008) No.,pp.111-119 The First Integral Method for Solving a System of Nonlinear Partial Differential Equations Ahmed
More informationDept. of Computer Science. Raleigh, NC 27695, USA. May 14, Abstract. 1, u 2 q i+1 :
Anti-Leture Hall Compositions Sylvie Corteel CNRS PRiSM, UVSQ 45 Avenue des Etats-Unis 78035 Versailles, Frane syl@prism.uvsq.fr Carla D. Savage Dept. of Computer Siene N. C. State University, Box 8206
More informationDifferential Equations 8/24/2010
Differential Equations A Differential i Equation (DE) is an equation ontaining one or more derivatives of an unknown dependant d variable with respet to (wrt) one or more independent variables. Solution
More informationSome recent developments in probability distributions
Proeedings 59th ISI World Statistis Congress, 25-30 August 2013, Hong Kong (Session STS084) p.2873 Some reent developments in probability distributions Felix Famoye *1, Carl Lee 1, and Ayman Alzaatreh
More informationCoefficients of the Inverse of Strongly Starlike Functions
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malaysian Math. S. So. (Seond Series) 6 (00) 6 7 Coeffiients of the Inverse of Strongly Starlie Funtions ROSIHAN M. ALI Shool of Mathematial
More informationAfter the completion of this section the student should recall
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 3 I. SETS, NUMERS, COORDINTES, FUNCTIONS Objetives: fter the ompletion of this setion the student should reall - the definition
More informationAppendix A Market-Power Model of Business Groups. Robert C. Feenstra Deng-Shing Huang Gary G. Hamilton Revised, November 2001
Appendix A Market-Power Model of Business Groups Roert C. Feenstra Deng-Shing Huang Gary G. Hamilton Revised, Novemer 200 Journal of Eonomi Behavior and Organization, 5, 2003, 459-485. To solve for the
More informationTHE computation of Fourier transforms (see Definition
IAENG International Journal of Applied Mathematis, 48:, IJAM_48 6 The Computation of Fourier Transform via Gauss-Regularized Fourier Series Qingyue Zhang Abstrat In this paper, we onstrut a new regularized
More informationMaxmin expected utility through statewise combinations
Eonomis Letters 66 (2000) 49 54 www.elsevier.om/ loate/ eonbase Maxmin expeted utility through statewise ombinations Ramon Casadesus-Masanell, Peter Klibanoff, Emre Ozdenoren* Department of Managerial
More informationMA2331 Tutorial Sheet 5, Solutions December 2014 (Due 12 December 2014 in class) F = xyi+ 1 2 x2 j+k = φ (1)
MA2331 Tutorial Sheet 5, Solutions. 1 4 Deember 214 (Due 12 Deember 214 in lass) Questions 1. ompute the line integrals: (a) (dx xy + 1 2 dy x2 + dz) where is the line segment joining the origin and the
More informationLecture 7: Sampling/Projections for Least-squares Approximation, Cont. 7 Sampling/Projections for Least-squares Approximation, Cont.
Stat60/CS94: Randomized Algorithms for Matries and Data Leture 7-09/5/013 Leture 7: Sampling/Projetions for Least-squares Approximation, Cont. Leturer: Mihael Mahoney Sribe: Mihael Mahoney Warning: these
More information