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Topic: Limsup

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	Help with limsup and liminf (Site not responding. Last check: 2007-10-09)
		My only problem = is proving = limsup ((a_n)/n) <= inf {(a_(nk))/(nk)} = = There is a definition I found : limsup (x_n) = inf { sup = {x_(n+1),x_(n+2),...}
		So first I want to show that limsup ((a_n)/n) <= inf((a_n)/n) Well, I proved that (a_(nk))/(nk) <= (a_n)/n by the first supposition by just noting that a_(kn) = a_(n+n+...+n k-times) <= k*a_n.
		My only problem is proving limsup ((a_n)/n) <= inf {(a_(nk))/(nk)} There is a definition I found : limsup (x_n) = inf { sup {x_(n+1),x_(n+2),...}
	www.forum-one.org /new-5366815-4346.html (645 words)

	Adept Scientific plc - The Technical Computing People
		LimSup, but I do not know of a case where they are not.
		> LimSup, but I do not know of a case where they are not.
		> > LimSup, but I do not know of a case where they are not.
	lists.adeptscience.co.uk /mug/mug_Nov_2002/thid_9ab24b8e0789c737f6ba0f28ad9ef3ab.html (313 words)

	MA 711 Assignments (Site not responding. Last check: 2007-10-09)
		This material should be mostly review material, and I am not going to lecture on it, except to say a few things about the important notions of limsup and liminf of a sequence.
		You can simply treat the case limsup = + infinity by way of example.
		(2) Do 14, 15, 16 using using Royden's definitions of limsup and liminf with the additional notation introduced by me rather than the epsilon characterization (sometimes used as definition) contained in #13, which is unassigned and so cannot be invoked.
	math.bu.edu /people/twh/711hw.html (1159 words)

	big-Oh notation... (Site not responding. Last check: 2007-10-09)
		I can easily show that, if the big-Oh condition is satisfied, then limsup E
		bounded (as it's often called) and construct a sequence whose limsup is
		construct a sequence now that has limsup precisely equal to 1....
	www.groupsrv.com /science/about29780.html (349 words)

	MUG: Limit Superior & Limit Inferior (9.11.02) (Site not responding. Last check: 2007-10-09)
		In some cases, "limit" returns a range and you can take the left or right endpoint.
		But it will not be very powerful, because in many cases limit just returns undefined even though there may be a lim sup.
		The documentation does not claim that these numbers are the LimInf and LimSup, but I do not know of a case where they are not.
	www.math.rwth-aachen.de /mapleAnswers/html/1627.html (215 words)

	homework3_solutions (Site not responding. Last check: 2007-10-09)
		There are no other limit points besides the limsup and liminf.
		Since the sup of a set is less than or equal to any upper bound for the set, we get
		if both limsups are finite, and give an example where equality does not hold.
	xena.hunter.cuny.edu /thompson/math351/homework3_solutions (213 words)

	► » Help with limsup and liminf (Site not responding. Last check: 2007-10-09)
		limsup ((a_n)/n) = inf {(a_(nk))/(nk)} and then I will have proved that
		If I can do this I am sure the other half is very similar.
		There is a definition I found : limsup (x_n) = inf { sup
	www.science-chat.org /detail-5366815.html (387 words)

	Seminar on Stochastics: Irina Grabovsky's Abstract (Site not responding. Last check: 2007-10-09)
		Many random sets such as the set of fast points of Brownian motion, the set of thick points of the sojourn measure of spatial Brownian motion and the set of exceptional growth of the branching measure on a Galton-Watson tree can be well approximated by limsup random fractals.
		In this talk, we will show that, under some mild conditions, the hitting probabilities of a limsup random fractal $A$ are determined by the packing dimension of the target set $E$, rather than its Hausdorff dimension.
		When $A \cap E \ne \emptyset$, we give results on the Hausdorff dimension and packing dimension of $A \cap E$.
	www.math.utah.edu /~davar/seminars/abstracts/sp98/xiao.html (154 words)

	[No title]
		The limsup, abbreviation for limit superior is a refined and generalized notion of limit, being the largest dependent-variable subsequence limit.
		That is, among all subsequences of independent-variable values tending to some independent-variable value, usually infinity, there will be a corresponding dependent-variable subsequence.
		Some of these dependent-variable sequences will have limits, and among all these, the largest is the limsup.
	www.math.unl.edu /~sdunbar1/Teaching/MathematicalFinance/Lessons/CoinTossing/IteratedLog/iteratedlog.shtml (1199 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		In a paper from 1985, M. Tsukada proposed notions of liminf and limsup for a net of von Neumann algebras acting on a fixed Hilbert space.
		We used these notions in a recent paper to study endomorphisms and subfactors.
		A subsequent talk (9/4) will survey the rest of the joint work mentioned above, in particular the connection with the classification of von Neumann algebras, and modular theory.
	www.math.ku.dk /~eilers/abstracts/winslabs3.txt (95 words)

	Chapter 7 Preview (Site not responding. Last check: 2007-10-09)
		lim s limsup a = Join s:(subsequences of a, convergent).
		You will see that although the "upper frontier" oscillates quite a bit, it slowly tends to 1, the limsup.
		For mean payoff game, form 2 is the most useful because we can use it to rewrite the winning condition to something extremely familiar: (here '
	www.cs.toronto.edu /~trebla/ch07-preview.html (418 words)

	Mathematics 2KF Complex Analysis 2000: Status of week 6 (Site not responding. Last check: 2007-10-09)
		The theorem about termwise differentiation of power series was recalled and I showed that the the differentiated series has the same radius of convergence.
		We also recalled the definition of the radius of convergence in terms of limsup.
		I showed that if b_n converges to a positive number b then limsup(a_nb_n)=b limsup(a_n).
	www.math.ku.dk /ma/kurser/mat2kf/status6.html (173 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Let T_n denote the time spent at the most frequently visited site by time n, and let C_n denote the time needed to cover completely the disc of radius n; Erd\"os and Taylor (1960) proved that 1/4pi \le liminf T_n/(log n)^2 \le limsup T_n/(log n)^2 \le 1/pi, a.s.
		Kesten and R\'{e}v\'{e}sz (separately, around 1990) proved that e^{-4/t} \le liminf P(log C_n \le t (log n)^2) \le limsup P(log C_n \le t (log n)^2) \le e^{-1/t} and conjectured the lower bound is sharp.
		The latter probability estimate is strongly related to limits of the cover time of the discrete two dimensional torus by simple random walk.
	www.math.technion.ac.il /~techm/20001114113020001114zei (233 words)

	MATH 251 EXAM 2 INFORMATION (Site not responding. Last check: 2007-10-09)
		Be able to determine the liminf and limsup for concrete examples
		Be able to use the definitions of liminf and limsup to derive additional properties
		Be able to use the various tests for convergence of an infinite series to determine whether a given series converges or diverges
	www.math.wvu.edu /~sherm/m251/exam2.remarks.html (723 words)

	Conical limit set and Poincaré exponent for iterations of rational functions (ResearchIndex)
		We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere.
		We define the Poincar'e exponent ffi(f; z) = inffff 0 : P(z; ff) 0g where P(z; ff) := limsup n!1 1 n log X f n (x)=z j(f n) 0 (x)j \Gammaff : We prove that ffi(f; z) and P(z; ff) do not depend on z, provided z is non-exceptional.
		P plays the role of pressure, we prove that it coincides with Denker-Urba'nski's pressure if ff ffi(f).
	citeseer.ist.psu.edu /przytycki98conical.html (490 words)

	Journal of Zhejiang University Science020515 (Site not responding. Last check: 2007-10-09)
		Some limsup results for increments of stable processes in random scenery
		In this paper, we prove some limsup results for increments and lag increments of G(t), which is a stable processe in random scenery.
		The proofs rely on the tail probability estimation of G(t).
	www.zju.edu.cn /jzus/2002/0205/020515.htm (141 words)

	Advanced Calculus (Site not responding. Last check: 2007-10-09)
		Topics include the Completeness axiom, cardinality and Cantor's Theorem; sequences, subsequences, monitonicity and boundedness; the Bolzano-Weierstrass Theorem, LimSup and LimInf; topology of R1 and R2; open sets and limit points.
		Topics include the Completeness axiom, cardinality, and Cantor's theorem, Limsup, and Liminf; the topology of R1 and R2, open sets, and limit points as well as compactness and the Heine-Borel theorem; the properties of continuous functions, uniform continuity, the Mean-Value theorem, inverse functions and differentiability; the Riemann integral, and Lebesgue measure.
		The role of Set Theory as one of the foundations common to all branches of mathematics is shown in the course.
	www.nu.edu /Academics/Schools/COLS/MathematicsSciencesa/Courses/MTH432Syllabus.html (2462 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		R$ the power series diverges.} \gap If $R=0$ then $\limsup c_n^{1/n}=+\infty.$ Then for any $z\ne 0$ we have $\limsup c_nz^n^{1/n}=+\infty$ and thus $\limsup c_nz^n=+\infty$ as well, so the terms of the power series do not tend to zero if $z\ne 0.$ The conclusion holds in this case.
		R,$ $\limsup c_n^{1/n}z=z/R>1.$ Hence there is some $r$ \st $\limsup c_n^{1/n}z=z/R>r>1.$ Thus for every $N$ there is $n\ge N$ \st $c_n^{1/n}z>r$ so that $c_nz^n>r^n$ and thus there is a sub\seq $\{n_k\}$ of values of $n$ \st $c_{n_k}z^{n_k}\to+\infty$ and the series diverges rather badly!
		\gap {\bf Note: } This solution made free use of the note {\it On $\limsup$ and radius of convergence} that is linked to the class Web page.
	www.math.umn.edu /~jodeit/course/SP3solnF04 (150 words)

	Thick Points for Spatial Brownian Motion: Multifractal Analysis of Occupation Measure - Dembo, Rosen, Peres, Zeitouni ...
		Abstract: this paper, log 2 stands for the logarithm to the base 2.
		ffl Theorem 2.1, which is formulated for random fractals of limsup type in [0; 1], has an obvious generalization to random `fractals of limsup type' in [0; 1] d.
		10: Limsup random fractals (context) - Khoshnevisan, Peres et al.
	citeseer.ist.psu.edu /447954.html (691 words)

	Limsup from LiveJournal (Site not responding. Last check: 2007-10-09)
		Results 1-4 of about 4 for the Limsup (0.09 sec)
		...extremum leukemias Evanston quo polytropic Lipschitzian Hadamard's interpolant nonblocking affinely unidimensionality pseudoinverse superlinear motivic Deligne cocycle Beilinson polylogarithm Quasiregular limsup isoperimetric Fredholmness bornologies abscissae PDEs Noether widehat Lusztig's Oligomorphic tuples overdiagonal superalgebras progenerator cogenerator coresolution pseudomonotone subdifferential...
		For example, when it comes to sup's, inf's, limits, (and their combos limsup [ limbsoup =] and liminf) and convergence (the bread and butter of analysts!), it was only in this sequence when I'd really become able to handle these things with considerable dexterity...
	www.ljseek.com /search/Limsup (307 words)

	MathLinks Math Forum :: View topic - limsup whitout diophantine approximation (Site not responding. Last check: 2007-10-09)
		MathLinks Math Forum :: View topic - limsup whitout diophantine approximation
		Posted: Thu 28 Apr 2005, 22:31 Post subject: limsup whitout diophantine approximation
		Here is a nice problem that was discussed today and to which Fedja gave a nice proof using diophantine approximations:
	www.mathlinks.ro /Forum/post-218957.html (304 words)

	% Use % to comment
		(b) By inspection, $\limsup s_n=1=\sup S,\liminf s_n=0=\inf S$.
		The last inequality holds because $\limsup$ of every sequence is
		of $\limsup$ and the product limit theorem, we have
	www.math.unl.edu /~bdeng/Teaching/math425/f02/TeX_Samples/hwksolutions.htm (1866 words)

	Re: Analysis question (correct question this time)
		Therefore, >>limsup f'(x) at p and liminf f'(x) at p are both real numbers.
		>For this function f'(0) is indeed the average of the limsup (1) and >liminf (-1) but you can tweak >the function a bit so that the limsup becomes 2, say, and the liminf >becomes -1.
		To expand a bit on that: Imagine an example where f(x)
	www.usenet.com /newsgroups/sci.math/msg20984.html (287 words)

	MAT 360 course notes
		Corollary (of the above 2 facts): A sequence converges iff it is bounded and has a unique cluster point
		The concept of limsup and liminf for a sequence in R; examples
		Basic properties of limsup and liminf; in particular, the fact that a sequence converges iff its limsup and liminf coincide.
	www.math.sunysb.edu /~zakeri/mat360/mat360cn.html (783 words)

	MathLinks Math Forum :: View topic - limsup for complex numbers (Site not responding. Last check: 2007-10-09)
		MathLinks Math Forum :: View topic - limsup for complex numbers
		Posted: Sun 03 Jul 2005, 18:05 Post subject: limsup for complex numbers
		Posted: Sun 03 Jul 2005, 19:32 Post subject:
	www.mathlinks.ro /Forum/topic-43323.html (203 words)

	AoPS Math Forum :: View topic - limsup and cosinus (Site not responding. Last check: 2007-10-09)
		AoPS Math Forum :: View topic - limsup and cosinus
		The USAMTS Round 2 Math Jam takes place this coming
		Posted: Fri Jan 07, 2005 1:11 pm Post subject: limsup and cosinus
	www.artofproblemsolving.com /Forum/ptopic-23115.html (204 words)

	Aupair Profile - Au Pair Connect-Worldwide aupair, nanny and au pair job database.
		Benjamas Limsup, Female, 23, Thai, Member ID : 32971
		Only Premium members can see the contact information.
		To access contact information for Benjamas Limsup, click
	www.aupairconnect.com /aupairfinddetails.asp?id=19716 (227 words)

	MathLinks Math Forum :: View topic - limsup and cosinus
		MathLinks Math Forum :: View topic - limsup and cosinus
		Posted: Fri 07 Jan 2005, 23:11 Post subject: limsup and cosinus
		Posted: Sun 09 Jan 2005, 12:29 Post subject:
	www.mathlinks.ro /Forum/topic-22987.html (180 words)

	Third Year (Special Degree) Subjects (Site not responding. Last check: 2007-10-09)
		AM 3004: Mathematical modeling in Economics and Business (45L, 3C)
		Syllabus: Sequences: Subsequences; Cluster point of a sequence; Limsup and liminf; Proof of Cauchy condition; Bolzano-weierstrass theorem.
		Series: d’Alembert’s ratio test and Cauchy’s root test in terms of limsup and liminf; Existence of radius of convergence of a power series.
	www.cmb.ac.lk /academic/Science/Departments/Mathematics/Degree_Programmes/third_year_Syllabus3(Special)(new).html (797 words)

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