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text:
# Time complexity of linear iteration bounded by logarithmic iteration?

I was wondering about the time complexity of something like the following loop (where work() is O(1)):

for (i = 0 to N; i = i * 2) do
for(j = 0 to i; j = j + 1) do
work()
end
end

The outer loop takes $$\lg(N)$$ steps, and the inner loop takes (on the final iteration) $$N$$ steps. So, a simple bound would be $$O(N\lg(N))$$. However, the inner loop doesn't fully reach $$N$$ until the last iteration, so I was wondering if the bound could be narrowed any further.

Derp, nevermind. I observed that on the last iteration, the inner loop takes $$N$$ steps. On the previous iteration, the inner loop would take $$\frac{N}{2}$$ steps. Overall, this would take $$N, \frac{N}{2}, \frac{N}{4}, \frac{N}{8},...$$ steps, which has a geometric sum to $$2N$$ steps. Therefore, the loop is $$O(N)$$.

format: plain


text:
Now showing items 21-40 of 56

• #### 1130 - Real Analysis, Harmonic Analysis and Applications

[OWR-2011-36] (2011) - (24 Jul - 30 Jul 2011)
The workshop has focused on important developments within the last few years in the point of view and methods of real and harmonic Analysis as well as significant concurrent progress in the application of these to various ...
• #### 1129 - Explicit Methods in Number Theory

[OWR-2011-35] (2011) - (17 Jul - 23 Jul 2011)
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics.
• #### 1128 - Dynamische Systeme

[OWR-2011-34] (2011) - (10 Jul - 16 Jul 2011)
This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the ”Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of ...
• #### 1127 - Differentialgeometrie im Großen

[OWR-2011-33] (2011) - (03 Jul - 09 Jul 2011)
The meeting continued the biannual conference series Differentialgeometrie im Großen at the MFO which was established in the 60’s by Klingenberg and Chern. Global Riemannian geometry with its connections to topology, ...
• #### 1126 - Mathematical Methods in Quantum Chemistry

[OWR-2011-32] (2011) - (26 Jun - 02 Jul 2011)
The field of quantum chemistry is concerned with the analysis and simulation of chemical phenomena on the basis of the fundamental equations of quantum mechanics. Since the ‘exact’ electronic Schrödinger equation for a ...
• #### 1125 - Algebraische Zahlentheorie

[OWR-2011-31] (2011) - (19 Jun - 25 Jun 2011)
The workshop brought together researchers from Europe, Japan and the US, who reported on various recent developments in algebraic number theory and related fields. Dominant topics were Shimura varieties, automorphic forms ...
• #### 1123 - Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences

[OWR-2011-30] (2011) - (05 Jun - 11 Jun 2011)
Arithmetic groups arise naturally in many fields such as number theory, algebraic geometry, and analysis. Mapping class groups arise in both low dimensional topology and geometric group theory. They have been studied ...
• #### 1122 - Stochastic Analysis

[OWR-2011-29] (2011) - (29 May - 04 Jun 2011)
The meeting took place on May 30-June 3, 2011, with over 55 people in attendance. Each day had 6 to 7 talks of varying length (some talks were 30 minutes long), except for Thursday: the traditional hike was moved to Thursday ...
• #### 1121 - Schnelle Löser für Partielle Differentialgleichungen

[OWR-2011-28] (2011) - (22 May - 28 May 2011)
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This ...
• #### 1120b - New Directions in Algebraic K-Theory

[OWR-2011-27] (2011) - (15 May - 21 May 2011)
This meeting brought together algebraic geometers, algebraic topologists and geometric topologists, all of whom use algebraic K-theory. The talks and discussions involved all the participants.
• #### 1120a - Finite-dimensional Approximations of Discrete Groups

[OWR-2011-26] (2011) - (15 May - 21 May 2011)
The main objective of this workshop was to bring together experts from various fields, which are all interested in finite and finite-dimensional approximations of infinite algebraic and analytic objects, such as groups, ...
• #### 1119 - Billiards, Flat Surfaces, and Dynamics on Moduli Spaces

[OWR-2011-25] (2011) - (08 May - 14 May 2011)
This workshop brought together people working on the dynamics of various flows on moduli spaces, in particular the action of SL$_2(\mathbb R)$ on flat surfaces. The new results presented covered properties of interval ...
• #### 1118 - Random Matrices, Geometric Functional Analysis and Algorithms

[OWR-2011-24] (2011) - (01 May - 07 May 2011)
The workshop gathered close to 50 participants on the topics of random matrix theory, high dimensional convex geometry and probabilistic methods in theoretical computer science. It favored cooperation between researchers ...
• #### 1117c - Mini-Workshop: Random Trees, Information and Algorithms

[OWR-2011-23] (2011) - (24 Apr - 30 Apr 2011)
The subject of this Mini-Workshop is the probabilistic analysis of random tree models that originate from applications in Computer Science. Emphasis is put on their connections to algorithms and information theory. Trees ...
• #### 1117b - Mini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture

[OWR-2011-22] (2011) - (24 Apr - 30 Apr 2011)
The mathematical foundation of fracture mechanics has seen considerable advances in the last fifteen years. While this progress has been substantial, it has been largely limited to quasi-static evolutions based on global ...
• #### 1117a - Mini-Workshop: Modular Representations of Symmetric Groups and Related Objects

[OWR-2011-21] (2011) - (24 Apr - 30 Apr 2011)
The mini-workshop focussed on the modular representation theory of the symmetric group and other closely related objects, including Hecke algebras and Schur algebras. The topics and problems discussed include computations ...
• #### 1116 - Real Enumerative Questions in Complex and Tropical Geometry

[OWR-2011-20] (2011) - (17 Apr - 23 Apr 2011)
The workshop Real Enumerative Questions in Complex and Tropical Geometry was devoted to a wide discussion and exchange of ideas between the best experts representing various points of view on the subject. Enumeration of ...
• #### 1115 - Geometric Methods of Complex Analysis

[OWR-2011-19] (2011) - (10 Apr - 16 Apr 2011)
The purpose of this workshop was to discuss recent results in Several Complex Variables, Complex Geometry and Symplectic/Contact Geometry with a special focus on the interplay and exchange of ideas among these areas, which ...
• #### 1114 - Arbeitsgemeinschaft: Rational Homotopy Theory in Mathematics and Physics

[OWR-2011-18] (2011) - (03 Apr - 08 Apr 2011)
This Arbeitsgemeinschaft focused on the interplay among rational homotopy theory, differential geometry and the physics of string topology. The talks centered on one hand on how geometry and string topology make use of ...
• #### 1113 - Operator Algebras and Representation Theory: Frames, Wavelets and Fractals

[OWR-2011-17] (2011) - (27 Mar - 02 Apr 2011)
The central focus of the workshop was Kadison-Singer conjecture and its connection to operator algebras, harmonic analysis, representation theory and the theory of fractals. The program was intrinsically interdisciplinary ...


text:
# Convolution and Correlation

Started by December 25, 2003
Hello all,

When the two sequances are symmetrical then the correlation and the
convolution are same from the classical definitions.

My query is that, the correlation (if not ergodic) has joint pdf term
in its definition which is not there in convolution. So can I assume
that the above definition as limited to ergodicity or there is
something more to it.

Thanks,

Merry Christmas and a Happy New Year

Best regards,
Vimal
PS: One of the classical confussion 'conv and corr' has again caught
me!!

Vimal wrote:
> Hello all,
>
> When the two sequances are symmetrical then the correlation and the
> convolution are same from the classical definitions.

Actually this is true if only one sequence is (even) symmetric.

> My query is that, the correlation (if not ergodic) has joint pdf term
> in its definition which is not there in convolution. So can I assume
> that the above definition as limited to ergodicity or there is
> something more to it.

There are two definitions for autocorrelation: deterministic and
probabilistic. The deterministic autocorrelation can use either
a deterministic function or a random process (at a particular
realization), x(t), as its argument, and is defined as

R_xx(\tau) = \int_{-\infty}^{+\infty} x(\tau)x(t-\tau) dtau

The probabilistic definition of autocorrelation requires a random
process {X(t)} as its argument and is defined as

R_XX(t, s) = E[X(t)X(s)],

where E[Y] is the probabilistic expectation operator on the random
variable Y, i.e.,

E[Y] = \int_{-\infty}^{+\infty} y*f_Y(y) dy,

where f_Y(y) is the pdf of the random variable Y. If the random
process is wide-sense stationary, its autocorrelation is a function
only of time difference \tau = t - s, i.e., R_XX(t, s) = R_XX(0, t - s).

If the random process is ergodic, then the deterministic autocorrelation
approximates the probabilistic autocorrelation.
--
%% Fuquay-Varina, NC            %       'cause no one knows which side
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO

vimal_bhatia2@yahoo.com (Vimal) wrote in message
> Hello all,
>
> When the two sequances are symmetrical then the correlation and the
> convolution are same from the classical definitions.
>
> My query is that, the correlation (if not ergodic) has joint pdf term
> in its definition which is not there in convolution. So can I assume
> that the above definition as limited to ergodicity or there is
> something more to it.
>
> Thanks,
>
> Merry Christmas and a Happy New Year
>
> Best regards,
> Vimal
> PS: One of the classical confussion 'conv and corr' has again caught
> me!!

The formal definition of correlation includes the absolute time t as
well as the lag tau,

Rxy(t,tau) = E[x(t)y^*(t+tau)].                     [1]

Only for stationary (and ergodig) processes does this simplify to the
familiar lag-only expression where Rxy(t,tau) = Rxy(tau).

The reason for the joint PDF to appear with the correlation is that the
correlation function is defined in terms of the statistical Expectation
operator

E[x] = integral xp(x) dx                           [2]

for some PDF p(x). This PDF does not appear in any definitions of the
convolution operator and the familiar convolution operator for Linear
Time Invariant (LTI) systems holds for any t when applied to a non-
stationary input signal.

In the case of a Linear Time Varying (LTV) system, the impulse response
of the system changes with time, and the convolution operator must be
specified as

x(t) (*) h(t,tau) = integral x(t)*h(t,t-tau) dt    [3]

even for stationary x(t).

The non-stationary Rxy(t,tau) finds its counterpart in the LTV system
convolution [3].

Rune


text:
# Mechanics new question just need a slight help

• Nov 10th 2012, 11:32 PM
abdulrehmanshah
Mechanics new question just need a slight help
Attachment 25634i couldnt solve the last part where u have to find the resultant i know that the forumla to find the resultant but yet cant find it. I know that horizantal forces is 192cos50 but what would be the sum of vertical forces help
• Nov 11th 2012, 08:50 AM
skeeter
Re: Mechanics new question just need a slight help
$T_1 \cos(40) = T_2 \cos(50)$

$T_1 = \frac{T_2 \cos(50)}{\cos(40)}$

$T_1 \sin(40) + T_2 \sin(50) + 150 = 400$

$T_1 \sin(40) + T_2 \sin(50) = 250$

$\frac{T_2 \cos(50)\sin(40)}{\cos(40)} + T_2 \sin(50) = 250$

$T_2\left[\cos(50)\tan(40) + \sin(50)] = 250$

$T_2 = 192 \, N$

$T_1 = 161 \, N$

forces on the pulley in the x-direction (leftward) ...

$T_2 \cos(50)$

forces on the pulley in the y-direction (downward) ...

$T_2 \sin(50) + T_2$

Resolving forces acting on the pulley ...

$|F| = \sqrt{[T_2 \cos(50)]^2 + [T_2 \sin(50) + T_2]^2} = 360 \, N$

$\theta = \arctan\left[\frac{ T_2 \sin(50) + T_2}{T_2 \cos(50)}\right] = 70^\circ$ relative to the horizontal ... same as $20^\circ$ relative to the vertical
• Nov 11th 2012, 09:46 AM
abdulrehmanshah
Re: Mechanics new question just need a slight help
u deserve a million thanks well i just wanted to know what would be the vertical forces well thanks anyway u did a too much writing
• Nov 11th 2012, 09:54 AM
skeeter
Re: Mechanics new question just need a slight help
Quote:

Originally Posted by abdulrehmanshah
u deserve a million thanks well i just wanted to know what would be the vertical forces well thanks anyway u did a too much writing

Note that I have to do the problem from the start so that I understand how the entire problem develops.

It's like coming into a movie after it has been running a while ... you really do not understand what is going on.


text:
## Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole

# Chapter 2 - Systems of Linear Equations and Inequalities - 2.5 Absolute Value Equations and Inequalities - 2.5 Exercises - Page 188: 62

no solution

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|x+3|+15\lt4 ,$ isolate first the absolute value expression. Then use the definition of absolute value to analyze the solution. $\bf{\text{Solution Details:}}$ Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} |x+3|+15\lt4 \\\\ |x+3|\lt4-15 \\\\ |x+3|\lt-11 .\end{array} The absolute value of $x,$ written as $|x|,$ is the distance of $x$ from zero. Hence, it is always a nonnegative number. In the same way, the left side of the inequality above is a nonnegative number. This can never be $\text{ less than }$ the negative number at the right. Hence, there is $\text{ no solution .}$

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