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Performance Optimization
What is it and what should it do?
Its from codechef.com, I'm eager to learn different approaches. I came up with this after a lot of tries but still the performance is slow.
It is required to find the minimum n such that Fn % p = c.
Fn is the nth fibonacci term, ct is n in this code.
Range of c, 0 ≤ c ≤ p-1
p is a prime number <2000000000 and also p % 10 is guaranteed to be a perfect square. Can this property be used in any way to optimize instead of just brute-forcing?
Last edited on
I observed the results of mod p on Fibonacci series and noticed that they repeat themselves after a certain count (the count most of the times being p-1) and so wrote this code but it still isn't
optimal, for the prime 1999999871 it loops until 1999999870.
Does p%10 being a perfect square help??
1) you are doing it all wrong.
From equation F[n] % p = c → F[n] = x·p + c where x ∈ N
From numbers less than p you need to check only c itself. Next one will be p+c, then 2·p+c etc.
You might want for each t = x·p + c to check if it is a fibonacci number using known formulas, calculate n of said number using other formulas and output that number.
Topic archived. No new replies allowed.
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User verret
bio website
visits member for 2 years
seen 9 hours ago
stats profile views 184
22h awarded Promoter
22h comment $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients
I'm not sure I understand your example. Could you maybe be more explicit? (Keep in mind that I am only considering finite groups.)
14 awarded Yearling
14 awarded Student
14 asked $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients
Mar Max order for which connected Cayley Graphs are known to be Hamiltonian
26 comment It has been checked to 1280 that, in the cubic vertex-transitive case, only the four well-known exceptions occur. The question asked for Cayley graphs of arbitrary valency.
21 answered Max order for which connected Cayley Graphs are known to be Hamiltonian
Feb On the Groups of Order $(p^2+1)/2$
24 comment ADDENDUM: there are three more candidates for $p$ between 3 and 4 million: $p=3319597,3456127,3636443$, and then none up to 10 million.
On the Groups of Order $(p^2+1)/2$
Feb A small observation : since the group has odd order, it is soluble. In particular, it has SOME (minimal) abelian normal subgroup. Moreover, by "Groups of Cube-Free Odd Order", by Curran,
24 comment we may assume that the group is not cube-free. Anyway, I checked the conjecture up to $p=3000000$. I was only checking that $n=(p^2+1)/2$ was not squarefree and that Sylow's theorem
would not force a normal $q$-Sylow subgroup of order at most $q^2$ for some prime $q$. Up to $p=3000000$, the only exceptions are for $p=239$, when we get n=$13^4$ and $p=2905807$ when
we get $n=5^4∗13∗61∗97∗137∗641$.
23 answered Upper bound on the number of vertex transitive graphs
15 answered Fantastic properties of Z/2Z
Sep Presentation of the Monster Group
15 revised deleted 344 characters in body
15 answered Presentation of the Monster Group
Jul Maximum automorphism group for a 3-connected cubic graph
17 comment In the cubic vertex-transitive case and n twice an odd number, it immediately follows from the same paper that you get a polynomial rather than exponential upper bound. For example
Corollary 4 yields that, for large enough n, we have |G|<n^2. This is not best possible, but is not far off, at least for some values of n. If you need more precise estimates, I can show
you a few more references that deal with this. (By the way, the link in your "added" section has a typo.)
Jul Maximum automorphism group for a 3-connected cubic graph
17 comment The family of vertex-transitive graphs you have in mind are in fact best possible among large enough cubic vertex-transitive graphs. (n=100 or so should already suffice.) This is shown
in the paper : "Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs", arxiv.org/abs/1010.2546. Therefore, a counter-example to
your conjecture will necessarily be not vertex-transitive.
May frobenius group
26 comment Actually, the Wikipedia page answers both queries (it gives more examples and mentions the odd dihedral groups) and also point towards the fact that being a semidirect product is not
enough. It's clear that the author has not even glanced at that page and I would close this question as not being research level.
May frobenius group
26 comment en.wikipedia.org/wiki/Frobenius_group Not all semidirect products are Frobenius.
29 awarded Citizen Patrol
25 awarded Yearling
Mar comment Smallest non-isomorphic strongly regular graphs
21 The links appear to be broken. In the meantime, win.tue.nl/~aeb/graphs/srg/srgtab.html has some information.
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Expected value of sum of fractions
up vote 1 down vote favorite
Suppose $r$ is a set of attributes with probabilities while $p$ is a set of attributes without probabilities. For example, say that $r$ = {$a$:0.4, $b$:1.0} and $p$ = {$a$, $c$}. (Here, $a.prob =
0.4$ and $b.prob = 1.0$.) There are two possible worlds for $r$: {$a$, $b$} with probability 0.4*1 = 0.4 and {$b$} with probability (1-0.4)*1 = 0.6.
I would like to efficiently compute the following equation:
$F(r,p)$ = $\sum_{r' \in 2^r} (\Pi_{a \in r'} a.c)(\Pi_{a \in r-r'} 1 - a.c)\frac{|W(r')\cap p|}{|W(r')|}$
where $W(r')$ strips off the probabilities from $r'$ (e.g., $W$({$b$:1.0}) = {$b$}). Hence, $F(r,p)$ = 0.4 * 1/2 + 0.6 * 0 = 0.2.
The above equation is clearly inefficient to compute because of the exponential size of the power set $2^r$. The approximation
$F'(r,p)$ = $\frac{\sum_{a \in r \cap p} a.prob}{\sum_{a \in r} a.prob}$
does not seem accurate because $F'(r,p)$ = 0.4 / 0.4+1 = 0.285 $\neq$ 0.2 = $F(r, p)$.
Is $F'$ still an theoretically approximate result or is there some better way to efficiently compute $F$ (say in polynomial time)? What are the related math problems?
st.statistics computational-complexity
How do evaluate that when $|W(r')|=0$? Also, is this notation standard anywhere? Is it ok to change it? – Douglas Zare May 11 '11 at 23:36
Good point, I only want to use non-empty r' values. The notation is not standard and can be changed. Thanks. – Steven May 12 '11 at 3:47
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Assignment for Friday, 10-30-09
Log onto a computer with your group.
On the Novell Screen go to:
Interactive Journey Through Physics
–Circular Motion 1
–Circular Motion 2
Do all parts except MCAT Questions
Submit one set of answers for the group. You can submit by paper, or copy/paste to the comment section below (just do it as an anonymous poster).
1 comment:
1. Kevin I. Hazel M. Yoshika W.
Circular Motion 1.
problem 1:
A conical pendulum consists of a 380-g bob attached to a string 35-cm long that makes an angle of 15 degrees with the verticle. What is the centripetal force acting on the bob?
Answer: .998
Problem 2:
a ball of mass .1kg moves in a circle of radius .2m with a constant speed of 4m/s. Acceleration?
Problem 3:
A ball moves in a circle of radius 1 m at a constant speed. The period of motion is 2 seconds. What is the speed of the ball?
Answer:π m/s
Problem 4:
The ride in question has a 44.5 N chiar which hangs freely from a 9.14 m long chain attached to a pivot on the top of a tall tower. When a child enters the ride, the chain is hanging straight
down. The child is then attached to the chair with a seat belt and shoulder harness. When the ride starts up,the chain rotates about the tower. Soon the chain reaches its max. speed and remains
rotating at that speed. It rotates about the tower once every 3.0 seconds. When you ask the operator, he says that the ride is perfectly safe. He demonstrates this by sitting in the stationary
chair. The chain creaks but holds and he weighs 890 N. Has the operator shown that this ride is safe for a 222.5 N child?
Answer: No, he has not shown that it is safe and the child should not ride.
Circular motion 2
Problem 1:
a wheel is set into motion so that it experiences a constant angular acceleration of .2rad/s. If it starts from rest, what is its angular displacement after 5 seconds?
Answer: 143.2 degrees
Problem 2:
A ball moves in a circle with a radius of 15cm. Its angular displacement is given by θ=2t^2. Assume that it starts from rest. What is its angular acceleration?
Answer:4 rad/s^2
Problem 3:
A solid rotating ball has an angular velocity about an axis through its center of 8 rad/s when it is acted upon by a force. This force gives the ball an angular acceleration of 1 rad/s for 4
seconds. By what factor is its kinetic energy increased?
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a. Find the surface area of the container. Show your work. b. Find the volume of the container. Show your work. c. The company will manufacture the container if the surface area to volume ratio
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The triangular prism is made up of 5 sides. We have two congruent triangles, and three rectangles (all of which are different in dimensions). Remember that the area of a triangle is\[A_{\rm
triangl e} = {b*h \over 2}\]and for a rectangle\[A_{\rm rectangle} = b*h\] The surface area is just the sum of all the areas of the sides. The easiest way to find the volume of the triangular
prism is using the triangle side as the "base" and 15 in. side as the "height." The volume equation is\[V = B*h\]where B is the area of the base (the triangle side). The surface area to volume
ratio is simply\[r = {SA \over V}\]
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Probability Abstract
Probability Abstracts 68
This document contains abstracts 1907-1959. They have been mailed on April 30, 2002.
1907. BROWNIAN EXCURSIONS AND PARISIAN BARRIER OPTIONS: A NOTE
Michael Schr\"{o}der
This note re-addresses the Paris barrier options proposed by Yor and
collaborators and their valuation using the Laplace transform approach. The
notion of Paris barrier options, based on excursion theory and using the
Brownian meander, is extended such that their valuation is now possible at any
point during their lifespan. The pertinent Laplace transforms are modified when
1908. EXISTENCE OF QUASI-STATIONARY MEASURES FOR ASYMMETRIC ATTRACTIVE PARTICLE SYSTEMS ON $\ZZ^D$
A. Asselah, F. Castell
We show the existence of non-trivial quasi-stationary measures for
conservative attractive particle systems on $\ZZ^d$ conditioned on avoiding an
increasing local set $\A$. Moreover, we exhibit a sequence of measures
$\{\nu_n\}$, whose $\omega$-limit set consists of quasi-stationary measures.
For zero range processes, with stationary measure $\nur$, we prove the
existence of an $L^2(\nur)$ nonnegative eigenvector for the generator with
Dirichlet boundary on $\A$, after establishing a priori bounds on the
Aureli Alabert, Marco Ferrante
We consider linear n-th order stochastic differential equations on [0,1],
with linear boundary conditions supported by a finite subset of [0,1]. We study
some features of the solution to these problems, and especially its conditional
independence properties of Markovian type.
1910. ASYMPTOTIC OF THE HEAT KERNEL IN GENERAL BENEDICKS DOMAINS
P.Collet, S.Martinez, J.San Martin
Using a new inequality relating the heat kernel and the probability of
survival, we prove asymptotic ratio limit theorems for the heat kernel (and
survival probability) in general Benedicks domains. In particular, the
dimension of the cone of positive harmonic measures with Dirichlet boundary
condition can be derived from the rate of convergence to zero of the heat
kernel (or the survival probability).
Wlodek Bryc
In this paper we study empirical measures which can be thought as a decoupled
version of the empirical measures generated by random matrices. We prove the
large deviation principle with the rate function, which is finite only on
product measures and hence is non-convex. As a corollary, we derive a large
deviations principle for (univariate) average empirical measures with the rate
function that superficially resembles the rate function of random matrices, but
may be concave.
1912. RANDOM MATRICES, NON-COLLIDING PROCESSES AND QUEUES
Neil O'Connell
This is survey of some recent results connecting random matrices,
non-colliding processes and queues.
Neil O'Connell
In [O'Connell and Yor (2002)] a path-transformation G was introduced with the
property that, for X belonging to a certain class of random walks on the
integer lattice, the transformed walk G(X) has the same law as that of the
original walk conditioned never to exit a type-A Weyl chamber. In this paper,
we show that G is closely related to the Robinson-Schensted algorithm, and use
this connection to give a new proof of the above representation theorem. The
new proof is valid for a larger class of random walks and yields additional
information about the joint law of X and G(X). The corresponding results for
the Brownian model are recovered by Donsker's theorem. These are connected with
Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix
theory. The connection we make between the path-transformation G and the RS
algorithm also provides a new formula and interpretation for the latter. This
can be used to study properties of the RS algorithm and, moreover, extends
easily to a continuous setting.
1914. THE BROWNIAN WEB
L. R. G. Fontes, M. Isopi, C. M. Newman, K. Ravishankar
Arratia, and later T\'oth and Werner, constructed random processes that
formally correspond to coalescing one-dimensional Brownian motions starting
from every space-time point. We extend their work by constructing and
characterizing what we call the {\em Brownian Web} as a random variable taking
values in an appropriate (metric) space whose points are (compact) sets of
paths. This leads to general convergence criteria and, in particular, to
convergence in distribution of coalescing random walks in the scaling limit to
the Brownian Web.
R. van der Hofstad, F. den Hollander, W. Koenig
In this paper we present a new and flexible method to show that, in one
dimension, various self-repellent random walks converge to self-repellent
Brownian motion in the limit of weak interaction after appropriate space-time
scaling. Our method is based on cutting the path into pieces of an
appropriately scaled length, controlling the interaction between the different
pieces, and applying an invariance principle to the single pieces. In this way
we show that the self-repellent random walk large deviation rate function for
the empirical drift of the path converges to the self-repellent Brownian motion
large deviation rate function after appropriate scaling with the interaction
parameters. The method is considerably simpler than the approach followed in
our earlier work, which was based on functional analytic arguments applied to
variational representations and only worked in a very limited number of
We consider two examples of a weak interaction limit: (1) vanishing
self-repellence, (2) diverging step variance. In example (1), we recover our
earlier scaling results for simple random walk with vanishing self-repellence
and show how these can be extended to random walk with steps that have zero
mean and a finite exponential moment. Moreover, we show that these scaling
results are stable against adding self-attraction, provided the self-repellence
dominates. In example (2), we prove a conjecture by Aldous for the scaling of
self-avoiding walk with diverging step variance. Moreover, we consider
self-avoiding walk on a two-dimensional horizontal strip such that the steps in
the vertical direction are uniform over the width of the strip and find the
scaling as the width tends to infinity.
1916. LARGE DEVIATIONS FOR THE ONE-DIMENSIONAL EDWARDS MODEL
R. van der Hofstad, F. den Hollander, W. Koenig
In this paper we prove a large deviation principle for the empirical drift of
a one-dimensional Brownian motion with self-repellence called the Edwards
model. Our results extend earlier work in which a law of large numbers,
respectively, a central limit theorem were derived. In the Edwards model a path
of length $T$ receives a penalty $e^{-\beta H_T}$, where $ H_T$ is the
self-intersection local time of the path and $\beta\in(0,\infty)$ is a
parameter called the strength of self-repellence. We identify the rate function
in the large deviation principle for the endpoint of the path as $\beta^{\frac
23} I(\beta^{-\frac 13}\cdot)$, with $I(\cdot)$ given in terms of the principal
eigenvalues of a one-parameter family of Sturm-Liouville operators. We show
that there exist numbers $0<b^{**}<b^*<\infty$ such that: (1) $I$ is linearly
decreasing on $[0,b^{**}]$; (2) $I$ is real-analytic and strictly convex on
$(b^{**},\infty)$; (3) $I$ is continuously differentiable at $b^{**}$; (4) $I$
has a unique zero at $b^*$. (The latter fact identifies $b^*$ as the asymptotic
drift of the endpoint.) The critical drift $b^{**}$ is associated with a
crossover in the optimal strategy of the path: for $b\geq b^{**}$ the path
assumes local drift $b$ during the full time $T$, while for $0\leq b<b^{**}$ it
assumes local drift $b^{**}$ during time $\frac{b^{**}+b}{2b^{**}}T$ and local
drift $-b^{**}$ during the remaining time $\frac{b^{**}-b}{2b^{**}}T$. Thus, in
the second regime the path makes an overshoot of size $\frac{b^{**}-b}{2}T$ in
order to reduce its intersection local time.
1917. FLOWS, COALESCENCE AND NOISE
Yves Le Jan and Olivier Raimond
We are interested in stationary ``fluid'' random evolutions with independent
increments. Under some mild assumptions, we show they are solutions of a
stochastic differential equation (SDE). There are situations where these
evolutions are not described by flows of diffeomorphisms, but by coalescing
flows or by flows of probability kernels.
In an intermediate phase, for which there exists a coalescing flow and a flow
of kernels solution of the SDE, a classification is given: All solutions of the
SDE can be obtained by filtering the coalescing motion with respect to a
sub-noise containing the Gaussian part of its noise. Thus, the coalescing
motion cannot be described by a white noise.
1918. CONVERGENCE IN ENERGY-LOWERING (DISORDERED) STOCHASTIC SPIN SYSTEMS
Emilio De Santis, Charles M. Newman
We consider stochastic processes, S^t \equiv (S_x^t : x \in Z^d), with each
S_x^t taking values in some fixed finite set, in which spin flips (i.e.,
changes of S_x^t) do not raise the energy. We extend earlier results of
Nanda-Newman-Stein that each site x has almost surely only finitely many flips
that strictly lower the energy and thus that in models without zero-energy
flips there is convergence to an absorbing state. In particular, the assumption
of finite mean energy density can be eliminated by constructing a
percolation-theoretic Lyapunov function density as a substitute for the mean
energy density. Our results apply to random energy functions with a
translation-invariant distribution and to quite general (not necessarily
Markovian) dynamics.
1919. FREE L\'EVY PROCESSES ON DUAL GROUPS
Uwe Franz
We give a short introduction to the theory of L\'evy processes on dual
groups. As examples we consider L\'evy processes with additive increments and
L\'evy processes on the dual affine group.
1920. SCALED BOOLEAN ALGEBRAS
Michael Hardy
Scaled Boolean algebras are a category of mathematical objects that arose
from attempts to understand why the conventional rules of probability should
hold when probabilities are construed, not as frequencies or proportions or the
like, but rather as degrees of belief in uncertain propositions. This paper
separates the study of these objects from that not-entirely-mathematical
problem that motivated them. That motivating problem is explicated in the first
section, and the application of scaled Boolean algebras to it is explained in
the last section. The intermediate sections deal only with the mathematics. It
is hoped that this isolation of the mathematics from the motivating problem
makes the mathematics clearer.
Itai Benjamini, Gil Kalai, Oded Schramm
Let $0<a<b<\infty$, and for each edge $e$ of $Z^d$ let $\omega_e=a$ or
$\omega_e=b$, each with probability 1/2, independently. This induces a random
metric $\dist_\omega$ on the vertices of $Z^d$, called first passage
percolation. We prove that for $d>1$ the distance $dist_\omega(0,v)$ from the
origin to a vertex $v$, $|v|>2$, has variance bounded by $C |v|/\log|v|$, where
$C=C(a,b,d)$ is a constant which may only depend on $a$, $b$ and $d$. Some
related variants are also discussed
1922. FUNCTIONAL CENTRAL LIMIT THEOREMS FOR VICIOUS WALKERS
Makoto Katori and Hideki Tanemura
We consider the diffusion scaling limit of the vicious walkers, which is a
model of nonintersecting random walks. We show a functional central limit
theorem for the model and derive two types of nonintersecting Brownian motions,
in which we impose nonitersecting condition in the finite time interval $(0,T]$
(resp. in the infinite time interval $(0,\infty)$) for the first-type (resp.
second-type). The first-type is a temporally inhomogeneous diffusion, and the
second-type is a temporally homogeneous diffusion called Dyson's model of
Brownian motions. We also study the vicious walkers with wall restriction and
prove the functional central limit theorem in the diffusion scaling limit.
Nils Berglund and Barbara Gentz
We consider slow-fast systems of differential equations, in which both the
slow and fast variables are perturbed by additive noise. When the deterministic
system admits a uniformly asymptotically stable slow manifold, we show that the
sample paths of the stochastic system are concentrated in a neighbourhood of
the slow manifold, which we construct explicitly. Depending on the dynamics of
the reduced system, the results cover time spans which can be exponentially
long in the noise intensity squared (that is, up to Kramers' time). We give
exponentially small upper and lower bounds on the probability of exceptional
paths. If the slow manifold contains bifurcation points, we show similar
concentration properties for the fast variables corresponding to
non-bifurcating modes. We also give conditions under which the system can be
approximated by a lower-dimensional one, in which the fast variables contain
only bifurcating modes.
Alexander R. Pruss
Let $X_1,X_2,...$ be a sequence of independent and identically distributed
random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the
positive sequence $\tau_n$ and the positive increasing sequence $a_n$, we give
necessary and sufficient conditions for the convergence of $\sum_{n=1}^\infty
\tau_n P(|S_n|\ge \epsilon a_n)$ for all $\epsilon>0$, generalizing Baum and
Katz's (1965) generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large
numbers, also allowing us to characterize the convergence of the above series
in the case where $\tau_n=n^{-1}$ and $a_n=(n\log n)^{1/2}$ for $n\ge 2$,
thereby answering a question of Spataru. Moreover, some results for
non-identically distributed independent random variables are obtained by a
recent comparison inequality. Our basic method is to use a central limit
theorem estimate of Nagaev (1965) combined with the Hoffman-Jorgensen
inequality (1974).
1925. A CHAOTIC DECOMPOSITION FOR L\'EVY PROCESSES ON MANIFOLDS WITH AN APPLICATION TO PROCESSES OF MEIXNER'S TYPE
E. Lytvynov, D. Mierzejewski
It is well known that between all processes with independent increments,
essentially only the Brownian motion and the Poisson process possess the
chaotic decomposition property (CRP). Thus, a natural question appears: What is
an appropriate analog of the CRP in the case of a general L\'evy process. At
least three approaches are possible here. The first one is due to It\^o. It
uses the CRP of the Brownian motion and the Poisson process, as well as the
representation of a L\'evy process through those processes. The second approach
is due to Nualart and Schoutens and consists in representing any
square-integrable random variable as a sum of multiple stochastic integrals
constructed with respect to a family of orthogonalized centered power jumps
processes. Finally, the third approach uses the idea of orthogonalization of
polynomials with respect to a probability measure defined on the dual of a
nuclear space. The main aims of the present paper are to develop the three
approaches in the case of a general (${\mathbb R}$-valued) L\'evy process on a
Riemannian manifold and (what is more important) to understand a relationship
between these approaches. We apply the obtained results to the processes of
Meixner's type--the gamma, Pascal, and Meixner processes. In this case, the
analysis related to the orthogonalized polynomials becomes essentially simpler
and reacher than in the general case.
1926. A SINGULAR PARABOLIC ANDERSON MODEL
Carl Mueller and Roger Tribe
We give a new example of a measure-valued process without a density, which
arises from a stochastic partial differential equation with a multiplicative
noise term. This process has some unusual properties. We work with the heat
equation with a random potential: u_t=Delta u+kuF. Here k>0 is a small number,
and x lies in d-dimensional Euclidean space with d>2. F is a Gaussian noise
which is uncorrelated in time, and whose spatial covariance equals |x-y|^(-2).
The exponent 2 is critical in the following sense. For exponents less than 2,
the equation has function-valued solutions, and for exponents higher than 2, we
do not expect solutions to exist. This model is closely related to the
parabolic Anderson model; we expect solutions to be small, except for a
collection of high peaks. This phenomenon is called intermittency, and is
reflected in the singular nature of our process. Solutions exist as singular
measures, under suitable assumptions on the initial conditions and for
sufficiently small k. We investigate various properties of the solutions, such
as dimension of the support and long-time behavior. As opposed to the
super-Brownian motion, which satisfies a similar equation, our process does not
have compact support, nor does it die out in finite time. We use such tools as
scaling, self-duality and moment formulae.
1927. AN ASYMPTOTIC LINK BETWEEN LUE AND GUE AND ITS SPECTRAL INTERPRETATION
Yan Doumerc
We use a matrix central-limit theorem which makes the Gaussian Unitary
Ensemble appear as a limit of the Laguerre Unitary Ensemble together with an
observation due to Johansson in order to derive new representations for the
eigenvalues of GUE. For instance, it is possible to recover the celebrated
equality in distribution between the maximal eigenvalue of GUE and a
last-passage time in some directed brownian percolation. Similar identities for
the other eigenvalues of GUE also appear.
N. Anantharaman
We study the approximation of action minimizing measures, for a mechanical
Lagrangian on the torus, by the probability densities given by quasi-periodic
eigenfunctions of the Schrodinger operator.
1929. HAUSDORFF DIMENSIONS FOR $SLE_6$
Vincent Beffara
We prove that the Hausdorff dimension of the trace of $SLE_6$ is almost
surely 7/4 and give a more direct derivation of the result (due to
Lawler-Schramm-Werner) that the dimension of its boundary is 4/3. We also prove
that for all $\kappa<8$ the $SLE_\kappa$ trace has cut-points.
Applying the technique of characteristic functions developped for
one-dimensional regular surfaces (curves) with compact support, we obtain the
distribution of hitting probabilities for a wide class of finite membranes on
square lattice. Then we generalize it to multi-dimensional finite membranes on
hypercubic lattice. Basing on these distributions, we explicitly construct the
Brownian self-transport operator which governs the Laplacian transfer. In order
to verify the accuracy of the distribution of hitting probabilities, numerical
analysis is carried out for some particular membranes.
1931. ON EXPONENTIAL STABILITY OF WONHAM FILTER
P. Chigansky, R. Liptser
We give elementary proof of a stability result concerning an exponential
asymptotic ($t\to\infty$) for filtering estimates generated by wrongly
initialized Wonham filter. This proof is based on new exponential bound having
independent interest.
David Gamarnik
Our model is a constrained homogeneous random walk in a nonnegative orthant
Z_+^d. The convergence to stationarity for such a random walk can often be
checked by constructing a Lyapunov function. The same Lyapunov function can
also be used for computing approximately the stationary distribution of this
random walk, using methods developed by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly is an undecidable problem: no algorithm can exist to achieve this task.
We then prove that computing large deviation rates for this model is also an
undecidable problem. We extend these results to a certain type of queueing
systems. The implication of these results is that no useful formulas for
computing stationary probabilities and large deviations rates can exist in
these systems.
1933. ON THE SCALING LIMIT OF PLANAR SELF-AVOIDING WALK
Gregory F. Lawler, Oded Schramm, Wendelin Werner
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in
the square lattice with no self-intersection. A planar self-avoiding polygon
(SAP) is a loop with no self-intersection. In this paper we present conjectures
for the scaling limit of the uniform measures on these objects. The conjectures
are based on recent results on the stochastic Loewner evolution and
non-disconnecting Brownian motions. New heuristic derivations are given for the
critical exponents for SAWs and SAPs.
N. V. Krylov, R. Liptser
Convergence of stochastic processes with jumps to diffusion processes is
investigated in the case when the limit process has discontinuous coefficients.
An example is given in which the diffusion approximation of a queueing model
yields a diffusion process with discontinuous diffusion and drift coefficients.
1935. ON THE UNIVERSALITY OF THE PROBABILITY DISTRIBUTION OF THE PRODUCT $B^{-1}X$ OF RANDOM MATRICES
Joshua Feinberg
Consider random matrices $A$, of dimension $m\times (m+n)$, drawn from an
ensemble with probability density $f(\rmtr AA^\dagger)$, with $f(x)$ a given
appropriate function. Break $A = (B,X)$ into an $m\times m$ block $B$ and the
complementary $m\times n$ block $X$, and define the random matrix $Z=B^{-1}X$.
We calculate the probability density function $P(Z)$ of the random matrix $Z$
and find that it is a universal function, independent of $f(x)$. Universality
of $P(Z)$ is, essentially, a consequence of rotational invariance of the
probability ensembles we study. More generally, $P(Z)$ must be independent, of
course, of any common scale of the distribution functions of $B$ and $X$. As an
application, we study the distribution of solutions of systems of linear
equations with random coefficients, and extend a classic result due to Girko.
1936. OCCUPATION DENSITIES FOR SPDE'S WITH REFLECTION
Lorenzo Zambotti
We consider the solution (u,\eta) of the white-noise driven stochastic
partial differential equation with reflection on the space interval [0,1]
introduced by Nualart and Pardoux. First, we prove that at any fixed time t>0,
the measure \eta([0,t]\times d\theta) is absolutely continuous w.r.t. the
Lebesgue measure d\theta on (0,1). We characterize the density as a family of
additive functionals of u, and we interpret it as a renormalized local time at
0 of (u(t,\theta))_{t\geq 0}. Finally we study the behaviour of \eta at the
boundary of [0,1]. The main technical novelty is a projection principle from
the Dirichlet space of a Gaussian process, vector-valued solution of a linear
SPDE, to the Dirichlet space of the process u.
1937. A NOTE ON EDGE ORIENTED REINFORCED RANDOM WALKS AND RWRE
N. Enriquez and C. Sabot
This work introduces the notion of edge oriented reinforced random walk which
proposes in a general framework an alternative understanding of the annealed
law of random walks in random environment.
Russell Lyons and Jeffrey E. Steif
We study a class of stationary processes indexed by $\Z^d$ that are defined
via minors of $d$-dimensional Toeplitz matrices. We obtain necessary and
sufficient conditions for the existence of a phase transition (phase
multiplicity) analogous to that which occurs in statistical mechanics. The
absence of a phase transition is equivalent to the presence of a strong $K$
property, a particular strengthening of the usual $K$ (Kolmogorov) property. We
show that all of these processes are Bernoulli shifts (isomorphic to i.i.d.\
processes in the sense of ergodic theory). We obtain estimates of their
entropies and we relate these processes via stochastic domination to product
Russell Lyons
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We initiate a detailed study of the
discrete analogue, the most prominent example of which has been the uniform
spanning tree measure. Our main results concern relationships with matroids,
stochastic domination, negative association, completeness for infinite
matroids, tail triviality, and a method for extension of results from
orthogonal projections to positive contractions. We also present several new
avenues for further investigation, involving Hilbert spaces, combinatorics,
homology, and group representations, among other areas.
G.G. Amosov
We introduce Markovian cocycle perturbations of the groups of transformations
associated with the classical and quantum stochastic processes with stationary
increments, which are characterized by a localization of the perturbation to
the algebra of events of the past. It is namely the definition one needs
because the Markovian perturbations of the Kolmogorov flows associated with the
classical and quantum noises result in the perturbed group of transformations
which can be decomposed in the sum of a part associated with deterministic
stochastic processes lying in the past and a part associated with the noise
isomorphic to the initial one. This decomposition allows to obtain some analog
of the Wold decomposition for classical stationary processes excluding a
nondeterministic part of the process in the case of the stationary quantum
stochastic processes on the von Neumann factors which are the Markovian
perturbations of the quantum noises. For the classical stochastic process with
noncorrelated increaments it is constructed the model of Markovian
perturbations describing all Markovian cocycles up to a unitary equivalence of
the perturbations. Using this model we construct Markovian cocyclies
transformating the Gaussian state $\rho $ to the Gaussian states equivalent to
$\rho $.
1941. ON NODAL LINES OF NEUMANN EIGENFUNCTIONS
Krzysztof Burdzy
We present a new method for locating the nodal line
of the second eigenfunction for the Neumann problem
in a planar domain. The technique is based on the
`mirror coupling' of reflected Brownian motions.
1942. DUALITIES FOR THE DOMANY-KINZEL MODEL
Makoto Katori, Norio Konno, Aidan Sudbury and Hideki Tanemura
We study the Domany-Kinzel model, which is
a class of discrete-time Markov processes in one-dimension with two parameters
$(p_{1}, p_{2}) \in [0,1]^{2}.$ When $p_{1}=\alpha \beta$ and $p_{2}=\alpha
( 2 \beta-\beta^{2})$ with $(\alpha, \beta) \in [0,1]^{2}$, the process
can be identified with the mixed site-bond oriented percolation model
on a square lattice with probabilities $\alpha$ of a site being open and
$\beta$ of a bond being open.
This paper treats dualities for the Domany-Kinzel model $\xi^A _t$
and the DKdual $\eta^A _t$ starting from $A.$ We prove that
(i) $E(x^{|\xi^A _t \cap B|}) = E(x^{|\xi^B _t \cap A|})$ if $x=1 - (2p_1-p_2)/p_1 ^2$,
(ii) $E(x^{|\xi^A _t \cap B|}) = E(x^{|\eta^B _t \cap A|})$ if $x=1 - (2p_1-p_2)/p_1$,
(iii) $E(x^{|\eta^A _t \cap B|}) = E(x^{|\eta^B _t \cap A|})$ if $x=1 - (2p_1-p_2),$
as long as one of $A, B$ is finite and $p_2\leq p_{1}$.
katori@phys.chuo-u.ac.jp norio@mathlab.sci.ynu.ac.jp
Aidan.Sudbury@sci.monash.edu.au tanemura@math.s.chiba-u.ac.jp
1943. TWO WORKLOAD PROPERTIES FOR BROWNIAN NETWORKS
M. Bramson and R. J. Williams
As one approach to dynamic scheduling problems for open
stochastic processing networks, J. M. Harrison has proposed
the use of formal heavy traffic approximations known as
Brownian networks. A key step in this approach is a
reduction in dimension of a Brownian network, due to
Harrison and Van Mieghem, in which the `queue length' process
is replaced by a `workload' process.
In this paper, we establish two properties of these workload
processes. Firstly, we derive a formula for
the dimension of such processes. For a given Brownian
network, this dimension is unique. However, there are
infinitely many possible choices for the workload process.
Harrison has proposed a `canonical' choice, which
reduces the possibilities to a finite number.
Our second result provides sufficient conditions for this
canonical choice to be valid and for it to yield a
non-negative workload process. The assumptions and proofs
for our results involve only first-order model parameters.
williams@math.ucsd.edu bramson@math.umn.edu
Steven N. Evans
The $(n-1)$-dimensional simplex is the collection of
probability measures on a set with $n$ points. Many applied
situations result in simplex-valued data or in stochastic
processes that have the simplex as their state space.
In this paper we study a large class of simplex-valued
diffusion processes that are constructed by first
``coordinatising'' the simplex with the points of a smooth
hypersurface in such a way that several points on the
hypersurface may correspond to a given point on the simplex,
and then mapping forward the canonical Brownian
motion on the hypersurface. For example, a particular
instance of the Fleming-Viot process on $n$ points
arises from Brownian motion on the $(n-1)$-dimensional
sphere. The Brownian motion on the hypersurface has the
normalised Riemannian volume as its equilibrium
distribution. It is straightforward to compute the
corresponding distribution on the simplex, and this provides
a large class of interesting probability measures on the
Alexey V. Lebedev
We consider maximal branching processes (MBP) introduced by
J.Lamperti (1970) which resemble the Galton-Watson branching processes,
but with one difference: every time we get maximum of offspring numbers
instead of their sum. We generalize MBP from $Z_+$ to $R_+$
(by analogy with Jirina processes). Ergodic theorem is proved, some
properties are studied, examples are given. Limit theorems for stationary
distributions are obtained. Applications in the queueing theory are shown
(namely, for gated infinite-server queues researched by S.Browne et all.(1992)).
Lancelot F. James
This article discusses the usage of a partition based
Fubini Calculus for Poisson processes. The approach is an
amplification of Bayesian techniques developed in Lo and
Weng for gamma/Dirichlet processes. Applications to models
are considered which fall within an inhomogeneous spatial
extension of the size biased framework used in Perman,
Pitman and Yor. Among some of the results; an explicit
partition based calculus is then developed for such models,
which also includes a series of important exponential
change of measure formulae. These results
are then applied
to solve the mostly unknown calculus for spatial Levy-Cox
moving average models.
The analysis then proceeds to exploit a structural feature
of a scaling operation which arises in
Brownian excursion theory.
From this a series of new mixture representations
and posterior characterizations for large classes of
random measures, including probability measures, are
given.These results are applied to yield new
results/identities related to the large class of
two-parameter Poisson Dirichlet models.
The results also yields easily perhaps
the most general and certainly quite
informative characterizations of extensions of the Markov-Krein
correspondence exhibited by linear functionals of
Dirichlet processes. This article then defines a natural
extension of Doksum's Neutral to the Right priors (NTR)
to a spatial setting. NTR models are practically
synonymous with exponential functions of subordinators and
arise in Bayesian nonparametric survival models. It is shown
that manipulation of the exponential formulae makes
what has been formidable analysis transparent. Additional
interesting identities related to the the Dirichlet process
and other measures are developed. Based on practical
considerations, computational procedures which are
extensions of the Chinese restaurant process are also
1947. INTERSECTIONS OF BROWNIAN MOTIONS
Davar Khoshnevisan
This article presents a survey of the theory of the
intersections of Brownian motion paths. Among
other things, we present a truly elementary
proof of a classical theorem of A. Dvoretzky,
P. Erdös and S. Kakutani. This proof is motivated
by old ideas of P. Lévy that were originally used
to investigate the curve of
planar Brownian motion.
1948. STABILITY OF PARABOLIC HARNACK INEQUALITIES
Richard Bass and Martin Barlow
Let $(G,E)$ be a graph with weights $\{a_{xy}\}$ for which a
parabolic Harnack inequality holds with space-time scaling
exponent $\beta\ge 2$.
Suppose $\{a'_{xy}\}$ is another
set of weights that are comparable to $\{a_{xy}\}$.
We prove that this parabolic Harnack inequality
also holds for $(G,E)$ with the weights $\{a'_{xy}\}$.
We also give necessary and sufficient conditions
for this parabolic Harnack inequality to hold.
bass@math.uconn.edu barlow@math.ubc.ca
1949. EINSTEIN RELATION FOR A CLASS OF INTERFACE MODELS
Roberto H. Schonmann
A class of SOS interface models which can be seen as simplified
stochastic Ising model interfaces is studied.
In the absence of an external field the long-time fluctuations
of the interface are shown to behave as Brownian motion with
diffusion coefficient $(\sigma^{\text{GK}})^2$ given by a Green-Kubo
When a small external field $h$ is applied, it is shown that
the shape of the interface converges exponentially fast to a stationary
distribution and the interface moves with an asymptotic velocity $v(h)$.
The mobility is shown to exist and to satisfy the
Einstein relation: $(dv/dh)(0) = \beta (\sigma^{\text{GK}})^2$,
where $\beta$ is the inverse temperature.
Yosef Rinott and Vladimir Rotar
Let $W$ be the sum of dependent random variables, and $h(x)
$ be a function. This paper provides an Edgeworth expansion
of an arbitrary ``length'' for $E\{h(W)\}$ in terms of
certain characteristics of dependency, and of the
smoothness of $h$ and/or the distribution of $W$. The core
of the class of dependency structures for which these
characteristics are meaningful is the local dependency, but
in fact, the class is essentially wider. The remainder is
estimated in terms of Lyapunov's ratios. The proof is based
on a Stein's method. The most typical example for the
results of this paper is mixing on graphs, that is when the
parameter indexing the summands, which is usually thought
of as a ``time'' or ``space'' parameter, has values which
may be identified with vertices of a graph. If the graph is
a usual integer valued lattice in $\QTR{Bbb}{Z}^{k}$, with
edges connecting only nearest vertices, we deal with the
usual mixing scheme for random fields, and for $k=1$ - with
a process on a line. If the graph is arbitrary, the scheme
is more complicated. This is especially true when the graph
is random, and its structure may depend on the values of
summands. In this case the above dependency neighborhoods
may be random too.
Tatiana Belkina and Vladimir Rotar
The paper concerns both controlled diffusion processes, and
processes in discrete time. We establish conditions under
which the strategy minimizing the expected value of a cost
functional has a much stronger property; namely, it minimizes
the random cost functional itself for all
realizations of the controlled process from a set, the
probability of which is close to one for large time
horizons. The main difference of the conditions mentioned
from those obtained earlier is that the former do not deal
with strategies optimal in the mean themselves but concern
a possibility of transition of the controlled process from
one state to another in a time with a finite expectation.
It makes the verification of these conditions in a number
of situations much easier.
Ben M. Hambly and Takashi Kumagai
A fractal field is a collection of fractals with, in general, different
Hausdorff dimensions, embedded in ${\bf R}^2$. We will construct diffusion
processes on such fields which behave as Brownian motion in ${\bf R}^2$
outside the fractals and as the appropriate fractal diffusion within
each fractal component of the field. We will discuss the properties of
the diffusion process in the case where the fractal components tile
${\bf R}^2$. By working in a suitable shortest path metric we will
establish heat kernel bounds and large deviation estimates which
determine the trajectories followed by the diffusion over short times.
hambly@maths.ox.ac.uk kumagai@kurims.kyoto-u.ac.jp
S. Cerrai, M. Roeckner
Following classical work by M.I. Freidlin and subsequent works
by R. Sowers and S. Peszat, we prove large deviation estimates
for the small noise limit of systems of stochastic
reaction-diffusion equations with globally Lipschitz but
unbounded diffusion coefficients, however,
assuming the reaction terms to be only locally Lipschitz
with polynomial growth. This generalizes results of the above
mentioned authors.
Our results apply, in particular, to systems of stochastic
Ginzburg-Landau equations with multiplicative noise.
V.I. Bogachev, M. Roeckner
It is proved that the symmetric diffusion operator
$\mathcal{L}f = \Delta f + <b, \nabla>$ on a complete
Riemannian Manifold of dimension $d$ is $L^p$-unique
provided that the vector field $b$ is locally in $L^p$
with respect to the Riemannian volume and $p>d$.
An analogous statement is proved for for elliptic
operators with non constant second order part.
P. Lescot, M. Roeckner
In this paper we solve the Kolmogorov equation and, as a
consequence, the martingale problem corresponding to a stochastic
differential equation of type
\[ dX_t=AX_t\,dt+b(X_t)\,dt+dY_t, \]
on a Hilbert space $E$, where $(Y_t)_{t\ge0}$ is a Levy process on
$E$, $A$ generates a $C_0$-semigroup on $E$ and $b:E\to E$. Our main
point is to allow unbounded $A$ and also singular (in particular,
non-continuous) $b$. Our approach is based on perturbation theory
of $C_0$-semigroups, which we apply to generalized Mehler semigroups
considered on $L^2(\mu)$, where $\mu$ is their respective invariant
measure. We apply our results, in particular, to stochastic heat
equations with Levy noise and singular drift.
M. Grothaus, Y.G. Kondratiev, E. Lytvynov, M. Roeckner
We investigate a scaling limit of gradient
stochastic dynamics associated to Gibbs states
in classical continuous systems on $\RR^d$,
$d \geq 1$. For these dynamics several scalings
have already been studied, see e.g. [Bro80].
The aim is to derive macroscopic quantities
from a given micro- or mesoscopic system. The
scaling we consider has been investigated in
[Bro80] and [Ros81]. Assuming that the underlying
potential is smooth, compactly supported and
positive, convergence of the generators of the
scaled stochastic dynamics, averaged with respect
to time, has been analyzed in [Spo86].
Another approach has been proposed in [GP85],
where the idea has been to prove convergence of
the corresponding resolvents.
We prove that the Dirichlet forms of the scaled
stochastic dynamics converge on a core of
functions to the Dirichlet form of a generalized
Ornstein-Uhlenbeck process. The proof is based
on the analysis and geometry on the configuration
space which was developed in [AKR98a],[AKR98b],
and works for general Gibbs measures of Ruelle
type. Hence, the underlying potential may have
a singularity at the origin, only has to be
bounded from below, and may not be compactly
supported. Therefore, singular interactions of
physical interest are covered, as e.g. the one
given by the Lennard-Jones potential, which is
studied in the theory of fluids. Furthermore,
using the Lyons-Zheng decomposition we give a
simple proof for the tightness of the scaled
processes. We also prove that the corresponding
generators, however, do not converge in the $L^2$-
sense. This settles a conjecture formulated in
[Bro80], [Ros81], [Spo86].
G.d. Prato, M. Roeckner
Existence of solutions to martingale problems
corresponding to singular dissipative stochastic
equations in Hilbert spaces are proved for any
initial condition. The solutions for the single
starting points form a conservative diffusion
process whose transition semigroup is shown to be
strong Feller. Uniqueness in a generalized sense
is proved also, and a number of applications is
1958. ON THE SPECTRUM OF A CLASS OF (NONSYMMETRIC) DIFFUSION OPERATORS
M. Roeckner, F.-Y. Wang
In terms of the upper bounds of a second order
elliptic operator acting on specific Lyapunov-type
functions with compact level sets, sufficient
conditions are presented for the corresponding
Dirichlet form to satisfy the Poincar\'e and
the super-Poincar\'e inequalities. Here the
elliptic operator is assumed to be symmetric on
$L^2(\mu)$ for some probability measure $\mu$.
As applications, we prove for a class of (non-
symmetric) diffusion operators generating
$C_0$-semigroups on $L^1(\mu)$ that their
$L^p(\mu)$-essential spectrum is empty. This
follows since we prove that their $C_0$-semigroups
are compact.
1959. SURFACE MEASURES AND TIGHTNESS OF CAPACITIES ON POISSON SPACE
V.I. Bogachev, O.V. Pugachev, M. Roeckner
We prove tightness of $(r,p)$--Sobolev capacities
on configuration spaces equipped with Poisson measure.
By using this result we construct surface measures
on configuration spaces in the spirit of
the Malliavin calculus.
A related Gauss--Ostrogradskii formula is obtained.
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Geometric Image Transformations
The functions in this section perform various geometrical transformations of 2D images. They do not change the image content but deform the pixel grid and map this deformed grid to the destination
image. In fact, to avoid sampling artifacts, the mapping is done in the reverse order, from destination to the source. That is, for each pixel
In case when you specify the forward mapping
The actual implementations of the geometrical transformations, from the most generic remap() and to the simplest and the fastest resize() , need to solve two main problems with the above formula:
• Extrapolation of non-existing pixels. Similarly to the filtering functions described in the previous section, for some BORDER_TRANSPARENT . This means that the corresponding pixels in the
destination image will not be modified at all.
• Interpolation of pixel values. Usually interpolation methods , where a polynomial function is fit into some neighborhood of the computed pixel resize() for details.
Converts image transformation maps from one representation to another.
C++: void convertMaps(InputArray map1, InputArray map2, OutputArray dstmap1, OutputArray dstmap2, int dstmap1type, bool nninterpolation=false )
Python: cv2.convertMaps(map1, map2, dstmap1type[, dstmap1[, dstmap2[, nninterpolation]]]) → dstmap1, dstmap2
• map1 – The first input map of type CV_16SC2 , CV_32FC1 , or CV_32FC2 .
• map2 – The second input map of type CV_16UC1 , CV_32FC1 , or none (empty matrix), respectively.
Parameters: • dstmap1 – The first output map that has the type dstmap1type and the same size as src .
• dstmap2 – The second output map.
• dstmap1type – Type of the first output map that should be CV_16SC2 , CV_32FC1 , or CV_32FC2 .
• nninterpolation – Flag indicating whether the fixed-point maps are used for the nearest-neighbor or for a more complex interpolation.
The function converts a pair of maps for remap() from one representation to another. The following options ( (map1.type(), map2.type()) (dstmap1.type(), dstmap2.type()) ) are supported:
• remap() ) are converted to a more compact and much faster fixed-point representation. The first output array contains the rounded coordinates and the second array (created only when
nninterpolation=false ) contains indices in the interpolation tables.
• Reverse conversion. Obviously, the reconstructed floating-point maps will not be exactly the same as the originals.
Calculates an affine transform from three pairs of the corresponding points.
The function calculates the
Calculates a perspective transform from four pairs of the corresponding points.
The function calculates the
Retrieves a pixel rectangle from an image with sub-pixel accuracy.
C++: void getRectSubPix(InputArray image, Size patchSize, Point2f center, OutputArray patch, int patchType=-1 )
Python: cv2.getRectSubPix(image, patchSize, center[, patch[, patchType]]) → patch
C: void cvGetRectSubPix(const CvArr* src, CvArr* dst, CvPoint2D32f center)
• src – Source image.
• patchSize – Size of the extracted patch.
Parameters: • center – Floating point coordinates of the center of the extracted rectangle within the source image. The center must be inside the image.
• dst – Extracted patch that has the size patchSize and the same number of channels as src .
• patchType – Depth of the extracted pixels. By default, they have the same depth as src .
The function getRectSubPix extracts pixels from src :
where the values of the pixels at non-integer coordinates are retrieved using bilinear interpolation. Every channel of multi-channel images is processed independently. While the center of the
rectangle must be inside the image, parts of the rectangle may be outside. In this case, the replication border mode (see borderInterpolate() ) is used to extrapolate the pixel values outside of the
Calculates an affine matrix of 2D rotation.
C++: Mat getRotationMatrix2D(Point2f center, double angle, double scale)
Python: cv2.getRotationMatrix2D(center, angle, scale) → retval
C: CvMat* cv2DRotationMatrix(CvPoint2D32f center, double angle, double scale, CvMat* map_matrix)
• center – Center of the rotation in the source image.
Parameters: • angle – Rotation angle in degrees. Positive values mean counter-clockwise rotation (the coordinate origin is assumed to be the top-left corner).
• scale – Isotropic scale factor.
• map_matrix – The output affine transformation, 2x3 floating-point matrix.
The function calculates the following matrix:
The transformation maps the rotation center to itself. If this is not the target, adjust the shift.
Inverts an affine transformation.
C++: void invertAffineTransform(InputArray M, OutputArray iM)
Python: cv2.invertAffineTransform(M[, iM]) → iM
Parameters: • M – Original affine transformation.
• iM – Output reverse affine transformation.
The function computes an inverse affine transformation represented by M :
The result is also a M .
Remaps an image to log-polar space.
C: void cvLogPolar(const CvArr* src, CvArr* dst, CvPoint2D32f center, double M, int flags=CV_INTER_LINEAR+CV_WARP_FILL_OUTLIERS )
The function cvLogPolar transforms the source image using the following transformation:
□ Forward transformation (CV_WARP_INVERSE_MAP is not set):
□ Inverse transformation (CV_WARP_INVERSE_MAP is set):
The function emulates the human “foveal” vision and can be used for fast scale and rotation-invariant template matching, for object tracking and so forth. The function can not operate in-place.
• An example using the geometric logpolar operation in 4 applications can be found at opencv_source_code/samples/cpp/logpolar_bsm.cpp
Applies a generic geometrical transformation to an image.
C++: void remap(InputArray src, OutputArray dst, InputArray map1, InputArray map2, int interpolation, int borderMode=BORDER_CONSTANT, const Scalar& borderValue=Scalar())
Python: cv2.remap(src, map1, map2, interpolation[, dst[, borderMode[, borderValue]]]) → dst
C: void cvRemap(const CvArr* src, CvArr* dst, const CvArr* mapx, const CvArr* mapy, int flags=CV_INTER_LINEAR+CV_WARP_FILL_OUTLIERS, CvScalar fillval=cvScalarAll(0) )
• src – Source image.
• dst – Destination image. It has the same size as map1 and the same type as src .
• map1 – The first map of either (x,y) points or just x values having the type CV_16SC2 , CV_32FC1 , or CV_32FC2 . See convertMaps() for details on converting a floating point
representation to fixed-point for speed.
Parameters: • map2 – The second map of y values having the type CV_16UC1 , CV_32FC1 , or none (empty map if map1 is (x,y) points), respectively.
• interpolation – Interpolation method (see resize() ). The method INTER_AREA is not supported by this function.
• borderMode – Pixel extrapolation method (see borderInterpolate() ). When borderMode=BORDER_TRANSPARENT , it means that the pixels in the destination image that corresponds to the
“outliers” in the source image are not modified by the function.
• borderValue – Value used in case of a constant border. By default, it is 0.
The function remap transforms the source image using the specified map:
where values of pixels with non-integer coordinates are computed using one of available interpolation methods. convertMaps() . The reason you might want to convert from floating to fixed-point
representations of a map is that they can yield much faster (~2x) remapping operations. In the converted case, (cvFloor(x), cvFloor(y)) and
This function cannot operate in-place.
Resizes an image.
C++: void resize(InputArray src, OutputArray dst, Size dsize, double fx=0, double fy=0, int interpolation=INTER_LINEAR )
Python: cv2.resize(src, dsize[, dst[, fx[, fy[, interpolation]]]]) → dst
C: void cvResize(const CvArr* src, CvArr* dst, int interpolation=CV_INTER_LINEAR )
• src – input image.
• dst – output image; it has the size dsize (when it is non-zero) or the size computed from src.size(), fx, and fy; the type of dst is the same as of src.
• dsize –
output image size; if it equals zero, it is computed as:
Either dsize or both fx and fy must be non-zero.
• fx –
scale factor along the horizontal axis; when it equals 0, it is computed as
Parameters: • fy –
scale factor along the vertical axis; when it equals 0, it is computed as
• interpolation –
interpolation method:
□ INTER_NEAREST - a nearest-neighbor interpolation
□ INTER_LINEAR - a bilinear interpolation (used by default)
□ INTER_AREA - resampling using pixel area relation. It may be a preferred method for image decimation, as it gives moire’-free results. But when the image is zoomed, it is
similar to the INTER_NEAREST method.
□ INTER_CUBIC - a bicubic interpolation over 4x4 pixel neighborhood
□ INTER_LANCZOS4 - a Lanczos interpolation over 8x8 pixel neighborhood
The function resize resizes the image src down to or up to the specified size. Note that the initial dst type or size are not taken into account. Instead, the size and type are derived from the src,
``dsize``,``fx`` , and fy . If you want to resize src so that it fits the pre-created dst , you may call the function as follows:
// explicitly specify dsize=dst.size(); fx and fy will be computed from that.
resize(src, dst, dst.size(), 0, 0, interpolation);
If you want to decimate the image by factor of 2 in each direction, you can call the function this way:
// specify fx and fy and let the function compute the destination image size.
resize(src, dst, Size(), 0.5, 0.5, interpolation);
To shrink an image, it will generally look best with CV_INTER_AREA interpolation, whereas to enlarge an image, it will generally look best with CV_INTER_CUBIC (slow) or CV_INTER_LINEAR (faster but
still looks OK).
Applies an affine transformation to an image.
The function warpAffine transforms the source image using the specified matrix:
when the flag WARP_INVERSE_MAP is set. Otherwise, the transformation is first inverted with invertAffineTransform() and then put in the formula above instead of M . The function cannot operate
cvGetQuadrangleSubPix is similar to cvWarpAffine, but the outliers are extrapolated using replication border mode.
Applies a perspective transformation to an image.
C++: void warpPerspective(InputArray src, OutputArray dst, InputArray M, Size dsize, int flags=INTER_LINEAR, int borderMode=BORDER_CONSTANT, const Scalar& borderValue=Scalar())
Python: cv2.warpPerspective(src, M, dsize[, dst[, flags[, borderMode[, borderValue]]]]) → dst
C: void cvWarpPerspective(const CvArr* src, CvArr* dst, const CvMat* map_matrix, int flags=CV_INTER_LINEAR+CV_WARP_FILL_OUTLIERS, CvScalar fillval=cvScalarAll(0) )
• src – input image.
• dst – output image that has the size dsize and the same type as src .
• M –
Parameters: • dsize – size of the output image.
• flags – combination of interpolation methods (INTER_LINEAR or INTER_NEAREST) and the optional flag WARP_INVERSE_MAP, that sets M as the inverse transformation (
• borderMode – pixel extrapolation method (BORDER_CONSTANT or BORDER_REPLICATE).
• borderValue – value used in case of a constant border; by default, it equals 0.
The function warpPerspective transforms the source image using the specified matrix:
when the flag WARP_INVERSE_MAP is set. Otherwise, the transformation is first inverted with invert() and then put in the formula above instead of M . The function cannot operate in-place.
Computes the undistortion and rectification transformation map.
The function computes the joint undistortion and rectification transformation and represents the result in the form of maps for remap() . The undistorted image looks like original, as if it is
captured with a camera using the camera matrix =newCameraMatrix and zero distortion. In case of a monocular camera, newCameraMatrix is usually equal to cameraMatrix , or it can be computed by
getOptimalNewCameraMatrix() for a better control over scaling. In case of a stereo camera, newCameraMatrix is normally set to P1 or P2 computed by stereoRectify() .
Also, this new camera is oriented differently in the coordinate space, according to R . That, for example, helps to align two heads of a stereo camera so that the epipolar lines on both images become
horizontal and have the same y- coordinate (in case of a horizontally aligned stereo camera).
The function actually builds the maps for the inverse mapping algorithm that is used by remap() . That is, for each pixel
In case of a stereo camera, this function is called twice: once for each camera head, after stereoRectify() , which in its turn is called after stereoCalibrate() . But if the stereo camera was not
calibrated, it is still possible to compute the rectification transformations directly from the fundamental matrix using stereoRectifyUncalibrated() . For each camera, the function computes
homography H as the rectification transformation in a pixel domain, not a rotation matrix R in 3D space. R can be computed from H as
where cameraMatrix can be chosen arbitrarily.
Returns the default new camera matrix.
C++: Mat getDefaultNewCameraMatrix(InputArray cameraMatrix, Size imgsize=Size(), bool centerPrincipalPoint=false )
Python: cv2.getDefaultNewCameraMatrix(cameraMatrix[, imgsize[, centerPrincipalPoint]]) → retval
• cameraMatrix – Input camera matrix.
Parameters: • imgsize – Camera view image size in pixels.
• centerPrincipalPoint – Location of the principal point in the new camera matrix. The parameter indicates whether this location should be at the image center or not.
The function returns the camera matrix that is either an exact copy of the input cameraMatrix (when centerPrinicipalPoint=false ), or the modified one (when centerPrincipalPoint=true).
In the latter case, the new camera matrix will be:
where cameraMatrix , respectively.
By default, the undistortion functions in OpenCV (see initUndistortRectifyMap(), undistort()) do not move the principal point. However, when you work with stereo, it is important to move the
principal points in both views to the same y-coordinate (which is required by most of stereo correspondence algorithms), and may be to the same x-coordinate too. So, you can form the new camera
matrix for each view where the principal points are located at the center.
Transforms an image to compensate for lens distortion.
C++: void undistort(InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray newCameraMatrix=noArray() )
Python: cv2.undistort(src, cameraMatrix, distCoeffs[, dst[, newCameraMatrix]]) → dst
C: void cvUndistort2(const CvArr* src, CvArr* dst, const CvMat* camera_matrix, const CvMat* distortion_coeffs, const CvMat* new_camera_matrix=0 )
• src – Input (distorted) image.
• dst – Output (corrected) image that has the same size and type as src .
Parameters: • cameraMatrix – Input camera matrix
• distCoeffs – Input vector of distortion coefficients
• newCameraMatrix – Camera matrix of the distorted image. By default, it is the same as cameraMatrix but you may additionally scale and shift the result by using a different matrix.
The function transforms an image to compensate radial and tangential lens distortion.
The function is simply a combination of initUndistortRectifyMap() (with unity R ) and remap() (with bilinear interpolation). See the former function for details of the transformation being performed.
Those pixels in the destination image, for which there is no correspondent pixels in the source image, are filled with zeros (black color).
A particular subset of the source image that will be visible in the corrected image can be regulated by newCameraMatrix . You can use getOptimalNewCameraMatrix() to compute the appropriate
newCameraMatrix depending on your requirements.
The camera matrix and the distortion parameters can be determined using calibrateCamera() . If the resolution of images is different from the resolution used at the calibration stage,
Computes the ideal point coordinates from the observed point coordinates.
C++: void undistortPoints(InputArray src, OutputArray dst, InputArray cameraMatrix, InputArray distCoeffs, InputArray R=noArray(), InputArray P=noArray())
C: void cvUndistortPoints(const CvMat* src, CvMat* dst, const CvMat* camera_matrix, const CvMat* dist_coeffs, const CvMat* R=0, const CvMat* P=0 )
• src – Observed point coordinates, 1xN or Nx1 2-channel (CV_32FC2 or CV_64FC2).
• dst – Output ideal point coordinates after undistortion and reverse perspective transformation. If matrix P is identity or omitted, dst will contain normalized point coordinates.
• cameraMatrix – Camera matrix
Parameters: • distCoeffs – Input vector of distortion coefficients
• R – Rectification transformation in the object space (3x3 matrix). R1 or R2 computed by stereoRectify() can be passed here. If the matrix is empty, the identity transformation is
• P – New camera matrix (3x3) or new projection matrix (3x4). P1 or P2 computed by stereoRectify() can be passed here. If the matrix is empty, the identity new camera matrix is
The function is similar to undistort() and initUndistortRectifyMap() but it operates on a sparse set of points instead of a raster image. Also the function performs a reverse transformation to
projectPoints() . In case of a 3D object, it does not reconstruct its 3D coordinates, but for a planar object, it does, up to a translation vector, if the proper R is specified.
// (u,v) is the input point, (u', v') is the output point
// camera_matrix=[fx 0 cx; 0 fy cy; 0 0 1]
// P=[fx' 0 cx' tx; 0 fy' cy' ty; 0 0 1 tz]
x" = (u - cx)/fx
y" = (v - cy)/fy
(x',y') = undistort(x",y",dist_coeffs)
[X,Y,W]T = R*[x' y' 1]T
x = X/W, y = Y/W
// only performed if P=[fx' 0 cx' [tx]; 0 fy' cy' [ty]; 0 0 1 [tz]] is specified
u' = x*fx' + cx'
v' = y*fy' + cy',
where undistort() is an approximate iterative algorithm that estimates the normalized original point coordinates out of the normalized distorted point coordinates (“normalized” means that the
coordinates do not depend on the camera matrix).
The function can be used for both a stereo camera head or a monocular camera (when R is empty).
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Challenge Algebra Equations
Wait, are the two equations separate questions or does x have to satisfy both equations?
Last edited by Wilmer; August 8th 2012 at 06:58 PM.
$\text{Hi, I realized I made a mistake in posting the equation. I am so sorry about that. The solution is posted below.}$ $\textbf{Problem 2:}\quad \text{Solve}\mspace{7mu} x^5-15x^3+45x-27=0 \quad\
quad (1)$ $\text{From the Rational Root Theorem,}\mspace{5mu} x=-3,\mspace{5mu} \text{is one of the many solution}.$ $\text{Thus}\mspace{7mu} (1)\, \Leftrightarrow\, (x+3)(x^4-3x^3-6x^2+18x-9)=0$ $x^
4-3x^3-6x^2+18x-9=0\, \Leftrightarrow\, x^4-3x^2(x-1)-9(x-1)^2=0$ $\text{Let}\mspace{7mu} y=x-1\, \Rightarrow\, x^4-3yx^2-9y^2=0\quad \quad (2)$ $\text{Solve for \emph{x} in terms of \emph{y}, we
get:}\mspace{7mu} 2x^2=3y\pm3y\sqrt{5}$ $\text{\underline{Case 1.}}\quad 2x^2=3y+3y\sqrt{5}\, \Leftrightarrow\, 2x^2=3(x-1)+3\sqrt{5}(x-1)$ $\Leftrightarrow\, 2x^2-3\left(\sqrt{5}+1\right)x+3\left(\
sqrt{5}+1\right) =0$ $\text{Solve the quadratic equation, we get:}$ $x=\dfrac{1}{4} \left(3+3\sqrt{5} \pm \sqrt{30-6\sqrt{5}}\mspace{1mu} \right)$ $\text{\underline{Case 2.}}\quad 2x^2=3y-3y\sqrt{5}
\, \Leftrightarrow\, 2x^2=3(x-1)-3\sqrt{5}(x-1)$ $\Leftrightarrow\, 2x^2+3\left(\sqrt{5}-1\right)x-3\left(\sqrt{5}-1\right)=0$ $\text{Solve the quadratic equation, we get:}$ $x=\dfrac{1}{4} \left(3-3
\sqrt{5} \pm \sqrt{30+6\sqrt{5}}\mspace{1mu} \right)$ $\text{So (1) has five real solutions:}$ $x=-3$ $x=\dfrac{1}{4} \left(3+3\sqrt{5} \pm \sqrt{30-6\sqrt{5}}\mspace{1mu} \right)$ $x=\dfrac{1}{4} \
left(3-3\sqrt{5} \pm \sqrt{30+6\sqrt{5}}\mspace{1mu} \right)$
It's looking right and but to be sure we have to wait thread starter to see if he has another method to solve this.algebra word problem solver free
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Patent US6097824 - Continuous frequency dynamic range audio compressor
Publication number US6097824 A
Publication type Grant
Application number US 08/870,426
Publication date Aug 1, 2000
Filing date Jun 6, 1997
Priority date Jun 6, 1997
Fee status Paid
Also published as DE69804096D1, DE69804096T2, EP0986933A1, EP0986933B1, WO1998056210A1
Publication number 08870426, 870426, US 6097824 A, US 6097824A, US-A-6097824, US6097824 A, US6097824A
Inventors Eric Lindemann, Thomas Lee Worrall
Original Assignee Audiologic, Incorporated
Export Citation BiBTeX, EndNote, RefMan
Patent Citations (12), Non-Patent Citations (30), Referenced by (33), Classifications (8), Legal Events (6)
External Links: USPTO, USPTO Assignment, Espacenet
Continuous frequency dynamic range audio compressor
US 6097824 A
An improved multiband audio compressor is well behaved for both wide band and narrow band signals, and shows no undesirable artifacts at filter crossover frequencies. The compressor includes a
heavily overlapped filter bank, which is the heart of the present invention. The filter bank filters the input signal into a number of heavily overlapping frequency bands. Sufficient overlapping of
the frequency bands reduces the ripple in the frequency response, given a slowly swept sine wave input signal, to below about 2 dB, 1 dB, or even 0.5 dB or less with increasing amount of overlap in
the bands. Each band is fed into a power estimator, which integrates the power of the band and generates a power signal. Each power signal is passed to a dynamic range compression gain calculation
block, which calculates a gain based upon the power signal. Each band is multiplied by its respective gain in order to generate scaled bands. The scaled bands are then summed to generate an output
We claim:
1. An improved multiband audio compressor of the type having a filter bank including a plurality of filters for filtering an audio signal, wherein said filters filter the audio signal into a
plurality of frequency bands, and further including a plurality of power estimators for estimating the power in each frequency band and generating a power signal for each band, and further including
a plurality of gain calculators for calculating a gain to be applied to each frequency band based upon the power signal associated with each frequency band, and further including means for applying
each gain to its associated band and for summing the gain-applied bands, wherein the improvement includes an improved, heavily overlapped, filter bank comprising:
a plurality of filters, said filters having sufficiently heavily overlapped frequency bands to reduce the ripple in the frequency response of the filter bank, given a slowly swept sine wave input
signal, to to below 2 dB.
2. The apparatus of claim 1 wherein the compression ratio of said filter bank is at least about 4.
3. The apparatus of claim 2 wherein said filter bank is implemented as a Short Time Fourier Transform system wherein the narrow bins of the Fourier transform are grouped into overlapping sets to form
the channels of the filter bank.
4. The apparatus of claim 2 wherein said filter bank is implemented as an IIR filter bank.
5. The apparatus of claim 2 wherein said filter bank is implemented as an FIR filter bank.
6. The apparatus of claim 2 wherein said filter bank is implemented as a wavelet filter bank.
7. The apparatus of claim 1 wherein the compression ratio of said filter bank is at between about 1.5 and about 4 and the ripple is below about 1 dB.
8. The apparatus of claim 7 wherein said filter bank is implemented as a Short Time Fourier Transform system wherein the narrow bins of the Fourier transform are grouped into overlapping sets to form
the channels of the filter bank.
9. The apparatus of claim 7 wherein said filter bank is implemented as an IIR filter bank.
10. The apparatus of claim 7 wherein said filter bank is implemented as an FIR filter bank.
11. The apparatus of claim 7 wherein said filter bank is implemented as a wavelet filter bank.
12. A continuous frequency dynamic range compressor comprising:
a filter bank including a plurality of filters for filtering an input signal into a plurality of frequency bands;
a plurality of power estimators, each power estimator connected to a filter, each power estimator for estimating the power in the frequency band of its associated filter and generating a power signal
related to the power in the frequency band of its associated filter;
a plurality of gain calculators, each gain calculator connected to a power estimator, each gain calculator for calculating a gain related to the power estimated by its associated power estimator;
a plurality of gain applying means, each gain applying means connected to a gain calculator, each gain applying means for applying the gain calculated by its associated gain calculator to the
frequency band associated with its associated gain calculator; and
means for summing the gain-applied frequency bands;
wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the frequency response, given a slowly swept sine wave input signal and a
compression ratio of at least about 4, to below about 2 dB.
13. The continuous frequency dynamic range compressor of claim 12, wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the
frequency response, given a slowly swept sine wave input signal, to below about 1 dB.
14. The continuous frequency dynamic range compressor of claim 13, wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the
frequency response, given a slowly swept sine wave input signal, to below about 0.5 dB.
15. A continuous frequency dynamic range compressor comprising:
a filter bank including a plurality of filters for filtering an input signal into a plurality of frequency bands;
a plurality of power estimators, each power estimator connected to a filter, each power estimator for estimating the power in the frequency band of its associated filter and generating a power signal
related to the power in the frequency band of its associated filter;
a plurality of gain calculators, each gain calculator connected to a power estimator, each gain calculator for calculating a gain related to the power estimated by its associated power estimator;
a plurality of gain applying means, each gain applying means connected to a gain calculator, each gain applying means for applying the gain calculated by its associated gain calculator to the
frequency band associated with its associated gain calculator; and
means for summing the gain-applied frequency bands;
wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the frequency response, given a slowly swept sine wave input signal and a
compression ratio of between about 1.5 and about 4, to below about 1 dB.
16. The continuous frequency dynamic range compressor of claim 15, wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the
frequency response, given a slowly swept sine wave input signal, to below about 0.5 dB.
17. The continuous frequency dynamic range compressor of claim 16, wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the
frequency response, given a slowly swept sine wave input signal, to below about 0.25 dB.
18. A hearing aid comprising:
a microphone for detecting sound and generating an electrical signal relating to the detected sound;
an analog to digital converter for converting the electrical signal into a digital signal;
means for digitally processing the digital signal;
a digital to analog converter for converting the processed digital signal to a processed analog signal; and
means for converting the processed analog signal into a processed sound signal;
wherein the digital processing means includes a continuous frequency dynamic range compressor including:
a filter bank including a plurality of filters for filtering the digital signal into a plurality of frequency bands;
a plurality of power estimators, each power estimator connected to a filter, each power estimator for estimating the power in the frequency band of its associated filter and generating a power signal
related to the power in the frequency band of its associated filter;
a plurality of gain calculators, each gain calculator connected to a power estimator, each gain calculator for calculating a gain related to the power estimated by its associated power estimator;
a plurality of gain applying means, each gain applying means connected to a gain calculator, each gain applying means for applying the gain calculated by its associated gain calculator to the
frequency band associated with its associated gain calculator; and
means for summing the gain-applied frequency bands;
wherein said filters filter the input signal into sufficiently heavily overlapped frequency bands to reduce the ripple in the frequency response of the filter bank, given a slowly swept sine wave
input signal, to less than 2 dB.
19. The hearing aid of claim 18 wherein the compression ratio of said filter bank is at least about 4 and the ripple is below about 2 dB.
20. The hearing aid of claim 18 wherein the compression ratio of said filter bank is between about 1.5 and about 4 and the ripple is below about 1 dB.
1. Field of the Invention
The present invention relates to apparatus and methods for multiband compression of sound input.
2. Description of the Prior Art
Multiband dynamic range compression is well known in the art of audio processing. Roughly speaking, the purpose of dynamic range compression is to make soft sounds louder without making loud sounds
louder (or equivalently, to make loud sounds softer without making soft sounds softer). One well known use of dynamic range compression is in hearing aids, where it is desirable to boost low level
sounds without making loud sounds even louder.
The purpose of multiband dynamic range compression is to allow compression to be controlled separately in different frequency bands. Thus, high frequency sounds, such as speech consonants, can be
made louder while loud environmental noises--rumbles, traffic noise, cocktail party babble--can be attenuated.
The pending patent filed Oct. 10, 1995, Ser. No. 08/540,534 (herein incorporated by reference), entitled Digital Signal Processing Hearing Aid, inventors Melanson and Lindemann, gives an extended
summary of multiband dynamic range compression techniques with many references to the prior art.
FIG. 1 (prior art) shows a block diagram of a conventional multiband compressor. The input signal from a microphone 104 or other audio source is divided into frequency bands using a filter bank 106
made up of a plurality of band pass filters, of which three are shown here: 108, 110, and 112. Most multiband compressors in analog hearing aids have two or three frequency bands.
A power estimator (122, 124, 126) estimates the power of each frequency band (114, 116, 118) at the output of each band pass filter. These power estimates are input to a plurality of gain calculation
blocks (130, 132, 134) which calculate a gain (138, 140, 142 ) which will be applied to the frequency bands 114, 116, 118. In general, gains 138, 140, and 142 provide more gain for low power signals
and less gain for high power signals. The gain is multiplied with the band pass signal and the gain scaled band pass signals 146, 148, 150 are summed by adder 154 to form the final output. This
output will generally be provided to a speaker or receiver 158.
When dividing an audio signal into frequency bands, it is desirable to design the filter bank in such a way that, if equal gain is applied to every frequency channel, the sum of the frequency
channels is equal to the original input signal to within a scalar gain factor. The frequency response of the sum of the frequency channels should be nearly constant. In practice we can tolerate phase
distortion better than amplitude distortion so we will say that the magnitude frequency response of the sum of frequency channels should be nearly constant. Less than 1 dB of ripple is desirable.
FIG. 2 shows the magnitude frequency response of the band pass channels 201 and the magnitude frequency response of the sum of band pass channels 202 of a filter bank designed in the manner described
above. In U.S. Pat. No. 5,500,902, Stockham Jr. et al. propose just such a filter bank as the basis of a multiband compressor. The band centers and bandwidths of the filter bank are spaced roughly
according to the critical bands of the human ear. This is a quasi-logarithmic spacing--linear below 500 Hz and logarithmic above 500 Hz. It is suggested in U.S. Pat. No. 5,500,902 in column 5 lines
8-9 that the audio band pass filters should preferably have a band pass resolution of 1/3 octave or less. In other words, the band pass filters should be reasonably narrow as indicated in FIG. 2 so
that the compression is controlled independently in each band with little interaction between bands.
FIG. 3 shows the magnitude frequency response of the sum of frequency channels 202 for the same filter bank as FIG. 2, but with higher resolution on the Y axis. We can see that the residual ripple is
considerably less than 1 dB.
When a multiband compression system, based on such a filter bank, is presented with a broadband signal, such as white noise, it will adjust the gain similarly in each frequency channel. The gains may
be weighted so that the wider bands at high frequency, which measure more power because of their increased width, produce gains equivalent to the narrow low frequency bands. The result is a smooth,
flat output frequency response.
However, when such a filter bank is presented with a narrow band stimulus, such as a sinusoid slowly swept across frequency, the resulting output response is entirely different, as shown in FIG. 4.
The sine wave is swept slowly enough so that the time constants of the compressor are not a factor. We see a pronounced 4.5 dB ripple in the output 401. Here the stimulus is a -20 dB sinusoid
sweeping across frequency. The compression ratio in this example is 4 to 1 and the unity gain point of the compressor is 0 dB. Under these conditions, we would expect the compressor to generate 15 dB
of gain so that the resulting output is a constant -5 dB. This is clearly not the case.
As we recall, the filter bank is designed to sum to a constant response. This means at the filter crossover frequencies, where the response of adjacent band pass filters is the same, the band pass
response is -6 dB. Since the responses are the same at this point they will sum, giving a total of 0 dB which preserves the overall flat response. However, when a sinusoid is presented at a crossover
frequency the power measurement is also -6 dB relative to the band center. The compressor in each band sees this -6 dB output and, since the compression ratio is 4 to 1, generates a gain of 4.5 dB
which appears on the output as shown in FIG. 4. Note that the ripple would be smaller for a system having a lower compression ratio. For a compression ratio of 1.5, the ripple would be around 2 dB,
which is still quite significant.
For narrow band signals which change frequencies this will generate an undesirable audible warble. This would certainly be the case for musical sounds--flutes, violins, etc. It would also be the case
for high pitched speech sounds from women and children where the individual harmonics of voiced speech are relatively far apart and will appear as individual stimuli. As the formants of the voiced
speech sweep across frequency they will become distorted by the narrow band ripple shown in FIG. 4.
In addition, audiologists often test the frequency response of hearing aids with pure tone sinusoids of different frequencies. The results of their tests will clearly be compromised given the
response of FIG. 4.
For illustrative reasons, in FIG. 5 we have decreased the number of bands to three bands, 501, 502, and 503. This is considerably fewer bands than the FIG. 2 configuration, but the filter bands are
conventionally overlapped, and the ripple or warble problem remains the same as in the FIG. 2 configuration. In FIG. 5, the filter transfer functions are plotted using different symbols for each
filter. Thus, frequency band 501 is plotted with squares, frequency band 502 is plotted with triangles, and frequency band 503 is plotted with asterisks. The band transitions in the FIG. 5
configuration are relatively sharp and there is just enough overlap to guarantee that the sum of the magnitude frequency responses of the filters is constant, as shown by 504, which indicates the
broadband frequency response of the configuration. However, as shown in FIG. 6, the slowly swept sine response 601 of the 4 to 1 compressor manifests a 4.5 dB ripple, just as was seen in FIG. 4.
This poor response to narrow band inputs is true for any compressor with relatively narrow transition bands (conventional overlap) between band pass filters. In particularly it is true for both
digital and analog hearing aids with two or more frequency channels.
A need remains in the art for a multiband dynamic range compressor which is well behaved for narrow band and broad band signals.
An object of the present invention is to provide a multiband dynamic range compressor (also called a continuous frequency multiband compressor) which is well behaved for narrow band and broad band
signals. The present invention is a new type of multiband compressor called a continuous frequency compressor which is well behaved for both wide band and narrow band signals, and shows no
undesirable artifacts at filter crossover frequencies.
The continuous frequency multiband compressor of the present invention includes an improved filter bank comprising a plurality of filters having sufficiently overlapped frequency bands to reduce the
ripple in the frequency response given a slowly swept sine wave to below about 2 dB, and down to arbitrarily low sub dB levels depending on amount of overlap.
The invention is an improved multiband audio compressor of the type having a filter bank including a plurality of filters for filtering an audio signal, wherein the filters filter the audio signal
into a plurality of frequency bands, and further including a plurality of power estimators for estimating the power in each frequency band and generating a power signal for each band, and further
including a plurality of gain calculators for calculating a gain to be applied to each band based upon the power signal associated with each band, and further including means for applying each gain
to its associated band and for summing the gain-applied bands, wherein the improvement includes an improved, heavily overlapped, filter bank comprising a plurality of filters, the filters having
sufficiently overlapped frequency bands to reduce the ripple in the frequency response, given a slowly swept sine wave input signal, to less than half the dB's of a conventionally overlapped filter
As an example, when the compression ratio of the filter bank is at least about 4, the ripple is below about 2 dB. When the compression ratio is between 1.5 and 4, the ripple is reduced to below about
1 dB.
The filter bank may be implemented as a Short Time Fourier Transform system wherein the narrow bins of the Fourier transform are grouped into overlapping sets to form the channels of the filter bank.
Alternatively, the filter bank may be implemented as an IIR filter bank, an FIR filter bank, or a wavelet filter bank.
The invention may be used in a digital hearing aid, as part of the digital signal processing portion of the hearing aid.
FIG. 1 (prior art) shows a block diagram of a prior art multiband dynamic range compressor having conventionally overlapped band pass filters.
FIG. 2 (prior art) shows the filter bank structure and the performance (or magnitude frequency response of the sum of frequency channels) of an embodiment of the conventional compressor of FIG. 1,
having a large number of conventionally overlapped filters.
FIG. 3 shows the broadband performance of the conventional compressor of FIG. 2 at a higher resolution than FIG. 2.
FIG. 4 shows the performance of the conventional compressor of FIG. 2, given a narrow band swept input signal.
FIG. 5 (prior art) shows the filter bank structure and the performance of an embodiment of the conventional compressor of FIG. 1, having three filters, given a broadband input signal.
FIG. 6 shows the performance of the conventional compressor of FIG. 5, given a narrow band swept input signal.
FIG. 7 shows a block diagram of a multiband dynamic range compressor having heavily overlapped band pass filters according to the present invention.
FIG. 8 shows the filter bank structure and the performance of an embodiment of the compressor of FIG. 7, having a somewhat overlapped filters, given a broadband input signal.
FIG. 9 shows the performance of the embodiment of FIG. 8, given a narrow band swept input signal.
FIG. 10 shows the filter bank structure and the performance of an embodiment of the compressor of FIG. 7, having heavily overlapped filters, given a broadband input signal.
FIG. 11 shows the performance of the embodiment of FIG. 10, given a narrow band swept input signal.
FIG. 12 shows a digital hearing aid which utilizes the multiband dynamic range compressor having heavily overlapped band pass filters of FIG. 7.
FIGS. A1 through A7 provide graphical illustration of the mathematical principles illustrated in the appendix.
The attached Appendix presents a detailed mathematical analysis of the frequency response to narrow band input signals in conventional multiband compressors. This analysis was used to find a solution
to the problem shown in FIGS. 4 and 6, wherein conventionally overlapped filter banks produce a large ripple in the frequency response to a narrow band signal, such as a swept sine wave. The solution
involves increasing the amount of overlap between band pass filters by a considerable amount. The precise amount of overlap required is a function of the bandwidth and sharpness of the transition
bands of the band pass filters.
FIGS. 7 through 11 illustrate the effects of increasing filter band overlap. FIG. 7 shows an improved multiband dynamic range compression device (or continuous frequency dynamic range audio
compressor) 10 according to the present invention. An audio input signal 52 enters microphone 12, which generates input signal 54. In the preferred embodiment, signal 54 is converted to a digital
signal by analog to digital converter 15, which outputs digital signal 56. This invention could be implemented with analog elements as an alternative. Digital signal 56 is received by filter bank 16,
which is the heart of the present invention. In the preferred embodiment the filter bank is implemented as a Short Time Fourier Transform system, where the narrow bins of the Fourier Transform are
grouped into overlapping sets to form the channels of the filter bank. However, a number of techniques for constructing filter banks including Wavelets, FIR filter banks, and IIR filter banks, are
well documented in the literature and it would be obvious to one skilled in the art that any of the techniques could be used as the foundation for filter bank design in this invention.
Filter bank 16 filters signal 56 into a large number of heavily overlapping bands 58. The theory behind the selection of the number of frequency bands and their overlap is given in detail in the
Appendix at the end of this section.
Each band 58 is fed into a power estimation block 18, which integrates the power of the band and generates a power signal 60. Each power signal 60 is passed to a dynamic range compression gain
calculation block, which calculates a gain 62 based upon the power signal 60 according to a predetermined function. Power estimation blocks 18 and gain calculation blocks 20 are conventional and well
known in the art.
Multipliers 22 multiply each band 58 by its respective gain 62 in order to generate scaled bands 64. Scaled bands 64 are summed in adder 24 to generate output signal 68. Output signal 68 may be
provided to a receiver in a hearing aid (not shown) or may be further processed.
FIG. 8 shows the filter bank structure and the performance of an embodiment of the compressor of FIG. 7, having a somewhat overlapped filters, given a broadband input signal. In FIG. 8, the number of
filter bands has been increased over the number in the FIG. 5 configuration, to five filters 801-805. The bandwidths of the filters have not changed, so the filters are significantly more overlapped
than the FIG. 5 configuration. In other words, the original filters of FIG. 5 are still as they were, and there is a new set of filters interleaved with the originals, resulting in considerably more
overlap between adjacent filters. Filter 801 is plotted with diamonds, filter 802 is plotted with x's, filter 803 is plotted with circles, filter 804 is plotted with pluses, and filter 805 is plotted
with asterisks.
In FIG. 9 we see the swept sine response 901 of the 4 to 1 compressor for the more overlapped filter set of FIG. 8. The ripple has been reduced from 4.5 dB to approximately 2 dB. If the FIG. 8
configuration used a compression ratio of 1.5, the ripple would be reduced from around 2 dB to less than 1 dB.
In FIG. 10 we have increased the number of filters over the FIG. 5 and FIG. 8 configurations, to eleven filters, still without changing the filter bandwidths. Filter 1001 is plotted with diamonds.
Filter 1002 is plotted with left-pointing triangles. Filter 1003 is plotted with down-pointing triangles. Filter 1004 is plotted with x's. Filter 1005 is plotted with circles. Filter 1006 is plotted
with x's again. Filter 1007 is plotted with squares. Filter 1008 is plotted with pluses. Filter 1009 is plotted with left-pointing triangles again. Filter 1010 is plotted with asterisks. Filter 1011
is plotted with pluses again.
FIG. 11 shows the swept sine response 1101 of the compressor configuration of FIG. 10. We see that the ripple has been reduced to less than one half dB for the 4 to 1 compressor. In the case of a
compression ratio of 1.5, the ripple would be reduced to less than one quarter of a dB.
FIG. 12 shows a digital hearing aid which utilizes the continuous frequency dynamic range audio compressor 10 having heavily overlapped filter bank 16 of FIG. 7. The hearing aid of FIG. 12 includes a
microphone 1202 for detecting sounds and converting them into analog electrical signals. Analog to digital (A/D) converter 1204 converts these analog electrical signals into digital signals. A
digital signal processor (DSP) 1206 may accomplish various types of processing on the digital signals. It includes audio compressor 10 having heavily overlapped filter bank 16, as shown in FIG. 7.
The processed digital signals from DSP 1206 are converted to analog form by digital to analog (D/A) converter 1208, and delivered to the hearing aid wearer as sound signals by speaker 1210.
In the Appendix we analyze in depth the reasons for the dramatic reduction in ripple with increase in filter overlap. We will briefly summarize these reasons here. We can think of calculating the
gain for a multiband compressor as kind of black box filter, which takes as input the power spectrum of the input signal and generates as output a frequency dependent gain. We can think of the input
and output of this black box as continuous functions of frequency. Inside the black box we estimate power in a number of discrete frequency bands. In other words, we reduce the continuous power
spectrum to a number of sampled points. We then calculate a gain value corresponding to each one of these discrete power spectrum samples, resulting in a discrete set of gain points. Since we must
apply gain to every frequency, we interpolate these discrete gain values over the entire frequency range to generate the continuous gain function. This gain interpolation is implicit in the process
of applying gain to the output of band pass filters and summing these outputs.
This interpretation of multiband compression in terms of sampling the power spectrum and interpolating gain gives us insight into the problems of narrow band response. We know that when we sample a
time domain function we must first band limit the function in frequency to one half the sampling frequency. Since we are sampling the power spectrum in the frequency domain, it is reasonable to
assume that we must first limit the time domain representation of the frequency domain power spectrum. This is exactly the dual of limiting the frequency domain bandwidth of a time domain function
before sampling.
When we band limit the frequency response of a time domain function we convolve the function in the time domain with the impulse response of a low pass filter. When we time limit the power spectrum
we convolve it in the frequency domain with the impulse response of a low pass filter. When we sample the power spectrum, by measuring power at the output of a band pass filter, we are effectively
integrating the power spectrum over frequency but first multiplying or windowing the power spectrum with the magnitude squared frequency response of the band pass filter. When we repeat the operation
for the next frequency band, it as if we are moving the band pass window in the frequency domain to a new center point and repeating the integration operation. This act of placing a window on the
power spectrum, integrating, then moving the window, integrating again, and so on, is, in fact, convolving the power spectrum in the frequency domain by the band pass window and sampling the result
of this convolution. It is the same thing as low pass filtering before sampling.
The fact that we vary the width and displacement of the band pass window as we move it across the power spectrum because we use band pass filters with quasi-logarithmic spacing, means that we are
continually changing the sample rate and low pass filter response of our sampling system. Nevertheless, the rules of sampling still apply.
In the Appendix we show that the frequency domain sampling interval, that is the band spacing of the band pass filters in Hz, should be less than or equal to one divided by the length in samples of
the inverse transform of the magnitude squared frequency response of the band pass filter. This is the same as one divided by the autocorrelation of the band pass impulse response. The impulse
response naturally reduces in magnitude towards its extremities and so does its autocorrelation. The length of the autocorrelation is the length comprising all values above some arbitrary minimum
values--e.g. 60 dB down from the peak value. This shows that the band pass filter frequency response determines the number of bands required to eliminate narrow band ripple in the compression system.
If this criterion is strictly obeyed the resulting ripple in narrow band response can, in theory, be completely eliminated. In practice we do not need to completely eliminate this ripple so we can
compromise. Nevertheless, as we have seen with a typical three band filter bank in FIG. 5, it is not until we increase the number of bands greatly--to eleven bands--without changing the bandwidths of
the filters, that we reduce the ripple to sub dB levels as shown in FIG. 10.
Thus, starting with a conventional filter bank whose band pass responses sum to a constant with conventional overlap between band pass filters, we must increase the number of bands by a factor of
about three to guarantee sufficiently low ripple for narrow band stimuli. If f(k) for k=1 . . . N are the -6 dB crossover frequency points of a set of band pass filters in a filter bank such as shown
in FIGS. 2 and 5, then we define a conventionally overlapped filter bank as one in which each band pass filter, with -6 dB crossover point at f(k), reaches its stopband attenuation at or before f
We have defined the criterion for reducing narrow band ripple in a multiband compression system in terms of sampling theory applied to the input power spectrum. When we correctly sample a band
limited continuous time domain signal we say that there is no loss of information because we can reconstruct the continuous time domain signal from its samples. What's more, any linear filtering
which we perform on the sampled signal will appear as linear filtering of the continuous reconstructed signal. Therefore we do not see the effect of sample boundaries in the output signal and can
think of the system as the implementation of a continuous time filter.
Similarly, when we correctly time limit and sample the continuous power spectrum in a multiband compression system we do not see the effect of band edges in the compressed signal and can think of the
system as a system which is continuous in frequency. It is a continuous frequency compressor.
While the exemplary preferred embodiments of the present invention are described herein with particularity, those skilled in the art will appreciate various changes, additions, and applications other
than those specifically mentioned, which are within the spirit of this invention. ##SPC1##
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Remember those wooden crystal models?
mpollock September 4th, 2012
WOOSTER, OH – I’m always awed by the beautiful and perfect symmetry of crystals. I can think of no better way to teach external symmetry than with wooden crystal models. The wooden crystal models are
a common experience in geology, across generations and continents, although it seems they may be on the endangered list. I’ve chosen to continue using the models in Mineralogy because they allow
students to see and “feel” the symmetry operations and our structural geologist thinks the blocks help students with their spatial reasoning skills.
Most people develop a love-hate relationship with the models, but I have to admit I’ve always been infatuated. They’re like logic puzzles with a million different secrets all wrapped up in what seems
like a simple wooden block. Spend some time with the block and it will reveal a wealth of information.
This combination of symmetry elements belongs to the point group (or crystal class) 4/m 2/m 2/m. In Hermann-Mauguin notation, 4/m refers to a four-fold rotation axis perpendicular to a mirror. The
second and third 2/m terms refer to two-fold rotation axes that are perpendicular to mirrors, one set of axes that exits the crystal in the middle of the faces and another set of axes that exits the
crystal on the edges.
The 4/m 2/m 2/m point group belongs to the tetragonal crystal system, which has three mutually perpendicular crystallographic axes. The two horizontal axes (a1 and a2) are equal in length and
coincide with the two-fold rotation axes. The vertical axis (c) is longer than the horizontal axes and coincides with the four-fold rotation axis.
Once the crystallographic axes have been determined, we can describe the orientation of the crystal faces using Miller Indices. In short, Miller Indices consist of three numbers (four in the case of
hexagonal crystals) that are derived from the intercepts of crystal faces. The crystal face that intersects the a1 axis but parallels the a2 and c axes is assigned a Miller Index of 100. The crystal
face that never intersects a1 but cuts both a2 and c is assigned a Miller Index of 011.
We could keep going with this…describing forms, measuring angles, plotting on stereonets, but we won’t. Making it through Miller Indices this week will be enough for the Mineralogy students. Here’s
the big secret: this is one of crystals we’re working on in class. I guess we’ll find out which students read the blog on a regular basis!
# Mark Wilsonon 04 Sep 2012 at 6:15 pm
I had those very same models when I was a student. Your explanations are much better than the ones I had then!
# Stephanie Jarvison 04 Sep 2012 at 6:19 pm
This post made me smile
# Candice Thorntonon 04 Sep 2012 at 7:14 pm
Hey Dr. Pollock! Does the four fold/two fold correspond to the subscript in the axes of rotation notation (A_4/A_2)? Or is the subscript supposed to be the total number of axes present in the
crystal? And can there not be a horizontal plane of symmetry in the crystal? Or would that be the center of symmetry? See you in class tomorrow!
# Candice Thorntonon 04 Sep 2012 at 8:05 pm
Nevermind, did the reading for tomorrow and figured it out. Thanks for the post! ^_^
# Jesse Davenporton 05 Sep 2012 at 3:58 pm
I’ve been revisiting this stuff, although not quite in the same way. Inorganic Chemistry is fun, but not as fun as mineralogy was!
# Kiton 05 Sep 2012 at 8:38 pm
I miss the wooden blocks, they were
# Elyssa Krivicichon 09 Sep 2012 at 9:27 pm
I wish that I had you as my prof…..:-)
# Wooster Geologists » Blog Archive » Early Morning Powdering Sessionon 02 Oct 2012 at 9:14 am
[...] have been able to describe the physical properties of their sample, but only some can observe the crystallography. Some students are starting to use optical methods while others are using the
XRD. Eventually, a [...]
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FOM: Kreisel's "unwinding" program
Solomon Feferman sf at Csli.Stanford.EDU
Mon Mar 2 15:49:13 EST 1998
In his posting of yesterday (March 1, 19:00) Steve Simpson reviewed
Angus MacIntyre's essay, "The strength of weak systems". The review
concerns many matters, but since I have not read the essay itself, I just
want to comment on one of these. Apparently MacIntyre spoke favorably of
the work of Kreisel in the 1950s demonstrating the formalizability of
analytic number theory in PA (Peano Arithmetic) and using his
results on the unwinding of proofs in PA to obtain concrete information
about results obtained by analytic methods. In particular, Kreisel
claimed in various places to have obtained three or four level
(depending on how you count) exponential bounds for the first change of
sign in pi(x) - li(x), proved without bounds by Littlewood using
arguments [it went one way if R.H., the Riemann Hypothesis, is true,
another if false]. I have examined Kreisel's arguments in detail in a
paper entitled "Kreisel's 'unwinding' program", and have found them to be
seriously wanting if not suspicious as to their grounds. As I say in my
article, which was published in the volume "Kreiseliana" (P. Odifreddi,
ed., A.K. Peters, Ltd., 1996), and which can also be found at my ftp
under the rubric 'unwind', the "expected details are either problematic
or simply missing". This is why I called specifically for a
re-examination of the formalization of results like Littlewood's in weak
subsystems of PA, in a recent posting (referred to by Simpson). What the
foundational significance of such work may be is another matter that I will
try to take up on another occasion.
Let me add some more personal remarks vis a vis the evaluation of
Kreisel's work. At the beginning of my article I said that Kreisel "did
perhaps more than anyone else to promote the development of proof theory
and the metamathematics of constructivity in the last forty-odd years"
through "his own contributions (individual and collaborative) and his
extraordinary personal influence." In particular, he and I were very
close colleagues for close to twenty years, and he greatly influenced my
own development as a logician during that period. But his increasingly
critical views of foundational work in general, and of
individual logicians in particular, led to his estrangement from
practically all of his former collaborators, myself included. Since the
1970s, his views have had a baleful impact on our field and have led some
on the outside to a serious mis-evaluation of its achievements. There is
no need to formulate this in terms of general intellectual interest vs.
mathematical interest, but what has apparently resulted in such essays as
MacIntyre's (if Simpson's report is accurate) is an unfortunate snobbism
or dismissal of foundational work, and that only applications of logic to
"real", "hard" mathematics is to be valued. Let me make clear that this
has nothing to do with my own great appreciation of the latter through the
work of MacIntyre, van den Dries, Wilkie, Zilber, Hrushovski and others,
which had its roots in the pioneering model-theoretic work of Tarski and
Abraham Robinson. The excitement and drive of their research programs is
palpable (as is evidenced by the current convocations at MSRI) and cannot
be gainsaid. Let us hope the non-logical mathematicians value their work
as much as these logicians admire current hard-core mathematics.
--Sol Feferman
More information about the FOM mailing list
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[SOLVED] Another lim inf/lim sup proof
February 24th 2010, 05:18 PM #1
[SOLVED] Another lim inf/lim sup proof
Let $(s_{n})$ be a sequence of nonnegative numbers, and for each n define $a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$. Show that $\lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_
{n} \le \lim \,sup \,s_{n}$. Also show that if $\lim s_{n}$ exists, then $\lim a_{n}$ exists and $\lim a_{n} = \lim s_{n}$.
For this one I am completely stuck. Any help would be appreciated.
Let $(s_{n})$ be a sequence of nonnegative numbers, and for each n define $a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$. Show that $\lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_
{n} \le \lim \,sup \,s_{n}$. Also show that if $\lim s_{n}$ exists, then $\lim a_{n}$ exists and $\lim a_{n} = \lim s_{n}$.
For this one I am completely stuck. Any help would be appreciated.
What is the definition of $\limsup$?
$lim sup = \lim_{N\to \infty}\, sup\{s_{n} : n > N\}$
Any suggestions?
Apparently, for part of the proof I need to show that $M > N$ implies $sup\{a_{n} : n > M\} \le \frac{1}{M}(s_{1}+s_{2}+...+s_{N}) + sup\{s_{n} : n > N\}$ (and I don't even how to show that or
why that's true...
Sorry for all the consecutive posts, but I am still absolutely stuck on this.
February 24th 2010, 05:53 PM #2
February 24th 2010, 06:13 PM #3
February 24th 2010, 09:01 PM #4
February 25th 2010, 06:45 PM #5
February 27th 2010, 04:00 PM #6
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Partial fraction
April 6th 2009, 05:41 AM #1
Sep 2008
Partial fraction
$<br /> \int\frac{x+4}{x^2+5x-6}\,dx<br />$
so factoring the bottom we get
$<br /> \int\frac{x+4}{(x-1)(x+6)}\,dx<br />$
so we have
A / x-1
B / x+6
now what? I know you have to do this
x+4 = A(x+6) + B(x-1)(x+6)/(x-6)
now im stuck because of the x-6 in denominator
ok so Ive tried to work it a little further.
a= 5/7
is this correct?
then I get
$<br /> 1/7 \int\frac{5}{x-1}+\int\frac{2}{x+6}\,dx<br />$
ok now I know this is a du/u but the book has this answer
1/7 ln|(x+6)^2(x-1)^5| + C
is there some log property they used to get to this form?
All I meant with "by design" was that the denominator will always cancel because of the way partial fractions work.
Those values for a and b are correct.
$\frac{1}{7} \int\frac{5}{x-1}+\frac{2}{x+6}~dx$
This is as you say du/u which integrates to ln(u).
So we have $\frac{1}{7}(5ln(x-1)+2ln(x+6))$ which is what you have I hope?
The following are true for logs:
$ln(x) + ln(y) = ln(xy)$
$xln(y) = ln(y^x)$
April 6th 2009, 06:05 AM #2
Sep 2006
April 6th 2009, 06:12 AM #3
Sep 2008
April 6th 2009, 06:25 AM #4
Sep 2008
April 6th 2009, 01:18 PM #5
Sep 2006
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Math 121, Section 201 - Honours Integral Calculus - Fall 2009
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika AT math DOT ubc DOT ca
Lectures: Note the different times and locations. Click on the links below for detailed location information.
Office hours: Monday, Wednesday 12:30-1:30 or by appointment. (There may be one or two weeks during the semester when the office hours would have to be rescheduled. You will be notified in this case.
Course information and class handouts
The first day handout .
Midterms and exams
• The first midterm will be held in class on Friday February 12 and will cover sections 5.1--7.4, including endpoints. Calculators and formula sheets are not allowed in the exam.
• The second midterm will be held in class on Friday March 26 and will cover sections 7.1--9.4, including endpoints. Calculators and formula sheets are not allowed in the exam.
• Formula sheet for the final. A copy of this sheet will be attached to the final exam. You should not bring your own formula sheet or other aids, including calculators.
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Are the numbers of elements of two distinct prime orders not equal in finite groups?
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Are the numbers of elements of two distinct prime orders not equal in finite groups?
add comment
The answer is "no":
One can construct a counterexample of the form $G\times H$. Say $p$ and $q$ are distinct primes, and for some integer $m$ one can find two groups $G, H$ with with $q\nmid|G|$ and $m$
elements of order $p$ in $G$, and $p\nmid|H|$ and $m$ elements of order $q$ in $H$. Then $G\times H$ has $m$ elements of order $p$, and $m$ of order $q$.
Now take $p=5$ and $q=3$. Then $G=C_{11}\rtimes C_5$ has $44$ elements of order $5$ (every element not in $C_{11}$), and $H=C_{21}\rtimes C_3$ has $44$ elements of order $3$ ($2$ in $C_
up vote 14 {21}$ plus all $42$ outside it). So $G\times H$ has 44 elements of order $3$ and of order $5$.
down vote
To check this in Magma:
assert #[g: g in P | Order(g) eq 5] eq 44;
assert #[g: g in P | Order(g) eq 3] eq 44;
This is very clever.
Todd Trimble♦ Dec 18 '10 at 12:46
add comment
Consider $C_6$. It has 1 element of order 1, 1 element of order 2, 2 elements of order 3, and 2 elements of order 6.
up vote 0 down
vote Notice here that the numbers of elements of order 2 and of order 3 differ.
I interpret the question as wanting a finite group G and distinct primes p and q where the number of elements of order p and of order q in G are the same and nonzero. Gerhard "Ask
Me About System Design" Paseman, 2010.12.17
Gerhard Paseman Dec 18 '10 at 7:22
Or a proof that there is no such finite group. Gerhard "Then The Other Shoe Drops" Paseman, 2010.12.17
Gerhard Paseman Dec 18 '10 at 7:24
add comment
Not the answer you're looking for? Browse other questions tagged gr.group-theory or ask your own question.
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Index Calculation and Cutoff Points
I found the following website which purports to give the index calculation for several law schools:
Although this list includes the method of calculation for many schools it does not give the various cutoff points for each school (e.g. automatic accept and decline scores).
Does anyone know where I might find a list correlating one's index score at a law school with the chance of being accepted to said law school?
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Re: polcompositum() vs factornf() performance comparison
Karim BELABAS on Thu, 9 Oct 2003 14:48:19 +0200 (MEST)
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
Re: polcompositum() vs factornf() performance comparison
On Wed, 8 Oct 2003, Bill Allombert wrote:
> On Wed, Oct 08, 2003 at 02:19:53PM -0400, Igor Schein wrote:
>> Basically, I am verifying in 2 different ways that 2 number fields are
>> non-isomorphic, but it turns out factornf ( for which nfisisom is a
>> wrapper ) is twice slower!!
> Karim and I have spent quite some time making the nfgcd routine fast,
> but it still often slower than the actual factorisation. This is what
> happens here.
That's because we spent even more time trying to make the polynomial
factorization routines fast over number fields [ over finite fields, it's
still slow ].
> I have two remarks: nfisisom use nfroots and not nffactor. Unfortunately
> there is no equivalent of nfroots for plain polynomial, so it uses
> factornf when the input are not given as nfinit() output. This make
> things much slower when the field are *not* isomorphic
This problem will disappear when nffactor / nfroots are able to work in
arbitrary orders. factornf() will not actually disappear since the method it
implements (Trager) is still optimal when factoring small polynomials over
fields of large degree, but it will in most cases default back to nffactor().
I need to find one or two hours to implement and debug this ( 99% of the code
is there ). Not likely in the weeks to come, though...
> Second, I see that factornf use indexpartial instead of plain
> discriminant. While it take 0 ms for most examples, this is wasteful
> since nfgcd() care only about `bad' primes dividing the denominator,
> not about the exponents which is what indexpartial reduce.
No. nfgcd calls matratlift(..., den), which calls lift_to_frac(..., den)
whose (heuristic) result must divide 'den'. Lowering den by reducing
exponents yields faster and more stringent tests.
I've benchmarked it, and it was (barely:-) faster.
Karim Belabas Tel: (+33) (0)1 69 15 57 48
Dép. de Mathématiques, Bât. 425 Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud http://www.math.u-psud.fr/~belabas/
F-91405 Orsay (France) http://www.parigp-home.de/ [PARI/GP]
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11 infinity
To Infinity And Beyond!
Lecture 11 CS 15-251
The Ideal Computer:
no bound on amount of memory
Whenever you run out of memory, the
computer contacts the factory. A
maintenance person is flown by
helicopter and attaches 100 Gig of
RAM and all programs resume their
computations, as if they had never been
An Ideal Computer Can Be
Programmed To Print Out:
: 3.14159265358979323846264…
2: 2.0000000000000000000000…
e: 2.7182818284559045235336…
1/3: 0.33333333333333333333….
: 1.6180339887498948482045…
Computable Real Numbers
A real number r is computable if there
is a program that prints out the decimal
representation of r from left to right.
Thus, each digit of r will eventually be
printed as part of an infinite sequence.
Are all real numbers
Describable Numbers
A real number r is describable if it can
be unambiguously denoted by a finite
piece of English text.
2: “Two.”
: “The area of a circle of radius one.”
Theorem: Every computable real
is also describable
Proof: Let r be a computable real that
is output by a program P. The following
is an unambiguous denotation:
“The real number output by:“P
MORAL: A computer
program can be viewed
as a description of its
Are all real numbers
To INFINITY ….
and Beyond!
Correspondence Principle
If two finite sets can be
placed into 1-1 onto
correspondence, then
they have the same size.
Correspondence Definition
Two finite sets are
defined to have the
same size if and only if
they can be placed into 1-1
onto correspondence.
Georg Cantor (1845-1918)
Cantor’s Definition (1874)
Two sets are defined to have
the same size if and only if
they can be placed into 1-1
onto correspondence.
Cantor’s Definition (1874)
Two sets are defined to have
the same cardinality if and
only if they can be placed
into 1-1 onto correspondence.
Do N and E have the same
N = { 0, 1, 2, 3, 4, 5, 6, 7, …. }
E = The even, natural numbers.
E and N do not have the
same cardinality! E is a
proper subset of N with
plenty left over.
The attempted
correspondence f(x)=x
does not take E onto N.
E and N do have the
same cardinality!
0, 1, 2, 3, 4, 5, ….…
0, 2, 4, 6, 8,10, ….
f(x) = 2x is 1-1 onto.
Cantor’s definition only
requires that some 1-1
correspondence between the
two sets is onto, not that all 1-1
correspondences are onto.
This distinction never arises
when the sets are finite.
If this makes you feel
TOUGH! It is the price that
you must pay to reason
about infinity
Do N and Z have the same
N = { 0, 1, 2, 3, 4, 5, 6, 7, …. }
Z = { …, -2, -1, 0, 1, 2, 3, …. }
N and Z do have the
same cardinality!
0, 1, 2, 3, 4, 5, 6 …
0, 1, -1, 2, -2, 3, -3, ….
f(x) = x/2 if x is odd
-x/2 if x is even
Transitivity Lemma
If f: A->B 1-1 onto, and g: B->C 1-1 onto
Then h(x) = g(f(x)) is 1-1 onto A->C
Hence, N, E, and Z all have the same
Do N and Q have the same
N= { 0, 1, 2, 3, 4, 5, 6, 7, …. }
Q = The Rational Numbers
No way!
The rationals are dense:
between any two there is
a third. You can’t list
them one by one without
leaving out an infinite
number of them.
Don’t jump to
There is a clever way
to list the rationals,
one at a time, without
missing a single one!
The point at x,y represents x/y
The point at x,y represents x/y
We call a set
countable if it can be
placed into 1-1 onto
correspondence with
the natural numbers.
So far we know that N,
E, Z, and Q are
Do N and R have the same
N = { 0, 1, 2, 3, 4, 5, 6, 7, …. }
R = The Real Numbers
No way!
You will run out of
natural numbers long
before you match up
every real.
Don’t jump to
You can’t be sure that
there isn’t some clever
correspondence that you
haven’t thought of yet.
I am sure!
Cantor proved it.
He invented a very
important technique
Theorem: The set I of reals
between 0 and 1 is not countable.
Proof by contradiction:
Suppose I is countable. Let f be the 1-1
onto function from N to I. Make a list L
as follows:
0: decimal expansion of f(0)
1: decimal expansion of f(1)
k: decimal expansion of f(k)
Theorem: The set I of reals
between 0 and 1 is not countable.
Proof by contradiction:
Suppose I is countable. Let f be the 1-1
onto function from N to I. Make a list L
as follows:
0: .3333333333333333333333…
1: .3141592656578395938594982..
k: .345322214243555345221123235..
L 0 1 2 3 4 …
L 0 1 2 3 4 …
0 d0
1 d1
2 d2
3 d3
… …
L 0 1 2 3 4
0 d0
1 d1
2 d2
3 d3
… …
ConfuseL =
. C0 C 1 C2 C3 C4 C5 …
L 0 1 2 3 4
0 d0
5, if dk=6
1 d1 C k=
6, otherwise
2 d2
3 d3
… …
ConfuseL =
. C0 C 1 C2 C3 C4 C5 …
L 0 1 2 3 4
0 d0
5, if dk=6
1 d1 C k=
6, otherwise
2 d2
3 . C 0 C1 C2 C
d3 3 C4 C5 …
… …
By design, ConfuseL can’t be on the list!
ConfuseL differs from the kth element on the
list in the kth position. Contradiction of
assumption that list is complete.
The set of reals
is uncountable!
Hold it!
Why can’t the same
argument be used to
show that Q is
The argument works
the same for Q until
the punchline.
CONFUSEL is not
necessarily rational,
so there is no
contradiction from the
fact that it is missing.
Standard Notation
S = Any finite alphabet
Example: {a,b,c,d,e,…,z}
S* = All finite strings of symbols
from S including the empty
string e
Theorem: Every infinite subset S
of S* is countable
Proof: List S in alphabetical order. Map
the first word to 0, the second to 1,
and so on….
Stringing Symbols Together
S = The symbols on a standard
The set of all possible Java
programs is a subset of S*
The set of all possible finite
pieces of English text is a
subset of S*
The set of all possible
Java programs is
The set of all possible
finite length pieces of
English text is
There are countably
many Java program and
uncountably many reals.
There are countably
many descriptions and
uncountably many reals.
ARE NOT
Is there a real
number that can
be described, but
not computed?
We know there are
at least 2 infinities.
Are there more?
There are many, many,
many, many, many more!
So many infinities that
the number of infinities
goes beyond any infinity!
Power Set
The power set of S is the set of all
subsets of S. The power set is denoted
Proposition: If S is finite, the power
set of S has cardinality 2|S|
Theorem: S can’t be put into 1-1
correspondence with P(S)
Suppose f:S->P(S) is 1-1 and ONTO.
Let CONFUSE =All x in S such that
x is not contained in f(x)
There is some y such that f(y)=CONFUSE
IS Y in CONFUSE?
YES: definition of CONFUSE implies NO
NO: definition of CONFUSE implies YES
This proves that there
are at least a countable
number of infinities.
The first infinity is
0,, 1, 2,
Cantor wanted to
show that the number
of reals was 1
Cantor couldn’t prove
that 1 was the
number of reals. This
helped feed his
depression. He called
it The Continuum
The Continuum
Hypothesis can’t be
proved or disproved!
This has been proved!
How Many Infinities?
Suppose there are q infinities.
For all i, let Si be a set of size i.
S = union of Si for i q
Easy to prove that S is bigger than q
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|
Summary: " Computing [Suppl] 15, 21-28 (2001)
@ Springer- Verlag 2001
Printed in Austria
On the Existence Theorems of Kantorovich, Moore and Miranda
G. E. Alefeld,Karlsruhe,F. A. Potra, Baltimore,and Z. Shen,Nanjing
Dedicated to Professor Tetsuro Yamamoto on the occasion of his 65th birthday
We show that the assumptions of the well-known Kantorovich theorem imply the assumptions of
Miranda's theorem, but not vice versa.
AMS Subject Classifications: 65HlO.
Key Words: Nonlinear systems, existence theorems, interval analysis.
1. Introduction
The purpose of this paper is to show that the assumptions of the well-known
Kantorovich theorem imply the assumptions of Miranda's theorem. This sur-
prising fact can be conc1udedfrom aseries of well-known results on the existence
of solutions of a nonlinear system of equations in Rn. At the end of the paper we
give a short direct proof of this fact.
2. Some Existence Theorems and their Relations
Proving the existence of a solution of a system of nonlinear equations is a fun-
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Measuring Pressure of Gas and Manometers with Examples
Manometers with Examples
Pressure of gas in a closed container is equal in everywhere. Manometers are used for measure pressure of gas in closed container. There are two types of manometer, they are in U shape and filled
with mercury. If one of the end is open to the atmosphere, we call this type open manometer, and if it is closed, then we cal it closed manometer. We will examine them in detail. Let me begin with
closed end manometers.
1. Closed End Manometers:
As you can see from the picture given below one end of manometer is open to gas container and one of them is closed. x and y points in the picture are at same level, thus pressures acting on these
points are equal. Pressure at point x is the pressure of gas and pressure at point y is the pressure of mercury at h height.
In this system pressure of gas is equal to;
2. Open End Manometers:
As you can see from the pictures given below, one end of manometer is open to container filled wit gas end one end of it is open to atmosphere. There are three situation we should learn in measuring
pressure of gas by the help of atmospheric pressure.
In first situation; pressures at points x and y are equal. Px=pressure of gas and Py=h+P[0] thus;
In second situation, pressures at point x and y are also equal and Px=pressure of gas and Py=P[0] thus;
In third situation; pressures at point x and y are also equal and Px=pressure of gas+h and Py=P[0] thus;
Example: Find the relation between gases X, Y, Z in the manometers given below.
Relation between densities of water and mercury is; d[water]<d[mercury] and P[0]=75 cm Hg.
X gas in open end manometer;
P[X]=75 cm Hg+30 cm Hg
Y gas in open end manometer;
P[Y]=75 cm Hg+30 cm H[2]O
Z gas in closed end manometer;
P[Z]=75 cm Hg
Since d[water]<d[mercury] pressure of Hg is larger than pressure of H[2]O. Thus;
Example: Find pressure of Y gas.(Atmospheric pressure is 75cm Hg)
we can write;
P[X]=15 cm Hg
P[Y]=45 cm Hg
Gases Exams and Problem Solutions
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(Co)Induction and (Co)Recursion
Papers on (Co)Induction and (Co)Recursion
People expect a proof assistant to support definition by recursion, but deriving recursion from fundamental logical principles is surprisingly difficult. Only recently could we say that support for
recursive datatype and function definitions in proof assistants is starting to be good enough to support the main idioms of functional programming.
Inductive definitions are fundamental in mathematics and during the 1990s became increasingly prominent in proof assistants as well. The well-known inductive method for verifying security protocols
in Isabelle only became possible because of the work described in papers such as those below.
1. L. C. Paulson.
Deriving Structural Induction in LCF. In: G. Kahn, D. B. MacQueen, G. Plotkin, editors, Semantics of Data Types (Springer, 1984), 197–214.
2. L. C. Paulson.
Proving termination of normalization functions for conditional expressions. J. Automated Reasoning 2 (1986), 63–74.
3. L. C. Paulson.
Constructing recursion operators in Intuitionistic Type Theory. J. Symbolic Computation 2 (1986), 325–355.
4. L. C. Paulson.
Set theory for verification: II. Induction and recursion. J. Automated Reasoning 15 (1995), 167–215.
5. L. C. Paulson.
Mechanizing coinduction and corecursion in higher-order logic. J. Logic and Computation 7 (March 1997), 175–204.
6. L. C. Paulson.
Final coalgebras as greatest fixed points in ZF set theory. Mathematical Structures in Computer Science 9 (1999), 545–567.
7. L. C. Paulson.
A fixedpoint approach to (co)inductive and (co)datatype definitions In: G. Plotkin, C. Stirling, and M. Tofte (editors), Proof, Language, and Interaction: Essays in Honour of Robin Milner (MIT
Press, 2000), 187–211.
8. L. C. Paulson.
A simple formalization and proof for the mutilated chess board.
Logic J. of the IGPL 9 3 (2001), 499–509.
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The dChip survival analysis module for microarray data.
Jump to Full Text
MedLine PMID: 21388547 Owner: NLM Status: MEDLINE
Abstract/ BACKGROUND: Genome-wide expression signatures are emerging as potential marker for overall survival and disease recurrence risk as evidenced by recent commercialization of gene
OtherAbstract: expression based biomarkers in breast cancer. Similar predictions have recently been carried out using genome-wide copy number alterations and microRNAs. Existing software packages for
microarray data analysis provide functions to define expression-based survival gene signatures. However, there is no software that can perform survival analysis using SNP array data or
draw survival curves interactively for expression-based sample clusters.
RESULTS: We have developed the survival analysis module in the dChip software that performs survival analysis across the genome for gene expression and copy number microarray data.
Built on the current dChip software's microarray analysis functions such as chromosome display and clustering, the new survival functions include interactive exploring of Kaplan-Meier
(K-M) plots using expression or copy number data, computing survival p-values from the log-rank test and Cox models, and using permutation to identify significant chromosome regions
associated with survival.
CONCLUSIONS: The dChip survival module provides user-friendly way to perform survival analysis and visualize the results in the context of genes and cytobands. It requires no coding
expertise and only minimal learning curve for thousands of existing dChip users. The implementation in Visual C++ also enables fast computation. The software and demonstration data are
freely available at http://dchip-surv.chenglilab.org.
Authors: Samir B Amin; Parantu K Shah; Aimin Yan; Sophia Adamia; Stéphane Minvielle; Hervé Avet-Loiseau; Nikhil C Munshi; Cheng Li
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Publication Type: Journal Article; Research Support, N.I.H., Extramural; Research Support, Non-U.S. Gov't Date: 2011-03-09
Journal Title: BMC bioinformatics Volume: 12 ISSN: 1471-2105 ISO Abbreviation: BMC Bioinformatics Publication Date: 2011
Date Detail: Created Date: 2011-04-01 Completed Date: 2011-05-25 Revised Date: 2013-06-30
Medline Nlm Unique ID: 100965194 Medline TA: BMC Bioinformatics Country: England
Journal Info:
Other Details: Languages: eng Pagination: 72 Citation Subset: IM
Affiliation: Department of Biostatistics and Computational Biology, Dana-Farber Cancer Institute and Harvard School of Public Health, 450 Brookline Ave, Boston, MA 02215, USA.
Export APA/MLA Format Download EndNote Download BibTex
MeSH Terms
Descriptor/ Breast Neoplasms / genetics
Qualifier: Cluster Analysis
Gene Expression Profiling / methods*
Kaplan-Meier Estimate
Oligonucleotide Array Sequence Analysis / methods*
Polymorphism, Single Nucleotide
Proportional Hazards Models
Grant Support
ID/Acronym/ 1R01GM077122/GM/NIGMS NIH HHS
From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine
Full Text
Journal Information Article Information
Journal ID (nlm-ta): BMC Bioinformatics Download PDF
ISSN: 1471-2105 Copyright ©2011 Amin et al; licensee BioMed Central Ltd.
Publisher: BioMed Central open-access:
Received Day: 27 Month: 9 Year: 2010
Accepted Day: 9 Month: 3 Year: 2011
collection publication date: Year: 2011
Electronic publication date: Day: 9 Month: 3 Year: 2011
Volume: 12First Page: 72 Last Page: 72
ID: 3068974
Publisher Id: 1471-2105-12-72
PubMed Id: 21388547
DOI: 10.1186/1471-2105-12-72
The dChip survival analysis module for microarray data
Samir B Amin12 Email: samir_amin@dfci.harvard.edu
Parantu K Shah1 Email: parantu.shah@gmail.com
Aimin Yan1 Email: jlfmssm@gmail.com
Sophia Adamia2 Email: Sophia_Adamia@dfci.harvard.edu
Stéphane Minvielle34 Email: stephane.minvielle@chu-nantes.fr
Hervé Avet-Loiseau34 Email: herve.avetloiseau@chu-nantes.fr
Nikhil C Munshi25 Email: Nikhil_Munshi@dfci.harvard.edu
Cheng Li1 Email: lch3000@gmail.com
1Department of Biostatistics and Computational Biology, Dana-Farber Cancer Institute and Harvard School of Public Health, 450 Brookline Ave, Boston, MA, 02215, USA
2Department of Medical Oncology, Dana-Farber Cancer Institute, 450 Brookline Ave, Boston, MA, 02215, USA
3Hematology Department, University Hospital, Nantes, France
4Centre de Recherche en Cancérologie, INSERM U892, Nantes, France
5Veterans Administration Boston Healthcare System and Harvard Medical School. 1400 VFW Pkwy, West Roxbury, MA, 02132, USA
In cancer clinical practice, predicting patient survival based on traditional tumor staging systems using clinical, histopathological and molecular markers remains an integral component in the
treatment decision for patients. For example, patients with advanced disease and poor survival prognosis are subjected to more aggressive treatments. However, this conventional approach is
non-specific and has limited success in the cancer treatment. Many patients have recurrence despite having aggressive therapy based on survival risk score [^1,^2].
With high-throughput cancer genomics data, we and others have reported using genome-wide expression signatures to predict survival risk, and these signatures are now increasingly being used in
treatment decision for several cancer types [^3^-^6]. Survival predictions have also been carried out using genome-wide copy number alterations [^7,^8] and microRNAs [^9,^10]. Encouraged by these
results, researchers routinely analyze large sets of microarray data in relation to survival information. Common analysis tasks and endpoints include gene signatures that predict survival risk,
survival difference between sample groups defined by unsupervised clustering, and survival analysis using the copy-number data of local genomic regions. Such survival analysis on a high-dimensional
data requires statistical programming and command-line skills, or the use of the existing software packages such as BRB-ArrayTools, Survival Online tool and Prediction Analysis for Microarrays (PAM)
However, there is no specific utility that can perform survival analysis using SNP array data or draw survival curves interactively for expression-based sample clusters. We have developed the
widely-used dChip software that can efficiently process and derive gene expression and copy number data from microarray datasets (http://www.dchip.org) [^14,^15], and have pioneered using SNP arrays
to find chromosomal alterations such as amplification, deletion, and loss of heterozygosity (LOH) [^16]. Thus, the addition of survival functions will be helpful for researchers to query and
correlate chromosomal regions of interest with associated survival data.
Here, we describe the survival analysis module in the dChip software that performs survival analysis across the genome for gene expression and copy number microarray data. The new survival functions
include interactive exploring of Kaplan-Meier (K-M) plots using both expression and copy number data, computing survival p-values from the log-rank test and Cox models, and using permutation to
assess the survival significance of copy numbers genome-wide. Researchers can also compare survival curves between sample clustering groups derived from expression data. The dChip survival module
enables user-friendly, interactive survival analysis and visualization of microarray data in the context of genes and cytobands. It requires no need for coding and minimal learning curve for existing
dChip users. The implementation in Visual C++ also enables fast computation for processing large data sets from studies such as the Cancer Genome Atlas (TCGA).
Implementation and analysis examples
The survival analysis functions are implemented in dChip using Visual C++ and optimized for fast computation. The computed log-rank test and Cox model statistics and p-values are confirmed using R
code. Figure 1 summarizes the preliminary raw data analysis and new workflow functions in two categories. a) those for SNP copy number data, and b) those for expression-based sample clustering
Example data sets
Here we will use two example data sets to demonstrate the functions: 1) performing survival analysis using SNP data, and 2) drawing K-M plots using expression-based sample clustering groups. For the
first dataset [^7], we will discuss the following dChip analysis steps: SNP data input and normalization, plotting copy number data in the chromosome view, carrying out survival analysis using the
log-rank and Cox model, and the permutation function to adjust for multiple testing and assess the genome-wide significance of the survival scores. For the second dataset, we will use a gene
expression dataset consisting of 170 uniformly treated patients with multiple myeloma with clinical follow-up of more than five years (Munshi et al., manuscript in preparation). We will first perform
unsupervised hierarchical clustering and define gene signatures that classify the samples into sub-groups, and then compare K-M curves by the log-rank test among these sub-groups.
Preparing an example dataset with survival outcome for analysis
We will use a 192-sample microarray dataset processed on Affymetrix SNP 500 K microarray platform [^7] to illustrate the usage of the new analysis and visualization functions. This data set
represents 192 uniformly treated patients with multiple myeloma (MM). MM is a common type of hematological cancer and characterized by malignant clonal transformation of plasma cells in bone marrow
with excess production of a monoclonal immunoglobulin. Chromosomal aberrations are a hallmark of MM with specific changes (del-13 and t(4;14)) giving poor prognosis and other changes (hyperdiploidy
and t(11;14)) conferring better survival [^17].
We use the dChip software to normalize all the arrays of 192 myeloma samples and additional 10 normal blood samples to compute model-based signal values. Normalization and model-based signals are
calculated for each of the two sub-arrays and combined. The median genotype call rates are 96.77% and 97.35% respectively for the 250 K Nsp and Sty sub-arrays. A tab-delimited sample information file
is prepared with columns specifying survival outcome including survival time and event indicator (0 = alive, 1 = death). Numerical columns are marked in the column header such as "Survival(numeric)",
and will be standardized and displayed above the samples in the clustering or chromosome data views. After loading the normalized data into dChip using the menu function "Analysis > Open group", we
use the "Tools > Array List File" menu function to create an array list to order samples by the values of a particular sample variable. Array list files are also useful when doing survival analysis
on a subset of samples, leaving other samples out of the downstream analysis.
The next step displays the data along chromosomes using the menu function "Analysis > Chromosome" to specify analysis parameters for copy number and LOH analysis (Figure 2). The copy number analysis
functions are explained in detail in the dChip manual [^14]. When the ploidy of samples are unknown (i.e. tumor samples), we can check the option "Scale copy number mode to 2 copy" to adjust for
ploidy effect sample-wise, so that in hyperdiploid samples the chromosomes with normal copy numbers are estimated to have two copies rather than deletion events.
Once the data is displayed by chromosomes, we can toggle between showing individual and all chromosomes using the "Chromosome > Show All" menu. Figure 3 shows the gain and loss of copy numbers in the
chromosome view, with SNPs on the rows and samples on the columns. It also uses an array list file to sort all the samples by survival time irrespective of the event indicator. We can browse the
genome and quickly observe whether copy gain and loss events are associated with survival. Figure 3 shows that hyperdiploid samples with three copies in chromosome 5 tend to locate on the right side
(thus longer survival time) rather than on the left side, suggesting hyperdiploidy is associated with better survival outcome of myeloma patients.
Genome-wide log-rank test using sample groups defined by SNP copy number
In the first survival analysis function, we use the menu function "Chromosome > Compute Score" to perform genome-wide, single-SNP survival analysis by selecting the "Survival Log-Rank p-value" option
under "Scoring method" (Figure 4). It will perform SNP-wise log-rank test for association between copy numbers and survival outcome. The data variables of either event free survival or overall
survival can be selected by clicking "Response variable" to open the "Select factors" dialog (Figure 4). For a SNP, all the samples are divided into three groups based on this SNP's inferred copy
number: Deletion (≤ 1.5 copy), Gain (≥ 2.5 copy), and "No change" (between 1.5 and 2.5 copy). This copy thresholds are adjustable at the "Options > Score" dialog. The log-rank test is then applied to
the three sample groups, testing the null hypothesis that there is no survival difference between the groups.
Across the genome, we compute the log-rank test score for one of every 10 consecutive SNPs to speed up the computation. This is because the inferred copy numbers are correlated for nearby SNPs when
computed from 10-SNP local median smoothing, leading to correlated log-rank test scores for nearby SNPs. The window size of 10 is user-adjustable at the dialog option "Tools > Options > Score > Use
one of every 10 markers". Once the log-rank test is applied to all the SNPs, dChip will display the survival scores genome-wide (Figure 3). The blue curve on the right displays the negative log10
transformed p-values from the log-rank test, with larger values indicating stronger association between copy numbers and survival outcome. The vertical red line in the gray box indicates the score
threshold for the genome-wide significance at the 0.05 level by the MaxT permutation method (see the section below on permutation). One or more regions across the genome could have scores exceeding
the threshold. Inspecting the genes within these regions could identify candidate survival-associated genes (Figure 5).
To draw a Kaplan-Meier plot (K-M plot) for an individual SNP, we first click on either a SNP nearby a gene of interest (the menu function "Chromosome > Find Gene" can search for genes) or a SNP from
the regions with significant survival scores. Then, we use the "View > K-M plot" function to draw a K-M plot using the copy number data of the SNP to form Deletion, Gain, and "No change" groups as
above (Figure 6). The p-value to test the different survival rates between the groups is computed using the log-rank test when there are two or more groups each with 5 or more expected number of
events. The SNP (rs16915104) in this K-M plot is within the transcribed region of the PLEKHA5 (PEPP2), an X-linked human homeobox gene at chromosome 12p12.3, which encodes transcription factor with
known oncogenic role in cancer and drug resistance via phosphoinositide-mediated signal pathways [^18,^19]. Using this approach, we can first perform genome-wide screen of SNPs to locate chromosome
regions of significant log-rank scores, and then zoom into the peak regions and use K-M plots to check the survival correlation of the SNPs and genes in the region.
Genome-wide Cox regression using SNP copy numbers
The second analysis function performs univariate Cox regression for survival outcome using the inferred copy numbers of SNPs. After calculating and displaying the inferred copy numbers in the
chromosome view, we use the menu function "Chromosome > Compute Score" and select the "Survival Cox Regression" option under "Scoring method" to perform univariate Cox regression for individual SNPs
across the genome (Figure 4). The computing time will vary depending on the number of samples and SNPs; it takes less than 10 minutes to process the 500 K SNP, 192 sample data on a 2.4 GHz CPU with 3
GB RAM.
This function displays the absolute z-scores computed from the Cox model in the gray box on the right of the copy number view (Figure 3 and 5). Negative z-scores are displayed in blue, indicating
less hazard or longer survival as copy number increases: copy number gains are associated with longer survival, or copy number deletions are associated with shorter survival. Positive z-scores are
displayed in red, indicating higher hazard or shorter survivals as copy number increases. As before, ordering samples based on survival time visualizes and confirms the association between copy
number alteration events and survival variables. Other continuous or binary sample variables can also be selected as "Covariates" to perform multivariate Cox regression to assess the explanatory
power of a SNP's copy number to survival in the context of these variables (Figure 4). Permutation can be applied to determine the genome-wide threshold to call chromosome regions significantly
associated with survival outcome.
Permutation to identify significant chromosome regions associated with survival
We also extend the existing permutation function in dChip to assess the genome-wide significance of the survival scores derived from the log-rank test or Cox regression. The number of permutation
runs is specified at the "Chromosome > Compute Score" dialog (Figure 4). The permutation is computationally intensive but also benefits from computing for only a subset of SNPs across the genome.
The permutation tests the null hypothesis that there is no chromosome region in the cancer genome whose copy number is associated with survival, and therefore any observed association is due to
random chance. To simulate data sets under the null hypothesis, we can either permute survival times with censoring indicators across samples or permute chromosome region blocks within every sample.
The survival scores from the simulated data sets are then compared to those from the original data set. Specifically for permuting chromosome region blocks, for each sample, whose SNPs are ordered
first by chromosomes and then by positions within chromosome, we randomly partition the whole genome into K (≥ 2) blocks, and randomly switch the order of these blocks while preserving the order of
SNPs within each block. In this way, the SNPs and their copy numbers in a sample are randomly relocated in blocks to new positions in the genome, while only minimally perturbing the dependence of the
copy number data of neighboring SNPs. The same permutation applies to all samples using a different random partition for each sample. The survival score at each SNP locus can then be computed for the
permuted data set, and the MaxT method can be applied to assess the significance of the original scores [^20]. The maximal survival scores from every permuted data set form the score distribution,
whose 95^th largest value is the genome-wide threshold at the 0.05 significance level to determine the chromosome regions significantly associated with survival in the original dataset. Similarly,
clinical variables such as survival times and censoring indicators can be permuted together among all the samples and this achieves similar genome-wide significance threshold (Additional file 1:
Figure S1 and Additional file 2: Figure S2).
Kaplan-Meier plots for expression-based sample clustering groups
Unsupervised hierarchical clustering is frequently used to discover novel sample sub-groups from microarray data or inspect the expression-based samples clusters in relation to clinical variables. In
particular, it is interesting to know whether sample clusters correspond to differential survival outcomes.
We develop a third analysis function in dChip to facilitate drawing K-M plots based on sample clusters. We first use the "Analysis > ANOVA & Correlation" menu to specify the survival variable and
event indicator, similar to Figure 4. Next, we use "Analysis > Clustering & Enrichment" menu to cluster samples using a variation-filtered gene list. Interested in whether sample clusters correspond
to different survival outcomes, we can click to select a main sample cluster branch (in blue color) and use Control-click to select and color additional sample clusters (Figure 7A). We then use the
menu "View/K-M Plot" to display the Kaplan-Meier plot and log-rank p-value based on the specified sample clusters (Figure 7B). The dChip Analysis View will also show the details of log-rank test and
the number of samples omitted due to missing survival data.
Discussion and conclusions
Gene expression changes and copy-number alterations are known to influence cancer progression and patient survival. Currently, the survival analysis of genome-wide copy number data using SNP arrays
and the assessment of significance requires specialized statistical software and skills. We have developed the survival analysis module within the dChip software to streamline the survival analysis
and interactive visualization of SNP copy number data and expression-based sample clusters. We use data analysis examples to show how dChip can interactively explore K-M plots and find survival
associated genomic regions of interest. The easy user-interface and fast computing make these dChip functions accessible to biomedical researchers.
Survival analysis using copy number data provides options to use either the log-rank test or Cox regression model to compute survival association. The genome-wide view after both analysis methods
displays a side-curve showing survival scores for consecutive SNPs. The results from the dChip survival functions agree with previously published results on the same dataset [^7]. The amplifications
in chromosome 1q and deletions in 1p and 16q are associated with poor survival, whereas the copy number gain of chromosomes 5 and 11 predicts a good prognosis.
We can change the size of SNP window for copy-number smoothing from the default 10 SNPs to other numbers. Increasing the window size will speed up the computation when sample size is large. But we
need to be cautious that large window sizes may miss potential survival association of SNPs, especially those SNPs near or within the gene coding regions.
Unsupervised hierarchical clustering is a standard approach to analyzing expression profiling. Researchers are often interested in whether samples clusters correspond to different event-free
survival, overall survival, or treatment response outcomes. We have also developed functions in dChip to draw sample cluster-based K-M plots. A user can select two or multiple cluster nodes to
compute, display, and export high-quality survival figures. To our knowledge, few software packages provide such graphical interface to ease the survival analysis without coding. We find Survival
Online (SO) tool by Corradi et al. [^12] a useful online portal for Cox regression and survival analysis using gene expression data. At present, dChipSurv provides similar analysis for SNP array
data. Together, these two applications will provide complementary set of features to users in need of survival analysis using expression and copy-number microarray data.
In summary, the dChip survival module addresses the frequent need of many researchers to integrate survival data analysis under a single microarray analysis package with minimal learning curve, fast
computing, and no requirement of programming skills. We will add more survival functions as well as extend them to microRNA and RNA-seq data in future dChip versions.
Availability and requirements
Project name: dChip survival analysis module
Project home page: http://dchip-surv.chenglilab.org/
Operating system(s): Windows 2000 or after
Programming language: Visual C++ 2005
Source code: Available on request.
License: The software is freely available.
Authors' contributions
CL designed and implemented the software module, S Amin and CL wrote the manuscript, S Amin, PKS and AY contributed to the data analysis and manuscript preparation, S Adamia, SM, HA and NCM designed
and performed the microarray experiments.
All authors read and approved the final draft.
We thank Cheng Li Lab members for constructive discussion. This work has been supported by NIH grant 1R01GM077122. P.K.S. has been supported by the Claudia Adams Barr Program in Innovative Basic
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Zhang Y,Martens J,Yu J,Jiang J,Sieuwerts A,Smid M,Klijn J,Wang Y,Foekens J,Copy number alterations that predict metastatic capability of human breast cancerCancer ResYear: 20096993795380110.1158/
Schetter A,Leung S,Sohn J,Zanetti K,Bowman E,Yanaihara N,Yuen S,Chan T,Kwong D,Au G,et al. MicroRNA expression profiles associated with prognosis and therapeutic outcome in colon adenocarcinomaJAMA
Year: 2008299442543610.1001/jama.299.4.42518230780
Ji J,Shi J,Budhu A,Yu Z,Forgues M,Roessler S,Ambs S,Chen Y,Meltzer P,Croce C,et al. MicroRNA expression, survival, and response to interferon in liver cancerN Engl J MedYear:
Simon R,Lam A,Li MC,Ngan M,Menenzes S,Zhao Y,Analysis of gene expression data using BRB-ArrayToolsCancer InformYear: 20073111719455231
Corradi L,Mirisola V,Porro I,Torterolo L,Fato M,Romano P,Pfeffer U,Survival Online: a web-based service for the analysis of correlations between gene expression and clinical and follow-up dataBMC
BioinformaticsYear: 200910Suppl 12S1010.1186/1471-2105-10-S12-S1019828070
Tibshirani R,Hastie T,Narasimhan B,Chu G,Diagnosis of multiple cancer types by shrunken centroids of gene expressionProc Natl Acad Sci USAYear: 200299106567657210.1073/pnas.08209929912011421
Li C,Amin S,Fan J, Lin X, Liu JSAnalysis of cancer genome alterations using single nucleotide polymorphism (SNP) microarraysNew developments in biostatistics and bioinformaticsYear: 2009Singapore:
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Hofmann O,Caballero OL,Stevenson BJ,Chen YT,Cohen T,Chua R,Maher CA,Panji S,Schaefer U,Kruger A,et al. Genome-wide analysis of cancer/testis gene expressionProc Natl Acad Sci USAYear:
Wittig R,Nessling M,Will RD,Mollenhauer J,Salowsky R,Munstermann E,Schick M,Helmbach H,Gschwendt B,Korn B,et al. Candidate genes for cross-resistance against DNA-damaging drugsCancer ResYear:
Westfall PH,Young SS,Resampling-based multiple testing: examples and methods for P-value adjustmentYear: 1993New York: Wiley
Figure 1
ID: F1] The overview of dChip survival functions for microarray data. MBEI: model-based expression index.
Figure 2
ID: F2] The "Analysis > Chromosome" menu dialog and options to view copy number data along chromosome. More description of the options is at the dChip website (http://www.dchip.org).
Figure 3
ID: F3] The dChip chromosome view of copy numbers and survival association scores. The SNP copy numbers in all the chromosomes are displayed in log2 ratios (red for gains and blue for losses), with
SNPs on rows ordered by chromosome and positions, and samples on columns ordered by survival time. Hyperdiploid samples (copy number > 2.5 for a number of chromosomes) tend to locate towards
the right, where samples have longer survival. The first blue curve on the right side is SNP-wise, -log10 transformed p-value from the log-rank test, e.g. 3 correspond to a p-value of 0.001.
The second blue curve on the right is the Cox model z-scores. The absolute z-scores are displayed SNP-wise: the red color for positive and blue color for negative scores. The vertical red
lines indicate the genome-wide score threshold at the significance 0.05 level using permutation analysis (log-rank: 5.71; Cox model: 5.31).
Figure 4
ID: F4] The "Chromosome > Compute Score" menu dialog for log-rank analysis and permutation function. (A) The survival response variables are selected from the data columns in a sample information
file. The event and event-free survival (EFS) variables can also be selected to perform survival analysis. (B) Additionally, the "Permutation methods" option provides permuting scheme by
either "Chromosome blocks" or "Clinical variables".
Figure 5
ID: F5] An enlarged chromosome region. The enlarged chromosome 12 region has significant survival scores (indicated by the red arrow in Figure 3). The gene names are displayed on the left with bars
representing the transcribed region. The Cox z-scores are displayed on the right as the blue curve. One of the SNP in the gene region of PLEKHA5 is used to draw K-M plot (Figure 6).
Figure 6
ID: F6] The Kaplan-Meier plot for a SNP, using its copy numbers to group samples. The copy number deletion at SNP_A-1882232 (rs16915104 at chromosome 12p12.3) is associated with poor survival. The
chromosome view above the K-M plot shows a score value in the right-sided gray box, crossing the significant score threshold (red) derived from permutation analysis. Also, the neighboring
SNPs are showing similar survival scores. These SNPs are in the transcribed region of gene PLEKHA5 (PEPP2) - a known oncogenic transcription factor (see text for details). The legend for each
sample group shows the number of samples, the Expected events (Exp), and the Observed events (Obs).
Figure 7
ID: F7] The Kaplan-Meier survival plots from expression-based sample clusters. (A) In the sample clustering view from the "Analysis > Clustering & Enrichment" function, we can click and select sample
clusters of interest for making K-M plots. The blue and red clusters are selected and highlighted. (B) The K-M plot is drawn by the "View > K-M plot" function. The legend for each sample
group shows the number of samples, the Observed events (Obs) and the Expected events (Exp).
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The Hamilton-Jacobi Equation for a Relativistic Particle.
Next: Solution to the Hamilton-Jacobi Up: Laser-driven particle mechanics Previous: The Dynamical Phase Contents
Being defined in terms of the action integral, the dynamical phase satisfies a differential equation which one obtains by a simple argument:
both satisfying Lagrange's equation of motion, but having slightly different termination points
as in Figure 4. Then the (principal linear part of the) difference in the value of the dynamical phase at these termination points is
The fact that 4, one sees that at the two termination points one has
Consequently, the principal linear part of the difference between the two
are the momentum components and
is the superhamiltonian of the charged particle at the termination point of its worldline. Equation (11) is the expression for the differential of
Thus the differential equation for the dynamical phase function
or explicitly
This is the Hamilton-Jacobi for a charged particle in an electromagnetic vector potential
Next: Solution to the Hamilton-Jacobi Up: Laser-driven particle mechanics Previous: The Dynamical Phase Contents Ulrich Gerlach 2005-11-07
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Rota's Conjecture: Researcher solves 40-year-old math problem
(Phys.org) —A Victoria University mathematician has experienced his own eureka moment, solving a 40 year-old mathematical problem.
Professor Geoff Whittle, from Victoria's School of Mathematics, Statistics and Operations Research, has been working with colleagues Professor Jim Geelen (Canada) and Professor Bert Gerards
(Netherlands) to solve a problem posed by the famous mathematician and philosopher Gian-Carlo Rota in 1970.
Earlier this year the trio realised that, after more than 15 years of work, they had achieved all the essential ingredients to prove Rota's Conjecture.
Geoff visited the United Kingdom last month to break news of the discovery to mathematics colleagues at a conference where he was a guest speaker.
Rota's Conjecture relates to a specialised area of mathematics known as matroid theory, a modern form of geometry, which Geoff specialises in.
Rather than focusing on distance and angles, matroid theory investigates properties of structures which don't change under projection—for example, whether or not three points are always on a line, or
four points are on a plane.
The theory investigates geometric structures that can be completely different from those in our world, and Rota's Conjecture is a way of using mathematics to recognise these alternative structures.
"I like to compare it to Kafka's Metamorphosis story, where a man wakes up and realises he has transformed into an insect—the way he views the world changes entirely," says Geoff.
This video is not supported by your browser at this time.
"Matroid theory is all about visualising a world of new geometrical structures and developing ways of describing the big, overarching structures which would emerge."
Although he has been working on proving Rota's Conjecture for a long period of time, Geoff says the hard work will really begin now, as the team starts writing up the results of its work.
"Resolving Rota's Conjecture is really special, and the product of many, many years of collaborative work.
"Now, we have a lot of writing to do, which I expect to take several more years—as well as many hundreds of pages of journal articles."
"It's a little bit like discovering a new mountain—we've crossed many hurdles to reach a new destination and we have returned scratched, bloodied and bruised from the arduous journey—we now need to
create a pathway so others can reach it."
Head of School Dr Peter Donelan is also excited about the news, which he believes will attract international recognition.
"This will be regarded as one of the outstanding mathematical achievements in recent years," he says.
Geoff has been based at Victoria University since 1992, when he joined the School of Mathematics, Statistics and Operations Research as a lecturer. He was promoted to Reader in 1997 and to Professor
in 2001.
Geoff is recognised as a world leader in the field of discrete mathematics. He has spent time as a visiting research fellow at Merton College, Oxford and is a Fellow of the Royal Society of New
Zealand. In 1996, his achievements were recognised with the New Zealand Mathematical Society's Research Award.
1 / 5 (7) Aug 15, 2013
"...my carbon footprint's no good..."
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Parametric RMT, discrete symmetries, and cross-correlations between
I will describe numerical and analytical results on cross-correlations between zeros of different L-functions. By analogy with parametric spectral correlations in random matrix theory and in
dynamical systems, these cross-correlations can be used to establish the concept of a "distance" in the space of (conjectural) generalised Riemann operators, and to gain some insight into their
overall structure.
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MathGroup Archive: January 2000 [00372]
[Date Index] [Thread Index] [Author Index]
RE: Making a function dynamically define another conditional function...
• To: mathgroup at smc.vnet.net
• Subject: [mg21766] RE: [mg21733] Making a function dynamically define another conditional function...
• From: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
• Date: Thu, 27 Jan 2000 22:56:46 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com
Paul Howland wrote:
How can I make a function dynamically define a conditional function?
Given a list of arguments {{a,A}, {b,B}, ...} I want to write a function
that will take these arguments, and generate a new function, f say,
which is defined as (for example):
f[x_] := x+a /; x==A
f[x_] := x+b /; x==B
So, the obvious solution is to define a function as follows:
In[1] := TestFn[data_List] := Module[{args},
args = data[[i]];
f[x_] = x + args[[1]] /; x==args[[2]],
{i, Length[data]}
and call it using something like TestFn[{{1,2},{3,4},{5,6}}].
But this doesn't work
I don't feel like figuring out why your method didn't work, but I do have a
However, are you sure you don't really want to define
and likewise for other conditions.
Well I will show how to get the result that I think you were trying to get.
Your definitions of (f) are stored in DownValues[f]. The next line shows the
first and only DownValue in this situation.
HoldPattern[f[x_]] :> x + x1 /; x == XX1
DownValues such as the one above are expressions just like anything else in
Mathematica, and you can change parts of this expression. In the next line
ReplacePart does this very nicely. Check the documentation for ReplacePart
to see what's going on here.
HoldPattern[f[x_]] :> x + a /; x == A
The line above changes (x1) to (a) and (XX1) to (A).
The next line does this for each sublist in (data).
Map[ReplacePart[dv,#,{{2,1,2},{2,2,2}},{{1},{2}}]& ,data]
{HoldPattern[f[x_]] :> x + a /; x == A, HoldPattern[f[x_]] :> x + b /; x ==
HoldPattern[f[x_]] :> x + c /; x == C}
In the next line I give you (TestFn) which wraps it all together.
DownValues[f]=Map[ReplacePart[dv,#,{{2,1,2},{2,2,2}},{{1},{2}}]& ,data]
Next I make sure TestFn works.
In[9]:= ?f
Ted Ersek
On 12-18-99 Mathematica tips, tricks at
had a major update
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Game Theory
What is Game Theory? Description
Game Theory is a special branch of mathematics which has been developed for studying decision-making in complex circumstances. Game theory tries to predict outcomes based on interactive models in
which the decisions of each party affect the decisions of the other parties. The meaning of "Game" here is: a move by one player will result in moves by others. The idea historically dates back to
the Talmud and Sun Tzu's writings. However, the contemporary codification is attributed to John von Neumann and Oskar Morgenstern. They published the Theory of Games and Economic Behavior in 1944. In
the early 1950s, John Nash generalized their results and provided the basis of the modern field of Game Theory. A rapid rise in theoretical developments led to the founding of the first academic
magazine devoted to the field by Oskar Morgenstern in 1972. Few corporations nowadays think about their strategy without adding some game theory models or game elements into their strategy process.
Game theory can be defined as the study of how people interact and make decisions. This broad definition applies to most of the social sciences, but game theory applies mathematical models to this
interaction under the assumption that each person's behavior impacts the well-being of all other participants in the game. These models are often quite simplified abstractions of real-world
interactions. While many game theorists certainly enjoy playing games, a "game" is an abstract representation of many serious situations and has a serious purpose.
Usage of Game Theory. Applications
• Preparing business negotiations.
• Analyzing future market conditions.
• Strategic decision-making.
• Assess the viability of a new venture, business model, program, project, product, service or technology.
Assumptions in Game Theory
A major issue with game theory is: it is necessary to make assumptions. Any model of the real world must make assumptions that simplify the reality, because the real world is too complex to analyze
with any precision. There is a constant tradeoff between realism and the technical capability to solve problems. Even if one could write down a model that accurately describes how people make
decisions in general, no amount of computers would be able to calculate it.
What assumptions are made normally? The usual assumptions are:
• Rationality. People take whatever actions are likely to make them more happy. And they know what makes them happy.
• Common knowledge. We know that everyone is trying to make himself as happy as possible, potentially at our expense.
These assumptions take many mathematical forms, from very strong (and likely unrealistic) towards much weaker forms in the study of behavioral game theory.
Experimental economics examines the validity of these assumptions by seeing how real people act in controlled environments.
Example of Game Theory
The most widely known example of game theory is probably the Prisoner's Dilemma: A zero-sum game cooperation game that got its name from the following hypothetical situation: imagine two criminals
arrested under the suspicion of having committed a crime together. However, the police does not have sufficient proof to have them convicted. The two prisoners are being isolated from each other, and
the police offers each of them a deal: the person that offers evidence against the other one will be freed. If none of them accepts the offer, they are in fact cooperating against the police, and
both of them will get only a small punishment because of lack of proof. They will both win. However, if one person betrays the other, by confessing to the police, he will gain more, since he is
freed. The one who remained silent, on the other hand, will receive the full punishment, since he did not help the police, and there is sufficient proof. If both betray, both will be punished, but
less severely than if they had refused to talk. The dilemma resides in the fact that each prisoner has a choice between only two options. But they can not make a good decision, without knowing what
the other person will do.
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D'Arcy and the Gnomons.
What does a mollusk mollusk: see Mollusca. mollusk
or mollusc
Any of some 75,000 species of soft-bodied invertebrate animals (phylum Mollusca), many of which are wholly or partly enclosed in a calcium carbonate shell secreted by the mantle, a soft called
'head-foot' and a Scotsman with a penchant for ice-cream have in common? The answer lies in a sunflower. Or the path of an insect as it flies toward a light. Or a Greek column. Or a pineapple. Is
mathematics to be found in a textbook or a shell? When scientists model nature, are we inventing something new, or simply discovering relationships as old as the universe itself? And how can all
this help design hearing implants?
Some see mathematics as a product of the mind. As a thought, a human conception that lives in the mind and sleeps in a textbook. Surely the most abstract and esoteric corners of mathematics have no
place in the world of ants and rocks--after all, where can we point to find a Picard Group (1) or a hypercube A parallel processing architecture made up of binary multiples of computers (4, 8, 16,
etc.). The computers are interconnected so that data travel is kept to a minimum. For example, in two eight-node cubes, each node in one cube would be connected to the counterpart node in the other.
The peculiarly Pythagorean view of the world as number sees mathematics as the key to understanding the universe--a view implicitly shared by physicists of our day, albeit without the same
mysticism. Pythagoras saw mathematics as an intrinsic part of nature, music and the stars; something magical to be discovered. Pythagoras' thinking clearly inspired Plato's Forms and Solids, (2) for
he conceived a model of the universe based on the dodecahedron dodecahedron: see polyhedron. . Some 2000 years later, Kepler's revolutionary elliptic el·lip·tic or el·lip·ti·cal
1. Of, relating to, or having the shape of an ellipse.
2. Containing or characterized by ellipsis.
a. planetary model was directly influenced by Pythagoras' Spheres, (3) which has more recently inspired the spatial design of the [C.sub.60] Buckminsterfullerene buckminsterfullerene (bŭk'mĭnstərf
molecule. (4)
Western philosophy and mathematics have been intimately entwined from their beginnings. Plato, Descartes and even Kant were foundationalists who sought purity and 'transcendent perfection' in
mysteries both mathematical and natural. (5) The earliest philosophers were deeply concerned with geometry, astronomy and logic. Gottlob Frege challenged the dichotomous di·chot·o·mous
1. Divided or dividing into two parts or classifications.
2. Characterized by dichotomy.
di·chot notion of insisting mathematical concepts were either physical or mental by describing them as 'abstract objects', which is taken a step further in some modern thinking by describing them in
terms of a quasi-collective consciousness.
But while Platonism would say we cannot 'invent' mathematics, but only discover it, an empiricist em·pir·i·cism
1. The view that experience, especially of the senses, is the only source of knowledge.
a. Employment of empirical methods, as in science.
b. An empirical conclusion.
3. would insist we only learn about mathematics through our senses as we interact with the world. (6) Reuben Hersh offers us an intriguing perspective: that of mathematics as humanistic pursuit, a
socio-historic phenomenon, (7) invoking the Fregean notion of the abstract. He points out that mathematics as we know it today is a product of history and human activity, including its mistakes,
false starts, incompleteness and limitations.
So while there is no clear consensus on the pure universality of mathematics, the question has not impeded the pursuit. On the contrary, modern society is literally built on the sometimes shaky
foundations of mathematics. Despite Kurt Godel, (8) we still teach 'incomplete' algebra, and despite Georg Cantor and Bertrand Russell (9) we still make heavy use of set theory. And while some
invoke a deity in rapturous rap·tur·ous
Filled with great joy or rapture; ecstatic.
rap adv. wonder at mathematics, as quoth quoth
tr.v. Archaic
Uttered; said. Used only in the first and third persons, with the subject following: "Quoth the Raven, 'Nevermore!'" Edgar Allan Poe. Mary Somerville:
Nothing has afforded me so convincing a proof of the unity
of the Deity as these purely mental conceptions of numerical
and mathematical science which have by slow degrees been
vouchsafed to man. (10)
other giants, such as logicist Russell, are content to do without when he states that he
like[s] mathematics because it is not human and has nothing
particular to do with this planet or with the whole accidental
universe--because, like Spinoza's God, it won't love us in
return. (11)
Perhaps it is human nature to invoke a higher power when faced with something so awesome and wondrous that we struggle to conceive of it 'just happening'.
The seafaring mollusk known as nautilus nautilus, in zoology
nautilus, cephalopod mollusk belonging to the sole surviving genus (Nautilus) of a subclass that flourished 200 million years ago, known as the nautiloids. (Cephalopoda Nautiloidea, or 'head-foot
sailor') carried a calcified Calcified
Hardened by calcium deposits.
Mentioned in: Heart Valve Repair rendering of a mathematical wonder for its shell millions of years before Descartes anointed it with the first of its many names. The logarithmic spiral is also
known as the growth spiral or the equiangular spiral. The eminent mathematician Jacob Bernoulli was so enamoured enamoured or US enamored
enamoured of
a. in love with
b. very fond of and impressed by: he is not enamoured of Moscow [Latin amor love] with this spira mirabilis, the miraculous spiral, he wished it inscribed on his gravestone. (12)
The logarithmic spiral can be defined as a curve that exhibits a constant angle between the radius vector (a line from the centre to a point on the curve) and the tangent vector (a line oriented
along the path of travel). Try this: put a bin in the middle of your office, stand next to it, then stretch out your left arm so that it is pointing approximately at 10 o'clock. (13) Walk backwards
around the bin, moving further away from it while keeping your arm pointing directly at the bin. Stop before you run into your desk, and you will have traced out a logarithmic spiral. The tangent
vector is pointing out your back, while your sore arm (14) is the radius vector.
The logarithmic spiral has many special properties that make it very useful in both nature and engineering: it is self-similar, in that its shape remains unaltered by scaling and angular growth; the
distance between arms increases in a geometric progression; (15) any straight line passing through the origin makes a constant angle with the curve (Figure 1); a degenerate logarithmic spiral is a
straight line at one extreme and a circle at the other; it can be produced using incredibly simple rules, such as 'move forward a bit, turn left 30 degrees'. If instead of moving forward, we simply
'grow and turn', we enter the domain of Lindenmeyer Systems: a formal set of simple rules capable of generating remarkably complex fractal figures, such as trees and ferns. (16)
[FIGURE 1 OMITTED]
Nature provides a cornucopia cornucopia (kôr'ny, in Greek mythology, magnificent horn that filled itself with whatever meat or drink its owner requested. (17) of examples of the logarithmic spiral
in action. The arrangement of seeds in a sunflower--the optimal arrangement for efficient packing. The path of an insect as it flies toward a light--arising from the structure of its compound eyes.
The path of an eagle as it swoops on its prey--so it can keep a constant eye on the target. The arrangement of scales on a pineapple. The swirling rage of a tropical cyclone. Or the swirling mist of
stars in a galactic spiral. And the shell of a mollusk--due to the accretive mode of its construction. You may even find one in your own backyard.
So what does a logarithmic spiral have to do with rabbits? For rabbits were the etude e·tude
n. Music
1. A piece composed for the development of a specific point of technique.
2. A composition featuring a point of technique but performed because of its artistic merit. of the great Fibonacci: how fast could they breed in ideal circumstances? The sweat of his brow (and that
of his rabbits) produced a fascinating result, and thus his eponymous (18) sequence. (19) That is, the total population of this idealised colony after each generation is given by 1, 1, 2, 3, 5, 8,
13, 21, 34, 55 ..., where each successive number is the sum of the previous two.
[FIGURE 2 OMITTED]
While many people are understandably riveted by this ingenious model of bunny proliferation, the fun doesn't stop there. Fibonacci numbers crop up in a surprising number of places in nature around
us: the number of petals on a flower; the number of spirals of seeds in a flower head or a pine-cone; the number of leaves per turn around a stem; cauliflower florets; and romanesque broccoli. (20)
But there are some things even more interesting than rabbits and cauliflower. For example, we can geometrically construct an approximation of our miraculous logarithmic spiral with rectangles sized
according to the Fibonacci sequence.
[FIGURE 3 OMITTED]
The Golden Mean is another very special number with many names. It is a natural relationship, a special proportion revered by artists and mathematicians alike. Many consider it not only
aesthetically pleasing but almost mystical, due in no small part to its intriguing properties and history. (21) Between two measures, it is the ratio of the whole to the larger, as the larger is to
the smaller. That is, (see Figure 4). Also known by the Greek symbol phi or [phi], the
[FIGURE 4 OMITTED]
a + b / a = a / b
Golden Mean or Divine Proportion is associated with a somewhat controversial area of study known as 'Sacred Geometry', a particularly Pythagorean concept. Phi is ancient, and may be found variously
in nature, architecture, art and music. It is frequently claimed that the facade of the Parthenon is laid out according to the Golden Mean, so too the dimensions of the pyramids, (22) yet these
claims (and many others) have been debunked as 'wishful thinking'. There are, however, many genuine examples, (23) such as the work of Paul Serusier (1864-1927), Gino Severini (1883-1966) and the
Cubists. The renowned architect Le Corbusier invented a comprehensive system of proportions called Modulor, which was based on a combination of the Fibonacci sequence, the Golden Mean, and the
proportions of da Vinci's Vitruvian Man, designed to provide a 'range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanical things'. (24)
Unfortunately, while there are many authentic manifestations of the Golden Mean to be found across human pursuits, it is often difficult to distinguish these from overly optimistic seekers who
retrospectively juxtapose jux·ta·pose
tr.v. jux·ta·posed, jux·ta·pos·ing, jux·ta·pos·es
To place side by side, especially for comparison or contrast. the ratio onto older works with insufficient evidence insufficient evidence n. a finding (decision) by a trial judge or an appeals court
that the prosecution in a criminal case or a plaintiff in a lawsuit has not proved the case because the attorney did not present enough convincing evidence. or justification.
Now it was in fact our friend Kepler who discovered that the ratio of successive Fibonacci numbers converges to [phi], the Golden Mean. (25) It wasn't until as recently as 1994, however, that two
French mathematicians, Stephane Douady and Yves Couder, proved (26) that the optimum growth pattern for the most efficient packing in a plant occurs when the angle between successive primordia
(points of growth) is the irrational Golden Angle, 2[pi](1-[phi]). Thus these three mathematical concepts--logarithmic spirals, Fibonacci numbers and the Golden Mean--are intimately related, and
appear to form part of nature itself. It is perhaps no wonder that many mathematicians are moved to poetry when beholding such elegance, depth and beauty.
Now D'Arcy Thompson (1860-1948) liked to walk around St Andrews, Scotland with a parrot on his shoulder. He was also very fond of ice-cream, and translating classical Latin and Greek texts. But more
than for his parrot, he will be remembered for his opus magnum, On Growth and Form (1917), wherein an entire chapter is devoted to the logarithmic spiral. (27) In these pages we find a poetic
treatise on natural development, a thorough analysis of patterns of growth and models that became a cornerstone of modern morphometry mor·phom·e·try
Measurement of the form of organisms or of their parts.
mor (the study of the shape of living organisms). Thompson's unifying theory is that ontogeny ontogeny: see biogenetic law. Ontogeny
The developmental history of an organism from its origin to maturity. It starts with fertilization and ends with the attainment of an adult state, usually expressed in terms of both maximal body
(individual growth) is determined by the physical forces of nature, and shaped by environment. This powerful statement, to some extent at odds to Darwinian theory (and pre-dating the discovery of
DNA DNA: see nucleic acid. DNA
or deoxyribonucleic acid
One of two types of nucleic acid (the other is RNA); a complex organic compound found in all living cells and many viruses. It is the chemical substance of genes. ), is now generally accepted as
complementary to our modern notions of growth, development and biological processes.
Shells, such as the nautilus discussed earlier, grow by accretion; matter is gradually accumulated or deposited at the opening, enlarging as it winds around a central axis. (28) This means that only
the opening ever grows; the original form never changes, so that a young shell has the exact same form as a larger one of the same species. Thompson marvels that 'this remarkable property ... is
characteristic of the equiangular spiral, and of no other mathematical curve.' Such a shell can be described as a gnomon gnomon (nō`mŏn): see sundial. , which in its most general sense is 'any
figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original.' (29) The word has its origins nearly 2000 years ago with the mathematician Hero of
Alexandria, in a day when people were more interested in the design of sundials; (30) the gnomon was the 'indicator', the part of the sundial that cast the shadow. Gnomons can be found in many areas
of both mathematics and nature, shells being the most obvious exemplar.
In exploring just this one particular aspect of mathematics, we find richly intertwining relationships, and more examples than could possibly be surveyed in this manuscript. It is clear that these
relationships are there to be discovered and applied to new areas of enquiry: all we need is the right mathematical language, and a way to determine how well our views fit our observations. We need
Model (n): a system or thing used as an example to follow or imitate; a simplified description, esp. a mathematical one, of a system or process, to assist calculations and predictions.
--Oxford English Dictionary
Mathematical modelling is the judicious art of oversimplification o·ver·sim·pli·fy
v. o·ver·sim·pli·fied, o·ver·sim·pli·fy·ing, o·ver·sim·pli·fies
To simplify to the point of causing misrepresentation, misconception, or error.
v.intr. .
As Rodney Brooks is fond of pointing out, 'the world is its own best model.' This philosophy, espoused in his 'subsumption architecture' for robotics, (31) argues that the classical symbol-system of
artificial intelligence (32) is 'fundamentally flawed'. He posits that any attempt to recreate an internal model of the external world is inherently lacking in every way, and is therefore pointless.
Brooks explicitly rejects a reductionist re·duc·tion·ism
An attempt or tendency to explain a complex set of facts, entities, phenomena, or structures by another, simpler set: "For the last 400 years science has advanced by reductionism ... approach to
artificial intelligence, and argues strongly for a biologically inspired approach. Despite this, many roboticists today opt for a pragmatic hybrid approach. For example, a robot does not need to
distinguish the fine grain of the hardwood, or the sheen of the lacquer to avoid running into a chair; recognising it merely as an obstacle is perfectly sufficient. A purposive pur·po·sive
1. Having or serving a purpose.
2. Purposeful: purposive behavior.
pur model is one that is tailored specifically for the task at hand, with no more detail or complexity than is needed to satisfy the goals.
This serves to highlight the distinction between 'modelling perfectly' and a 'perfect model'. It may be that modelling perfectly, as Brooks' earlier observation alludes to, would require a model
equivalent to the subject itself, since anything less would necessarily be imprecise or incomplete, and therefore imperfect. The 'perfect model', however, is a model that completely satisfies all
the requirements of the task at hand, and is therefore achievable. In all practical scenarios, with finite time and resources, we will most likely have to satisfy ourselves with a 'good model'; one
which is as accurate as we can practically make it given all the constraints, and for which the limitations and shortcomings are understood.
The father of computer science, Alan Turing, shared Thompson's fascination with nature and numbers. Indeed, Turing was greatly influenced by On Growth and Form; as a child he spent hours sketching
flowers, poring over pine-cones, examining seeds and searching for relationships and patterns. His unfinished work in the latter period of his life explored plant growth and the relationship between
the Fibonacci sequence and sunflowers. (33) In his seminal 1952 paper on chemical morphogenesis morphogenesis /mor·pho·gen·e·sis/ (mor?fo-jen´e-sis) the evolution and development of form, as the
development of the shape of a particular organ or part of the body, or the development undergone by individuals who attain the type to , (34) Turing rather modestly described his proposed model as
'a simplification and idealization idealization /ide·al·iza·tion/ (i-de?il-i-za´shun) a conscious or unconscious mental mechanism in which the individual overestimates an admired aspect or attribute
of another person. , and consequently a falsification'. While it has since become 'the definitive basis for modelling biological growth', (35) Turing rightly (and succinctly) questions the nature of
the modelling process, and just how much faith we can place in these idealised forms. Fortunately, all is not lost.
So what makes a good model? Ideally our model would be:
* General: can be applied to a wide variety of normal subjects.
* Expressive: captures essential aspects of the subject in a meaningful way.
* Simple: no more complex than necessary to carry out the task.
* Verifiable: can be validated against observation.
Are mathematical models appropriate for modelling nature? If we take the word of the Platonists, most definitely. In practice, engineers and physicists use them all the time, cognisant of their
inherent limitations. But is there an intrinsic mathematical relationship there for us to find in nature? An idealist would likely say there is, but whether or not we can find it is another
question. How would we recognise it if we found it? That is not so easy to answer. When evaluating a model, we can only increase our confidence the more we successfully validate it against different
Are we not inherently limited by our mathematical toolbox? Mathematicians are constantly expanding the horizons of theory in pure and applied domains. Our knowledge is assuredly finite, but growing
rapidly. We may not yet have the tools at our fingertips in order to find the 'perfect model', so we make do with what we have. Sometimes, we really do just see every problem as a (linear function)
nail, and bang on it with our (linear algebra) hammer.
Modelling as an activity is after all purposive, in that we have a goal, a subject and a set of constraints. We should also have some means to measure how well our model fits the subject, and an
idea of just what is 'good enough'. There is in general no single best solution--the efficacy of a model can be measured only in terms of its fitness for purpose. Good practice would have us apply
Occam's Razor to all models; that we employ the simplest explanation. In the same vein, the principle of parsimony Noun 1. principle of parsimony - the principle that entities should not be
multiplied needlessly; the simplest of two competing theories is to be preferred
law of parsimony, Occam's Razor, Ockham's Razor eliminates from a model any parameter that does not contribute significantly to the explanatory power of the model. Linear is better than non-linear,
fewer parameters are better than more parameters.
With this simplicity and generality in mind, we turn our attention to a particularly interesting problem at the nexus of the above, seemingly disparate topics: producing a mathematical model of a
very small spiral organ spiral organ
See organ of Corti. .
ON THE COCHLEA cochlea (kŏk`lēə): see ear.
Millions of people worldwide suffer from profound sensorineural hearing loss Sensorineural hearing loss
Hearing loss caused by damage to the nerves or parts of the inner ear governing the sense of hearing.
Mentioned in: Tinnitus
sensorineural hearing loss . One in a thousand babies is born with congenital hearing defects, while around forty percent of people over the age of seventy-five develop progressive loss. The
cochlear cochlear
pertaining to or emanating from the cochlea.
cochlear duct
the coiled portion of the membranous labyrinth located inside the cochlea; contains endolymph.
cochlear nerve
see Table 14. ear implant has become the standard clinical intervention for nerve-impaired deafness, with over fifty thousand recipients in 120 countries worldwide. The cochlear implant cochlear
An electronic device that stimulates auditory nerve fibers in the inner ear in individuals with severe or profound bilateral hearing loss, allowing them to recognize some sounds, especially speech
sounds. restores hearing by sending tiny electrical impulses to the residual nerves (36) (Figure 5).
[FIGURE 5 OMITTED]
Your humble author is engaged in the development of a three-dimensional shape model of the cochlea that may ultimately assist clinicians in providing better treatment for people so affected. The
research is aimed at analysing cochlea morphometry, deriving metrics to help quantitatively describe cochleae (where there are now predominantly qualitative descriptions), and constructing a
mathematical model that can capture its essential characteristics. Such anatomical shape models will give us insight into normal and abnormal shape variation. They may ultimately be used for a
variety of purposes: diagnosis, by providing quantitative measures that can be compared to known normal forms; simulation and training, by enabling clinicians to generate a variety of plausible
forms; and surgery planning, by highlighting regions of unusual shape that may require attention.
The cochlea (Latin for 'snail shell') is the organ of hearing Noun 1. organ of hearing - the part of the ear that is responsible for sensations of sound
sense organ, sensory receptor, receptor - an organ having nerve endings (in the skin or viscera or eye or ear or nose or mouth) that respond to stimulation , a tiny 2[cm.sup.3] shell-like structure
in the inner ear, embedded in the temporal bone of the skull. It is spiral in shape, and is often described as approximating a logarithmic spiral. A normal cochlea revolves through two and a half
turns, from the basal turn (lower turn) up to the helicotrema (top of spiral). Three channels run the length of the cochlea: the scala tympani, scala media and scala vestibuli. The cross-sectional
shape resembles a cardioid (rounded 'B' shape).
While detailed studies have been carried out on the morphology of the bony labyrinth (incorporating the cochlea), (37) the metrics are typically linear measurements or relative orientations--for
example, the width of the basal turn of the cochlea, or the angle of the lateral semicircular canal in the sagittal plane, all of which are most relevant to phylogenetic phy·lo·ge·net·ic
1. Of or relating to phylogeny or phylogenetics.
2. Relating to or based on evolutionary development or history. studies. Most of the cochlea models described are spacecurves that either do not take into account cross-sectional shape at all, (38)
or use an approximation such as a circle. Clearly there is a need for clinically relevant models of the cochlea, beyond what is currently described in the literature.
It may be tempting to apply Raup's shell models (39) to the cochlea. These biological models of shell shape are simple yet effective parametric models that reproduce a wide variety of shapes found
in nature, as well as a wide variety that are not. This may simply imply that the parameter space of the model only partially overlaps with the parameter space of the system, or it may imply that
the natural model is better described in some other way. Or it may simply imply that the natural model has implicit constraints on its parameters, which is consistent with Thompson's theory of
phylogeny. Since it is simple and works very well for a wide variety of examples, we can say it is a good model.
However, assuming the Raup model was an accurate representation of the actual biological growth process for shells, there is still the question of whether it is appropriate, or semantically and
biologically valid to apply an accretion growth model to the cochlea. First is the fact that the curve of Raup's shells grows from top to bottom (apex to base), growing outward. However, the cochlea
itself grows from base to apex, curving inward. Indeed, it is fully formed by the age of seven months in the womb. Second is the fact that the cochlea does not grow by accretion, and its development
is linked to the growth of the temporal bone itself. Since so much of our knowledge of the growth and development in utero is incomplete, we must focus on what we can best observe and measure, which
is primarily CT scans of fully-developed adult temporal bones.
L T Cohen cohen
or kohen
(Hebrew: “priest”) Jewish priest descended from Zadok (a descendant of Aaron), priest at the First Temple of Jerusalem. The biblical priesthood was hereditary and male. describes a 2D spiral model
that was originally designed to model the path of the implant electrode itself (40) (and not of the otic capsule, the bone that encapsulates the cochlea). Sun Yoo et al. describe an extension of
Cohen's model, reinterpreting it as defining the centreline of the cochlea. Yoo added an exponential height function to the 2D spiral, and described a method of fitting the spacecurve model to CT
data. (41) Until recently it was believed that the shape of the cochlea was primarily due to an economy of space; however, a new theory suggests that the spiral shape of the cochlea is highly
significant, and can have an amplifying effect of up to 20dB for low frequencies. (42) The mechanism is believed to be similar to the 'whispering walls' effect in St Paul's Cathedral This article is
about the cathedral church of the diocese of London. For other cathedrals consecrated to Saint Paul, see Cathedral of Saint Paul.
St Paul's Cathedral in London, where one can whisper at one side of the gallery and be heard on the other. (43)
This author is investigating a 3D parametric shape model, based on the models of Cohen and Yoo, since they are well-known in the literature and provide a simple and convenient starting point. My
model extends these into 3D, adding a full outer surface and an elliptical el·lip·tic or el·lip·ti·cal
1. Of, relating to, or having the shape of an ellipse.
2. Containing or characterized by ellipsis.
a. cross-section. From the outset, the goal of this work is to produce a model that features a clinically relevant parameter space, and incorporates or yields metrics that are directly relevant in
the clinical domain. (44) The explanatory power lies in the parameter space and its embedding in the real world; otherwise any numbers derived from such a model would have no basis for biological
interpretation. Metrics such as the average diameter of the basal turn or the curvature of the centreline of the otic capsule are far more clinically relevant for the purposes of planning for
cochlear implant surgery than (for example) the linear measurements described above, or the coefficients of some convenient but effectively arbitrary trigonometric function.
The rigid model currently being explored (45) is useful as a first approximation as it incorporates the gross-scale metrics that can uniquely describe the shape of the cochlea. However, it is also
inherently unable to capture fine-scale detail, such as local pathologies, deformities and so on, and may even be incapable of deforming to normal shapes (only extensive fitting to normal samples
can confirm this). So ultimately this model may not be the most pure, elegant or abstract, or even the most accurate. But in terms of fitness for purpose, the model satisfies our criteria for
simplicity (seven parameters), expressiveness (the parameters are clinically relevant), general, and verifiable (it is currently being fitted to a range of human CT scans). Knowing the properties
and limitations of this parametric model, the model fitting process is being extended to incorporate a fully deformable model, which will not only provide a means for validation but can also be used
to quantify the regions of the model that poorly fit the organ, so that we may further improve the model and its ability to be generalised.
Here then is the intersection of nature, spirals, mathematics and models: can we represent the cochlea with a logarithmic spiral in a clinically meaningful way? Does a logarithmic spiral actually
capture the natural growth and form of the cochlea?
The research described herein invokes a certain degree of Pythagorean idealism in its search for mathematical truth and beauty in nature. But it is necessarily tempered with pragmatism, for a clear
goal stands before us. The path ahead is long, but the early results are encouraging.
More than a model of nature, mathematics can be seen as the language of relationships in nature. As we add to our vocabulary, our mathematical toolbox becomes richer, allowing us to capture, express
and explore more profoundly the theoretical aspects of the patterns of life before us. The limits of our models only serve to highlight the limits of our finite understanding of the universe, which
is at once humbling and challenging.
(1) This of course refers to a particular type of multiplicative Abelian group in Number Theory, and not the television series Star Trek.
(2) Forms: perfect, abstract entities that exist independently of the world. Solids: five perfectly symmetrical three-dimensional geometric shapes composed of non-planar points.
(3) Robert Mankiewicz, The Story of Mathematics, Cassell & Co, London, 2000.
(4) Martin Kemp, 'Intimations and intuitions', New Scientist, no. 2568, 9 September 2006, 48-9.
(5) Robert Solomon, A Short History of Philosophy, Oxford University Press, New York, 1996.
(6) Ted Honderich (ed.), The Oxford Companion to Philosophy, Oxford University Press, Oxford and New York, 1995.
(7) Reuben Hersh, What is Mathematics, Really?, Vintage, Cary, North Carolina, 1998.
(8) Godel shook the foundations of mathematics when he proved that no axiomatic system (one built up from a series of rules) was complete, in that it was capable of proving all algebraic truths
within that system. Fortunately our buildings still stand.
(9) Cantor's set theory got tied up in knots when Russell pointed out a paradox involving sets containing themselves, rather like the way in which boxes cannot.
(10) Hersh, 9.
(11) Nicholas Griffin (ed.), The Selected Letters of Bertrand Russell: I, The Private Years (1884-1914), Routledge, London, 2002, 404.
(12) Tragically, the engraver mistakenly produced an Archimedean spiral (resembling a coiled rope) instead.
(13) Where 12 o'clock is straight ahead, 9 o'clock is to your left, and 3 o'clock is about tea time.
(14) Hope you didn't trip over anything!
(15) A constant ratio between successive measures, and hence the origin of one of its names.
(16) Przemyslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic Beauty of Plants, Springer-Verlag, New York, 1996.
(17) Cornu cornu /cor·nu/ (kor´noo) pl. cor´nua [L.] horn.
cornu ammo´nis hippocampus.
cornu cuta´neum cutaneous horn. copiae, the 'horn of plenty', was a mythical horn able to produce whatever was wished for, a gift from Zeus to Amalthea from the goat upon whose milk he was raised.
Horns, of course, are shaped like a cone twisted by a helical spiral.
(18) Despite the sequence being attributed to Fibonacci, it was in fact first described by Indian mathematicians in 1150, calculating the optimal way to pack a cart with boxes of unit sizes one and
two. Donald Knuth, The Art of Computer Programming: Volume 1, Addison-Wesley, Reading, Mass., 1997.
(19) Ian Stewart, Nature's Numbers, Weidenfeld & Nicolson, London, 1995.
(20) Stewart, 138.
(21) Mario Livio, The Golden Ratio, Broadway Books, New York, 2002.
(22) Midhat Gazale, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999.
(23) Livio, 168.
(24) Livio, 174.
(25) Livio, 101.
(26) Stewart, 139.
(27) D'Arcy Thompson, On Growth and Form, Cambridge University Press Cambridge University Press (known colloquially as CUP) is a publisher given a Royal Charter by Henry VIII in 1534, and one of the
two privileged presses (the other being Oxford University Press). , Cambridge, 1992.
(28) This parallels the growth of plants, whereby the meristem meristem (mĕr`istĕm'), a specialized section of plant tissue characterized by cell division and growth. Much of the mature plant's
growth is provided by meristems. (the growing cells at the tip of a plant) grows and turns at a constant angle.
(29) Thompson, 182.
(30) Gazale, 125.
(31) Rodney Brooks, 'Elephants don't play chess', Robotics and Autonomous Systems, vol. 6, 1990, 315-327.
(32) The idea that intelligence is a central reasoning engine operating on a set of symbols that represents the world.
(33) Alan Hodges, Alan Turing: The Enigma, Walker & Company, New York, 2000.
(34) As a complement to modelling the physical shape of a shell, this work described reaction-diffusion partial differential equations that might explain the pigments and patterns on the shell
(35) Diran Basmadjian, The Art of Modelling in Science and Engineering, Chapman & Hill, Boca Raton, Fl., 1999.
(36) Philipos Loizou, 'Introduction to cochlear implants', IEEE (Institute of Electrical and Electronics Engineers, New York, www.ieee.org) A membership organization that includes engineers,
scientists and students in electronics and allied fields. Engineering in Medicine and Biology, January 1999, 32-442.
(37) Fred Spoor spoor
The track or trail of an animal, especially a wild animal.
v. spoored, spoor·ing, spoors
tr. & intr.v.
To track (an animal) by following its spoor or to engage in such tracking. , 'A comparative review of the human bony labyrinth', Yearbook of Physical Anthropology, vol. 41, 1998, 211 -251.
(38) LT Cohen et al., 'Improved and simplified methods for specifying positions of the electrode bands of a cochlear implant array', Journal of Otology otology /otol·o·gy/ (o-tol´ah-je) the branch
of medicine dealing with the ear, its anatomy, physiology, and pathology.otolog´ic
The branch of medicine that deals with the ear. , 1996, vol. 17, 859-865; Sun Yoo et al, 'Three-dimensional geometric modelling of the cochela using helico-spiral approximation', IEEE Transactions
on Biomedical Engineering, vol. 47, no. 10, 2000, 13921402.
(39) David Raup, 'Computer as aid in describing form in gastropod gastropod, member of the class Gastropoda, the largest and most successful class of mollusks (phylum Mollusca), containing over
35,000 living species and 15,000 fossil forms. shells', Science, vol. 138, 1962, 150-152.
(40) Cohen, 859.
(41) Yoo, 1392.
(42) Daphne Manoussaki et al., 'Cochlea's graded curvature effect on low frequency waves', Physical Review Letters Physical Review Letters is one of the most prestigious journals in physics.^[1]
Since 1958, it has been published by the American Physical Society as an outgrowth of The Physical Review. , vol. 96, no. 8, 2006, 1-4.
(43) A E Bate bate^ 1
tr.v. bat·ed, bat·ing, bates
1. To lessen the force or intensity of; moderate: "To his dying day he bated his breath a little when he told the story" , 'Note on the whispering gallery of St Paul's Cathedral, London',
Proceedings of the Royal Physics Society, vol. 50, 1938, 293-297.
(44) Gavin Baker, Stephen O'Leary, Nick Barnes and Ed Kazmierczak, Cochlea modelling: Clinical challenges and tubular extraction, 17th Australian Joint Conference on Artificial Intelligence, Cairns,
(45) Gavin Baker, Nick Barnes, 'Model-image registration of parametric shape models: Fitting a shell to the cochlea', The Insight Journal, October 2005. Available: www.insight-journal.org.
Reader Opinion
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Student Learning Outcomes,
Mathematics Planning and Documentation Template - Course Student Learning Outcomes, Methods of Student Evaluation, And Method of Course SLO Assessment
MATH-151 Algebra I
Student Learning Outcomes
The student will:
1. Demonstrate basic skills in algebra up through quadratic equations through tests and problem solving.
2. Set up stated problems algebraically and solve the resulting equations.
3. Solve problems presented via formulas or procedures.
4. Graph linear equations.
5. Solve systems of linear equations using graphing, substitution, and elimination methods.
6. Simplify exponential expressions with integer exponents.
7. Identify polynomials and perform operations with polynomials.
8. Factor polynomials using grouping, FOIL, special products formulas, and trial and error methods.
9. Solve quadratic equations using factoring and their applications.
10. Simplify rational expressions and complex fractions and solve applications of rational equations.
Methods of Evaluation:
1. Tests pertaining to course outline.
2. Quizzes on solving linear equations in one and two variables, graphing, and solving word problems.
3. Homework from text pertaining to course outline.
MATH-151A Algebra I (part 1)
Blog: MATH-151A Algebra I (part 1).
Student Learning Outcomes
The student will:
1. Demonstrate basic skills in algebra.
2. Solve linear equations and inequalities.
3. Plot points and graph linear equations in Cartesian coordinate system.
4. Solve systems of linear equations by graphing, substitution, and addition methods.
5. Set up stated problems algebraically and solve the resulting equations.
6. Solve problems presented via formulas or procedures.
Methods of Evaluation:
1. Homework from text pertaining to course outline.
2. Quizzes on solving linear equations in one and two variables, graphing, and solving word problems.
3. Tests on solving linear equations, graphing linear equations, setting up and solving one and two variable word problems.
4. Projects
MATH-151B Algebra I (part 2)
Blog: MATH-151B Algebra I (part 2).
Student Learning Outcomes
The student will:
1. Simplify exponential expressions with integer exponents.
2. Identify polynomials and perform operations with polynomials.
3. Factor polynomials using grouping, FOIL, special products formulas, and trial and error methods.
4. Solve quadratic equations using factoring and their applications.
5. Simplify rational expressions and complex fractions and solve applications of rational equations.
Methods of Evaluation:
1. Tests
2. Quizzes
3. Homework
MATH-190 Basic Mathematics
Blog: MATH-190 Basic Mathematics.
Student Learning Outcomes
The student will:
1. Solve numerous problems in order to gain mastery of the arithmetic skills needed for everyday situations.
2. Demonstrate a systematic and logical approach to solving arithmetic problems.
3. Demonstrate the knowledge and skills required to select the correct introductory formulas and apply the correct procedures to solve word problems related to practical situations.
Methods of Evaluation:
1. Homework.
2. Quizzes on fractions, decimal, ratio, percent and the metric system.
3. Tests on various math topics on the chapters.
MATH-190A Basic Mathematics (self-paced)
Blog: MATH-190A Basic Mathematics (self-paced).
Student Learning Outcomes
The student will:
1. Solve numerous problems in order to gain a mastery of the arithmetic skills needed to work with whole numbers, integers, fractions, and decimals.
2. Demonstrate a systematic and logical approach to solving arithmetic problems.
3. Demonstrate the knowledge and skills required to select the correct introductory formulas, procedures, and concepts from arithmetic and apply them to practical situations as described in word
MATH-190B Basic Mathematics (self-paced)
Blog: MATH-190B Basic Mathematics (self-paced).
Student Learning Outcomes
The student will:
1. Solve numerous problems in order to gain a mastery of the arithmetic skills needed to solve proportion and percentage problems.
2. Demonstrate a mastery of the skills needed to work on problems employing the U.S. Customary system and the metric system of measurement and conversion between the two.
3. Demonstrate the ability to use bar, line, and circle graphs along with the mean and median to analyze data.
4. Solve various real world problems involving proportions, percents, and measurement.
Methods of Evaluation:
1. Homework from the textbook and the computer disc
2. Classwork
3. Quizzes
4. Tests
MATH-191 Pre-Algebra
Student Learning Outcomes
The student will:
1. Gain mastery of the skills needed to use signed numbers.
2. Demonstrate a systematic and logical approach to solving arithmetic problems.
3. Demonstrate the knowledge and skills required to select the correct introductory formulas, procedures and concepts from algebra and geometry and use them to calculate and problem solve.
Methods of Evaluation:
1. Homework from the textbook and the computer disc
2. Classwork
3. Quizzes
4. Tests
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Introduction to Mathematical Logic
The Propositional Calculus
Propositional Connectives. Truth Tables
Adequate Sets of Connectives
An Axiom System for the Propositional Calculus
Independence. Many-Valued Logics
Other Axiomatizations
First-Order Logic and Model Theory
First-Order Languages and Their Interpretations. Satisfiability and Truth. Models
First-Order Theories
Properties of First-Order Theories
Additional Metatheorems and Derived Rules
Rule C
Completeness Theorems
First-Order Theories with Equality
Definitions of New Function Letters and Individual Constants
Prenex Normal Forms
Isomorphism of Interpretations. Categoricity of Theories
Generalized First-Order Theories. Completeness and Decidability
Elementary Equivalence. Elementary Extensions
Ultrapowers: Nonstandard Analysis
Semantic Trees
Quantification Theory Allowing Empty Domains
Formal Number Theory
An Axiom System
Number-Theoretic Functions and Relations
Primitive Recursive and Recursive Functions
Arithmetization. Gödel Numbers
The Fixed-Point Theorem. Gödel’s Incompleteness Theorem
Recursive Undecidability. Church’s Theorem
Nonstandard Models
Axiomatic Set Theory
An Axiom System
Ordinal Numbers
Equinumerosity. Finite and Denumerable Sets
Hartogs’ Theorem. Initial Ordinals. Ordinal Arithmetic
The Axiom of Choice. The Axiom of Regularity
Other Axiomatizations of Set Theory
Algorithms. Turing Machines
Partial Recursive Functions. Unsolvable Problems
The Kleene–Mostowski Hierarchy. Recursively Enumerable Sets
Other Notions of Computability
Decision Problems
Appendix A: Second-Order Logic
Appendix B: First Steps in Modal Propositional Logic
Answers to Selected Exercises
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higher gauge transformation
higher gauge transformation
Differential cohomology
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Fiber integration
Application to gauge theory
In gauge theory two configurations $\phi_1, \phi_2$ of gauge fields may be different and still be equivalent: there may be a gauge transformation $\lambda \colon \phi_1 \to \phi_2$ between them.
In higher gauge theory also gauge transformations themseves may be different but still equivalent: if there is a gauge-of-gauge transformation $\rho \colon \lambda_1 \to \lambda_2$ between them.
These higher order gauge transformations are maybe best known in the physics literature in terms of their infinitesimal approximation, the BRST complex: here the gauge transformations correspond to
ghost fields and the gauge-of-gauge transformations to ghost-of-ghost fields.
A basic example of a gauge field that has higher order gauge transformations is the B-field. But also magnetic current, if described properly, exhibits higher gauge transformations, see at Dirac
charge quantization.
For more see at geometry of physics.
Revised on December 28, 2013 14:29:09 by
Urs Schreiber
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A Particle Of Mass M Collides With A Second Particle ... | Chegg.com
A particle of mass m collides with a second particle of mass m. Before the collision, the rest particle is moving in the x-direction with a speed 2v and the second particle is at rest. After the
collision,the second particle is moving in the direction 45 below the x-axis and with a speed sqr(2v) , as shownin gure(1)
(a) Find the velocity of the rest particle after the collision.
(b) Find the total kinetic energy of the two particles before and after the collision.
(c) Is the collision elastic or inelastic? Justify your answer mathematically.
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Buena Park Trigonometry Tutor
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Positive and Negative Numbers - Waldomaths
Basic Arithmetic with Positive and Negative Numbers
Learn how to do simple arithmetic with positive and negative numbers - adding, subtracting, multiplying, dividing, squaring, cubing.
UK Years 5-9, KS2, KS3, KS4, Foundation GCSE - Number
US - Pre-Algebra
Instructions below See also: Fractions Standard Index Form (Scientific Notation)
How to Use this Applet
This applet investigates adding, subtracting, multiplying, dividing, squaring and cubing with positive and negative numbers. Every time you click "New Problem" a new number appears. You can add to,
subtract from, multiply or divide this number by other numbers using the 4 sliders at the bottom of the page. You can also square and cube the number by clicking the buttons at the top of the page.
Sometimes dividing will result in a fraction, and you can do the same things to the fraction that you see.
As you drag a slider left and right, you'll see some writing appear on the screen. This tells you what will happen when you release the slider. In other words, you must decide what you want to do,
drag the slider to the correct position, and then release it. You will then see the number change. You will learn much from this applet if you try to predict the answer to each calculation before you
release the mouse and see it happen. Play around and good luck!
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Linear Recurrence Relations in 2 Variables with Variable Coefficients
up vote 4 down vote favorite
Consider the following recurrence relation: $$-2a_{n,m} +a_{n-1,m}+a_{n,m-1}=0,$$ where $a_{n,m} \in \mathbb{C}.$I would like a purely combinatorial way to understand the subspace of solutions to
this equation which have tempered growth. There is an obvious solution given by setting all $a_{n,m}=C$ for any constant $C,$ and I have reasons (coming from topology) to believe that these are the
only solutions with tempered growth.
Now consider the similar but probably much harder recurrence relation: $$-2a_{n,m} +a_{n-1,m}+e^{2 \pi i n \theta}a_{n,m-1}=0$$ where $\theta$ is a fixed irrational. Note that the relation now
depends on $n.$ I haven't even been able to come up with a solution to this that has tempered growth. I am hoping that it also has a 1 dimensional (or at least finite dimensional) space of solutions
with tempered growth.
Is there a general combinatorial method for attacking either of these recurrence relations? Is there a general way to attack any linear recurrence relation like these?
EDIT: Let me also give my "proof" (I think it is correct) that any solution to the first relation with tempered growth must be constant. Consider the 2 dimensional torus thought of as $S^1\times S^
1$, where $S^1$ is the unit circle. Now consider the function $z_1+z_2-2,$ thought of as a (finite) Fourier series. This has only 1 zero, at $(1,1).$
Now consider distributions $D$ on the torus, also thought of as Fourier series $D=\sum_\mathbb{Z^2} a_{n,m}z_1^nz_2^m$ where the $a_{n,m}$ now have only tempered growth. The first recurrence above is
exactly the condition that $(z_1,+z_2 -2)D=0$ as a distribution. Since $z_1+z_2-2$ has only a single zero, the only solution with tempered growth should be a multiple of the Dirac distribution which
is given by $a_{n,m}=1.$ I want a combinatorial proof or understanding of this phenomenon, since the second relation does not have this kind of topological interpretation. Ideally I would like to
prove that the vector space of solutions to the second relation is also dimension 1, or is at least in some way related to the first relation.
2nd EDIT: WillSawin's answer shows that my initial proof is wrong. The space of tempered growth solutions to the first recurrence relation should be spanned (as a vector space) by the delta function
and some linear combinations of its partial derivatives. Does the second recurrence have the same property? I.e. is there one "basic solution" $B$ to the second recurrence such that all other
solutions can be expressed as linear combinations of the formal derivatives of $B?$
recurrences co.combinatorics
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3 Answers
active oldest votes
Polynomial growth is tempered growth, right?
I think you're failing to consider Dirac delta deriviative distributions, like the one that sends a smooth function $f(x)$ to $\frac{df}{dx} (0)$. Their Fourier series will be polynomials,
like the ones that satisfy your recurrence:
$a_{n,m}=(n-m)^2 + (n+m)$
$a_{n,m}= (n-m)^3 + 3( n^2-m^2)$
Because the operation $a_{n,m} \to 2a_{n,m} - a_{n-1,m} - a_{n,m-1}$ sends polynomials of degree $\leq d$ to polynomials of degree $\leq d-1$, its kernel among the polynomials of degree $\
leq d$ must be at least the dimension of the space of polynomials of degree $\leq d$ minus the dimension of the space of polynomials of degree $\leq d-1$, or $d+1$. Moreover, it is clear
that the leading term of anything in the kernel must have the form $k (n-m)^d$. This gives a basis for the polynomial solutions.
I'm pretty sure one can show any solution to the second equation with $\theta\neq 0$ must have exponential growth, and explicitly find how large the exponential growth can be. First
\[a_{n,m} = \frac{ a_{n-1,m-1}+a_{n,m-1} e^{2 \pi i n \theta} }{2}\]
Now we can think of this as the dynamics of an operator from functions on $\mathbb Z$ to functions on $\mathbb Z$:
\[Ta_{n} = \frac{ a_{n-1}+a_{n} e^{2 \pi i n \theta} }{2}\]
up vote 3
down vote Consider the Hilbert space of functions $a_n$ that satisfy $\sum_n |a_n|^2 e^{ - 2\alpha |n|} <\infty$. $T$ is a bounded operator on this Hilbert space. Let $\rho$ be its spectral radius.
Since $\rho= \lim\sup_k |T^k|^{1/k}$ We have:
\[ \lim\sup_k \left(\sum_n ||a_{n,m-k}||^2 e^{ - 2\alpha |n|} \right)^{1/k} \geq 1/\rho\]
so $a_{n,m}$ has exponential growth of at least $e^{\alpha |n|}$ or $\rho^{-|m|}$ in some direction. We will show that for $\alpha$ sufficiently small, $\rho<1$, so there is always
exponential growth.
Suppose $\lambda$ is in the spectrum. Then $T- \lambda$ is not invertible in the Hilbert space. But we can explicitly write an inverse:
\[ ((T-\lambda)^{-1} a )n = \sum_{m=n}^\infty a_m \prod_{k=n}^m \frac{1}{2\lambda - e^{2 \pi i n \theta}} \]
$\prod_{k=n}^m \frac{1}{2\lambda - e^{2 \pi i n \theta}}$ decreases rapidly. It's bounded by a multiple of $e^{-(m-n)\int_0^{1} \ln |2 \lambda - e^{2\pi i x}|dx} $ . Since $\ln | \cdot | $
is harmonic, this is just $e^{-\ln |2\lambda|}=1/2\lambda$. As long a $\alpha < \ln |2 \lambda|$, this inverse is a bounded operator, so $\lambda$ is not in the spectrum. Thus $\rho \leq e^
So it either has growth of at least $e^{\alpha |n|}$ or $e^{(\ln 2 - \alpha)|m|}$.
We can easily show that this is in fact exactly the spectral radius by finding an eigenvalue with $\alpha > 2 \ln |\lambda|$. By continuing the product in the opposite direction we get an
eigenvalue, and by the same estimate the growth rate is less than $e^{\alpha |n|}$. Thus, there are solutions with exponential growth.
One can do a similar argument for $\theta$ a rational number, using a finite sum instead of an integral. This will give a slightly different spectral radius that is still less than $1$ for
$\alpha$ small enough.
Thanks! These solutions correspond to the fact that the partial derivatives agree at $(1,1).$ – mkreisel Apr 6 '13 at 21:28
How does your first step work? The original recurrence gives $$a_{n,m}=\frac{a_{n-1,m}+a_{n,m-1}e^{2\pi i n\theta}}{2}$$ but you've written $$a_{n,m}=\frac{a_{n-1,m-1}+a_{n,m-1}e^{2\pi i
n\theta}}{2}.$$ – mkreisel Apr 7 '13 at 14:52
Replace $m$ with $m+n$. – Will Sawin Apr 7 '13 at 16:38
Sorry if I'm being obtuse, but I'm still having trouble understanding your solution. I now understand how the operator $T$ helps; if $a_{n,m}$ satisfies the recurrence relation I can
take $b_n=a_{n,-n}$ to be my "initial data" and then applying $T$ yields the values $Tb_n=a_{n,1-n}.$ So understanding the dynamics of $T$ tells me about the growth of the sequence as we
move up and to the right in the $\mathbb{Z}^2$ lattice. However it cannot tell us about the lower left portion of the grid... – mkreisel Apr 8 '13 at 17:22
1 My answer certainly isn't the clearest. We have the identity $T^n a_{-n,m}=a_{0,m}$, which implies the inequality $|| a_{0,m}|| \leq ||T^n|| ||a_{-n,m}||$, so $||a_{-n,m}|| \geq ||a_
{0,m}|| /||T^n||$. Thus if $1/||T^n||$ grows exponentialy, $a_{n,m}$ does as well. – Will Sawin Apr 8 '13 at 18:11
show 2 more comments
I don't think constant is the only tempered growth solution to your 2d recurrence. Essentially the recurrence needs a 1-dimensional subspace in $\mathbb{Z}^2$ of boundary conditions. Then
your recurrence is saying the value at any point $(x,y)$ is the average of values at $(x-1,y)$ and $(x,y-1)$, which has the effect of smoothing things out as we move to the upper right
corner of the 2d lattice.
For instance, take the subspace $S = \{(i,-i): i \in \mathbb{Z}\}$. Then you can prescribe any values on points in $S$ and get a consistent solution first on the part of the integer lattice
to the up-left of $S$. If you choose the values on $S$ to be say bounded, then the solution on the upper left side of $S$ will also be bounded, hence certainly "tempered". Now to extend this
up vote solution below and to the left of $S$, just assign $0$ to all points in $T =\{(i,i): i < 0\}$ and all the other points in the bottom left corner of the lattice are uniquely determined. Again
2 down it's easy to see they are all bounded.
Your second recurrence can be treated in the same way. One trivial solution is to assign all points constant at $0$. On the other hand I imagine it can be quite interesting to consider 2d
recurrences where the coefficients do not add up to 0 (or cannot be arranged on the right and left of the equality so that the moduli of two sides add up to the same number) hence can
exhibit exponential growth. But maybe that case can be reduced to 1d recurrence.
I do not think this is quite right. I agree that if you specify $S=\{(-i,i)\}_{i \in \mathbb{Z}}$ then the region above and to the right of $S$ is determined and bounded. However your
extension to the lower left does not make sense to me. It is not clear to me how specifying $T=\{(i,i)_{i<0}\}$ determines the rest of the grid. So instead consider this example. Let $a_
{n,m} = 1$ on $S$, and thus $a_{n,m}=1$ whenever $n+m>1$ as well. Now let $a_{0,m}=0$ for all $m<0,$ similar to what you have described. This fully determines the grid as we can now work
down the lower diagonals one at a time... – mkreisel Apr 6 '13 at 18:57
However, in this case we have $a_{-n,0} = 2^n,$ so this solution clearly cannot be tempered since it is growing exponentially in $n$ in this direction. To see that $a_{-n,0}=2^n,$ note
that because we have placed zeros on the half-column $a_{0,-m},$ the strict lower left quadrant $m,n<0$ will be entirely zero, and in particular the half-row $a_{m,-1} = 0$ for $m<1.$ Thus
since we placed a 1 at $a_{0,0},$ we get $a_{-1,0}=2,$ and since the entire half-row is zero we get $a_{-n,0} = 2^n.$ So just specifying a bounded initial data does not mean the whole
sequence remains bounded, nor tempered. – mkreisel Apr 6 '13 at 19:06
@mkreisel: yes you are right. The lower half of the plane is the more interesting part. There is more structure to the problem than I thought. – John Jiang Apr 7 '13 at 1:03
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This recurrence brings to mind the wave equation. Denote by $L$ the lattice $\newcommand{\bZ}{\mathbb{Z}}$ $L=\bZ^2$ and $t:L\to\bZ$ the "time" function
$$ t(x,y)=x+y,\;\; (x,y)\in L. $$
The "initial hypersurface" $S_0$ is defined by the equation $T=0$. A point $p=(x,y)\in L$, $t(p)>0$, is uniquely determined by its time $t(p)=x+y$ and its position $u(p)=x-y$. $\newcommand
Consider a function $a:L\to\bC$ satisfying your recurrence relation. Its value at the point $(x,y)$ with time $t=(x_0,y_0)$ is only affected by its values in the region $|x-x_0|\leq t$ on
initial hypersurface $S_0$.
For any $R>0$ define
$$S_0(R,t)= \lbrace (x,y)\in L;\;\;x+y=t,\;\;|x|\leq R\rbrace, $$
$$ m(R, t) :=\max_{(x,y)\in S(R, t)}|a(x,y)|. $$
Your recurrence implies
$$ |a(x,y|\leq \frac{1}{2}\bigl(\; |a(x-1,y)|+|a(x,y)|\;\bigr),\;\;\forall (x,y)\in R. $$
This implies immediately that
$$ m(R,t) \leq m(R+1, t-1). $$
In particular, for $t>0$ we have
$$ m(R,t)\leq m(R+t,0). $$
This controls the size of the "future" values of $a$, i.e., the values of $a$ in the region $t\geq 0$. If we assume that
$$ m(R,0)\leq Ma^R, $$
for some $c\geq 1$, $M>0$ then we deduce that
$$ M(R,t)\leq M a^{R+t},\;\;\forall t\geq 0. $$
In particular, if $a$ is bounded along $S_0$, it will stay bounded in the future. The past values seem a bit more difficult to control. I have to think more about this.
I thought more about this and I reached a conclusion: the past cannot be determined from the initial conditions at $t=0$.
Suppose that we have a function $a: \bZ\to \bC$ satisfying your recurrence conditions and such that, at $t=0$ is zero. What could be the values on the time slice $t=-1$?. Denote by $A$ the
restriction of $a$ to the slice $t=-1$. We set $\newcommand{\ii}{\boldsymbol{i}}$
$$ A_n = a(n,-n-1),\;\;n\in\bZ. $$
Set $c:=e^{\ii\theta}$. The recurrence relation implies
$$A_n +c^n A_{n-1}= 0,\forall n\in\bZ $$
so that
$$ A_n = A_0 (-c)^{\ell(n)},\;\;\ell(0)=0,\;\;\ell(n+1)-\ell(n)=n,\;\;\forall n\in\bZ $$
This shows that we can generate solutions of your recurrence that are far from temperate for $t<0$. Here is how you do it.
Fix a function $f_0 :\bZ\to\bC$. This is the initial condition
$$ a(x,-x)=f_0(x),\;\;\forall x\in \bZ. $$
Define $g_{-1}:\bZ\to \bC$ by requiring
$$ g_{-1}(0)=0,\;\;g_{-1}(x)+cg_{-1}(x-1) = 2f_0(x),\;\;\forall x\in \bZ. $$
Pick a constant $M_1>0$ and then set
$$ f_{-1}(x) = M_1 (-c)^{\ell(x)} +g_{-1}(x),\;\;\forall x\in \bZ. $$
Observe that $f_{-1}(0)= M_1+1$ and
$$ f_{-1}(x)+c^xf_{-1}(x-1)=2f_0(x). $$
Proceed inductively. Suppose we have produced functions $f_{-1},\dotsc, f_{-k}:\bZ\to \bC$,
$$ f_{-j}(0)= M_j+1,\;\;j=1,\dotsc, k. $$
up vote 2 We determine $g_{-k-1}:\bZ\to \bC$ by requiring that
down vote
$$ g_{-k-1}(0)=0,\;\; g_{-k-1}(x)+c^xg_{-k-1}(x-1)= 2f_{-k}(x),\;\;\forall x\in \bZ. $$
Pick a positive constant $M_{k+1}$ and then set
$$ f_{-k-1}(x) := M_{k+1}(-c)^{\ell(x)}+ g_{-k-1}(x),\;\;\forall x\in\bZ. $$
For $k>0$ we define inductively $f_k:\bZ\to \bC$
$$ f_k(x)=\frac{1}{2}\bigl(\; f_{k-1}(x)+c^xf_{k-1}(x-1)\;\bigr). $$
Finally define $ a:\bZ^2\to\bC $ by setting
$$ a(x,y)= f_{x+y}(x),\;\;\forall (x,y)\in\bZ^2. $$
The function $a$ satisfies the recurrence relation and
$$ a(0, -k) = M_k+1,\;\;\forall k\in \bZ_{>0}. $$
By choosing the sequence $M_k$ suitably, e.g. $M_k = k^{k!}$, your guaranteed a non-temperate behavior for $a$.
Remark 1. Suppose we are given a function $f_0:S_0\to\bC$, a sequence of points $p_n=(x_n,y_n)$, $n>0$, so that $t(p_n)=x_n+y_n)=-n$ and a sequence of complex numbers $C_n$, $n>0$. Then
there exists a unique solution $a$ of the recurrence equation satisfying the "initial conditions"
$$ a(p_n)= C_n,\;\;\forall n>0, $$
$$ a(x,y)= f_0(x,y),\;\;\forall (x,y)=S_0. $$
Remark 2. To obtain estimates for the growth of this solutions in the past one needs to understand the growth of the solution of the following initial value problem. Given $f:\bZ\to \bC$
let $u:\bZ\to\bC$ be the solution of the initial value problem
$$ u(x)+c^xu(x-1) = 2f(x),\;\; x\in \bZ, $$
$$u(0) =0. $$
Note that
$$ u(1)= 2f(1),\;\; u(2)= -c^2 u(1) + 2f(2)=-2c^{\ell(2)} f(1)+2f(2), $$
$$ u(3)= -c^3u(2)+2f(3)= 2f(3) -2c^3f(2) +2c^{\ell(3)} f(1) $$
$$ = 2(-c)^{\ell(3)}\bigl(\; (-c)^{-\ell(3)} f(3)+(-c)^{-\ell(2)}f(2)+(-c)^{-\ell(1)} f(1)\;\bigr). \tag{1} $$
The pattern is now clear and one can see that
$$ u(n)\leq 2\bigl( |f(1)|+ \cdots +2|f(n)|\;\bigr). $$
This is an optimal bound which is achieved. Suppose for example that
$$ f(n)= (-c)^{-\ell(n)} r_n,\;\; r_n>0 $$
$$u(n)= 2(-c)^n (r_1+\cdots + r_n), \;\; n>0. $$
The solution with the temperate initial condition
$$ a(n,-n)= (-c)^{\ell(n)} ,\;\;n\in\bZ, $$
$$ a(0,-n)=0,\;\; n>0, $$
is non-temperate because $|a(1,1-t)|=2^t$, $\forall t\geq 0$.
In general is satisfies an estimate of of the type
$$ a(n,n-t)\sim 2^t \frac{n^t}{t!},\;\; t>0,\;\; n>n(t). $$
Remark 3. If the "initial" condition for $a$ is the $\delta$ function concentrated at the orgin, then $a$ has exponential growth in the past.
More precisely, if $a$ satisfies the recurrence and the initial conditions
$$a(x,-x)=0,\;\;\forall x\in\bZ\setminus 0,\;\;a(0,0)=1, $$
$$ a(0,-t)=0,\;\;\forall t<0, $$
then $$|a(1,1-t)|=2^t. $$
As I noted in the comment above, specifying the $S_0$ plus the values of $a(0,n), n<0$ will completely determine the grid for either relation. Clearly if we make either of these choices
in a way that does not have tempered growth then the solution they determine will not have tempered growth. I think the solutions you describe above were essentially chosen, as I
described, not to have tempered growth. What is more interesting to me is whether choosing bounded or temperate growth initial data leads to a bounded/temperate growth solution. –
mkreisel Apr 6 '13 at 19:37
Isn't it more like the heat equation than the wave equation? – Will Sawin Apr 6 '13 at 20:31
It is rather strange. The future behavior is like the wave equation. If the initial condition is supported at the origin, say $a(0,0)=1$ and $a(x,-x)=0$, for $x\neq 0$, then on the slice
$t=1$ there are only two points where $a\neq 0$, namely the points $a(1,0)$ and $a(0,-1)$. The heat propagates instantaneously so that if the initial condition is Dirac concentrated at
$0$ the temperature after $\epsilon$ seconds at a point $x$ is not $0$ it is $\frac{1}{\sqrt{4\pi\epsilon}} e^{-\frac{x^2}{4\epsilon}}$. – Liviu Nicolaescu Apr 6 '13 at 21:41
@ mkreisel See the update to my answer where I construct a non-temperate solution with temperate initial data. – Liviu Nicolaescu Apr 6 '13 at 22:19
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Not the answer you're looking for? Browse other questions tagged recurrences co.combinatorics or ask your own question.
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Formal group
From Encyclopedia of Mathematics
An algebraic analogue of the concept of a local Lie group (cf. Lie group, local). The theory of formal groups has numerous applications in algebraic geometry, class field theory and cobordism theory.
A formal group over a field [1], [4], [6], [7]). Let group scheme
satisfying the following conditions:
and converts the functor [2], [5]). Sometimes by a formal group one means just a formal Lie group or even a formal group law.
Just as for local Lie groups (cf. Lie group, local) one can define the Lie algebra of a formal Lie group. Over fields [1], while all one-dimensional Lie algebras are isomorphic [3]. Over perfect
fields of finite characteristic, commutative formal Lie groups are classified by means of Dieudonné modules (see [1], [6]).
The theory of formal groups over fields can be generalized to the case of arbitrary formal ground schemes [7].
[1] Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 (1963) pp. 1–80 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90 MR157972 Zbl
[2] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[4] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[5] M. Lazard, "Commutative formal groups" , Springer (1975) MR0393050 Zbl 0304.14027
[6] J.-M. Fontaine, "Groupes Astérique , 47–48 (1977) MR498610
[7] B. Mazur, J.T. Tate, "Canonical height pairings via biextensions" J.T. Tate (ed.) M. Artin (ed.) , Arithmetic and geometry , 1 , Birkhäuser (1983) pp. 195–237 MR0717595 Zbl 0574.14036
A universal formal group law (for
For commutative formal group laws explicit formulas are available for the construction of universal formal group laws, cf. [a3]. The underlying ring
A homomorphism of formal group laws
It follows that for every commutative formal group law
The formal group law of complex cobordism is a universal one-dimensional formal group law (Quillen's theorem) and its logarithm is given by Mishchenko's formula
Combined with the explicit construction of a one-dimensional universal group law these facts yield useful information on the generators of the complex cobordism ring Cobordism for more details.
where Witt vector for the latter. Cartier's second and third theorems on formal group laws say that the
Let Witt vector). Let
There exists a Pontryagin-type duality between commutative formal groups and commutative affine (algebraic) groups over a field [a3], [a4] for more details. Correspondingly, Dieudonné modules are
also important in the classification of commutative affine (algebraic) groups. Essentially, Cartier duality comes from the "duality" between algebras and co-algebras; cf. Co-algebra.
Then [a3], [a5] for more details.
Formal groups also are an important tool in algebraic geometry, especially in the theory of Abelian varieties. This holds even more so for a generalization: .
Lazard's theorem on one-dimensional formal group laws says that all one-dimensional formal group laws over a ring without nilpotents are commutative.
be the logarithm of
The result extends to the case that
[a1] J.T. Tate, "Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 158–183 MR0231827 Zbl 0157.27601
[a2] J.-P. Serre, "Groupes Sem. Bourbaki , 19, Exp. 318 (1966–1967) MR1610452 MR0393040
[a3] M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020
[a4] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson & North-Holland (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[a5] J. Lubin, J. Tate, "Formal complex multiplication in local fields" Ann. of Math. , 81 (1965) pp. 380–387 MR0172878 Zbl 0128.26501
How to Cite This Entry:
Formal group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Formal_group&oldid=21866
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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There has been much discussion of the exorbitant price increases over the past dozen years or so of journals owned by publishing houses such as Reed-Elsevier, Kluwer, and Springer-Verlag/Birkhauser
(as can be seen in some current journal pricing data). At the beginning of 2007 entire editorial board of Topology quit in protest of these pricing practices, drawing some press attention. In the
summer of 2007, a similar turnover occured in the area of K-theory.
For many years Topology was the signature journal of our subfield, but it has been given competition by Geometry and Topology, a low-cost alternative founded with some foresight in 1997. In response
to this resignation, one of the representatives of Elsevier posted a letter to the largest topology mailing list. Below are my responses to this letter.
To: "Ross, Robert \(ELS\)"
Cc: Topology E-mail List
Subject: response (for the list) to letter about Topology
From: "Dev P. Sinha"
Date: Fri, 10 Nov 2006 17:23:52 -0800
Dear Mr. Ross,
Be advised that mathematicians can be a skeptical audience, especially when presented with cherry-picked facts and statistics. A couple instances:
"Our price increase is among the lowest in the industry"
Well, increasing 5% on a $2000 title hurts a lot more than a 7% increase on a $300 title. We'd need to see a full picture here for this sentence to be meaningful.
..."more available"...
Maybe, with the advent of predominant use of electronic sources. But compare your usage totals to those of open access journals after they get established and the comparison is usually not favorable.
"invested $160 million in digitizing and maintaining the digital archive of our entire program"
Yes, we appreciate efforts of publishers here. Our issue isn't with what you are doing, but what you are charging for it (and reaping in profits). The Annals of Mathematics is having all of its
articles where they have the (La)Tex source moved to the arXiv, and have (most? all?) of its further-back issues on JStor. The Annals cost much less for libraries than Topology. They may be
subsidized by Princeton University, but almost certainly not the tune where the discrepancy between what they charge and what you charge can be accounted for.
Given the tenor of your letter, I am not surprised that the former editors of Topology have severed their relationship with Elsevier.
Dev Sinha
Associate Professor and University Library Committee Member,
University of Oregon
Subject: Posting Elsevier Response
From: "Ross, Robert \(ELS\)"
Date: Fri, 10 Nov 2006 21:08:55 -0600
As the Pure Maths Journal Publisher at Elsevier, I welcome your comments and alternate views on the Topology Editor's resignation and Elsevier's journal publishing program. Please feel free to
contact me at anytime.
Subject: More on Journals
From: "Dev P. Sinha"
Date: Mon, 13 Nov 2006 15:00:19 -0800
(For the list: Mr. Ross replied to me, claiming that access to Topology was not being restricted by price. He would prefer not to engage in a listserv discussion, but I would like to clarify a few
points and then will let the matter rest.)
Dear Mr. Ross,
You said to the topology e-mail list that you welcomed alternate views. The alternate view here at the University of Oregon is that access to journals is being restricted by price. We cancelled our
subscription to "Topology" last year because it was too pricey - $1558 in 2004 (vs. for example $170 for the same year of "Geometry&Topology"). This wasn't an easy decision - roughly 1/4 of our
faculty and graduate students have a strong interest in topology. While we may be in the minority in having to do without "Topology", may I ask you how "Topology and Its Applications" is doing? Are
universities lined up around the block to buy it at $3000/yr? Perhaps once it is packaged with other journals it costs $2800 or even $2000 (six times the price of G&T!) - is it selling like hotcakes
yet? Will it continue to do so?
You claim that you are not attempting to compete with university-subsidized journals, so the only relevant comparison is with other private publishers. But you _are_ in competition with
university-subsidized journals, as well as society journals (which generally run a profit), and non-profit publishers such as MathSciPublishers. This is your business climate. Moreover, in
mathematics you are selling the work of the mathematical community back to itself. You can say what you will about value added, but do you really think it is worth it for us to pay six times as much
for one of your journals than for one we can produce ourselves?
Given these facts about the business climate, it is not surprising to see a high-cost titles such as "Topology" in decline while titles such as "Geometry&Topology" gain in both prestige and pages
published. If it weren't for the facts that mathematicians (especially those good enough to be editors) prefer just to do math, that we are fairly sentimental about the history of our journals, and
that time is needed for "prestige transfer" for purposes of promotion and tenure, we would have already moved much faster in this direction.
From the nature of your responses, you seem to think that you should be able to continue your pricing practices indefinitely. To summarize what I am saying: I seriously doubt that. You have gotten
fat (see Wall-Street Journal clip) on mathematics library budgets for too long.
(Note about link: of course math titles are a drop in the bucket of Reed-Elsevier's profits, but this old clip gives you an idea of the company's mission. It's great that they are so into making
money; as a community we mathematicians should be more into saving money.)
Subject: RE: Posting Elsevier Response
From: "Ross, Robert \(ELS\)"
Date: Sun, 12 Nov 2006 11:55:00 -0600
We appreciate and respect the concerns of the Topology Editors and we are taking the issues they presented very seriously. To be fair, they were not presented with the numbers in our response before
they resigned. They did not make demands or requests of us and we do not want to make is seem as if there was any indication their decision was reversible. We were interested in retaining the Editors
but again we respect their decision.
We do greatly appreciate their long standing contributions to the journal. The numbers in our response indicate they were very successful in building and maintaining high quality and widespread
We intend to engage our other editors and the community more frequently. We also appreciate the alternatives to our program. Much here comes down to consumer choice.
If there is an Elsevier journal you would like to subscribe to but cannot due to price, please let me know and I will try to assist.
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Triangle centers quiz & similarity review
Editing needed:
explicitly state applications of similar figures (and prerequisite skills for being able to solve for missing side lengths)
- amysaundersbigelow WARM UP: For our warm up we went over a few problems from the practice quiz we got the night before.
Next we took our quiz, which looked like this.
"HOW TO DO THIS" SECTION
How to draw the altitude -
To draw the altitude of triangle ABC from point C first you would get a compass and put one end on point C, and then draw a semi-circle along line segment AB, if the semi-circle doesnt touch both
parts of the line you may need to extend the line segment. Next you should take the compass and draw a small semi-circle from where the other simi-circle touched one point of the line segment, then
do the same to the other point of intersection. Then you will draw a line from point C to where the two smaller semi-circles cross.
How to write the equation of a line
- First take the quardinates from poin C then plug them into point slope formula which is (y-y) = slope (x-x), then to find the slope, you take the neagative reciprocal of line segment AB, and plug
that into the formula, and then you have your equation of a line.
How to find the -
Circumcenter -
To find the circumcenter first you take a compass and put it at any one of the points, then you open the compass so that it is larger then half the segment, then you draw a semi-circle. After that
you take the compass and move the point to the other end of that same segment, and you draw another semi-circle. Finally you take your straight edge and you make a line conecting where the two half
circles crossed in two different points. After all that you repeat the steps on another segment. Then where the two points cross is the circumcenter. You only need to do this to two segments, not all
Orthocenter -
To find the orthocenter you will need to draw the altitude twice, you can find how to draw the altitude above in the how to draw the altitude section. The point where the two altitudes cross is the
Incenter -
To draw the incenter you will first need to draw at least two angle bisectors. To do this you will need take your compass and put it where two of the lines meet. Then draw a semi-circle touching both
lines. After that you will need to take the compass and draw two semi circles one from where the first semi-circle hits a line, and the other on the other side. Then you will need to take your
straight edge and make a line from the original point to where the two smaller semi-circles meet. After you have done this twice the point where the two lines you drew cross is the incenter.
- To draw the centroid you will need to first find the midpoint of at least two of the lines. Then once you have the midpoint you need to connect the midpoint to the point on the oposite side of the
line. Then do this at least one more time, then where the two points connect is where the centroid is.
After we took our quiz, we talked about similar triangles:
Similar Triangles
Please make a note that the text in red represent the verticies of the triangle, and that the text in green represent the lengths of the triangle's sides. Also note that the triangles are not drawn
to scale.
DF FE DE
<A is congruent to <D
<B is congruent to <F
<C is congruent to <E
Iff (if and only if):
Triangle ABC is similar to Triangle DEF (in geometry, you'd write it as "(Triangle) ABC ~ (Triangle) DEF")
There are two ways to find similar triangles
Way #1: "long-to-long", "short-to-short":
long one
short one
long two short two
Way #2: "long-to-short", "long-to-short":
long 1
long 2
short 1 short 2
Here's how you'd do this in Way #1
And here's how you'd do this in Way #2
$\frac{8}{x}=\frac{20}{6}$ Some common mistakes -
are confusing the centroid and the cercumcenter. One way to not get them confused is by thinking of the centroid as the center mass meaning that this point is always going to be in the exact center
of the triangle.
Links: http://www.mathopenref.com/constorthocenter.html
In this link you can figure out how to construct many of the centers of a triangle.
In this link you can practice some problems that have to do with similar triangles.
Learning how to solve some of these simpler similar triangle will help in the long run to solve some of the harder problems.
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When is Sobolev space a subset of the continuous functions?
up vote 4 down vote favorite
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several
1. If $d=1$ then $\mathcal{H}^1(\Omega)\subset\mathcal{C}^0(\Omega)$.
2. If $d=2$ then $\mathcal{H}^2(\Omega)\subset\mathcal{C}^0(\Omega)$ but $\mathcal{H}^1(\Omega)\not\subset\mathcal{C}^0(\Omega)$.
3. If $d=3$ then $\mathcal{H}^3(\Omega)\subset\mathcal{C}^0(\Omega)$ but $\mathcal{H}^2(\Omega)\not\subset\mathcal{C}^0(\Omega)$.
I was interested in trying to show these relationships. Does anyone know any references that would be useful.
Thanks in advance.
differential-equations fa.functional-analysis
2 The classical reference is of course Adams and Fournier: Sobolev spaces. That book is not easy going, though. – Harald Hanche-Olsen May 21 '10 at 11:44
See also mathoverflow.net/questions/17736/… – Willie Wong May 21 '10 at 17:54
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4 Answers
active oldest votes
I understand from your post that you'd like to show those facts by yourself first, and not necessarily to approach the whole theory now (I like your approach). Trivial hint: start with
smooth functions with compact support in $\Omega$, and try to bound their $L^\infty$ norm in terms of the $H^d$ norm. Also, I suggest that you try building counter-examples by yourself
up vote 7 down for the case of non-inclusions. Reference: Brezis' book of Functional Analysis may give you nice hints.
vote accepted
2 In the Hilbert space setting the easiest way to see the whole thing is to fo on the Fourier side and see what you can get from Cauchy-Schwartz (just localize first by multiplying by
a smooth cutoff). The second part of Claim 3 is wrong by the way... – fedja May 22 '10 at 1:52
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If you just want the answer, then not surprisingly you can find it at: http://en.wikipedia.org/wiki/Sobolev_inequality
up vote 5 down vote If you want a careful introduction to and derivation of the Hilbert space case, see: "Seminar on the Atiyah-Singer Index Theorem" (Princeton Univ. Press)
add comment
By now, I can't remember precisely where the best places to learn this is. Here are some rather vague suggestions:
1. I don't know if this stuff is in any of Nirenberg's writings, but if it is, it's sure to be a clear and easy approach.
2. Look in books about nonlinear elliptic PDE's by, say, Craig Evans, Gilbarg and Trudinger, or Thierry Aubin.
up vote 1
down vote 3. Ideally, there should be a proof that involves integration over cubes. Differential geometers such as Aubin tend to prove such results by integrating over balls because that's what
generalizes more easily to Riemannian manifolds. That works fine but the formulas are messier than for a cube. In the end, after you get the idea of what's going on, just write out
your own proof.
add comment
I'll give you a hint for the first one $d=1$. Consider first the case that your function $f \in H^1([0,1])$ was smooth. Then we could say $f(x) - f(y) = \int_{x}^y f'(s)ds$. Apply Cauchy
Schwartz now and you'll be able to see immediately that $f$ is $1/2$ Hölder continuous.
up vote 1
down vote For higher dimensions you actually proceed similarly but you need to use the co-area formula.
Cheers! :) The first one $d=1$ is simple doing it this way. Now off to find out about the co-area formula... – alext87 Sep 6 '10 at 19:19
Also, for the $d=2$ case, consider the function $u(x) = \log |x|$. Then $|Du(x)| = \frac{1}{|x|}$ and so $\int |Du(x)|^2 = 2\pi \int_0^r r^2/r$ which is finite obviously. So $\log|x|$
is in $H_0^1(\Omega)$ but not continuous. A similar example can be made for $d=3$ with $1/|x|^{\alpha}$ for an appropriate choice of $\alpha > 0$. – Dorian Sep 6 '10 at 19:46
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Not the answer you're looking for? Browse other questions tagged differential-equations fa.functional-analysis or ask your own question.
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The continuous limit of random planar maps
Seminar Room 1, Newton Institute
Planar maps are graphs embedded in the two-dimensional sphere $S^2$, considered up to continuous deformation. They have been studied extensively in combinatorics, but they also have significant
geometrical applications. Random planar maps have been used as models of random geometry in theoretical physics. Our goal is to discuss the convergence of rescaled random planar maps viewed as random
metric spaces. More precisely, we consider a random planar map $M(n)$, which is uniformly distributed over the set of all planar maps with $n$ vertices in a certain class. We equip the set of
vertices of $M(n)$ with the graph distance rescaled by the factor $n^{-1/4}$. We then discuss the convergence in distribution of the resulting random metric spaces as $n\to\infty$, in the sense of
the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we
first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit can be written as a quotient space of the so-called Continuum Random Tree
(CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. This limiting random metric space had been introduced by Marckert and
Mokkadem and called the Brownian map. It can be viewed as a ``Brownian surface'' in the same sense as Brownian motion is the limit of rescaled discrete paths. We show that the Brownian map is almost
surely homeomorphic to the sphere $S^2$, although it has Hausdorff dimension $4$. Furthermore, we are able to give a complete description of the geodesics from a distinguished point (the root) of the
Brownian map, and in particular of those points which are connected by more than one geodesic to the root. As a key tool, we use bijections between planar maps and various classes of labeled trees.
Related Links
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MathGroup Archive: May 2004 [00220]
[Date Index] [Thread Index] [Author Index]
Re: Selecting by first element of each list
• To: mathgroup at smc.vnet.net
• Subject: [mg48167] Re: Selecting by first element of each list
• From: Bill Rowe <readnewsciv at earthlink.net>
• Date: Fri, 14 May 2004 20:59:57 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com
On 5/14/04 at 12:12 AM, camartin at snet.net wrote:
>I'm trying to learn to handle a flat file kind of database using
>Mathematica. An example is the list of lists below:
>It's actually more complicated than this but good enough for
>illustrative purposes. The first element in each list is actually
>an identifier, like an account number. I want to create a list of
>lists with each account, that is,
>When I use Select with an anonymous function such as
>Select[list1,#1[[1]] = = #2[[1]]&]
There are two problems with the syntax above. First, there is a space between the two "=" characters. Second, Select checks each element of the list against the specified criteria. Select does not compare one element with another.
One way to achieve what you want would be to create your own selection function, i.e.,
accountSelect[data_,account_]:=Select[data,First@# == account&]
will do what you want.
FWIW, I've written a package for my use containing similar functions. I've thought about submitting it to MathSource but haven't taken time to write examples showing usage of the functions I've created. If you are interested, I would be willing to send it to you offline as is.
In my case, the first row of my data arrays is a list of undefined symbols. Using your example above, I might add as the first row {id, value1, value2, value3, value4}.
This does two things for me. First when I display the data array in either TraditionalForm or TableForm, I get a nice display with something that identifies the purpose of each column. Second, I can use the symbols in the first row to refer a particular column, i.e., I can write things such as
SelectRows[data, id, 2]
to select all rows of data with a 2 in the column headed by id. With this I find it much easier to manipulate the data and determine what my code is doing with the data.
To reply via email subtract one hundred and four
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Prolog Programming
A First Course
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Prolog Programming A First Course
Paul Brna
The course for which these notes are designed is intended for undergraduate students who have some programming experience and may even have written a few programs in Prolog. They are not assumed to
have had any formal course in either propositional or predicate logic.
At the end of the course, the students should have enough familiarity with Prolog to be able to pursue any undergraduate course which makes use of Prolog.
This is a rather ambitious undertaking for a course of only twelve lectures so the lectures are supplemented with exercises and small practical projects wherever possible.
The Prolog implementation used is SICStus Prolog which is closely modelled on Quintus Prolog (SICS is the Swedish Institute of Computer Science). The reference manual should also be available for
consultation [SICStus, 1988]. © Paul Brna 1988
Paul Brna
Mon May 24 20:14:48 BST 1999
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finding angles using bearings
February 27th 2008, 04:10 PM
finding angles using bearings
A man starts walking from point A at a bearing of 56°. He walks for 400m and stops at point B. He then walks a bearing of 300° for 200m and stops at point C. From there he returns to point A.
Could I have some help finding the interior angles of this triangle, using theorems or trig or something, I am completely lost...
February 27th 2008, 09:57 PM
A man starts walking from point A at a bearing of 56°. He walks for 400m and stops at point B. He then walks a bearing of 300° for 200m and stops at point C. From there he returns to point A.
Could I have some help finding the interior angles of this triangle, using theorems or trig or something, I am completely lost...
1. Make a sketch.
2. The greyed angle must be 56°. Then you can calculate the red angle.
3. Now use Cosine rule to calculate $\overline{AC}$
4. Now you can use either Sine rule or Cosine rule to calculate the missing angles.
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Here's the question you clicked on:
At a fair i bought 6 erasers and 2 pens for $15. If i could have bought 4 more erasers for $14 than pens for $9, the price of a pen was????
• one year ago
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i don't understand the last part of your question. is it saying 4 erasers cost $14??
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thats the problem its an aptitude question
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Is it saying 4 more erasers (plus the ones she bought) which would make it 10 erasers = $ 14 ???
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i am not able to understand the ques
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@damien @e.cociuba @firefightere9 @gjhfdfg @Hayatcakes @HB0315
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6e + 2p = 14 I am sorry...I can't get past this :(
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it's ok
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@Mimi_x3 @myko @n14r96
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the question's worded weird
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yes it is....confusing
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thank u ppl jus got the answer by my own
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cool. can you tell us the answer?
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just curious
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hmm.. yea
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the ques says that 6 erasers and 2 pens is for $15 so let us consider the cost of eraser and pen be E and P respectively => 6E+2P = 15 then , it says 4 more erasers for $14 than pens for $9 that
is , if the number of pens is 'n' then the number of erasers is '(4+n)' nP=9 ,(4+n)E=14
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Spatial Analysis Along Networks: Statistical and Computational Methods
ISBN: 978-0-470-77081-8
306 pages
August 2012
Read an Excerpt
In the real world, there are numerous and various events that occur on and alongside networks, including the occurrence of traffic accidents on highways, the location of stores alongside roads, the
incidence of crime on streets and the contamination along rivers. In order to carry out analyses of those events, the researcher needs to be familiar with a range of specific techniques. Spatial
Analysis Along Networks provides a practical guide to the necessary statistical techniques and their computational implementation.
Each chapter illustrates a specific technique, from Stochastic Point Processes on a Network and Network Voronoi Diagrams, to Network K-function and Point Density Estimation Methods, and the Network
Huff Model. The authors also discuss and illustrate the undertaking of the statistical tests described in a Geographical Information System (GIS) environment as well as demonstrating the
user-friendly free software package SANET.
Spatial Analysis Along Networks:
• Presents a much-needed practical guide to statistical spatial analysis of events on and alongside a network, in a logical, user-friendly order.
• Introduces the preliminary methods involved, before detailing the advanced, computational methods, enabling the readers a complete understanding of the advanced topics.
• Dedicates a separate chapter to each of the major techniques involved.
• Demonstrates the practicalities of undertaking the tests described in the book, using a GIS.
• Is supported by a supplementary website, providing readers with a link to the free software package SANET, so they can execute the statistical methods described in the book.
Students and researchers studying spatial statistics, spatial analysis, geography, GIS, OR, traffic accident analysis, criminology, retail marketing, facility management and ecology will benefit from
this book.
See More
Chapter 1 Introduction
1.1 What is network spatial analysis?
1.1.1 Network events: events on and alongside networks
1.1.2 Planar spatial analysis and its limitations
1.1.3 Network spatial analysis and its salient features
1.2 Review of studies of network events
1.2.1 Snow’s study on cholera around Broad Street
1.2.2 Traffic accidents
1.2.3 Road-kills
1.2.4 Street crimes
1.2.5 Events on river networks and coastlines
1.2.6 Other events on networks
1.2.7 Events alongside networks
1.3 Outline of the book
1.3.1 Structure of chapters
1.3.2 Questions solved by network spatial methods
1.3.3 How to study this book
Chapter 2 Modeling events on and alongside networks
2.1 Modeling the real world
2.1.1 Object-based model
2.1.1.1 Spatial attributes
2.1.1.2 Nonspatial attributes
2.1.2 Field-based model
2.1.3 Vector data model
2.1.4 Raster data model
2.2 Modeling networks
2.2.1 Object-based model for networks
2.2.1.1 Geometric networks
2.2.1.2 Graph for a geometric network
2.2.2 Field-based model for networks
2.2.3 Data models for networks
2.3 Modeling entities on and alongside networks
2.3.1 Objects on network space
2.3.2 Field functions on network space
2.4 Stochastic processes on network space
2.4.1 Object-based model for stochastic spatial events on network space
2.4.2 Binomial point processes on network space
2.4.3 Edge effects
2.4.4 Uniform network transformation
Chapter 3 Basic computational methods for network spatial analysis
3.1 Data structures for one-layer networks
3.1.1 Planar networks
3.1.2 Winged-edge data structures
3.1.3 Efficient access and enumeration of local information
3.1.4 Attribute data representation
3.1.5 Local modifications of a network
3.1.5.1 Inserting new nodes
3.1.5.2 New nodes resulting from overlying two networks
3.1.5.3 Deleting existing nodes
3.2 Data Structures for nonplanar networks
3.2.1 Multiple-layer networks
3.2.2 General nonplanar networks
3.3 Basic Geometric Computations
3.3.1 Computational methods for line segments
3.3.1.1 Right-turn test
3.3.1.2 Intersection test for two line segments
3.3.1.3 Enumeration of line segment intersections
3.3.2 Time complexity as a measure of efficiency
3.3.3 Computational methods for polygons
3.3.3.1 Area of a polygon
3.3.3.2 Center of gravity of a polygon
3.3.3.3 Inclusion test of a point with respect to a polygon
3.3.3.4 Polygon-line intersection
3.3.3.5 Polygon intersection test
3.3.3.6 Extraction of a subnetwork inside a polygon
3.3.3.7 Set-theoretic computations
3.3.3.8 Nearest point on the edges of a polygon from a point in the polygon
3.3.3.9 Frontage interval
3.4. Basic computational methods on networks
3.4.1 Single-source shortest paths
3.4.1.1 Network connectivity test
3.4.1.2 Shortest-path tree
3.4.1.3 Extended shortest-path tree
3.4.1.4 All nodes within a prespecified distance
3.4.1.5 Center of a network
3.4.1.6 Heap data structure
3.4.2 Shortest path between two nodes
3.4.3 Minimum spanning tree on a network
3.4.4 Monte Carlo simulation for generating random points on a network
Chapter 4 Network Voronoi diagrams
4.1 Ordinary network Voronoi diagram
4.1.1 Planar versus network Voronoi diagrams
4.1.2 Geometric properties of the ordinary network Voronoi diagram
4.2 Generalized network Voronoi diagrams
4.2.1 Directed network Voronoi diagram
4.2.2 Weighted network Voronoi diagram
4.2.3 k-th nearest point network Voronoi diagram
4.2.4 Line and polygon network Voronoi diagram
4.2.5 Point-set network Voronoi diagram
4.3 Computational methods for network Voronoi diagrams
4.3.1 Multi-start Dijkstra method
4.3.2 Computational method for the ordinary network Voronoi diagram
4.3.3 Computational method for the directed network Voronoi diagram
4.3.4 Computational method for the weighted network Voronoi diagram
4.3.5 Computational method for the
4.3.6 Computational method for the line and polygon network Voronoi diagrams
4.3.7 Computational method for the point-set network Voronoi diagram
Chapter 5 Network nearest-neighbor distance methods
5.1 Network auto nearest-neighbor distance method
5.1.1 Network local auto nearest-neighbor distance method
5.1.2 Network global auto nearest-neighbor distance method
5.2 Network cross nearest-neighbor distance method
5.2.1 Network local cross nearest-neighbor distance method
5.2.2 Network global cross nearest-neighbor distance method
5.3 Network nearest-neighbor distance method for lines
5.4 Computational methods for network nearest-neighbor distance methods
5.4.1 Computational methods for network auto nearest-neighbor distance methods
5.4.1.1 Computational methods for network local auto nearest-neighbor distance method
5.4.1.2 Computational methods for network global auto nearest-neighbor distance method
5.4.2 Computational methods for network cross nearest-neighbor distance methods
5.4.2.1 Computational methods for network local cross nearest-neighbor distance method
5.4.2.2 Computational methods for network global cross nearest-neighbor distance method
Chapter 6 Network K function methods
6.1 Network auto K function methods
6.1.1 Network local auto K function method
6.1.2 Network global auto K function method
6.2 Network cross K function methods
6.2.1 Network local cross K function method
6.2.2 Network global cross K function method
6.2.3 Network global Voronoi cross K function method
6.3 Network K function methods in relation to geometric characteristics of a network
6.3.1 Relationship between the shortest-path distance and the Euclidean distance
6.3.2 Network global auto K function in relation to the level-of-detail of a network
6.4 Computational methods for the network K function methods
6.4.1 Computational methods for the network auto K function methods
6.4.1.1 Computational methods for the network local auto K function method
6.4.1.2 Computational methods for the network global auto K function
6.4.2 Computational methods for the network cross K function methods
6.4.2.1 Computational methods for the network local auto K function method
6.4.2.3 Computational methods for the network global cross K function method
6.4.2.3 Computational methods for the network global Voronoi cross K
function method
Chapter 7 Network spatial autocorrelation
7.1 Classification of spatial autocorrelations
7.2 Spatial randomness of the attribute values of network cells
7.2.1 Permutation spatial randomness
7.2.2 Normal variate spatial randomness
7.3 Network Moran’s I statistics
7.3.1 Network local Moran’s I statistic
7.3.2 Network global Moran’s I statistic
7.4 Computational methods for network Moran’s I statistics
Chapter 8 Network point cluster analysis and clumping method
8.1 Network point cluster analysis
8.1.1 General hierarchical point cluster analysis
8.1.2 Hierarchical point clustering methods with specific intercluster distances
8.1.2.1 Network closest-pair point clustering method
8.1.2.2Network farthest-pair point clustering method
8.1.2.3 Network average-pair point clustering method
8.1.2.4 Network point clustering methods with other interclaster distances
8.2 Network clumping method
8.2.1 Relation to network point cluster analysis
8.2.2 Statistical test with respect to the number of clumps
8.3 Computational methods for network point cluster analysis and clumping method
8.3.1 General computational framework
8.3.2 Computational methods for individual intercluster distances
8.3.2.1 Computational methods for the network closest-pair point clustering
8.3.2.1 Computational methods for the network farthest-pair point clustering
8.3.2.3 Computational methods for the network average-pair point clustering
8.3.3 Computational aspects of the network clumping method
Chapter 9 Network point density estimation methods
9.1 Network histograms
9.1.1 Network cell histograms
9.1.2 Network Voronoi cell histograms
9.1.3 Network cell-count method
9.2 Network kernel density estimation methods
9.2.1 Network kernel functions
9.2.2 Equal-split discontinuous kernel functions
9.2.3 Equal-split continuous kernel functions
9.3 Computational methods for network point density estimation
9.3.1 Computational methods for network cell histograms with equal-length network cells
9.3.2 Computational method for equal-split discontinuous kernel density functions
9.3.3 Computational method for equal-split continuous kernel density functions
Chapter 10 Network spatial interpolation
10.1 Network inverse-distance weighting
10.1.1 Concepts of neighborhoods on a network
10.1.2 Network inverse-distance weighting predictor
10.2 Network kriging
10.2.1 Network kriging models
10.2.2 Concepts of stationary processes on a network
10.2.3 Network variogram models
10.2.4 Network kriging predictors
10.3 Computational methods for network spatial interpolation
10.3.1 Computational methods for network inverse-distance weighing
10.3.2 Computational methods for network kriging
Chapter 11 Network Huff model
11.1 Concepts of the network Huff model
11.1.1 Huff models
11.1.2 Dominant market subnetworks
11.1.3 Huff-based demand estimation
11.1.4 Huff-based locational optimization
11.2 Computational methods for the Huff-based demand estimation
11.2.1 Shortest-path tree distance
11.2.2 Choice probabilities in terms of shortest-path tree distances
11.2.3 Analytical formula for the Huff-based demand estimation
11.2.4 Computational tasks and their time complexities for the Huff-based demand estimation
11.3 Computational methods for the Huff-based locational optimization
11.3.1 Demand function for a newly entering store
11.3.2 Topologically invariant shortest-path trees
11.3.3 Topologically invariant link sets
11.3.4 Numerical method for the Huff-based locational optimization
11.3.5 Computational tasks and their time complexities for the Huff-based locational optimization
Chapter 12 GIS-based tools for spatial analysis along networks and their application
12.1 Preprocessing tools in SANET
12.1.1 Tool for testing network connectedness
12.1.2 Tool for assigning points to the nearest points on a network
12.1.3 Tool for computing shortest-path distances between points
12.1.4 Tool for generating random points on a network
12.2 Statistical tools in SANET and their applications
12.2.1 Tools for network Voronoi diagrams and their application
12.2.2 Tools for network nearest neighbor distance methods and their application
12.2.2.1 Network global auto nearest-neighbor distance method
12.2.2.2 Network global cross nearest-neighbor distance method
12.2.3 Tools for network K function methods and their application
12.2.3.1 Network global auto K function method
12.2.3.2 Network global cross K function method
12.2.3.3 Network global Voronoi cross K function method
12.2.3.4 Network local cross K function method
12.2.4 Tools for network cluster analysis and their application
12.2.5 Tools for network kernel density estimation methods and their application
12.2.6 Tools for network spatial interpolation methods and their application
See More
“Students and researchers studying spatial statistics, spatial analysis, geography, GIS, OR, traffic accident analysis, criminology, retail marketing, facility management and ecology will benefit
from this book.” (Zentralblatt MATH, 1 May 2013)
See More
Buy Both and Save 25%!
Spatial Analysis Along Networks: Statistical and Computational Methods (US $109.00)
-and- Simplicity, Complexity and Modelling (US $119.00)
Total List Price: US $228.00
Discounted Price: US $171.00 (Save: US $57.00)
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geometric sequence problem
October 25th 2008, 01:25 AM #1
Oct 2008
Perth, Australia (i.e desert land)
geometric sequence problem
Here's the question I need a hand with:
When satellites are put into orbit, they travel around the globe, relying on the earth's gravitational pull to slingshot it around. Scientists discovered that every 100 orbits the satellite does,
its distance is 98% of what it was previously. They have also discovered that when the satellite is 50% of its original distance, the gravitational pull would be too much and it will fall to the
surface of the planet.
If each orbit of the earth lasts for 7 days:
Find how many orbits it would take for the satellite to be 75% of it's original distance from the earth (answer to the nearest one orbit)
Also, find how long a satellite will last in orbit (answer to the nearest day)
Thanks in advance
let $n$ = number of orbits
$R_0$ = initial orbital radius
$R$ = orbital radius for any number of orbits $n$
$R = R_0 (.98)^{100n}$
determine the number of orbits it takes for $\frac{R}{R_0} = .75$
October 25th 2008, 05:31 AM #2
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South Grafton Precalculus Tutor
Find a South Grafton Precalculus Tutor
...I had to model my teaching techniques during these training sessions, with my student every time I met with my trainer. This was to show my understanding, fluency, and proficiency in the Wilson
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Chatting about REAL numbers
The term "
real number
" confuses many who are not immersed in mathematics. For these, to whom 1, 2, 3 and the other counting numbers seem most
, the identification of the real numbers as all infinite decimals (i.e., all numbers representable by points on a number line) seems at first to go beyond intuition. But, upon further reflection, the
idea of a number as "real" iff it can represent a distance on a line to the right or left of a central origin,
, indeed seems reasonable.
Professor Fred Richman of Florida Atlantic University takes on the questions of computability and enumerability of the real numbers in his poem, "Dialogue Between Machine and Man": Dialogue between
Machine and Man by Fred Richman
Hey man! Do you have time, would you agree,
To chat about real numbers now with me?
My friend, you've never seen a real real number.
You cut them off before they're halfway done.
Nor twenty decimal places, nor a thousand,
Are adequate to hold a single one.
I think you underestimate me, man.
My software package is the best in town.
See here's a little program that computes
The first n places of the number pi.
You choose the n, and if you have the time
You'll get as much of pi as you can stomach.
Such strings of digits are my cup of tea,
I mind not that they go on endlessly.
You've barely scratched the surface none the less.
The set of numbers you can calculate
With programs, like the one you wrote for pi,
Can be enumerated one by one,
And Cantor showed that, given such a list,
There is at least one number that is missed.
Insanity! What number has been seen
In all the world that I can't calculate?
Nor can you list the programs that compute
Each digit in a number's decimal string.
A child of ten can write the code to list
The programs my compiler will accept.
But here the programs must consist of those
That endless strings of digits do produce.
Write such a code, man, and the pigs will fly.
There's no such code. Look here, I'll show you why.
I know why, Mac, you just use Cantor's proof
To show that there is no recursive list
That itemizes all recursive functions.
The problem is that every list you know
Is general recursive---that's your world.
It's a paradox you'll never understand,
That I can count your numbers one by one
Despite your proof that it cannot be done.
1 comment:
1. Hi JoAnne,
I recently published a post that asks the question "what are real numbers?" Your explanation is a much simpler and more succinct version of my entire post! For something that seems so simple now,
we forget how strange this concept was when we first learned about it.
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Briny Breezes, FL Math Tutor
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Live Simulation: Satellite Rescue | Teacher's Toolkit | Lesson Plan
3: Ratios, Proportions, and Cross-multiplication
During space shuttle missions astronauts sometimes have to perform extravehicular activities (EVAs), also called spacewalks. They do EVAs to develop techniques, test procedures, and capture
satellites for repair. NASA has to closely monitor the vital signs of astronauts during an EVA to make sure the astronauts don’t use up their oxygen supplies too quickly.
In this activity your students will use cross-multiplication to find the unknown respiration rate per minute of an astronaut during several EVA tasks. They will use this same mathematical process to
find missing values in proportions during the mission.
60 minutes
cross-multiplication: the process of multiplying the numerator of one ratio by the denominator of the other ratio in a proportion; the cross-products of two equal ratios are always equal. You can
use cross-multiplication to find a missing value in a proportion.
proportion: a pair of equal ratios.
rate: a comparison of two quantities with different units of measure.
ratio: a comparison of two quantities, often written as a fraction.
NCTM Standards for Mathematics
Grades 6-8; Number and Operations Standard
Understand numbers, ways of representing numbers, relationships among numbers, and number systems:
• Work flexibly with fractions, decimals, and percentages to solve problems.
• Understand and use ratios and proportions to represent quantitative relationships.
• Copies of Student Worksheet: Ratios, Proportions, and Cross-multiplication for each student
• Pencils
Teacher Preparation
1. Make copies of student worksheets.
2. Gather calculators (optional) if you will allow students to use them for this activity or for the mission.
1. Decide how you want the students to work on the activity. You may choose to have the students work individually or in cooperative groups. Another option would be to assign this activity for
homework and review the answers together in class.
2. Discuss all vocabulary in the lesson, including ratio, rate, proportion, and cross-multiplication.
3. Preview the activity with the students. Read the student version of the instructions out loud in class. Allow time for questions and discussion.
4. Make sure that your students understand the relevance of this activity to their mission work. During the Satellite Rescue live simulation, students will use proportions to calculate the heart
rates, respiration rates, and oxygen usage of the astronauts.
Discussion Tips
Students must be able to use cross-multiplication to problem solve during the mission. Make sure you allow enough time for student questions and to check for student understanding.
Special Comments
You may want to use this lesson to assess student skill with math application. Even if a student is not a particularly strong math student, he/she can sometimes do well when the math is tied to a
mission scenario.
Download Microsoft Active Accessibility.
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#REF Error in Named Range with INDIRECT, SUMPRODUCT, and OFFSET
Hi All!
I have run into an odd issue using a Named Range with INDIRECT, SUMPRODUCT,
and OFFSET.
I have a worksheet that has a formula which will look for other worksheets
with names that are listed on the worksheet and once it finds the worksheet,
the formula then looks for a Project name on that worksheet and returns the
number in the corresponding cell.
The formula uses SUMPRODUCT, INDIRECT, OFFSET, AND IF(ISERROR).
The formula is as follows:
Range1= "'"$A$5:$A$9&"'!$a$50:$a$60" (A5:A9 = a list of other worksheet
names; A50:A60 = a list of project on those worksheets)
Range2= "'"$A$5:$A$9&"'!$e$50:$e$60" (A5:A9 = a list of other worksheet
names, as above;E50:E60 = the hours total for each project - per row - for a
A10 is the project name on the current worksheet; I want the formula to look
for this name on the other sheets, to return the corresponding value for the
month in question.
OFFSET is in the formula to allow the formula to be used across muliple
columns representing a year, and thus returning the hour total for the same
month on the other worksheets. The month columns are in the same columns on
all sheets.
Sooooooooooo....when A5:A9 are all filled with names, the formula works like
a charm (many thanks to the Excel gurus here for helping me learn about these
complex formulae)...but when any of the cells in A5:A9 are blank, the formula
does not return any values. Using the Formula Auditing function and going
through the evaluation, as I step thorugh the formula, I get #REF errors for
the blank cells.
As soon as I put in any data, the formula works great.
Is there way to make this formula work and ignore the blank cells? I have a
cell range there, as some worksheets will have several worksheets listed
(filling the range) and others will have only 1 subordinate worksheet listed.
I did not want to create a custom sheet each time the number of subordinate
worksheets is different.
All help is appreciated!!!!!!!
The folks here have been great and the info I have found without even having
to post questions has been super!
Thanks again!
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Graph theory questionLet G be a graph with no cycles. Prove that
V = E+C where V is the number of verticies, E is... - Homework Help - eNotes.com
Graph theory question
Let G be a graph with no cycles. Prove that
V = E+C where V is the number of verticies, E is the number of edges
and C is the number of components of the graph.
To prove this formula, you must prove the Euler's formula.
Remember which is the Euler's characteristic for connected planar graphs:
`chi` = V-E+F
F is the number of faces in the graph
V - vertices of the graph
E- edges
In case of any planar connected graph, chi = 2. This value is similar to Euler's polyhedron formula for the surfaces of any polyhedron.
Using induction on the number of faces in the graph => V = E-F+C+1.
C is the number of components of the graph.
For a tree graph, F=1 => V = E+C
Join to answer this question
Join a community of thousands of dedicated teachers and students.
Join eNotes
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How to Convert Ccf to Mmbtu | The Classroom | Synonym
CCF is shorthand for 100 cubic feet, usually referring to either the volume of water or of natural gas. MMBTU is 1 million BTUs, which are British thermal units and are a measurement of energy. The
measurement of cubic feet of natural gas can be converted into BTU representation. Converting from CCF of natural gas to MMBTUs is the same as converting from cubic feet to BTUs, with a few added
Step 1
Multiply the number of CCF by 100 to obtain the amount of natural gas in cubic feet. For example, if the amount of natural gas is 15 CCF, then multiply by 100 to obtain the amount in cubic feet,
which is 1,500 cubic feet.
Step 2
Multiply that number by 1,027, which is the conversion constant from cubic feet to BTUs. For the example, 1,500 multiplied by 1,027 equals 1,540,500 BTUs.
Step 3
Divide that number by 1 million to obtain the number of MMBTUs. For the example, 1,540,500 divided by 1,000,000 equals 1.540500, or approximately 1.55 MMBTUs.
Style Your World With Color
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Growing rate
September 23rd 2009, 02:30 AM #1
A drop of water is a perfect sphere and by condensation, the droplet picks up moisture at a rate proportional to its surface area. Show that the drops radius increases at a constant rate.
What sort of calculus would I use to tackle this problem? Thanks
Differential Calculus! More specifically, "related rates". Use the formula for volume of a sphere- $V= \frac{4}{3}\pi r^3$- and the formula for area of the surface- $A= 4\pi r^2$. Differentiate
the volume formula, with respect to t, using the chain rule, and set it equal to a constant times the area. Solve for dr/dt.
Rates of change
Hello RAz
We can re-write the phrase
the droplet picks up moisture at a rate proportional to its surface area
as "the rate of change of its volume is proportional to its surface area"; or, with the usual notation:
$\frac{dV}{dt}= kS$, for some constant $k$.
So, using $V = \tfrac43\pi r^3$ and $S = 4\pi r^2$, we get:
$\frac{d}{dt}\Big(\tfrac43\pi r^3\Big)=4k\pi r^2$
$\Rightarrow \frac{d}{dr}\Big(\tfrac43\pi r^3\Big)\times\frac{dr}{dt}=4k\pi r^2$
$\Rightarrow 4\pi r^2 \frac{dr}{dt}=4k\pi r^2$
$\Rightarrow \frac{dr}{dt}=k$
$\Rightarrow r$ is increasing at a constant rate.
September 23rd 2009, 05:31 AM #2
MHF Contributor
Apr 2005
September 23rd 2009, 05:39 AM #3
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Founding cryptography on oblivious transfer
Results 1 - 10 of 179
- JCSS
"... Private Information Retrieval (PIR) schemes allow a user to retrieve the i-th bit of an n-bit data string x, replicated in k 2 databases (in the information-theoretic setting) or in k 1
databases (in the computational setting), while keeping the value of i private. The main cost measure for suc ..."
Cited by 105 (19 self)
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Private Information Retrieval (PIR) schemes allow a user to retrieve the i-th bit of an n-bit data string x, replicated in k 2 databases (in the information-theoretic setting) or in k 1 databases (in
the computational setting), while keeping the value of i private. The main cost measure for such a scheme is its communication complexity.
- In Birgit Pfitzmann, editor, Advances in Cryptology — EUROCRYPT 2001, volume 2045 of Lecture Notes in Computer Science , 2001
"... Abstract. We consider the question of protecting the privacy of customers buying digital goods. More specifically, our goal is to allow a buyer to purchase digital goods from a vendor without
letting the vendor learn what, and to the extent possible also when and how much, it is buying. We propose s ..."
Cited by 95 (5 self)
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Abstract. We consider the question of protecting the privacy of customers buying digital goods. More specifically, our goal is to allow a buyer to purchase digital goods from a vendor without letting
the vendor learn what, and to the extent possible also when and how much, it is buying. We propose solutions which allow the buyer, after making an initial deposit, to engage in an unlimited number
of priced oblivioustransfer protocols, satisfying the following requirements: As long as the buyer’s balance contains sufficient funds, it will successfully retrieve the selected item and its balance
will be debited by the item’s price. However, the buyer should be unable to retrieve an item whose cost exceeds its remaining balance. The vendor should learn nothing except what must inevitably be
learned, namely, the amount of interaction and the initial deposit amount (which imply upper bounds on the quantity and total price of all information obtained by the buyer). In particular, the
vendor should be unable to learn what the buyer’s current balance is or when it actually runs out of its funds. The technical tools we develop, in the process of solving this problem, seem to be of
independent interest. In particular, we present the first one-round (two-pass) protocol for oblivious transfer that does not rely on the random oracle model (a very similar protocol was independently
proposed by Naor and Pinkas [21]). This protocol is a special case of a more general “conditional disclosure ” methodology, which extends a previous approach from [11] and adapts it to the 2-party
setting. 1
, 1989
"... Abstract. We address the problem of performing a multiparty computation when more than half of the processors are cooperating Byzantine faults. We show how to compute any boolean function of n
inputs distributively, preserving the privacy of inputs held by nonfaulty processors, and ensuring that fau ..."
Cited by 75 (4 self)
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Abstract. We address the problem of performing a multiparty computation when more than half of the processors are cooperating Byzantine faults. We show how to compute any boolean function of n inputs
distributively, preserving the privacy of inputs held by nonfaulty processors, and ensuring that faulty processors obtain the function value “if and only if ” the nonfaulty processors do. If the
nonfaulty processors do not obtain the correct function value, they detect cheating with high probability. Our solution is based on a new type of verifiable secret sharing in which the secret is
revealed not all at once but in small increments. This slow-revealing process ensures that all processors discover the secret at roughly the same time. Our solution assumes the existence of an
oblivious transfer protocol and uses broadcast channels. We do not require that the processors have equal computing power. 1
, 2001
"... We present session-key generation protocols in a model where the legitimate parties share only a human-memorizable password. The security guarantee holds with respect to probabilistic
polynomial-time adversaries that control the communication channel (between the parties), and may omit, insert and ..."
Cited by 75 (7 self)
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We present session-key generation protocols in a model where the legitimate parties share only a human-memorizable password. The security guarantee holds with respect to probabilistic polynomial-time
adversaries that control the communication channel (between the parties), and may omit, insert and modify messages at their choice. Loosely speaking, the effect of such an adversary that attacks an
execution of our protocol is comparable to an attack in which an adversary is only allowed to make a constant number of queries of the form “is w the password of Party A”. We stress that the result
holds also in case the passwords are selected at random from a small dictionary so that it is feasible (for the adversary) to scan the entire directory. We note that prior to our result, it was not
clear whether or not such protocols were attainable without the use of random oracles or additional setup assumptions.
, 1992
"... We describe a protocol for quantum oblivious transfer , utilizing faint pulses of polarized light, by which one of two mutually distrustful parties ("Alice") transmits two one-bit messages in
such a way that the other party ("Bob") can choose which message he gets but cannot obtain information about ..."
Cited by 73 (12 self)
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We describe a protocol for quantum oblivious transfer , utilizing faint pulses of polarized light, by which one of two mutually distrustful parties ("Alice") transmits two one-bit messages in such a
way that the other party ("Bob") can choose which message he gets but cannot obtain information about both messages (he will learn his chosen bit's value with exponentially small error probability
and may gain at most exponentially little information about the value of the other bit), and Alice will be entirely ignorant of which bit he received. Neither party can cheat (ie deviate from the
protocol while appearing to follow it) in such a way as to obtain more information than what is given by the description of the protocol. Our protocol is easy to modify in order to implement the
All-or-Nothing Disclosure of one out of two string messages, and it can be used to implement bit commitment and oblivious circuit evaluation without complexity-theoretic assumptions, in a way that
remains secure e...
- In EUROCRYPT 2007, Springer-Verlag (LNCS 4515 , 2007
"... We show an efficient secure two-party protocol, based on Yao’s construction, which provides security against malicious adversaries. Yao’s original protocol is only secure in the presence of
semi-honest adversaries, and can be transformed into a protocol that achieves security against malicious adver ..."
Cited by 70 (10 self)
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We show an efficient secure two-party protocol, based on Yao’s construction, which provides security against malicious adversaries. Yao’s original protocol is only secure in the presence of
semi-honest adversaries, and can be transformed into a protocol that achieves security against malicious adversaries by applying the compiler of Goldreich, Micali and Wigderson (the “GMW compiler”).
However, this approach does not seem to be very practical as it requires using generic zero-knowledge proofs. Our construction is based on applying cut-and-choose techniques to the original circuit
and inputs. Security is proved according to the ideal/real simulation paradigm, and the proof is in the standard model (with no random oracle model or common reference string assumptions). The
resulting protocol is computationally efficient: the only usage of asymmetric cryptography is for running O(1) oblivious transfers for each input bit (or for each bit of a statistical security
parameter, whichever is larger). Our protocol combines techniques from folklore (like cut-andchoose) along with new techniques for efficiently proving consistency of inputs. We remark that a naive
implementation of the cut-and-choose technique with Yao’s protocol does not yield a
- SIGKDD Explorations , 2002
"... Research in secure distributed computation, which was done as part of a larger body of research in the theory of cryptography, has achieved remarkable results. It was shown that non-trusting
parties can jointly compute functions of their different inputs while ensuring that no party learns anything ..."
Cited by 63 (0 self)
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Research in secure distributed computation, which was done as part of a larger body of research in the theory of cryptography, has achieved remarkable results. It was shown that non-trusting parties
can jointly compute functions of their different inputs while ensuring that no party learns anything but the defined output of the function. These results were shown using generic constructions that
can be applied to any function that has an ecient representation as a circuit. We describe these results, discuss their efficiency, and demonstrate their relevance to privacy preserving computation
of data mining algorithms. We also show examples of secure computation of data mining algorithms that use these generic constructions.
, 2004
"... We consider the following problem: a user wants to store his files in an encrypted form on a remote file server S. ..."
- In CRYPTO 2000: 20th International Cryptology Conference , 2000
"... Abstract. In this work we use cryptography to solve a game-theoretic problem which arises naturally in the area of two party strategic games. The standard game-theoretic solution concept for
such games is that of an equilibrium, which is a pair of “self-enforcing ” strategies making each player’s st ..."
Cited by 61 (1 self)
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Abstract. In this work we use cryptography to solve a game-theoretic problem which arises naturally in the area of two party strategic games. The standard game-theoretic solution concept for such
games is that of an equilibrium, which is a pair of “self-enforcing ” strategies making each player’s strategy an optimal response to the other player’s strategy. It is known that for many games the
expected equilibrium payoffs can be much higher when a trusted third party (a “mediator”) assists the players in choosing their moves (correlated equilibria), than when each player has to choose its
move on its own (Nash equilibria). It is natural to ask whether there exists a mechanism that eliminates the need for the mediator yet allows the players to maintain the high payoffs offered by
mediator-assisted strategies. We answer this question affirmatively provided the players are computationally bounded and can have free communication (so-called “cheap talk”) prior to playing the
game. The main building block of our solution is an efficient cryptographic protocol to the following Correlated Element Selection problem, which is of independent interest. Both Alice and Bob know a
list of pairs (a1, b1)... (an, bn) (possibly with repetitions), and they want to pick a random index i such that Alice learns only ai and Bob learns only bi. Our solution to this problem has constant
number of rounds, negligible error probability, and uses only very simple zero-knowledge proofs. We then show how to incorporate our cryptographic protocol back into a game-theoretic setting, which
highlights some interesting parallels between cryptographic protocols and extensive form games. 1
, 2003
"... We consider the problem of extending oblivious transfers: Given a small number of oblivious transfers \for free," can one implement a large number of oblivious transfers? Beaver has shown how to
extend oblivious transfers given a one-way function. However, this protocol is inecient in practice, ..."
Cited by 57 (1 self)
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We consider the problem of extending oblivious transfers: Given a small number of oblivious transfers \for free," can one implement a large number of oblivious transfers? Beaver has shown how to
extend oblivious transfers given a one-way function. However, this protocol is inecient in practice, in part due to its non-black-box use of the underlying one-way function.
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Backpropagation-Decorrelation: online recurrent learning with O(N) complexity
Backpropagation-Decorrelation: online recurrent learning with O(N) complexity
Download Links
author = {Jochen J. Steil},
title = { Backpropagation-Decorrelation: online recurrent learning with O(N) complexity},
year = {}
We introduce a new learning rule for fully recurrent neural networks which we call Backpropagation-Decorrelation rule (BPDC). It combines important principles: one-step backpropagation of errors and
the usage of temporal memory in the network dynamics by means of decorrelation of activations. The BPDC rule is derived and theoretically justified from regarding learning as a constraint
optimization problem and applies uniformly in discrete and continuous time. It is very easy to implement, and has a minimal complexity of 2N multiplications per time-step in the single output case.
Nevertheless we obtain fast tracking and excellent performance in some benchmark problems including the Mackey-Glass time-series.
1490 Independent Component Analysis - Hyvärinen, Karhunen, et al. - 2001
243 Long short-term memory - Hochreiter, Schmidhuber - 1997
134 Gradient calculations for dynamic recurrent neural networks: A survey - Pearlmutter - 1995
116 An efficient gradient-based algorithm for on-line training of recurrent network trajectories - Williams, Peng - 1990
115 Gradient-based learning algorithms for recurrent networks and their computational complexity - Williams, Zipser - 1995
58 Adaptive nonlinear system identification with Echo State Networks - Jaeger - 2003
33 New results on recurrent network training: unifying the algorithms and accelerating convergence. Neural Networks - Atiya, Parlos - 2000
22 Learning with Recurrent Neural Networks - Hammer - 2000
20 The “liquid computer”: A novel strategy for real-time computing on time series - Natschläger, Maass, et al. - 2002
17 A fixed size storage O(n 3 ) time complexity learning algorithm for fully recurrent continually running networks - Schmidhuber - 1992
10 The vanishing gradient problem during learning recurrent neural nets and problem solutions - Hochreiter - 1998
9 Analyzing the weight dynamics of recurrent learning algorithms,” Neurocomputing - Schiller, Steil
8 Recurrent learning of input-output stable behaviour in function space: A case study with the Roessler attractor - Steil, Ritter - 1999
7 A Conjugate Gradient Learning Algorithm for Recurrent Neural - Chang, Mak - 1999
5 A fixed size storage o(n3) time complexity learning algorithm for fully recurrent continually running networks - Schmidhuber - 1992
5 Natural gradient learning for spatio-temporal decorrelation: recurrent network - Choi, Amari, et al.
4 A learning rule for dynamic recruitment and decorrelation - Körding, König - 2000
3 Local structural stability of recurrent networks with timevarying weights - Steil - 2002
2 Analysis and comparison of algorithms for training recurrent neural networks - Schiller - 2003
1 On the weight dynamcis of recurrent learning - Schiller, Steil
1 A conjugate gradient learning algorith for recurrent neural networks - Chang, Mak - 1999
1 Recurrent Neural Networks: Design and Applications - Santos, Zuben - 1999
1 Attractive periodic sets in descrete-time recurrent networks (with emphasis on fixed-point stability and bifurcations in two-neuron networks - Tiňo, Horne, et al. - 2001
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Engine Pressure
This is a complex question, and the pressure doesn't stay constant at any point during the combustion cycle.
Mean Effective Pressure is calculated by:
MEP = 1000 * (P * Nr) / (V * N)
Where MEP is in kPa, P is power in kW, Nr is the number of crank revolutions per power stroke, V the displaced volume per cylinder, and N the number of cylinders.
Note that the peak cylinder pressure will be much higher than this, but the calculation of this is far more involved.
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Factor and Simplify
Date: 4/10/96 at 21:41:35
From: Anonymous
Subject: Complex Fractions
How would you do the following problem and if possible, please
give tips for how to do similar problems. Thank you for the help.
Factor whenever possible.
Date: 5/28/96 at 20:33:24
From: Doctor Pete
Subject: Re: Complex Fractions
Let a=1/x, b=1/y. Then your expression is equivalent to
which you can factor as (a+b)(a-b)/(a+b)^2, so this is
Substituting back, we get (1/x-1/y)/(1/x+1/y) =
((y-x)/xy)/((x+y)/xy), so your expression is equal to
The trick in this case was to look at your x's and y's, and notice
that they were all reciprocals, so you can do a substitution that
makes the factorization much easier to spot. Here's one in a
similar vein, but a bit trickier:
(x-y)/(sqrt(x)-sqrt(y)) .
-Doctor Pete, The Math Forum
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Re-Creating AdWords Campaign Performance Charts in Excel, Part 3: Making the Scorecard
By Chad Summerhill [2] Posted In: AdWords Tips [3] Comments: 1 [4]
In part two [6] of this series, you learned how to transform the data we will be using to create our dashboard. In this article, I will show you:
• How to build the scorecard.
• How to use conditional formatting.
• How to use the camera tool.
Presentation (the “P” in “DTP”)
Again, we are using the DTP framework [7] created by the smart folks at Juice Analytics. The presentation layer for our exercise consists of all the pieces of our dashboard:
● Dynamic charts and scorecards
● Dynamic controls for date ranges and campaign filtering
Building dynamic dashboards in Excel can be a challenge, but if you’ve already completed the steps from the first two posts in this series you’ve done the hard part. Now we are ready for the fun
The Dashboard Tab
First, we need to create a new worksheet and rename it “dashboard.” This is where we will put our charts, scorecard, and the dynamic controls for our dashboard. Please highlight all cells and resize
the column widths to 18px to match the default row heights. This will give us a uniform canvas to work on.
Building the Scorecard
Creating our scorecard, like the one in the AdWords dashboard, is very easy, because we took the time to prepare our data properly.
For example, to get our sum of current impressions, you simply use this formula: =sum(curr_impr). Of course, “curr_impr” is the name we assigned our “Sum of Impressions” column in our pivot table, so
using the formula above will result in the summed total for the “Sum of Impressions” column. Here’s how you put the scorecard together:
1. Somewhere to the right of your pivot table, located in the “transform” tab of your worksheet, create a small table similar to the one shown below. Include all of the column headers and row labels.
2. Use the following formulas for each “Current” metric:
a. =sum(curr_impr)
b. =sum(curr_clicks)
c. =sum(curr_conv)
d. =sum(curr_cost)
3. Use the following formulas for each “Prev_Year” metric:
a. =sum(prv_impr)
b. =sum(prv_clicks)
c. =sum(prv_conv)
d. =sum(prv_cost)
4. Now we need to create our calculated metrics referencing the cells that contain the formulas above. For example, CTR = sum(curr_clicks)/sum(curr_impr). Calculate the rest of the metrics shown in
the image above using this method for both your current and previous year.
5. To calculate your percent change use the following formula = current/previous-1 and copy the formula down the #Chg column.
Next, we want to use conditional formatting to add our colored arrows.
1. Highlight all of the values in the %Chg column.
2. Select “3 Arrows (colored)” from the “Conditional Formatting” drop-down located in the “Home” ribbon.
3. Next, highlight all of the values in your “%Chg” column again and select “Manage rules…” from the “Conditional Formatting” drop-down in the “Home” ribbon.
4. Then highlight your rule and click “Edit Rule…”
5. Now you can edit your rules to change arrow types based on whatever thresholds you wish. Here’s the way I set it up:
6. Format the rest of your table to your liking. I made the background white, bolded my header row, and bolded my current values.
Once you have your formatting the way you like it, you should have a finished scorecard, but it’s not where we need it. In order to get our scorecard on our dashboard tab, we are going to use the
Excel camera tool.
The camera tool allows you to take a snapshot of any area in your workbook and create a moveable picture. This gives us the flexibility to put our scorecard anywhere we want on our dashboard without
it being tied directly to cells. It will also update automatically after we had our dynamic controls.
Here’s how you add the camera tool to your quick access toolbar:
After you’ve added the camera tool to your quick access toolbar, highlight your new scorecard and click on the camera tool, then click anywhere in your dashboard worksheet to paste the image of your
scorecard. You can then position it anywhere you want.
In part 4 of this series we will learn how build the charts we need for our dashboard. You will learn some best practices to use when creating multiple charts and we will prepare them to change
dynamically when we adjust our date ranges and campaign filters.
This is a guest post by Chad Summerhill, author of the blog PPC Prospector, provider of free PPC tools [8] and PPC tutorials [9], and in-house AdWords Specialist [10] at Moving Solutions, Inc. (
UPack.com [11] and MoveBuilder.com [12]).
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Question about E field
The reason the B field is zero outside is because its an infinite solenoid and the current enclosed is zero in my amperian loop.
Ok yes sometimes accelerating charges radiate but is this true for a solenoid.
And is this true for an infinite uniformly charged rod.
ah, duh! I forgot it is like a solenoid, because I normally think of a solenoid as a coil of wire. It is an interesting problem. In practice, the pole would need to be spinning pretty fast to be able
to create a significant magnetic field. Also, there is the problem of what happens in the centre of the pole. Maybe the example of a pole which is hollow in the middle is a bit simpler/easier to work
Right, so if we're going to assume that there is zero magnetic field outside, then there won't be any energy emitted in the form of electromagnetic waves. Because there must be a non-zero magnetic
field to allow energy to be propagated (simple use of Poynting vector). So if you want to find something which is going to emit EM energy, then its going to have to be not strictly a solenoid in this
edit: out of curiosity, is this machine going to be used for some science fiction writing? or is it just general curiosity?
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Is it true that ? |b-2|+|b+8|=10 (1) b less than or equal to
Author Message
Is it true that ? |b-2|+|b+8|=10 (1) b less than or equal to [#permalink] 20 Aug 2007, 15:14
5% (low)
Question Stats:
Joined: 04 May 2007
Posts: 111
(00:00) correct
Followers: 1
0% (00:00)
Kudos [?]: 0 [0], given: 0
based on 0 sessions
Is it true that ? |b-2|+|b+8|=10
(1) b less than or equal to 2
(2) b is greater than or equal to negative 8
Last edited by
on 21 Aug 2007, 07:26, edited 2 times in total.
Re: DS absolute value [#permalink] 21 Aug 2007, 08:13
gmatiscoming wrote:
Is it true that ? |b-2|+|b+8|=10
(1) b less than or equal to 2
FN (2) b is greater than or equal to negative 8
Current Student OK I get E...
Joined: 28 Dec 2004 here is how...
Posts: 3411 1) lets first see the break points they are 2 and -8
Location: New York City lets set a boundry condition
Schools: Wharton'11 HBS'12 -8<b<2>0 then yes works out...10=10
Followers: 13 if b<0 then (pick -6)
Kudos [?]: 148 [0], given: 2 -6+2=-4 -6+8=2 -4+2=-2 which is not equal to 10 Insuff
2) b < or equal to -8
then b-2+b+8 is equal to 10.
if b <-8 then its not!
Re: DS absolute value [#permalink] 21 Aug 2007, 10:38
fresinha12 wrote:
gmatiscoming wrote:
Is it true that ? |b-2|+|b+8|=10
(1) b less than or equal to 2
(2) b is greater than or equal to negative 8
OK I get E...
gixxer1000 here is how...
Director 1) lets first see the break points they are 2 and -8
Joined: 26 Jul 2007 lets set a boundry condition
Posts: 541 -8<b<2>0 then yes works out...10=10
Schools: Stern, McCombs, if b<0 then (pick -6)
Marshall, Wharton
-6+2=-4 -6+8=2 -4+2=-2 which is not equal to 10 Insuff
Followers: 4
2) b < or equal to -8
Kudos [?]: 90 [0], given: 0
then b-2+b+8 is equal to 10.
if b <-8 then its not!
-6-2=-8 -6+8=-2 You added instead of subtracted
The break points are -8 and 2
Stmt 1 gives you the upper point 2
Stmt 2 gives you the lower point -8
Ans is C
Re: DS absolute value [#permalink] 21 Aug 2007, 10:48
gmatiscoming wrote:
Is it true that ? |b-2|+|b+8|=10
bkk145 (1) b less than or equal to 2
(2) b is greater than or equal to negative 8
Joined: 10 Jun 2007
These type of questions, it is much faster if you realize the critical points of the equation. In this case, critical points are b=2 and b=-8.
Posts: 1469
From this, you know that there are three intervals that the equation must satisfy.
Followers: 5
A interval: b<-8
B interval: -8<b<2
C interval: b>2
If you find that b fall into one of these interval, then it must be right. (1) doesn't since it falls into intervals A and B, so INSUFFICIENT. (2) doesn't either because
it falls into interval B and C, INSUFFICIENT. Together, it falls into interval B. SUFFICIENT.
I am not sure how C could be correct..
There are only 2 possible values of B that could satisfy the equation
Sumithra Sen
|b-2|+|b+8|=10 , B is either2 or -8
Combining C
Joined: 18 Jun 2007
Posts: 88
Followers: 2 Combining both
Kudos [?]: 1 [0], given: 0
B could have a value between -8......2 ?
Why are we not considering the values between ?
Thanks for clarrifying..
VP |b-2|+|b+8|=10
Joined: 10 Jun 2007 Plug in any number between -8 and 2 will satisfy the equation.
Posts: 1469
Followers: 5
Re: DS absolute value [#permalink] 21 Aug 2007, 11:21
gmatiscoming wrote:
Is it true that ? |b-2|+|b+8|=10
(1) b less than or equal to 2
bkk145 (2) b is greater than or equal to negative 8
VP C.
Joined: 10 Jun 2007 These type of questions, it is much faster if you realize the critical points of the equation. In this case, critical points are b=2 and b=-8.
Posts: 1469 From this, you know that there are three intervals that the equation must satisfy.
Followers: 5 A interval: b<-8
B interval: -8<b<2
C interval: b>2
If you find that b fall into one of these interval, then it must be right. (1) doesn't since it falls into intervals A and B, so INSUFFICIENT. (2) doesn't either because
it falls into interval B and C, INSUFFICIENT. Together, it falls into interval B. SUFFICIENT.
*Additional comment: for this method to be useful, you should at least plug in a value from those interval to see if which one equals 10.
(C) for me too
I suggest to analyse quickly what is going on by replacing the absolutes.
o If b < -8 then
= -(b-2) - (b+8)
= -2*b - 6 >>> Not 10 and could not be
Fig o If -8 =< b =< 2 then
SVP = -(b-2) + (b+8)
= 10 >>> Sounds good
Joined: 01 May 2006
o If b > 2 then
Posts: 1818 |b-2|+|b+8|
= +(b-2) + (b+8)
Followers: 8 = 2*b + 6 >>> Not 10 and could not be
Kudos [?]: 83 [0], given: 0 From 1
b <= 2 >>>> So, we cannot conclude as it could be that -8 =< b =< 2 or b < -8
From 2
b >= -8 >>>> So, we cannot conclude again as it could be that -8 =< b =< 2 or b > 2.
Both (1) & (2)
Bongo.... It's -8 =< b =< 2 and we have seen that |b-2|+|b+8| = 10.
Current Student
If I can only freakin add on D day..arrrgggggggh...
Joined: 28 Dec 2004
guys any suggestions on how to avoid such mistakes????
Posts: 3411
Location: New York City
Schools: Wharton'11 HBS'12
Followers: 13
Kudos [?]: 148 [0], given: 2
fresinha12 wrote:
If I can only freakin add on D day..arrrgggggggh...
Joined: 10 Jun 2007
guys any suggestions on how to avoid such mistakes????
Posts: 1469
It happens to me too when I try to do it too fast or do it at work. I suggest writing it out step by step or get a good sleep. Or maybe it is just rainy today :D
Followers: 5
Re: DS absolute value [#permalink] 21 Aug 2007, 18:19
Whatever bkk145 wrote:
Manager These type of questions, it is much faster if you realize the critical points of the equation. In this case, critical points are b=2 and b=-8.
From this, you know that there are three intervals that the equation must satisfy.
Joined: 03 Sep 2006
A interval: b<-8
Posts: 233 B interval: -8<b<2>2
Followers: 1 If you find that b fall into one of these interval, then it must be right. (1) doesn't since it falls into intervals A and B, so INSUFFICIENT. (2) doesn't either because
it falls into interval B and C, INSUFFICIENT. Together, it falls into interval B. SUFFICIENT.
Kudos [?]: 5 [0], given: 0
Many thanks!
I also got C on this question =( But your explanation is great, it'll be definitely a shortcut for me!
gmatclubot Re: DS absolute value [#permalink] 21 Aug 2007, 18:19
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Definition of Indirect Proof | Chegg.com
An indirect proof is the same as proving by contradiction, which means that the negation of a true statement is also true. Indirect proof is often used when the given geometric statement is NOT true.
Start the proof by assuming the statement IS true. Then reason correctly from the given information until a contradiction of a known postulate, theorem, or given fact is reached. For example,
Given: is not a right angle.
Prove: .
Indirect Proof:
Assume .
If , then by definition of a right angle, is a right angle. However, this contradicts the given statement. Because the assumption leads to a contradiction, the assumption must be false. Therefore, .
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Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material,
please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 27
27 Upper benefit cutoff value Friction number AREADN-TOT(FRICTION) Lower benefit cutoff value 5 10 15 20 Age, years Upper benefit cutoff value AREADN-TOT(RUTTING) Rutting Lower benefit cutoff value
set to zero 5 10 15 20 Age, years International Roughness Upper benefit cutoff value AREADN-TOT(ROUGHNESS) Index (IRI) Lower benefit cutoff value 5 10 15 20 Age, years Figure 4. Total areas
associated with individual condition indicators for the do-nothing options. Step 6: Computation of Areas Associated over time (as shown in Figure 7). The final post-treatment with the Post-Treatment
Case condition area for a given condition indicator is only deter- mined after applying the following area boundary conditions: The sixth step in the benefit calculation process is deter- mining the
important post-treatment condition curve areas that are used to compute benefit. As with the do-nothing · Y-axis limits--in the y direction, the pertinent area is condition area calculations, the
individual condition indicator bounded by the defined upper and lower benefit cutoff areas are computed by taking the integrals of the specific per- values. formance equations that define the
post-treatment performance · X-axis limits--in the x direction, the pertinent area is curves. As mentioned previously, the important benefit-related bounded by an age of zero on the lower end and the
area is the area below condition indicator curves that decrease overall determined post-treatment case expected service over time or above condition indicator curves that increase life (from step 5)
on the upper end.
OCR for page 27
28 Upper benefit cutoff value Friction number Lower benefit cutoff value Do-nothing curve intersects benefit cutoff value at 15 years 5 10 15 20 Age, years Do-nothing curve intersects benefit Upper
benefit cutoff value at 14 years cutoff value This 14-year life is is the shortest of all expected condition indicator lives. The Rutting expected service life of the do-nothing condition is,
therefore, set to 14 years. Lower benefit cutoff value set to zero 5 10 15 20 Age, years Do-nothing curve intersects benefit cutoff value at 17 years International Roughness Upper benefit cutoff
value Index (IRI) Lower benefit cutoff value 5 10 15 20 Age, years Figure 5. Determination of the overall do-nothing condition expected service life. Figure 7 illustrates the total benefit-related
areas associated curve equation and the upper and lower benefit cutoff values. with both decreasing and increasing post-treatment curves. It is important to note that the post-treatment performance
Also illustrated in Figure 7 are the different intersection points equations are expressed in terms of the treatment age rather used to define the x-axis boundary conditions required for the than the
pavement age. For example, for a linear treatment different parts of the area-calculation equations. performance equation such as y = mx + b, the x values are The area calculation details are
different, depending on treatment age values (i.e., time after treatment application) whether the post-treatment performance equation is decreas- rather than pavement age values (i.e., time since
original con- ing or increasing. Equations 4 and 5 are used to compute struction). Therefore, some of the x-axis values associated these post-treatment benefit-related areas associated with with
computing the area after the treatment application age decreasing and increasing equations, respectively. Both of are adjusted to account for this difference in age (e.g., X4 - XA these equations are
functions of the actual post-treatment and X3 - XA in equation 4).
OCR for page 27
29 Upper benefit cutoff value This 20-year life is is the shortest of all Friction number Do-nothing curve expected condition indicator lives. The Post-treatment curve post-treatment condition
analysis period is, therefore, set to 20 years. Lower benefit cutoff value 5 10 15 20 25 Age, years Upper benefit cutoff value Rutting Do-nothing curve Lower benefit cutoff value set to zero 5 10 15
20 25 Age, years Post-treatment curve intersects governing benefit cutoff value at 24 years Upper benefit International Roughness cutoff value Index (IRI) Post-treatment curve Do-nothing curve Lower
benefit cutoff value 5 10 15 20 25 Age, years Figure 6. Determination of the overall post-treatment condition expected service life (analysis period). X2 where: AREA PT( - ) = (EQ DN - LBC) AREAPT(-)
= Computed post-treatment area associated X0 X1 with a decreasing condition indicator rela- tionship (i.e., area from time zero to the end - (EQ DN - UBC) of the post-treatment analysis period). X0 (
X4 - X A ) (Eq. 4) EQDN = Equation defining the do-nothing condition + (EQ PT - LBC) indicator relationship. EQPT = Equation defining the post-treatment condi- 0 ( X3 - X A ) tion indicator
relationship (i.e., treatment per- - (EQ PT - UBC) formance curve). Note that the post-treatment 0 equation is a function of the treatment age
OCR for page 27
30 DECREASING RELATIONSHIP Do-nothing Treatment performance performance curve curve UBC = Upper benefit cutoff value Condition Indicator Decreasing AREA PT() LBC = Lower benefit cutoff value X0 X1
XA X3 X4 Age, years Note: for this case, X2 = XA as XA is less than the projected intersection of the do-nothing curve and the LBC. INCREASING RELATIONSHIP UBC = Upper benefit cutoff value Condition
Indicator AREA PT(+) Treatment Increasing performance curve Do-nothing performance curve LBC = Lower benefit cutoff value X0 X1 XA X3 X4 Age, years Note: for this case, X2 = XA as XA is less than the
projected intersection of the do-nothing curve and the LBC. Figure 7. Determination of total areas associated with decreasing and increasing individual condition indicators. (i.e., time since
application age, expressed in X3 = One of the following: (1) overall pavement years) rather than the overall pavement age. age at which the treatment performance curve UBC = Upper benefit cutoff
value associated with intersects the UBC value, or (2) XA if the ini- the condition indicator. tial treatment condition is less than the UBC, LBC = Lower benefit cutoff value associated with or (3)
X4 if the treatment condition is greater the condition indicator. than the UBC at the determined X4 age. X0 = Lower age boundary (equal to zero). X4 = The overall post-treatment analysis period X1 =
One of the following: (1) pavement age (in (in terms of pavement age). years) at which the do-nothing curve inter- X2 sects the UBC, or (2) zero if the do-nothing condition at pavement age zero is
less than AREA PT( + ) = (UBC - EQ DN ) X0 the UBC, or (3) the pavement age at treat- X1 ment application (XA) if the do-nothing con- dition is greater than the UBC at the treat- - (LBC - EQ DN ) X0
ment application age. (Eq. 5) ( X4 - X A ) X2 = Minimum of (1) the pavement age at treat- ment application and (2) the pavement age + (UBC - EQ PT ) 0 at which the do-nothing curve intersects the (
X3 - X A ) XA = lower benefit cutoff value. Pavement age at treatment application. - (LBC - EQ PT ) 0
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Norridge, IL Prealgebra Tutor
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[R] 5 binary_class models vs one 5-class model
Weiwei Shi helprhelp at gmail.com
Tue Sep 26 23:09:09 CEST 2006
I apologize this question is not very r-related, but believe many
people using R are expertised at or interested to know the answer to
the following question.
I am having a problem in classification. In bioinformatics study, we
always ends with a limited size of samples. While in algorithms, some
specific algorithm cannot handle modeling with more than 2 classes
problem. For the time being, not considering those limitations, I just
have a general question like this:
suppose I have a problem for classification, which involves 5 classes.
I am wondering if there is a general research comparison on which
approach is more accurate: building 5 binary_class models or building
one 5-class model (suppose cost (penalty) is same when accuracy is
An extended or more practical question, in bioinformatics, if you do
not have many samples but you are having such problem, what approach
will you take?
Weiwei Shi, Ph.D
Research Scientist
GeneGO, Inc.
"Did you always know?"
"No, I did not. But I believed..."
---Matrix III
More information about the R-help mailing list
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What Is an Electric Field?
An electric field can be regarded as the sphere of influence of an electrically charged object. Anything that has an electrical charge will affect, and be affected by, other charged bodies. If two
charged objects are placed sufficiently close to one another, each will experience a measureable force acting upon it. The field is theoretically infinite in extent, but its magnitude diminishes with
distance from the source according to the inverse square law. This means that if the distance is doubled, the strength of the field is divided by four, and at three times the distance, the strength
is divided by nine, and so on; the field therefore becomes negligible at large distances.
Since an electric charge can be positive or negative, the electric field is a vector field, which means that it has a direction as well as a magnitude. Two electrically charged objects will
experience a repulsive force if they have the same type of charge and an attractive force if they have different types of charge. The force experienced by a charged object in an electric field can be
calculated as F = Eq, where F is the force in Newtons, E is the electric field in volts per meter (v/m) and q is the charge in Coulombs. This equation can be rearranged to give the strength of the
field, E, in volts per meter: E = F/q. These examples apply to small, point-like, objects; for more complex, or multiple, charged bodies, the calculations are more complicated.
The direction of an electric field is defined as the direction in which the electric force would be felt by an object with a positive charge placed in the field. Thus, the field would point away from
a positive charge and toward a negative charge, since like charges repel and unlike charges attract. In the case of two bodies with the same type of charge, each would experience a force — calculable
by the F = Eq equation — directed away from the other object. Conversely, for two oppositely charged bodies, each would experience a force directed toward the other object.
An electric field line can be drawn with an arrow pointing away from a positive charge and pointing toward a negative charge. Thus, a positively charged object would be depicted with field lines
pointing away from it in all directions, and a negatively charged object with field lines converging upon it. This, however, is just a convention and does not indicate that there is anything physical
pointing in a particular direction.
The concept of an electric field as described above is part of “classical” physics. The classical description works well for everyday applications, but does not explain what is actually happening
when charged objects attract or repel one another. A branch of quantum theory known as quantum electrodynamics (QED), attempts to do this in terms of the exchange of photons, the carriers of the
electromagnetic force.
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flipcode | Daily Game Development | Programming 3D Games and Graphics | Message Center - Moveable cameras
Archive Notice: This thread is old and no longer active. It is here for reference purposes. This thread was created on an older version of the flipcode forums, before the
site closed
in 2005. Please keep that in mind as you view this thread, as many of the topics and opinions may be outdated.
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Random Number Generator Pro
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Random Number Generator to create random number sequences software, random integers and random floating point numbersCreate a list of random numbers using this great random number generator! Some of
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Summary: On the Union of #Round Objects
in Three and Four Dimensions #
Boris Aronov + Alon Efrat # Vladlen Koltun § Micha Sharir ¶
April 1, 2004
A compact body c in R d is #round if for every point p # #c there exists a closed
ball that contains p, is contained in c, and has radius # diam c. We show that, for any
fixed # > 0, the combinatorial complexity of the union of n #round, not necessarily
convex objects in R 3 (resp., in R 4 ) of constant description complexity is O(n 2+# ) (resp.,
O(n 3+# )) for any # > 0, where the constant of proportionality depends on #, #, and
the algebraic complexity of the objects. The bound is almost tight.
1 Introduction
Given a set C of n geometric objects in R d , let U = U(C) :=
# c#C c denote their union, and
let A = A(C) denote the arrangement [39] of the (boundaries of the) objects in C. The
(combinatorial) complexity of U is defined to be the number of faces of A of all dimensions
on the boundary #U of the union. The study of the complexity of the union of objects in
two dimensions has a long and rich history in computational and combinatorial geometry,
starting with the results of Kedem et al. [29] and Edelsbrunner et al. [19], who have shown
# Work on this paper by B.A. and M.S. has been supported by a grant from the U.S.Israeli Binational
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1. What are the essential ingredients of a symmetric cipher?
A symmetric cipher encryption has five ingredients. They are:
Encryption algorithm
Secret key
Cipher text
Decryption algorithm
2. What are the two basic functions used in encryption algorithms?
The two basic functions used in encryption algorithms are
3. How many keys are required for two people to communicate via a cipher?
If both sender and receiver use the same key, the system is referred to as symmetric,
single key, secret key, or conventional encryption. If the sender and receiver each use a
different key, the system is referred to as asymmetric, two-key, or public-key encryption.
4. What is the difference between a block cipher and a stream cipher?
A block cipher processes the input one block of elements at a time, producing an output
block for each input block.
A stream cipher processes the input elements continuously, producing output one element
at a time, as it goes along.
5. What are the two approaches to attacking a cipher?
The two approaches to attack a cipher are:
Brute-force attack
6. What is the difference between an unconditionally secure cipher and a
computationally secure cipher?
An unconditionally secure cipher is a scheme such that if the cipher text generated
by the scheme does not contain enough information to determine uniquely the
corresponding plain text, no matter how much cipher text is available.
A computationally secure scheme is such that the cost of breaking the cipher
exceeds the value of the encrypted information and the time required to break the
cipher exceeds the useful lifetime of the information.
7. Briefly define the Caesar cipher.
The Caesar cipher involves replacing each letter of the alphabet with the letter standing
three places further down the alphabet. For example:
Plain: meet me after the toga party
Cipher: PHHW PH DIWHU WKH WRJD SDUWB
8. Briefly define the monoalphabetic cipher?
A monoalphabetic cipher maps from a plain alphabet to cipher alphabet. Here a single
cipher alphabet is used per message.
9. Briefly define the playfair cipher.
The best-known multiple-letter encryption cipher is the playfair, which treats diagrams in
the plain text as single units and translates these units into cipher text diagrams.
10. What are the two problems with one-time pad?
1.It makes the problem of making large quantities of random keys.
2.It also makes the problem of key distribution and protection.
11. What is a transposition cipher?
Transposition cipher is a cipher, which is achieved by performing some sort of
permutation on the plaintext letters.
12. What is Steganography?
This conceals the existence of the message.
13. Why is it important to study feistel cipher?
This cipher can be used to approximate the simple substitution cipher by utilizing the
concept of a product cipher, which is the performing of two or more basic ciphers in
sequence in such a way that the final result or product is cryptographically stronger than
any of the component ciphers.
14. Why is it not practical to use an arbitrary reversible substitution cipher?
An arbitrary reversible cipher for a large block size is not practical, however, from an
implementation and performance point of view. Here the mapping itself is the key.
15. What is the difference between diffusion and confusion?
In diffusion, the statistical structure of the plain text is dissipated into long-range statistics
of the cipher text. This is achieved by permutation.
In confusion, the relationship between the statistics of the cipher text and the value of the
encryption key is made complex. It is achieved by substitution.
16. Which parameters and design choices determine the actual algorithm of a feistel
Block size
Key size
Number of rounds
Sub key generation algorithm
Round functions
Fast software encryption or decryption
Ease of analysis
17. What is the purpose of the S-boxes in DES?
Each row of a S-box defines a general reversible substitution. It consists of a set of eight
S-boxes, each of which accepts 6 bits as input and produces 4 bits as output.
18. Explain the avalanche effect.
It is that a small change in either the plaintext or the key should produce a significant
change in the cipher text.
A change in one of the bit of the plaintext or one bit of the key should produce a change
in many bits of the cipher text.
19. What is the difference between a mono alphabetic cipher and a poly alphabetic
Mono alphabetic cipher: Here a single cipher alphabet is used.
Poly alphabetic cipher: Here a set of related mono alphabetic substitution rules is used.
20. List the types of cryptanalytic attacks.
Cipher text only
Known plaintext
Chosen plaintext
Chosen cipher text
Chosen text
1. What is the difference between differential and linear cryptanalysis?
In differential cryptanalysis, it breaks the DES in less 255 complexities.
In cryptanalysis, it finds the DES key given 247 plaintexts.
2. Define product cipher.
Product cipher performs two or more basic ciphers in sequence in such a way that the
final result or product is cryptologically stronger than any of the component ciphers.
3. What was the original set of criteria used by NIST to evaluate candidate AES
The original set of criteria used by NIST to evaluate candidate AES cipher was:
Actual Security
Other security factors
Licensing Requirements
Computational Efficiency
Memory Requirements
Algorithm And Implementation Characteristics
Hardware and software suitability
4. What was the final set of criteria used by NIST to evaluate candidate AES
The final set of criteria used by NIST to evaluate candidate AES ciphers was:
General Security
Software Implementations
Restricted-Space Environments
Hardware Implementations
Attacks On Implementations
Encryption vs. Decryption
Key Agility
Other Versatility And Flexibility
Potential for Instruction-Level Parallelism
5. What is power analysis?
Power analysis is the power consumed by the smart card at any particular time during the
cryptographic operation is related to the instruction being executed and to the data being
Eg) Multiplication consumes more power than addition and writing 1s consumes ore
power than writing 0s.
6. What is the purpose of the State array?
A single 128-bit block is depicted as a square matrix of bytes. This block is copied into
the State array, which is modified at each stage of encryption or decryption. After the
final stage, State is copied to an output matrix.
7. How is the S-box constructed?
The S-box is constructed in the following fashion:
Initialize the S-box with the byte values in ascending sequence row by row. The
first row contains {00}, {01}, {02}, ……….., {0F}; the second row contains
{10},{11},etc; and so on. Thus, the value of the byte at row x, column y is {x y}.
Map each byte in the S-box to its multiplicative inverse in the finite field GF (28);
the value {00} is mapped to itself.
Consider that each byte in the S-box consists of 8 bits labeled
(b7,b6,b5,b4,b3,b2,b1,b0).Apply the following transformation to each bit of each byte in
the S-box.
8. Briefly describe Sub Bytes.
Sub byte uses an S-box to perform a byte-by-byte substitution of the block. The left most
4 bits of the byte are used as row value and the rightmost 4 bits are used as a column
value. These row and column values serve as indexes into the S-box to select a unique 8-
bit value.
9. Briefly describe Shift Rows.
In shift row, a row shift moves an individual byte from one column to another, which is a
linear distance of a multiple of 4 bytes. In Forward Shift Row, each row perform circular
left shift. Second Row a 1-byte circular left shift is performed. Third Row a 2-byte
circular left shift is performed. For the Fourth Row a 3-byte circular left shift is
performed. In Inverse Shift Row, each row perform circular right shift.
10. How many bytes in State are affected by Shift Rows?
Totally 6-bytes in state are affected by Shift Rows.
11. Briefly describe Mix Columns.
Mix Column is substitution that makes use of arithmetic over GF(28).Mix Column
operates on each column individually. Each byte of a column is mapped into a new value
that is a function of all four bytes in the column. The Mix Column Transformation
combined with the shift row transformation ensures that after a few rounds, all output bits
depend on all input bits.
12. Briefly describe Add Round Key.
In Add Round Key, the 128 bits of State are bit wise XORed with the 128 bits of the
round key. The operation is viewed as a column wise operation between the 4 bytes of a
State column and one word of the round key; it can also be viewed as a byte-level
operation. The Add Round Key transformation is as simple as possible and affects every
bit of State.
13. Briefly describe the Key Expansion Algorithm.
The AES key expansion algorithm takes as input a 4-word(16-byte) key and produces a
linear array of 44 words(156 bytes). This is sufficient to provide a 4-word round key for
the initial Add Round Key stage and each of the 10 rounds of the cipher.
14. What is the difference between Sub Bytes and Sub Word?
Sub Bytes:
Sub Bytes uses an S-box to perform a byte-by-byte substitution of the block.
Sub Word:
Sub Word performs a byte substitution on each byte of its input word,using the Sbox.
15. What is the difference between Shift Rows and Rot Word?
Shift Rows:
Shift Row is simple permutation. It shifts the rows circularly left or right.
Rot Word:Rot word performs a one-byte circular left shift on a word. This means that an
input word [b0,b1,b2,b3] is transformed into [b1,b2,b3,b0].
16. Why do some block cipher modes of operation only use encryption while others
use both encryption and decryption?
Some block cipher modes of operation only use encryption because the input is set to
some initialization vector and the leftmost bits of the output of the encryption function
are XORed with the first segment of plain text p1 to produce the first unit of cipher text
C1 and it is transmitted. While in decryption, the cipher text is XORed with the output of
the encryption function to produce the plain text.
17. What is triple encryption?
Tuchman proposed a triple encryption method that uses only two keys [TUCH79].
The function follows an encrypt – decrypt – encrypt (EDE) sequence.
There is no cryptographic significance to the use of decryption for the second
stage. Its only advantage is that it allows users of 3DES to decrypt data encrypted by
users of the older single DES:
C=Ek1[Dk2[Ek1[P]]] = Ek1[P]
18. What is a meet-in-the-middle attack?
Meet-in-the-middle attack, was first described in [DIFF77]. It is based on the
observation that, if we have
Given a known pair, (P,C), the attack proceeds as follows. First, encrypt P for all
256 possible values of K1. Store these results in a table and then sort the table by the
values of X. Next, decrypt C using all 256 possible values of K2. As each decryption is
produced, check the result against the table for a match. If a match occurs, then test the
two resulting keys against a new known plaintext-ciphertext pair. If the two keys produce
the correct ciphertext, accept them as the correct keys.
19. How many keys are used in triple encryption?
Tuchman proposed a triple encryption method that uses only two keys [TUCH79].
20. What is the key size for Blowfish?
Blowfish makes use of a key that ranges from 32 bits to 448 bits (one to fourteen
32-bit words). That key is used to generate 18 32-bit subkeys and four 8*32 S-boxes
containing a total of 1024 32-bit entries. The total is 1042 32-bit values, or 4168 bytes.
1. What primitive operations are used in Blowfish?
Blowfish uses two primitive operations:
• Addition: Addition of words, denoted by +, is performed modulo 232.
• Bit wise exclusive-OR: This operation is denoted by Å.
2. What common mathematical constants are used in RC5?
W :Word size in bits. RC5 encrypts 2-word
r: Number of rounds. 0,1,….,255
B Number of 8-bit bytes (octets) in the secret
key K.
3. What primitive operations are used in RC5?
RC5 uses three primitive operations (and their inverse):
• Addition: Addition of words, denoted by +, is performed modulo 2w. The inverse
operation, denoted by -, is subtraction modulo 2w.
• Bitwise exclusive-OR: This operation is denoted by “Å”.
• Left cicular rotation: The cyclic rotation of word x left by y bits is denoted by
4. List important design considerations for a stream cipher.
The encyption sequence should have a large period. The keystream should
approximate the properties of a true random number stream as close as possible. The
output of the pseudorandom number generator is conditioned on the value of the input
5. Why is it not desirable to reuse a stream cipher key?
If two plaintexts are encrypted with the same key using a stream cipher then
cryptanalysis is often quite simple. If the two ciphertext streams are XORed together the
result is the XOR of the original plaintexts. So it is not desirable to reuse a stream cipher
6. What primitive operation is used in RC4?
The primitive operation used in RC4 is bit wise Exclusive-OR (XOR) operation.
7. For a user workstations in a typical business environment, list potential locations
for confidentiality attacks.
• LANs in the same building that are interconnected with bridges and routers.
• The wiring closet itself is vulnerable.
• Twisted pair and coaxial cable can be attacked using either invasive taps or
inductive devices that monitor electromagnetic emanation.
• In addition to the potential vulnerability of the various communications links, the
various processors along the path are themselves subject to attack.
8. What is the difference between link and end-to-end encryption?
Link Encryption End-to-end Encryption
Applied by sending host
Transparent to user
Host maintains encryption facility
One facility for all users
Can be done in hardware
All or no messages encrypted
Applied by sending process
User applies encryption
User must determine algorithm
User selects encryption scheme
Software implementation
User chooses to encrypt,
or not, for each message
9. What types of information might be derived from a traffic analysis attack?
The following types of information can be derived from traffic analysis attack:
• Identities of partners
• How frequently the partners are communicating
• Message pattern, message length, or quantity of messages that suggest important
information is being exchanged
• The events that correlate with special conversations between particular partners.
10. What is traffic padding and what is its purpose?
Traffic padding produces ciphertext output continuously, even in the absence of
plaintext. A continuous random data stream is generated. When plaintext is available, it is
encrypted and transmitted. When input plaintext is not present, random data are
encrypted and transmitted.
11. List ways in which secret keys can be distributed to two communicating parties.
• A can select a key and physically deliver it to B.
• A third party can select the key and physically deliver it o A and B
• If A and B have previously and recently used a key, one party can transmit the
new key to the other, encrypted using the old key
• If A and B each has an encrypted connection to a third party C, C can deliver a
key on the encrypted links to A and B
12. What is the difference between a session key and a master key?
Session key
Master key
Communication between end systems is
encrypted using temporary key, often
referred to as a session key.
Session keys are transmitted in encrypted
form, using master key that is shared by the
keys distribution center and an end system.
The session key is used for the duration of
a logical connection, such as a frame relay
connection or transport connection, and
then discarded.
For each end system or user, there is a
unique master key that it shares with the
key distribution center. These master keys
must be distributed in some fashion.
13. What is nonce?
Consider A issues a request to the KDC for a session key to protect a logical
connection to B. The message includes the identity of A and B and a unique identifier,
N1, for this transaction, which we refer to as nonce. The nonce may be a timestamp, a
counter, or a random number.
14. What is key distribution center?
A key distribution center is responsible for distributing keys to pairs of users such
as hosts, processes, applications. Each user must share a unique key with the key
distribution center for purposes of key distribution.
15. What is the difference between statistical randomness and unpredictability?
In applications such as reciprocal authentication and session key generation the
requirement is not so much that the sequence of numbers be statistically random but that
the successive numbers of the sequence are unpredictable. With true random sequences
each number is statistically independent of other numbers in the sequence and therefore
16. What is the difference between Rijndael and AES?
AES was developed by NIST .AES is a symmetric block cipher that is intended to replace
DES.NIST selected rijndael as the proposed AES algorithm. The two researchers who
developed and submitted Rijndael for the AES are the both cryptographers from
17. Why is the middle portion of 3DES a decryption rather than an encryption?
Decryption requires that the keys be applied in reverse order:
This results in a dramatic increase in cryptographic strength.The use of DES results in a
mapping that is not equivalent to a single DES encryption.
18. What is the difference between the AES decryption algorithm and the equivalent
inverse cipher?
In AES decryption, we use inverse shift rows inverse sub bytes, add round key, inverse
mix columns. But in equivalent inverse cipher, we interchange inverse shift rows and
inverse sub bytes.
Unit IV
1. Differentiate public key encryption and conventional encryption.
Conventional Encryption Public key Encryption
1. Same algorithm with 1.Same algorithm is used for
same key used for encryption & decryption with
encryption and decryption. a pair of keys.
2. Sender & receiver must 2.Sender & receiver have one of
share the algorithm and key. the matched pair keys.
3. Key must be kept secret. 3. Any one of the key must be
kept secret.
2. Specify the application of public key cryptography.
Digital signature.
Key exchange.
3. Determine the gcd(24140,16762) using Euclid’s algorithm.
We know, gcd(a,b)=gcd(b,a mod b)
gcd(24140,16762) = 34.
4.Perform encryption and decryption using RSA alg. For the following.
P=7; q=11; e=17; M=8.
ö(n)=(p-1) (q-1)
=6*10 = 60
d =27
C= Me mod n
C = 817 mod 77
= 57
M = Cd mod n
= 5727 mod 77
= 8
5. User A & B exchange the key using Diffie Hellman alg. Assume
á=5 q=11 XA=2 XB=3. Find YA, YB, K.
YA = áXA mod q
=52 mod 11
= 3
YB = áXB mod q
= 53 mod 11
= 4
KA = YBXA mod q
= 42 mod 11
= 5
KB = YAXB mod q
= 33 mod 11
= 5
6.What is message authentication?
It is a procedure that verifies whether the received message comes from assigned
source has not been altered.
7. Define the classes of message authentication function.
• Message encryption: The entire cipher text would be used for authentication.
• Message Authentication Code: It is a function of message and secret key produce
a fixed length value.
• Hash function: Some function that map a message of any length to fixed length
which serves as authentication.
8. What you meant by MAC?
MAC is Message Authentication Code. It is a function of message and secret key
which produce a fixed length value called as MAC.
9. Specify the techniques for distribution of public key.
Public announcement.
Publicly available directory.
Public key authority.
Public key certificate.
10. Specify the requirements for message authentication.
i. Disclosure.
ii. Traffic analysis.
iii. Masquerade.
iv. Content Modification.
v. Sequence Modification.
vi. Timing modification.
vii. Repudiation.
11. Differentiate internal and external error control.
Internal error control:
In internal error control, an error detecting code also known as
frame check sequence or checksum.
External error control:
In external error control, error detecting codes are appended after
12. What you meant by hash function?
Hash function accept a variable size message M as input and produces a fixed size
hash code H(M) called as message digest as output. It is the variation on the message
authentication code.
13. Differentiate MAC and Hash function?
MAC: In Message Authentication Code, the secret key shared by sender
and receiver. The MAC is appended to the message at the source
at a time which the message is assumed or known to be correct.
Hash Function: The hash value is appended to the message at the source
at time when the message is assumed or known to be correct. The
hash function itself not considered to be secret.
14.Define Kerberos.
Kerberos is an authentication service developed as part of project Athena at
MIT.The problem that Kerberos address is, assume an open distributed environment in
which users at work stations wish to access services on servers distributed throughout the
15.In the content of Kerberos, what is realm?
A full service Kerberos environment consisting of a Kerberos server, a no. of
clients, no.of application server requires the following:
The Kerberos server must have user ID and hashed password of all participating
users in its database.
The Kerberos server must share a secret key with each server. Such an
environment is referred to as “Realm”.
17.Assume the client C wants to communicate server S using Kerberos procedure.
How can it be achieved?
a) C ® AS: [IDC|| PC || IDV]
b) AS ® C: Ticket
c) C ® V: [IDC || ADC || IDV]
Ticket = EKV [IDC ||ADC || IDV]
18. Any three hash algorithm.
• MD5 (Message Digest version 5) algorithm.
• SHA_1 (Secure Hash Algorithm).
• RIPEMD_160 algorithm.
19. Specify the four categories of security threats
• Interruption
• Interception
• Modification
• Fabrication
20. Differentiate symmetric and Asymmetric Encryption
Symmetric Encryption
Sender and receiver use the same key.
Sender and receiver uses different key.
Unit V
1. What are the services provided by PGP services
• Digital signature
• Message encryption
• Compression
• E-mail compatibility
• Segmentation
2. Explain the reasons for using PGP?
a) It is available free worldwide in versions that run on a variety of platforms,
including DOS/windows, UNIX, Macintosh and many more.
b) It is based on algorithms that have survived extensive public review and are
considered extremely secure.
E.g.) RSA, DSS and Diffie-Hellman for public key encryption, CAST-128,
IDEA, 3DES for conventional encryption, SHA-1for hash coding.
c) It has a wide range of applicability from corporations that wish to select and
enforce a standardized scheme for encrypting files and communication.
d) It was not developed by nor is it controlled by any governmental or standards
3. Why E-mail compatibility function in PGP needed?
Electronic mail systems only permit the use of blocks consisting of ASCII text.
To accommodate this restriction PGP provides the service converting the row 8-
bit binary stream to a stream of printable ASCII characters. The scheme used for
this purpose is Radix-64 conversion.
4. Name any cryptographic keys used in PGP?
a) One-time session conventional keys.
b) Public keys.
c) Private keys.
d) Pass phrase based conventional keys.
5. Define key Identifier?
PGP assigns a key ID to each public key that is very high probability unique
with a user ID. It is also required for the PGP digital signature. The key ID
associated with each public key consists of its least significant 64bits.
6. List the limitations of SMTP/RFC 822?
a) SMTP cannot transmit executable files or binary objects.
b) It cannot transmit text data containing national language characters.
c) SMTP servers may reject mail message over certain size.
d) SMTP gateways cause problems while transmitting ASCII and EBCDIC.
e) SMTP gateways to X.400 E-mail network cannot handle non textual data
included in X.400 messages.
7. Define S/MIME?
Secure/Multipurpose Internet Mail Extension(S/MIME) is a security enhancement
to the MIME Internet E-mail format standard, based on technology from RSA Data
8. What are the elements of MIME?
Five new message header fields are defined which may be included in an
RFC 822 header.
A number of content formats are defined.
Transfer encodings are defined that enable the conversion of any content
format into a form that is protected from alteration by the mail system.
9. What are the headers fields define in MME?
• MIME version.
• Content type.
• Content transfer encoding.
• Content id.
• Content description.
10. What is MIME content type &explain?
It is used to declare general type of data. Subtype define particular format
for that type of the data. It has 7 content type & 15 subtypes. They are,
1. Text type
• Plain text.
• Enriched.
2. Multipart type
• Multipart/mixed.
• Multipart/parallel.
• Multipart/alternative.
• Multipart/digest.
3. Message type
• Message/RFC822.
• Message/partial.
• Message/external.
4. Image type
• JPEG.
• CIF.
5. Video type.
6. Audio type.
7. Application type
• Post script.
• Octet stream.
11.What are the key algorithms used in S/MIME?
• Digital signature standards.
• Diffi Hellman.
• RSA algorithm.
12. Give the steps for preparing envelope data MIME?
• Generate Ks.
• Encrypt Ks using recipient’s public key.
• RSA algorithm used for encryption.
• Prepare the ‘recipient info block’.
• Encrypt the message using Ks.
13. What you mean by versioned certificate?
Mostly used issue X.509 certificate with the product name” versioned
digital id”. Each digital id contains owner’s public key, owner’s name and serial number
of the digital id.
14. What are the function areas of IP security?
• Authentication
• Confidentiality
• Key management.
15. Give the application of IP security?
• Provide secure communication across private & public LAN.
• Secure remote access over the Internet.
• Secure communication to other organization.
16. Give the benefits of IP security?
• Provide security when IP security implement in router or firewall.
• IP security is below the transport layer is transparent to the
• IP security transparent to end-user.
• IP security can provide security for individual user.
17. What are the protocols used to provide IP security?
• Authentication header (AH) protocol.
• Encapsulating Security Payload(ESP).
18. Specify the IP security services?
• Access control.
• Connectionless interpretty.
• Data origin authentication
• Rejection of replayed packet.
• Confidentiality.
• Limited traffic for Confidentiality.
19. What do you mean by Security Association? Specify the parameters
that identifies the Security Association?
• An association is a one-way relationship between a sender and receiver that
affords security services to the traffic carried on.
• A key concept that appears in both the authentication and confidentiality
mechanism for ip is the security association (SA).
A security Association is uniquely identified by 3 parameters:
• Security Parameter Index (SPI).
• IP Destination Address.
• Security Protocol Identifier.
20. What does you mean by Reply Attack?
• A replay attack is one in which an attacker obtains a copy of an
authenticated packet and later transmits it to the intended destination.
• Each time a packet is send the sequence number is incremented .
21. Explain man in the middle attack?
If A and B exchange message, means E intercept the message and receive
the B’s public key and b’s userId,E sends its own message with its own public key and
b’s userID based on the private key and Y.B compute the secret key and A compute k2
based on private key of A and Y
22. Steps involved in SS L required protocol?
1. SSL record protocol takes application data as input and fragments it.
2. Apply lossless Compression algorithm.
3. Compute MAC for compressed data.
4. MAC and compression message is encrypted using conventional alg.
23. What is mean by SET? What are the features of SET?
Secure Electronic Transaction (SET) is an open encryption and security
specification designed to protect credit card transaction on the internet.
Features are:
1. Confidentiality of information
2. Integrity of data
3. Cardholder account authentication
4. Merchant authentication
24. What are the steps involved in SET Transaction?
1. The customer opens an account
2. The customer receives a certificate
3. Merchants have their own certificate
4. The customer places an order.
5. The merchant is verified.
6. The order and payment are sent.
7. The merchant requests payment authorization.
8. The merchant confirm the order.
9. The merchant provides the goods or services.
10. The merchant requests payment.
25. What is dual signature? What it is purpose?
The purpose of the dual signature is to link two messages that intended for
two different recipients.
To avoid misplacement of orders
26. List the 3 classes of intruder?
Classes of Intruders
1) Masquerader
2) Misfeasor
3) Clandestine user
27. Define virus. Specify the types of viruses?
A virus is a program that can infect other program by modifying them the
modification includes a copy of the virus program, which can then go on to infect other
1) Parasitic virus
2) Memory-resident virus
3) Boot sector virus
4) Stealth virus
5) Polymorphic virus
28. What is application level gateway?
An application level gateway also called a proxy server; act as a relay of
application-level traffic. The user contacts the gateway using a TCP\IP application,
such as Telnet or FTP, and the gateway asks the user for the name of the remote host to
be accessed.
29. List the design goals of firewalls?
1. All traffic from inside to outside, and vise versa, must pass through
the firewall.
2. Only authorized traffic, as defined by the local security policy, will
be allowed to pass.
3. The firewall itself is immune to penetration.
UNIT 1
1.Explain the OSI Architecture.
• Notes
• Diagram
• Illustration
2.Explain Classical Encryption Techniques.
Symmetric Ciphers
Caesar Cipher
Mono alphabetic
Poly alphabetic
3.Explain DES Algorithm.
• Notes
• Diagram
• Illustration
• Algorithm
4.Explain AES.
• Notes
• Diagram
• Illustration
• Algorithm
5.Describe about Traffic Confidentiality.
Cryptanalytic methods in traffic analysis and ciphers achieving
UNIT 2
1.Explain about the various Key management techniques.
public announcement
Publicly available directory
public-key authority
public-key certificates
2.Describe Diffie-Hellman Key Exchange.
3.Explain RSA algorithm.
4.Describe Public Key Cryptography.
Two Keys
Private key
public key
5.Explain Elliptic Curve Architecture.
• Architecture
• Algebraic description
• Geometric description
UNIT 3
1.Explain Authentication Functions.
• Message Encryption
• MAC
• Hash function
2.Describe HMAC algorithm.
o Algorithm
o Diagram
o Notes
3.Describe RIPEMD-160.
4.Explain Hash Functions.
Security features
Algorithms used
5.Explain Digital Signature Standard.
• Algorithm
• Analysis
• Diagram
UNIT 4
1.Explain Kerberos.
2.Explain X.509 Authentication Services.
o Algorithm
o Explanation
o Diagram
3.Describe Electronic Mail Security.
4.Explain about PGP services.
5.Describe S/MIME.
UNIT 5
1.Explain Intrusion Detection.
Audit records
Statistical Anomaly Detection
Rule Based Intrusion Detection
Base-Rate Valley
Honey pot
Exchange format
2.State and Explain Password Management.
o Password Protection
o Password Selection Strategies
3.Explain the Firewall Design Principles.
Firewall characteristics
Firewall Configuration
4.Describe about Trusted Systems.
Data Access Control
Trojan Horse Defense
5.Name some Viruses and Explain it.
Malicious Programs
Macro viruses
E-mail Viruses
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Introduction to orthogonal polynomials Michael Anshelevich
Summary: Introduction to orthogonal polynomials
Michael Anshelevich
November 6, 2003
µ = probability measure on R with finite moments
mn(µ) =
xn dµ(x) < .
Induces a functional on polynomials C[x],
[P(x)] =
P(x) dµ(x).
On the polynomials C[x], define the sesquilinear inner
xn, xk
= µ xn+k = mn+k(µ).
The set {xn}
n=0 is a basis for C[x]. Gram-Schmidt with
respect to the inner product ·, · µ, get a family of polyno-
Source: Anshelevich, Michael - Department of Mathematics, Texas A&M University
Collections: Mathematics
Summary: Introduction to orthogonal polynomials Michael Anshelevich November 6, 2003 µ = probability measure on R with finite moments mn(µ) = R xn dµ(x) < . Induces a functional on polynomials C[x],
[P(x)] = R P(x) dµ(x). On the polynomials C[x], define the sesquilinear inner product xn, xk µ = µ xn+k = mn+k(µ). The set {xn} n=0 is a basis for C[x]. Gram-Schmidt with respect to the inner product
·, · µ, get a family of polyno- mials
Source: Anshelevich, Michael - Department of Mathematics, Texas A&M University
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Engine Pressure
This is a complex question, and the pressure doesn't stay constant at any point during the combustion cycle.
Mean Effective Pressure is calculated by:
MEP = 1000 * (P * Nr) / (V * N)
Where MEP is in kPa, P is power in kW, Nr is the number of crank revolutions per power stroke, V the displaced volume per cylinder, and N the number of cylinders.
Note that the peak cylinder pressure will be much higher than this, but the calculation of this is far more involved.
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alculation -
Mr Srinivas
Mr Madhu has clearly and correctly answered to your qestion. The point that Mr Madhu has tried to clear is that if any person is having salary not more than 10,000 then he will get bonus and to
calculate bonus we will take Rs. 3,500 only as basic + DA no matter his salary is 6000 or 10000.
Rs. 3500/- is for calculation of bonus and its not a limit for bonus value after calculation. I hope I have put the same thing into more simpler manner :)
Also minimum and maximum %age for bonus calculation i.e. 8.33% and 20% respectively leads to range of bonus from Rs. 3499/- to Rs. 8400/-
Also if person is getting basic + DA less than 3500/- (which is quite unlikely) then bonus will be calculated on that lesser amount only.
Always Wishing Good for You...
Silky Bhardwaj
Dear Madhu,
Thank you very much. I still have a doubt about the calculation.
Say an employee Basic is Rs 9000/- who joined 27.02.2006. For calculation of bonus for 2006-07 (ie from April-06 to March 07) i have taken Rs 3500*12*8.33%=Rs3499/-
Is that the method correct?
or Rs 9000*12*8.33%=Rs8997/- which exceeds Rs3500/- .so Rs 3500 should be taken as suggested by one of my collugue.
Which one is correct?
In my view the limit Rs 3500/- is not for reached after calculation, but for the calculation purpose , in no case the amount of basic should not cross Rs 3500/- for each month.
Please suggest me the correct method.
Thanking you in advance...
For Bonus Rs 3500 means basic + DA for a month.
An employee who is drawing a monthly salary of not more than Rs 10000 will be eligible for bonus. But those whose salary exceeds Rs 3500 will be paid bonus as if his salary is Rs 3500. That means one
who gets salary of Rs 7500 or Rs 9000 will be paid bonus calculated on the basis of a salary of Rs 3500 per month only.
The rate of bonus is declared by the employer as per available profit with a minimum of 8.33% and a maximum of 20%.
srini.kaduru Started The Discussion:
I need a clarification on the below clause of Payment of Bonus Act,1965(amendment)
ii) Amendment to section 12 of the Payment of Bonus Act, 1965 to raise the ceiling for calculation purpose from the salary or wage of Rs. 2500/- per month to Rs. 3500/- per month;
Does the word "ceiling for calculation purpose " replicate the meaning that if the Basic+DA of an employee exceed Rs 3500/- for calculation purpose we need to take 3500/-for each month and sum the
same for 12 months and apply the %.
Or any other method for calculating the Bonus.
Please help me out at the earliest.
Thanx and Regards,
kindly help to me,I left from my previous company one year past,i had applied for pf withdrawals 2 times but there is no response from them.I checked out with them,they told to me didn't recieve any
forms.presently am in bangalore,prevoius company is mumbai based,i had worked at hyderabad.kindly help wat should i do now because i need amount.
with regards
Dear Madhu, Greetings of the day, please clarifay me wheather we should deduct PF and ESI on bonus amount? thanks and regards, Rajendra Kumar Sahoo,
Bonus amount paid will not come under the definition of salary/ wages and therefore, PF or ESI should not be deducted from the amount of bonus paid. Regards, Madhu.T.K
If someone's Gross Salary is 6000 and basic is 3500 and earned salary of 25 days in March'09. then at wt amount will taken for calculation for Bonus.
Kindly send me revert asap
This will be great help for me.
Thanks & Rgds
Dear Sunny,
Ceiling of monthly salary for calculation of Bonus payment is Rs. 3500. i.e Rs. 3500 is for one month. If he has not full attendace in a month, pro-rata basis calculation is to be applied. Accordigly
for 25 days in March will be equal to 3500x25/31 = 2822.58
Also if the salary for one year is 3500x12 = Rs. 42000, the minimum bonus will be Rs. 3500. (not 3,499). Because 8.33% is the percentage form of 1/12. Accordigly 20% bonus will be 2.4 times of
average of one month admissible salary.
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The Amelia Sequence
reply to post by charliesmudgedfudge
Interesting post, and sounds like you share a bit in common with the creator Tim Kring, as I discovered his son is autistic. Just watching this series now while I'm working in Australia, and saw the
Amelia sequence episode 6 this morning, thoroughly enjoying it, although it seems I'm about 7 months behind you guys on this forum.
I like how you've arranged the numbers there.
I presume you mean 8 being representative of the symbol for infinity?
I noticed though with your 1+6+1 = 8 that you are showing the first 4 numbers of the number Phi, the golden ratio, which is expressed as a rational number as (1 + squareroot(5))/2)
Back in about 1998, I was playing with a calculator, while I was working in Kuwait. I had read Peter Lemesurier's book, "The Great Pyramid Decoded" when I was a student in Scotland. His observation
that the tan of the angle of the sides of the great pyramid of Giza was close to equal to the root(phi). i.e. tan (51.85 deg) = root(phi).
I was wondering what this number could be used for, except for symmetry, since it is known as the golden ratio. Anyway, while I was playing with the calculator and the number phi (I was using 1.618),
I tried squaring it. I noticed that the square of phi looked like 1+ phi. So I ended up writing an equation phi^2 = 1 + phi. Then re-writing it as a quadratic equation phi^2 - phi - 1 = 0. Using the
formula to solve quadratic equations gave me two values for phi. (1+ root(5))/2 or (1-root(5))/2. So phi is approx 1.618 or -0.618. Unfortunately, this is not a new discovery I found out later, once
I'd read some maths book on phi, but I did discover this equation on my own without reading a book.
These numbers are used in financial charting, I later discovered, and funnily enough if you square -0.618 you get another one of the numbers used, 0.381966.
I think this episode is a play on the Fibonacci sequence of numbers. Worth a look on Wikipedia.
What many people report noticing is certain numbers appearing frequently in their daily lives. I was in Adelaide about 3 weeks ago, and kept seeing the number 22 (time 22:22)and 23 along with 11. In
Numerology, 11, is the number of spirituality. 22 is called the Master number, and 23 represents a person balanced in mind and spirituality from what I can remember from a reading given by the late
Edgar Cayce.
I liked the episode with the guy with 22:22 tattooed on his neck. 22 is referred to as a master number.
Daniel Goldman wrote a book on numerology called the life you were born to lead. I discovered I have the birth number of 38, which reduces to 11, which alleges an influence towards spiritual things.
Tim Kring I discovered five minutes ago when I thought about it, also is a 38. So perhaps an explanation for his creations being biased towards spiritual-like things such as psychic powers,
numerology, etc.
Great show though, and off to enjoy episode #7. Hmmmmm, 1 divided by 7 has a repeating sequence of numbers 0.142857142857...142857 7 chakras, 7 endocrine glands in the body, 7 days of the week....
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The Fibonacci Numbers: Part Two - In NatureThe Fibonacci Numbers: Part Two – in Nature
The Fibonacci Numbers: Part Two – in Nature
The Fibonacci sequence of numbers, an infinite series, was defined mathematically, in Part One, to be
Fn = Fn˗₁ + Fn˗₂.
Their sequence begins, 0, 1, 1, 2, 3, 5, 8, 13….
The question arises, how do these numbers manifest themselves in nature, around us? What is their significance, their métier?
Fibonacci on Land
There are numerous manifestations of the Fibonacci sequence in nature. Quite a few are listed at the University of Surrey mathematics web-site. The site explains Fibonacci’s problem of the
multiplying rabbits that lead, in the year 1202, to his namesake sequence.
Although the logic behind that problem contains a number of flaws, it did introduce Fibonacci and his sequence to us. A real occurrence of Fibonacci numbers in nature can be seen in the bottom of a
pine cone.
Fibonacci in the Sea
Perhaps most inspiring is the occurrence of the Fibonacci numbers is seen in a sea-dwelling creature. The Fibonacci sequence, if visualized as a sequence of squares drawn in clockwise fashion (as in
these images) can be seen to form a spiral. Compare that spiral with the cutaway shell of the nautilus.
The concluding, Part Three, of this series of articles will discuss the significance of these occurrences of Fibonacci’s numbers in nature.
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Zariski geometries: from classical to quantum
Seminar Room 1, Newton Institute
The notion of a Zariski geometry has been developed in Model Theory in the search for ``logically perfect'' structures. Zariski geometries are abstract topological structures with a dimension,
satisfying certain assumptions. Algebraic varieties over algebraically closed fields and compact complex manifolds are Zariski geometries and for some time it was thought that all Zariski geometries
are of this kind. In fact the classical examples are only ``limit'' cases of the general pattern, where one necessarily comes to a coordinatisation by non-commutative algebras. Conversely, we show in
particular that any quantum algebra at roots of unity of certain type coordinatises a geometric object which is a Zariski geometry. This includes quantum groups.
Related Links
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-resveratrol before (red) and after (green) UVR with 5 MED.
Figure 6. The antioxidant activity of trans-resveratrol before (red) and after (green) UVR with 5 MED.
The steeper the curve, the greater the antioxidant activity. As the graph shows, UVR has reduced the antioxidant activity of the trans-resveratrol solution. Also shown are the trend lines
(logarithmic regression), their equations, and the “R squared,” a statistical measure of how well the trend line matches the actual data. A perfect fit is R2 = 1. In both cases the trend lines are
very close fits to the data.
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Solving an equation by substitution
June 15th 2007, 05:59 AM #1
Jun 2007
Solving an equation by substitution
Hello, I've come across a problem in my textbook that really baffles me.
Task: to use substitution $u=x+1/x$ to solve the equation for x:
It was easy to make the quadratic equation,
$u1= (-3+sqrt(17))/2$
but solving for x was a bit "ugly" (quoted because math is NOT ugly
I've spent quite a time over this, any pointers?
Hello, I've come across a problem in my textbook that really baffles me.
Task: to use substitution $u=x+1/x$ to solve the equation for x:
It was easy to make the quadratic equation,
$u1= (-3+sqrt(17))/2$
but solving for x was a bit "ugly" (quoted because math is NOT ugly
I've spent quite a time over this, any pointers?
Multiply through by $x$ and rearrange, to get quadratics.
Thanks for the reply CaptainBlack.
That's exactly what I did, but I ended up with a complex solution (i.e. "wrong") for $u1$ and square roots of square roots of whatever (rightfully wrong) for $u2$. Could you be a bit more
The solution is supposed to be $-2-sqrt(3),-2+sqrt(3)$
Thanks for the reply CaptainBlack.
That's exactly what I did, but I ended up with a complex solution (i.e. "wrong") for $u1$ and square roots of square roots of whatever (rightfully wrong) for $u2$. Could you be a bit more
The solution is supposed to be $-2-sqrt(3),-2+sqrt(3)$
I don't like the look of your roots to that quadratic in $u$.
I think that:
$<br /> x^2+3x-2+3/x+1/x^2=0<br />$
$<br /> u^2+3u-4=0<br />$
under the proposed substitution, which has roots $(u=-4) \vee (u=1)$
CaptainBlack, you're absolutely right. I always forget the rules for squaring polynomials
Thank you very much.
Thanks for the reply CaptainBlack.
That's exactly what I did, but I ended up with a complex solution (i.e. "wrong") for $u1$ and square roots of square roots of whatever (rightfully wrong) for $u2$. Could you be a bit more
The solution is supposed to be $-2-sqrt(3),-2+sqrt(3)$
First off: There are four solutions to this equation. Two complex and two real. You have listed only the two real solutions.
Second: Your quadratic in u is incorrect.
$u = x + \frac{1}{x}$
$u^2 = x^2 + 2 \frac{1}{x^2}$
So we wish to have
$au^2 + bu + c = ax^2 + bx + c + \frac{b}{x} + \frac{b^2}{x^2} = x^2 + 3x - 2 + \frac{3}{x} + \frac{1}{x^2}$
a = 1
b = 3
c = -4
$u^2 + 3u - 4 = 0$
$(u + 4)(u - 1) = 0$
So u = -4 or u = 1.
$x + \frac{1}{x} = -4$
$x^2 + 4x + 1 = 0$ ==> $x = -2 \pm \sqrt{3}$
$x + \frac{1}{x} = 1$
$x^2 - x + 1 = 0$ ==> $x = \frac{1 \pm \sqrt{3}}{2}$
(Rats! FOILed again.
Last edited by topsquark; June 15th 2007 at 06:53 AM. Reason: I was beaten!!
June 15th 2007, 06:08 AM #2
Grand Panjandrum
Nov 2005
June 15th 2007, 06:32 AM #3
Jun 2007
June 15th 2007, 06:45 AM #4
Grand Panjandrum
Nov 2005
June 15th 2007, 06:50 AM #5
Jun 2007
June 15th 2007, 06:52 AM #6
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Weekly Problem 34 - 2009
Copyright © University of Cambridge. All rights reserved.
'Weekly Problem 34 - 2009' printed from http://nrich.maths.org/
I am standing behind five pupils who are signalling a five-digit number to someone on the opposite side of the playground. From where I am standing the number looks like 23456. What number is
actually being signalled?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.
View the previous week's solutionView the current weekly problem
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Mastering Physics Solutions: Using X-ray Diffraction
Using X-ray Diffraction
Part A = 0.2130nm
Part B = 69.9°
Part C = No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.
Solution Below:
When an x-ray beam is scattered off the planes of a crystal, the scattered beam creates an interference pattern. This phenomenon is called Bragg scattering. For an observer to measure an interference
maximum, two conditions have to be satisfied:
The angle of incidence has to be equal to the angle of reflection.
The difference in the beam’s path from a source to an observer for neighboring planes has to be equal to an integer multiple of the wavelength; that is,
2dsin(θ) = mλ (for m = 1, 2, …)
The path difference 2dsin(θ) can be determined from the diagram. The second condition is known as the Bragg condition.
Part A
An x-ray beam with wavelength 0.200nm is directed at a crystal. As the angle of incidence increases, you observe the first strong interference maximum at an angle 28.0°. What is the spacing d between
the planes of the crystal?
Express your answer in nanometers to four significant figures.
This is very straightforward. Just use the formula given in the intro:
2dsin(θ) = mλ
2d * sin(28) = 1 * 0.200
d = 0.2130 nm
0.2130 nm
Part B
Find the angle at which you will find a second maximum.
Express your answer in degrees to three significant figures.
Just use the same formula, only change m to “2″ and use the value of d that you found in Part A:
2d * sin(28) = 1 * 0.200
2 * 0.2130 * sin(θ) = 2 * 0.200
sin(θ) = 0.938967
θ = 69.9°
Part C
Will you observe a third maximum?
• Yes, because all crystals have at least three planes.
• Yes, because the diffraction pattern has an infinite number of maxima.
• No, because the angle of a third maximum is greater than 180°.
• No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.
The answer is “No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.”
Note that this depends on the inputs for your particular version of the problem. You can use the calculator in Part B to solve for the 3rd maxima, just make “m” equal to 3
No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.
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Interval Operators - GNU Gnulib
14.3.4.4 Interval Operators ({ ... } or \{ ... \})
If the syntax bit RE_INTERVALS is set, then Regex recognizes interval expressions. They repeat the smallest possible preceding regular expression a specified number of times.
If the syntax bit RE_NO_BK_BRACES is set, ‘{’ represents the open-interval operator and ‘}’ represents the close-interval operator ; otherwise, ‘\{’ and ‘\}’ do.
Specifically, supposing that ‘{’ and ‘}’ represent the open-interval and close-interval operators; then:
matches exactly count occurrences of the preceding regular expression.
matches min or more occurrences of the preceding regular expression.
{min, max}
matches at least min but no more than max occurrences of the preceding regular expression.
The interval expression (but not necessarily the regular expression that contains it) is invalid if:
• min is greater than max, or
• any of count, min, or max are outside the range zero to RE_DUP_MAX (which symbol regex.h defines).
If the interval expression is invalid and the syntax bit RE_NO_BK_BRACES is set, then Regex considers all the characters in the would-be interval to be ordinary. If that bit isn't set, then the
regular expression is invalid.
If the interval expression is valid but there is no preceding regular expression on which to operate, then if the syntax bit RE_CONTEXT_INVALID_OPS is set, the regular expression is invalid. If that
bit isn't set, then Regex considers all the characters—other than backslashes, which it ignores—in the would-be interval to be ordinary.
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Kids.Net.Au - Encyclopedia > Absolute value
, the
absolute value
, or
), of a number is that number without a
negative sign
. So, for example, 3 is the absolute value of both 3 and -3.
It can be defined as follows: For any real number a, the absolute value of a (denoted |a|) is equal to a itself if a ≥ 0, and to -a, if a < 0 (see also: inequality). |a| is never negative, as
absolute values are always either positive or zero. In other words, the solution to |a| < 0 is that a is equal to the empty set, as there is no quantity which has a negative absolute value.
The absolute value can be regarded as the distance of a number from zero; indeed the notion of distance in mathematics is a generalisation of the properties of the absolute value.
The absolute value has the following properties:
1. |a| ≥ 0
2. |a| = 0 if and only if a = 0.
3. |ab| = |a||b|
4. |a/b| = |a| / |b| (if b ≠ 0)
5. |a+b| ≤ |a| + |b|
6. |a-b| ≥ ||a| - |b||
7. <math>\left| a \right| = \sqrt{a^2}</math>
8. |a| ≤ b if and only if -b ≤ a ≤ b
This last property is often used in solving inequalities; for example:
|x - 3| ≤ 9
-9 ≤ x-3 ≤ 9
-6 ≤ x ≤ 12
The absolute value function f(x) = |x| is continuous everywhere and differentiable everywhere except for x = 0.
For a complex number z = a + ib, one defines the absolute value or modulus to be |z| = √(a^2 + b^2) = √ (z z^*) (see square root and complex conjugate). This notion of absolute value shares the
properties 1-6 from above. If one interprets z as a point in the plane, then |z| is the distance of z to the origin.
It is useful to think of the expression |x - y| as the distance between the two numbers x and y (on the real number line if x and y are real, and in the complex plane if x and y are complex). By
using this notion of distance, both the set of real numbers and the set of complex numbers become metric spaces.
The operation is not reversible[?] because either negative or non-negative number or becomes the same non-negative number.
If the absolute value would not be a standard function Abs in Pascal it could be easily computed using the following code:
program absolute_value;
var n: integer;
read (n);
if n < 0 then n := -n;
writeln (n)
All Wikipedia text is available under the terms of the GNU Free Documentation License
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Math books for young children
Posted by: Dave Richeson | April 14, 2011
Math books for young children
I have a child in first grade and another who will be in elementary school in a couple years. So I’m on the lookout for good children’s books about mathematics. Below is a collection of books that
I’ve read or that have been recommended to me. (I got some of these suggestions from people on Twitter.) I’d really appreciate it if you would add your own suggestions in the comments (if you want to
give age-ranges, descriptions, or links, that would be great too). I’ll add more to the list as I find them.
Again, I’d say that the primary focus would be books for kids ages 5-12. Thanks!
This list is awesome! Thanks.
By: Dana Ernst on April 14, 2011
at 4:22 pm
I can recommend Kjartan Poskitt’s Murderous Maths series:
They’re very well and humorously written, but as the ‘Maths’ indicates, are UK rather than US based, which may or may not be an issue for you!
By: Jon Ingram on April 14, 2011
at 6:32 pm
My kids all loved “G is for Googol” when they were in the 4-7 range.
The partner book “Q is for Quark” was also quite good.
By: Mark James on April 14, 2011
at 7:16 pm
I posted my favorites here, and a longer list (less detailed reviews) here. Some of these are great for kids under 5. I so love Quack and Count, but 5 is getting on the old side for that. Some of the
Anno books are for younger kids, too.
By: Sue VanHattum on April 14, 2011
at 7:20 pm
The How and Why Book of Mathematics. I read it when I was three, in 1960. It’s where I learned what Infinity, Googol, and Googleplex were. I was 3 years old. I didn’t understand what exponents were,
but the book listed Googol with all 100 digits. I thought it weird that same number could be represented with 5 symbols. The COOL thing at three was, BIG numbers … LISTED! Aw, what are you going to
do? Kids!
The BEST things your kids have is having YOU as a Dad. The 5 best books for 8-yr-olds to 80, IMO, are in order: 50 Things You Really Should Know About Mathematics, The MaTH bOOK, Mathematics 1001,
Euler’s Gem, and The Princeton Companion.
And guess who wrote one of those?
Take em fishing, would be my advice. Lots of Math in fishing. And as an added bonus … time with Dad. If Mom comes along too, a plus. :-)
By: Steven Colyer on April 14, 2011
at 10:54 pm
Recommend the Phantom Tollbooth as well, there’s lots of mathematics in it. Nice list!
By: dwees on April 14, 2011
at 11:00 pm
Daniel very much enjoys The Number Devil. When he was bit younger he loved the Anno book as well, and Can You Count to a Googol? – http://www.albertwhitman.com/content.cfm/bookdetails/
By: Ed Webb on April 15, 2011
at 12:44 am
My own children are (almost) grown, but I love Rabbits Rabbits Everywhere: A Fibonacci Tale by Ann McCallum, ages 4-8.
Great Estimations by Bruce Goldstone, grades 1-4
This is a book with beautiful photographs that teaches estimating.
One Grain of Rice: A Mathematical Folktale by Demi, grades 1-4
A lovely book that combines morals, math, and Indian culture to illustrate the doubling function.
The Grapes of Math by Greg Tang, probably 1st-2nd grade is a book of math riddles in rhyme.
My kids absolutely LOVED these next two and still remember them fondly. They both deal with numbers and counting.
Bunny Money by Rosemary Wells, pre-school to grade 2
We made “bunny money” and a pocket in the back of the book to keep it in and my kids loved counting out the money along with the story.
Too Many Eggs, by M. Christina Butler, ages 4-8
This book is put of print, but I think it’s well worth searching out. There are eggs to punch out and count into mixing bowls along with the story.
Have fun and thanks for helping me recall some great memories of reading these to/with my sons.
By: bonny on April 15, 2011
at 10:13 am
Here’s another goodie oldie, which can easily be purchased from Amazon.com: “The Man Who Counted: A Collection of Mathematical Adventures” by Malba Tahan
Here’s the URL: http://tinyurl.com/3g3ur8r
Have a pleasant reading!
By: Yan Kow Cheong on April 17, 2011
at 7:31 am
Posted in Links, Math | Tags: books, Math, suggestions
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A Curve Is Banked At An Angle Of 30 Degrees With ... | Chegg.com
A curve is banked at an angle of 30 degrees with respect to the horizontal. Assume that its radius is 100 m and that the acceleration due to gravity is 10 m/s^2, what is the ideal speed for a
motorist to take the curve when it is covered by glare ice?
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stats 121
Bivariate data
two measurements on a single individual during a study (response variable, explanatory variable)
Direction relationship
positive if while X increases Y increases
Negative if while X increases Y decreases
Correlation coefficient
denoted by r, a number that gives the direction and strength of a linear relationship between two Quantitative variables
properties for r
both variables must be quantitive
sign of r denotes direction
r is between -1 and +1
no unit of measure
is affected by outliers
Statistical model
an equation that fits the pattern between a response variable and explanatory variable, accounting for deviations in the model
prediction errors
y-(y-hat)=prediction error
vertical distance from observed y to the line
facts about regression line
a change in one standard deviation x accounts for a r change in standard deviation y...
regression line passes through point (Xbar, Ybar)
it tells us the percentage of variation in Y that is explained by the least-squares regression line...
or.... it is a measure of how successfully the regression explains the response y
residual plot diagnostics
smile or frown shape-means there is a non-linear relationship
Megaphone-indicates constant variation (variation in y is dependant on x)
shoe-boX: point outside indicates outlier in x or Y direction
influential observation
an observation that if removed would change the regression line slope and y-intercept noticeably
-otliers in x direction are often influential
-influential observations may have small residuals
-not all outliers are influential observations
drawbacks of observational studies
-cannot systematically change x to observe y
-cannot randomize
-cannot establish causation; on correlation or association
Simpson's paradox
demonstrates that a great deal of care has to be taken when combining small data sets into a large one. Sometimes conclusions from the large data set are exactly the opposite of conclusion from the
smaller sets. Unfortunately, the conclusions from the large set are also usually wrong.
rules of data analysis
-always plot your data
-always describe shape, center, spreadof distributions
random phenomenon
the outcome of one play is unpredictable, but the outcome of many plays forms a distribution and then we can make a prediction
probability of an outcome
portion of how many times an out come occurs based on repition of plays (total)
probability rules
-between 0-1
-sum of all probabilities must equal 1
- the probe that event will NOT occur = 1-prob that it will occur
disjoint events
two events that have no outcomes in common and, thus, cannot both occur simultaneously.
Law of large numbers
If population has a finite mean mu or if x-bar is used to estimate mu,
as sample sixe increases, x-bar gets closer to mu
sampling distribution of x-bar
the distribution of all x-bar values from all possible samples of the same size from a population - (x-bar=mu)
-standard deviation of x-bar= standard deviation of pop/square root of n
-standard deviation of x-bar is always less than pop standard deviation where n>1
Central limit theorem
if you take a large srs of size n from any population shape gets more normal as n increases
out-of-control signals
-one point above or below control limits
-9 points in a row on the same side of the centerline
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Complex Analysis: Simplifying Polar Form
September 12th 2011, 05:45 AM #1
Oct 2009
United States
Complex Analysis: Simplifying Polar Form
I am having trouble simplifying the following equations which are the solutions to the equation $(z+1)^5=z^5$.
Any help you can give me will be greatly appreciated!
Re: Complex Analysis: Simplifying Polar Form
You could expand the fractions with $\exp\left[-i \frac{1}{5} \pi\right]$, $\exp\left[-i \frac{2}{5} \pi\right]$ and so on.
Re: Complex Analysis: Simplifying Polar Form
I am afraid I really don't understand how this would help.
Re: Complex Analysis: Simplifying Polar Form
After expanding, the demoninator will be a 2 i Sin[~], which is a multiple of Pi. The numerator is Exp[i~], which could you rewrite in a+ ib if you want to do.
Just try to expand the first solution with $\exp\left[-i \frac{1}{5} \pi\right]$ and you will see it works.
Re: Complex Analysis: Simplifying Polar Form
Sorry... I really appreciate your help, but I am just not seeing it. I still have both these items in the denominator and am back to 1 in the numerator.
Re: Complex Analysis: Simplifying Polar Form
I think it might be as simple as it gets for my purpose! Thank you.
Re: Complex Analysis: Simplifying Polar Form
What is the original statement of the problem?
Where do you get those answers?
It might help to know that
Re: Complex Analysis: Simplifying Polar Form
The original problem statement was to find all the solutions of $(z+1)^5=z^5$. I simplified this equation to equal $(1+\frac{1}{z})^5=1$ and then wrote $\frac{1}{z}$ in polar form as $(\frac{1}
I then simplified to the equation $e^{i(2/5)n\pi}$ for $n=0,\cdots,4$.
Obviously, $n=0$ isn't a solution, so I came up with the other 4 using that format, but I wasn't able to simplify them. But then I wondered if they need to be simplified. Thanks for your help.
Re: Complex Analysis: Simplifying Polar Form
Okay, I will explain again
$\frac{1}{e^{i\frac{2}{5}\pi}-1} = \frac{1}{e^{i\frac{2}{5}\pi}-1} \cdot \frac{e^{-i\frac{1}{5}\pi}}{e^{-i\frac{1}{5}\pi}}= \frac{e^{-i\frac{1}{5}\pi}}{e^{i\frac{1}{5}\pi}- e^{-i\frac{1}{5}\pi}}
= \frac{e^{-i\frac{1}{5}\pi}}{2 i \sin\left(\frac{1}{5}\pi\right)} = \frac{ e^{-i\frac{1}{10}\pi}}{2 \sin\left(\frac{1}{5}\pi\right)}$
Re: Complex Analysis: Simplifying Polar Form
Oh, sorry, yes I see that. Thank you.
September 12th 2011, 05:55 AM #2
Jul 2011
September 12th 2011, 06:54 AM #3
Oct 2009
United States
September 12th 2011, 06:59 AM #4
Jul 2011
September 12th 2011, 07:19 AM #5
Oct 2009
United States
September 12th 2011, 07:22 AM #6
Oct 2009
United States
September 12th 2011, 07:27 AM #7
September 12th 2011, 12:44 PM #8
Oct 2009
United States
September 12th 2011, 01:48 PM #9
Jul 2011
September 12th 2011, 04:16 PM #10
Oct 2009
United States
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Posts by
Posts by sk
Total # Posts: 29
What is the final pressure gas sample that is subjected to a temperature change from 22.0 C to 30.0 C and a volume change from 85.4 L to 37.0 L with an initial pressure 105 kPa?
7th grade science
It has come to my attention that you have violated the Honor Code of the class. Your credentials on the site have been removed. Thank you for your time.
Find the volume generated by rotating the area between y = cos( 3 x ) and the x axis from x = 0 to x = π/ 12 around the x axis
A man steps on his bathroom scale and obtains a reading of 243 lb. The spring in the scale is compressed by a displacement of -0.0480 inches. Calculate the value of its spring constant in (a) pounds
per inch (b) newtons per meter.
A student applies an upward force of 56 N to a 5.6-kg physics textbook. Find the book s acceleration (positive for up, negative for down). answer in miles/seconds squared.
pls show answers and work with the questions i have posted.
A juggler throws a ball from height of 0.950 m with a vertical velocity of +4.25 m/s and misses it on the way down. What is its velocity when it hits the ground?
Sam tosses a ball horizontally off a footbridge at 3.1 m/s. How much time passes after he releases it until its speed doubles?
Use the double angle formula to verify the identity cotxsin2x=1+cos2x
Find the value of tangent of angle A, given cos2A = 4/5, and A terminates in quadrant 2
find sin(s+t), given that sin t=1/3, t in quadrant 2, and cos s = -2/5, s in quadrant 3
the angle of depression from the top of a building to a point on the ground is 32degrees30', how far is the point of the ground from the top of building if the building is 252m high?
The angle of depression from the top of the building to a point on the ground is 29degrees30', the point of the ground is 36 meters from the top of the building. Find the height of the building
A plane flies 1.3 hr at 100mph on a bearing of N40degreesE. It then turns and flies 1.5hr at the same speed on a bearing of S50degreesE. How far is the plane from its starting point?
A ship leaves its homeport and sails on a bearing of N28degrees10'E. Another ship leaves the same port at the same time and sails on a bearing of S61degrees50'E. If the first ship sails at 24.0 mph
and second sails at 28mph, find the distance between the two ships afte...
assume a pulley has a radius of 12.96 cm. If it takes 18 seconds for 56 cm of belt to go around the pulley, find the angular velocity in radians per seconds
In 1992, the average cost of a ticket on a privately-owned airline was $145. This amount has increased by approximately $51 yearly. How much should you expect to pay for a ticket on this airline in
the year 2005?
5th grade Math
Thank you. I never thought about looking at them like money.
5th grade Math
Having problems with less than/greater than decimals. Wanted to double check some answers. a)0.5_1.0 My Answer is < b)3.2_3.02 My answer > c)4.83_4.8 My answer < d)6.25_6.4 My answer < e)0.7_0.07 My
Answer >
physics 12- dynamics
A 5kg block slides down an incline of 25
A 5kg block slides down an incline of 25 degrees to the horizontal at a constant speed. a) what is the net force on the block? b) what is the magnitude of the friction force acting on the block? c)
If friction was negligible and the block was accelerating down , what would be ...
A block is launched up the incline(coefficient of friction is 0.3) with an initial speed of 12m/s. What is the maxximum displacement of the block up the incline?
What is institutional discrimination? Give an example
How would you purify decanoic acid contaminated with biphenyl? what method and solvent.Please help me
Large atom goes on equatorial becasue equatorial is more stable than axial. You always want to have stable shape.
what is the good of morality This site will give you some good ideas about that topic. http://en.wikipedia.org/wiki/Morality Morality is a guide; it helps us to decide what is "good" and what is
"evil." In other words, it helps us to choose between those th...
Ethics in business
What do you think is the best way of doing business ethics and why 1) all transactions are above board: nothing under the table. 2) Honesty in all transactions. No misleading advertising, or terms.
3) Follow the law, if it conflicts with the two above, then state it publicly i...
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Westchester, FL SAT Math Tutor
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Posts from December 12, 2008 on The Unapologetic Mathematician
The next obvious things to consider are the kernel and the image of an intertwining map. So let’s say we’ve got a representation $\rho:A\rightarrow\mathrm{End}(V)$, a representation $\sigma:A\
rightarrow\mathrm{End}(W)$, and an intertwiner $f:\rho\rightarrow\sigma$ defined by the linear map $f:V\rightarrow W$ which satisfies $\left[\sigma(a)\right]\left(f(v)\right)=f\left(\left[\rho(a)\
right](v)\right)$ for all $v\in V$.
Now the linear map $f$ immediately gives us two subspaces: the kernel $\mathrm{Ker}(f)\subseteq V$ and the image $\mathrm{Im}(f)\subseteq W$. And it turns out that each of these is actually a
subrepresentation. Showing this isn’t difficult. A subrepresentation is just a subspace that gets sent to itself under the action on the whole space, so we just have to check that $\rho(a)$ always
sends vectors in $\mathrm{Ker}(f)$ back to this subspace, and that $\sigma(a)$ always sends vectors in $\mathrm{Im}(f)$ back into this subspace.
First off, $v\in V$ is in the kernel of $f$ if $f(v)=0$. Then we calculate
which shoes that $\left[\rho(a)\right](v)$ is also in the kernel of $f$.
On the other hand, if $w\in W$ is in the image of $f$, then there is some $v\in V$ so that $w=f(v)$. We calculate
And so $\left[\sigma(a)\right](w)$ is also in the image of $f$.
So we’ve seen that the image and kernel of an intertwining map have well-defined actions of $A$, and so we have subrepresentations. Immediately we can conclude that the coimage $\mathrm{Coim}(f)=V/\
mathrm{Ker}(f)$ and the cokernel $\mathrm{Cok}(f)=W/\mathrm{Im}(f)$ are quotient representations.
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Unsolved Puzzle
September 29th 2006, 10:16 AM #1
Unsolved Puzzle
In the year 3141, seven astronauts are forced to land their spaceship on a mysterious planet. To their surprise they find an abundance of fruit resembling the Earth’s apple. They decide to call
it the kuja fruit. On their first day, they collect as many kujas as they can find and place them into a single pile. As the planet’s sun begins to set over the horizon, the astronauts decide
that they will wait until the
next day to divide the kujas among themselves.
That night while the others slept, each astronaut took a turn watching for the alien creatures that they had seen earlier that day. The first watcher decided to divide the kujas into seven equal
piles. When he did this, he found that he had one kuja left over which he gave to one of the alien creatures. He decided to take one of the piles and hide it for himself and then combined the
remaining six piles back into one
big pile for the next astronaut to watch over.
It turns out that throughout the night, each of the seven astronauts did the exact same thing. They took the single pile, divided it equally into seven piles and had one kuja left over which they
gave to an alien creature. They took and hid a pile for themselves
and combined the remaining six piles back
into a single pile.
What is the smallest number of kujas that could have been in the original pile?
This is unsolvable puzzle for my group. We tried to figure out the solution for serveral months, but can't find any good. This is quite a big difficult than last puzzle that i posted.
Challenge increases talents
talents bases on intelligent
intelligent needs pateint
Do puzzlers have pateint ?
In the year 3141, seven astronauts are forced to land their spaceship on a mysterious planet. To their surprise they find an abundance of fruit resembling the Earth’s apple. They decide to call
it the kuja fruit. On their first day, they collect as many kujas as they can find and place them into a single pile. As the planet’s sun begins to set over the horizon, the astronauts decide
that they will wait until the
next day to divide the kujas among themselves.
That night while the others slept, each astronaut took a turn watching for the alien creatures that they had seen earlier that day. The first watcher decided to divide the kujas into seven equal
piles. When he did this, he found that he had one kuja left over which he gave to one of the alien creatures. He decided to take one of the piles and hide it for himself and then combined the
remaining six piles back into one
big pile for the next astronaut to watch over.
It turns out that throughout the night, each of the seven astronauts did the exact same thing. They took the single pile, divided it equally into seven piles and had one kuja left over which they
gave to an alien creature. They took and hid a pile for themselves
and combined the remaining six piles back
into a single pile.
What is the smallest number of kujas that could have been in the original pile?
This is unsolvable puzzle for my group. We tried to figure out the solution for serveral months, but can't find any good. This is quite a big difficult than last puzzle that i posted.
Challenge increases talents
talents bases on intelligent
intelligent needs pateint
Do puzzlers have pateint ?
This is a classic puzzle with monkeys and coconuts.
See this
September 29th 2006, 10:38 AM #2
Grand Panjandrum
Nov 2005
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search results
Expand all Collapse all Results 1 - 8 of 8
1. CMB Online first
On an Exponential Functional Inequality and its Distributional Version
Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb R$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions
$f\colon G\to \mathbb K$ satisfying the inequality $ \Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G, $ where $\phi\
colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots
\otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions,
respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$.
Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability
Categories:46F99, 39B82
2. CMB Online first
Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems
We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by
Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.
Keywords:functional inequalities, Brownian motion
Categories:39B62, 60J65
3. CMB 2011 (vol 56 pp. 218)
Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -
the d'Alembert equation, the Wilson equation, and the d'Alembert long equation - on compact groups.
Keywords:functional equations, Fourier analysis, representation of compact groups
Categories:39B52, 22C05, 43A30
4. CMB 2011 (vol 55 pp. 214)
Positive Solutions of Impulsive Dynamic System on Time Scales
In this paper, some criteria for the existence of positive solutions of a class of systems of impulsive dynamic equations on time scales are obtained by using a fixed point theorem in cones.
Keywords:time scale, positive solution, fixed point, impulsive dynamic equation
Categories:39A10, 34B15
5. CMB 2011 (vol 55 pp. 424)
Convergence Rates of Cascade Algorithms with Infinitely Supported Masks
We investigate the solutions of refinement equations of the form $$ \phi(x)=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\:\phi(Mx-\alpha), $$ where the function $\phi$ is in $L_p(\mathbb R^s)$$(1\le p\le\
infty)$, $a$ is an infinitely supported sequence on $\mathbb Z^s$ called a refinement mask, and $M$ is an $s\times s$ integer matrix such that $\lim_{n\to\infty}M^{-n}=0$. Associated with the mask
$a$ and $M$ is a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by $Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\phi_0(M\cdot-\alpha)$. Main results of this paper are related to
the convergence rates of $(Q_{a,M}^n \phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial
function $\phi_0$, $Q_{a,M}^n \phi_0$ converges in $L_p(\mathbb R^s)$ with an exponential rate.
Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergence
Categories:39B12, 41A25, 42C40
6. CMB 2011 (vol 54 pp. 580)
Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales
Consider the second order superlinear dynamic equation \begin{equation*} (*)\qquad x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$} \end{equation*} where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb
{T}$ is a time scale, $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for $x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear
condition, which includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly known as the Emden--Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for
arbitrarily large values of $t$. In addition to extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we obtain analogues in the difference equation and
$q$-difference equation cases.
Keywords:Oscillation, Emden-Fowler equation, superlinear
Categories:34K11, 39A10, 39A99
7. CMB 2008 (vol 51 pp. 161)
Wirtinger's Inequalities on Time Scales
This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue $\Delta$-integral on an arbitrary time scale $\T$. We prove a general inequality for a class of absolutely
continuous functions on closed subintervals of an adequate subset of $\T$. By using this expression and by assuming that $\T$ is bounded, we deduce that a general inequality is valid for every
absolutely continuous function on $\T$ such that its $\Delta$-derivative belongs to $L_\Delta^2([a,b)\cap\T)$ and at most it vanishes on the boundary of $\T$.
Keywords:time scales calculus, $\Delta$-integral, Wirtinger's inequality
8. CMB 2005 (vol 48 pp. 505)
On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group
Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^
{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes
d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.
Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.
Categories:39B32, 39B42, 22D10, 22D12, 22D15
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Jupiter Math Tutor
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yntax for
This is one of those mundane little activities I found myself doing frequently enough that I finally just had to automate it.
When working in VBA I frequently need to range enter a formula across multiple cells. It's not particularly difficult to convert the cell-based formula to the appropriate VBA R1C1 format and
adjusting for double quotes if necessary, but if you do it enough, it gets old; especially when converting formulas with lots of double quotes.
So, here's the new gadget I just added to my personal macro toolbar that will convert the activecell formula to the proper VBA text string, and write it to the clipboard so I can then just paste it
in my code.
I've even seen some questions posted here along the lines of, "What's the proper syntax for inputting my formula in VBA so the range refs are incremented properly?" This pretty much instantly answers
that type of question.
Simple little code, but I find it rather helpful.
Sub Get_VBA_Formula()
Dim VBA_Formula As String, n$, x$
Dim i As Integer
Dim MyData As DataObject
VBA_Formula = ActiveCell.FormulaR1C1Local
'double quote substitution
For i = 1 To Len(VBA_Formula)
n$ = Mid(VBA_Formula, i, 1)
If n$ = """" Then
x$ = x$ & """"""
x$ = x$ & n$
End If
Next i
'post formula to clipboard
Set MyData = New DataObject
VBA_Formula = """" & x$ & """"
MyData.SetText VBA_Formula
MsgBox VBA_Formula, vbOKOnly, "VBA Formula Copied to Clipboard"
End Sub
If you like these VB formatting tags please consider sponsoring the author in support of injured Royal Marines
I believe it works for any formula (array or not). If you can think of a character other than the double quote that I may not be accounting for properly in VBA, or some other issue I've overlooked,
please add your comments to the thread.
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Here's the question you clicked on:
When 155 mL of water at 26 C is mixed with 75 mL of water at 85 C, what is the final temperature? (Assume that no heat is lost to the surroundings; d of water 1.00 g/mL.)
• one year ago
• one year ago
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Someone please help, I really don't know where to begin or how to solve this. :( Even just an equation will do.
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I'm not that familiar with this exact calculation, so excuse me if I'm wrong. The equation for this should be : ( Volume1 * Temp1) + (Volume2 * Temp2) / (Volume1 + Volume2) = Final temp
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The amount of heat a substance gains or loses is governed by the following formula: \[Q=mc \Delta T\] Where Q is the heat lost or gained, m is the mass in kg, c is the specific heat of the
substance (c-1.00 for water), and deltaT is the change in temperature. Since the heat gained by the cold water is = to the heat lost by the warm water the two Qs can are equal. Thertefore: \[m _
{c} \Delta T _{c }=m _{w}\Delta T _{w}\] The c and w subscripts are for the cold and warm water. Since both waters are at the same final temperature: \[m _{c}(T _{f}-26)=m _{w}(85-T _{f})\] where
T sub f is the final temp for both masses of water. The expressions in the parentheses were switched because you always want a positive delta T.
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Thank you so much!
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.
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Got Homework?
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Here's the question you clicked on:
How do you simplify:
• one year ago
• one year ago
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\[\sqrt{72} \]
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Okay so some easy steps to start simplifying.
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First you're going to want to try to split that 72 into something you know can factor.
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Any idea what 72 can be split into?
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and what?
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ohhhhh. Do you mean like 8 times 9?
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yes, or like tyou said 36 times what? Your first guess was correct.
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so then what?
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or is that the answer? lol
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\[\sqrt{36 * ?}\]
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\[\sqrt{36 * 2}\]
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now can you simplify sqrt(36 or sqrt(2)?
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WE don't split it up, so sorry if that last comment is confusing....
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WE can just look at each piece now, and determine what's needed....
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You can also use 8 and 9...
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yes, we can abb0t.... but this is easier. I was going to show 8 and 9 after.
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i can simplify 36 into
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\[\sqrt{6 * 6}\]
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nono, we don't need to simplify it that way. I meant does sqrt(36) or sqrt(2)= anything?
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oh yeah, 36= 6 and 2= 1.414214562. do i have to simplify 2?
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correct so we get \[6 \sqrt{2}\] which is our answer.
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ahhhhhhh thanks a lot
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we don't need to simplify 2, we just need to get it into it's simpliest form.
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so for example, lets use 8 and 9.
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\[\sqrt{72} = \sqrt{8 * 9} = 3 \sqrt{ 4 * 2} = 2 *3 \sqrt{2} = 6 \sqrt{2}\]
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Its' really just trying to take some ugly sqrt thart we cannot solve and split it into things that we can solve for :).
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i liked the 36 and 2 one better lol. Thanks again!
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Yeah, it's easier, but you can see how it can be applied with any set :). Good luck, I hope you understand the concepts.
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so lets see if i have this:
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Sure, throw up a random # and lets see if we can solve.
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as long as it's not prime... AHAH..
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\[\sqrt{75} = 5\sqrt{3}\]
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ok thanks cya later!
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\[\sqrt{75} = \sqrt{25 * 3} = 5 \sqrt{3}\]
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np good luck.
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Hey here is a new type of problem:
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\[\sqrt{5} \times \sqrt{35}\]
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Think you can do that?
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hmm Idk man you're asking a lot here... :p
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lets start off basic. Multiply them together, what do we get?
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or, if you want, you can split sqrt(35) into 2 parts, and go that way :).
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well i dont need to get the answer to the equation, i think i just need to simplify it. Its under the section of simplifying
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so do i just simplify each part?
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simplifying is the same thing.
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you basically are given an eq, to solve. IT could be multiplication, addition, etc, etc. Then you want to use the elements to combine and form something you are able to simplify and take
something out of the sqrt. That is our main objective, to get a real number.
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We are doing exactly what we did before, exept we are adding an additional step. Instead of 35 and 5, we had 36 and 2. or 9 and 8(9 * 4 *2)
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I guess above I shouldn't have said you shouldn't split them out, because in the end you can, and it doesn't matter.... sqrt(36) * sqrt(2) is the same thing
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6 * sqrt(2)
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so would it be sqrt(5) and 35= 7 * 5
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\[\sqrt{5} \times \sqrt{7} \times \sqrt{5}\]?
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but those are primes....
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yessir, but you can combine them in other ways, no?
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sqrt(5) * sqrt(5) = ?
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i dont know. So then what do i do?
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oh didnt see your comment
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Always remember when we have 2 of the same sqrt() we get the actual number
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oops 5
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and yes.
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we now have our answer.
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so how do i write down this answer?
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ohhhh ok
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Math is awesome that you can combine and move things around to find whatever you need. Lets say we have sqrt(3) sqrt(3) and sqrt(4) the 2 sqrt(3) become a 3. and the sqrt(4) becomes a 2. Our
answer is 6.
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But if we had lets say sq(15) and sqrt(3) we will have to split up the sqrt(15) into sqrt(3) sqrt(5) and then another sqrt(3) which we end up getting 3sqrt(5)
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ahhhhhhhh that makes sense! Are you a math teacher or something?
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haha na...
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I just know how much it sucks to not get proper help.....
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yeah man. Ok well you saved my grade! Thanks again
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No problem good luck ,anymore questions you can do @konradzuse to get me.
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ok lets hope i dont need to do that :P
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There are lots more math in your future :).
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true that.
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:) I just got done with Linear Algebra and Calculus 2 :)
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im half way through geometry...
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hey would the answer to sqrt of 9 over 5 = sqrt of 45 over 5?
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nope... what's sqrt(9)?
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would it be:
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nope 3/sqrt(5)...
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so just:
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or should it have a sqrt sign over the entire thing?
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think about it. sqrt(9)/sqrt(5) is the same thing as sqrt(9/5) sqrt(9) is 3.
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ok thanks. Have a good day
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you can also double check everything with a calculator...
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Subring Elements
September 25th 2010, 12:35 AM #1
Dec 2009
Subring Elements
Let $R$ be a ring and $a$ an element of $R$. Let $S = \{ x \in R: ax = 0_R\}$. $S$ is a subring of $R$.
Let $R = M_2(\mathbb{Z}_5)$ (the ring of 2×2 matrices with entries from the field $\mathbb{Z}_5$) and
$a= \begin{pmatrix}4 & 2 \\ 3 & 4 \end{pmatrix}$. Determine the elements of the subring S defined previously. (I) How many elements are in S? (II) Show that S is not an ideal of R.
(I) the elements of S are 2x2 matrices x such that $ax=0_R$.
let $x=\begin{pmatrix}a & b \\ c & d \end{pmatrix}$
$ax=\begin{pmatrix}4 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix}a & b \\ c & d \end{pmatrix} = \begin{pmatrix}4a+2c & 4b+2d \\ 3a+4c & 3b+4d \end{pmatrix}= \begin{pmatrix}0 & 0 \\ 0 & 0 \end
If I solve the homogeneous system
I get a=b=c=d=0.
So does this mean the only element in the subring S is the zero matrix?
(II) I will use the "ideal test":
Since the zero matrix belongs to S, $S eq \emptyset$.
Then $S \triangleleft R \iff (x-y) \in S$ and $rS, Sr \subseteq S \forall r \in R$. (x,y are in S).
The question says show that S is NOT an ideal. But the problem is that S passes the ideal test since (x-y) is equal to the zeo matrix and therefore $a(x-y)=0_R \in S$. The same is true with
multipication. So, is something wrong with the question or did I make a mistake?!
Let $R$ be a ring and $a$ an element of $R$. Let $S = \{ x \in R: ax = 0_R\}$. $S$ is a subring of $R$.
This may depend on the definition. Many authors require that if R is unitary then in order to be considered a
candidate to be a subring a subset must contain the unit of R. In this case, though, it
must be not so since then a = 0.
Let $R = M_2(\mathbb{Z}_5)$ (the ring of 2×2 matrices with entries from the field $\mathbb{Z}_5$) and
$a= \begin{pmatrix}4 & 2 \\ 3 & 4 \end{pmatrix}$. Determine the elements of the subring S defined previously. (I) How many elements are in S? (II) Show that S is not an ideal of R.
(I) the elements of S are 2x2 matrices x such that $ax=0_R$.
let $x=\begin{pmatrix}a & b \\ c & d \end{pmatrix}$
$ax=\begin{pmatrix}4 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix}a & b \\ c & d \end{pmatrix} = \begin{pmatrix}4a+2c & 4b+2d \\ 3a+4c & 3b+4d \end{pmatrix}= \begin{pmatrix}0 & 0 \\ 0 & 0 \end
If I solve the homogeneous system
I get a=b=c=d=0.
Even without checking this cannot be right since the matrix x is singular, and indeed:
$3a+4c=2(5a+2c)\,,\,\,3b+4d=2(4b+2d)\Longrightarrow c=-2a=3a\,,\,\,d=-2b=3b\Longrightarrow$
any matrix of the form $\begin{pmatrix}a&b\\3a&3b\end{pmatrix}$ belongs to $S$
So does this mean the only element in the subring S is the zero matrix?
(II) I will use the "ideal test":
Since the zero matrix belongs to S, $S eq \emptyset$.
Then $S \triangleleft R \iff (x-y) \in S$ and $rS, Sr \subseteq S \forall r \in R$. (x,y are in S).
The question says show that S is NOT an ideal. But the problem is that S passes the ideal test since (x-y) is equal to the zeo matrix and therefore $a(x-y)=0_R \in S$. The same is true with
No, it's not. Try to find a counterexample to $x\in S\,,\,r\in R\Longrightarrow rx\in S$ now that you know the general
form of elements in $S$
So, is something wrong with the question or did I make a mistake?!
So if the elements are of the form $\begin{pmatrix}a&b\\3a&3b\end{pmatrix}$, then $a \in \{0,1,2,3,4\}$, $b \in \{0,1,2,3,4\}$ in $\mathbb{Z}_5$. 5x5=25 possible combinations of a and b.
Therefore there are 25 elements in S?
Try to find a counterexample to $x\in S\,,\,r\in R\Longrightarrow rx\in S$ now that you know the general
form of elements in S
Let a=b=1 then $x= \begin{pmatrix}1&1\\3&3\end{pmatrix}$, and $r= \begin{pmatrix}1&1\\1&1\end{pmatrix}$.
rx= $\begin{pmatrix}4&4\\4&4\end{pmatrix}$, and
$a(rx)= \begin{pmatrix}4&2\\3&4\end{pmatrix} \begin{pmatrix}4&4\\4&4\end{pmatrix} = \begin{pmatrix}4&4\\3&3\end{pmatrix} eq R_0$
Therefore $rx otin S$. Is this okay?
So if the elements are of the form $\begin{pmatrix}a&b\\3a&3b\end{pmatrix}$, then $a \in \{0,1,2,3,4\}$, $b \in \{0,1,2,3,4\}$ in $\mathbb{Z}_5$. 5x5=25 possible combinations of a and b.
Therefore there are 25 elements in S?
Let a=b=1 then $x= \begin{pmatrix}1&1\\3&3\end{pmatrix}$, and $r= \begin{pmatrix}1&1\\1&1\end{pmatrix}$.
rx= $\begin{pmatrix}4&4\\4&4\end{pmatrix}$, and
$a(rx)= \begin{pmatrix}4&2\\3&4\end{pmatrix} \begin{pmatrix}4&4\\4&4\end{pmatrix} = \begin{pmatrix}4&4\\3&3\end{pmatrix} eq R_0$
Therefore $rx otin S$. Is this okay?
Yes and yes, but it is S, not R_o.
September 25th 2010, 05:29 AM #2
Oct 2009
September 26th 2010, 03:15 AM #3
Dec 2009
September 26th 2010, 05:41 AM #4
Oct 2009
September 26th 2010, 02:24 PM #5
Dec 2009
September 26th 2010, 02:27 PM #6
Oct 2009
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Eddington on the Expanding Universe
On Friday 22 January 1932 Arthur Stanley Eddington addressed the Friday Evening Discourse in Physical Science at the Royal Institution in London. He addressed the audience on The Expanding Universe.
In the tradition of the lectures he assumed that members of the audience had no expert scientific knowledge. It is worth commenting that Eddington puts forward his own theories in the latter part of
the lecture. Below is a version of Eddington's lecture:
The Expanding Universe
Sir Arthur Eddington, M.A., D.Sc., F.R.S.
Plumian Professor of Astronomy and Director of the Observatory, University of Cambridge
In recent years the line-of-sight velocities of about go of the spiral nebulae have been measured. The distances of some of the nearest of them have been determined by a fairly trustworthy method,
and for others rude estimates depending on statistical methods are available. When we survey these data a remarkable state of affairs is revealed. The spiral nebulae are almost unanimously running
away from us; moreover, the greater the distance the greater the speed of recession. The law of increase is found to be fairly regular, the speed being simply proportional to the distance. The
progression has been traced up to a distance of more than 100 million light-years, where the recession is 20,000 km. per sec. - about the speed of an
At first sight this looks as though they had a rather pointed aversion to our society; but a little consideration will show that the phenomenon is merely a uniform dilation of the system and is not
specially directed at us. If this room were suddenly to expand to twice its present size, the seats separating in proportion, you would notice that every one had moved away from you. Your neighbour
who was 3 feet away has become 6 feet away; the man over yonder who was 20 feet away is now 40 feet away. Each has moved proportionately to his distance from you, which is precisely what the spiral
nebulae are observed to be doing. The motion is not directed from any one centre, but is a general expansion such that each individual observes every other individual to be receding.
In 1917, before any hint of this phenomenon had been obtained from observation, Professor de Sitter was on the look out for something of the kind. He found that, on one of two alternative hypotheses
arising out of Einstein's relativity theory, the light of very remote objects should be displaced to the red as though they were moving away from us; and he suggested the observed motions of the
spiral nebulae (by far the most remote objects known) as a discriminating test. At that time only three radial velocities had been published, and they rather lamely supported his hypothesis by a
majority of 2 to 1. The majority has now become about 85 to 5, and the exceptions are nebulae close to us, which in any case should have had only small receding velocities. De Sitter's theory has
been developed and modified by Friedmann and Lemaitre; the modern view of it is as follows :-
Einstein's law of gravitation contains a term called the "cosmical term" which is extremely small in ordinary applications to the solar system, etc., and is generally neglected. The term, however,
actually represents a repulsive force directly proportional to the distance; so that however small it may be in ordinary applications, if we go to distances sufficiently great it must ultimately
become important. It is this cosmical repulsion which is, we believe, the cause of the expansion of the great system of the nebulae. The repulsion may be to some extent counterbalanced by the
ordinary gravitational attraction of the nebulae on one another. This countervailing attraction will become weaker as the expansion increases and the nebulae become farther apart. It seems likely
that the universe started with a balance between gravitational attraction and cosmical repulsion; this equilibrium state is called an "Einstein universe." But it can be shown that the Einstein
universe is unstable; and the slightest disturbance will cause either the repulsion or the gravitation to gain the upper hand so as to topple the system into a state of continually increasing
expansion or continually increasing contraction. Apparently expansion won the initial struggle, and as the nebulae spread apart the opposition of gravitation became less and less, until now it is
comparatively insignificant.
We see, then, that according to observation the system of the spiral nebulae is expanding, and that relativity theory had foreseen just such an expansion (except that as an alternative it would have
been content with an equally regular contraction). What better agreement could we desire? Nevertheless there were some misgivings which I would not by any means condemn as unreasonable. It is true
that theory predicted an effect of the kind observed, but it did not say how rapid the expansion would be. It expressed it in terms of an unknown "cosmical constant"
I think, however, that we shall have to accept the expansion. My reason is that it now seems possible to calculate the cosmical constant
I have been tracing the effects of the cosmical constant in the behaviour of the great system of galaxies - phenomena on the grandest scale we have yet imagined. Now I want to turn to the other end
of the scale and look into the interior of the atom, where I think we shall find that the same cosmical constant turns up again. It is, in fact, the main key to the mystery of protons and electrons.
I cannot take you very far into this part of the theory, but I will try to show why I am convinced that the cosmical constant comes into the theory of the atom. I must premise one thing. It is well
known that in Einstein's theory gravitation has an interpretation not only as a force but as a geometrical property - a curvature - of space-time. So also the cosmical constant has an interpretation
not only as a repulsive force but as a measure of curvature. The constant R^2, R being the radius of the world in the equilibrium (Einstein) state from which we suppose it to have started.
Length is necessarily relative. That is one of the results of Einstein's theory which has become almost a commonplace of physics; but it was a rather complicated kind of relativity that Einstein
considered - relativity to the motion of our frame of reference. I am going to refer to another much more elementary relativity of length, viz. that length always implies comparison with a standard
of length. It is only the ratio of lengths that enters into our experience. Suppose that every length and every distance in the universe were suddenly to be doubled; nothing would seem altered. I do
not think we could attach any meaning to the change. Intrinsically Brobdingnag and Lilliput are precisely the same; it needs an intruding Gulliver - an extraneous standard of length - to make them
appear different.
Now, it is commonly stated in physics that all normal hydrogen atoms have the same size, or have the same spread of electric charge. We have a very fundamental equation (the wave equation)
determining the spread, which is supposed to apply to any hydrogen atom and, of course, gives the same result for all. But what do we mean by their having the same size? Or it may be better to put
the question negatively - what would it mean if we said that two hydrogen atoms were of different sizes, i.e. similarly constructed but on different scales? It would be Brobdingnag and Lilliput over
again. To give any meaning to the difference we need a Gulliver. Now, the Gulliver of physics is always supposed to be a certain bar of metal called the International Metre. He is anything but a
traveller; I think he has never been away from Paris. It was Professor Weyl who first called attention to the very big hiatus involved, when we speak of a length such as the radius of a hydrogen atom
being a certain fraction of the standard metre. We have, as it were, our Gulliver, but have left out his travels. The travels are (as Weyl showed) the interesting part of the story, and are not to be
glossed over as irrelevant. Weyl went farther and pointed out that there is a natural standard of comparison which is always on the spot, namely, the radius of curvature of the world at that spot. We
can thus give a direct meaning to the statement that two hydrogen atoms in any part of the universe have the same size; we mean that each of them is the same fraction of the radius of curvature of
space-time at the place where it lies. The atom here is a particular fraction of the radius of curvature here; the atom on Sirius is the same fraction of the radius of curvature at Sirius. Whether
the radius of curvature here is the same as at Sirius does not arise, and I do not think there is any meaning in trying to compare them absolutely.
The above definition of equality, and the use of Weyl's standard, may seem a dangerous innovation; but indirectly we have been using it all along without knowing that we were doing so. Some years ago
I pointed out that Einstein's law of gravitation can be stated in the form, "What we call a metre at any place and in any direction is a constant fraction of the radius of curvature of space-time for
that place and direction." That is simply a translation of the law from symbols into words. The law is verified by observation, so that the statement gives us not only an ideal definition of the
metre, but one which we know will accord with the reckoning of metres that is actually used. Thus measurement in terms of the metre is equivalent to measurement in terms of the world-radius, since
the two standards are always in a constant ratio. Practically it is more convenient to employ the metre, but in pursuing the theory we must go direct to the world radius; for obviously a particular
bar of metal at Paris can have no fundamental status in physics and is altogether irrelevant to equations describing the mechanism of the atom. The world curvature, on the other hand, is on the spot,
and is directly reacting with the atom. I now return to the wave equation which professes to determine how large an atom will be. That, as we have seen, means that it finds the ratio of the various
intervals in the atom to the world radius there; so the world radius must come into the equation. But the world radius is the cosmical constant in another form. The cosmical constant has cropped up
again inside the atom.
My task now was to spot the cosmical constant or the world radius in the current form of the wave-equation, which is known by experiment to be substantially correct. It is very much disguised because
the current equation introduces the standard metre and all sorts of irrelevancies. But one knows the sort of effect that curvature can have; and the way it will appear in the equation is pretty well
dictated by the quantum laws which make a speciality of the properties of "closed circuits" such as are introduced by curved space. I think I succeeded, and I arrived at the identification
mc^2/e^2 = √N/R
The left side is a term in the current equation, and its value is known experimentally. The right side is the way that we write it now that we have penetrated its disguise. R is the Einstein radius
of the world, equal to the inverse square root of the cosmical constant; N is the number of electrons (or protons) in the universe.
This additional equation, combined with other equations already known, gives all the information required. We deduce, for example, that the number of electrons in the universe is 1.29 ^79; and that
the original radius of the universe, before it started to expand, was 1070 million light-years. Most important of all, we find that the consequent rate of expansion of the universe is 528 kilometres
per second per megaparsec distance. The. observational determinations from the recession of the spiral nebulae (which might be a little lower since they include any countervailing gravitational
attraction) range from 430 to 550 km per sec per megaparsec. We can feel little doubt therefore that the observed motions of the nebulae are genuine and represent the expansion effect predicted by
relativity. We must reconcile ourselves to this alarming rate of expansion, which plays havoc with older ideas as to the time-scale.
However interesting may be the application of this theory to the universe, the application to the interior of the atom seems likely to be still more fruitful. Now that we know the magnitude of the
radius of curvature, we can set aside the arbitrary metre and use this natural unit in our equations. The big uninformative coefficients disappear; and the equations are so much simplified that I
think I have a fair idea of what they really mean and how they work. In particular the relation of the proton to the electron is now apparent, and the theoretical ratio of their masses is found to be
1847.6; this is certainly very near to the observed value. I do not want to stress too much the accuracy or finality of these first results. I cannot see how anything can possibly be wrong with them;
but then one never does see these faults until some new circumstance arises or some ingenious person comes forward to show us how blind we have been. At least a way of progress has been found. I
think that some day, when electrons and protons have come to order, we shall look back and see that the key which unlocked the mystery was lying somewhere in intergalactic space and was picked up by
astronomers who measured the velocities and distances of nebulae ten million light-years away.
JOC/EFR April 2007
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