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integer proof
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integer proof
I've been trying to solve this for a while, but without any good result. I could definitely need some help :]
k = (3*9^(p+1)+25^(p+1))/4
p is any number larger or equal to 0. How can I be 100% sure that k is an integer? I've tried different values to test it, and it looks like it is an integer, but I don't know how to actually proof
Re: integer proof
I hope you get your answer here. I was looking for a similar definition in an earlier thread, but a clear one never emerged.
Re: integer proof
Edit: Ah, spotted a large error in my proof, fixing it right now.
Last edited by Ricky (2006-02-04 09:41:21)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: integer proof
p is any number? Only integers, right?
I guess it can be prove that 3*9^(p+1)+25^(p+1) is always a multiple of 4.
You could try an inductive proof:
1) prove the base case (p=1),
2) prove that whenever you add 1 to p, the result will still be a multiple of 4 (use p=n → p=n+1)
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
Re: integer proof
I've tried the induction proof, but I didn't get too far with it (I'm still not too comfortable with it btw). I might try something similar to what Ricky posted before he edited his post, though I
was confused with one of the steps he provided.
Re: integer proof
I was skeptical at first, but it seems that induction is the way to go on this one.
Glad you responded, I wasn't sure if you knew what an inductive proof was, so that saved me a lot of typing.
Inductive assumption:
Inductive assumption, 216, and 600 are all divisible by 4.
Last edited by Ricky (2006-02-04 10:24:26)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: integer proof
I think Enfy said any number greater than or equal to zero. I dont think that there is anything to prove otherwise. If p is an integer, then p + 1 would also be an integer the equation would only
produce a series like;
(27n + 25n)/4 = 54n/4 = 13n, which of course would all be integers.
This is obviously not a proof, because I mistakenly used an arithmetic series example instead of a geometric series, but I think that it can be shown to be similar since the relationship of multiples
still holds. I will try it that way.
Last edited by irspow (2006-02-04 10:30:04)
Re: integer proof
If p is an integer, then p + 1 would also be an integer the equation would only produce a series like;
(27n + 25n)/4 = 54n/4 = 13n, which of course would all be integers.
Let n = 1:
So you are saying there is an integer p ≥ 1 such that (3*9^(p+1)+25^(p+1))/4 = 13?
Furthermore, you are saying that 9^(p+1) = 25^(p+1) for all p, since n = n
Last edited by Ricky (2006-02-04 10:22:42)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: integer proof
Thanks Ricky, I got it now :]
I need to learn those tricks with adding more terms, very clever.
Re: integer proof
Oh, and I meant to say integer larger or equal to 0 in my post, since this was about integers and nothing else. Sorry if that confused you, irspow.
Re: integer proof
I am always very confused, but thanks for the sympathy.
Re: integer proof
When you have experience with proving statements like this, you know what the poster means even if (s)he doesn't say it.
That's really all it is, experience, nothing else irspow.
Oh, and I meant to say integer larger or equal to 0 in my post, since this was about integers and nothing else.
Was this a question that a teacher/professor gave you? It should be worded 9^p and 25^p for p ≥ 1, aka, natural numbers.
It's not wrong the way it is, just...weird.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: integer proof
how do i solve the following?
write down the integer value of n which satisfy the inequality -2<n1
solve these inequalities: 2x - 5 < 10
x/3 > 6
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Incomplete Data by Design: Bringing Machine Learning to Marketing Research
May 6, 2013
By Joel Cadwell
Survey research deals with the problem of question wording by always asking the same question. Thus, the
Gallup Daily Tracking
is filled with examples of moving averages for the exact same question asked precisely the same way every day. The concern is that even small changes in the wording can have an impact on how people
perceive and respond to the question. The question itself becomes the construct and our sole focus.
Yet, such a practice severely limits how much marketing research can learn about its customers. Think of all the questions that a service provider, such as a mobile phone company, would want to ask
its customers about both their personal and work-related usage: voice, text, and data performance, reliability, quality, speed, internet, email, product features, cost, billing, support, customer
service, bundling, multitasking, upgrades, promotions, and much more. Moreover, if we want actionable findings, we must "drill" down into each of these categories seeking information about specific
attributes and experiences. Unfortunately, we never reach the level of specificity needed for action. The respondent burden quickly becomes excessive because we believe that every respondent must be
shown every item. Our fallback position is a reliance on summary evaluative judgments, for example, satisfaction ratings asked at a level of abstraction far removed from actual customer experiences.
We find the same constraints in many customer surveys, whether those questionnaires ask about usage, performance, perceptions, or benefits.
What if we did not require every respondent to complete every item?
Obviously, we could no longer compare respondents question by question. This would be a problem only if each question were unique and irreplaceable. But few believe this to be the case. Most of us
accept some version of latent variable modeling and hold that answers to questions are observed or manifest indicators of a smaller number of latent constructs. In educational testing, for example,
we have an item pool from which we can sample individual items. Actually, with
computer adaptive testing
one can tailor the item selection process to match the respondent's ability level. If we have many items measuring the same construct, then we are free to sample items as we would respondents.
Generalizability theory
is a formal expression of this concept. The construct should be our focus. The individual ratings serve a secondary role as indicators.
In a previous post on
reducing respondent burden
, I outlined an approach using item response theory as the basis for item sampling. If you recall, I used a short airline satisfaction questionnaire with 12 items as my example. The ratings were the
usual summary evaluative judgments measuring performance on the typical five-point scale with only the endpoints labeled with the words "poor" and "excellent." As I showed in another
, such rating scales tend to yield a strong first principal component or halo effect. When the ratings are context-free, respondents can answer without needing to recall any encounter or usage
occasion. However, this is precisely what we are trying to avoid by using ratings of specific experiences or needs in the actual usage context.
For example, I might be reluctant to ask you to rate the importance of call quality without specifying explicit criterion because it would require far too much respondent interpretation. Without
specificity, the respondent is forced to "fill-in-the-blanks" and the researcher has no idea what the respondent was thinking when they responded to the question. Instead, I would ask about dropping
calls or needing to repeat what was said to a client while traveling on business. We are not as likely to find a single dimension underlying ratings at such high levels of specificity.
Multidimensional item response theory (
) is one alternative, but I would like to suggest the R package
as a flexible and comprehensive solution to the matrix completion problem. Perhaps we should visit the machine learning lab to learn more about matrix completion and recommender systems.
The Matrix Completion Problem and Recommender Systems
In 2006 Netflix released a data set with almost 500,000 respondents (rows) and almost 18,000 movies (columns). The entries were movie ratings on a scale from 1 to 5 stars, although close to 99% of
the matrix was empty. The challenge was to complete the matrix or fill-in the missing values, since an individual rates only a small proportion of all the movies. That is, in order to make personal
recommendations for an individual respondent, we need to estimate what their ratings would have been for all the movies they did not rate. One solution was to turn to
matrix factorization
and take advantage of the structure underlying both movies and viewers preferences. Matrix completion can be run in R using the new package softImpute.
Although the mathematics behind
can be challenging, the rationale should sound familiar. Movies are not unique. You can see this clearly in the maps from the earlier link to
matrix factorization
. If nothing else, genre creates similarity, as does leading actor and director. When asked to sort movies based on their similarity, most viewers tend to make the same kinds of distinctions using
many of the same features. In fact, we could create a
content-based recommender system
based solely on movie similarity.
Moreover, customers are not unique. The same features that we use to describe similarity among movies will serve to define preference heterogeneity among respondents. This reciprocity flows from the
mutual co-evolution of movies and viewer preferences. One successful film release gives rise to many clones, which are seen by the same viewers because they are "like" the original movie.
Consequently, these large matrices are not as complicated as they seem at first. They appear to be of high dimension, but they have low rank, or alternatively we might say that the observed ratings
can be explained by a much smaller number of latent variables. An example from marketing research might help.
My example is a study where 1603 users were persuaded with incentives to complete a long battery of 165 importance ratings. The large number of ratings reflected the client's need to measure interest
in specific features and services in actual usage occasions. The complete matrix contains 1603 rows and 165 columns with no missing data. As you can see in the scree plot below, the first principal
component accounted for a considerable amount of the total variation (35%). There is a clear elbow, although the eigenvalues are still greater than one for the first twenty or so components. The
substantial first principal component reflects the considerable heterogeneity we observe among users. This is common for product categories with customers who run the spectrum from basic users with
limited needs to premium users wanting every available feature and service.
To illustrate how the R package softImpute works, I will discard 85% of the observed ratings. That is, I will randomly select for each respondent 140 of the 165 ratings and set these ratings to
missing (NA). Thus, the
complete matrix
had 165 ratings for every respondent, and the
matrix to be completed
has 25 randomly selected ratings for each respondent.
In order to keep this analysis simple, I will use the defaults from softImpute R package. The function is also called softImpute(), and it requires only a few arguments. It needs a data matrix, so I
have used the function as.matrix() to convert my data frame into a matrix. It needs a rank.max value to restrict the rank of the solution, which I gave a value of 25 because of the location of the
elbow in the scree plot. And finally the function requires a value for lambda, the nuclear-norm regularization parameter. Unfortunately, it would take some time to discuss
and how best to set the value of this parameter. It is an important issue, as is the question of the conditions under which matrix completion might or might not be successful. But you will have to
wait for a later post. For this analysis, I have set lambda to 15 because the authors of softImpute, Hastie and Mazumder, recommend that lambda should be slightly less than rank.max and most of the
examples in the manual set lambda to value of about 60% of the size of rank.max. As you will see, these values appear to do quite well, so I accepted this solution.
• fit <- softImpute(importance_missing, rank.max=25, lambda=15)
• importance_impute<-complete(importance_missing, fit)
This is the R-code needed to run softImpute. The matrix with the missing importance ratings is called importance_missing. The object called importance_impute is a 1603 x 165 completed matrix with the
missing value replaced by "recommendations" or imputed values.
How good is our completed matrix? At the aggregate level we are able to reproduce the item means almost perfectly, as you can see from the plot below. The x-axis shows the item means from the
completed matrix after the missing values have been imputed. The item means calculated from the complete data matrix are the y-values that we want to reproduce. The high R-squared tells us that we
learn a considerable amount about the importance of the individual items even when each respondent is shown only a small randomly selected subset of the item pool.
It is useful to note the spread in the mean item ratings across the 7-point scale. We do not see the typical negative skew with a large proportion of respondents falling in the top-box of the rating
scale, which is so common when importance is measured using a smaller number of more abstract statements. Of course, this was one of the reasons our client wanted to get "specific" and needed to ask
so many items.
As might be expected, the estimation is not as accurate at the level of the individual respondent. We started with complete data, randomly sampled 15% of the items, and then completed the matrix
using softImpute. The average correlation between the original and imputed ratings for individual respondents across the 165 items was 0.65. Given that we deleted 85% of each respondent's ratings, an
average R-squared of 0.42 seems acceptable. But is it important to reproduce every respondent's rating to every question? If this were an achievement test, we would not be concerned about the
separate items. Individual respondent ratings on each item contain far too much noise for this level of analysis.
On the other hand, the strong first principal component that I reported earlier in this post justifies the calculation of a total score for each respondent, and our imputed total score performs well
as a predictor of the total score calculated from the complete data matrix (R-squared = 0.93). In addition to the total score, it is reasonable to expect to find some specific factors or latent
variables from which we can calculate factor scores. I am speaking of a
bifactor structure
where covariation among the items can be explained by the simultaneously effects of a general factor plus one or more specific factors. I was able to identify the same four specific factors for the
complete data and the imputed data from the softImpute matrix completion. The average correlation among the four factor scores was 0.79, suggesting that we were able to recover the underlying factor
structure even when 85% of the data were missing.
So it seems that we can have it all. We can have long batteries of detailed questions that require customers to recall their actual experiences with specific features and services in real usage
occasions. In fact, given an underlying low-dimensional factor structure, we need randomly sample only a small fraction of the item pool separately for each respondent. Matrix factorization will
uncover that structure and use it to fill-in the items that were not rated and complete the data matrix. Redundancy is a wonderful thing. Use it!
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Author’s Note: This is part 2 of a series of posts about my adventures in building a “large”, in-memory hash table. Part 1 introduced our goals and our approach to the task at hand. This post is a
summary of some of the research I’ve done to familiarize myself with the problem.
Our current approach to custom attribution models is very simple: pay Amazon thousands of dollars a month and “do it in the cloud” with Elastic MapReduce. In Hadoop, we partition the data by user,
sort by time, identify their conversion events, and run an attribution model on these conversion-terminated “chains” of events. This is both costly and more cumbersome than we’d like.
A faster, cheaper, and arguably more transparent approach might be to pipe the live events to a service that could buffer and assemble these chains in memory and output “completed” chains (when a
conversion event arrived) to a separate service to do the model computation.
We’ve come to the conclusion that a large in-memory hash table could be suitable to the task. Our specifications for said hash table are:
• 1.5 billion 64-bit keys, uniformly and randomly distributed
• values between 16 bytes and 16 kilobytes
• deployed to one machine, all in-memory
• sustained 200,000 updates per second over the course of a 14-hour “internet day”
Before jumping into building one of these, I thought I’d learn a bit about hash tables themselves.
The Research
Naturally, my research began with Wikipedia. The article on hash tables is a fairly comprehensive overview. From there, I read handful of papers and articles to dig a little deeper. Below are a
selection that helped me immensely.
Dynamic Hashing
Hash Functions
• Jenkins Hash
A solid general-purpose hash whose source and documentation are a masterwork of explication and thoroughness.
• (Minimal) Perfect Hashing: some theory, some practice
For when you have all of your keys ahead of time and want 100% occupancy.
Collision Resolution
After my academic explorations, I started to look for candidate data stores. In doing this, I began digging into the history of key-value stores and hash table implementations. A few things jumped
out immediately:
• Engineering effort seems to have been diverted from hash table development to distributed hash table development, in the past 5 years.
• Dynamic hashing innovation seems to have stopped at linear and extendible hashing.
• No benchmarks I ran into exceeded 100M insertions. In fact, this benchmark is the only one that I found that exceeded 10M insertions.
The first seems obvious given the meteoric rise in data captured from the web and the relatively fixed decrease in RAM price and increase in density. With dozens or hundreds of terabytes of “online”
data, one can hardly be blamed for steering toward mid-range commodity servers en masse. However, this approach comes at a cost: coordinating and maintaining a cluster of servers is no mean feat. In
fact, I consider the consensus and commitment protocols that make said coordination possible significantly more challenging to understand, let alone implement, than any of the hashing subjects
mentioned above. (Just look at the Wikipedia entry for Paxos!) Similarly, hot-node issues and debugging distributed systems strike me as being an order of magnitude harder to solve than the problem
of building a “better” hash table. To be clear, I’m not arguing that these two things solve the same problems. Rather, given the choice of implementing a “huge”, performant hash table in memory or
the algorithms to support a clustered solution, I would choose the former.
Despite the fact that the progress of Dynamo, Cassandra, Riak, and Voldemort took most of the headlines from 2005 to 2010, work still progressed on in-memory and disk-based non-distributed hash
tables like Tokyo Cabinet and Kyoto Cabinet, Redis, and even the venerable Berekely DB. (If you’re at all interested in the history of “NoSQL” data stores, you should check out this handy timeline.)
That said, little in terms of novel hash table technology came from these efforts. As far as I know, BDB still uses a variant of linear hashing, Redis uses standard chaining, and Kyoto Cabinet falls
back on std::unordered_map for its in-memory hash table.
This brings us to the other two points: indeed, how could traditional hash table development cease (practically) in light of the advances of DHTs? With “web-scale” data sets even a single node’s data
storage needs should easily exceed anything seen 5 years prior, right? In fairness, some work has been done in the last few years to add concurrency to linear hashing as well as some work on
optimizing hash table algorithms for modern cache hierarchies, but this doesn’t feel like the same kind of fundamental, basic result as, say, the introduction of extendible hashing.
I suppose the fact that there has been little visible engineering progress on this front is a testament to the quality of the existing algorithms and code. Either that or existing workloads have not
yet exceeded that high watermark of 100M entries and we’re just waiting for the next jump to inspire new work in the field.
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Electronic Journal of Biotechnology
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On-line version ISSN 0717-3458
Electron. J. Biotechnol. vol.9 no.4 Valparaíso July 2006
Biotechnology Industry Process Biotechnology
Electronic Journal of Biotechnology ISSN: 0717-3458 Vol. 9 No. 4, Issue of July 15, 2006
© 2006 by Pontificia Universidad Católica de Valparaíso -- Chile Received August 18, 2005 / Accepted December 29, 2005
DOI: 10.2225/vol9-issue4-fulltext-7
Optimization of culture conditions for exopolysaccharides production in Rhizobium sp. using the response surface method
Flávia Pereira Duta*
Escola de Química
Dep. de Engenharia Bioquímica
Universidade Federal do Rio de Janeiro
Ilha do Fundão, Rio de Janeiro
RJ, 21949-900, Brasil
Tel: 55 21 25627621
Fax: 55 21 25627667
E-mail: fpduta@hotmail.com
Francisca Pessôa de França*
Escola de Química
Dep. de Engenharia Bioquímica
Universidade Federal do Rio de Janeiro
Ilha do Fundão, Rio de Janeiro
RJ, 21949-900, Brasil
Tel: 55 21 25627621
Fax: 55 21 25627667
E-mail: fpfranca@eq.ufrj.br
Léa Maria de Almeida Lopes
Instituto de Macromoléculas Professora Eloísa Mano
Universidade Federal do Rio de Janeiro
Ilha do Fundão, Rio de Janeiro
RJ, 21945-970, Brasil
Tel: 55 21 25627232
Fax: 55 21 25627232
E-mail: lealopes@ima.ufrj.br
*Corresponding authors
Financial support: CNPq (Conselho Nacional do Desenvolvimento Científico e Tecnológico), and CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior).
Keywords: exopolysaccharides, experimental design, response surface method, Rhizobium sp.
Abbreviations: EPS: exopolysaccharides
HPLC: high performance liquid chromatography
YMA: yeast mannitol agar
Abstract Reprint (BIP) Reprint (PDF)
The combined effects of the processing parameters for exopolysaccharides production by Rhizobium sp. was studied using the experimental design and response surface methodology. The experiments were
carried out using a fermenter with 20 L capacity, as the reactor. All processing parameters were online monitored. The temperature [(30 ± 1)ºC] and pH value (7.0 ± 0.1) were kept constant throughout
the experimental time. As statistical tools, a complete 2^3 factorial planning with central point and response surface were used to study the interactions among three relevant variables of the
fermentation process: calcium carbonate concentration, aeration and agitation. The processing parameters setup for reaching a maximum response for exopolysaccharides production was obtained when
applying the highest values for calcium carbonate concentration (1.1 g/L), aeration (1.3 vvm) and agitation (800 rpm). In addition, the combination of these optimum processing parameters yielded Y[P/
S] (g/g) = 0.35.
Bacterial exopolysaccharides are extensively used as thickening and gelling agents in a wide range of industrial products and processes due to their structural and physical properties diversity (
Copetti et al. 1997; Rinaudo, 2001). The structure, composition and viscosity of the microbial polysaccharides depend on several factors, such as the composition of the culture medium, carbon and
nitrogen source, mineral salts, trace elements, type of strain, and fermentation conditions (pH, temperature, oxygen concentration, agitation) (Moreira et al. 1998; Weuster-Botz, 2000; Pinto et al.
2002; Duta et al. 2004).
For many microbial species, calcium is essentially required in small amounts: it is essential in maintaining cell wall rigidity, it stabilizes oligomeric proteins and covalently bound protein
peptidoglycan complexes in the outer membrane, as well as have a requirement for chemotaxis (Macció et al. 2002).
The reach of optimized fermentation conditions, particularly associated to physical and chemical parameters, is of primary and great importance for the development of any process, due to their impact
upon its economics and practicability. The diversity of combinatory interactions among medium components, metabolism of cells and the large number of chemical requirements for processing metabolic
products, do not allow satisfactory detailed modelling. The one-dimensional search with successive changes on variables conditions is still employed, even trough it is well accepted that it is
practically impossible for the one-dimensional search to accomplish an appropriate optimum combination in a finite number of experiments. Single variable optimization methods are not only tedious,
but can also lead to misinterpretation of results, especially taking into account that the interaction between different factors is overlooked (Abdel-Fattah et al. 2005). Statistical experimental
designs have been used for many decades and can be adopted on several steps of an optimization strategy, such as for screening experiments or searching for the optimal conditions of a targeted
response (Kim et al. 2005; Lee and Gilmore, 2005; Nawani and Kapadnis, 2005; Senthilkumar et al. 2005; Wang and Lu, 2005). Recently, the results analyzed by a statistical planned experiment are
better acknowledged than those carried out by the traditional one-variable-at-a-time method. Some of the popular choices, applying statistical designs to bioprocessing, include the Plackett-Burman
design (Liu et al. 2003; Wang and Lu, 2005) and response surface methodology with various designs (Abdel-Fattah, 2002; Abdel-Fattah and Olama, 2002; Tanyildizi et al. 2005). The response surface
methodology is an empirical modelling system that assesses the relationship between a group of variables, which can be controlled experimentally, and the observed response. This methodology is
applied mainly both in food science and in the optimization of fermentative processes. It is a 2-level factorial design, where contour plots are generated by linear or quadratic effects of key
variables, and a model equation is derived, fitting the experimental data to the calculate system's optimal response (Lakshman et al. 2004; Cazetta et al. 2005; Khanna and Srivastava, 2005).
Fast growing rhizobia synthesize different extracellular polysaccharides, like acid exopolysaccharides (EPS) of high molecular weight (Zevenhuizen, 1986). Those microorganisms can also produce
neutral glucans, formed by β-1,4 bonds, which are found as cellulosic microfibrils of low molecular weight (Zevenhuizen, 1986; Breedveld et al. 1990; Jain et al. 1990; Breedveld et al. 1993).
Many rhizobia stains produced polysaccharides are not found freely dispersed in the medium, but attached upon the microbial cell as an amorphous viscous material. Among these, the curdlana
homo-polymer is outstanding, in which the derived sulphated sites show anticoagulant and anti-thrombosisactivities. Besides, they also present an inhibiting effect against the HIV-1 virus in vitro
infection (Jagodzinski et al. 1994). Curdlana sulpho-alkiled sites present anti-tumour activity, inhibiting the development of the Sarcoma tumours 180 (Bohn, 1995).
The present study investigated the relationship among three variables, that is, calcium carbonate concentration, agitation, and aeration, in exopolysaccharide production by Rhizobium sp.
The Rhizobium EQ1 strain was used in the experiments. This was isolated from a variety of beans culture, native from an arid region of the Northeast of Brazil, named as "Caupi". This strain has been
kept in yeast mannitol agar (YMA), for several months, being catalogued, after its characterization, at the culture bank of the Biochemical Engineering Department from Federal University of Rio de
Janeiro, under the code EQ1.
The microorganism was grown under (30 ± 1)ºC for 48 hrs, using laboratory tubes filled with YMA medium. After growth, the cultures were stored at (5 ± 1)ºC.
Culture media
The microorganism stock culture was maintained in modified yeast mannitol agar extract, which presented the following composition (g/L): mannitol (10.0); K[2]HPO[4] (0.1); KH[2]PO[4] (0.4); MgSO
[4].7H[2]O (0.2); NaCl (0.1); yeast extract (0.4); agar (15.0). The medium pH was adjusted to 7.0 (Jordan, 1984).
The stock culture was used for preparing the inoculum, using 500 mL Erlenmeyer flasks, containing 100 mL of the YMA medium free from agar. Incubation was carried out in a rotary shaker at 200 rpm and
(30 ± 1)ºC, for 48 hrs, with the cells in the final of exponential growth phase.
The exopolysaccharides production assays were carried out using a similar culture medium, where the YMA medium was: supplemented with manganese ions (MnCl[2]. 4H[2]O 0.12 g/L); free from agar; and
the calcium carbonate concentration ranged in accordance to the experimental design (CaCO[3] 0.5-1.0 g/L). The pH of this medium was adjusted to 7.0 by the addition of NaOH (50% w/v). All culture
media were sterilized at 121ºC for 20 min.
The inoculation of the production medium was made in order to obtain an average cells concentration of 0.77 ± 0.02 mg/mL.
Production of exopolysaccharides in bioreactors
These experiments were conducted in a fermenter (Model BioFlow IV, New Brunswick Scientific), with 20 L capacity, equipped with disc impeller, oxygen and pH electrodes. The equipment also monitored
temperature, agitation speed, gas purging flow rate, pumping rates, antifoam addition and the vessel level. All processing parameters were online monitored, with the aid of AFS 3.0 software (Advanced
Fermentation Software, New Brunswick Scientific). The temperature (30 ± 1ºC) and pH value (7.0 ± 0.1) were kept constant during the experiments. Other parameters, like substrate concentration,
aeration and agitation, were chosen as the most significant ones, considering the experimental design. After selecting those parameters, experiments were done in duplicate, for superior (+) and lower
(-) levels of the experimental design, and in triplicate, for the central point (0).
For each experiment, 1000 mL of the inoculum was used, that is, 10% (v/v) of the initial working volume (10 L). The process was conducted throughout 48 hrs.
Analytical methods
During the process, microscopic examinations, using Gram method, were performed in order to detect possible microbial contaminations in the medium.
Prior for the quantitative determination of mannitol, the fermented broth was filtered through 0.2 µm Millipore membranes, in order to remove microbial cells. In the filtered fluid, the substrate was
analyzed by high performance liquid chromatography (HPLC), in a Waters chromatograph, equipped with SHODEX SC1011 ion-exchange columns, at 75ºC. Reagent water type I (ASTM, 2001) was used as eluent,
and the elution rate applied was 0.8 mL/min.
The amount of fermented exopolysaccharide was determined by dry-weight measurements. The fermented broth was heated at (80 ± 1)ºC, for 10 min, to ensure microbial inactivation. Afterwards, the
microbial cells were removed by a filtration step. In order to precipitate the exopolysaccharides, a solution of ethanol P.A. and reagent water type I (ASTM, 2001) (3:1) was added to the fermented
broth. After the exopolysaccharides total precipitation, the suspended material was filtered through 0.2 µm Millipore membrane, using Gouche crucible previously weighed. The obtained product was
dried at (80 ± 1)ºC until constant weight. All determinations were done in triplicate.
The exopolysaccharides extracted from the fermented broth was purified through successive washings with solutions of ethanol P.A. and reagent water type I (ASTM, 2001) at 70, 80 and 90% (v/v),
respectively. The product was finally dried by a nitrogen gas purging flow, under controlled heating.
Experimental design and statistical analysis
This statistical technique is widely used as a tool to verify the efficacy of several processes. In the present work, it has been used for obtaining pieces of information about the exopolysaccharides
production process; thus, a reduction in operational costs can be expected. A 2^3 factorial planning, in duplicate, with central point in triplicate, was used (Box et al. 1978; Neto et al. 1995).
Additionally, each experiment was repeated once, around the central point neighbourhood, while the central point was repeated twice, leading to a set of 19 experiments (Table 1). Three central points
were added to estimate the experimental error and to investigate the suitability of the proposed model. Table 2 lists the independent variables studied, X[1] calcium carbonate concentration (g/L), X
[2] aeration (vvm), X[3] agitation (rpm). The manipulation responses of the input variables were evaluated as a function of the substrate conversion into exopolysaccharide, coded by Y[p/s] (g/g).
A mathematical model, describing the relationships among the process dependent variable, Yp/s, and the independent variables in a second-order equation, was developed. Design-based experimental data
were matched according to the following second-order polynomial equation:[
][equation 1]
where, Y = substrate conversion into exopolysaccharides (Y[p/s]),
β[0] = constant,
β[i] = linear terms coefficients,
β[ii] = quadratic terms coefficients,
β[ij] = interaction coefficients.
All the calculations involved as well as the drawing of all three-dimensional surface (3D) have been obtained using the Statistica^TM Software for Windows, Version 5.5 computer package, produced by
Stat Soft. The model allowed the evaluation of the effects of linear, quadratic and combined effects of the independents variables upon the variable dependent variable. The Student's t-test was
employed in order to check the statistical significance of the regression coefficients. The Fisher's F-test for analysis of variance (ANOVA) was performed on experimental data to evaluate the
statistical significance of the model.
Three-dimensional surface (3D) plots were drawn to illustrate the main and interactive effects of the independent variables on exopolysaccharides production. The optimum values of the selected
variables were obtained both by solving the regression equation and also by analyzing the response surface contour plots (Myers and Montgomery, 2002).
The statistical technique is widely used as a tool for checking the efficiency of several processes. In the present work it has been used with the purpose of obtaining information about the
exopolysaccharides production process; consequently, a reduction in the operational variability and a cut down in operational costs can be expected.
The experimental results (Y[p/s]), associated to the processing set up of each independent variables are listed in Table 3. Using the designed experimental data presented in Table 3, the polynomial
proposed model for Y[p/s ]was regressed by only considering the significant terms. The expanded equation is shown below:
Y[p/s] = -0.361612 + (1.147835 X[1]) + (-0.714687 X[2]) + (-0.000081 X[3]) + (0.205141 X[1]X[2]) + (0.000153 X[1]X[3]) + (0.000052 X[2]X[3])
Besides the linear effect of the substrate/ exopolysaccharides factor, Y[p/s], the response surface method also gives an insight about the parameters quadratic and combined effects. These analyses
were done by using both Fisher's F- test and Student t-test statistical tools. The student t-test was used to determine the significance of the parameters regression coefficients. The p-values were
used as a tool to check the significance of the interaction effects, which in turn may indicate the patterns of the interactions among the variables. In general, larger magnitudes of t and smaller of
p, indicates that the corresponding coefficient term (Myers and Montgomery, 2002).
The regression coefficient, t and p values for all the linear, quadratic and combined effects are given in the Table 4, with a 95% significance level. It was observed that the coefficients for the
linear and quadratic effects of the factor calcium carbonate concentration parameter, the linear effect of the aeration parameter, the combined effects between calcium carbonate concentration with
the aeration and agitation parameters (p = 0.000 for all) were highly significant. The value observed for the factor agitation was slightly less significant (p = 0.014).
The statistical significance of the ratio, between the of mean square variation, due to regression, and the mean square residual error, was tested using analysis of variance (ANOVA). ANOVA is a
statistical technique that subdivides the total variation of a set of data into component associated to specific sources of variation for the purpose of testing hypotheses for the modelled
parameters. According to the ANOVA (Table 5), the F-values for all regressions were high, what indicates that most of the variations on the response variable can be explained by the regression
equation. The associated p-value is used to estimate whether F is large enough to indicate statistical significance. A p-value lower than 0.01 indicates that the model is considered to be
statistically significant (Kim et al. 2003).
The p values of all of the regression were lower than 0.01. This means that at least one of the terms in the regression equation has a significant correlation with the response variable. The ANOVA
table also shows a term for residual error, which measures the amount of variation in the response data left unexplained by the model. The type of the model, chosen to explain the relationship
between the factors and response, is correct.
The analysis of variance (ANOVA), indicates that the second-order polynomial model (Equation 1) was highly significant and adequate to represent the actual relationship between the response and input
variables, with very small p values (p = 0.0000).
Pareto chart (Figure 1) corroborates the data shown in Table 5, and also enhances the understanding of this table. Figure 1 shows that all the linear terms of the model were significant for the set
confidence level, as well as the quadratic term of X[1] variable (Table 5). It can also be seen that the interaction term among the three variables (X[1]X[2]X[3]), was not statistically significant (
Figure 1 and Table 5). Statistical data analysis shows that calcium carbonateconcentration is the most important variable for the production process.
The matching quality, of the data obtained by the model proposed in Equation 1, was evaluated considering the correlation coefficient, R^2, between the experimental and modelled data. The
mathematical adjust of those values generated a R^2 = 0.9884, revealing that the model could not explain only 1.16% of the overall effects, showing that it is a robust statistical model.
Figure 2 shows the regression plot of the Y[p/s] experimental values against those predicted by Equation 1, revealing a linear mathematical relation among them. In addition, the mismatching analysis
and further error terms found in Table 5, agree the adequacy of the predicted data, thus the reliability of the model proposed by Equation 1.
The 3D response surface plots described by the regression model were drawn to illustrate the effects of the independent variables, and combined effects of each independent variable upon the response
variable. Figure 3 illustrates 3D response surface based on the Y[p/s] response against the variation of calcium carbonate concentration (X[1]) and aeration (X[2]) independent variables upon Y[p/s],
while the third independent variable, agitation (X[3]), was kept constant level (800 rpm). It can be observed that the maximum estimated Y[p/s] 0.3431 (g/g) was obtained using calcium carbonate
concentration of 1.0 g/L and aeration of 1.3 vvm.
The data obtained by varying calcium carbonate concentration (X[1]) and agitation (X[3]), fixing aeration at 1.3 vvm, can be observed in Figure 4. The analysis of Figure 4 reveals that the maximum
substrate conversion into exopolysaccharides was also obtained under the following condition: at calcium carbonate concentration of 1.0 g/L and agitation of 800 rpm.
Figure 3 and Figure 4 also show that an increase on agitation and aeration parameters promotes an increase on Y[p/s]. For each combination of agitation and aeration values, the model generates a
maximum value for the concentration of calcium carbonate. Thus, for the maximum values used in the experimental design, 1.3 vvm for the aeration and 800 rpm for agitation, the predicted value for of
calcium carbonate concentration is 1.1 g/L.
The application of the regression model (Equation 1) for the substrate/exopolysaccharides factor, Y[p/s], was tested using calcium carbonate concentration of 1.1 g/L, agitation of 800 rpm and
aeration of 1.3 vvm, with triplicate experiments. For this experimental condition, Y[p/s] mean value was of 0.3574 ± 0.008448 (g/g), which agrees with the predicted value, 0.3471 (g/g). This
verification revealed a high accuracy of the model, that is, 97.12%, which is an evidence of the model validation, under the investigated conditions.
The process for exopolysaccharides production is carried out under aeration and agitation. The control of such parameters is of great importance for adequately conducting the fermentation process.
According to Brock and Madigan (1991), for a rise in biomass from aerobic microorganisms, a vigorous aeration is required, what should be reached by forced aeration. This induced aeration is
essential for getting high performance responses form the process, since oxygen is slightly soluble in water, not being quickly replaced by air diffusion, and worthwhile for microbial growth.
Zevenhuizen (1986), using a mannitol-rich culture medium, has directed the polysaccharide synthesis towards exopolysaccharides by applying forced aeration.
The efficiency in conducing forced aeration is linked to agitation, which favours oxygen diffusion in the medium and its transfer to cells. Agitation also promotes a reduction in nutrient particles,
favouring the nutrient homogenization in the culture medium, providing additionally a rise in mass transfer rates, this favouring microbial growth.
In the production medium used for obtaining biopolymers, several ions are added to propitiate the exopolysaccharides, and these shall be in appropriate amounts. The metallic ions perform catalytic
and essential structural functions in proteins, being accumulated inside the cell by active transport (Macció et al. 2002).
The literature tells about the use of calcium carbonate to prevent the acidification of the bacterial broth (Macció et al. 2002). Jordan (1984) suggested the use of 4.0 g/L of calcium carbonate in
the culture media, for controlling pH in the Rhizobium sp culture. O' Hara et al. (1989) previously reported that 1 and 2 mM of calcium was necessary for cytoplasmic pH maintenance in Rhizobium
meliloti acid-sensitive strain. The use of calcium ions can be intimately related to the stabilization of proteins involved in the exopolysaccharides synthesis process (Soto et al. 2004).
In agreement with the data presented in this study the yield income values of the product (Y[p/s]), in a medium containing 1.0 g/L of calcium carbonate was 0.35 g/g, in average. These results are
consistent with the biochemical processes involving polymerization reactions forming high-viscosity products.
Most researches developed using Rhizobium genus bacteria are related to genetics and bacteria-host plant symbiotic interactions issues. Little is known about the production of extra-cellular
polysaccharides by Rhizobium, as well as their properties in solution. In addition, no studies on monitoring medium composition, agitation and aeration parameters effects, on exopolysaccharides
production, were found.
The analysis of the response surfaces obtained by the experimental design, with central and having Y[p/s ]as response variable, shows the existence of a point of maximum production when agitation of
800 rpm, aeration of 1.3 vvm and calcium carbonate concentration of 1.1 g/L are applied.
The introduction of calcium carbonate, in the composition of the culture media, associated with high agitation and aeration, promoted a significant rise in the substrate/product yield (Y[p/s]) (Duta
et al. 2004).
Based on the present study, it is evident that the use of statistical optimization tools, response surface methodology, has helped to locate the optimum levels of the most significant parameters for
exopolysaccharides production, with minimum effort and time.
ABDEL-FATTAH, Yasser Refaat. Optimization of thermostable lipase production from a thermophilic Geobacillus sp. using Box-Behnken experimental design. Biotechnology Letters, July 2002, vol. 24, no.
14, p. 1217-1222. [ Links ][CrossRef]
ABDEL-FATTAH, Yasser Refaat and OLAMA, Zakia A. L-asparaginase production by Pseudomonas aeruginosa in solid-state culture: evaluation and optimization of culture conditions using factorial designs.
Process Biochemistry, September 2002, vol. 38, no. 1, p. 115-122. [ Links ][CrossRef]
ABDEL-FATTAH, Yasser Refaat; SAEED, Hesham M.; GOHAR, Yousry M. and EL-BAZ, Mohamed A. Improved production of Pseudomonas aeruginosa uricase by optimization of process parameters through statistical
experimental designs. Process Biochemistry, April 2005, vol. 40, no. 5, p. 1707-1714. [ Links ][CrossRef]
AMERICAN SOCIETY FOR TESTING MATERIAL (ASTM). Standard specification for reagent water. ASTM D-1193, October 2001, p. 1-3. [ Links ]
BOHN, John A. and BEMILLER, James N. (1 >3)-b-D-glucans as biological response modifiers: a review of strutucture-funcional activity relationships. Carbohydrate Polymers, January 1995, vol. 28, no.
1, p. 3-14. [ Links ][CrossRef]
BOX, G.E.P.; HUNTER, W.G. and HUNTER, J.S. Statistics for experimenters. An introduction to design, data analysis and model building. NY, USA. 1978. 653 p. ISBN 0-471-09315-7. [ Links ]
BREEDVELD, M.W.; ZEVENHUIZEN, L.P.T.M. and ZEHNDER, A.J.B. Excessive excretion of cyclic β-(1,2)-glucan by Rhizobium trifolii TA-1. Applied and Environmental Microbiology, July 1990, vol. 56, no. 7,
p. 2080-2086. [ Links ]
BREEDVELD, M.W.; ZEVENHUIZEN, L.P.T.M.; CANTER CREMERS, H.C.J. and ZEHNDER, A.J.B. Influence of growth conditions on production of capsular and extracellular polysaccharides by Rhizobium
leguminosarum. Antonie van Leeuwenhoek, March 1993, vol. 64, no. 1, p. 1-8. [ Links ][CrossRef]
BROCK, T.D. and MADIGAN, M.T. Biology of Microorganisms. New Jersey: Prentice-Hall International Editions, 6 ed., 1991. 874 p. [ Links ]
CAZETTA, M.L.; CELLIGOI, M.A.P.C.; BUZATO, J.B.; SCARMINO, I.S. and DA SILVA, R.S.F. Optimization study for sorbitol production by Zymomonas mobilis in sugar cane molasses. Process Biochemistry,
February 2005, vol. 40, no. 2, p. 747-751. [ Links ][CrossRef]
COPETTI, Giuliano; GRASSI, Mario; LAPASIN, Romano and PRICL, Sabrina. Synergic gelation of xanthan gum with locust bean gum: a rheological investigation. Glycoconjugate Journal, January 1997, vol.
14, no. 8, p. 951-961. [ Links ][CrossRef]
DUTA, Flávia Pereira; DA COSTA, Antonio Carlos Augusto; LOPES, Léa Maria De Almeida; BARROS, Ana; SÉRVULO, Eliana Flávia Camporese and de FRANÇA, Francisca Pessôa. Effect of Process Parameters on
Production of a Biopolymer by Rhizobium sp. Applied Biochemistry and Biotechnology, April 2004, vol. 114, no. 1-3, p. 639-652. [ Links ]
GORE, R.S. and MILLER, K.J. Cell surface carbohydrates of microaerobic, nitrogenase-active, continuous culture of Bradyrhizobium sp. strain 32H1. Journal of Bacteriology, December 1992, vol. 174, no.
23, p. 7838-7840. [ Links ]
JAGODZINSKI, Paul P.; WIADERKIEWICZ, Richard; KURZAWSKI, Grzegorz; KLOCZEWIAK, Marek; NAKASHIMA, Hideki; HYJEK, Elizabeth; YAMAMOTO, Naoki; URYU, Toshiyuki; KANEKO, Yutaro; POSNER, Marshall R. and
KOZBOR, Danuta. Mechanism of the inhibitory effect of curdlan sulfate on HIV-1 infection in vitro. Virology, August 1994, vol. 202, no. 2, p. 735-745. [ Links ][CrossRef]
JAIN, D.K.; PREVOST, D. and BORDELEAU, L.M. Role of bacterial polysaccharides in the derepression of ex-planta nitrogenase activity with rhizobia. FEMS Microbiology Letters, February 1990, vol. 73,
no. 2, p. 167-174. [ Links ][CrossRef]
JORDAN, D.C. Family III. Rhizobiaceae. In: KRIEG, N.R. and HOLT, J. G. eds. Bergey`s Manual of Systematic Bacteriology, Willians and Wilkins, Baltimore, vol. I, 1984, p. 234-342. [ Links ]
KHANNA, Shilpi and SRIVASTAVA, Ashok K. Statistical media optimization studies for growth and PHB production by Ralstonia eutropha. Process Biochemistry, May 2005, vol. 40, no. 6, p. 2173-2182. [
Links ][CrossRef]
KIM, H.O.; LIM, J.M.; JOO, J.H.; KIM, S.W.; HWANG, H.J.; CHOI, J.W. and YUN, J.W. Optimization of submerged culture condition for the production of mycelial biomass and exopolysaccharides by Agrocybe
cylindracea. Bioresource Technology, July 2005, vol. 96, no. 10, p. 1175-1182. [ Links ][CrossRef]
KIM, H.M.; KIM, J.G.; CHO, J.D. and HONG, J.W. Optimization and characterization of UV-curable adhesives for optical communications by response surface methodology. Polymer Testing, December 2003,
vol. 22, no. 8, p. 899-906. [ Links ][CrossRef]
LAKSHMAN, Kshama; RASTOGI, N.K. and SHAMALA, T.R. Simultaneous and comparative assessment of parent and mutant strain of Rhizobium meliloti for nutrient limitation and enhanced polyhydroxyalkanoate
(PHA) production using optimization studies. Process Biochemistry, October 2004, vol. 39, no. 12, p. 1977-1983. [ Links ][CrossRef]
LEE, Kwang-Min and GILMORE, David F. Formulation and process modeling of biopolymer (polyhydroxyalkanoates: PHAs) production from industrial wastes by novel crossed experimental design. Process
Biochemistry, January 2005, vol. 40, no. 1, p. 229-246. [ Links ][CrossRef]
LIU, Chuanbin; LIU, Yan; LIAO, Wei; WEN, Zhiyou and CHEN, Shulin. Application of statistically-based experimental designs for the optimization of nisin production from whey. Biotechnology Letters,
June 2003, vol. 25, no. 11, p. 877-882. [ Links ][CrossRef]
MACCIÓ, Daniela; FABRA, Adriana and CASTRO, Stella. Acidity and calcium interaction affect the growth of Bradyrhizobium sp. and attachment to peanut roots. Soil Biology and Biochemistry, February
2002, vol. 34, no. 2, p. 201-208. [ Links ][CrossRef]
MOREIRA, A. da S.; SOUZA, A. da S. and VENDRUSCOLO, Claire T. Determinação da composição de biopolímero por cromatografia em camada delgada: metodologia. Revista Brasileira de Agrociência,
September-December 1998, vol. 4, no. 3, p. 222-224. [ Links ]
MYERS, R.H. and MONTGOMERY, D.C. Response surface methodology: process and product optimization using designed experiments. New York: Wiley, 2002. 824 p. ISBN 0-471-41255-4. [ Links ]
NAWANI, N.N. and KAPADNIS, B.P. Optimization of chitinase production using statistics based experimental designs. Process Biochemistry, February 2005, vol. 40, no. 2, p. 651-660. [ Links ][CrossRef]
NETO, B.B.; SCARMÍNIO, I. and BRUNS, R.E. Planejamento e otimização de experimentos, Editora UNICAMP, Campinas, Brasil, 1995. 299 p. ISBN 8-526-80336-0. [ Links ]
O' HARA, Graham W.; GOSS, Thomas J.; DILWORTH, Michael J. and GLENN, Andrew R. Maintenance of intracellular pH and acid tolerance in Rhizobium meliloti. Applied and Environmental Microbiology, August
1989, vol. 55, no. 8, p. 1870-1876. [ Links ]
PINTO, Ellen; MOREIRA, Angelita and VENDRUSCOLO, Claire T. Influence of pH, addition of salts and temperature in the viscosity of biopolimers produced by Beijerinckia sp. 7070 and UR4. Revista
Brasileira de Agrociência, September-December 2002, vol. 8, no. 3, p. 247-251. [ Links ]
RINAUDO, Marguerite. Relation between the molecular structure of some polysaccharides and original properties in sol and gel states. Food Hydrocolloids, July 2001, vol. 15, no. 4-6, p. 433-440. [
Links ][CrossRef]
SENTHILKUMAR, S.R.; ASHOKKUMAR, B.; RAJ, K. Chandra and GUNASEKARAN, P. Optimization of medium composition for alkali-stable xylanase production by Aspergillus fischeri Fxn 1 in solid-state
fermentation using central composite rotary design. Bioresource Technology, August 2005, vol. 96, no. 12, p. 1380-1386. [ Links ][CrossRef]
SOTO, María José; VAN DILLEWIJIN, Pieter; MARTÍNEZ-ABARCA, Francisco; JIMÉNEZ-ZURDO, José I. and TORO, Nicolás. Attachment to plant roots and nod gene expression are not affected by pH or calcium in
the acid-tolerant alfalfa-nodulating bacteria Rhizobium sp. LPU83. FEMS Microbiology Ecology, April 2004, vol. 48, no. 1, p. 71-77. [ Links ][CrossRef]
TANYILDIZI, M. Saban; OZER, Dursun and ELIBOL, Murat. Optimization of α-amylase production by Bacillus sp. using response surface methodology. Process Biochemistry, June 2005, vol. 40, no. 7, p.
2291-2296. [ Links ][CrossRef]
WANG, Yun-Xiang and LU, Zhao-Xin. Optimization of processing parameters for the mycelial growth and extracellular polysaccharide production by Boletus spp. ACCC 50328. Process Biochemistry, March
2005, vol. 40, no. 3-4, p. 1043-1051. [ Links ][CrossRef]
WEUSTER-BOTZ, Dirk. Experimental design for fermentation media development: statistical design or global random search? Journal of Bioscience and Bioengineering, September-October 2000, vol. 90, no.
5, p. 473-483. [ Links ][CrossRef]
ZEVENHUIZEN, L.P.T.M. Selective synthesis of polysaccharides by Rhizobium trifolli, strain TA-1. FEMS Microbiology Letters, June 1986, vol. 35, no. 1, p. 43-47. [ Links ][CrossRef]
Note: Electronic Journal of Biotechnology is not responsible if on-line references cited on manuscripts are not available any more after the date of publication. | {"url":"http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0717-34582006000400007&lng=en&nrm=iso&tlng=en","timestamp":"2014-04-19T09:41:48Z","content_type":null,"content_length":"83033","record_id":"<urn:uuid:2855747e-8da6-4630-9b94-aff3fc7799c3>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00116-ip-10-147-4-33.ec2.internal.warc.gz"} |
Testing computability by width two OBDDs
"... Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for a prespecified distance measure) from every object with that
property. In this work we design and analyze an algorithm for testing functions for the property of being c ..."
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Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for a prespecified distance measure) from every object with that
property. In this work we design and analyze an algorithm for testing functions for the property of being computable by a read-once width-2 Ordered Binary Decision Diagram (OBDD), also known as a
branching program, where the order of the variables is not known to us. That is, we must accept a function f if there exists an order of the variables according to which a width-2 OBDD can compute f.
The query complexity of our algorithm is Õ(log n)poly(1/ɛ). In previous work (in Proceedings of RANDOM, 2009) we designed an algorithm for testing computability by an OBDD with a fixed order, which
is known to the algorithm. Thus, we extend our knowledge concerning testing of functions that are characterized by their computability using simple computation devices and in the process gain some
insight concerning these devices. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=10297409","timestamp":"2014-04-18T08:48:19Z","content_type":null,"content_length":"12657","record_id":"<urn:uuid:7a54b6f2-5bf2-4226-9525-b230cd4f73d9>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00299-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Speed of Sound
Lesson 2: Sound Properties and Their Perception
The Speed of Sound
A sound wave is a pressure disturbance which travels through a medium by means of particle interaction. As one particle becomes disturbed, it exerts a force on the next adjacent particle, thus
disturbing that particle from rest and transporting the energy through the medium. Like any wave, the speed of a sound wave refers to how fast the disturbance is passed from particle to particle.
While frequency refers to the number of vibrations which an individual particle makes per unit of time, speed refers to the distance which the disturbance travels per unit of time. Always be cautious
to distinguish between the two often confused quantities of speed (how fast...) and frequency (how often...).
Since the speed of a wave is defined as the distance which a point on a wave (such as a compression or a rarefaction) travels per unit of time, it is often expressed in units of meters/second
(abbreviated m/s). In equation form, this is
speed = distance/time
The faster which a sound wave travels, the more distance it will cover in the same period of time. If a sound wave is observed to travel a distance of 700 meters in 2 seconds, then the speed of the
wave would be 350 m/s. A slower wave would cover less distance - perhaps 600 meters - in the same time period of 2 seconds and thus have a speed of 300 m/s. Faster waves cover more distance in the
same period of time.
The speed of any wave depends upon the properties of the medium through which the wave is traveling. Typically there are two essential types of properties which effect wave speed - inertial
properties and elastic properties. The density of a medium is an example of an inertial property. The greater the inertia (i.e., mass density) of individual particles of the medium, the less
responsive they will be to the interactions between neighboring particles and the slower the wave. If all other factors are equal (and seldom is it that simple), a sound wave will travel faster in a
less dense material than a more dense material. Thus, a sound wave will travel nearly three times faster in Helium as it will in air; this is mostly due to the lower mass of Helium particles as
compared to air particles.
Elastic properties are those properties related to the tendency of a material to either maintain its shape and not deform whenever a force or stress is applied to it. A material such as steel will
experience a very small deformation of shape (and dimension) when a stress is applied to it. Steel is a rigid material with a high elasticity. On the other hand, a material such as a rubber band is
highly flexible; when a force is applied to stretch the rubber band, it deforms or changes its shape readily. A small stress on the rubber band causes a large deformation. Steel is considered to be a
stiff or rigid material, whereas a rubber band is considered a flexible material. At the particle level, a stiff or rigid material is characterized by atoms and/or molecules with strong attractions
for each other. When a force is applied in an attempt to stretch or deform the material, its strong particle interactions prevent this deformation and help the material maintain its shape. Rigid
materials such as steel are considered to have a high elasticity (elastic modulus is the technical term). The phase of matter has a tremendous impact upon the elastic properties of the medium. In
general, solids have the strongest interactions between particles, followed by liquids and then gases. For this reason, longitudinal sound waves travel faster in solids than they do in liquids than
they do in gases. Even though the inertial factor may favor gases, the elastic factor has a greater influence on the speed (v) of a wave, thus yielding this general pattern:
v[solids] > v[liquids] > v[gases]
The speed of a sound wave in air depends upon the properties of the air, namely the temperature and the pressure. The pressure of air (like any gas) will effect the mass density of the air (an
inertial property) and the temperature will effect the strength of the particle interactions (an elastic property). At normal atmospheric pressure, the temperature dependence of the speed of a sound
wave through air is approximated by the following equation:
v = 331 m/s + (0.6 m/s/C)*T
where T is the temperature of the air in degrees Celsius. Using this equation is used to determine the speed of a sound wave in air at a temperature of 20 degrees Celsius yields the following
v = 331 m/s + (0.6 m/s/C)*T
v = 331 m/s + (0.6 m/s/C)*20 C
v = 331 m/s + 12 m/s
v = 343 m/s
At normal atmospheric pressure and a temperature of 20 degrees Celsius, a sound wave will travel at approximately 343 m/s; this is approximately equal to 750 miles/hour. While this speed may seem
fast by human standards (the fastest humans can sprint at approximately 11 m/s and highway speeds are approximately 30 m/s), the speed of a sound wave is slow in comparison to the speed of a light
wave. Light travels through air at a speed of approximately 300 000 000 m/s; this is nearly 900 000 times the speed of sound. For this reason, humans can observe a detectable time delay between the
thunder and lightning during a storm. The arrival of the light wave from the location of the lightning strike occurs in so little time that it is essentially negligible. Yet the arrival of the sound
wave from the location of the lightning strike occurs much later. The time delay between the arrival of the light wave (lightning) and the arrival of the sound wave (thunder) allows a person to
approximate his/her distance from the storm location. For instance if the thunder is heard 3 seconds after the lightning is seen, then sound (whose speed is approximated as 345 m/s) has traveled a
distance of
distance = v * t = 345 m/s * 3 s = 1035 m
If this value is converted to miles (divide by 1600 m/1 mi), then the storm is a distance of 0.65 miles away.
Another phenomenon related to the perception of time delays between two events is the phenomenon of echolation. A person can often perceive a time delay between the production of a sound and the
arrival of a reflection of that sound off a distant barrier. If you have ever made a holler within a canyon, perhaps you have heard an echo of your holler off a distant canyon wall. The time delay
between the holler and the echo corresponds to the time for the holler to travel the round-trip distance to the canyon wall and back. A measurement of this time would allow a person to estimate the
one-way distance to the canyon wall. For instance if an echo is heard 2.2 seconds after making the holler, then the distance to the canyon wall can be found as follows:
distance = v * t = 345 m/s * 1.1 s = 380 m
The canyon wall is 380 meters away. You might have noticed that the time of 1.1 seconds is used in the equation. Since the time delay corresponds to the time for the holler to travel the round-trip
distance to the canyon wall and back, the one-way distance to the canyon wall corresponds to one-half the time delay.
While the phenomenon of echolation is of relatively minimal importance to humans, it is an essential trick of the trade for bats. Being merely blind, bats must use sound waves to navigate and hunt.
They produce short bursts of ultrasonic sound waves which reflect off their surroundings and return. Their detection of the time delay between the sending and receiving of the pulses allows a bat to
approximate the distance to surrounding objects. Some bats, known as Doppler bats, are capable of detecting the speed and direction of any moving objects by monitoring the changes in frequency of the
reflected pulses. These bats are utilizing the physics of the Doppler effect discussed in an earlier unit (and also to be discussed later in Lesson 3). This method of echolation enables a bat to
navigate and to hunt.
Like any wave, a sound wave has a speed which is mathematically related to the frequency and the wavelength of the wave. As discussed in a previous unit, the mathematical relationship between speed,
frequency and wavelength is given be the following equation.
Speed = Wavelength * Frequency
Using the symbols v, f, the equation can be re-written as
v = f *
The above equations are useful for solving mathematical problems related to the speed, frequency and wavelength relationship. However, one important misconception could be conveyed by the equation.
Even though wave speed is calculated using the frequency and the wavelength, the wave speed is not dependent upon these quantities. An alteration in wavelength does not effect (i.e., change) wave
speed. Rather, an alteration in wavelength effects the frequency in an inverse manner. A doubling of the wavelength results in a halving of the frequency; yet the wave speed is not changed. The speed
of a sound wave depends on the properties of the medium through which it moves and the only way to change the speed is to change the properties of the medium.
Check Your Understanding
1. An automatic focus camera is able to focus on objects by use of an ultrasonic sound wave. The camera sends out sound waves which reflect off distant objects and return to the camera. A sensor
detects the time it takes for the waves to return and then determines the distance an object is from the camera. If a sound wave (speed = 340 m/s) returns to the camera 0.150 seconds after leaving
the camera, how far away is the object?
2. The annoying sound from a mosquito is produced when it beats its wings at the average rate of 600 wingbeats per second.
a. What is the frequency in Hertz of the sound wave?
b. Assuming the sound wave moves with a velocity of 340 m/s, what is the wavelength of the wave?
3. Doubling the frequency of a wave source doubles the speed of the waves.
a. True
b. False
4. Playing middle C on the piano keyboard produces a sound with a frequency of 256 Hz. Assuming the speed of sound in air is 345 m/s, determine the wavelength of the sound corresponding to the note
of middle C.
5. Humans can detect frequencies as high as 20 000 Hz. Assuming the speed of sound in air is 345 m/s, determine the wavelength of the sound corresponding to the upper range of audible hearing.
6. An elephant produces a 10 Hz sound wave. Assuming the speed of sound in air is 345 m/s, determine the wavelength of this infrasonic sound wave.
7. Determine the speed of sound on a cold winter day in Glenview (T=3 C).
8. Miles Tugo is camping in Glacier National Park. In the midst of a glacier canyon, he makes a loud holler. He hears an echo 2.0 seconds later. The air temperature is 20-degrees C. How far away are
the canyon walls.
9. Two sound waves are traveling through a container of nitrogen gas. Wave A has a wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The velocity of wave B must be __________ the velocity of
wave A.
a. one-ninth
b. one-third
c. the same as
d. three times larger than
10. Two sound waves are traveling through a container of nitrogen gas. Wave A has a wavelength of 1.5 m. Wave B has a wavelength of 4.5 m. The frequency of wave B must be __________ the frequency of
wave A.
a. one-ninth
b. one-third
c. the same as
d. three times larger than | {"url":"http://www.mwit.ac.th/~physicslab/applet_04/physics_classroom/Class/sound/u11l2c.html","timestamp":"2014-04-21T05:15:49Z","content_type":null,"content_length":"28335","record_id":"<urn:uuid:0bbd4fea-1c87-4644-ad6a-4b93c3043b62>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00389-ip-10-147-4-33.ec2.internal.warc.gz"} |
need to check logical equivalence
September 4th 2011, 11:47 AM #1
Oct 2010
Mumbai, India
I am reading "Introduction to Set theory" by J.Donald Monk and while going through
some theorem there, I needed the following equivalence.
$(p\Leftrightarrow q)\wedge (r \Leftrightarrow q)\equiv (p\Leftrightarrow r)$
I want to know if this is true ? I think so. Any hints about proving this without using
truth tables. I would just convert this into combination of $\wedge$ and
$\vee$ .I tried to do that but couldn't get rid of $r$ .
Any pointers ?
Re: need to check logical equivalence
You can use a truth table, or you can just argue that both sides have the same truth value. Here's a sketch:
The left is true
(p iff q) and (r iff q) are both true
p, q have the same truth value AND r,q have the same truth value
p,q and r have the same truth value
p, r have the same truth value
the right hand side is true.
Remarks: (1) iff stands for "if and only if"
(2) I did this very quickly, so make sure that all these steps are reversible. If there is an error it should be easy to correct - the basic idea is sound (I'm running out of the house now or I
would check it more carefully).
Re: need to check logical equivalence
Thanks for the reply, DrSteve , but how do I use different logical equations to go from left to the right ? If they are logically equivalent , shouldn't q
drop somewhere ? I will briefly write what I did .... I have already proven that
$(P\Leftrightarrow Q) \equiv (P\wedge Q)\vee(eg P \wedge eg Q)$
so we can also write
$(R\Leftrightarrow Q) \equiv (R\wedge Q)\vee(eg R \wedge eg Q)$
so using this I tried to simplify
$(P\Leftrightarrow Q)\wedge(R\Leftrightarrow Q)$
doing some logical algebra I arrived at
$[P\wedge Q\wedge R]\vee[eg P\wedge eg Q\wedge eg R]$
so how do I make this
$(P\Leftrightarrow R)$ ?
Re: need to check logical equivalence
EDIT: forget it.
Last edited by ModusPonens; September 4th 2011 at 05:44 PM.
Re: need to check logical equivalence
Hello, issacnewton!
I believe I have a valid proof . . .
Prove or disprove: . $(p\Leftrightarrow q)\wedge (r \Leftrightarrow q)\;\equiv\; (p\Leftrightarrow r)$
Start with the left side:
. . $\begin{array}{cc cc cc}1. & (p \Leftrightarrow q) \:\wedge\: (r \Leftrightarrow q) && 1. & \text{Given} \\ \\[-3mm] 2. & [(p \to q) \wedge (q \to p)] \wedge [(r\to q) \wedge (q \to r)] && 2.
&\text{Def. bicond'l} \\ \\[-3mm] 3. & [(p\to q) \wedge (q \to r)] \wedge [(r\to q) \wedge (q \to p)] && 3. & \text{Comm. Assoc.} \\ \\[-3mm] 4. & (p \to r) \wedge (r \to p) && 4. & \text
{Syllogism} \\ \\[-3mm] 5. & p \Leftrightarrow r && 5. & \text{Def. bicond'l} \end{array}$
Re: need to check logical equivalence
@Soroban: I think that you only showed the left side implies the right. Most of your steps are reversible, but I don't think you can show (4) implies (3) using a single law.
@isaac: It looks like you're trying to convert everything to disjunctive normal form. Is this a requirement that your teacher has imposed? If so you should state it in the problem. If not, it
doesn't seem like the best way to go about it.
Re: need to check logical equivalence
Re: need to check logical equivalence
I was thinking about your derivation and how you got from step 3 to 4. In
Daniel Velleman's "How to prove it" , on page 54 , there is an exercise to prove that
$(P\Rightarrow Q)\wedge(Q\Rightarrow R)\equiv (P\Rightarrow R )\wedge[(P\Leftrightarrow Q)\vee(R\Leftrightarrow Q)]$
I have not been able to prove it , but I could see that this can be used to go from step 3 to 4. Since we have been given ,
$(P\Leftrightarrow Q)\wedge(Q\Leftrightarrow R)$
it follows from here that
$(P\Leftrightarrow Q)\vee(R\Leftrightarrow Q)$
and using the equivalence I quoted , we can say
$(P\Rightarrow Q)\wedge(Q\Rightarrow R)\equiv (P\Rightarrow R )$
is it the right conclusion ?
But as Drsteve and emakarov have pointed out, we can prove only one direction. So may be there is something wrong in the reasoning just provided by me.
Can anybody point out the flaw in my reasoning ?
September 4th 2011, 12:13 PM #2
Senior Member
Nov 2010
Staten Island, NY
September 4th 2011, 12:33 PM #3
Oct 2010
Mumbai, India
September 4th 2011, 05:23 PM #4
September 4th 2011, 05:31 PM #5
Super Member
May 2006
Lexington, MA (USA)
September 5th 2011, 08:28 AM #6
Senior Member
Nov 2010
Staten Island, NY
September 5th 2011, 01:17 PM #7
MHF Contributor
Oct 2009
September 5th 2011, 10:49 PM #8
Oct 2010
Mumbai, India | {"url":"http://mathhelpforum.com/discrete-math/187276-need-check-logical-equivalence.html","timestamp":"2014-04-17T11:09:42Z","content_type":null,"content_length":"56457","record_id":"<urn:uuid:08f55315-fdae-45c2-975a-085c9ab403b2>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00201-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Regulation and Efficiency of Transmission Line Help for - Transtutors
Regulation and Efficiency of a Transmission Line
Regulation. When the load is supplied, there is a voltage drop in the line due to resistance and inductance of the line and, therefore, receiving end voltage V[R] is usually less than sending end
voltage V[s]. The voltage drop i.e. difference of sending end voltage and receiving end voltage expressed as a percentage of receiving end voltage is called the regulation.
When the load is thrown off i.e. when the line is supplying no load, the receiving end voltage becomes equal to sending end voltage and, therefore, regulation can be defined as below:-
Regulation is defined as the change in voltage at the receiving and when full load is thrown off, the sending end voltage remaining the same. It is usually expressed as a percentage of receiving end
Mathematically %age regulation of transmission line is given by
%age regulation = (Vs - Vr)/Vr × 100
Where Vs is the voltage at sending end and V[R] is the voltage ate receiving end.
Sending end voltage and sending end power factor for a short transmission line in terms of receiving end voltage VR, load current, I, phase angle Ø, line resistance R and inductive reactance XL (2
πfL) is given as
V[s] = √((V[r] COSØ + I[R] )^2 )+ (V[R] SINØ + IX[L] )^2
= VR + IR COSØ + I X sin Ø.
Knowledge of regulation helps in maintaining the voltage at the load terminals within prescribed limits (± 5% of declared voltage) by employing suitable voltage control equipment.
Efficiency. When the load is supplied there are line losses due to ohmic resistance of line conductors and power delivered at the load end of a transmission line is less than the power supplied at
the sending end. Efficiency of a transmission line is defined as below:-
Efficiency of a transmission line is defined as the ratio of power received to the power sent out. Mathematically transmission line efficiency is given by
η[r] = (V[R] I[R] cosØ[R])/(V[s] I[s] cosØ[s]) × 100
Where VR, IR and cosØR are the receiving end voltage, current and power factor while Vs, Is and cosØS are the sending end voltage, current and power factor.
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b^2+16b+1=0 is the answer b=8+3root7?
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Cryptology ePrint Archive: Report 2006/444
Lattices that Admit Logarithmic Worst-Case to Average-Case Connection FactorsChris Peikert and Alon RosenAbstract: We demonstrate an \emph{average-case} problem which is as hard as finding $\gamma(n)
$-approximate shortest vectors in certain $n$-dimensional lattices in the \emph{worst case}, where $\gamma(n) = O(\sqrt{\log n})$. The previously best known factor for any class of lattices was $\
gamma(n) = \tilde{O}(n)$.
To obtain our results, we focus on families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to \emph{ideals} in the ring of integers of an algebraic
number field. The worst-case assumption we rely on is that in some $\ell_p$ length, it is hard to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of
the number field. Our results build upon prior works by Micciancio (FOCS 2002), Peikert and Rosen (TCC 2006), and Lyubashevsky and Micciancio (ICALP 2006).
For the connection factors $\gamma(n)$ we achieve, the corresponding \emph{decisional} promise problems on ideal lattices are \emph{not} known to be NP-hard; in fact, they are in P. However, the \
emph{search} approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to
perform \emph{no better} than the best known algorithms for general lattices.
To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant \emph{root discriminant}. Such families are known to exist and
are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to $O(n^{2/3-\
epsilon})$ would yield connection factors better than the current best of~$\tilde{O}(n)$.
Category / Keywords: foundations / lattices, worst-case to average-case reductions, number fieldsDate: received 26 Nov 2006Contact author: cpeikert at alum mit eduAvailable format(s): Postscript (PS)
| Compressed Postscript (PS.GZ) | PDF | BibTeX Citation Version: 20061204:101659 (All versions of this report) Discussion forum: Show discussion | Start new discussion[ Cryptology ePrint archive ] | {"url":"http://eprint.iacr.org/2006/444","timestamp":"2014-04-20T20:59:35Z","content_type":null,"content_length":"3744","record_id":"<urn:uuid:149d0aa5-b8f3-4898-b18e-357d9215e25c>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00486-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
The capacity of the gas tank in Jennifer"s car is x gallons. She can drive 28 miles for each gallon of gas in the tank. Jennifer filled the tank completely and drove 98 miles. She then stopped and
added 2 more gallons of gas. Part A: How many gallons of gas did Jennifer use to drive 98 miles ? Part B: Write an equation to find how far Jennifer could drive with the gas in her gas tank. Part C:
Use the equation from Part B to find how far Jennifer could drive if her gas tank has a capacity of 18.5 gallons.
• one year ago
• one year ago
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well for a, divide 98 by 28
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so a is 3.5 gallons
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what about part b and part c
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y = 28(x - 1.5)
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do you know how i got that?
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what about part c
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so x is the tank's capacity. and we know she only used 3.5 gallons. so x - 3.5. but then she added 2 gallons. so x - 3.5 + 2 which simplified is x - 1.5. and she can get 28 miles per gallon, so
we multiply that by 28
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with part c you just substitute x with 18.5
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csTriangleVertexCost Class Reference
[Geometry utilities]
The representation of a vertex in a triangle mesh. More...
#include <csgeom/trimeshlod.h>
Inheritance diagram for csTriangleVertexCost:
Public Attributes
float cost
Precalculated minimal cost of collapsing this vertex to some other.
bool deleted
True if already deleted.
int to_vertex
Vertex to collapse to with minimal cost.
Detailed Description
The representation of a vertex in a triangle mesh.
This is basically used as a temporary structure to be able to calculate the cost of collapsing this vertex more quickly.
Definition at line 40 of file trimeshlod.h.
Member Data Documentation
Precalculated minimal cost of collapsing this vertex to some other.
Definition at line 47 of file trimeshlod.h.
Vertex to collapse to with minimal cost.
Definition at line 49 of file trimeshlod.h.
The documentation for this class was generated from the following file:
Generated for Crystal Space 2.0 by | {"url":"http://www.crystalspace3d.org/docs/online/api-2.0/classcsTriangleVertexCost.html","timestamp":"2014-04-16T17:16:14Z","content_type":null,"content_length":"7923","record_id":"<urn:uuid:b6f6cad8-9af7-4629-bf65-1ffa375fbb65>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00574-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SciPy-dev] genetic algorithm, number theory, filter design, zero finding
pearu at scipy.org pearu at scipy.org
Tue Apr 9 04:24:21 CDT 2002
On Mon, 8 Apr 2002, Chuck Harris wrote:
> I've written a number of python routines over the past half year for
> my own use, and wonder if it might be appropriate to include some of
> them in scipy. They break down into the general categories:
I would suggest that you first make them modules so that we can look at
them whether they can be included to SciPy and how. Since SciPy itself is
quite short from documentation and unit testing (fixing this has a high
priority level in forthcoming SciPy development) I would expect that any
new module to be considered for inclusion to SciPy should be (more or
less) fully documented and has a (more or less) complete testing site.
> Number Theory : These were used for analysing arrays of antennas used
> in radar interferometry. They are also useful in integer programming,
> cryptography, and computational algebra.
> Reduction of a matrix to Hermite normal form
> Reduction of a matrix to Smith normal form
> LLL basis reduction
> LLL basis reduction - deep version
> Gram-Schmidt orthogonalization
I am not sure where these should go when considering the current scipy
state. As you mention, they are parts of different fields from what we
have in scipy now. I think they should be parts of the corresponding
packages that may not exist as of yet. Personally, I am very interested in
CA stuff.
> Filter Design: Routines used for designing Complex Hermitian digital
> filters :
> Remez exchange algorithm - for arbitrary Chebychev systems.
Would signal be an appropiate place for this?
> Zero finders: General use and fun. Bisection best for some special
> cases, Ridder is middle of the pack, Brent is generally best, with the
> two versions basically a wash, although the hyperbolic version is
> simpler. Whether or not there is any virtue to these as opposed to
> solve, I don't know.
> Bisection
> Illinois version of regula falsa
> Ridder
> Brent method with hyperbolic interpolation
> Brent method with inverse quadatic interpolation
Can you compare these zero finders with ones in
scipy? Performance? Robustness to initial conditions? Etc. Are they any
> Genetic algorithm: Used in digital filter design to optimize for
> coefficient truncation error. I looked at galib and found it easier to
> roll my own, but didn't try for any great generality. I think it would
> be good to include uniform crossover and to pull the fitness function
> out of the genome --- in a tournament, fitness can depend on the
> population. Perhaps it can all be made simpler.
galib seems to be developed more than 6 years and I would expect it to be
rather mature, though, I have not used it myself. May be a
wrapper to such a library would be more appropiate for a longer term.
Though the licence may be an issue, galib seems to be GPL compatible.
More information about the Scipy-dev mailing list | {"url":"http://mail.scipy.org/pipermail/scipy-dev/2002-April/000854.html","timestamp":"2014-04-18T20:47:47Z","content_type":null,"content_length":"5892","record_id":"<urn:uuid:937943ad-80c7-48aa-9467-f34a3657c211>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00610-ip-10-147-4-33.ec2.internal.warc.gz"} |
Standard Model particle spectrum from String Theory
Dimitri Terryn
It seems that a group of researchers has constructed a Calabi-Yau compactification that reproduces that particles of the standard model.
A bit exagerated the claim!
1) There is no possibility for deriving the compactification from firts principles. In fact, string theory cannot predict anything.
2) Nobody has obtained the full standard model. As far i know only partial ideas like hypotetical relations of CY holes with the
number of families, etc. This work appears to be based in some topological 'conjetures', etc.
3) The only physical states in string theory are masless supersimmetric states. Experiments claim just the contrary. Even if in the future high-energy experiment (e.g. HLC), supersimmetry is found,
that does NOT imply that string theory was correct -in fact, supersimmetry is previous to string hype and appears in several alternative theories- . Since the standard model continues to be
non-superssimetric and with no massless particles -and this is
independent of future experiments
-, string theory continues to be wrong.
P.S.1: Any theory predicting contrary to experiment is wrong.
P.S2: The CY is not the last 'geometry' of string theory. The CY is only valid as approximation in the asymptotic regimes of the 'branescan' just when 'string theory general relativity' does not hold
-string theory is background dependent, GR is NOT-. In M-theory -the full quantum gravity regime-, the manifold is still unknown, but some people is researching in G2 manifolds. | {"url":"http://www.physicsforums.com/showthread.php?t=95758&page=2","timestamp":"2014-04-16T13:50:47Z","content_type":null,"content_length":"43995","record_id":"<urn:uuid:a706844f-1557-419e-9ef5-b679e69e870e>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00432-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
Simplify. Use long division to solve. Subtract your answers, NOT add 10. (x^3 + 13x^2 - 12x - 8) / (x + 2)
• one year ago
• one year ago
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My answer is -72, but I don't know what to do next
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Setup the long division as follows, solve by dividing . (x^3 + 13x^2 - 12x - 8) with (x + 2)
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it teaches you long division of polynomials
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what ever remainder you have left you have to keep it divided by x + 2
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I know how to do it, it's just I don't know if I put it over x + 2 ----> \[\frac{ x + 2 }{ -72 }\]
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What do you have for your long division?
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x+2 remains in the denominator and it should be 60
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I'm pretty sure based on wolfram alphas answer
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@.Sam. so far
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The answer goes with|dw:1351049782957:dw|
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I know that; it's just I need help with where the -72 goes and @Australopithecus it's not 60
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@Firejay5 the remainder is 60
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No it's not it's -72 I did my work
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Try check your work with this,
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It's -76
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Like when you are subtracting 11x^2 - 12x - (11x^2 + 22x)
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you are are trying to cancel and get rid of the 11x^2
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yes, then you get -34 x
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It should be -34x - 8 - (34x + 68)
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-34 x 2 = -68
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its not +68 but its -68
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what's -8 - 68
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You should be careful there, its actually -8-(-68)
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Are you getting confused??? @.Sam.
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don't memorize the facts, just remember that it is top minus the bottom, you look back the previous steps, all you do is to subtract the top one with the bottom one, the steps that I gave you is
computer generated, its completely true.
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you have to apply the negative -(34x + 68) to all number s in parentheses
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a remainder is just part of the equation that you cannot divide, so it is left with the denominator so in your case 60/(x+2)
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The actual answer is 60, why are you ignoring it? Take a look at the picture below.
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I'm basing my 60/(x+2) on the wolfram alpha simplification
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It's not -34x - 68 it's -(34x + 68) the minus/negative in front of the parentheses goes with everything in it
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Real Analysis: Is n-binary digit expansion of a number in [0,1] measurable?
November 20th 2010, 09:35 PM #1
Nov 2010
Real Analysis: Is n-binary digit expansion of a number in [0,1] measurable?
If I have a function $X_n(x)$ where $x \in [0,1]$ and $X_n(x)$ is the n-binary digit expansion of $x$, how can I show that $X_n(x)$ is measurable?
The full question I'm looking at asks to show that
$\limsup_n_\rightarrow_\infty \frac{1}{n} \sum_{1}^{k} X_n(x)$ is measurable.
I know that when we have a sequence of measurable functions, their sum will be measurable, and dividing a measurable function by n will still leave it measurable as well. Finally, when dealing
with a sequence of measurable functions than their $\limsup_n_\rightarrow_\infty$ will also be measurable.
I have most of the puzzle figured out, but have never dealt with n-binary digits before. If I can just show that this expansion is measurable, everything else will fall into place.
My textbook tells me that if a sequence of functions converges, then it's limit will be measurable. Doesn't that apply in this case, since the n-binary digit expansion is converging to some point
in [0,1]?
The set $\{x\in[0,1)\mid X_2(x)=1\}=[1/4,1/2)\cup[3/4,1)$. In general, the $X_n^{-1}(0)$ and $X_n^{-1}(1)$ are some unions of parts that you get by splitting [0, 1] into $2^n$ equal segments.
Thanks for the reply! So the preimages of the sequence of $X_n (x)$ for any $x \in [0,1]$ will be unions of measurable sets, thus making the function itself measurable? This explanation was much
simpler than what I had in mind.
Since the only empty set occurs when $x$=0, does this mean that the function is continuous as long as $x eq 0$ (almost everywhere)?
Wouldn't this also mean that $\int_0^1 f(x)dx$ is just 1/2? That seems to make sense, but I'm still very new to binary digit expansions and want to make certain.
I am not sure I understand. What do you mean that the empty set occurs? What function is continuous? The function $X_n(x)$ is not continuous; it is equal to 1 and 0 on alternating segments. I
believe $X_n(x)$ has n - 1 points on [0, 1) where it is not continuous.
Wouldn't this also mean that $\int_0^1 f(x)dx$ is just 1/2?
I think so.
November 21st 2010, 12:50 AM #2
MHF Contributor
Oct 2009
November 21st 2010, 01:18 PM #3
Nov 2010
November 22nd 2010, 06:39 AM #4
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[Numpy-discussion] Array indexing and repeated indices
Nicolas Rougier Nicolas.Rougier@inria...
Fri Mar 1 01:30:52 CST 2013
I'm trying to increment an array using indexing and a second array for increment values (since it might be a little tedious to explain, see below for a short example).
Using "direct" indexing, the values in the example are incremented by 1 only while I want to achieve the alternative behavior. My question is whether there is such function in numpy or if there a re better way to achieve the same result ?
(I would like to avoid the while statement)
I found and adapted the alternative solution from: http://stackoverflow.com/questions/2004364/increment-numpy-array-with-repeated-indices but it is only for a fixed increment from what I've understood.
# ------------------------
import numpy as np
n,p = 5,100
nodes = np.zeros( n, [('value', 'f4', 1)] )
links = np.zeros( p, [('source', 'i4', 1),
('target', 'i4', 1)])
links['source'] = np.random.randint(0, n, p)
links['target'] = np.random.randint(0, n, p)
targets = links['target'] # Indices can be repeated
K = np.ones(len(targets)) # Note K could be anything
# Direct indexing
nodes['value'] = 0
nodes['value'][targets] += K
print nodes
# "Alternative" indexing
nodes['value'] = 0
B = np.bincount(targets)
while B.any():
I = np.argwhere(B>=1)
nodes['value'][I] += K[I]
B = np.maximum(B-1,0)
print nodes
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IMPACT Grant Lesson Plans
A. Unit Overview:
1. Content Area: Math
2. Unit Title: Variables and Patterns Introducing Algebra
3. Target Course/Grade Level: Resource Room Math/Grade 8
4. Instructor’s Name: Kim Fassett
5. School: Lakeside Middle School
6. Date: July 21, 2011
7. Unit Summary: Variables and Patters, the first unit in the Connected Mathematics algebra strand is a unit that develops students’ ability to explore a variety of situations in which change
occurs. Students will be collecting, organizing and representing data in charts, graphs and formulas and will be able to use these concepts in everyday real-world situations.
8. Primary interdisciplinary connections: Science, History, Health/PE, Literacy
21st century themes:
21st century skills:
Unit Rationale:
To discover order, analyze, construct, and predict how and why we use patterns and variables using data and their uses in everyday real-world and mathematical situations. Learning to observe,
describe, and record changes is the first step in analyzing and searching for patterns in a real-world situation. We will be reviewing several concepts that students have had over the last few years
organizing and using data, graphing patterns and basic algebra skills. However, the student levels are at a below average level so it is important to be sure that we build their confidence as well
refresh their skills. Variables and patterns are terms used within the entire 8th grade interdisciplinary curriculum so this will allow students to continue to use their vocabulary skills knowledge.
Another high-quality rationale for this review is to explore the use of a new more advanced calculator for the 8th grade group.
B. Learning Targets :
7.EE.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply
properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For
example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches
long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an
algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is
its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in
the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of
sales you need to make, and describe the solutions.
2. Standards for Mathematical Practice
• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.
3. Unit Essential Questions:
• How or when can one use variables and/or patterns to make decisions and solve real-world problems?
• What is the relationship between variables and patterns?
□ How can we use the relationship between patterns and variables with data we have collected or with data we are provided?
4. Unit Enduring Understandings:
• What did you learn today about variables and patterns and how could you explain or show someone at home what you learned?
• How could you explain the relationship of variables and patterns to someone at home in the future if you needed to?
• How can you show someone at home relationship between variables and patterns with data you collect or with data that’s provided?
C. Evidence of Learning:
• Formative Assessments
• Informal observations
• Daily Homework and review
• Classwork/ review
• Daily warm-up (4 Square) and review
• Pre-Quiz and review (1Day)
D. Equipment / Resources
• Equipment / Technology needed throughout the Unit:
• Internet
• laptop
• SmartBoard/Ti84 Calculator software
• Ti84 Calculator
• SmartDocument Camera
• Stop watch (showing seconds)
• Paper cut outs of hexagons
• Copies of student lab sheets
• Graph paper
• Colored pencils
• pencil
• Blackline masters
• Teacher Resources
• Connected Math Teacher Planner pgs. 71-77
• Graphing calculator and linking cable
• Internet
• laptop
• SmartBoard/Ti84 Calculator software
• SmartDocument Camera
• Stop watch (showing seconds)
• Blackline Masters
• EZschool http://www.ezschool.com/EZSheets/
• Kuta http://www.kutasoftware.com/free.html
E. Lesson Plan Topics / Titles
Lesson 1: Variables and Coordinate Graphs
Lesson 2: Graphing Change
Lesson 3: Check Up
Lesson 4: Analyzing Graphs and Tables
Lesson 5: Patterns and Rules
Lesson 6: Quiz
Lesson 7: Using Graphing Calculators
Lesson 8: Looking Back and Looking Ahead: Unit reflections
F. Teacher Notes about Lesson Plans
Each lesson:
• Begins with an activity to connect to Previous Knowledge and to Student Interest / Preferences.
• Includes High Quality Curriculum –
• Connected Math-
• Variables and Patterns
• -Investigation 4 Patterns and Rules
• Includes Flexible Grouping via – (based on activity or student need or both)
• Homogeneous
• Heterogeneous
• Cooperative learning groups
• Peer buddies
• Pairs
• Whole Class
• Independent work
• Provides Respectful Tasks within a Supportive Learning Environment via Multiple Means.
G. Universal Design for Learning Options
• Multiple Means of Representation
□ Guideline 1: Provide options for perception
The AIM Explorer is designed to be used by a reader working collaboratively with an educator, tutor, parent, or assistive technology specialist as a guide. The guide may create a student account in
order to re-assess reader preferences at a later time, or lead the reader on an exploration without creating an account. In both cases a reader summary profile will be created at the end of the
exploration, but student preferences will only be saved if an account is created. Alternatively, a reader could initiate an exploration independently. The AIM Explorer allows users to explore their
preferences for customizable features such as: magnification, text and background colors, and layout, TtS voice and speed, and more.
□ Guideline 2: Provide options for language, mathematical expressions, and symbols
OMA is Tucson's remarkable school transformation program that is increasing student achievement through arts integration. The OMA Program, developed in the Tucson Unified School District (TUSD), uses
the arts to teach academic standards in math, science, reading, writing and social studies and is designed around state and federal standards. In OMA schools, all ethnic backgrounds, regardless of
socioeconomic status, showed improvement on mandated tests (AIMS, Terra Nova, and Stanford 9) in all tested areas. The quality of the OMA Program and the documented student achievement results have
gained national recognition from the U.S. Department of Education, Harvard Project Zero, Arts Education Partnership, and others. Pay special attention to how the integration of the arts benefits
English Language Learners. Linking to content that crosses language barriers, while taking steps to develop vocabulary and build communication skills are effective examples of promoting
cross-linguistic understanding!
□ Guideline 3: Provide options for comprehension
Illuminations was designed to support the NCTM standards for mathematics. This website offers interactive tools to facilitate exploration of math concepts.
2. Multiple Means of Action and Expression
Guideline 4: Provide options for physical action
http://aim.cast.org/learn/accessiblemedia/allaboutaimThis site serves as a resource to state- and district-level educators, parents, publishers, conversion houses, accessible media producers, and
others interested in learning more about and implementing AIM and NIMAS. AIM are specialized formats of curricular content that can be used by and students with print-disabilities. They include
formats such as Braille, audio, large print, and electronic text. The audio and the electronic text formats are excellent examples of providing options in the mode of physical response for students
who have difficulty turning pages or holding a book.
Guideline 5: Provide options for expression and communication
http://scratch.mit.edu/Scratch is designed to help young people (ages 8 and up) develop 21st century learning skills. As they create and share Scratch projects, young people learn important
mathematical and computational ideas, while also learning to think creatively, reason systematically, and work collaboratively. Scratch provides students with an array of ways to demonstrate learning
- through creating interactive stories, animations, games, art. http://scratch.mit.edu/projects/Calliope23/423530
Guideline 6: Provide options for executive functions
http://www.studygs.net/timman.htm This site provides study guides and strategies in areas of learning such as thinking, studying, planning and communication. The section on time management offers
helpful tips on planning and prioritizing.
3. Multiple Means of Engagement
Guideline 7: Provide options for recruiting interest
http://www.pbis.org/school/what_is_swpbs.aspxThe TA Center on Positive Behavioral Interventions and Supports has been established by the Office of Special Education Programs, US Department of
Education to give schools capacity-building information and technical assistance for identifying, adapting, and sustaining effective school-wide disciplinary practices. PBIS's focus on environmental
aspects that lead to problem behavior is reflective of the importance of varying threats and distractions.
Guideline 8: Provide options for sustaining effort and persistence
"Skype allows you to make free calls over the internet to other people on Skype for as long as you like, to wherever you like." Skype is another powerful example of a tool that can be used to foster
collaboration and communication among students across classrooms, districts, states, and countries!
Guideline 9: Provide options for self-regulation
http://worksheetplace.com/index.php?function=DisplayCategory&showCategory=Y&links=2&id=279&link1=31&link2=279 Find templates for goal-setting worksheets to use with your students to support their
organizational skills. These organizational worksheets are great examples of strategies that guide students' goal-setting.
H. Online Resources :
The following websites can be used in preparing additional strategies, accommodations, and modifications in the above lessons: | {"url":"http://www.rowan.edu/open/mcsiip/impact/impactlessonplans/math/variables_and_patterns_introducing_algebra.html","timestamp":"2014-04-16T19:58:57Z","content_type":null,"content_length":"19367","record_id":"<urn:uuid:9433ae61-ab98-4573-916e-f276c265bd05>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00454-ip-10-147-4-33.ec2.internal.warc.gz"} |
stable pro-object
stable pro-object
Given any category $C$, we can form the corresponding category of pro-objects in $C$, which is denoted by $\mathrm{pro}$-$C$. Since the category $\mathrm{\pi }$ with one morphism is a coflitered
category, within $\mathrm{pro}$-$C$, we have all pro-objects of the form $X:\mathrm{\pi }\beta C$. Clearly such a functor is completely determined by the single object, $X\left(*\right)$, of $C$
to which it corresponds. This gives a functor:
$c:C\beta \mathrm{pro}\beta C$c: C\to pro-C
which embeds the category $C$ in $\mathrm{pro}$-$C$. (This is really the Yoneda embedding in disguise.)
Any pro-object isomorphic in $\mathrm{pro}$-$C$ to one of the form, $c\left(X\right)$, for $X$ an object of $C$, is called stable or essentially constant.
Stability problem
In any given categorical context, the so-called stability problem is the problem of deciding what internal criteria can be applied to check if a given pro-object in that context, is or is not stable,
If $C$ is an abelian category, it is relatively simple to give necessary and sufficient β internalβ conditions for a given pro-object to be essentially constant. It must be both essentially
epimorphic? and essentially monomorphic?.
Revised on March 12, 2010 07:41:43 by
Tim Porter | {"url":"http://www.ncatlab.org/nlab/show/stable+pro-object","timestamp":"2014-04-21T02:01:02Z","content_type":null,"content_length":"15092","record_id":"<urn:uuid:f008e057-a9dd-4d34-bb7e-55d7db1c0655>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00361-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
5 different games are to be distributes among 4 children randomly. The probabilty that each child get altleast one game is A)1/4 B)15/64 c) 21/64 d) none of these
• one year ago
• one year ago
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none of these
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the chance a child gets a game is more than one, all the fractions are less than one
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@Bdude999 , probability is always less than or equal to 1 :)
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@satellite73 , help please :)
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@precal , any idea??
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Lemme look.
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Sorry, not completely sure how to do this with permutations and combinations. I could do a listing, but that would be quite unefficient.
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no problem. I have also faced difficulties in this problem :)
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So, I can do listing?
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Go ahead
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But I warn that it would be too hectic. Permutation and Combination would be the right way
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Ok, I think I have an idea.
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First find the probability that the 4 games would be distributed evenly.
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Now, find the probability that ou of the 4 games, one got repeated.
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Actually, find the probability, sorry.
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I am trying..
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sorry out of my league
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@precal , out of my league too. Do you know who can solve this or who is master of permutations and combinations here??
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sorry not at the moment
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Thanks :) Let me keep trying myself :)
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@amistre64 , any help please ?
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once a child gets a game, are they out of rotation until they all have games? or can it be that 1 child gets all 5 games?
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It is like there are 5 different toys and 4 children and we have to find the probability that each child should get atleast one toy
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or can it be that 1 child gets all 5 games --->not possible
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It can be that 1 child gets 2 toys and rest of them get one each
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if the toys are handed out at random, then i dont see why 1 child couldnt get all the toys
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Since it is given in the question that we have to find the probability that each child should atleast get one game.
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if each child gets a toy and there is one left over; then the prob that each child gets at least 1 toy is: 1 right?
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otherwise there has to be the possibility that one child gets all the toys and such
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Yes you are right. But the question is not that . It is what is the probability that each boy has atleast one toy. Which means like 21111 , 12111, 11211,11121,11112 --> These are the favourable
cases(accd to question). Now the problem is I need to know total no of cases. to get my probability
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well, 1112 1121 1211 2111 is your favoured cases and i we bruting out the total cases
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@amistre64 , I have found a pre-written solution to this problem. But I am not able to understand the solution
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Here is the solution n(S) = 4^5 Total ways of distribution so that each child gets atleast one game \[=4^{5}-4 C _{1} * 3^{5} + 4C_{2} * 2^{5}-4C _{3}\] = 1024- 4*243 + *32-4 = 240 Reqd
probability = 240 / 4^5 = 15/64 Now I don't understand the total ways of distribution in the solution??
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@shivam_bhalla Read about Stirling number of second kind
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5000 0500 0050 0005 : 4 4100 3200 4010 ... 4001 0410 0401 1400 0041 1040 0140 1004 0104 0014 : 24 0122 0113 0212 ... 0221 1022 1202 1220 2012 2021 2102 2120 2201 2210 : 24 1112 1121 1211 2111 :4
56 altogether? you see any i missed?
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The ways of distribution is consistent to dividing r distinct things into n distinct groups \( n! \times S(n,r) \). If you expand this you will find the closed form which is used in your
solution. REF:http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Definition
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@FoolForMath , can you suggest a better source because I am totally new to this. Thanks :) @amistre64 ,Thanks for helping a lot mate :)
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@FoolForMath , thanks a lot mate :) You saved my day :) If I have any doubts regarding this, can I message you ??
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@FoolForMath , Thanks bro, I got it :)
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Glad to help.
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If this question was diving 5 common things among four different children, then??
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Read about "Stars and bars" combinatorics
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Will read about Stars and bars" combinatorics and report back here :)
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@FoolForMath if this question was dividing 5 common things among four different children, then the answer would be \[(5-1)C _{4-1}= 4C _{3} = 4\] ?? And what should be the total no of cases ??
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@FoolForMath Or the answer is \[(5+4-1) C _{4-1}= 8 C _{3}=56\]
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with the at-least one constraint it will be the first one.
Your question is ready. Sign up for free to start getting answers.
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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.
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Let A Be An Arbitrary Binary Number. B Is Obtained ... | Chegg.com
Let A be an arbitrary binary number. B is obtained by flipping all bits of A and then adding 1. C is obtained by subtracting 1 from A and then flipping bits of the resulting number. Is it true that B
= C?
Electrical Engineering | {"url":"http://www.chegg.com/homework-help/questions-and-answers/let-arbitrary-binary-number-b-obtained-flipping-bits-adding-1-c-obtained-subtracting-1-fli-q3535771","timestamp":"2014-04-19T18:01:38Z","content_type":null,"content_length":"20415","record_id":"<urn:uuid:d62ddad5-76e9-4176-8201-3f8fd3c07dd6>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00165-ip-10-147-4-33.ec2.internal.warc.gz"} |
Computer Science MCQ (Multiple Choice Questions)
1. Basic geometric transformation include
a. Translation
b. Rotation
c. Scaling
d. All of these
2. Some additional transformation are
a. Shear
b. Reflection
c. Both a & b
d. None of these
3. The transformation in which an object is moved in a minimum distance path from one position to another is called
a. Translation
b. Scaling
c. Rotation
d. Reflection
3. The transformation in which an object is moved from one position to another in circular path around a specified pivot point is called
a. Translation
b. Scaling
c. Rotation
d. Reflection
5. The transformation in which the dimension of an object are changed relative to a specified fixed point is called
a. Translation
b. Scaling
c. Rotation
d. Reflection
6. The selection and separation of a part of text or image for further operation are called
a. Translation
b. Shear
c. Reflection
d. Clipping
7. The complex graphics operations are
a. Selection
b. Separation
c. Clipping
d. None of these
8. In computer graphics, a graphical object is known as
a. Point
b. Segment
c. Parameter
d. None of these
9. An object can be viewed as a collection of
a. One segment
b. Two segment
c. Several segments
d. None of these
10. Every segment has its own attributes like
a. Size, visibility
b. Start position
c. Image transformation
d. All of these
11. By using the attributes of segment , we can________ any segment
a. Change
b. Control
c. Print
d. None of these
11. A two-dimensional array contain the details of all the segment are called
a. Segmentation table
b. Segment name
c. Operation
d. None of these
12. We assign all the attributes of segment under this
a. Segment name
b. Segment size
c. Array
d. None of these
14. The initial size of segment will be_______
15. The removal of a segment with its details are called
a. Alter the segments
b. Deletion of segments
c. Closing the segment
d. None of these
16. Deletion of any segment is much________ than creation of any new segment
a. Easier
b. Difficult
c. Higher
d. None
17. _______is very important in creating animated images on the screen
a. Image transformation
b. Morphing
c. Clipping
d. None of these
18. Which attributes of image transformation change the size of an image corresponding to the x-axis and y-axis
a. SCALE-X
b. SCALE-Y
c. Both a & b
d. None of these
19. Which attributes of image transformation change the position of image corresponding to the x-axis and y-axis
a. TRANSLATE-X
b. TRANSLATE-Y
c. Both a & b
d. None of these
19. Which attributes of image transformation rotate the image by a given angle
a. TRANSLATE-X
b. TRANSLATE-Y
c. Both a & b
d. None of these
21. Which attributes of image transformation rotate the image by a given angle
a. ROTATE-X
b. ROTATE-Y
c. Both a & b
d. None of these
22. The graphics method in which one object is transformed into another object are called
a. Clipping
b. Morphing
c. Reflection
d. Shear
a. Oil takes the shape of a car
b. A tiger turns into a bike
c. Both a & b
d. None of these
23. A many sided figure is termed as
a. Square
b. Polygon
c. Rectangle
d. None
25. The end point of polygon are called as
a. Edges
b. Vertices
c. Line
d. None of these
26. The line segment of polygon are called as
a. Edges
b. Vertices
c. Line
d. None of these
27. How many types of polygon are
28. What are the types of polygon
a. Convex polygon
b. Concave polygon
c. Both a & b
d. None of these
29. If a line joining any of its two interior points lies completely within it are called
a. Convex polygon
b. Concave polygon
c. Both a & b
d. None of these
30. If a line joining any two of its interior points lies not completely inside are called
a. Convex polygon
b. Concave polygon
c. Both a & b
d. None of these
30. In which polygon object appears only partially
a. Convex polygon
b. Concave polygon
c. Both a & b
d. None
32. If the visit to the vertices of the polygon in the given order produces an anticlockwise loop are called
a. Negatively oriented
b. Positively oriented
c. Both a & b
d. None of these
32. If the visit to the vertices of the polygon in the given order produces an clockwise loop are called
a. Negatively oriented
b. Positively oriented
c. Both a & b
d. None of these
34. Which things are mainly needed to make a polygon and to enter the polygon into display file
a. No of sides of polygon
b. Vertices points
c. Both a & b
d. None of these
35. Two types of coordinates are
a. Positive and negative coordinates
b. Absolute and relative coordinates
c. Both a & b
d. None
36. Which approaches are used for determine whether a particular point is inside or outside of a polygon
a. Even-odd method
b. Winding number method
c. Both a & b
d. None of these
36. The transformation that produces a parallel mirror image of an object are called
a. Reflection
b. Shear
c. Rotation
d. Scaling
38. The transformation that disturbs the shape of an object are called
a. Reflection
b. Shear
c. Rotation
d. Scaling
39. The process of mapping a world window in world coordinate system to viewport are called
a. Transformation viewing
b. View Port
c. Clipping window
d. Screen coordinate system
40. In which transformation the shape of an object can be modified in x-direction ,y-direction as well as in both the direction depending upon the value assigned to shearing variables
a. Reflection
b. Shearing
c. Rotation
d. Scaling
41. The process of extracting a portion of a database or a picture inside or outside a specified region are called
a. Translation
b. Shear
c. Reflection
d. Clipping
42. The rectangle portion of the interface window that defines where the image will actually appear are called
a. Transformation viewing
b. View port
c. Clipping window
d. Screen coordinate system
43. The space in which the image is displayed are called
a. Screen coordinate system
b. Clipping window
c. World coordinate system
d. None of these
44. The rectangle space in which the world definition of region is displayed are called
a. Screen coordinate system
b. Clipping window or world window
c. World coordinate system
d. None of these
45. The object space in which the application model is defined
a. Screen coordinate system
b. Clipping window or world window
c. World coordinate system
d. None of these
45. The process of cutting off the line which are outside the window are called
a. Shear
b. Reflection
c. Clipping
d. Clipping window
47. Some common form of clipping include
a. curve clipping
b. point clipping
c. polygon clipping
d. All of these
48. A composite transformation matrix can be made by determining the ________of matrix of the individual transformation
a. Addition
b. Subtraction
c. Product
d. None of these
49. Each successive transformation matrix _________ the product of the preceding transformation
a. pre-multiples
b. post-multiples
c. both a & b
d. none of these
50. Forming products of transformation matrices is often referred as
a. Composition of matrix
b. Concatenation of matrix
c. Both a & b are same
d. None of these
51. The alteration of the original shape of an object, image, sound, waveform or other form of information are called
a. Reflection
b. Distortion
c. Rotation
d. None of these
52. Two consecutive translation transformation t1 and t2 are
a. Additive
b. Subtractive
c. Multiplicative
d. None of these
53. Two consecutive rotation transformation t1 and t2 are
a. Additive
b. Subtractive
c. Multiplicative
d. None of these
53. Two consecutive scaling transformation t1 and t2 are
a. Additive
b. Subtractive
c. Multiplicative
d. None of these | {"url":"http://csmcq.blogspot.com/2012/03/computer-graphics-mcq-03.html","timestamp":"2014-04-19T12:16:30Z","content_type":null,"content_length":"170205","record_id":"<urn:uuid:c1f74593-2eae-4fbe-95f3-bb6551c3d570>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00588-ip-10-147-4-33.ec2.internal.warc.gz"} |
Julianthemath's discovery 5
1. Think of a number between 1 and 1000.
2. Divide that by 6. Don't care about the remainder, if any.
3. Divide it again by 6. Also don't mind the remainder.
4. Get the decimal of your username (the 36-bit).
5. Multiply that by your number in step 3.
6. If it's bigger than 777,777,777,777, divide it by 123.
7. If bigger still than 777,777,777,777, divide it by 246.
8. If still, divide by 369. If it is still bigger than 777,777,777,777, just add 123 to the dividend, then divide it to the number.
9. If it is now smaller than 777,777,777,777, and not sure if it is a Squimintrius number, apply the second sentence of Rule 8.
10. If it is smaller than 1000, subtract it by 50. Then, 75, 100, 125... till you reach 50 below. If it is negative, remove the + sign.
If you got a Squimintrius number, square it.
That is called a Squisquaretrius number.
If not, don't do anything. Try again until you get one.
Good luck in solving! | {"url":"http://www.mathisfunforum.com/viewtopic.php?id=19267","timestamp":"2014-04-20T16:54:02Z","content_type":null,"content_length":"12377","record_id":"<urn:uuid:a6d11e5f-7466-49df-bce4-ef9b45840ec8>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00628-ip-10-147-4-33.ec2.internal.warc.gz"} |
1/4 wave TL design help
06-13-2009 #1
1/4 wave TL design help
so i did all the math for a pair of 07CVR10's.. Sd=(58.7x2)=118sq.in Fs=33.2hz
1120/4=280/33.2=8.4ft long ~102"... port:10"x12"
externals are 38"x22"x13", using .5"mdf with a 2x baffle on the front..
box i got out of it is 8.75ft. long which is about 32hz
lemme know if i got any calculations wrong and i did a sketchup..will this work with the second sub so far down the line already?
line measurements are 26.5", 20.5", 37", 21.5" from front to back
2013 Scion FRS
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Re: 1/4 wave TL design help
Doesn't look long enough to me
Edit: and let me guess, "That's what she said..."
Re: 1/4 wave TL design help
thats not a 1/4 wave box...its looks more like a spl box for 2 10's
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Re: 1/4 wave TL design help
x2 and I don't see 102" of line in that thing. If that is even the correct measurement. I'm too tired to work through a quarterwave right now.
Re: 1/4 wave TL design help
I've done plenty of T-lines with the 2nd sub partly down the line, works fine (can actually smooth the response a little). It's not untill the sub is over 1/3 of the way down the line that you
start having significant negative effects.
Just give the open end plenty of room to breathe in the car, these lines are particularly sensitive to loading.
[edit] and yes it does look too short for a 32hz line though I have done similar length lines for customers and they liked em so "shrug".
Re: 1/4 wave TL design help
well the box is 38" wide and almost 2 ft deep..and i only need 8 some feet..so 6 ft both directions and almost 2 ft in both back and fowarth directions..its a quarter wave so its not a full TL..
2013 Scion FRS
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Re: 1/4 wave TL design help
2013 Scion FRS
Infinity Kappa FIVE - Pheonix Gold RSD65cs's - Audio Technix AT80 (40 ft^2) - KnuKonceptz 4ga 4ch install kit - Subaruaudio.net speaker spacers
Re: 1/4 wave TL design help
I just modeled it in my horn simulation software (which will also model TL's), your design just needs to add another 10'' to the total length of the line for a correct response.
[edit] or, you can keep the current enclosure dimentions and taper the line instead, 12''x11.75'' at the closed end down to 12''x7.75'' at the open end, the response is practically identical
either way.
Re: 1/4 wave TL design help
I just modeled it in my horn simulation software (which will also model TL's), your design just needs to add another 10'' to the total length of the line for a correct response.
[edit] or, you can keep the current enclosure dimentions and taper the line instead, 12''x11.75'' at the closed end down to 12''x7.75'' at the open end, the response is practically identical
either way.
2013 Scion FRS
Infinity Kappa FIVE - Pheonix Gold RSD65cs's - Audio Technix AT80 (40 ft^2) - KnuKonceptz 4ga 4ch install kit - Subaruaudio.net speaker spacers
Re: 1/4 wave TL design help
2013 Scion FRS
Infinity Kappa FIVE - Pheonix Gold RSD65cs's - Audio Technix AT80 (40 ft^2) - KnuKonceptz 4ga 4ch install kit - Subaruaudio.net speaker spacers
Re: 1/4 wave TL design help
The 7.75'' width is the line end, not an internal measurement, imagine the line as a straight tube that's 12'' high, the closed end where the subs are is 11.75'' wide, the open end is 7.75''
Fold this up in your enclosure space, start at the port end, make it 7.75'' wide then increase the width as you get to the closed end, your drawing is on the right track as a concept (other than
the 7.75'' measurement)
Re: 1/4 wave TL design help
Quick question... Where can you find the length needed for a transline route for a said tuning frequency. I've been really interested in trying one of these. Also, if you can't quite achieve the
full quarter wave length, you just line the enclosure with polyfill to achieve the correct tuning right?
----My Two 10's Flexin'
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I design and build boxes.
Re: 1/4 wave TL design help
Quick question... Where can you find the length needed for a transline route for a said tuning frequency. I've been really interested in trying one of these. Also, if you can't quite achieve the
full quarter wave length, you just line the enclosure with polyfill to achieve the correct tuning right?
First off, polyfill doesn't really effect T-lines like that, it will damp the line and this effects higher frequencies more than lower ones, so when you listen to the line before and after, the
"after" sounds like it's going lower than the "before", when it's really just filtering out the higher frequencis more.
Not that it can't be used to good effect though, and I often partially stuff lines that have to be made "too small" to fit in the vehicle, it just won't lower the tuning frequency by any
meaningfull amount. (essentially I can make a small line have a similar response to a larger line with carefull design and stuffing, but the large line will have far greater output)
Now, design, all though tapered lines are harder to fold and build, they are the most forgiving acoustically.
Try this, get the sub/s you intend to use, measure the cone area, make the start of the line (where the subs are) 1.5 x total cone area, make the total line length 70''- 80'', make the open end
of the line 0.7 x line area.
That will give you a pretty balanced response with a majority of the subs on the market, especially high power/stiff suspension (spl ect) subs, anything that's normally designed to handle abuse
in a ported box.
I make "cash'n'carry" enclosures based on those ratios, if you want more punch, make it shorter, if you want a lower response, make it longer, simple
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06-15-2009 #13 | {"url":"http://www.caraudio.com/forums/enclosure-design-construction-help/411969-1-4-wave-tl-design-help.html","timestamp":"2014-04-16T18:21:49Z","content_type":null,"content_length":"84251","record_id":"<urn:uuid:4fac7c40-ffdc-49e3-8f2f-104c8e809be5>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00089-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lecture Series in Black Hole Astrophysics
Dr. Charles Dermer, Code 7653, NRL, and Prof. Govind Menon, Troy University, Troy, Alabama, will present a lecture series on black hole physics and astronomy during the months of May and June, 2007.
The lectures will be presented on parallel tracks. Chuck Dermer will present the observational background and radiation physics to understand and model gamma rays, cosmic rays, and neutrinos from
black holes. In alternating lectures, Govind Menon will develop the fundamentals of general relativity, leading to an understanding of black hole electrodynamics.
Location: Bldg. 209 (Space Science Division), Rm. 321A.
Chapters 1 - 8 and Appendix (rev. 6/25/07)
Lecture 13B. The Penrose Process
(Menon) Monday, July 2, 2 pm
Lecture 13A. High Energy Radiation from Black Holes
Complete Blazar Model
(Dermer) Friday, June 29, 3 pm (NOTE TIME)
Revised Compton Scattering chapter
Lecture 12B. The Ergosphere and the Event Horizon
(Menon) Thursday, June 28, 2 pm
Revised Kerr Black Hole chapter
Lecture 12A. Compton Scattering. IV.
Gamma Rays from Blazars
(Dermer) Wednesday, June 27, 2 pm
Lecture 11B. Geodesics of the Kerr Black Hole
(Menon) Tuesday, June 26, 2 pm
Lecture 11A. Compton Scattering. III.
Scattering in the Klein-Nishina Regime
(Dermer) Monday, June 25, 2 pm
Lecture 10B. Covariant Form of Maxwell's Equations
(Menon) Friday, June 22, 2 pm
Revised GR chapters
Lecture 10A. Compton Scattering. II.
Formation of Compton Spectra in Jet Sources; Energy Loss Rates
(Dermer) Thursday, June 21, 2 pm
Lecture 9B. Geometry of Spacetime. II.
(Menon) Wednesday, June 20, 3 pm (NOTE TIME)
Lecture 9A. Compton Scattering. I.
Compton Effect, Compton Cross Section, Transformation of Cross Section
(Dermer) Tuesday, June 19, 2 pm
Chapter 6, Pt. 1 (rev. 6/19/07)
Lecture 8B. Geodesics in Schwarzschild Geometry. II.
(Menon) Monday, June 18, 2 pm
Lecture 8A. Radiation Physics of Relativistic Flows. II.
Superluminal motion, shell framework, curvature relation.
(Dermer) Friday, June 12, 2 pm
Chapter 5 (rev. 6/15/07)
Lecture 7B. Geodesics in Schwarzschild Geometry. I.
(Menon) Thursday, June 14, 2 pm
Geometry of Spacetime (rev. 6/14/07)
Lecture 7A. Radiation Physics of Relativistic Flows. I.
Transformations of energy densities, fluxes of cosmological sources, blob framework
(Dermer) Tuesday, June 12, 3 pm
Lecture 6B. Geometry of Spacetime I.
(Menon) Monday, June 11, 2 pm
Revised General Relativity Chapters
Lecture 6A. Physical Cosmology II. Friedmann Cosmologies and Event Rates of Cosmological Sources
(Dermer) Friday, June 8, 2 pm
Chapter 4 (rev. 6/08/07)
Lecture 5B. The Covariant Derivative and the Einstein Equation II.
(Menon) Wednesday, June 6, 2 pm
Lecture 5A. Physical Cosmology I. Robertson-Walker Metric, Expansion Scale Factor, and Diffuse Intensity
(Dermer) Tuesday, June 5, 2 pm
Lecture 4B. The Covariant Derivative and the Einstein Equation I.
(Menon) Monday, June 4, 2 pm
Lecture 4A. Relativistic Kinematics III. Reaction Rate and Secondary Production Spectra
(Dermer) Wednesday, May 30, 2 pm
Lecture 3B. Tensor Calculus III.
(Menon) Tuesday, May 29, 2 pm
Lecture 3A. Relativistic Kinematics II. Application of Invariants
(Dermer) Wednesday, May 23, 2 pm
Chapter 2 (rev. 5/27/07)
Lecture 2B. Tensor Calculus II.
(Menon) Tuesday, May 22, 2 pm
Lecture 2A. Relativistic Kinematics I. Special Relativity
(Dermer) Monday, May 21, 2 pm
Lecture 1B. Introduction to Tensor Calculus
(Menon) Friday, May 18, 2 pm
General Relativity Lectures
Lecture 1A. Introduction. Black Holes in Nature
(Dermer) Thursday, May 17, 2 pm
Chapter 1 (rev. 5/25/07)
Chapter 1 Lecture
Chuck Dermer (202) 767-2965
Code 7653, NRL, Washington, DC 20375-5352 | {"url":"http://heseweb.nrl.navy.mil/gamma/~dermer/lectures/Book/index.htm","timestamp":"2014-04-18T20:42:54Z","content_type":null,"content_length":"6209","record_id":"<urn:uuid:1dd363aa-ec0c-4fa8-8a48-ccfbd2dc255e>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00026-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
lee is taking some friends on a picnic. They'll need to follow a path to get to the picnic spot, a map of the scale is based on a scale of 1:20,000 in cm. If the path is 12 cm on the map how many
kilometers is the actual path?
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Best Response
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\[\frac{ 1 }{ 20,000 } = \frac{ 12 }{ x }\]-------------- Cross multiply. \[20,000 \times 12 = 240,000\]So\[1 \times x = 240,000\] 1 times 240,000 equals 240,000. x = 240,000 There are 240,000
kilometers in the actual path.
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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.
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The Compton frequency
Left Arthur H. Compton (1892-1962)
In 1923, Arthur H. Compton determined a wavelength by observing the scattering of x-rays. The Compton frequency of the electron is:
l = h / Mc
The relationship between frequency and wavelength is.
f l = v
The phase velocity of the matter wave is c. The combination of these two formulas with the luminal velocity c produces the Compton frequency.
f = Mc^2 / h
The Compton Frequency of the single electron = 1.2356 x 10 ^20 Hertz
The electron rings like a bell at its Compton frequency. This author employes the Compton frequency throughout this text. | {"url":"http://www.angelfire.com/scifi2/zpt/compton.html","timestamp":"2014-04-21T15:14:51Z","content_type":null,"content_length":"1748","record_id":"<urn:uuid:28ce3ad7-5acb-4af5-9d31-b57ff3c57e36>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00402-ip-10-147-4-33.ec2.internal.warc.gz"} |
Random Walk Coin Simulation
Date: 10/27/2005 at 11:56:04
From: Ravi
Subject: Probability "Random walk"
I start at x = 0 and flip a coin, then carry out one of the following
If coin lands with tails: x = x - 1
If coin lands with heads: x = x + 1
I continue to flip the coin until I reach x = +n or x = -n.
Is there a way to calculate how many times I will have to toss the
coin? If yes, please suggest the way. I tried applying probability
and can prove that since p(head) = p(tail) = 0.5 the probability of
reaching +n/-n is also = 0.5.
I know that the answer to this question is n squared, but I have not
been able to reason out the solution.
Date: 10/28/2005 at 04:45:51
From: Doctor Jacques
Subject: Re: Probability
Hi Ravi,
Here is the outline of a way to find the solution; I will let you fill
in the missing steps.
I assume that you want to know the average number of steps required--
there is no maximum number of steps; indeed, if you are out of luck,
you could get alternately heads and tails infinitely often and the
game would never end.
We let y[k] be the average number of steps required to end the game
when x = k (we are interested in values of k between -n and +n). As
we start with x = 0, we must find y[0].
Note first that, because of the symmetry of the problem, we have:
y[k] = y[-k] [1]
If x = n or x = -n, the game is over, and we have therefore:
y[n] = y[-n] = 0 [2]
Assume now that -n < x < n; we have to toss the coin at least once
more. If we get a head, x will become x + 1, and there will be, on
the average, y[k+1] moves remaining after this move (this is the
definition of y[k]). If you take into account the next move, this
means that, if you get a head, the total number of moves left is:
y[k;head] = 1 + y[k+1] [3]
In the same way, if you get a tail, you have:
y[k;tail] = 1 + y[k-1] [4]
Each of these two cases can happen with probability 1/2; this means
that y[k] is the average of the above two values:
y[k] = 1/2(1 + y[k+1]) + 1/2(1 + y[k-1])
and this can be written as:
2*y[k] - y[k+1] - y[k-1] = 2 [5]
You could write all the equations [5] for k between -n+1 and n-1,
together with the two equations [2]. This would give you a system of
(2n+1) linear equations in the (2n+1) unknowns y[-n] ... y[n]. You
could solve that system to find y[0]. Although this does work, there
is a nice shortcut available.
We define:
d[k] = y[k-1] - y[k] [6]
so that d[k] is the difference between consecutive values of y.
Note that, because of [1], we have:
d[1] = y[0] - y[1] = 1 [7]
We can also rewrite [5] as:
(y[k] - y[k+1]) - (y[k-1] - y[k]) = 2
d[k+1] = d[k] + 2 [8]
Because of the symmetry, we only need to consider the equation [7]
and equations [8] for k > 0. These equations show that the d's are
just the odd integers:
d[1] = 1
d[2] = 1 + 2 = 3
d[3] = 3 + 2 = 5
d[k] = 2k-1 [9]
It is easy to prove by induction that:
1 + 3 + ... + 2k-1 = k^2 [10]
and, therefore:
d[1] + ... + d[k] = k^2 [11]
On the other hand, if you add together the equations [6], the
intermediate y[k] terms cancel out, and you have:
d[1] + ... + d[k] = y[0] - y[k]
which means, because of [11], that:
y[k] = y[0] - k^2 [12]
Because of [2], if we let k = n, we see that we must have y[0] = n^2,
which is the result we are looking for.
Does this help? Write back if you'd like to talk about this some
more, or if you have any other questions.
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Math Help
October 3rd 2011, 06:40 PM #1
Junior Member
Sep 2011
Hey again, I need a lot of help here.
Is $O_n$ isomorphic to the product group $SO_n \times \{\pm I \}$?
I can show that for $n=2$, then No it is not, and for $n=3$ it is isomorphic but how do I show it in general.
Thank for your help.
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Fundamentals of Programming using C++ by Baldwin
Passing parameters
You will normally want to make the function general.
For example, when designing and writing a function that calculates the square root of a number, it is desirable to write it so that it can calculate the square root of any number (as opposed to only
one specific number).
This is accomplished through the use of something called parameters. | {"url":"http://www.dickbaldwin.com/Cosc1315/Slides/Pf00108ak.htm","timestamp":"2014-04-18T15:53:06Z","content_type":null,"content_length":"1204","record_id":"<urn:uuid:22ec4d6f-4c26-45f1-b891-ef74ade1f117>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00084-ip-10-147-4-33.ec2.internal.warc.gz"} |
Penn Ctr, PA Science Tutor
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Understanding Recursive Functions
This example uses a recursive function to find the factorial of the number 4. I used a 'cout' command to try to understand what is going on. But I don't know why the first output is '2'. I tried
really hard to create my own logic to undstand why it's doing what it's doing lol. And also read the page in the book a few times but I still don't get it.
#include "stdafx.h"
#include <iostream>
using namespace std;
int factr(int n);
int fact(int n);
int main()
// Use recursive version.
cout << "factorial is " << factr(4);
cout << "\n";
// Use iterative version.
//cout << "4 factorial is " << fact(4);
cout << "\n";
return 0;
// Recursive version.
int factr(int n)
int answer;
if(n==1) {return(1);}
answer = factr(n-1)*n; //Execute a recursive call to factr().
cout << answer << " ";
//Iterative version.
int fact(int n)
int t, answer;
answer = 1;
for(t=1; t<=n; t++) {answer = answer*(t);} | {"url":"http://www.dreamincode.net/forums/topic/145146-understanding-recursive-functions/","timestamp":"2014-04-18T09:16:38Z","content_type":null,"content_length":"133432","record_id":"<urn:uuid:f784bd7e-6b3f-4aa9-9691-d4ba18887c5c>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00658-ip-10-147-4-33.ec2.internal.warc.gz"} |
Eckmann-Hilton argument
Higher category theory
Basic concepts
Basic theorems
Universal constructions
Extra properties and structure
1-categorical presentations
The Eckmann–Hilton argument
In its usual form, the Eckmann–Hilton argument shows that a monoid or group object in the category of monoids or groups is commutative. In other terms, if a set is equipped with two monoid
structures, such that one is a homomorphism for the other, then the two structures coincide and the resulting monoid is commutative.
From the nPOV, we may want to think of the statement in this way:
Let $C$ be a 2-category and $x \in C$ an object. Write $Id_x$ for the identity morphism of $X$ and $End(Id_x)$ for the set of endo-2-morphisms on $X$. Then:
On the face of it, this is a special case of the general situation, although in fact every case is an example for appropriate $C$.
A more general version is this: If a set is equipped with two binary operations with identity elements, as long as they commute with each other in the sense that one is (with respect to the other) a
homomorphism of sets with binary operations, then everything else follows:
1. the other is also a homomorphism with respect to the first;
2. each also preserves the other's identity;
3. the identities are the same;
4. the operations are the same;
5. the operation is commutative;
6. the operation is associative.
This can also be internalised in any monoidal category.
A pasting diagram-proof of 1 is depicted in Cheng below. Here we prove the $6$-element general form in $Set$.
The basic equation that we have (that one operation $*$ is a homomorphism with respect to another operation $\circ$) is
$(a \circ b) * (c \circ d) = (a * c) \circ (b * d) .$
In $End(Id_x)$, this is the exchange law.
We prove the list of results from above in order:
1. Simply read the basic equation backwards to see that $\circ$ is a homomorphism with respect to $*$.
2. Now if $1_*$ is the identity of $*$ and $1_\circ$ is the identity of $\circ$, we have
$1_\star \circ 1_\star = (1_\star \circ 1_\star) * 1_\star = (1_\star \circ 1_\star) * (1_\star \circ 1_\circ) = (1_\star * 1_\star) \circ (1_\star * 1_\circ) = 1_\star \circ 1_\circ = 1_\star .$
A similar argument proves the other half.
3. Then
$1_\star = 1_\star * 1_\star = (1_\star \circ 1_\circ) * (1_\circ \circ 1_\star) = (1_\star * 1_\circ) \circ (1_\circ * 1_\star) = 1_\circ \circ 1_\circ = 1_\circ ,$
so the identities are the same; we will now write this identity simply as $1$.
4. Now
$a * b = (a \circ 1) * (1 \circ b) = (a * 1) \circ (1 * b) = a \circ b ,$
so the operations are the same; we will write them both with concatenation.
5. Then
$a b = (1 a) (b 1) = (1 b) (a 1) = b a ,$
so this operation is commutative.
6. Finally,
$(a b) c = (a b) (1 c) = (a 1) (b c) = a (b c) ,$
so the operation is associative.
If you start with a monoid object in $Mon$, then only (4&5) need to be shown; the others are part of the hypothesis. This classic form of the Eckmann–Hilton argument may be combined into a single
$a * b = (a \circ 1) * (1 \circ b) = (a * 1) \circ (1 * b) = a \circ b = (1 * a) \circ (b * 1) = (1 * b) \circ (a * 1) = b * a ,$
where the desired results involve the first, middle, and last expressions.
A $2$-tuply monoidal $0$-category, if defined as a pointed simply connected bicategory, is also the same as an abelian monoid.
A $2$-tuply monoidal $1$-category, if defined as a pointed simply connected tricategory, is the same as a braided monoidal category.
Every homotopy group $\pi_n$ for $n \geq 2$ is abelian.
The beautiful and powerful Eckmann-Hilton argument is due to Beno Eckmann and Peter Hilton.
An expositions of the argument is given here:
The diagram proof is displayed here
and an animation of it is here
For higher analogues see within the discussion of commutative algebraic monads at: | {"url":"http://www.ncatlab.org/nlab/show/Eckmann-Hilton+argument","timestamp":"2014-04-21T07:33:09Z","content_type":null,"content_length":"43770","record_id":"<urn:uuid:4212c5e1-8395-475d-a65b-e8a146e7f7b5>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00654-ip-10-147-4-33.ec2.internal.warc.gz"} |
Linear Interpolation FP1 Formula
Re: Linear Interpolation FP1 Formula
Yes, the general form for a simple cf is
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
But how do you show that
are the same?
Re: Linear Interpolation FP1 Formula
They are not the same obviously but we might be able to prove they both converge to the same thing.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
They said something about evaluating that first fraction at every second term, to get the second fraction...
Re: Linear Interpolation FP1 Formula
And how does that prove they are the same?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
It doesn't but they are claiming there is a way to get the simple CF from the first one...
Or maybe just someone on Wikipedia typed it for the hell of it? Maybe that's why they didn't elaborate on what they meant exactly...
Re: Linear Interpolation FP1 Formula
I do not see any way right now of deriving one from the other.
But I can prove they both converge to the same thing.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
What methods of proof do you use?
Also, still nothing from adriana...
Re: Linear Interpolation FP1 Formula
Consider her dead meat.
It is possible to prove what they converge to by algebra and a little trickery.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
You may be right... this could venture into F territory. Something is wrong. Just a pity it ended so abruptly.
You mean trying to generate the CF?
Re: Linear Interpolation FP1 Formula
Take a look at the first cf and call it x
Last edited by bobbym (2013-02-26 06:27:21)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
Sorry, I had to take my sister to my grandmother's house.
I see, you just get the fraction on its own, invert it, and repeat, and you end up with a quadratic which has a root at x = √3. I imagine the same thing would work for the other CF.
Re: Linear Interpolation FP1 Formula
when you solve for x you get √3
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
Yes, that is what I meant -- eventually when you follow those steps, you get the original fraction (x again).
Re: Linear Interpolation FP1 Formula
Yes, somewhere in th cf there will be a repeat of x. You just replace it and solve for x.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
That seems pretty handy to check if these Wikipedia CFs converge as they're supposed to... but, I suppose this method might be hairy if you have a long period, for large n, say.
Re: Linear Interpolation FP1 Formula
It is easier to use numerical methods and arrive at an experimental result.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
We looked at continued fractions and applying them to solve Pell's equation, are there any applications of infinite nested square roots? Those seem interesting too.
Re: Linear Interpolation FP1 Formula
I do not know of any offhand apps but they are solved in the same way as a cf.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
Yes, I have read about them. The Wiki article is a lot shorter than the CFs one though.
Re: Linear Interpolation FP1 Formula
They are less useful apparently. CF's are very big in number theory and numerical analysis.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
What other things are they used for in number theory?
Re: Linear Interpolation FP1 Formula
Just for the Fermat - Pell equation as far as I know. But these are an important class of diophantine equations. They are more important I think in numerical analysis.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Linear Interpolation FP1 Formula
A girl asked me what I was doing today in chemistry, I was trying to solve a Pell equation using one of my CFs. When I tried to show her what a CF was, she said "I'd rather paint my nails".
23:53, nothing from adriana at all. Sigh...
Re: Linear Interpolation FP1 Formula
Yikes, an intellectual!
I got another method for CF's.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof. | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=255175","timestamp":"2014-04-18T21:52:28Z","content_type":null,"content_length":"36980","record_id":"<urn:uuid:353f3c61-8c72-4ba6-8a79-543530537402>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00450-ip-10-147-4-33.ec2.internal.warc.gz"} |
s, and
Mary Richardson, Phyllis Curtiss, John Gabrosek, and Diann Reischman
Department of Statistics
Grand Valley State University
1 Campus Drive
Allendale, MI 49401-9403
Statistics Teaching and Resource Library, September 1, 2002
© 2002 by Mary Richardson, Phyllis Curtiss, John Gabrosek, and Diann Reischman, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium
without express written consent from the authors and advance notification of the editor.
This article describes an interactive activity illustrating sampling distributions for means, properties of confidence intervals, properties of hypothesis testing, confidence intervals for means,
and hypothesis tests for means. Students generate and analyze data and through simulation explore these concepts. The activity is completed in three parts. The three parts of the activity can be
used in sequence or they can be used individually as “stand alone” activities. This allows the educator flexibility in utilizing the activity. Part I illustrates the sampling distribution of the
sample mean. Part II illustrates confidence intervals for the population mean. Part III illustrates hypothesis tests for the population mean. This activity is appropriate for use in an
introductory college or high school AP statistics course.
Key words: sampling distribution of a sample mean, confidence interval for a mean, hypothesis test on a mean, simulation
After completing the Rectangularity activity, students will understand:
How to construct and use the sampling distribution for the sample mean
How to construct and interpret a confidence interval for a mean
How to perform a hypothesis test on a mean
How to interpret the level of significance of a hypothesis test (type I error rate)
How to interpret the p-value of a hypothesis test
How to interpret the type II error rate of a hypothesis test
How to interpret the power of a hypothesis test
The relationship between type I and type II error rates and power
Materials and equipment
Each student needs a random number table or a calculator that generates random numbers, four sticky notes (for Parts II and III), and a copy of the activity (which includes statistical guides
containing relevant notation, formulas, and definitions). Included in the student’s version of the activity is a sheet with a population of 100 rectangles having different areas. Each square
counts as one unit towards a rectangle’s area.
Time involved
The activity is completed in three parts. The estimated completion time for each part is one class period (approximately one hour). The three parts of the activity can be used in sequence or they
can be used individually as “stand alone” activities. Part I illustrates the sampling distribution of the sample mean and involves calculations that should be completed using either a computer
software package or a graphing calculator. Part II illustrates confidence intervals for the population mean. Part III illustrates hypothesis tests for the population mean.
Activity description - Part 1: sampling distribution of the sample mean
To begin, the teacher draws a histogram of the population distribution of areas on the whiteboard. The population distribution of areas is skewed to the right (positively skewed).
Ten groups of two or three students are formed and the following tasks are assigned to each group.
Select two different random samples of n = 5 rectangles (with replacement)
Select two different random samples of n = 15 rectangles (with replacement)
Select two different random samples of n = 25 rectangles (with replacement)
Students calculate the average area of the rectangles for each sample drawn reinforcing the idea that the sample mean is a random variable.
To complete the data collection sheet, group results are combined to obtain 20 sample means for sample sizes n = 5, 15, and 25.
After data collection, students answer a series of questions based on the means and standard deviations of the sample means for the different sample sizes. Students discover properties of the
distribution of a sample mean; namely, (i) the distribution of sample mean values is centered at the population mean, (ii) the distribution of sample mean values approaches a normal distribution
as the sample size increases, (iii) the distribution of sample mean values has less variability than the original population, and (iv) the variability of sample mean values decreases as n
Activity description - Part 2:confidence interval for the population mean
Each student selects a simple random sample of 25 rectangles (with replacement). Note that the population of rectangle areas does not have a normal distribution, but the t confidence interval
procedure may be applied in this case since the sampling distribution of is approximately normal for samples of size 25. First, each student uses her sample to construct an 80% confidence
interval for the population mean rectangle area. Each student writes her result on a sticky note and gives it to the instructor. Each student’s confidence interval is sketched horizontally on an
overhead transparency leaving one blank horizontal line between intervals. The resulting overhead transparency displays all of the confidence intervals constructed by the students in the class.
Students see the results of drawing repeated samples from the same population and calculating 80% confidence intervals. Some of the confidence intervals will contain the population mean (6.26)
and some will not. After graphing the class confidence intervals, their meaning is discussed. We stress that if we claim that we are 80% confident that a mean lies within the endpoints of a
confidence interval, we are saying that the endpoints of the confidence interval were calculated by a method that gives correct results in 80% of all possible random samples. We are not saying
that there is an 80% chance that a calculated interval contains the population mean. Students are asked to write a statement explaining how an 80% level of confidence should be interpreted.
Students are then asked to construct a 99% confidence interval for the population mean rectangle area. As above, the class confidence intervals are graphed and the results are discussed. We
stress how to properly interpret a 99% confidence level and ask students to write a statement explaining how a 99% level of confidence should be interpreted. Students are asked to write a
statement explaining how increasing the confidence level from 80% to 99% changed the width of their confidence intervals.
Activity description - Part 3:hypothesis test on the population mean
Each student selects a simple random sample of 25 rectangles (with replacement) or uses the simple random sample selected for Part II. Note that the population of rectangle areas does not have a
normal distribution, but the t test may be applied in this case since the sampling distribution of is approximately normal for samples of size 25.
In question 1, students use their sample data to perform two hypothesis tests of H[o]:m=9 versus H[a]:m<9 with different levels of significance. Each student’s data is a different simulated
sample. Since the true population mean rectangle area is m=6.26, the null hypothesis H[o]:m=9 is false. Since H[o] is false, performing these tests provides an opportunity to use simulation to
illustrate properties of p-values, type II errors, and power.
The first test of H[o]:m=9 versus H[a]:m<9 is performed using level of significance a=.05. The instructor draws stems for a stem-and-leaf plot on the whiteboard. Each student writes her
calculated p-value on a sticky note and places it on the stem-and-leaf plot.
Assuming a class size of 30 students, the plot will contain 30 calculated p-values. The p-values are calculated under the assumption that H[o]:m=9 is true (when, in fact, m=6.26), so the p-values
will tend to be small. We discuss with students that small p-values contradict H[o]. Some students will not obtain small p-values. On the stem-and-leaf plot, a cut-off value is marked at a=.05.
Each p-value falling at or below this cut-off represents a rejection of H[o] (a correct decision). Each p-value falling above this cut-off represents a failure to reject H[o] (a type II error).
Since 30 samples are taken, and 30 tests are performed, students see that some samples result in a correct decision and other samples result in an incorrect decision (type II error). Students are
asked to calculate the fraction of incorrect decisions to obtain a simulated value for b, the probability of a type II error, and a simulated value for the power = 1-b. An explanation is then
given of how to interpret a type II error rate (and power) in terms of repeatedly performing the procedure of selecting a sample, then using the data to test a hypothesis about a population
parameter, when the null hypothesis is false.
The second test is performed using a=.20. The p-value is the same as for the first test; however, the type I error rate is increased to 20%. On the stem-and-leaf plot of p-values, a new cut-off
is marked at a=.20. Each p-value falling at or below this cut-off represents a rejection of H[o] (a correct decision). Each p-value falling above this cut-off represents a non-rejection of H[o]
(a type II error). Students are asked to calculate the fraction of non-rejections of H[o] out of the 30 tests to obtain a simulated value for b and a simulated value for the power. In examining
the class results, students note that an increase in the type I error rate results in a decrease in the type II error rate and thus an increase in the simulated power.
In question 2, students use their sample data to perform two hypothesis tests of H[o]:m=6.26 versus H[a]:m¹6.26 with different levels of significance. Under the assumption that m=6.26, performing
these tests provides an opportunity to illustrate properties of p-values and type I error.
The first test of H[o]:m=6.26 versus H[a]:m¹6.26 is performed using a=.05. The second test is performed using a=.20. As before, a stem-and-leaf plot of the class p-values is constructed.
The p-values are calculated under the assumption that H[o]:m=6.26 is true, so the p-values will tend to be large. We discuss with students that large p-values do not contradict H[o]. Some
students will not obtain large p-values. On the stem-and-leaf plot, a cut-off value is marked at a. Each p-value falling at or below this cut-off represents a rejection of H[o] (a type I error).
Each p-value falling above this cut-off represents a failure to reject H[o] (a correct decision). Since 30 samples are taken, and 30 tests are performed, students can see that some samples result
in a correct decision and other samples result in an incorrect decision (type I error). For a=.05 and a=.20, students are asked to calculate the fraction of rejections of H[o] out of the 30 tests
to obtain a simulated value for a. An explanation is then given of how to interpret a type I error rate in terms of repeatedly selecting a sample, then using the data to test a hypothesis about a
population parameter, when the null hypothesis is true.
Teacher notes
Students work with a population of 100 rectangles, drawing repeated simple random samples (with replacement). Prior to completing Part I, students should be familiar with descriptive statistics
and probability distributions. Prior to completing Part II, students should be familiar with the basic mechanics of how to construct confidence intervals. Prior to completing Part III, students
should be familiar with the basic mechanics of how to perform hypothesis tests, including the calculation of test statistics and p-values.
In this activity, we sample with replacement to preserve the independence of the sample observations. When sampling with replacement, it is possible for the same rectangle to be sampled more than
once. If sampled rectangles are not replaced in the population, then each time a rectangle is withdrawn the probability of selection for the remaining rectangles will increase. In practice, we
often either sample with replacement or we sample from a population that is so large that the withdrawal of successive items changes selection probabilities negligibly.
In this activity, we used the same data set to perform two different hypothesis tests at two different levels of significance. The instructor should emphasize that the level of significance, null
hypothesis, and alternative hypothesis should be determined prior to data collection. We use the same data for multiple hypothesis tests to save time. Technically, we should have collected four
separate data sets, one for each of the four tests conducted.
In addition, the instructor should stress to students that in reality one would not know the true value of the population mean m. If the parameter value were known, then there would be no point
in utilizing sample data to draw an inference about the parameter. The instructor should stress that we assume that we know the parameter so that we can investigate the properties of hypothesis
testing under different situations.
For Part I: Students should write about the effect of sampling variability on the center, spread, and shape of the sampling distribution of the sample mean. Students should write about the effect
of sample size on the shape and spread of the distribution of the sample mean.
The following questions can be used to assess student understanding or as challenge problems for students who complete the activity early.
1. What happens to the shape of the sampling distribution of the sample means for this non-normal population as the sample size increases?
2. How do you think the shape, mean, and standard deviation of the distribution of the sample means for samples of size 100 would compare to the shape, mean, and standard deviation for the
samples of size 25 that the class took?
3. Widgets produced by a machine are known to have a mean diameter of 12 mm with a standard deviation of 0.31 mm. Suppose that we take a random sample of 90 widgets and measure each widget’s
diameter. We calculate the mean diameter of the 90 widgets. We repeat this process every day for 365 days so that we have .
a. What would we expect the mean of the 365 daily means to be?
b. What would we expect the standard deviation of the 365 daily means to be?
c. What would we expect the shape of the histogram of the 365 daily means to be? Why?
d. Assuming that the machine continues to perform as it has in the past, what is the probability that for the next day the mean diameter of the 90 sampled widgets will be between 11.95 mm and
12.05 mm?
e. Why is simply looking at the mean diameter not enough to say that the machine is producing widgets with diameters close to the desired 12mm?
For Part II: Students should be able to explain how to interpret a confidence interval. Additionally, students should be able to describe the relationship between the confidence level and the
width of a confidence interval.
The following questions can be used to assess student understanding or as challenge problems for students who complete the activity early.
For all of these questions, assume that the samples are large enough so that the sampling distribution of the sample mean is approximately normal.
1. Suppose a simple random sample (SRS) of 20 rectangles has sample mean, = 7.3, and sample standard deviation, s = 6.1. Based on the sample, we wish to estimate the value of the population mean,
a. What is the point estimate for m?
b. What is the standard deviation of the point estimate?
c. The mean of the sample will not be exactly equal to the mean of the population, thus there is error associated with the point estimate. With 95% confidence, what is the maximum error
associated with the point estimate? (That is, what is the largest possible difference between and m) This value is often called the margin of error.
d. The margin of error in part (c) consists of how many estimated standard deviations of ?
2. Suppose the sample mean, from a SRS of 40 rectangles is used to estimate m.
a. How would you expect the standard deviation of the sampling distribution of the sample mean of 40 rectangles to compare to the standard deviation of the sampling distribution of the sample
mean of 20 rectangles? Explain.
b. How would you expect the 95% margin of error for the estimate of m for the 40 rectangles to compare to the 95% margin of error for the 20 rectangles in the previous problem? Explain.
c. Do you think using a sample mean from a sample of size 40 will give a more precise estimate of m than the sample mean from a sample of size 20? Explain.
3. In the activity, you selected a SRS of 25 rectangles and constructed an 80% confidence interval. Suppose you had selected a SRS of 40 rectangles and constructed an 80% confidence interval. How
would you expect the confidence interval constructed from 40 rectangles to compare to the confidence interval constructed from 25 rectangles? Explain.
4. For a large population, a 90% confidence interval for m is found to be 23.5 to 28.9. Why is the following statement incorrect? “There is a 90% chance that m is between 23.5 and 28.9.”
5. Suppose you select a SRS of size 30 from a large population and find a 95% confidence interval for m to be 17.30 to 23.47. Your friend selects a separate SRS of size 30 from the population and
finds a 95% confidence interval for m to be 18.64 to 24.81. Which confidence interval is better? Explain.
For Part III: Students should be able to explain type I error and type II error in a specific problem. Additionally, students should be able to describe the relationship between type I and type
II error rates and power.
The following questions can be used to assess student understanding or as challenge problems for students who complete the activity early.
1. A company is trying to decide whether to buy a new Widget machine that costs $1 million. It is decided it will be worth buying the machine if there is overwhelming evidence that the mean
number of defective Widgets will decrease from the current rate of 200 per day.
a. State the null and alternative hypotheses needed to test if the machine should be purchased.
b. Describe a type I error in the context of this problem.
c. Describe a type II error in the context of this problem.
d. Argue that a type I error is a more serious error in this problem.
e. For this situation, should the company run the test at the 1%, 5%, or 10% significance level? Explain.
2. Explain the fallacy in reasoning in the following statement. “I wanted to reduce the chance of committing an error, so I reduced the type I error rate to .001.”
3. A doctor claims that his patients wait an average of 10 minutes in his waiting room. A disgruntled patient claims it is really higher. For a random sample of patients, the sample mean is 10.8
with a standard deviation of 2.1.
a. If the sample consisted of 25 patients, perform the appropriate hypothesis test using a 1% level of significance (a=.01).
b. If the sample consisted of 50 patients, perform the appropriate hypothesis test using a 1% level of significance (a=.01).
c. Give an intuitive justification for why changing the sample size may result in changing the conclusion about a null hypothesis.
d. In general, what is the relationship between the sample size and the absolute value of the test statistic?
e. In general, what is the relationship between the sample size and the p-value? (To answer this question, refer to the t-curve.)
f. What do you think is the overall relationship between the sample size, the type II error rate, and the power, when H[o] is false?
Aliaga, M. and Gunderson, B. (1999). Interactive Statistics. New Jersey: Prentice Hall.
Scheaffer, R., Gnanadesikan, M., Watkins, A., and Witmer, J. (1996). Activity-Based Statistics: Instructor Resources. New York: Key Curriculum Press; Springer. | {"url":"https://www.causeweb.org/repository/StarLibrary/activities/richardson_curtiss_gabrosek_reischman2002/","timestamp":"2014-04-20T07:04:32Z","content_type":null,"content_length":"55377","record_id":"<urn:uuid:74cf19d8-df69-416e-9232-520238eaedaa>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00074-ip-10-147-4-33.ec2.internal.warc.gz"} |
CMU Theory Lunch
Runtime-Efficient Mesh Generation
Todd Phillips
Mar 31, 2010
The mesh generation problem in its simplest form is to construct a superset of an input set of points, such that the superset has good geometry, in particular so the vertices are well-spaced.
Existing algorithms can achieve an $O(1)$-minimal size superset with efficient runtime.
One can extend the notion of conforming beyond simply taking a superset; additionally conforming to input edges and facets that form a piecewise linear complex. Such a construction does not exist
unless we relax the well-spacing condition around acute angles between inputs.
This relaxation destroys known lower-bounds, so that $O(1)$-minimality can no longer be guaranteed, but some weak analyses on size exist.
I will present the first runtime-efficient algorithm for this problem in 3d, with worst-case runtime O(N log D + M), where N is the input size, D is the aspect ratio of the input (diameter to closest
pair ratio), and M is the output size. | {"url":"http://www.cs.cmu.edu/~theorylunch/20100331.html","timestamp":"2014-04-21T11:15:28Z","content_type":null,"content_length":"2368","record_id":"<urn:uuid:f25bae6b-7db7-4f3f-8368-be22923c5ee7>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00277-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Calculus - Chain Rule Derivative
October 10th 2007, 04:18 PM #1
Oct 2007
Calculus - Chain Rule Derivative
The question is find dy/dx of $\frac{1}{4}\sin^2(2x)$.
which I rewrote as $\frac{1}{4}(\sin2x)^2$
then, using the chain rule, $\frac{1}{2}\sin2x\cos2x$.
But the book gives the answer $\frac{1}{2}\sin4x$.
Is my answer wrong? Or do I just need to simplify it further?
Hello, Happy!
Find $\frac{dy}{dx}$ of: . $\frac{1}{4}\sin^2(2x)$.
which I rewrote as: $\frac{1}{4}(\sin2x)^2$ . . . . good!
then, using the chain rule, $\frac{1}{2}\sin2x\cos2x$ . . . . You dropped a "2"
$\frac{dy}{dx}\:=\;\frac{1}{4}\cdot2\sin2x\cos2x\cd ot2 \:=\:\sin2x\cos2x$
Then they used the identity: . $\sin2\theta \:=\:2\sin\theta\cos\theta$
Oh yes! Thank you, I forgot to "chain" the derivative of the 2x to the rest of the chained derivative.
October 10th 2007, 04:47 PM #2
Super Member
May 2006
Lexington, MA (USA)
October 10th 2007, 05:05 PM #3
Oct 2007 | {"url":"http://mathhelpforum.com/calculus/20348-calculus-chain-rule-derivative.html","timestamp":"2014-04-16T17:45:25Z","content_type":null,"content_length":"37060","record_id":"<urn:uuid:ee0ad580-9bdc-48d3-b0c7-6ac0107828f7>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00148-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Large collection of platform independent, interactive, java applets and activities for K-12 mathematics and teacher...
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Cannon, E.; Cannon, L.; Dorward, J.; Duffin, J.; Heal, R.; Stowell, D.; Susman, Z.; Wellman, R.; Youngberg, J.
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A huge list of resources in various topics of mathematics and math education.
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Solves and graphically displays the solutions to ordinary differential equations. The user can select from an extensive list...
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This JavaScript solves up to ten by ten systems of linear equations. It also allows the user to find the inverse of a matrix.
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the first resource for mathematics
Three-dimensional boundary detection for particle methods.
(English) Zbl 1173.76393
Summary: The three-dimensional exposure method for the detection of the boundary of a set of overlapping spheres is presented. Like the two-dimensional version described in a previous paper [
G. A. Dilts
, Int. J. Numer. Methods Eng. 48, No. 10, 1503–1524 (2000;
Zbl 0960.76068
)], the three-dimensional algorithm precisely detects void opening or closure, and is optimally suited to the kernel-mediated interactions of smoothed-particle hydrodynamics, although it may be used
in any application involving sets of overlapping spheres. The principle idea is to apply the two-dimensional method, on the surface of each candidate boundary sphere, to the circles of intersection
with neighboring spheres. As the algorithm finds the exact solution, the quality of detection is independent of particle configuration, in contrast to gradient-based techniques. The observed CPU
execution times scale as
$O\left(M{N}^{\epsilon }$
(Porson)), where
is the number of particles,
is the average number of neighbors of a particle, and
(Porson) is a problem-dependent constant between 1.6 and 1.7. The time required per particle is comparable to the amount of time required to evaluate a three-dimensional linear moving-least-squares
interpolant at a single point.
76M28 Particle methods and lattice-gas methods (fluid mechanics)
74S30 Other numerical methods in solid mechanics
65M99 Numerical methods for IVP of PDE | {"url":"http://zbmath.org/?q=an:1173.76393","timestamp":"2014-04-19T17:23:42Z","content_type":null,"content_length":"22114","record_id":"<urn:uuid:310a90fc-81f3-4779-9cd4-51b4bcaa3db4>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00062-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help on Finance!
May 2nd 2011, 01:38 PM #1
May 2011
I missed the lesson in class, I need help on one question then i should be able to figure out how to do the rest of my assignments! Please and thank you, it would be greatly appreciated.
The current value of a home is $180,000. There is an outstanding mortgage on the home for $130,000 at 7.5%. With 25 years remaning on the mortgage. The value of this particular home has been
increasing at a rate of 3% a year.
a) What is the estimated value of the home after 5 years, correct to the neartest $100.
b) how much is still owed on the home after 5 years?
(hint: find the monthly payment, than find future value after 5 years)
(capitals/bolded are where the answer should be in the tmv solver)
There is a reason why there are lectures. You should attend them.
Please learn to speak english, rather than jargon. Define all terms, no matter how common you think they are.
What are these?
I think you should use your hints and show us what you get.
TMV solver is how you solve questions like this on your calculator.
This sis the layout for the TMV solver.
This is what i get for A) 180,000 x 0.03 = 5400
5400 x 5 =$27,000
27,000 + 180,000 = $207,000.
.. I'm not sure if this is correct.
n= 25 x 12 (total number of payment periods) .. 25 years, 12 months a year since its paid for monthly
i%= 7.5 (interest)
pv= 180, 000 (present value)
PMT= (payments per month)
FV=0 (future value)
p/y=12 (payments per year)
c/p=2 (compound period)
With this format i get my PMT to be 1316.79885.
Are you asking HOW these solutions are arrived at,
or how to use your calculator?
If you can't keypunch correctly, can you not use the calculator's manual?
TMV solver is how you solve questions like this on your calculator.
This sis the layout for the TMV solver.
This is what i get for A) 180,000 x 0.03 = 5400
5400 x 5 =$27,000
27,000 + 180,000 = $207,000.
.. I'm not sure if this is correct.
n= 25 x 12 (total number of payment periods) .. 25 years, 12 months a year since its paid for monthly
i%= 7.5 (interest)
pv= 180, 000 (present value)
PMT= (payments per month)
FV=0 (future value)
p/y=12 (payments per year)
c/p=2 (compound period)
With this format i get my PMT to be 1316.79885.
this doesnt stack up .
House appreciates at 3% pa, one can assume its compunded annually so part a is wrong
the loan is 130k not 180k so part b is wrong.
In addition c/p dont stack either
As to punching this stuff into a calculator. GIGO
May 2nd 2011, 02:45 PM #2
MHF Contributor
Aug 2007
May 2nd 2011, 02:55 PM #3
May 2011
May 3rd 2011, 04:52 AM #4
MHF Contributor
Dec 2007
Ottawa, Canada
May 11th 2011, 12:47 PM #5
Apr 2011 | {"url":"http://mathhelpforum.com/business-math/179297-help-finance.html","timestamp":"2014-04-17T10:10:41Z","content_type":null,"content_length":"41922","record_id":"<urn:uuid:b02e87bc-e1af-437f-aff7-f288f701dd75>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00338-ip-10-147-4-33.ec2.internal.warc.gz"} |
Seating arrangements
February 19th 2009, 01:27 PM
Seating arrangements
The question has four parts:
A. Suppose there are 8 seats in a row with 8 people to seat in the seats, one per seat, how many seating arrangements are possible?
B. Suppose 2 of the 8 are twins and are indistinguishable, thus any arrangement should be labeled the same regardless of which twin in is which seat, how many arrangements are possible?
C. Same as B but with triplets instead of twins
D. Same as B but with triplets and a different set of twins, still 8 total.
For A it should be 8! right? idk what to do for B, C, and D though, any help would be appreciated, TIA.
February 19th 2009, 01:41 PM
If we rearrange the string "AAABCDEF" is that not the exact same as part C?
The number of possible arrangements is $\frac{8!}{3!}$.
Now you show us the rest.
February 19th 2009, 01:48 PM
so would it be like
part a is ABCDEFGH so 8! = 40320 combinations
part b is AABCDEFG so 8!/2! = 20160 combinations
part c is AAABCDEF so 8!/3! = 6720 combinations
part d is AAABBCDE so 8!/(3!+2!) = 3360 combinations?
February 19th 2009, 02:03 PM
February 19th 2009, 02:03 PM | {"url":"http://mathhelpforum.com/statistics/74531-seating-arrangements-print.html","timestamp":"2014-04-25T08:21:24Z","content_type":null,"content_length":"7998","record_id":"<urn:uuid:59c12402-4f7c-418c-90c3-d9cffad783de>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00258-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prime Factors as Bricks
Date: 10/26/2001 at 22:18:29
From: Andre Burrell
Subject: Prime Factorization
I have trouble with prime factorization. I need an easier way to do it
than making a tree.
Date: 10/27/2001 at 11:43:28
From: Doctor Sarah
Subject: Re: Prime Factorization
Hi Andre - thanks for writing to Dr. Math.
To begin with, it's helpful to have in your head the divisibility
rules for prime numbers like 2, 3, 5, and 7. You'll find them in the
Dr. Math FAQ:
Divisibility Rules
2 If the last digit is even, the number is divisible by 2.
3 If the sum of the digits is divisible by 3, the number is also.
5 If the last digit is a 5 or a 0, the number is divisible by 5.
7 Take the last digit, double it, and subtract it from the rest
of the number; if the answer is divisible by 7 (including 0),
then the number is also.
A good basic knowledge of the multiplication table is also a big help
(you knew there was a reason for learning all those multiplication
facts, right?).
Now let's take an example. How would you find the prime factorization
of 126? Well, one way you could start would be by noticing that 126
is even. 2 is the only even prime number, and it divides evenly into
every even number. So, if we divide 126 by 2 we get 63.
126 = 2 x 63
Next we know from our multiplication tables that 63 = 7 x 9.
126 = 2 x 7 x 9
We know that 7 is prime; what about 9? Nine is not prime: 9 = 3 x 3.
126 = 2 x 7 x 3 x 3
Now we can stop since we have reached only prime numbers. The prime
factors of 126 are: 2 x 3 x 3 x 7 or 2 x 3^2 x 7
Does this help?
- Doctor Sarah, The Math Forum
Date: 10/28/2001 at 10:28:17
From: Andre Burrell
Subject: Re: Prime Factorization
Thank you very much, but could you explain it a little easier?
Date: 10/28/2001 at 23:15:37
From: Doctor Peterson
Subject: Re: Prime Factorization
Hi, Andre.
Let's try taking the most basic approach I can think of, to see what
this is all about.
The prime numbers are the building blocks of which any whole number
can be built by multiplying them together. Think of them as bricks.
Some buildings might consist of a single brick (weird, but possible);
most will be built of a number of bricks. Some of those bricks might
be different sizes or colors, others might be identical. Suppose we
want to break a building down into a pile of bricks, to see what it is
made of. How do we do it? One brick at a time. That's what we want to
do with numbers: to break them down by pulling out one brick (prime
factor) at a time and putting them in piles.
So let's take the number 245. There are two main ways to find the
factors. One is to methodically go through all the possible prime
factors and see if they are there. So we get a list of small primes to
2, 3, 5, 7, 11, ...
Try one at a time, starting at the beginning of the list. Is there a 2
in this number? Divide by 2, and you find that it doesn't divide
evenly. How about 3? Again, it doesn't go in. (This is where knowing
those divisibility rules can save time, but you can just do the
divisions if you prefer not to take the time to learn them.)
Now we try 5:
5 ) 245
It goes evenly, so we know that
245 = 5 * 49
We've pulled one brick out of the wall, and what's left of the wall is
Now we can see what prime factors there are in 49. We first check
whether there is another 5 in there; taking one out doesn't mean there
isn't another! (On the other hand, we knew we didn't have to try 3
again, because we know there weren't any there.) But 49 is not
divisible by 5, so we continue through our list of primes. Is 49
divisible by 7? Yes, and the quotient is another 7:
49 = 7 * 7
245 = 5 * 7 * 7
Since we know 7 is a prime, we're finished; we have a pile of prime
"bricks." The only thing left to do, if we want, is to pile up
identical bricks by combining groups of the same prime as powers:
245 = 5 * 7^2
That's it!
The other approach is the opportunistic method: rather than go through
the primes in order, we often see an obvious prime to pick first.
(That's like seeing a loose brick sticking out and pulling that one
out first, rather than starting at the top.) In this case, since 245
ends with 5, we can tell immediately that it is divisible by 5, so we
would divide by that first. Then when we see 49, we should recognize
that as a square, and can just write it that way. All that comes with
experience. If you just want to take the slow route and make sure you
get the job done, that's fine. The important thing is that you are
getting to know how numbers are built.
If you'd like a different explanation, try going to our search page
and entering the phrase prime factorization .
- Doctor Peterson, The Math Forum
Date: 10/29/2001 at 08:37:07
From: Andre Burrell
Subject: Re: Prime Factorization
Thanks, that was much simpler. | {"url":"http://mathforum.org/library/drmath/view/58530.html","timestamp":"2014-04-16T05:54:13Z","content_type":null,"content_length":"10163","record_id":"<urn:uuid:765fa58f-e998-4e56-b801-d5b445bf1936>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00288-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quadratic Functions and Their Graphs
Learn the anatomy of the graph of a quadratic function.
Demonstrates how to graph a quadratic function using a table.
Determining if a Quadratic Function has a Maximum or Minimum
Graphing y = ax^(2 )Using a Table of Values |a| > 0
Graphing y = ax^(2 )Using a Table of Values |a| < 1
Graphing y = ax^(2) + c Using a Table of Values (c > 0)
Graphing y = ax^(2) + c Using a Table of Values (c < 0)
Graphing y = ax^(2) + bx + c Using a Table of Values
Graphing y = ax^(2) + bx + c Using the Vertex and Axis of Symmetry
Students will generate questions, activate prior knowledge and collect information to answer these questions.
Explore the relationship of between each of the variables in the quadratic function and how they affect the appearance of the graph in this simulation.
These flashcards help you study important terms and vocabulary from Quadratic Functions and Their Graphs.
You need to be signed in to perform this action. Please sign-in and try again. | {"url":"http://www.ck12.org/algebra/Quadratic-Functions-and-Their-Graphs/?difficulty=basic","timestamp":"2014-04-19T19:51:15Z","content_type":null,"content_length":"120911","record_id":"<urn:uuid:df08ba5c-3ef5-45de-a3de-68b2de5461ac>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00544-ip-10-147-4-33.ec2.internal.warc.gz"} |
Video Digitization Answer
1. Full scale is 62500 electrons, so even if there were no noise, the biggest useful number would be just short of 65536 = 2^16. The question is: what is the minimum useful number of electrons to
count? Dark current and read noise come to 45 e^-, the square root (shot noise limit) of which is ~ 7 electrons. Thus, each digitization has an uncertainty of 6-7 electrons. If we want to ensure
that we can see the noise, choose 1 bit amplitude = 6 electrons, which means the highest count we can use is 62500/6 = 10411 counts. That is between 2^13 = 8192 and 2^14 = 16384. If we chose 1
bit amplitude = 7 electrons, then the highest count would be 62500/7 = 8930, still greater than 8192, so we might as well say: 14 bits.
2. This is tricky only because it's so straight forward. Since the optimum clocking speed of the array is 40 MHz, each channel must digitize at that speed! Each channel looks at 80 × 1024 pixels =
81920 pixels. That means we can digitize the output of the array 4 × 10^7 / 81920 = 488 Hz.
3. This is harder than it looks. We have to find EITHER a 14 bit converter that can operate at 40 MHz OR a different resolution converter that is this fast OR we have to slow down the ouput.
Amazingly, there are several such ADCs, for example the AD10465 from Analog Devices (why not check Texas Instruments and other vendors for competitive products?).
4. If the signal is 0, S/N = 0. If the signal is full scale, then we're looking at 62500 electrons, and the shot noise is S^1/2 = 62500^1/2 = 250. Note that if we were only interested in high
intensities, 1 part in 250 would mean using a flash 8 bit ADC! If we were really rigorous, we'd include the influence of read noise, but since 45 electrons is <1/250 of full scale, that noise is
irrelevant to high intensity measurements. If we did want the overall S/N, since the read noise and dark current are additive offsets, σ[total] = (σ[full scale]^2 + σ[dark/read]^2)^1/2.
5. Uncertainty for a single, full scale read is 250 electrons. We get 488 reads per second. 488^1/2 = 22, and t for 95% confidence for large numbers of reads (>30) is about 2. Thus S/N is
approximately 250 * 2/22 = 23 electrons, or an improvement in S/N vs. a single read of about an order of magnitude. | {"url":"http://www.asdlib.org/onlineArticles/elabware/Scheeline_ADC/ADC_Applic_AK1.html","timestamp":"2014-04-21T09:38:01Z","content_type":null,"content_length":"4204","record_id":"<urn:uuid:90f4ca15-fe93-4fc2-8edb-bdbbd7794d51>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00618-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here's the question you clicked on:
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solve the qudratic 5x^2-20x-5=0
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x will have two solutions
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Were you able to get it to a quadratic form?
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5x^2-20x-5=0 is in the quadratic form, I need to factor 20x?
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just divide through by 5,and then solve the qudratic x^2-4x-1=0
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ans==>x=2-root5 x=2+root5
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Take the left part (left of the equal sign) and add using the LCD of 5x 2x + 3 x+5 ------- = ----- now cross multiply 5x 10 20x + 30 = 5x^2 +25x Consolidate \[5x ^{2}+5x-30=0\] Divide thru by 5
getting \[x ^{2}+x-6=0\] Do you know how to solve the quadratic?
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Hint, it factors easily. Good luck
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sorry bro resolved the problem,turned out to be 5x^2+5x-30=0 then divide through by 5 to get it in radar's form....radar is right
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ans x=-3 x= 2
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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.
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Which one of these is quadratic?
A) `y=x^2+3x`
B) `f(x) = x (4-x)-3`
C) `3x^2-2=5x` - Homework Help - eNotes.com
Which one of these is quadratic?
A) `y=x^2+3x`
B) `f(x) = x (4-x)-3`
C) `3x^2-2=5x`
A quadratic equation in x is an equation that can be written in the general form:
where, a, b and c are real numbers with ‘a’ having a non-zero value.
A quadratic equation is also defined as a univariate polynomial equation of the second degree.
Options A) and B) contains equations that are not univariate, hence these two do not represent quadratic equations.
The equation in option C) is `3x^2-2=5x`
Which can be rewritten as:
This is definitely a quadratic equation, in x.
Thus, the correct answer is option C).
c. because it has an a,b and c
Join to answer this question
Join a community of thousands of dedicated teachers and students.
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Math Help
December 6th 2009, 04:19 PM #1
Jan 2008
A pan of water (46 degrees) was put into a fridge. Ten min later, its temp was 39. Ten min after that, it was 33. Use newtons law of cooling. How cold was the fridge?
I know his law, but how do I find the rate?
Let $T_R$ be the temperature inside the refridgerator (which we can assume is kept constant) and let T(t) be the temperature of the water after t minutes. "Newton's law of cooling says that heat
will flow from a hotter object to a cooler at a rate proportional to the difference in temperature. Since the temperature is proportional to the amount of heat in the object, we can put the two
proportions together to say $\frac{dT}{dt}= k (T- T_R)$ where "k" is the, yet unknown, constant of proportion.
That is a relatively simple separable differential equation which you can solve by integrating both sides of $\frac{dT}{T- T_R}= k dt$.
That will, of course, depend upon three unknown constants, k, $T_R$, and the constant of integration. Fortunately, you have three equations, T(0)= 46, T(10)= 39, and T(20)= 33. You can solve
those three equations for the three constants, in particular $T_R$ which is what you are asked.
December 6th 2009, 06:50 PM #2
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/calculus/118949-temperature.html","timestamp":"2014-04-17T01:29:58Z","content_type":null,"content_length":"34481","record_id":"<urn:uuid:818a8557-6407-4f46-8f9d-21c2f710ab65>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
A good guessing method makes supercomputing more efficient
August 24th, 2012 in Physics / Condensed Matter
In his doctoral dissertation, Kurt Baarman, a researcher from Aalto University, developed methods for making electron density calculations more efficient. These methods can also be applied to
pharmaceutical development.
A computer is an increasingly important tool for both physicists and chemists. Computational physics can be used to model materials that cannot yet be produced experimentally. Even the electronic
structure surrounding an atom can be determined with numerical methods. These nonlinear computational problems, however, come with a particular difficulty: the solution has to be guessed in advance.
"Only then we can see how good our guess was. In my dissertation, I have looked at ways to minimise the number of guesses needed to achieve the best possible result", Kurt Baarman explains.
With the help of supercomputers and computational methods, the interactions between hundreds of molecules can be modelled. This has aroused interest in the pharmaceutical companies, electronics
industry and developers of new materials, because the electronic structure of a material determines its essential electric and chemical properties. Computational speed is of essence because a large
part of the supercomputer capacity in the world is used on calculating the electronic structure of materials.
In his dissertation, Baarman developed new methods for making electron density calculations more efficient. The first research paper in the dissertation demonstrates how quasi-Newton methods, which
have been long known to mathematicians, can be applied to calculations of materials' electronic structures using the density functional theory.
Earning the 1998 Nobel Prize in Chemistry to its inventor, density functional theory has become the most popular model for calculating the electronic structure of materials. "Based on the results,
when applied correctly, quasi-Newton is a very good method. For more complex problems it comes up with a solution more effectively than other methods. "
Medicines and new materials
The other important innovation to come from the dissertation is a novel update operator, which speeds up iterative electronic structure calculations.
"It shortens the computational times required. Now we are able to perform calculations on more challenging systems", Baarman states.
Baarman's methods work particularly well with computationally challenging materials, such as metals.
At the moment, the researcher from the Aalto University Department of Mathematics and Systems Analysis is visiting the biomolecular research group of the Fritz Haber Institute in Berlin where he is
implementing the computational methods for the code developed and used by the physicists of the group.
"It makes the physicists' job easier. They are interested in what a molecule looks like, its formation energy or how a drug reacts with another molecule, not in constructing a computational method."
Baarman hopes that in the future, the methods now being tested by chemists and physicists will be in wider use.
"The next step is for product developers to start using the tools, developing new medicines, materials or for instance catalytic converters for cars."
Provided by Aalto University
"A good guessing method makes supercomputing more efficient." August 24th, 2012. http://phys.org/news/2012-08-good-method-supercomputing-efficient.html | {"url":"http://phys.org/print265014988.html","timestamp":"2014-04-18T14:06:08Z","content_type":null,"content_length":"8255","record_id":"<urn:uuid:fa66d84a-1c23-45b3-8cf0-f3fa9a095414>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00601-ip-10-147-4-33.ec2.internal.warc.gz"} |
Advantix Calculator 2.0
Advantix Calculator 2.0 Ranking & Summary
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Date added: 2002-11-07
Advantix Calculator 2.0 description
A mathematical tool that assist students studying Algebra, Linear Algebra, Trigonometry, Calculus, Statistics, Finance, Engineering, or Discrete and Logic Mathematics. Advantix can compute
mathematical expressions involving complex numbers, polynomials, rational functions, vectors and matrices. In addition, Advantix can compute mathematical expressions involving binary, octal,
hexadecimal and logics (Boolean Algebra). Advantix provides many common mathematical functions such as integration, differentiation, matrix determinant and inversion, special functions, Fourier
transforms. Advantix also graph user defined functions in 2D, 3D and polar coordinates.
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Michael Lawler
Assistant Professor
Office: S2-258
Phone: 607-777-4331
E-mail: michael.lawler@binghamton.edu
Dr. Lawler received his PhD in physics in 2006 from the University of Illinois at Urbana-Champaign and BSc, in engineering physics in 1999 from Queen's University, Kingston, ON, Canada. Lawler joined
our department in August 2008.
Research Overview
My primary interests lie in the field of strongly correlated condensed matter physics. In this field, we seek to gain an understanding of the behavior of many strongly interacting particles. This is
a far from well understood subject. However, we are fortunately aided in our exploration of it by an intimate connection between experiment and theory. Below are some of the topics related to this
phenomenon that form my particular interest.
Frustrated Quantum Mechanical Systems
Suppose for the moment that we could turn off Heisenberg’s uncertainty principle so that we could know the exact position and momentum of our particles or that we could know the precise direction a
particle’s spin is pointing. Now suppose these classical particles are placed in a configuration in which Newton’s law cannot tell us what they will do next because there are many options available,
all equally likely. We say that these particles are frustrated.
This can happen, for example, when interacting classical spins are placed on certain lattices such that there are many spin configurations, each of which minimize the total energy. A central question
in the field is what happens when we turn on Heisenberg’s uncertainty principle? The spins can no longer point in a specific direction and it turns out that the laws of quantum mechanics then help
the spins decide what they want to do. Should this happen, then the spin system is actually governed by the laws of quantum mechanics at a macroscopic scale.
Recently, experimentalists have fabricated a number of materials (Herbertsmithite, NIO, κ-BETD, diamond lattice spinels, ...) that may shed much light on the above central question. For some time
now, we have had theories of quantum spin liquids based on the emergence of either fermions or bosons interacting via gluons, photons or “visons” as the new particles describing this exotic phase.
With these new materials, we can finally compare theory with experiment and so it is an exciting time for the field.
Recent papers:
1. "Gapless spin liquids on the three dimensional hyper-kagome lattice of Na4IR3O8", MJL, Arun Paramekanti, Yong Baek Kim and Leon Balents, arXiv:0806.4395
2. "Quantum order by disorder in frustrated diamond lattice antiferromagnets", Jean-Sebastien Bernier, MJL and Yong Baek Kim, arXiv:0801.0598 (to be published in Phys. Rev. Lett.)
Quantum Liquid Crystals
It stands to reason that increasing the strength of interactions between particles in a gas phase will cause them to want to crystalize, to form a solid phase. If the particles are electrons, Wigner
discovered in 1931 that this indeed happens. However, a transition to a Wigner crystal phase need not happen directly, but could in principle happen through a series of intermediate phases each
progressively more crystalline. Examples of such intermediate phases include the electron “nematic” phase, which spontaneously breaks rotational symmetry, and an electron “smectic” or stripe phase,
which involves the formation of an array of one-dimensional electronic “rivers”.
While symmetry provides a guiding principle in the theory of quantum liquid crystals, it alone cannot characterize a quantum world. As such, the general theory is fundamentally incomplete. It is very
exciting, therefore, to study quantum liquid crystals found in nature, such as the nematic states found in Sr3Ru2O7 and quantum Hall systems and the stripe phases in cuprate superconductors. An
interesting question, for example, is what happens at a continuous phase transition between two quantum liquid crystals? Or, can the stripes in a stripe phase slide freely next to each other? Can
electrons in quantum liquid crystals pair easily to form a superconductor? At the heart of these questions is the quantum nature of these new phases, and their answers rely on the deep connection
between experiment and theory achievable in condensed matter physics.
Recent papers:
1. "Fluctuating stripes in strongly correlated electron systems and the nematic-smectic quantum phase transition", Kai Sun, Benjamin Fregoso, MJL and Eduardo Fradkin, arXiv: 0805.3526 (to be
published in Phys. Rev. B)
2. "Theory of the nodal nematic quantum phase transition in supercondeuctors" Eun-Ah Kim, MJL, Paul Oreto, Eduardo Fradkin and Steven A. Kivelson, Phys. Rev. B 77, 184514 (2008) | {"url":"http://www2.binghamton.edu/physics/people/michael.html","timestamp":"2014-04-17T01:02:06Z","content_type":null,"content_length":"15636","record_id":"<urn:uuid:cc78d810-cfdc-4b8a-952d-7a661c35a39d>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00236-ip-10-147-4-33.ec2.internal.warc.gz"} |
Copyright © University of Cambridge. All rights reserved.
'Code to Zero' printed from http://nrich.maths.org/
Why do this problem?
To beat the spies at their own game!
It gives practice in working systematically to consider all cases. First however you need to cut down the number of cases to be checked, so think about the relation between $a$, $b$ and $c$ and what
you can deduce from it about the constraints on $a$, $b$ and $c$.
Possible approach
Ask the learners to find a relation between $a$, $b$ and $c$. Then have a class discussion about what can be deduced from this expression.
Key question
What do you know about $c^3-1$? | {"url":"http://nrich.maths.org/340/note?nomenu=1","timestamp":"2014-04-20T21:30:28Z","content_type":null,"content_length":"3761","record_id":"<urn:uuid:55c1d457-6e54-4c27-bf8a-cd434f9461cc>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00182-ip-10-147-4-33.ec2.internal.warc.gz"} |
the encyclopedic entry of partial differential equations
Numerical partial differential equations
is the branch of
numerical analysis
that studies the numerical solution of
partial differential equations
Numerical techniques for solving PDEs include the following:
The finite difference method is often regarded as the simplest method to learn and use. The finite element and finite volume methods are widely used in engineering and in computational fluid dynamics
, and are well suited to problems in complicated geometries. Spectral methods are generally the most accurate, provided that the solutions are sufficiently smooth.
See also
External links
• IMS, the Open Source IMTEK Mathematica Supplement (IMS) | {"url":"http://www.reference.com/browse/partial%20differential%20equations","timestamp":"2014-04-19T12:46:30Z","content_type":null,"content_length":"80530","record_id":"<urn:uuid:ccbebe65-27e1-48a5-82ec-3affa79e4720>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00597-ip-10-147-4-33.ec2.internal.warc.gz"} |
Redwood Estates Prealgebra Tutor
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RE: st: FW: ML for logit/ologit
Notice: On March 31, it was announced that Statalist is moving from an email list to a forum. The old list will shut down at the end of May, and its replacement, statalist.org is already up and
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RE: st: FW: ML for logit/ologit
From "Thomas Murray (Department of Economics)" <t.l.murray@bham.ac.uk>
To "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject RE: st: FW: ML for logit/ologit
Date Wed, 16 Nov 2011 11:18:26 +0000
Right yes - apologies for that - I was experimenting with different arguments and emailed a faulty program.
What I type is:
program drop logittest
program define logittest
version 12.0
args lnf b1 x rho
tempvar lng
qui {
gen double `lng' = ln(invlogit(`b1'*((`x'^(1-`rho')-1)/(1-`rho')))) if $ML_y1==1
replace `lng' = ln(invlogit(-`b1'*((`x'^(1-`rho')-1)/(1-`rho')))) if $ML_y1==0
replace `lnf' = `lng'
ml model lf logittest (happy =)
ml max
The trace returns:
- version 12.0
- args lnf b1 x rho
- tempvar lng
- qui {
- gen double `lng' = ln(invlogit(`b1'*((`x'^(1-`rho')-1)/(1-`rho')))) if $ML_y1==1
= gen double __000007 = ln(invlogit(__000006*((^(1-)-1)/(1-)))) if happy==1
unknown function ^()
replace `lng' = ln(invlogit(-`b1'*((`x'^(1-`rho')-1)/(1-`rho')))) if $ML_y1==0
replace `lnf' = `lng'
This is Brendan's comment from a bit earlier:
You have "((^(1-)" in there, so the ^ doesn't apply to a variable, and therefore isn't recognised as an operator.
I think this is because your `x' expands to "".
Could ^ be failing to pick up `rho' inside the bracket?
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Nick Cox
Sent: 16 November 2011 11:03
To: 'statalist@hsphsun2.harvard.edu'
Subject: RE: st: FW: ML for logit/ologit
That is not clear at all. The explanation is different. You are invoking a local rho, which you never define. Your earlier statement defines mu.
It now also seems that your earlier "along the lines of" means that you were not showing us the code that was buggy, which does help to explain why we couldn't see the bug.
Please do "Say exactly what you typed and exactly what Stata typed (or did) in response."
Thomas Murray (Department of Economics)
I have run the set trace and it is clear there is a problem with the including an argument within the squared term (I've pasted the relevant part below). I am sure this is the bug but I do not know why Stata doesn't like it.
I am confident all the quotations are correct in the program.
Many Thanks,
- mata: Mopt_search()
------------------------------------------------------------------------------------------------ begin logittest ---
- version 12.0
- args lnf b1 b2 x mu
- tempvar lng
- qui {
- gen double `lng' = ln(invlogit(`b1'*((`x'^(1-`rho')-1)/(1-`rho')))) if $ML_y1==1
= gen double __000007 = ln(invlogit(__000006*((^(1-)-1)/(1-)))) if educ1==1 unknown function ^()
replace `lng' = ln(invlogit(-`b1'*((`x'^(1-`rho')-1)/(1-`rho')))) if $ML_y1==0
replace `lnf' = `lng'
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Nick Cox
Sent: 16 November 2011 10:17
To: 'statalist@hsphsun2.harvard.edu'
Subject: RE: st: FW: ML for logit/ologit
I don't think the example calls for such a sweeping generalization. People write programs to maximise non-linear functions in Stata [sic] all the time.
There is a bug in Thomas' code. I can't spot it and there is a lingering doubt about use of quotation marks from previous posts. Using -set trace- might help him find where it is.
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2011-11/msg00795.html","timestamp":"2014-04-17T04:48:10Z","content_type":null,"content_length":"12739","record_id":"<urn:uuid:1cdf74f8-8ab9-4a83-a4ff-3c4d73ed302b>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00407-ip-10-147-4-33.ec2.internal.warc.gz"} |
Statistics & probabolity problems, please help
September 17th 2009, 07:14 AM #1
Sep 2009
Statistics & probabolity problems, please help
I have these problems to turn in for tomorrow. I don't know the best way to solve them.
Please see if you can solve them and post your results along with calculations.
Problem 1:
In bowl A there are 4 red balls, 3 blue balls and 2 green balls.
In bowl B there are 2 red, 3, blue and 4 green.
One ball is taken from bowl A and put into bowl B.
After this is done one ball is taken from bowl B.
What are the odds that the ball, taken from bowl B, is red?
Problem 2:
Given are:
S1 = {1,2,3,4}
S2 = {1,2,3,4,5,6}
S3 = {1,2,3,4,5,6,7,8}
We pick one number randomly from S1, where there is an equal chance of picking any one number. We do the same with S2 and S3.
What are the odds that the sum of the numbers we picked are equal to 5?
Thanks in advance for all your help.
Hello Lesarinn
Welcome to Math Help Forum!
I have these problems to turn in for tomorrow. I don't know the best way to solve them.
Please see if you can solve them and post your results along with calculations.
Problem 1:
In bowl A there are 4 red balls, 3 blue balls and 2 green balls.
In bowl B there are 2 red, 3, blue and 4 green.
One ball is taken from bowl A and put into bowl B.
After this is done one ball is taken from bowl B.
What are the odds that the ball, taken from bowl B, is red?
There are two different cases to consider:
□ (i) The ball taken from bowl A is red; and then the ball taken from bowl B is red.
□ (ii) The ball taken from bowl A is not red; and then the ball taken from bowl B is red.
Work out the probabilities $p_1$ and $p_2$ that the ball taken from A is (i) red, and (ii) not red. Then, by considering the number of red balls in bowl B in each case (i) and (ii), work out the
probabilities $q_1$ and $q_2$ that the second ball is red.
Finally, multiply $p_1$ by $q_1$; multiply $p_2$ by $q_2$; then add your answers to get the final answer. I reckon this comes out as $\frac{11}{45}$.
Problem 2:
Given are:
S1 = {1,2,3,4}
S2 = {1,2,3,4,5,6}
S3 = {1,2,3,4,5,6,7,8}
We pick one number randomly from S1, where there is an equal chance of picking any one number. We do the same with S2 and S3.
What are the odds that the sum of the numbers we picked are equal to 5?
Thanks in advance for all your help.
Work out all the ways in which the total could be 5; then work out the probability that each of these sequences of numbers occurs. Finally add all your answers together to get the overall
I'll start you off. We could have:
□ 1,1,3. The probability of this is $\tfrac14\times\tfrac16\times\tfrac18 = \tfrac{1}{192}$.
□ 1,2,2. The probability of this is exactly the same: $\tfrac14\times\tfrac16\times\tfrac18 = \tfrac{1}{192}$
□ ... and so on.
I reckon that the answer is $\tfrac{5}{192}$.
Can you complete these now?
This is exactly what I managed to come up with myself after I posted this thread.
I am pretty sure that this is the right way to solve Problem 1.
In Problem 2: Yeah I guess using the same method does seem to work.
Thank you Grandad
Problems are solved.
September 17th 2009, 07:54 AM #2
September 17th 2009, 08:06 AM #3
Sep 2009 | {"url":"http://mathhelpforum.com/advanced-statistics/102800-statistics-probabolity-problems-please-help.html","timestamp":"2014-04-21T16:54:07Z","content_type":null,"content_length":"41454","record_id":"<urn:uuid:df15df56-7f54-4183-8d5d-d5c2309fcfc4>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00172-ip-10-147-4-33.ec2.internal.warc.gz"} |
Determine the length of the inventory conversion period, receivables conversion and etc...
1. 188927
Determine the length of the inventory conversion period, receivables conversion and etc...
Please see the attached file.
A. Determine the length of the inventory conversion period.
B. Determine the length of the receivables conversion period.
C. Determine the length of the operating cycle.
D. Determine the length of the payables deferral period.
E. Determine the length of the cash conversion cycle.
F. What is the meaning of the number you calculated in question E.
PART 2
Calculate the effective annual percentage rate of forgoing the cash discounts under each of the following credit terms:
A. 2/10 NET 60
B. 2/10 NET 30 | {"url":"https://brainmass.com/business/management-accounting/188927","timestamp":"2014-04-16T05:07:12Z","content_type":null,"content_length":"29049","record_id":"<urn:uuid:705f0642-528e-4963-b966-6a997ff7526a>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00578-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculating Fabric Requirements for a Rag Quilt
"How much fabric do I need to make a rag quilt?"
I get asked that question a lot, so here's your answer!
You will need the following information: 1. What size of squares are you going to cut?
2. How many different fabrics are you going to use?
3. How large do you want the quilt to be?
4. How wide is your fabric?
For our example, here are our answers:
1. What size of squares are you going to cut? 6-inch (This is a pretty standard size. If you use bigger squares, the project will be easier and go faster.)
2. How many different fabrics are you going to use? three fabrics
3. How large do you want the quilt to be? 35” x 40”
4. How wide is your fabric? 43”
Now for the calculation!
Step 1: How many squares will you need?
Due to seam allowances, the finished quilt will have squares that are one inch smaller than what you cut. For example, since we are cutting 6-inch squares, the finished quilt will show 5-inch squares
. The additional one inch is in the fringe.
Our finished quilt is 35” x 40”. That means it will have 7 squares in each row, and it will have 8 rows.
Figuring out the number of squares needed for your quilt.
There are 56 squares total (7*8=56). However, we also need the same number of squares for the back of the quilt. So our total number of squares is 56*2=112.
(Note: if you are using batting as the middle layer in your quilt, you will need to cut 56 squares of batting. Make them 5" squares so they won't show in the fringe.)
Step 2: How many squares of each print will you need?
Draw a picture of your quilt and label where you will use each fabric. I chose a very simple design for this example, but you be as creative as you want. Then count up the number of squares for each
print. Don’t forget to multiply by 2 for the back of the quilt.
Calculating the number of squares needed for each fabric used.
Step 3: How much fabric will you need?
Now you need to figure out how many squares you can get out of the width of your fabric.
Our fabric is 43” wide, and we are cutting 6” squares. That means we can cut 7 squares out of the width of fabric.
In other words, for every 6” cut of our fabric, we can get 7 squares.
We need 38 squares of each print, so we need seven 6” strips of each of our fabrics. That is 42” of each of the three fabrics.
There you have it!
Do you have additional rag quilt questions? Leave a comment!
7 comments:
1. I've never made a rag quilt, but that one is so cute it makes me want to do one. I can follow the above for the number and amount of fabric, but now what happens. How much seam allowance, how do
you get the fringe? Please help.
Sue - schuster@enter.net
1. Sue - To answer your specific questions...I use a 1/2 inch seam allowance. Once your squares are all sewn together (with all of the raw edges facing up), you clip the raw edges about every 1/
2 inch. Then you wash the quilt to make the fringe.
There is a decent rag quilt tutorial here: http://www.knitandtonic.typepad.com/SimpleRagQuilt.pdf.
I am going to do a full rag quilt tutorial in the near future. I don't have it ready to go quite yet. Stay tuned!
2. i want to make a queen size rag quilt, how much fabric will that take?
3. Margaret, you will need to decide how big your squares will be, how many fabrics you will use, and how big the quilt needs to be. Then you can plug in those numbers to figure out the amount
of fabric needed.
2. Awesome thinking on making a chart for the amount of fabric needed!! Never would have thought to do something like that!! :) I just always buy fabric's a yard or more at a time as I make a lot of
the rag quilts to donate to the local NICU where my daughter stayed. You can check us out!! https://www.facebook.com/FortheLoveofGrace06
3. If I use your instructions of 3 prints and 6 inch squares and other measurement, how many yards of each fabric would I need? Thanks!
4. Thank you! I have looked at several tutorials for rag quilts and not one person ever said how much fabric you will need. This is so very helpful. | {"url":"http://my-fabric-obsession.blogspot.com/2012/11/calculating-fabric-requirements-for-rag.html","timestamp":"2014-04-21T15:08:47Z","content_type":null,"content_length":"103960","record_id":"<urn:uuid:f96c0ee8-475c-4b11-b87d-f1df82bf7bda>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00400-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need help with Bit manipulation programming
01-25-2008 #1
Registered User
Join Date
Dec 2007
Need help with Bit manipulation programming
I hope someone can give me some clues or tips to get started in solving the following problem.
I have an Integer array of size 5. Now I need to copy only the first 20 bits of each integer into a Character array. So finally, the character array has first 20 bits from first integer and
immediately followed by first 20 bits from second integer and so on..
I know this can be done using bit manipulation but I am not getting the right idea. I would really appreciate if someone can give me some clues.
This sounds rather ankward to do. When you say you want to store it in a char array, i guess you mean that you want to store it "somewhere" in memory. Also, i'm curious why you need to do this ?
If you have 5x20 bits to store, that mean you need 100 bits. 100 / 8 = 12.5, so it means you'll need, in fact, 13 bytes of memory, where the last 4 bits will contain some random stuff. Note that
each operation has to move k*8 bits at the same time, where k is an integer > 0, because that's how much of the computer works....
So... well... i don't have much time left... but from what i see, it's going to be easier to write a purely sequential program for treating 5 integer instead of using some kind of more elaborate
structure (like a loop).
Indeed. Unfortunately, the instructions can only work with data sizes 1, 2, 4 or 8 bytes. Processor instructions cannot work with other sizes, so what you propose will be cumbersome.
One thing that "might" work is that you treat one byte as one bit. That way you can allocate as many bytes of memory as you need for the number of bits and set them accordingly.
But then using that array will be cumbersome.
Perhaps you should explain a little more on what you're trying to do.
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
You put bits into an int in multiples of 20, and take them out in multiples of 8.
If any bits are left afterwards, you flush out the remainder.
Like this:
unsigned int bitsHeld=0, bits=0;
pack20BitsOf(int val)
bitsHeld += 20;
bits = (bits << 20) | val;
while (bitsHeld >= 8)
bitsHeld -= 8;
writeoutByte(bits >> bitsHeld);
bits ^= (bits >> bitsHeld) << bitsHeld;
if (bitsHeld > 0)
writeoutByte(bits << (8-bitsHeld));
bits = 0;
bitsHeld = 0;
or if you want to pack the bits in the other order (which may be a little bit simpler), then like this:
unsigned int bitsHeld=0, bits=0;
pack20BitsOf(int val)
bitsHeld += 20;
bits |= val << bitsHeld;
while (bitsHeld >= 8)
bitsHeld -= 8;
writeoutByte(bits & 0xFF);
bits >>= 8;
if (bitsHeld > 0)
writeoutByte(bits & 0xFF);
bits = 0;
bitsHeld = 0;
Last edited by iMalc; 01-25-2008 at 01:13 PM.
My homepage
Advice: Take only as directed - If symptoms persist, please see your debugger
Linus Torvalds: "But it clearly is the only right way. The fact that everybody else does it some other way only means that they are wrong"
Why do I see missing return type?
For information on how to enable C++11 on your compiler, look here.
よく聞くがいい！私は天才だからね！ ^_^
Rats! - Guess whether I tried compiling it or not.
void should do for the return type. However if writeoutByte were writing to disk and itself could return failure codes, then I'd make the return type mirror that of writeoutByte, with the failure
codes being passed back in the same manner.
It's the algorithm that I was more worried about. Make sure you take the time to understand how it works, as you'll have a better idea how to do such things in future.
If you would like the corresponding "unpacker" and have trouble implementing this yourself, then let me know. Or if you have trouble using the code I've posted, just ask.
flushBits is to only be called once, at the end, after all numbers have been processed.
My homepage
Advice: Take only as directed - If symptoms persist, please see your debugger
Linus Torvalds: "But it clearly is the only right way. The fact that everybody else does it some other way only means that they are wrong"
What are the "first bits?" The lowest 20 bits, or the highest 20 bits?
its the lowest 20 bits
01-25-2008 #2
01-25-2008 #3
01-25-2008 #4
01-25-2008 #5
01-25-2008 #6
01-26-2008 #7
01-28-2008 #8
Registered User
Join Date
Dec 2007 | {"url":"http://cboard.cprogramming.com/c-programming/98281-need-help-bit-manipulation-programming.html","timestamp":"2014-04-17T07:19:37Z","content_type":null,"content_length":"73699","record_id":"<urn:uuid:01b22393-b4f6-42f9-84a9-f5ac7f78cc9b>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00214-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wordless Wednesday: I Know She's There
Wordless Wednesday: I Know She’s There
21 thoughts on “Wordless Wednesday: I Know She’s There”
1. Way to go, Sage. You are starting to win her over!
2. Sage,
That’s so impressive buddy!! Just don’t look at her and she might go away!! Tee Hee
Your pal Snoopy :)
3. Maybe when you open your eyes she will be gone!!!
4. My kitty-cuz Mystic does NOT look happy. But neither would I if a D-O-G were that close to me!
5. I am so proud of you Sage!!!!!
6. Haha! I am impressed by your progress, Sage, but I kinda think you might be wishing a certain kitty cat wasn’t sitting so close to you and tempting you so badly. Great restraint!
7. Good job, Sage, – we don’t think we could pass that test at all.
Woos – Phantom, Thunder, Ciara, and Lightning
10. Well I’m very impressed, y’all are doing great!
11. Sage, just raise your head and give a big nudge. Ooops!
Don’t worry, they always land on their feet ;p
12. Good job sweet Sage!!! We are proud of you. Hugs and nose kisses
13. Hi Y’all,
I’d say that’s the only safe way to handle a cat.
Y’all come back now,
Hawk aka BrownDog
14. Sage, you are just remarkable. Such patience and control you have. What a good good girl. Such a cutie you are.
15. Mebbe you is hasing a puppymare.
16. Cute :) Sage . .you have excellent control . .that kitty is baiting you!!
17. Vof Sage, we get used to them! x Nero
18. What a good girl you are Sage, I think maybe the kitty is testing you :)
19. Hey Sage, Jet here. OMMMMM…. Doing this the Zen way… sending out good intentions for kitty to …….. ommmmm…. MOVE!
20. Your doing an excellent job sage, just sleep and ignore cat!!
21. Sage, that is impressive!
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Topic: Implementation of all steps of Music algorithm
Replies: 1 Last Post: Oct 1, 2013 2:59 AM
Messages: [ Previous | Next ]
Ste_ee Implementation of all steps of Music algorithm
Posted: Sep 30, 2013 2:47 PM
Posts: 7
Registered: 9/20/13 HI, i'm implementing all the steps of music algorithm, but i believe there are some problems.
This is m script:
N= 50; % number of antenna
K= 6; %number of signal
f0=10; %frequency of signal
fs=100; %sampling frequency
d = 0.5; %distance/lambda ratio
t = 0:1/fs:1;
x1= sin(2*pi*3*f0*t);
x2= 2*sin(2*pi*f0*t);
x3 = cos(2*pi*f0*t);
x = x1+x2+x3 + wgn(1,length(t),0); % signal + white noise
[t_matrix Rx]= corrmtx(x,N-1,'mod'); %Rx is autocorrelation matrix NxN.
[a_vector a_value] = eig(Rx); % find eigenvalues and eigenvector
[a_val_sort index]= sort(diag(a_value),1,'descend');
a_vett_sort=a_vector (:,index);
Qn = a_vett_sort(:,K+1:N); %noise subspace
angles = -90:1:90;
steering = exp(-i*2*pi*d*(0:N-1)'*sind(angles)); %steering matrix
for m= 1:length(angles)
spectrum(m)= 1/(steering(:,m)'*Qn*Qn'*steering(:,m));
grid on;
I'm not sure of value of K and its relation with signals x1,x2,x3.
The spectrum is very strange, i don't understand why it's simmetric!
I find always +/- angles.
If is use SVD insted eig?? Rx is always square, and i believe i don't need SVD.
Is there anyone who may help me?
Thanks very much to all!
Date Subject Author
9/30/13 Implementation of all steps of Music algorithm Ste_ee
10/1/13 Re: Implementation of all steps of Music algorithm Ste_ee | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2600431","timestamp":"2014-04-21T10:20:46Z","content_type":null,"content_length":"18242","record_id":"<urn:uuid:ee4a34c2-a1cc-4d76-9586-592af9b8ab5b>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00246-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prove x cosx < sinx on 0 < x < pi/2
July 17th 2010, 07:51 PM #1
Jun 2010
Prove x cosx < sinx on 0 < x < pi/2
I am asked to prove that $\frac{2}{\pi} < \frac{sin x}{x} < 1$ if $0 < x < \frac{\pi}{2}$, where the definition of $sin$ is given by $sin(x) = \frac{1}{2}(e^{ix} + e^{-ix})$. I think I have a bit
of the problem figured out. I reason that I can solve the whole problem if I know the derivative of $\frac{sin x}{x}$ is always negative, which it is if I am to trust a graphing calculator.
Once that's obtained, I think the rest follows from: $\displaystyle \lim_{x \rightarrow 0^{+}} \frac{\sin x}{x} = \lim_{x \rightarrow 0^{+}} cos(x) = 1$, and $\frac{sin(\frac{\pi}{2}}{\frac{\pi}
{2}} = \frac{2}{\pi}$.
So I need to show that $\frac{x cos x - sin x}{x^{2}}$ is negative within these bounds. But I don't see how to do so, or if I need to re-consider how I'm approaching the task.
I suppose I should add that we have defined $\pi$ as the smallest positive number such that $cos(\frac{\pi}{2})$ = 0.
You should know that
$-1 \leq \sin{x} \leq 1$ for all $x$.
So for $x > 0$, you that means
$-\frac{1}{x} \leq \frac{\sin{x}}{x} \leq \frac{1}{x}$.
Now see what happens as $x \to 0$ and $x \to \frac{\pi}{2}$...
I would use that $\frac{\sin x}{x} = \cos(x/2)\cos(x/2^2)\cos(x/2^3)\dots$. If $0 \leq x < y \leq \pi/2$, it is easy to see that $\cos(x/2) > \cos(y/2),\: \: \cos(x/2^2) >\cos(y/2^2)$, etc., so
that $\frac{\sin x}{x} > \frac{\sin y}{y}$.
Another way using the same idea : taking the derivative, we have $\frac{d}{dx}\frac{\sin x}{x} = -\frac{\sin x}{x}\sum_{j=1}^\infty \frac{\tan x/2^j}{2^j}$, which is $\leq 0$ at least for $0 \leq
x \leq \pi$.
In fact, the derivative of $\frac{\sin x}{x}$ is not always negative.
I can't say that I follow. You're seeing what the function does at infinity and at +/- pi/2. But if I don't know about the monotonicity of the function in the interval I'm considering, it seems
like the function could be absolutely anything in-between these points.
Use MVT on the interval $[0,x]$ where $0 < x < \frac{\pi}{2}$
let $f(x)=sin(x)$ then: $sin(x)-sin(0) = f'(c)(x-0)$ where $0 < c < x$.
So, $sin(x)=xf'(c)=xcos(c)$.
Now, note that the function cos(x) is decreasing on $[0,\frac{\pi}{2}]$ hence, $cos(c)>cos(x)$.
Therefore: $sin(x)=xcos(c)>xcos(x)$ if $0 < x < \frac{\pi}{2}$
July 17th 2010, 08:10 PM #2
Jun 2010
July 17th 2010, 08:15 PM #3
July 17th 2010, 08:24 PM #4
July 17th 2010, 08:30 PM #5
July 17th 2010, 08:40 PM #6
Jun 2010
July 18th 2010, 02:38 AM #7 | {"url":"http://mathhelpforum.com/differential-geometry/151245-prove-x-cosx-sinx-0-x-pi-2-a.html","timestamp":"2014-04-17T03:05:47Z","content_type":null,"content_length":"56125","record_id":"<urn:uuid:8dfc8f73-6e86-47d7-b6dc-4a21600d0046>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
MetalTabs.com Forum - The Awesome Science Thread
Chit Chat
far_beyond_sane 2006-02-13 08:22
Well, indeed. And I wasn't even that nasty to you.
Originally Posted by problematic
Jesus Fucking Christ, you elitist piece of shit.
Starts well!
I think you were brutally abused as a child to put you in your 'insane' (Lolzzzzzzz) state of mind. I was fucking voicing my opinion, like your fucking meant to do on a forum, for fucks sake, only to
be shut down by your PMS.
IM SOOO sorry that i'm not as smart as you, mr. astrophysicist, mr. cosmologist, mr. Hawking.
P.S. Go fuck yourself you homoerotic cunt.
...but goes downhill from there. Get a little more creative. That was so boring I think you gave me cancer.
Now, as you seem to be convinced I'm a fag, let's put a scenario on the following remark - imagine I am boring hard into your arsehole with my enormous Maddox-sized penis. On every word, I want you
to imagine a long finessed stroke smoking you open and bouncing off your little boy-prostate and a gravelly voice grunting the following at the back of your head.
*more lube*
far_beyond_sane 2006-02-13 08:45
All wah and no science...
...makes this thread fucking dull.
Right then. This one will do your head in - it concerns the first (that I'm aware of) testable hypothesis to start chiselling pieces off string theory. I read this a while ago (was published last
year) but thought it could use a write-up as I vaguely remember we used to have a string theory thread with you nerdier types being rather interested.
If you can deal, you can read the abstract
and a killer explanation of hierarchy theory
powersofterror 2006-02-13 10:15
"Of course, string theory hasn't been tested yet — experimental evidence is necessary. Additionally, Hewett, Lillie and Rizzo’s analysis can only disprove critical string theory; it cannot prove it."
Why not? Does this have something to do with the "theory" part?
far_beyond_sane 2006-02-13 10:51
Originally Posted by powersofterror
"Of course, string theory hasn't been tested yet — experimental evidence is necessary. Additionally, Hewett, Lillie and Rizzo’s analysis can only disprove critical string theory; it cannot prove it."
Why not? Does this have something to do with the "theory" part?
Long story short: it's possible for them to get an experimental result which falsifies the predictions of string theory (which is the discovery of many more dimensions than string theory predicts)
but not possible for them to confirm it with a congruent result. That is, they can get a consistent result (evidence of 10 dimensions, for example) but not confirmatory result.
brainsforbreakfast 2006-02-13 16:28
FBS, Your posts make me giggle like a little school girl :rofl:
far_beyond_sane 2006-02-14 09:09
powersofterror 2006-02-14 11:47
With all this new shit, I'll be in the same boat as an old cenile man in 20 years. Too bad I won't be in school all my life.
Can the theory of gravity be proven, as opposed to the string theory?
far_beyond_sane 2006-02-14 19:37
Proved, no. This isn't mathematics, where we prove theorems. Physics, as the name implies, is a physical science. Of course, we use what we could corasely call mathematical techniques to get theories
about different phenomena. What's different is then we have to make them fit environmental and experimental evidence... and sometimes 'good' theories fuck up spectacularly when it comes to mating
them with the available evidence.
So! Dark Matter is something that gives physicists the shits, and is highly confusing. If this solution to the gravitational theory is congruent with the cosmological data we get from galaxies, it
will be an excellent indication that it describes reality. But not proof.
The 'Dark Matter' theory reminds me this belief in 'aether', which was popular before the Michelson-Morley experiment in the 19th century. Both are pathetic attempts to connect raw theories with
experimental data.
OpethFan 2006-02-15 00:29
Originally Posted by far_beyond_sane
Now, as you seem to be convinced I'm a fag, let's put a scenario on the following remark - imagine I am boring hard into your arsehole with my enormous Maddox-sized penis. On every word, I want you
to imagine a long finessed stroke smoking you open and bouncing off your little boy-prostate and a gravelly voice grunting the following at the back of your head.
*more lube*
^ most brutal burn ever.
far_beyond_sane 2006-02-15 04:24
Originally Posted by nomad
The 'Dark Matter' theory reminds me this belief in 'aether', which was popular before the Michelson-Morley experiment in the 19th century. Both are pathetic attempts to connect raw theories with
experimental data.
This has often been said. But there is a great counterexample - the neutrino. Fermi said "this wacky little particle must exist". Many people said "Fuck off, wop". It took another twenty five years
to find them, using a wacky nuclear reactor.
Two further problems - dark matter theories explain some results a lot better than competing theories of non-Newtonian gravity, and some physicists think exactly the same kind of 'fudging' criticism
applies to arbitrarily farting around with the laws of gravity to explain cosmological anomalies.
I don't like the dark matter explanations very much myself. But then again, I don't understand them, either.
Yeah, but that was Fermi, whom we all, gifted with hindsight, know to have been awesome, even if he was a wop. These BM guys may or may not be Fermis. On the other hand, that gravity should be a
continuous function that changes over distance feels like it should be right to non-scientist me. Doesn't mean it is, but it seems logical. Anyway, let's see some falsifying.
johnmansley 2006-02-15 15:03
Originally Posted by far_beyond_sane
Proved, no. This isn't mathematics, where we prove theorems. Physics, as the name implies, is a physical science. Of course, we use what we could corasely call mathematical techniques to get theories
about different phenomena. What's different is then we have to make them fit environmental and experimental evidence... and sometimes 'good' theories fuck up spectacularly when it comes to mating
them with the available evidence.
Very true. The mathematics behind physics is based on a "modelling" concept - it models what we observe, sometimes good enough to derive very accurate predictions. But whether nature itself is
embodied and somehow, for want of a better word, "aware" of the special relationships within these mathematical models is very much a subject for philosophers.
Unfortuantely, even the most rigourous of theories will be chock full of assumptions that are highly unlikley to occur in reality, i.e. planets are assumed to be perfect spheres, many fields are
assumed to act uniformly, etc, etc. Basically, there's only so much detail that can be contained within a mathematical model, or else it becomes impractical to solve. I'd actually venture to say that
a perfect model of the universe would take an infinite amount of time to solve.
far_beyond_sane 2006-02-15 20:15
Originally Posted by johnmansley
Unfortuantely, even the most rigourous of theories will be chock full of assumptions that are highly unlikley to occur in reality, i.e. planets are assumed to be perfect spheres, many fields are
assumed to act uniformly, etc, etc. Basically, there's only so much detail that can be contained within a mathematical model, or else it becomes impractical to solve. I'd actually venture to say that
a perfect model of the universe would take an infinite amount of time to solve.
There's a Borges satire on the exactitude of science which addresses this problem.
A bunch of cartographers determine to make the most accurate map of the land ever built. It becomes larger and larger in scale until the bastard covers the entire country - scale 1:1. Then it becomes
impossible to maintain and gets run down and tore up, and goats ended up chewing bits off the side.
Moral of the story: anything that has to account for every variable in its entirety becomes as complicated as the system it's describing.
In conclusion, use the phrase "Borges' map" in conversation and watch the women flock to your feet in wonder.
Knowing Latin American fabulism is always the best route, to understanding as well as to getting laid.
Originally Posted by johnmansley
I'd actually venture to say that a perfect model of the universe would take an infinite amount of time to solve.
Even the simplest dynamical systems described by differential equations sets are pain in the neck to solve. In practice to find a solution they use one of the iteration methods which implies the
solution is never exact.
Another problem is that the universe is too imperfect for the perfect mathematical models we could build. Because of this, physicists will always have their jobs, even after all mathematical problems
are solved :)
Hopefully we'll all be dead before Mansley's math major is made useless by the solving of all possible mathematical problems. I don't want to live in a world where Mansley means nothing!
far_beyond_sane 2006-02-15 23:18
Originally Posted by nomad
Even the simplest dynamical systems described by differential equations sets are pain in the neck to solve. In practice to find a solution they use one of the iteration methods which implies the
solution is never exact.
And then, we have things like brains with their 2^100 possible synaptic pathways.
Brains make us cry, and pretend we know things.
far_beyond_sane 2006-02-16 07:37
fatdanny 2006-02-16 10:28
This week, I have been mostly learning about Multinuclear NMR (nuclear magnetic resonance) spectroscopy.
In solids, shielding anisotropy, dipolar interactions and quadrupolar interactions broaden the peaks in the spectrum - fucking it up.
However, all these interactions are dependent on cos^2(theta) - 1, so spinning your sample at 54.74 degrees (the magic angle) to the applied magnetic field reduces this term to zero, giving a
readable spectrum with sharp peaks.
There you have it - Magic angle spectroscopy - I think it's pretty nifty.
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Re: Test tstdiomisc fails when multiplication of NAN by -1 results in NAN
You have found a bug. The code should xor the sign bit when doing
negation. The existing code doesn't work for NANs. I'll try to
fix negdf2 and negsf2.
The problem is PA1.1 doesn't have a fneg instruction, so the above
takes a bit of work to implement. You will get the correct result
if you specify -march=2.0.
> Then it loads something which I assume *should* be -1, but isn't:
> .LC2:
> .word -1074790400
> .word 0
> What is this value, it's 0xbff0000000000000 e.g. -1.875. Should it be
> 0xbf80000000000000 e.g. -1.0 exactly, but it's not? Is this a mistake?
I believe 0xbff0000000000000 is the correct double representation
for -1.0. 0xbf800000 is the float representation for -1.0.
J. David Anglin dave.anglin@nrc-cnrc.gc.ca
National Research Council of Canada (613) 990-0752 (FAX: 952-6602)
Reply to: | {"url":"https://lists.debian.org/debian-hppa/2010/05/msg00017.html","timestamp":"2014-04-20T06:24:25Z","content_type":null,"content_length":"6616","record_id":"<urn:uuid:a10baabb-3b23-4db0-bf42-4d6a163c9ee6>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00003-ip-10-147-4-33.ec2.internal.warc.gz"} |
of Trapezoid
It is a quadrilateral in which one pair of opposite sides are parallel.
In the following trapezoid ABCD, AB // DC and AD and BC are non-parallel sides.
Isosceles trapezoid: A trapezoid in which the non-parallel sides are of equal length is called an isosceles trapezoid. In the isosceles trapezoid, the base angles are equal.
Let us study the following isosceles trapezoid.
In the trapezoid shown above, AB // DC and the non-parallel sides, AD = BC
The base angles are equal.
(i.e) $\angle$DAB = $\angle$ABC and $\angle$ADC = $\angle$BCD
Properties of Trapezoid
1. Only one pair of opposite sides are parallel. AB // DC.
2. The other pair of opposite sides are non-parallel BC is not parallel to AD.
3. The Diagonals are unequal.
4. No two angles are equal.
5. Pairs Interior angles on the same side of the non-parallel sides and between the parallel lines are supplementary.
6. The sum of all the interior angles = 360°.Properties of an Isosceles Trapezium: 1. Only one pair of opposite sides are parallel. AB // DC.
2. The other pair of opposite sides are non-parallel BC is not parallel to AD.
3. The Diagonals are equal.
4. Two pairs of angles are equal.[ pairs of acute angles and pairs of obtuse angles are equal ]
5. Pairs Interior angles on the same side of the non-parallel sides and between the parallel lines are supplementary.[ sum of an acute + obtuse angle = 180° .
6. The sum of all the interior angles = 360°.
Perimeter of Trapezoid
Perimeter of a trapezium is the boundary of the trapezium which is equal to the sum of all the four sides of the trapezium.
In the above trapezium, ABCD, Perimeter = AB + BC + CD + DA
IF AB = a units, DC = b units and the non-parallel sides BC = c, Ad = d,
then the Perimeter =
a + b + c + dExample:
Find the perimeter of the trapezium bounded by the sides, 10 cm, 5 cm, 8 cm and 6 cm.
Perimeter = 10 + 5 + 8 + 6
= 29 cm
Area of Trapezoid
Area of a trapezium =
[ sum of the parallel sides ] x height
In the above diagram Area of the Trapezium ABCD =
[ AB + CD ] x height
Formula Derivation:
Area of Trapezium ABCD = Area of $\Delta$ABD + Area of $\Delta$BDC
x AB x h +
x CD x h
[ since area of a $\Delta$=
x base x height ]
x h x ( AB + CD )[Factorising by taking
h outside ]
Area of Trapezium ABCD = $\frac{1}{2}$ ( sum of the parallel sides ) x height square units
Three Dimensional Trapezoid
It is a 3-dimensional solid which has a definite length whose cross section is a trapezium. It has definite length.
For example, the canal in which water flows, swimming pool.
Volume of Trapezoid:
Volume of the trapezoid = Area of cross section x length of the solid
= Area of the trapezium x length (l)
( sum of the parallel sides ) x height x length
( a + b ) x h x l cubic units | {"url":"http://math.tutornext.com/geometry/trapezoid.html","timestamp":"2014-04-18T15:52:38Z","content_type":null,"content_length":"23627","record_id":"<urn:uuid:ba8de68f-0cc6-469a-8987-885627938e27>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00382-ip-10-147-4-33.ec2.internal.warc.gz"} |
R-statistics blog
When analyzing a questionnaire, one often wants to view the correlation between two or more Likert questionnaire item’s (for example: two ordered categorical vectors ranging from 1 to 5).
When dealing with several such Likert variable’s, a clear presentation of all the pairwise relation’s between our variable can be achieved by inspecting the (Spearman) correlation matrix (easily
achieved in R by using the “cor.test” command on a matrix of variables).
Yet, a challenge appears once we wish to plot this correlation matrix. The challenge stems from the fact that the classic presentation for a correlation matrix is a scatter plot matrix – but scatter
plots don’t (usually) work well for ordered categorical vectors since the dots on the scatter plot often overlap each other.
There are four solution for the point-overlap problem that I know of:
1. Jitter the data a bit to give a sense of the “density” of the points
2. Use a color spectrum to represent when a point actually represent “many points”
3. Use different points sizes to represent when there are “many points” in the location of that point
4. Add a LOWESS (or LOESS) line to the scatter plot – to show the trend of the data
In this post I will offer the code for the a solution that uses solution 3-4 (and possibly 2, please read this post comments). Here is the output (click to see a larger image):
And here is the code to produce this plot: | {"url":"http://www.r-statistics.com/tag/correlation-matrix/","timestamp":"2014-04-17T12:40:12Z","content_type":null,"content_length":"40729","record_id":"<urn:uuid:9c4b8bd7-abdd-48f6-91f5-59805881715b>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00240-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help! Octagon
September 24th 2006, 05:59 PM #1
Feb 2006
Help! Octagon
The longest diagonal of a rectangular octagon varies directly as the length of a side. if a regular octagon with sides 6mm has a diagonal of lenth 23mm, how long is the diagonal of an octagon
with side of 15cm?
Is there a formula for this? I dont know where to start. thanks!
notice I highlighted "directly"
this means that as the length of a side increases, the diagonal of the octagon increases at a constant amount as well.
Think of a line: y=mx
where y is the diagonal, and x is the side.
thus: y/x=m
now substitute known values: 23/6=m
now that gives equation: y=(23/6)x
now substitute 15 for side length: y=(23/6)(15)=57.5
so the diagonal is 57.5
notice I highlighted "directly"
this means that as the length of a side increases, the diagonal of the octagon increases at a constant amount as well.
Think of a line: y=mx
where y is the diagonal, and x is the side.
thus: y/x=m
now substitute known values: 23/6=m
now that gives equation: y=(23/6)x
now substitute 15 for side length: y=(23/6)(15)=57.5
so the diagonal is 57.5
I am no engineer but one is in millemeters and the other in centimenters!
September 24th 2006, 06:08 PM #2
September 24th 2006, 06:15 PM #3
Global Moderator
Nov 2005
New York City
September 24th 2006, 06:16 PM #4
September 25th 2006, 06:20 AM #5 | {"url":"http://mathhelpforum.com/geometry/5789-help-octagon.html","timestamp":"2014-04-16T14:15:24Z","content_type":null,"content_length":"44072","record_id":"<urn:uuid:64222ee1-d4f4-42d2-856c-5ee82e0cc9e9>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00166-ip-10-147-4-33.ec2.internal.warc.gz"} |
Award Ceremony Speech
Presentation Speech by professor Stig Lundqvist of the Royal Academy of Sciences
Translation from the Swedish text
Your Majesty, Your Royal Highnesses, Ladies and Gentlemen,
The 1973 Nobel Prize for physics has been awarded to Drs. Leo Esaki, Ivar Giaever and Brian Josephson for their discoveries of tunnelling phenomena in solids.
The tunnelling phenomena belong to the most direct consequences of the laws of modern physics and have no analogy in classical mechanics. Elementary particles such as electrons cannot be treated as
classical particles but show both wave and particle properties. Electrons are described mathematically by the solutions of a wave equation, the Schrödinger equation. An electron and its motion can be
described by a superposition of simple waves, which forms a wave packet with a finite extension in space. The waves can penetrate a thin barrier, which would be a forbidden region if we treat the
electron as a classical particle. The term tunnelling refers to this wave-like property - the particle "tunnels" through the forbidden region. In order to get a notion of this kind of phenomenon let
us assume that you are throwing balls against a wall. In general the ball bounces back but occasionally the ball disappears straight through the wall. In principle this could happen, but the
probability for such an event is negligibly small.
On the atomic level, on the other hand, tunnelling is a rather common phenomenon. Let us instead of balls consider electrons in a metal moving with high velocities towards a forbidden region, for
example a thin insulating barrier. In this case we cannot neglect the probability of tunneling. A certain fraction of the electrons will penetrate the barrier by tunnelling and we may obtain a weak
tunnel current through the barrier.
The interest for tunnelling phenomena goes back to the early years of quantum mechanics, i.e. the late twenties. The best known early application of the ideas came in the model of alpha-decay of
heavy atomic nuclei. Some phenomena in solids were explained by tunnelling in the early years. However, theory and experiments often gave conflicting results, no further progress was made and
physicists lost interest in solid state tunnelling in the early thirties.
With the discovery of the transistor effect in 1947 came a renewed interest in the tunnelling process. Many attempts were made to observe tunnelling in semiconductors, but the results were
controversial and inconclusive.
It was the young Japanese physicist Leo Esaki, who made the initial pioneering discovery that opened the field of tunnelling phenomena for research. He was at the time with the Sony Corporation,
where he performed some deceptively simple experiments, which gave convincing experimental evidence for tunnelling of electrons in solids, a phenomenon which had been clouded by questions for
decades. Not only was the existence of tunnelling in semiconductors established, but he also showed and explained an unforeseen aspect of tunnelling in semiconductor junctions. This new aspect led to
the development of an important device, called the tunnel diode or the Esaki diode.
Esaki's discovery, published in 1958, opened a new field of research based on tunnelling in semiconductors. The method soon became of great importance in solid state physics because of its simplicity
in principle and the high sensitivity of tunnelling to many finer details.
The next major advance in the field of tunnelling came in the field of superconductivity through the work of Ivar Giaever in 1960. In 1957, Bardeen, Cooper and Schrieffer had published their theory
of superconductivity, which was awarded the 1972 Nobel Prize in physics. A crucial part of their theory is that an energy gap appears in the electron spectrum when a metal becomes superconducting.
Giaever speculated that the energy gap should be reflected in the current-voltage relation in a tunnelling experiment. He studied tunnelling of electrons through a thin sandwich of evaporated metal
films insulated by the natural oxide of the film first evaporated. The experiments showed that his conjecture was correct and his tunnelling method soon became the dominating method to study the
energy gap in superconductors. Giaever also observed a characteristic fine structure in the tunnel current, which depends on the coupling of the electrons to the vibrations of the lattice. Through
later work by Giaever and others the tunnelling method has developed into a new spectroscopy of high accuracy to study in detail the properties of superconductors, and the experiments have in a
striking way confirmed the validity of the theory of superconductivity.
Giaver's experiments left certain theoretical questions open and this inspired the young Brian Josephson to make a penetrating theoretical analysis of tunnelling between two superconductors. In
addition to the Giaever current he found a weak current due to tunelling of coupled electron pairs, called Coopers pairs. This implies that we get a supercurrent through the barrier. He predicted two
remarkable effects. The first effect is that a supercurrent may flow even if no voltage is applied. The second effect is that a high frequency alternating current will pass through the barrier if a
constant voltage is applied.
Josephson's theoretical discoveries showed how one can influence supercurrents by applying electric and magnetic fields and thereby control, study and exploit quantum phenomena on a macroscopic
scale. His discoveries have led to the development of an entirely new method called quantum interferometry. This method has led to the development of a rich variety of instruments of extraordinary
sensitivity and precision with application in wide areas of science and technology.
Esaki, Giaever and Josephson have through their discoveries opened up new fields of research in physics. They are closely related because the pioneering work by Esaki provided the foundation and
direct impetus for Giaever's discovery and Giaever's work in turn provided the stimulus which led to Jo- sephson's theoretical predictions. The close relation between the abstract concepts and
sophisticated tools of modern physics and the practical applications to science and technology is strongly emphasized in these discoveries. The applications of solid state tunnelling already cover a
wide range. Many devices based on tunneling are now used in electronics. The new quantum interferometry has already been used in such different applications as measurements of temperatures near the
absolute zero, to detect gravitational waves, for ore prospecting, for communication through water and through mountains, to study the electromagnetic field around the heart or brain, to mention a
few examples.
Drs. Esaki, Giaever and Josephson,
In a series of brilliant experiments and calculations you have explored different aspects of tunelling phenomena in solids. Your discoveries have opened up new fields of research and have given new
fundamental insight about electrons in semiconductors and superconductors and about macroscopic quantum phenomena in superconductors.
On behalf of the Royal Academy of Sciences I wish to express our admiration and convey to you our warmest congratulations. I now ask you to proceed to receive your prizes from the hands of his
Majesty the King.
From Nobel Lectures, Physics 1971-1980, Editor Stig Lundqvist, World Scientific Publishing Co., Singapore, 1992
Copyright © The Nobel Foundation 1973
To cite this page
MLA style: "Nobelprize.org". Nobelprize.org. Nobel Media AB 2013. Web. 20 Apr 2014. <http://www.nobelprize.org/nobel_prizes/physics/laureates/1973/presentation-speech.html>
Your mission is to arrange an amazing laser party!
Read more about the Nobel Prize in Physics.
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German Shepherd Dog Forums - View Single Post - I can't take it anymore!!!!
You are right, sloped and straight are not mutually exclusive. The slope is a property of a line.
" the slope or gradient of a line is a number that describes both the direction and the steepness of the line." - says the mathematical definition | {"url":"http://www.germanshepherds.com/forum/4749410-post7.html","timestamp":"2014-04-19T15:00:56Z","content_type":null,"content_length":"17725","record_id":"<urn:uuid:955d828e-36ec-4ed4-9ae9-731425d116de>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00640-ip-10-147-4-33.ec2.internal.warc.gz"} |
Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics
Results 1 - 10 of 59
, 2003
"... Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is
that of 123-avoiding permutations. The rewriting rule automatically gives a functional equation satis ..."
Cited by 33 (4 self)
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Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that
of 123-avoiding permutations. The rewriting rule automatically gives a functional equation satis ed by the bivariate generating function that counts the permutations by their length and the label of
the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series.
- J. Integer Seq , 2003
"... A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a
straightforward relationship between the ordinary generating function for simple permutations and that for all ..."
Cited by 25 (2 self)
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A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a
straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coe#cients of this series are not P -recursive, an asymptotic
expansion for these coe#- cients, and a number of congruence results.
"... To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is
isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity tha ..."
Cited by 24 (16 self)
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To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is
isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime
p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role
in the combinatorics underlying the proof of the functional equations.
, 801
"... Abstract. We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present
several intriguing open problems. 1. ..."
Cited by 22 (14 self)
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Abstract. We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several
intriguing open problems. 1.
, 2008
"... We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous’ beta-splitting model, which has an extended parameter
range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multif ..."
Cited by 10 (6 self)
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We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous’ beta-splitting model, which has an extended parameter range β>
−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter
Poisson–Dirichlet models for exchangeable random partitions of N, with an extended parameter range 0 ≤ α ≤ 1, θ ≥−2α and α<0, θ =−mα, m ∈ N.
- Electronic Journal of Combinatorics , 2005
"... We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative
description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to trea ..."
Cited by 9 (0 self)
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We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description
in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be
quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our
various classes of trees and more familiar combinatorial objects including certain Schröder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with
respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the
idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday’s
- Sém. Lothar. Combin
"... Abstract. We exhibit a weight-preserving bijection between semi-standard Young tableaux and skyline augmented fillings to provide the first combinatorial proof that the Schur functions decompose
into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads ..."
Cited by 9 (6 self)
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Abstract. We exhibit a weight-preserving bijection between semi-standard Young tableaux and skyline augmented fillings to provide the first combinatorial proof that the Schur functions decompose into
nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-Schensted-Knuth Algorithm for skyline augmented fillings. We also
prove that the nonsymmetric Schur functions are equal to the standard bases for Schubert polynomials introduced by Lascoux and Schützenberger. This provides a non-inductive construction of the
standard bases and a simple formula for the right key of a semi-standard Young tableau. 1.
, 2007
"... Let P n∈N d fnx n be a multivariate generating function that converges in a neighborhood of the origin of C d. We present a new, multivariate method for computing the asymptotics of the diagonal
coefficients fa1n,...,a dn and show its superiority over the standard, univariate diagonal method. ..."
Cited by 8 (3 self)
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Let P n∈N d fnx n be a multivariate generating function that converges in a neighborhood of the origin of C d. We present a new, multivariate method for computing the asymptotics of the diagonal
coefficients fa1n,...,a dn and show its superiority over the standard, univariate diagonal method.
, 2006
"... A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups
of free products of groups. In particular, we establish formulae for the Euler characteristics of: the ..."
Cited by 7 (4 self)
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A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of
free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms Wh( ∗ n i=1 Gi) when the Gi are of finite homological type; Aut( ∗
n i=1 Gi) and Out( ∗ n i=1 Gi) when the Gi are finite; and the palindromic automorphism groups of finite rank free groups.
- Probab. Surv , 2005
"... This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of
combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While ..."
Cited by 7 (4 self)
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This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of
combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by
Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=369067","timestamp":"2014-04-21T12:20:38Z","content_type":null,"content_length":"35376","record_id":"<urn:uuid:b6047e95-4918-4fc7-ad04-38c15f7bf812>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00186-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kodak Ektar Aero 2.5 radioactivity
07-27-2008, 03:15 AM
Marco B
But I would be extremely sceptical about interpolating anything to do with the bedrock of Norway. I've worked here as a geologist for 25 years, and there are daily surprises! I think Statens
Strålevern might need a little bit more geological competence in their numbers - if the oil boom ends, I'll send them a note. ;)
Please note this was several years ago, but:
That was exactly what I told him! Actually, my tool (a kriging spatial interpolator) showed that there was NO spatial relationship BETWEEN points with a 100% random nugget variance. This
essentially means that the dataset (a very good one with over 10.000 measurements) may NOT be interpolated.
However, that doesn't mean there isn't a spatial or better said statistical relationship between soil/rock types and Radon measurements. It just means that any two points close together have no
spatial relationship. Realizing this, I figured this is simply caused by the fact that soil / bedrock types tend to vary over relatively short distances with one bedrock type being replaced
completely by another, meaning that two houses separated by just 100 meter, might have totally different, and spatially unrelated, Radon levels.
Such data may never be interpolated...
I finally advised the guy to forget about a spatial interpolation, which simply is not allowed with such a dataset and 100% nugget variance (this was difficult for him to accept, like many other
people, spatial interpolation of point measurements is often considered as the "holy grail" in GIS. "If I can interpolate it, it will be good...")
I than advised him to try and statistically correlate soil / bedrock type with Radon levels in a normal statistical package, like SPSS. Simply by determining at what bedrock type, based on a
geological map of Norway, the measurements were made, and to input that data in the statistical package together with the Radon levels. And maybe even include possible other variables as well,
like building material of the houses or possible information about ventilation etc.
If that would show a statistical relationship between soil / bedrock type and Radon levels, he could than reclassify his geological map of Norway based on low / high Radon risk or levels to get
to a country wide map, which was what he was after...
I don't know if he finally did this, (he sounded actually a little bit desperate at the point I told him he wasn't allowed to interpolate the fantastic dataset collected over years), but I hope
so because with such a huge dataset, you can do some very interesting statistical analysis, just not interpolate it.
07-28-2008, 07:22 PM
Mark Layne
I had my Aero Ektar checked by the nuclear medicine department at the local hospital and got similar results. Just for curiousity they also tested the lens in the home made box I constructed and
it blocked all measurable radiation and so that is what I store it in.
What is it? A 6"x6" x6" cardboard postal carton lined with pieces of "Wonderboard" a cement impregnated fiberglass backing left over from when I installed a tile floor in the kid's bathroom.
Talk about opening 'Pandora's Box'
07-31-2008, 02:51 PM
Radioactive Ektar
I actually just bought 2 Aero Ektars (Went a little bid happy and won both on ebay !) The Bokeh is amazing...opens up some really creative opportunities.
I think you get more radiation sun bathing !
Very Cool Glass !
07-31-2008, 02:58 PM
Marco B
Yes, they do seem to be capable of some remarkable images. The Pandora box is actually quite empty, as you can read from my original post and all the comments thereafter, about the only thing you
probably better shouldn't do, is store your new lenses under your bed...
07-31-2008, 03:11 PM
Phillip P. Dimor
Great for portraits but I usually use mine one stop down from wide open, if not two. I know this probably negates the idea of using an aero-ektar in the first place but with the DOF being so
shallow, it's very easy to end up with a fuzzy portrait with a sharp nose!
07-02-2013, 02:31 AM
I ran into a seller at an antique fair this past weekend with a nice one for sale. It seemed to have some sort of hazing or filming on the rear element so I presume that was the effect of aging
spoken of here. The seller was asking $450 U.S. for it. Not knowing anything about it at the time, I thought it was a bit much for a lens sans shutter, even one that fast. He mentioned the
radioactivity as well. | {"url":"http://www.apug.org/forums/large-format-cameras-accessories/print-52650-kodak-ektar-aero-2-5-radioactivity-3.html","timestamp":"2014-04-16T09:02:53Z","content_type":null,"content_length":"14564","record_id":"<urn:uuid:cbdd2034-a6aa-45e9-8eb5-4e93e0464338>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00601-ip-10-147-4-33.ec2.internal.warc.gz"} |
2. METHOD OF SOLUTION
2.1. The equations of statistical equilibrium
In order to calculate the level populations of a given atom or ion, we make two basic assumptions :
1. The rates of processes involving ionization stages other than the atom or ion being considered (such as direct photoionization or recombination, charge exchange reactions, collisional ionization,
etc.) are slow compared to bound-bound rates.
2. All transitions considered are optically thin.
To calculate the population of some level i, we must take into account all possible processes that will (de)populate it:
where n[i] is the volume density of atoms or ions in level i.
Therefore, in steady state regime the sum over all processes that populate level i will be balanced by the sum over all processes that depopulate level i. Assuming that the two conditions listed
above are met, this can be written (see, for instance, Rybicki and Lightman [3]):
where we have considered all possible bound-bound processes, i.e., spontaneous, radiation-induced and collisionally-induced. The lefthand side of eq. (2) is the sum over all processes that populate
level i from the other levels j, whereas the righthand side is the sum over all processes that depopulate level i to levels j.
A[ij] is the transition probability of spontaneous decay from level i to level j. For i j, A[ij] = 0.
B[ij] are Einstein coefficients, related to the transition probabilities by:
for i > j, and B[ii] = 0; h is Plank's constant, E[i] is the energy of level i (expressed in cm^-1) and g[i] is the statistical weight of level i.
u[ij] is the spectral energy density of the radiation field integrated along the line profile [] of the transition from level i to level j:
with u[ii] = 0; [ij] is the frequency of the transition and we have assumed that the radiation field does not vary significantly along the line profile.
In eq. (2) we have also considered the effect of collisions; n^k is the volume density of the particle inducing the transition, the main collision partners usually being k = e^-, p^+, H^0, He^0, H
[2],... , depending whether the medium is primarily ionized or neutral.
q^k[ij] is the collision rate for the transition from level i to level j induced by the collision partner k . These coefficients are the cross-sections for the related process [ij] convolved with a
Maxwellian distribution of velocities f(v), making these quantities suitable for astrophysical applications (see, for instance, Osterbrock [4]):
for the deexcitation rates (i > j); k is Boltzmann's constant, T is the kinetic temperature, µ is the reduced mass of the system and
Excitation and deexcitation rates are related by the principle of detailed balance:
with q[ii] = 0. When the interaction is coulombian, as in collisions with electrons, it is convenient to express the cross-section in terms of the collision stregth [ij], defined by:
where m is the electron's mass. Substituting this in eq. (5) yields:
with T expressed in K and [ij] is defined by the integral in eq. (8) and is called Maxwellian-averaged collision stregth. Typically [ij] is a slowly varying function of T, of order unity. However,
for neutral atoms it may vary for several orders of magnitude.
These are the basic parameters needed to solve eq. (2). If we consider our model ion to be composed of n levels, then we must solve a linear system of n - 1 equations in order to calculate the
relative population ratios. | {"url":"http://ned.ipac.caltech.edu/level5/Silva/Silva2_1.html","timestamp":"2014-04-17T10:36:29Z","content_type":null,"content_length":"6940","record_id":"<urn:uuid:c3542871-e329-4977-923c-d6d854218c35>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00256-ip-10-147-4-33.ec2.internal.warc.gz"} |
USBR Water Measurement Manual - Chapter 14 - MEASUREMENTS IN
PRESSURE CONDUITS, Section 11. Point Velocity Area Methods
11. Point Velocity Area Methods
Computing discharge point velocities for open channel flow is discussed in chapter 10. In conduits, several point velocities can be obtained by traversing with a velocity measuring device such as a
pitot tube. Arrays of several velocity measuring devices, such as axial current meters, are sometimes fixed on racks that span the conduit to measure several velocities simultaneously. Some
flowmeters directly measure average velocity along lines through the flow.
The measurement of point velocities is relatively simple. However, partitioning the flow section relative to velocity points is complex, depending on the accuracy desired. The main problem in
determining proper partial areas is that each point velocity represents or determines meaningful velocity weighting factors related to each point location. Many schemes can be used to locate
measuring points on grids or diameters and assign weighting factors for each position. The procedures are further complicated when corrections are needed to account for the obstruction of rack
support systems and the size of the instruments themselves. If accuracies better than +/-3 percent are needed, then procedures set by codes such as ISO (1977) and ASME (1992) should be consulted.
Some methods of averaging velocity are done by selecting equal areas related to the shape of the flow cross section and measuring velocity at specific points within these areas. For pitot
measure-ments, the average of the square root of the velocity heads of the point measurements is multiplied by the flow section area.
The most common pressure conduit is the circular pipe. For a constant rate of flow, the velocity varies from point to point across the stream, gradually increasing from the walls toward the center of
the pipe. The mean velocity is obtained by dividing the cross-sectional area of the pipe into a number of concentric, equal area rings and a central circle. The standard (ASME, 1983) 10-point system
is shown on figure 1410a. More equal area divisions may be used if required by large flow distortions or other unusual flow conditions. Velocity measurements are taken at specific locations in these
subareas (figure 14-10a) and are adjusted in terms of average velocity head by the equation:
Figure 14-10 -- Locations for pitot tube measurements in circular and rectangular conduits (reproduced from British Standard 1042, Flow Measurement [1943], by permission of the British Standards
The mean velocity in rectangular ducts can be found by dividing the cross section into an even number (at least 16) of equal rectangles geometrically similar to the duct cross section and measuring
the velocity at the center of each area (figure 14-9b). Additional readings should be taken in the areas along the periphery of the cross section according to the diagram on figure 14-9c. Then, the
average velocity is determined from equation 14-5.
Acoustic devices, discussed in chapter 11, measure accurate average velocity along chords or diametral lines in planes across the flow section. The diametral arrangement uses the simple average of
the line velocities corrected for the angle of the plane across the conduit. The multiple chordal systems use a specific weighting factor for each line velocity to determine the average (AMSE, 1992).
The chord locations are specified to maximize accuracy. | {"url":"http://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/chap14_11.html","timestamp":"2014-04-19T14:33:58Z","content_type":null,"content_length":"6464","record_id":"<urn:uuid:d0f30e10-5516-4595-b227-5bbb9b7c2700>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00515-ip-10-147-4-33.ec2.internal.warc.gz"} |
Coexistence in a One-Predator, Two-Prey System with Indirect
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 625391, 13 pages
Research Article
Coexistence in a One-Predator, Two-Prey System with Indirect Effects
Pontificia Universidad Javeriana, Carrera 7 No. 43-82, Bogotá, Colombia
Received 26 October 2012; Accepted 27 February 2013
Academic Editor: Erik Van Vleck
Copyright © 2013 Renato Colucci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
We study the dynamics of a one-predator, two-prey system in which the predator has an indirect effect on the preys. We show that, in presence of the indirect effect term, the system admits
coexistence of the three populations while, if we disregard it, at least one of the populations goes to extinction.
1. Introduction
The importance of indirect effects is well established in Biology (see [1–5]), for example, in the case of predation (see [6]), the predator can alter the morphology (see [7]) or the behavior of the
preys. The preys, in order to reduce the possibility of contacts with the predators, could modify their normal conduct by reducing their activity or by hiding themselves for long time. There are many
types of indirect effects (see [3] for a detailed discussion); another interesting case is the refuge indirect effect (see [8], e.g.); anyway, it is of great interest trying to describe the indirect
interactions in population dynamics. Recently in [9] a model including indirect effects was proposed, modeling the effects of predator Daphnia over two groups of Phytoplankton of different morphology
(see [10]), having Phosphorous as a resource (see [11] or [12]). A general model describing indirect effects of predation can be written in the following form: where is the density of population of
the predator while and are the densities of population of the preys. The positive function represents the indirect effects of the predator on the prey generated by predation over the prey . In [9], a
system in which the functions are linear and the function is quadratic was proposed. The system takes the following form: where , , , , , are positive parameters and where .
In the above system represent the density of population of a predator (Daphnia-Zooplankton) that predates the preys (Phytoplankton) and that are of different size, in particular is of a smaller size
than . The variable represents the amount of resources (Phosphorous) for the preys and . We will show below that the system can be put in the form of system (1).
In absence of preys, the population of predators will extinguish exponentially while the presence of them brings a positive growth rate. In absence of predator, the populations of preys will increase
depending on the amount of available resources, while the presence of predators affects negatively their growth rate. Moreover, the population of predator has an effect on the prey which is described
by the term which produce a negative growth rate. This effect produces an indirect effect that is positive (the term ) for the population of prey . The growth rate of the amount of resources of the
system is effected by the presence of the preys while, for its renewable character, the presence of the predator lets it regenerate (the term ).
We first observe that the system is closed; in fact if we sum (2), we obtain and as a consequence we have the following proposition.
Proposition 1. The function defined as is a constant of motion.
By using the first integral we can reduce the degree of freedom of the problem; in fact if we fix a value of the the first integral, , then the system can be rewritten in the following way:
Since we are interested only in positive solution and using condition (4), we can limit our analysis to the following region:
First of all we show that the dynamics develops in the region (see Figure 1).
Theorem 2. The set is positively invariant for the solutions of the system (5).
Proof. In order to prove that is positively invariant, it is sufficient to compute the vector field of the system on the boundary of that is made up by 4 triangular regions.
First of all we consider the vertexes of , we set The vertexes are all fixed points except the point on which the vector field is .
We study the vector field on each triangular region separately.
The Face . In this case , then, in the interior of the triangle, the vector field points toward the interior of . We check the sides of the triangle : if we have and ; moreover, as . If we have that
and . If we consider the side on which then we have that . Finally we conclude that on the vector field is tangent or point inward .
The Face . In this region we have that , that is, the plane is invariant and as a consequence we have only to check the vector field on the boundary of . If we have that the vector field points
inward since , while in the side where we have that and so it is positively invariant. Finally, on the side where we have that and that as .
The Face . In this region we have that , then the plane is invariant and as a consequence we have only to check the vector field on the boundary of . On the side where we have that and as a
consequence , for all . Finally on the side where we have and and as , while on the side where we have that , , and when .
The Face . In this case we have that the vector field points inward . In fact, if we introduce the function that is the sum of the three variables and we sum the equations of the system, we obtain
the following differential equation depending on : From (9) we get that this concludes the proof since if the solutions start on then they enter the set or they remain on .
The rest of the paper is organized as follows: in Section 2 we analyze the stability of the fixed points while in Section 3 we study the dynamics on the boundary of . In Section 4 we analyze the
dynamics of the system in absence of indirect effects terms (that is ) while in Section 5 we study the asymptotic behavior of the solutions of the system for , in particular we prove the coexistence
of the three species under a proper choice of the parameters. In Section 6 we present some numerical simulations while in Section 7 we give some comments and remarks for future investigation.
2. Stability of the Fixed Points
The first step in studying the dynamics of the system (5) consists in finding all the fixed points in and analyzing their stability character. In Section 1 we have pointed out the existence of three
fixed points among the vertexes of : In order to find all possible further fixed points in the faces or in the interior of , we consider the intersections between the nullclines (see [13]). Since it
is a standard argument, we do not present the detailed computation. If , there exists a fixed point in the interior of the triangle , while if , then .
Moreover, there exists a segment of fixed points that we call : The segment is one of the sides of the face (on the plane ) and its extremes are the fixed points and .
We also have an interior fixed point if the following two conditions are fulfilled:
In details, the coordinate of the fixed point are where . As a summary of the previous analysis we can state the following theorem.
Theorem 3. The points , , and the points of the segment are equilibria of the system (5) for any (positive) value of the parameters. The system admits that the fixed point if and the interior fixed
point if (14) and (15) hold.
Remark 4. From the above analysis we notice that is a parameter of bifurcation for the fixed points. In particular we have that fixed points and collapse to a unique fixed point if , and the fixed
point if .
Then we pass to study the stability character of the fixed points. We consider the functional Jacobian of the vector field:
where .
2.1. The Fixed Point
We start with the analysis of the origin . The matrix of the linearized system at is The point is a saddle for any values of the parameters since we have one negative eigenvalue and two positive
eigenvalues and . The -axis is the stable space of the linearized system at while the instable space is the plane .
It is possible to distinguish two cases: three distinct eigenvalues if (that is ) and only two distinct eigenvalues if (that is ).
2.2. The Fixed Point
We analyze the point , and the matrix of the linearized system at is The eigenvalues of are
The eigenvalues are always negative, while if we have then the fixed point is instable. In the case in which , that is, , we would need to study the system on the center manifold (see [14]) of the
point in order to study its stability.
2.3. The Fixed Point
The matrix of the linearized system at is The eigenvalues of the matrix are
In this case we have at least one negative eigenvalue, that is, , along the axes , then the stability depends on the first two eigenvalues. In the case in which we can conclude that is instable. If
we have to study the system on the center manifold related to the eigenvalue (and if it is 0), in order to establish the stability of .
2.4. The Fixed Point
We analyze the point in the case in which that is if . The matrix of the linearized system at is where we have set . The eigenvalues of the matrix are where . Then if then are complex conjugate with
negative real part; otherwise they are both negative (coincident if the inequality holds in (28)). If the eigenvalue is positive, that is, we have that is instable.
2.5. The Segment of Fixed Point
On the fixed points that belong to the segment we have that and , then the matrix of the linearized system at any point of is of the following form:
where . The eigenvalues of the matrix are In this case we have that the eigenvalue is always negative since . In the case in which the eigenvalue and then the fixed points of the segments are all
instable. We note that if , then satisfies on the contrary satisfies the opposite inequality.
2.6. The Fixed Point
The matrix of the linearized system at is
where we have set . We consider the trace of the matrix :
Then, if we have that and then at least one of the eigenvalues of is negative.
We summarize the results of the previous analysis in the following statements.
Theorem 5. Suppose that the parameters of the system satisfy the following inequalities: Then the fixed points , , , and the fixed points that belong to the segment are all instable.
Remark 6. We note that the instability hypothesis for the point , that is, , implies the presence of the point . Then it would be interesting to prove if it is a necessary condition for the
instability of , that is, if the presence of the fixed point is necessary for the instability of all fixed points in .
Remark 7. We note that the hypothesis of the previous theorem are not compatible with the inequality that is, the condition for which .
In conclusion, the instability of the boundary fixed points implies the existence of the interior fixed point . The presence of the interior fixed point represents an obvious case of coexistence of
the three species; however, if is instable then it is not useful to describe real cases of coexistence.
Since we are not able to face the general problem of stability of point , we will follow another strategy in order to prove coexistence of the system (see Section 6 below).
Remark 8. It would be interesting to find if the interior fixed point is instable for some choice of the parameters that are compatible with the instability hypothesis for the boundary fixed points.
In that case it would be interesting to look for the existence of a limit cycle surrounding or of homoclinic cycle and chaotic attractors (see e.g., [15] or [16] in which the authors proved the
chaotic behavior of one-predator, two prey systems without indirect effects).
3. The Dynamics on the Boundary of
In this section we analyze the dynamics on the boundary of , in order to do that we first study the dynamics on the coordinate axes (see Figure 2) and then on the faces of .
3.1. Dynamics on the Axes
The three axes are invariant for the dynamics; in particular, on the -axes we have Then the solution is and we have that as . On the axes and we have logistic growth and in particular as .
In fact on the -axes we have while on the -axes we have For both axes we have that as and the solutions are of the form where and are the initial conditions.
3.2. The Dynamics on the Faces
In this subsection we analyze the dynamics on the faces of .
The Face is on the invariant plane . We have a system of competitive Volterra-Lotka equation: This is a degenerate case; in fact we have a segment (see definition (13)) of fixed points that connects
and . The -limit of every orbit starting inside (that is ) is a point on ; hence, the preys do not vanish and there are not periodic orbits. The behavior of the system on the face is represented in
Figure 3 above.
The Face is on the invariant plane . The system reduces to a system of Volterra-Lotka equations with intraspecific competitions between the preys: We recall that on we have the fixed point if . In
order to understand the behavior of the solution we analyze the nullclines. The -nullclines are and and the -nullclines are and . We distinguish two cases: with or without the fixed point .
In the case in which or (that is ), the -nullcline is above (or pass through) the fixed point , as a consequence inside . The -nullcline divide in two regions: below it we have while above it we have
. Then, all the solutions with tend to the fixed point according to the saddle character of . The behavior of the system is represented in Figure 4.
In the case in which (that is, ) the nullclines intersect at the point . The direction of the vector field is represented in Figure 5 below. Hence, the solutions wind in the clockwise direction
around . To be more specific we consider the linearization of (45) at . The eigenvalues of the matrix of the linearized system are Then the eigenvalues are both negative and in the complex case they
have negative real part. We conclude that is asymptotically stable.
In both cases there are no periodic orbit; in fact two dimensional Volterra-Lotka equations do not admit isolated periodic orbit (see [17]), and they admit a continuum of periodic orbits if and only
if the eigenvalues at the interior fixed point (in this case ) are purely imaginary (if and only if the trace of the matrix of the linearized system is zero, that is, ).
The Face is on the plane , and we have already shown (see equation (9)) that if the solutions start on it then they enter while if they remain on .
The Face is easy to analyze, if the solutions do not start on the axes (that is if ) we have that , as a consequence the solutions leave the plane and go inside .
Then we have proved the following.
Theorem 9. The system (5) admits neither limit cycles nor periodic orbits in the boundary of .
4. Analysis for
In this section we study the behavior of the system in absence of indirect effects that is in the case .
In this case we have the presence of a further boundary fixed points: in the case in which . If , then , while if we have that . Then we have the following.
Theorem 10. The points , , and the points of the segment are equilibria of the system (5) with , for any value of the other parameters. The system admits a further fixed point if and a fixed point if
The dynamics on the boundary is similar to that of the case . In particular on the faces and there are no differences between the two cases and .
To analyze the dynamics on the face we can use the same method of the analysis of the case and it is easy to see that we obtain the same conclusions.
The remarkable difference is that, in the case , the plane is invariant. The dynamics on it (the faces ) is similar to that on in the case in which (see Figures 6 and 7).
We can distinguish two cases: with or without the fixed point . The matrix of the linearized system at is the following:
This matrix has at least one real eigenvalue, that is, , while the other two eigenvalues cannot be purely imaginary since the trace, of the matrix formed by the first two columns and rows of , is
negative. Then, as in the case , we can conclude the following.
Theorem 11. If the system (5) admits neither limit cycles nor periodic orbits on the boundary of the set .
In the case the asymptotic behavior of the solution can be studied by using a well-known result (see [17, Theorem , page 43.]) that states that if in the interior of the forward invariant set there
are not fixed points then the interior of does not contain -limit or -limit sets of the solutions. In particular if there are not fixed points in the interior of then there are not periodic orbits
too (see [18], e.g., of system without periodic orbit). Then we can state the following.
Theorem 12. If , the system (5) admits neither limit cycles nor periodic orbit in the set ; that is, for each solution starting in at least one of the species goes to extinction.
Remark 13. It is interesting to note that three-dimensional predator-prey systems (without indirect effects) admit nontrivial cases of coexistence; see, for example, the paper [19] where the authors
studied oscillations for a class of singularly perturbed three-dimensional predator-prey systems.
5. Asymptotic Behavior for
In Section 4 we have shown that the asymptotic dynamics on does not admit coexistence of populations if . In this section we analyze the case and show that the system admits coexistence of the three
species by using persistence theory and in particular an acyclicity approach (see [20]).
First of all we give a remark on the evolution of volume elements and then we pass to the proof that the system admits coexistence of the three populations under a proper choice of the parameters.
5.1. Volume Evolution
We consider the evolution of volumes under the flow of the system (5). Let be a region of with regular boundary and define , that is, the image of under the flow at time . Let be the volume of , then
by Liouville's theorem we have that the evolution of is described by the following differential equation: where is the vector field of the system (5). In details we have that
The above expression is quite difficult to study and suggests a complex behavior. However, there exist values of the parameter for which is negative. In fact at least in the case in which we have
that so the 3-dimensional volume elements contract in .
In this case, if an attractor (or limit cycle) exists, we expect that it would be contained in a small subset of . On the contrary the attractor could be contained in a bigger subset of (see the
numerical simulations below).
Remark 14. From condition (53)-(55), we have that the volume element contracts in the whole region ; however, these conditions are incompatible with condition (37). Since (37) could be not necessary
conditions, it would be possible to have both volume decay and instability of the fixed points.
5.2. Coexistence of the Three Populations
The biological problem of the coexistence of the three species can be put in mathematical terms by looking for the conditions which prevent the positive solutions starting in the interior of from
converging to as . These ideas can be made rigorous in the context of persistence theory (see [20]). There are many definitions of persistence; the most useful for biological applications (see [21])
is uniform persistence.
Definition 15. The system (5) is uniformly -persistent if there exists such that with and where
In order to prove uniform persistence we use an acyclicity approach (see [20]), then (see Theorem 8.17 page 188, hypothesis (H) page 185 and Theorem 5.2 page 126 in [20]) the following conditions are
needed.(C1) There exists a compact attractor of bounded set.(C2) The invariant sets of are weakly -repelling.(C3) The invariant sets of are acyclic.In the following lines we give the rigorous
definitions of the above ideas and prove conditions (C1)–(C3).
Theorem 16. The system (5) admits a unique compact attractor that attracts all bounded sets of .
Proof. By Theorem 2.33 page 43 in [20], we get the existence of a compact attractor of bounded sets if the flow associated with the system (5) is point dissipative, asymptotically smooth, and
eventually bounded on every bounded set in .
In order to prove the thesis we consider Theorem 2; in particular we write again the system (5) in function of , that is, If with and , then and we conclude that there exists such that for . This
proves the point dissipative property of the flow.
We consider the following family of sets: where and . Every bounded set of is contained in a set for sufficiently large. The family of sets is forward invariant (see Theorem 2 for the set ), in fact,
again we have This proves that bounded sets have bounded orbits. Finally we get the asymptotically smoothness of the flow by noting that it is asymptotically compact on every forward invariant
bounded closed set (see remark 2.26, page 39 in [20]).
Before proving condition (C2) we consider the following definition (see [20, chapter 8]).
Definition 17. Let be the solution of (5) starting at . A set in is called weakly -repelling if there is no such that and as .
In order to prove weakly -repelling property it is sufficient to show that the stable manifolds of the invariant sets of are contained in . From the analysis of the stability of fixed points and of
the dynamics on we have that there are no periodic orbits on and the only invariant set are , , , , and .
Theorem 18. Suppose that hypothesis of Theorem 5 holds. Then the stable manifolds of all the fixed points of are contained in .
Proof. Following the results of Section 2 we have that the stable manifold of the fixed point is the axes: , while (see also Section 3) the stable manifold of the fixed point satisfies .
The points belonging to the segment have a stable manifold inside the face on the invariant plane . In particular the extremes and of the segment satisfy and . Finally all the points belonging to the
segment have the same center manifold and this concludes the proof.
In order to prove condition (C3) we first consider the following definitions (see [20, chapter 8]).
Definition 19. Let . is chained to in , written , if there exists a total trajectory in with , and as .
Definition 20. A finite collection of subsets of is called cyclic if after possibly renumbering or in for some . Otherwise it is called acyclic.
Theorem 21. Suppose that hypothesis of Theorem 5 holds. Then there set , of invariant sets of , is acyclic.
Proof. By the analysis of stability of the fixed points and of the dynamics on the boundary of it follows that there are no periodic orbits in . Moreover, the only invariant sets , , , , and are all
contained in . By the stability analysis of fixed points we get that only the following chains are admissible: Then the set is acyclic.
From Theorems 16, 18, and 21, we get the final result.
Theorem 22. Suppose that hypothesis of Theorem 5 holds. Then the system (5) is uniformly persistent.
From the previous theorem we obtain coexistence of the three populations under the hypothesis of Theorem 5 without analyzing the stability of the fixed point .
6. Numerical Simulations
The strategy used for proving coexistence of the three species does not exclude quasi-periodic or chaotic behavior of the system. The uniform persistence only lets us conclude the existence of a
bounded attractor contained in and with positive distance from . In this section we provide several numerical experiments in order to describe the structure of the attractor.
6.1. Experiment 1
We consider the following values of the parameters: In a first numerical experiment (represented in Figures 8, 12, 13, and 14 below) we considered the value for which the attractor is the fixed point
. In a second numerical experiment (represented in Figures 9, 15, 16, and 17) we considered the value . In this case the attractor appears to be bidimensional.
6.2. Experiment 2
We consider the following values of the parameters: Again, we perform two numerical experiments with different values of . In a first simulation we considered the value (the solution is represented
in Figure 10) and the attractor is the fixed point . In a second simulation we considered the value (the solution is represented in Figure 11) and the attractor appears to be a limit cycle.
6.3. Remarks on the Numerical Experiments
These experiments suggested that the parameter could be considered as a bifurcation parameter for the attractor. Moreover, we have effectuated many simulations with different choices of the function
with different values of parameters and in any cases a limit cycle arises. This suggests that the attractor may have some structural stability properties.
7. Conclusions
We have shown that in absence (that is ) of the terms that describe indirect effects, the system (5) does not admit coexistence of the three populations. The present work suggests the importance of
indirect effects in describing cases of coexistence (with or without an interior fixed point); in particular the parameter is a bifurcation parameter for coexistence. It has been already pointed out
(see for instance [22]) that a presence of limit cycle, oscillations, or chaotic fluctuations allows the coexistence of many species and could be beneficial for the functioning of the ecosystem (see
It would be interesting to consider indirect effects in different contexts such as spatial model (see [24]) or evolution problems (see [25]). In fact it is an important ecological problem (see [26])
to determine the consequences of indirect effects in evolution.
In the context of population dynamics, it would be interesting to consider a different form of the function ; in particular if we sum the second and the third equations we get then the indirect
effect terms have no effects on the whole population of preys; they effect only the proportion between them. In order to consider a different situation it would be sufficient to put different
constants and in the function for the two preys (see Figure 18).
Moreover, it would be interesting to consider the choice , or at least . This would be an improvement in description of indirect effects; in fact in the present work we have considered that the
indirect effect of the predator over the prey has the same order of magnitude of the direct effect (compare the terms and ). In the case considered, if , the form of the third equation is the most
effected while the second equation can be rewritten in the following way: that is simply a change of the coefficient of . By considering a term also the form of the second equation would be
drastically effected. Moreover, it is an interesting question (see [3]) to establish if direct and indirect effects are of the same magnitude. Then it would be useful to consider different values of
in (69) in order to describe cases in which direct effects are bigger (resp., less) than indirect effects.
Another possible improvement would be to consider the nonautonomous case for which is a function of time; for example, in the case of seasonal indirect effects, we could make the following choice:
where is a positive constant (see Figures 19 and 20). Some analysis of that data (see [3]) suggests that indirect effects could take a long time to become apparent since they occur at a slower time
scale than any direct effect (see [4]). For this reason it could be interesting to consider a delayed version (see [27], e.g., of existence of periodic solutions for a predator-prey model with delay)
of the system (5) where, for example, in the third equation we put Moreover, for more general forms of the function , it could be interesting to look for chaotic behavior (see [28]) of the solutions
(see [29], for example, in population dynamics and [30] for a discrete time case). In conclusion we can note that, beside the system (2) which is a simplification of the description of indirect
effects, from the numerical experiments, it seems that it shares the main features with more sophisticated models.
The author would like to thank Professor Piero Negrini (University Sapienza of Rome) and Professor Juan José Nieto (University of Santiago de Compostela) for their helpful suggestions and remarks.
This work was partially supported by project PPTA 4476 of Pontificia Universidad Javeriana, Bogotá.
1. B. Bolker, M. Holyoak, V. Křivan, L. Rowe, and O. Schmitz, “Connecting theoretical and empirical studies of trait-mediated interactions,” Ecology, vol. 84, no. 5, pp. 1101–1114, 2003. View at
2. D. Cariveau, R. E. Irwin, A. K. Brody, L. S. Garcia-Mayeya, and A. Von der Ohe, “Direct and indirect effects of pollinators and seed predators to selection on plant and floral traits,” Oikos,
vol. 104, no. 1, pp. 15–26, 2004. View at Publisher · View at Google Scholar · View at Scopus
3. B. A. Menge, “Indirect effects in marine rocky intertidal interaction webs: patterns and importance,” Ecological Monographs, vol. 65, no. 1, pp. 21–74, 1995. View at Scopus
4. W. E. Snyder and A. R. Ives, “Generalist predators disrupt biological control by a specialist parasitoid,” Ecology, vol. 82, no. 3, pp. 705–716, 2001. View at Scopus
5. J. T. Wootton, “Indirect effects, prey susceptibility, and habitat selection: impacts of birds on limpets and algae,” Ecology, vol. 73, no. 3, pp. 981–991, 1992. View at Scopus
6. J. Estes, K. Crooks, and R. Holt, Ecological Role of Predators. Enciclopedia of Biodiversity, vol. 4, Academic Press, New York, NY, USA, 2001.
7. V. Lundgren and E. Granéli, “Grazer-induced defense in Phaeocystis globosa (prymnesiophyceae): Influence of different nutrient conditions,” Limnology and Oceanography, vol. 55, no. 5, pp.
1965–1976, 2010. View at Publisher · View at Google Scholar · View at Scopus
8. M. F. Caruselaa, F. R. Momoa, and L. Romanellia, “Competition, predation and coexistence in a three trophic system,” Ecological Modelling, vol. 220, no. 10, pp. 2349–2352, 2009. View at Publisher
· View at Google Scholar
9. “Indirect effects affects ecosystem dynamics,” 2011, http://www.ictp-saifr.org/.
10. R. Margalef, “Life forms of Phytoplanktos as survival alternative in an unstable environment,” Oceanologica Acta, vol. 134, no. 1, pp. 493–509, 1978.
11. O. Sarnelle, “Daphnia as keystone predators: effects on phytoplankton diversity and grazing resistance,” Journal of Plankton Research, vol. 27, no. 12, pp. 1229–1238, 2005. View at Publisher ·
View at Google Scholar · View at Scopus
12. D. O. Hessen, T. Andersen, P. Brettum, and B. A. Faafeng, “Phytoplankton contribution to sestonic mass and elemental ratios in lakes: implications for zooplankton nutrition,” Limnology and
Oceanography, vol. 48, no. 3, pp. 1289–1296, 2003. View at Scopus
13. L. R. Devaney, M. W. Hirsh, and S. Smale, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elvesier, New York, NY, USA, 2004.
14. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, NY, USA, 2nd edition, 2010. View at MathSciNet
15. A. Klebanoff and A. Hastings, “Chaos in one-predator, two-prey models: general results from bifurcation theory,” Mathematical Biosciences, vol. 122, no. 2, pp. 221–233, 1994. View at Publisher ·
View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
16. S. Gakkhar and R. K. Naji, “Existence of chaos in two-prey, one-predator system,” Chaos, Solitons and Fractals, vol. 17, no. 4, pp. 639–649, 2003. View at Publisher · View at Google Scholar ·
View at Zentralblatt MATH · View at MathSciNet
17. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998. View at MathSciNet
18. P. van den Driessche and M. L. Zeeman, “Three-dimensional competitive Lotka-Volterra systems with no periodic orbits,” SIAM Journal on Applied Mathematics, vol. 58, no. 1, pp. 227–234, 1998. View
at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
19. W. Liu, D. Xiao, and Y. Yi, “Relaxation oscillations in a class of predator-prey systems,” Journal of Differential Equations, vol. 188, no. 1, pp. 306–331, 2003. View at Publisher · View at
Google Scholar · View at Zentralblatt MATH · View at MathSciNet
20. H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, RI, USA, 2011. View at MathSciNet
21. G. Butler, H. I. Freedman, and P. Waltman, “Uniformly persistent systems,” Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 425–430, 1986. View at Publisher · View at Google
Scholar · View at Zentralblatt MATH · View at MathSciNet
22. J. Huisman and F. J. Weissing, “Biodiversity of plankton by species oscillations and chaos,” Nature, vol. 402, no. 6760, pp. 407–410, 1999. View at Scopus
23. D. Li, J. Li, and Z. Zheng, “Measuring nonequilibrium stability and resilience in an n-competitor system,” Nonlinear Analysis. Real World Applications, vol. 11, no. 3, pp. 2016–2022, 2010. View
at Publisher · View at Google Scholar · View at MathSciNet
24. H. Nie and J. Wu, “Coexistence of an unstirred chemostat model with Beddington-DeAngelis functional response and inhibitor,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp.
3639–3652, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
25. R. A. de Assis, S. Bonelli, M. Witek et al., “A model for the evolution of parasite-host interactions based on the Maculinea-Myrmica system: numerical simulations and multiple host behavior,”
Nonlinear Analysis. Real World Applications, vol. 13, no. 4, pp. 1507–1524, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
26. M. R. Walsh and D. N. Reznick, “Interactions between the direct and indirect effects of predators determine life history evolution in a killifish,” Pnas, vol. 105, no. 2, pp. 594–599, 2008. View
at Publisher · View at Google Scholar
27. X. Liu and L. Huang, “Periodic solutions for impulsive semi-ratio-dependent predator-prey systems,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3266–3274, 2009. View at
Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
28. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1989. View at MathSciNet
29. J.-M. Ginoux, B. Rossetto, and J.-L. Jamet, “Chaos in a three-dimensional Volterra-Gause model of predator-prey type,” International Journal of Bifurcation and Chaos in Applied Sciences and
Engineering, vol. 15, no. 5, pp. 1689–1708, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
30. Z. He and X. Lai, “Bifurcation and chaotic behavior of a discrete-time predator-prey system,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 403–417, 2011. View at Publisher ·
View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet | {"url":"http://www.hindawi.com/journals/jam/2013/625391/","timestamp":"2014-04-17T19:33:03Z","content_type":null,"content_length":"693398","record_id":"<urn:uuid:1bd8b8cb-7ddb-4318-9278-93d56757eb94>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00066-ip-10-147-4-33.ec2.internal.warc.gz"} |
pgmac - written in most popular ciphers: caesar cipher, atbash, polybius square , affine cipher, baconian cipher, bifid cipher, rot13, permutation cipher
Caesar cipher
Caesar cipher, is one of the simplest and most widely known encryption techniques. The transformation can be represented by aligning two alphabets, the cipher alphabet is the plain alphabet rotated
left or right by some number of positions.
When encrypting, a person looks up each letter of the message in the 'plain' line and writes down the corresponding letter in the 'cipher' line. Deciphering is done in reverse.
The encryption can also be represented using modular arithmetic by first transforming the letters into numbers, according to the scheme, A = 0, B = 1,..., Z = 25. Encryption of a letter x by a shift
n can be described mathematically as
Plaintext: pgmac
┃ cipher variations: ┃
Decryption is performed similarly,
(There are different definitions for the modulo operation. In the above, the result is in the range 0...25. I.e., if x+n or x-n are not in the range 0...25, we have to subtract or add 26.)
Read more ... Atbash Cipher
Atbash is an ancient encryption system created in the Middle East. It was originally used in the Hebrew language.
The Atbash cipher is a simple substitution cipher that relies on transposing all the letters in the alphabet such that the resulting alphabet is backwards.
The first letter is replaced with the last letter, the second with the second-last, and so on.
An example plaintext to ciphertext using Atbash:
┃Plain: │pgmac ┃
┃Cipher: │ktnzx ┃
Read more ... | {"url":"http://easyciphers.com/pgmac","timestamp":"2014-04-18T13:23:59Z","content_type":null,"content_length":"21952","record_id":"<urn:uuid:7b2cb785-e000-4389-940f-c1dd68588e48>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00321-ip-10-147-4-33.ec2.internal.warc.gz"} |
Weekly Challenge 22: Combinations of Two
Copyright © University of Cambridge. All rights reserved.
'Weekly Challenge 22: Combinations of Two' printed from http://nrich.maths.org/
Can you use this diagram to prove that the number of different pairs of objects which can be chosen from six objects, $^6C_2$, is $$1 + 2 + 3 + 4 + 5?$$
Generalise this to show that the number of ways of choosing pairs from $n$ objects is
$$^nC_2 = 1 + 2 + ...+ (n-1) = \frac{1}{2}n (n - 1).$$
Did you know ... ?
The sum of the first $n$ whole numbers is called a triangle number because this sum can be represented geometrically by a triangular array of dots. The sum is easily found by working out the number
of dots in the parallelogram formed by putting two triangular arrays side by side. | {"url":"http://nrich.maths.org/7474/index?nomenu=1","timestamp":"2014-04-18T13:07:53Z","content_type":null,"content_length":"4123","record_id":"<urn:uuid:52ba6e8d-023d-4812-b1e3-a95490920068>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00427-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Excursions In Modern Mathematics 7th Edition Textbook Solutions | Chegg.com
The candidate with the most first-place votes wins. This method is called the plurality Method.
Thus, in an election conducted under the plurality method, one does not need each voter to rank the candidates.
From the preference schedule for the election given in Table 1, the following can be ascertained:
The candidate A gets eight first-place votes.
The candidate B gets three first-place votes.
The candidate C gets five first-place votes.
The candidate D gets five first-place votes.
The candidate E gets zero first-place votes.
Therefore, by the plurality Method, the candidate | {"url":"http://www.chegg.com/homework-help/excursions-in-modern-mathematics-7th-edition-solutions-9780321575197","timestamp":"2014-04-18T21:56:25Z","content_type":null,"content_length":"41118","record_id":"<urn:uuid:b1d2b23c-337b-432b-9636-1d1c496b169b>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00136-ip-10-147-4-33.ec2.internal.warc.gz"} |
Energy flow characteristics of vector X-Waves
The vector form of X-Waves is obtained as a superposition of transverse electric and transverse magnetic polarized field components. It is shown that the signs of all components of the Poynting
vector can be locally changed using carefully chosen complex amplitudes of the transverse electric and transverse magnetic polarization components. Negative energy flux density in the longitudinal
direction can be observed in a bounded region around the centroid; in this region the local behavior of the wave field is similar to that of wave field with negative energy flow. This peculiar energy
flux phenomenon is of essential importance for electromagnetic and optical traps and tweezers, where the location and momenta of micro-and nanoparticles are manipulated by changing the Poynting
vector, and in detection of invisibility cloaks.
© 2011 OSA
OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(350.5500) Other areas of optics : Propagation
(350.7420) Other areas of optics : Waves
(350.4855) Other areas of optics : Optical tweezers or optical manipulation
ToC Category:
Physical Optics
Original Manuscript: February 10, 2011
Revised Manuscript: March 26, 2011
Manuscript Accepted: March 29, 2011
Published: April 18, 2011
Virtual Issues
Vol. 6, Iss. 5 Virtual Journal for Biomedical Optics
Mohamed A. Salem and Hakan Bağcı, "Energy flow characteristics of vector X-Waves," Opt. Express 19, 8526-8532 (2011)
Sort: Year | Journal | Reset
1. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008). [CrossRef]
2. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989). [CrossRef] [PubMed]
3. R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997). [CrossRef]
4. E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron.
9, 59–73 (2003). [CrossRef]
5. A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004). [CrossRef]
6. A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44,
103–129 (2004). [CrossRef]
7. E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).
8. Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011). [CrossRef] [PubMed]
9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
[CrossRef] [PubMed]
10. B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009). [CrossRef]
11. H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011). [CrossRef]
12. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007). [CrossRef]
13. E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989). [CrossRef]
14. A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 9th ed. (Dover, 1964), chap. 15.
16. J. D. Jackson, Classical Electrodynamics 3rd ed. (Wiley, 1999).
17. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
39, 19–31 (1992). [CrossRef] [PubMed]
18. M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010). [CrossRef] [PubMed]
OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies.
In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.
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Psychology: Debates · Journals · Psychologists
A model is a representation of some aspect of reality that is created for a specific purpose, and if the representation involved mathematics, it is referred to as a mathematical model. Usually, the
model is a representation of a very complex real-life situation that is highly simplified to promote understanding, and there is no single "correct" model for any situation. For example, maps and
plans are models, which show a representation of reality that includes only the features relevant to the purpose of the map. By including different details, maps are produced for different purposes.
Models do not have to be absolutely correct; they only have to work well enough to be truly useful. For example, linear functions are often used in modelling over short ranges where linear
approximation to the true state of affairs is reasonable, because they are mathematically simpler than most other functions. One description of the main stages of the process of mathematical
modelling is the following: Specify the purpose (e.g. to make predictions, improve understanding); Create a model (e.g. identify relevant variables, specify assumptions, write the mathematical
function); Do the mathematics ; Interpret the results; Evaluate the model (e.g. revise assumptions and adjust the model). An abstract model (or conceptual model) is a theoretical construct that
represents physical, biological or social processes, with a set of variables and a set of logical and quantitative relationships between them. Models in this sense are constructed to enable reasoning
within an idealized logical framework about these processes and are an important component of scientific theories. Idealized here means that the model may make explicit assumptions that are known to
be false in some detail. Such assumptions may be justified on the grounds that they simplify the model while, at the same time, allowing the production of acceptably accurate solutions, as is
illustrated below.
Mathematical models
Main article: Mathematical model
• Model of a particle in a potential field. In this model we consider a particle as being a point of mass m that describes a trajectory modelled by a function x: R → R^3 given its coordinates in
space as a function of time. The potential field is given by a function V:R^3 → R and the trajectory is a solution of the differential equation
$m \frac{d^2}{dt^2} x(t) = - \operatorname{grad} V(x(t)).$
Note this model assumes that the particle is a point mass, which is certainly known to be false in many cases where we use the model, e.g. when we use it as a model of planetary motion.
• Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labelled 1,2,...,n each with a market price p[1], p[2],..., p[n]. The consumer is
assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x[1], x[2],..., x[n] consumed. The
model further assumes that the consumer has a budget M which she uses to purchase a vector x[1], x[2],..., x[n] in such a way as to maximize U(x[1], x[2],..., x[n]). The problem of rational
behavior in this model then becomes one of constrained maximization, that is maximize
$U(x_1,x_2,\ldots, x_n)$
subject to
$\sum_{i=1}^n p_i x_i = M.$
This model has been used in models of general equilibrium theory, particularly to show existence and Pareto optimality of economic equilibria. However, the fact that this particular formulation
assigns numerical values to levels of satisfaction is a source of criticism. But this is not an essential ingredient of the theory and again, the model is an idealization.
Other types of models
These two models are examples of mathematical models; following are examples of models that are not mathematical (or at least not numerical).
• Myers-Briggs personality type. Myers-Briggs Type Indicator® is a technique that claims to produce a representation of a person's preferences, using four scales. These scales can be combined in
various ways to produce 16 personality types. Types are typically denoted by four letters--for example, INTJ (introverted intuition with extraverted thinking)--to represent a person's
preferences. This model is claimed by CPP (formerly known as Consulting Psychologists Press, Inc.) to produce a good predictor of a person's career and marriage partner preference. It should be
pointed out, see [1], that there is considerable disagreement among psychologists on whether this assessment technique (and the implied idealized personality model) is of any value.
• Model of political contagion. Some versions of this model are sometimes referred to as the domino theory. In the broadest possible terms, according to this model, political movements that take
hold in one country are likely to spread to geographically neighboring ones. This model is surprisingly popular, although as it stands, it is extremely impoverished conceptually, saying nothing
about the type of political movement, the degree of geographical proximity, the time scale at which these events take place, etc.
Use of models
The purpose of a model is to provide an argumentative framework for applying logic and mathematics that can be independently evaluated (for example by testing) and that can be applied for reasoning
in a range of situations. Models are used throughout the natural and social sciences, psychology and the philosophy of science. Some models are predominantly statistical (for example portfolio models
used in finance); others use calculus, linear algebra or convexity, see mathematical model. Of particular political significance are models used in economics, since they are used to justify decisions
regarding taxation and government spending. This often leads to hotly contested debates in the academic world as well as in the political arena; see for instance supply side economics.
Abstract models are used primarily as a reusable tool for discovering new facts, for providing systematic logical arguments as explicatory or pedagogical aids, for evaluating hypotheses
theoretically, and for devising experimental procedures to test them. Reasoning within models is determined by a set of logical principles, although rarely is the reasoning used completely
In some cases, abstract models can be used to implement computer simulations that illustrate the behavior of a system over time. Simulations are used everywhere in science, especially in economics,
engineering, biology, ecology etc., to discover the effects of changing a variable. The validity of different simulation methodologies is a subject of debate in the philosophy and methodology of
Structure of models
Main article: Conceptual schema
A conceptual model is a representation of some phenomenon by logical and mathematical objects such as functions, relations, tables, stochastic processes, formulas, axiom systems, rules of inference
etc. A conceptual model has an ontology, that is the set of expressions in the model which are intended to denote some aspect of the modeled object. Here we are deliberately vague as to how
expressions are constructed in a model and particularly what the logical structure of formulas in a model actually is. In fact, we have made no assumption that models are encoded in any formal
logical system at all, although we briefly address this issue below. Moreover, the definition given here is oblivious about whether two expressions really should denote the same thing. Note that this
notion of ontology is different from (and weaker than) ontology as is sometimes understood in philosophy; in our sense there is no claim that the expressions actually denote anything which exists
physically or spatio-temporally (to use W. Quine's formulation).
For example, a stochastic model of stock prices includes in its ontology a sample space, random variables, the mean and variance of stock prices, various regression coefficients etc. Models of
quantum mechanics in which pure states are represented as unit vectors in a Hilbert space include in their ontologies observables, dynamics, measurement operators etc. It is possible that observables
and states of quantum mechanics are as physically real as the electrons they model, but by adopting this purely formal notion of ontology we avoid altogether this question.
Modeling, especially scientific modeling refers to the process of generating a model as a conceptual representation of some phenomenon as discussed above. Typically a model will refer only to some
aspects of the phenomenon in question, and two models of the same phenomenon may be essentially different, that is in which the difference is more than just a simple renaming. This may be due to
differing requirements of the model's end users or to conceptual or esthetic differences by the modellers and decisions made during the modeling process. Esthetic considerations that may influence
the structure of a model might be the modeller's preference for a reduced ontology, preferences regarding probabilistic models vis-a-vis deterministic ones, discrete vs continuous time etc. For this
reason users of a model need to understand the model's original purpose and the assumptions of its validity. Having found a model for some desired aspect of reality, it can serve as the basis for
simulation, the only way for non-invasive examination of physical reality besides real-world experiments.
See also
• I. Briggs Myers with P. Myers, Gifts Differing. Understanding Personality Type, CPP Books, 1993.
• K. Lancaster, Mathematical Economics, Dover Publications, 1968.
• W. Quine, From a Logical Point of View, Harper Torchbooks, 1961.
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Force due to velocity
It's more than just wording, shamrock, and we're just trying to be helpful. Some of the issues:
is there a way to calculate the force that an object will exert when it is moving at a certain velocity... force equals mass times acceleration, but if an object is travelling at a high rate of speed
it will exert a greater force than if it were at rest even if it is not accelerating. [emphasis added]
No, an object only exerts a force when it is accelerating.
The only way i can see to find the force is to find how long the impact takes which would be a very small amount of time and then divide your velocity by that to get your deceleration... so then your
deceleration/acceleration times the mass of the object is the force it will exert when it strikes something... is there a way to calculate the force without knowing how long the impact takes?
The first part is correct, which should tell you clearly that the answer to the question at the end is no.
so theres no way to figure out what the average force would be without the time duration of the impact?
Same question as above, same answer: no. | {"url":"http://www.physicsforums.com/showthread.php?p=1755960","timestamp":"2014-04-19T04:39:36Z","content_type":null,"content_length":"70618","record_id":"<urn:uuid:8608931c-786d-473f-a53f-6c27a464670a>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00405-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOCS 2010 Accepted Papers (with pdf files)
FOCS 2010 accepted paper list is here and list with abstracts is here. Following are PDF pointers to online versions.
1. Settling the Polynomial Learnability of Mixtures of Gaussians [arXiv]
Authors: Ankur Moitra (MIT) and Gregory Valiant (UC Berkeley)
2. Solving linear systems through nested dissection [pdf]
Authors: Noga Alon (Tel Aviv University) Raphael Yuster (University of Haifa)
3. Improved Bounds for Geometric Permutations [arXiv]
Authors: Natan Rubin, Haim Kaplan and Micha Sharir (Tel Aviv University)
4. Constructive Algorithms for Discrepancy Minimization [arXiv]
Authors: Nikhil Bansal (IBM Research)
5. An efficient test for product states, with applications to quantum Merlin-Arthur games [arXiv]
Authors:Aram W. Harrow: Department of Mathematics, University of Bristol & Departmenty of Computer Science and Engineering, University of Washington and Ashley Montanaro: Department of Computer
Science, University of Bristol and Department of Applied Mathematics and Theoretical Physics, University of Cambridge
6. Replacement Paths via Fast Matrix Multiplication [pdf]
Authors: Oren Weimann (Weizmann Institute of Science) Raphael Yuster (University of Haifa)
7. Logspace Versions of the Theorems of Bodlaender and Courcelle [ECCC]
Authors: Michael Elberfeld, Andreas Jakoby, Till Tantau, Institute of Theoretical Computer Science, University of Lubeck, Germany.
8. Impossibility of Differentially Private Universally Optimal Mechanisms [arXiv]
Authors : Kobbi Nissim (Microsoft AI, Israel, and Dept. of Computer Science, Ben-Gurion University) Hai Brenner (Dept. of Mathematics, Ben-Gurion University)
9. Determinant Sums for Undirected Hamiltonicity [arXiv]
Author: Andreas Bjorklund, Lund University, Department of Computer Science, P.O.Box 118, SE-22100 Lund, Sweden
10. A non-linear lower bound for planar epsilon-nets [pdf]
Author: Noga Alon, Tel Aviv University and IAS, Princeton
11. Pseudorandom generators for CC_0[p] and the Fourier spectrum of low-degree polynomials over finite fields [ECCC]
Authors: Shachar Lovett, The Weizmann Institute of Science, Partha Mukhopadhyay, Technion – Israel Institute of Technology, Amir Shpilka, Technion – Israel Institute of Technology
12. A lower bound for dynamic approximate membership data structures [ECCC]
Authors: Shachar Lovett and Ely Porat
13. A Decidable Dichotomy Theorem on Directed Graph Homomorphisms with Non-negative Weights [pdf]
Authors: Jin-Yi Cai: University of Wisconsin – Madison Xi Chen: University of Southern California
14. The Geometry of Manipulation – a Quantitative Proof of the Gibbard Satterthwaite Theorem [arXiv]
Authors: Marcus Isaksson and Guy Kindler and Elchanan Mossel
15. Optimal Testing of Reed-Muller Codes [ECCC]
Authors: Arnab Bhattacharyya (MIT) Swastik Kopparty (MIT) Grant Schoenebeck (UC Berkeley) Madhu Sudan (Microsoft Research New England) David Zuckerman (UT Austin)
16. Pseudorandom generators for regular branching programs. [ECCC]
Authors: Mark Braverman (MSR New England and University of Toronto), Anup Rao (University of Washington), Ran Raz (Weizmann Institute of Science) and Amir Yehudayoff (IAS and Technion)
17. Local list decoding with a constant number of queries [ECCC]
Authors: Avraham Ben-Aroya and Klim Efremenko and Amnon Ta-Shma
18. New Constructive Aspects of the Lovasz Local Lemma [arXiv]
Authors: Bernhard Haeupler (MIT) Barna Saha (University of Maryland) and Aravind Srinivasan (University of Maryland)
19. Matching vector codes [html]
Authors: Zeev Dvir, Princeton University Parikshit Gopalan, Microsoft Research Sergey Yekhanin, Microsoft Research
20. Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions
Author: Matthew Andrews, Alcatel-Lucent Bell Laboratories, 600-700 Mountain Avenue, Murray Hill, NJ 07974
21. The complexity of distributions [pdf]
Author: Emanuele Viola, Northeastern University
22. All-Pairs Shortest Paths in $O(n^2)$ Time With High Probability [video]
Authors: Yuval Peres, Microsoft Research, Redmond, Dmitry Sotnikov, Tel Aviv University, Benny Sudakov, UCLA, Uri Zwick, Tel Aviv University
23. Computational Transition at the Uniqueness Threshold [arXiv]
Author: Allan Sly, Microsoft Research, Redmond
24. Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature [arXiv]
Authors: David Doty, University of Western Ontario, Matthew J. Patitz, University of Texas — Pan American, Dustin Reishus, University of Southern California, Robert T. Schweller, University of
Texas — Pan American, Scott M. Summers, University of Wisconsin — Platteville
25. Subcubic Equivalences Between Path, Matrix, and Triangle Problems [pdf]
Authors: Virginia Vassilevska Williams, UC Berkeley and Ryan Williams, IBM Almaden Research Center
26. Minimum-Cost Network Design with (Dis)economies of Scale
Authors: Matthew Andrews and Spyridon Antonakopoulos and Lisa Zhang, Bell Laboratories, 600-700 Mountain Avenue, Murray Hill, NJ 07974
27. A Unified Framework for Testing Linear-Invariant Properties
Authors: Arnab Bhattacharyya (MIT), Elena Grigorescu (MIT), Asaf Shapira (Georgia Tech)
28. Information Cost Tradeoffs for Augmented Index and Streaming Language Recognition [arXiv]
Authors: Amit Chakrabarti, Dartmouth College, Graham Cormode, AT&T Labs — Research, Ranganath Kondapally, Dartmouth College, Andrew McGregor, University of Massachusetts, Amherst
29. Optimal stochastic planarization [arXiv]
Anastasios Sidiropoulos , Toyota Technological Institute at Chicago
30. Bounds on Monotone Switching Networks for Directed Connectivity [arXiv]
Author: Aaron Potechin, MIT
31. A Fourier-analytic approach to Reed-Muller decoding [html]
Parikshit Gopalan, Microsoft Research Silicon Valley.
32. Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP [arXiv]
Authors: Jin-Yi Cai, University of Wisconsin-Madison and Beijing University, Pinyan Lu, Microsoft Research Asia, and Mingji Xia, Institute of Software, Chinese Academy of Sciences
33. Backyard Cuckoo Hashing: Constant Worst-Case Operations with a Succinct Representation [arXiv]
Authors: Yuriy Arbitman and Moni Naor and Gil Segev, Weizmann Institute of Science.
34. Vertex Sparsifiers and Abstract Rounding Algorithms [arXiv]
Moses Charikar (Princeton University), Tom Leighton (MIT and Akamai Technologies, Inc), Shi Li (Princeton University), Ankur Moitra (MIT)
35. Deciding first-order properties for sparse graphs [pdf]
Authors: Zdenek Dvorak (Charles University, Prague), Daniel Kral (Charles University, Prague), Robin Thomas (Georgia Institute of Technology, Atlanta)
36. Clustering with Spectral Norm and the k-means Algorithm [arXiv]
Amit Kumar (IIT Delhi) and Ravindran Kannan (Microsoft Research India Lab, Bangalore)
37. Testing Properties of Sparse Images [pdf]
Authors: Dana Ron and Gilad Tsur, Department of Electrical Engineering – Systems, Tel-Aviv University.
38. Frugal Mechanism Design via Spectral Techniques [arXiv]
Authors: Ning Chen and Edith Elkind and Nick Gravin and Fedor Petrov
39. A separator theorem in minor-closed classes
Authors Ken-ichi Kawarabayashi (National Institute of Informatics, Japan) and Bruce Reed (McGill University, Canada, and Sophia Antipolis, France)
40. On the Insecurity of Parallel Repetition for Leakage Resilience [pdf]
Allison Lewko (University of Texas at Austin) and Brent Waters (University of Texas at Austin)
41. Approaching optimality for solving SDD linear systems [pdf]
Ioannis Koutis†, Gary L. Miller and Richard Peng, Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213
42. The subexponential upper bound for on-line chain partitioning problem
Authors: Bart\l{}omiej Bosek – Jagiellonian University, Theoretical Computer Science, ul. prof. Stanis\l{}awa \L{}ojasiewicza 6, Krak\’{o}w 30-348, Poland and Tomasz Krawczyk – Jagiellonian
University, Theoretical Computer Science, ul. prof. Stanis\l{}awa \L{}ojasiewicza 6, Krak\’{o}w 30-348, Poland.
43. One Tree Suffices: A Simultaneous O(1)-Approximation for Single-Sink Buy-at-Bulk [arXiv]
Authors: Ashish Goel (Stanford University) Ian Post (Stanford University)
44. On the Queue Number of Planar Graphs [pdf]
Authors: Giuseppe Di Battista, Roma Tre University, Italy, Fabrizio Frati, Roma Tre University, Italy, J\’anos Pach, EPFL Lausanne, Switzerland
45. Cryptography Against Continuous Memory Attacks [ePrint]
AUthors: Yevgeniy Dodis and Kristiyan Haralambiev and Adriana Lopez-Alt and Daniel Wich
46. Hardness of Finding Independent Sets in Almost 3-Colorable Graphs
Authors: Irit Dinur and Subhash Khot and Will Perkins and Muli Safra
47. Min st-Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time [arXiv]
Authors: Glencora Borradaile: School of Electrical Engineering and Computer Science, Oregon State University; Piotr Sankowski: Institute of Informatics, University of Warsaw and Department of
Computer and System Science, Sapienza University of Rome; Christian Wulff-Nilsen: Department of Computer Science, University of Copenhagen
48. Codes for Computationally Simple Channels: Explicit Constructions with Optimal Rate [arXiv]
Authors: Venkatesan Guruswami (Carnegie Mellon University) Adam Smith (Pennsylvania State University)
49. Polynomial Learning of Distribution Families [arXiv]
Mikhail Belkin (Ohio State University) and Kaushik Sinha (Ohio State University)
50. Overcoming the Hole in the Bucket: Public-Key Cryptography Resilient to Continual Memory Leakage [ePrint]
Authors: Zvika Brakerski and Yael Tauman Kalai and Jonathan Katz and Vinod Vaikuntanathan
51. Sequential Rationality in Cryptographic Protocols [arXiv]
Authors: Ronen Gradwohl, Kellogg School of Management, Northwestern University; Noam Livne, Weizmann Institute of Science; Alon Rosen, Herzliya IDC
52. Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures [pdf with different title]
Authors: Chandra Chekuri Dept. of Computer Science, Univ. of Illinois; Jan Vondrak IBM Almaden Research Center; Rico Zenklusen Dept. of Mathematics, MIT;
53. From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-box Identity Test for Depth-3 Circuits [ECCC]
Authors: Nitin Saxena, Hausdorff Center for Mathematics, Bonn; C. Seshadhri, IBM Almaden
54. The Geometry of Scheduling [arXiv]
Authors: Nikhil Bansal (IBM Research) and Kirk Pruhs (Univ. of Pittsburgh)
55. Stability yields a PTAS for k-Median and k-Means Clustering [pdf]
Authors: Pranjal Awasthi and Avrim Blum and Or Sheffet, Carnegie Mellon University
56. Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability
Authors:Konstantin Makarychev (IBM Research) and Yury Makarychev (TTIC).
57. Polylogarithmic Approximation for Edit Distance and the Asymmetric Query Complexity [arXiv]
Authors: Alexandr Andoni (Princeton University & Center for Computational Intractability) Robert Krauthgamer (Weizmann Institute) Krzysztof Onak (Massachusetts Institute of Technology)
58. Fast approximation algorithms for flow and cut-based problems in undirected graphs [pdf]
Author: Aleksander Madry, MIT
59. Efficient volume sampling for row/column subset selection [pdf]
Amit Deshpande, Microsoft Research India and Luis Rademacher, Computer Science and Engineering, Ohio State University
60. Position-Based Quantum Cryptography [arXiv]
Authors: Nishanth Chandran (UCLA), Serge Fehr (CWI), Ran Gelles (UCLA), Vipul Goyal (Microsoft Research, India), Rafail Ostrovsky (UCLA)
61. Estimating the longest increasing sequence in polylogarithmic time [pdf]
Authors: Michael Saks, Rutgers University and C. Seshadhri, IBM Almaden
62. The Monotone Complexity of k-Clique on Random Graphs [pdf]
Author: Benjamin Rossman, MIT
63. Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts [arXiv]
Authors: David Kempe and Mahyar Salek and Cristopher Moore
64. The Coin Problem, and Pseudorandomness for Branching Programs
Authors: Joshua Brody and Elad Verbin
65. Agnostically learning under permutation invariant distributions
Author: Karl Wimmer (Duquesne University)
66. Approximating Maximum Weight Matching in Near-linear Time [pdf]
Ran Duan, University of Michigan and Seth Pettie, University of Michigan
67. Bounded Independence Fools Degree-2 Threshold Functions [ECCC]
Authors: Ilias Diakonikolas (Columbia), Daniel M. Kane (Harvard), Jelani Nelson (MIT)
68. Boosting and Differential Privacy
Authors: Cynthia Dwork (Microsoft Research), Guy Rothblum (Princeton University), Salil Vadhan (Harvard University
69. Fighting Perebor: New and Improved Algorithms for Formula and QBF Satisfiability
Author: Rahul Santhanam, University of Edinburgh
70. Lower Bounds for Near Neighbor Search via Metric Expansion [arXiv]
Authors: Rina Panigrahy (Microsoft Research Silicon Valley), Kunal Talwar (Microsoft Research Silicon Valley), Udi Wieder (Microsoft Research Silicon Valley)
71. Black-Box Randomized Reductions in Algorithmic Mechanism Design [pdf]
Authors: Shaddin Dughmi and Tim Roughgarden (Stanford University)
72. Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction [pdf]
Authors: Renato Paes Leme and Eva Tardos (Cornell)
73. Learning Convex Concepts from Gaussian Distributions with PCA [pdf]
Authors: Santosh Vempala (Georgia Tech)
74. Sublinear Optimization for Machine Learning [pdf]
Authors: Kenneth L. Clarkson and Elad Hazan and David P. Woodruff (IBM Almaden Research Center)
75. The Limits of Two-Party Differential Privacy [html]
Authors: Andrew McGregor (University of Massachusetts, Amherst) Ilya Mironov (Microsoft Research Silicon Valley) Toniann Pitassi (University of Toronto) Omer Reingold (Microsoft Research Silicon
Valley) Kunal Talwar (Microsoft Research Silicon Valley) Salil Vadhan (Harvard)
76. Distance Oracles Beyond the Thorup-Zwick Bound
Authors: Mihai Patrascu – AT&T Labs; Liam Roditty – Bar Ilan University
77. A Multiplicative Weights Mechanism for Interactive Privacy-Preserving Data Analysis
Authors: Moritz Hardt (Princeton University); Guy Rothblum (Princeton University)
78. Subexponential Algorithms for Unique Games and Related problems [pdf]
Authors: Sanjeev Arora: Princeton University and Center for Computational Intractability; Boaz Barak: Microsoft Research and Princeton University; David Steurer: Princeton University and Center
for Computational Intractability.
79. Budget Feasible Mechanisms [arXiv]
Author: Yaron Singer, UC Berkeley
80. Black-Box, Round-Efficient Secure Computation via Non-Malleability Amplification
Hoeteck Wee (Queens College, CUNY)
81. On the Computational Complexity of Coin Flipping [pdf]
Authors: Hemanta K. Maji (UIUC) Manoj Prabhakaran (UIUC) Amit Sahai (UCLA)
82. Adaptive Hardness and Composable Security in the Plain Model from Standard Assumptions
Authors: Ran Canetti, Huijia Lin, and Rafael Pass.
11 thoughts on “FOCS 2010 Accepted Papers (with pdf files)”
1. Paper 50 is available here
2. Paper 35 is available here: http://iti.mff.cuni.cz/series/files/2009/iti484.pdf
3. Pingback: FOCS accepts… « the polylogblog
4. The author’s site has a preliminary version of paper 21.
5. Pingback: FOCS 2010 accepted papers « Constraints
6. 37 can now be found on Dana’s webpage:
7. Paper 40 has been posted on the Crypto ePrint archive http://eprint.iacr.org/2010/404
8. Hi,
The submitted version of Paper 21. (Emanuele Viola) can be found here:
9. Paper 54 is available here http://arxiv.org/abs/1008.4889 | {"url":"http://kintali.wordpress.com/2010/07/01/focs-2010-accepted-papers-with-pdf-files/","timestamp":"2014-04-21T02:00:21Z","content_type":null,"content_length":"91367","record_id":"<urn:uuid:8d59b495-f79f-4b55-8623-779495a869c8>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00542-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculus Course/Sequence
In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length
of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even
positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be
excluded depending on the context.
1. 1, 2, 3, ... , n
2. 1, 3, 5, ... , 2n + 1
3. 2, 4, 6, ... , 2n
Last modified on 26 February 2011, at 19:43 | {"url":"http://en.m.wikibooks.org/wiki/Calculus_Course/Sequence","timestamp":"2014-04-20T01:26:21Z","content_type":null,"content_length":"15556","record_id":"<urn:uuid:956bebb0-7e20-4029-b744-fd2f6311a8e6>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00339-ip-10-147-4-33.ec2.internal.warc.gz"} |
FIR Example
Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | Search
An example sinewave input signal is shown in Fig.5.12, and the output of a length filter is shown in Fig.5.12. These signals were computed by the following matlab code:
Nx = 1024; % input signal length (nonzero portion)
Nh = 128; % FIR filter length
A = 1; B = ones(1,Nh); % FIR "running sum" filter
n = 0:Nx-1;
x = sin(n*2*pi*7/Nx); % input sinusoid - zero-pad it:
zp=zeros(1,Nx/2); xzp=[zp,x,zp]; nzp=[0:length(xzp)-1];
y = filter(B,A,xzp); % filtered output signal
We know that the transient response must end decay-time lasts the same amount of time after the input signal switches back to zero.
Since the coefficients of an FIR filter are also its nonzero impulse response samples, we can say that the duration of the transient response equals the length of the impulse response minus one.
For Infinite Impulse Response (IIR) filters, such as the recursive comb filter analyzed in Chapter 3, the transient response decays exponentially. This means it is never really completely finished.
In other terms, since its impulse response is infinitely long, so is its transient response, in principle. However, in practice, we treat it as finished for all practical purposes after several time
constants of decay. For example, seven time-constants of decay correspond to more than 60 dB of decay, and is a common cut-off used for audio purposes. Therefore, we can adopt decay time (or ``ring
time'') for typical audio filters. See [84]^6.5 for a detailed derivation of dB.
Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | Search [How to cite this work] [Order a printed hardcopy] [Comment on this page via email] | {"url":"https://ccrma.stanford.edu/~jos/filters/FIR_Example.html","timestamp":"2014-04-23T10:34:56Z","content_type":null,"content_length":"13332","record_id":"<urn:uuid:c1214f3c-a193-44db-bb03-d3d52a8c00ed>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00188-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: POISON PILL OF CURRENT MATHEMATICS PUBLISHED, PLUS PRINCETON
Replies: 0
Posted: Jul 7, 2013 10:53 AM
International Journal of Applied Mathematical Research, 2 (3) (2013) 387-402
©Science Publishing Corporation
The poison pill of current mathematics theory, delivered
The concordance of numbers spirals and prime numbers spirals
Vinoo Cameron
Hope research, Athens, Wisconsin, USA
E-mail: Hope99@frontier.com
The author had receded to discover the calculus of spirals, and then this discovery hit him, cutting short the calculus. The findings of concordance between natural linear numbers and prime numbers
are so blatant in the mathematics, and clearly all prime numbers can be placed by spirals by their gaps and ascension of +2 and that linear ascension of prime numbers , is not mathematics in the
overall logic as shown here. This manuscript is about the basics of the correct spiral placement of prime numbers and completely rejects the current linear mathematics with regard to Prime numbers,
even though there is some abject work on prime number distribution over the last two centuries including the work of Riemann, but all that is irrelevant with regards to the reality of numbers
mathematics. The facts are even evident on a very special ,a novel Prime number sieve of Theo Denotter , who had done this for Hope research .
The author is a physician/surgeon, who in later life decided to take a fresh look into the circus of mathematics after his son was misdiagnosed because of an error in simple mathematics related to a
torsion deformity of the spine. The author in this short manuscript is concerned about mathematics, and not its current pedigree, and current writing modes. The author is recently published and
offers a fresh look at mathematics and clearly suggests that current mathematics is all wet in its pursuit of the final discovery in mathematics. The author points out for the sake of mathematics
this perpetuated obsession that Prime numbers are somehow random by linear ascension, is Poppy cock! And yet premier universities and journals peruse it. The author in very simple mathematics,
presents a simple evidence that by definition Prime numbers cannot be random (as is vastly proven in his publications), as their gaps are rational, divisible by 2 in several ways. The mathematical
readers can deduce that by examination of the evidence presented here and the readers are referenced to the much more complex papers recently published, the understanding of which (may) be beyond
the reach of current mathematicians.
Keywords: Random prime numbers, rational prime numbers, Pythagoras triangle1:3, revision | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2580907","timestamp":"2014-04-16T14:08:18Z","content_type":null,"content_length":"16408","record_id":"<urn:uuid:daeff9b0-7874-410d-9f92-1c8afd628216>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00029-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physical Quantities and Units
Table 1 gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood
pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions.
Table 1: Fundamental SI Units
Length Mass Time Electric Current
meter (m) kilogram (kg) second (s) ampere (A)
It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure
them. The units in which they are measured are thus called fundamental units. In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric current. (Note
that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric charge, can be expressed as algebraic combinations of length,
mass, time, and current (for example, speed is length divided by time); these units are called derived units. | {"url":"http://cnx.org/content/m42091/latest/?collection=col11406/latest","timestamp":"2014-04-21T14:58:31Z","content_type":null,"content_length":"359421","record_id":"<urn:uuid:749adb45-4767-45f1-8507-e0ca36d1bead>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00483-ip-10-147-4-33.ec2.internal.warc.gz"} |
Precalculus - A Fuctional Approach to Graphing and Problem Solving
• A rational function f is the quotient of polynomial functions defined by
P(x) and D(x) so that
f (x) = P(x)/D(x) where D(x) does not equal 0
• Discontinuities of rational functions are either deleted points or vertical asymptotes.
• Asymptotes
Let f (x) = P(x)/D(x) where P is a polynomial function with leading coefficient P and D is a polynomial function with leading coefficient d. Moreover, P(x) and D(x) and have no common factors
The vertical asymptote is the line x = c where D(c) = 0. That is, values that cause division by 0.
If P(x) has degree m and D(x) has degree n , then the horizontal asymptote is y = 0 if m < n and it is the horizontal line y = P /d
if m = n.
If the degree of P is one more than the degree of D, then carry out the division to find the oblique asymptote y = mx + b.
< Back to Section 1
© 2011 Karl J. Smith. All rights reserved. | {"url":"http://www.mathnature.com/precalc/chapter4/essential4-1.htm","timestamp":"2014-04-20T11:08:26Z","content_type":null,"content_length":"10857","record_id":"<urn:uuid:b4e1e659-393b-4832-99ea-6ef9f4be383f>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00057-ip-10-147-4-33.ec2.internal.warc.gz"} |
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a 20kg Block rests on a ramp incline 30 degrees upward from the horizontal. the surface of the ramp has coefficients of friction Us=.15 Uk=.11 someone attempts to pull the block up the rapm with a
150N paralell to the incline. Does the block move? if yes what is the acceleration?
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I know it moves and that the answer is supposed to be a=1.661 m/s^2 but i get 6.96 m/s^2 nevermind i figured it out. i was not subtracting mgcos(30) from my FT
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is replying to Can someone tell me what button the professor is hitting...
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Equations of Motion
Time Evolution In Macroscopic Systems.
... Department of Physics & Astronomy, University of Wyoming
... Laramie, Wyoming 82071
Abstract. The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found
that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in
physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for
nonequilibrium thermodynamics is also outlined.
1. Introduction
The empirical statement of the Second Law of thermodynamics by Clausius (1865) is
S(initial) £ S(final) , (1)
where S is the total entropy of everything taking part in the process under consideration, and the entropy for a single closed system is defined to within an additive constant by
ó 2 ó 2
S(2)-S(1) = õ dQ T = õ C(T) dT T , (2)
where C(T) is a heat capacity. The integral in (2) is to be taken over a reversible path, a locus of thermal equilibrium states connecting the macroscopic states 1 and 2, which is necessary because
the absolute temperature T is not defined for other than equilibrium states; dQ represents the net thermal energy added to or taken from the system in the process. As a consequence, entropy is
defined in classical thermodynamics only for states of thermal equilibrium. Equation (1) states that in the change from one state of thermal equilibrium to another along a reversible path the total
entropy of all bodies involved cannot decrease; if it increases, the process is irreversible. That is, the integral provides a lower bound on the change in entropy. This phenomenological entropy is
to be found from experimental measurements with calorimeters and thermometers, so that by construction it is a function only of the macroscopic parameters defining the macroscopic state of a system,
S(V,T,N), say, where V and N are the system volume and particle number, respectively. It makes no reference to microscopic variables or probabilities, nor can any explicit time dependence be
justified in the context of classical thermodynamics. Equation (1) is a statement of macroscopic phenomenology that cannot be proved true solely as a consequence of the microscopic dynamical laws of
physics, as appreciated already by Boltzmann (1895): ``The Second Law can never be proved mathematically by means of the equations of dynamics alone." (Nor, for that matter, can the First Law!)
Theoretical definitions of entropy were first given in the context of statistical mechanics by Boltzmann and Gibbs, and these efforts culminated in the formal definition of equilibrium ultimately
given by Gibbs (1902) in terms of his variational principle. In Part I (Grandy, 2003, preceding paper ^1) we observed that the latter is a special case of a more general principle of maximum
information entropy (PME), and in equilibrium it is that maximum subject to macroscopic constraints that is identified with the experimental entropy of (2). One of the dominant concerns in
statistical mechanics and thermodynamics has long been that of extending these notions unambiguously to nonequilibrium phenomena and irreversible processes, and it is that issue we shall address in
this work.
How is one to define a time-dependent S(t) for nonequilibrium states? Do there even exist sensible physical definitions of experimental and theoretical `entropy' analogous to those describing
equilibrium states? Other than S(t) and the density matrix r(t), what other key parameters might be essential to a complete description of nonequilibrium? These and other questions have been debated,
sometimes heatedly, for over a century without any broad consensus having been reached; perhaps a first step toward clarifying the issue should be to understand the source of the differences of
opinion that lead to so many different points of view. One problem, of course, is a lack of experimental guidance in determining those features of the phenomena that are really of fundamental
importance, and associated with this has been the necessary restriction of theories to linear departures from equilibrium, owing to enormous calculational difficulties with the appropriate nonlinear
forms. What happens, then, is that many theoretical descriptions of nonequilibrium systems tend to predict similar results in the linear domain and there is little to distinguish any fundamental
differences that get to deeper matters.
In view of these obstacles it may be useful to look at the problem from a different perspective. Such sharp disagreements would seem to arise from different hidden premises in various approaches to
nonequilibrium statistical mechanics, and we suggest here that these have much to do with differing views of the underlying probability theory and its precise role. We initiated an examination of
this point in I, which culminated in the expression (I-40) as the appropriate form of the equation of motion for the density matrix. Our purpose here is to apply the implications of that result to a
further study of time varying macroscopic systems.
As a preliminary step it might be helpful to note some overall features of entropy and nonequilibrium processes that have to be considered in any approach to the problem of generalizing S. Suppose we
prepare a system in a nonequilibrium state by applying an external force of some kind that substantially perturbs the equilibrium system, possibly increasing its energy and adding matter to it, for
example. This state is defined by removing the external source at time t=t[0], and at that instant it is described by a density matrix r(t[0]). Whether or not we can define a physical entropy at that
time, we can certainly compute the information entropy of that nonequilibrium state as S[I](t[0])=-kTr[r(t[0])lnr(t[0])]. Because the system is now isolated, r(t) can only evolve from r(t[0]) by
unitary transformation and S[I] remains constant into the future. ^2 What happens next?
At the cutoff t=t[0] the entropy S[I](t[0]) refers only to the nonequilibrium state at that time. In the absence of any other external influences we expect the system to relax into a new equilibrium
state, for no other reason than it is the one that can be realized in the overwhelmingly greatest number of ways subject to the appropriate macroscopic constraints. The interesting thing is that
often those constraints are already fixed once the external sources are removed, so that the total energy, particle number, volume, etc. at t[0] are now determined for t > t[0]. Density
inhomogeneities may remain, of course, which will relax to uniformity over the relaxation period, or to equilibrium distributions in a static external field. The entropy of the final equilibrium
state is definitely not S[I](t[0]), but it is in principle known well before equilibrium is reached: it is the maximum of the information entropy subject to constraints provided by the values of
those thermodynamic variables at t=t[0]. We may or may not know these values, of course, although a proper theory might predict them; but once the final macrostate is established they can be measured
and a new r[f] calculated by means of the PME, and hence a new entropy predicted for comparison with the experimental form of Clausius; indeed, in equilibrium the Clausius entropy (2) is an upper
bound for S[I]. Thus, in this relaxation mode we may not see a nice, continuous, monotonically increasing entropy function that can be followed into the equilibrium state; but that's not too
surprising, given that we know r(t[0]) cannot evolve unitarily into r[f]. (More about relaxation later.) There remains a significant dynamical evolution during this relaxation period, but it is
primarily on a microscopic level; its macroscopic manifestation is to be found in the relaxation time, and the possible observation of decaying currents. One thing we might compute and measure in
this mode is that relaxation time, which does not necessarily have an immediate connection with entropy. (There may, however, exist a `relaxation entropy' associated with the redistribution of
energy, say, during the relaxation period.)
Ironically, the equilibrium state described so well by classical thermodynamics is essentially a dead end; it is a singular limit in the sense discussed by Berry (2002). Equilibrium is actually a
very special, ideal state, for most systems are not usually in equilibrium, at least not completely. As external influences, and therefore time variations, become smaller, the system still remains in
a nonequilibrium state evolving in time. In the limit there is a discontinuous qualitative change in the macroscopic system and its description. That is, there is no longer either an `arrow of time'
or a past history ^3, and the main role of the theory is to compare neighboring states of thermal equilibrium without regard for how those states might have been prepared.
It has long been understood, though not widely, that entropy is not a property of a physical system per se, but of the thermodynamic system describing it, and the latter is defined by the macroscopic
constraints imposed. The above remarks, however, lead us to view entropy more as a property of the macrostate, or of the processes taking place in a system. In the equilibrium state these
distinctions are blurred, because the thermodynamic system and the macrostate appear to be one and the same thing and there are no time-dependent macroscopic processes. It will be our goal in the
following paragraphs to clarify these comments, as well as to provide an unambiguous definition of entropy in nonequilibrium systems, and to understand the possibly very different roles that the
entropy concept plays in the two states.
2. Some Preliminary Extensions of the Equilibrium Theory
In I we briefly outlined the variational method of constructing an initial density matrix that could then evolve in time via the appropriate equations of motion. A principal application of that
construction is to equilibrium systems, in which case the quantum form of (I-14) becomes, as in (I-10),
r = 1 Z e^-bH , Z(b)=Tre^-bH , (3)
where H is the system Hamiltonian and b = (kT)^-1. But, if there is no restriction to constants of the motion, the resulting state described by (3) could just as well be one of nonequilibrium based
on information at some particular time. Data given only at a single point in space and time, however, can hardly serve to characterize a system whose properties are varying over a space-time region,
so a first generalization of the technique is to information available over such regions. Thus, the main task in this scenario is to gather information that varies in both space and time and
incorporate it into a density matrix describing a nonequilibrium state. Given an arbitrary but definite thermokinetic history, we can look for what general behavior of the system can be deduced from
only this. The essential aspects of this approach were first expounded by Jaynes (1963, 1967, 1979).
To illustrate the method of information gathering, consider a system with a fixed time-independent Hamiltonian and suppose the data to be given over a space-time region R(x,t) in the form of an
expectation value of a Heisenberg operator F(x,t). We are reminded that the full equation of motion for such operators, if they are also explicitly time varying, is
i ℏ =[F,H]+¶[t] F . (4)
When the data vary continuously over R their sum becomes an integral and there is a distinct Lagrange multiplier for each space-time point. Maximization of the entropy subject to the constraint
provided by that information leads to a density matrix describing this macrostate:
é ó ù
r = 1 Z exp ë - õ l(x,t)F(x,t) d^3x dt û , (5)
é ó ù
Z[l(x,t)]=Trexp ë - õ l(x,t)F(x,t) d^3x dt û (6)
is now the partition functional. The Lagrange multiplier function is identified as the solution of the functional differential equation
áF(x,t)ñ º Tr[rF(x,t)]=- d dl(x,t) lnZ , (x,t) Î R , (7)
and is defined only in the region R. Note carefully that the data set denoted by áF(x,t)ñ is a numerical quantity that has been equated to an expectation value to incorporate it into a density
matrix. Any other operator J(x,t), including J=F, is determined at any other space-time point (x,t) as usual by
áJ(x,t)ñ = Tr é rJ(x,t) ù =Tr é r(t)J(x) ù . (8)
ë û ë û
That is, the system with fixed H still evolves unitarily from the initial nonequilibrium state (5); although r surely will no longer commute with H, its eigenvalues nevertheless remain unchanged.
Inclusion of a number of operators F[k], each with its own information-gathering region R[k] and its own Lagrange multiplier function l[k], is straightforward. If F is actually time independent an
equilibrium distribution of the form (3) results, and a further removal of spatial dependence brings us back to the canonical distribution of the original PME. But the full form (5) illustrates how r
naturally incorporates memory effects while placing no restrictions on spatial or temporal scales. Nor are there any issues of retardation, for example, since the procedure is a matter of inference,
not dynamics (at this point).
Some further discussion is required here. The density matrix r in (5) is not a function of space and time; it merely provides an initial nonequilibrium distribution corresponding to data áF(x,t)ñ Î R
. Lack of any other information outside R - in the future, say - may tend to render r less and less reliable, and the quality of predictions may deteriorate (fading memory). The maximum entropy
itself is a functional of the initial values áF[k](x,t)ñ Î R[k] and follows from substitution of (5) into the information entropy:
S[noneq][{áF[k]ñ}] º klnZ[{l[k]}]+k å õ l[k](x,t)áF[k](x,t)ñ d^3x dt . (9)
k R[k]
Although there is no obvious connection of S[noneq] with the thermodynamic entropy, it does provide a measure of the number of microscopic states consistent with the history of a system over the R[k]
(x,t); it might thus be interpreted as the physical entropy of the initial nonequilibrium state (5). If we visualize the evolution of a microstate as a path in `phase space-time', then S[noneq] is
the cross section of a tube formed by all paths by which the given history could have been realized, a natural extension of Boltzmann's S[B]=klnW, where W is a measure of the set of microscopic
states compatible with the macroscopic constraints on the system. In this sense S[noneq] governs the theory of irreversible processes in much the same way as the Lagrangian governs mechanical
processes. The role of entropy is thus greatly expanded to describe not only the present nonequilibrium state, but also the recent thermokinetic history leading to that state. We begin to see that
here, unlike the equilibrium situation, entropy is intimately related to processes.
If the information-gathering region R is simply a time interval we arrive at the initial state r(t[0]) considered in the previous section. Restriction of R to only a spatial region leads to a
description of inhomogeneous systems. For example, specifying the particle number density án(r)ñ throughout the system, in addition to H, constitutes a separate piece of data at each point in the
volume, and hence requires a corresponding Lagrange multiplier at each point. The distribution (3) is then replaced by
r = 1 Z exp é -bH+ ó l(r¢)n(r¢) d^3r¢ ù , (10)
ë õ û
which reduces to the grand canonical distribution if n(r) is in fact spatially constant throughout V, or if only the volume integral of n(r) is specified. A similar expression is obtained if, rather
than specifying or measuring án(r)ñ, the inhomogeneity is introduced by means of an external field coupled to n(r). In that case l(r) is given as a field strength and án(r)ñ is to be determined; that
is, l is taken as an independent variable. Extensive application of (10) to inhomogeneous systems is given in the review by Evans (1979).
To this point there has been no mention of dynamic time evolution; we have only described how to construct a single, though arbitrary, nonequilibrium macrostate based on data ranging over a
space-time region. A first step away from this restriction is to consider steady-state systems, in which there may be currents, but all variables are time independent. The main dynamical features of
the equilibrium state are that it deals only with constants of the motion, among which is the density matrix itself: [H,r]=0. These constraints characterize the time-invariant state of an isolated
(or closed) system, because the vanishing of the commutator implies that r commutes with the time-evolution operator, so that all expectation values are constant in time. The time-invariant state of
an open system is also stationary, but H almost certainly will not commute with the operators {F[k]} defining that state, and hence not with r. Nevertheless, we can add that constraint explicitly as
the definition of a steady-state probability distribution, and the requirement that [r,H]=0 leads to the result that only that part of F[k] that is diagonal in the energy representation is to be
included in r. ^4 It is reasonably straightforward to show (e.g., Grandy, 1988) that a representation for the diagonal part of an operator is given by
ó 0
=F- lim õ e^et ¶[t] F(x,t) dt
e® 0^+ -¥
ó 0
= lim e õ e^et F(x,t) dt
e® 0^+ -¥
ó 0
= lim 1 t õ F(x,t) dt ,
t®¥ -t
where the time dependence of F is determined by (4), and e > 0. The second line follows from an integration by parts; the third is essentially Abel's theorem and equates the diagonal part with a time
average over the past, which is what we might expect for a stationary process. That is, F^d is that part of F that remains constant under a unitary transformation generated by H.
Consider a number of operators F[k](x) defining a steady-state process. Then the steady-state distribution r[ss] is simply a modification of that described by (5) and (6):
é ó ù
r[ss] = 1 Z[ss] exp ë - å õ l[k](x)F^d[k](x) d^3x û , (12)
k R[k]
é ó ù
Z[ss][l(x)]=Trexp ë - å õ l[k](x)F^d[k](x) d^3x û . (13)
k R[k]
We illustrate some applications of these expressions further on, but note that in their full nonlinear form they present formidable difficulties in calculations.
This last caveat suggests that we first examine small departures from equilibrium, much in the spirit of Eqs.(I-17)-(I-19). Suppose the equilibrium distribution to be based on expectation values of
two variables, áfñ and ágñ, with corresponding Lagrange multipliers l[f], l[g]. We also suppose that no generalized work is being done on the system, so that only `heat-like' sources may operate. A
small change from the equilibrium distribution can be characterized by small changes in the Lagrange multipliers, which in turn will induce small variations in the expectation values. Thus,
= ¶áfñ ¶l[f] dl[f]+ ¶áfñ ¶l[g] dl[g] ,
= ¶ágñ ¶l[f] dl[f]+ ¶ágñ ¶l[g] dl[g] .
But from (I-13b) and (I-14)) the negatives of these derivatives are just the covariances of f and g,
so (14) reduce to the matrix equation
æ ö æ ö æ ö
ç ÷ ç ÷ ç ÷
ç ÷ =- ç ÷ ç ÷ . (16)
ç ÷ ç ÷ ç ÷
è ø è ø è ø
In references on irreversible thermodynamics (e.g., de Groot and Mazur, 1962) the quantities on the left-hand side of (16) are called fluxes, and the (-dl)s are thought of as the forces that drive
the system back to equilibrium. We can thus think of the ls as potentials that produce such forces. Linear homogeneous relations such as (16) were presumed by Onsager (1931), but here they arise
quite naturally, and in (15) we observe the celebrated Onsager reciprocity relations.
Suppose now that we add another constraint to the maximum-entropy construction by letting f be coupled to a weak thermal source. In addition, we shall specify that g is explicitly not driven, so that
any internal changes in it can only be inferred from the changes in f. We thus set dl[g]=0 in (16) and those equations reduce to
So for small variations the change in the coupled variable is essentially the source strength itself; the internal change in g is also proportional to that source strength, but modulated by the
extent to which g and f are correlated:
dágñ = K[gf] K[ff] dQ[f] , (18)
exhibiting what is sometimes referred to as mode-mode coupling. These expressions are precisely what one expects from a re-maximization of the entropy subject to a small change dáfñ. For example, if
dQ[f] > 0 and f and g are positively correlated, K[gf] > 0, then we expect increases in the expectation values of both quantities, as well as a corresponding increase in the maximum entropy.
Although this discussion of small departures from equilibrium is only a first step, it reinforces, and serves as a guide to, the important role of sources in any deeper theory. It also exhibits the
structure of the first approximation, or linearization of such a theory, which is often a necessary consideration. We return to the essential aspects of that approximation a bit later.
3. Sources and Thermal Driving
We seek a description of macroscopic nonequilibrium behavior that is generated by an arbitrary source whose precise details may be unknown. One should be able to infer the presence of such a source
from the data, and both the strength and rate of driving of that source should be all that are required for predicting reproducible effects. Given data - expectation values, say - that vary
continuously in time, we infer a source at work and expect r to be a definite function of time, possibly evolving principally by external means. In I we argued that, because all probabilities are
conditional on some kind of given information or hypothesis, P(A[i]|I) can change in time only if the information I is changing in time, while the propositions {A[i]} are taken as fixed. This then
served as the basis for an abstract model of time-dependent probabilities. With this insight we can see how the Gibbs algorithm might be extended to time-varying macroscopic systems in a
straightforward manner.
As in I, information gathered in one time interval can certainly be followed by collection in another a short time later, and can continue to be collected in a series of such intervals, the entropy
being re-maximized subject to all previous data after each interval. Now let those intervals become shorter and the intervals between them closer together, so that by an obvious limiting procedure
they all blend into one continuous interval whose upper endpoint is always the current moment. Thus, there is nothing to prevent us from imagining a situation in which our information or data are
continually changing in time. A rationalé for envisioning re-maximization to occur at every moment, rather than all at once, can be found by again appealing to Boltzmann's expression for the entropy:
S[B]=lnW. At any moment W is a measure of the phase volume of all those microstates compatible with the macroscopic constraints - and lnW is the maximum of the information entropy at that instant. As
Boltzmann realized, this is a valid representation of the maximized entropy even for a nonstationary state. It is essential to understand that W is a number representing the multiplicity of a
macrostate that changes only as a result of changing external constraints. It is not a descriptor of which microscopic arrangements are being realized by the system at the moment - there is no way we
can ascertain that - but only a measure of how many such states may be compatible with the macrostate defined by those constraints. In principle we could always compute a W for a set of values of the
macroscopic constraints without ever carrying out an experiment. Thus, we begin to see how an evolving entropy can possibly be related to the time-dependent process.
There may seem to be a problem here for someone who thinks of probabilities as real physical entities, since it might be argued that the system cannot possibly respond fast enough for W to readjust
its content instantaneously. But it is not the response of the system that is at issue here; only the set of possible microstates compatible with the present macroscopic constraints readjusts. Those
potentialities always exist and need no physical signal to be realized. A retardation problem might exist if we were trying to follow the system's changing occupation of microstates, but we are not,
because we cannot. The multiplicity W does not change just because the microstate occupied by the system changes; in equilibrium those changes go on continuously, but W remains essentially constant.
Only variations in the macroscopic constraints can change W, and those are instantaneous and lead to immediate change in the maximum information entropy S[B].
To introduce the notion of a general source let us consider a generic system described by a density matrix
and a process that drives the variable B such that an amount DB is transferred into the system. That is, B is driven by some means other than dynamically, with no obvious effective Hamiltonian. In
addition, the variable A is explicitly not driven, but can change only as a result of changes in B if A and B are correlated. Since there is no new information regarding A, even though it is free to
readjust when B is changed, the Lagrange multiplier a must remain unchanged. We also add the further constraint on the process that C is to remain unchanged under transfer of DB. This is a
generalization of the scenario described by (16), and can be summarized as follows:
=- K[CB] K[CC] db , áCñ® áCñ .
This is the most general form of a constrained driven process, except for inclusion of a number of variables of each kind. Any such driving not tied to a specific dynamic term in a Hamiltonian will
be referred to as thermal driving. A variable, and therefore the system itself, is said to be thermally driven if no new variables other than those constrained experimentally are needed to
characterize the resulting state, and if the Lagrange multipliers corresponding to variables other than those specified remain constant. As discussed in I, a major difference with purely dynamic
driving is that the thermally-driven density matrix is not constrained to evolve by unitary transformation alone.
Let us suppose that the system is in thermal equilibrium with time-independent Hamiltonian in the past, and then at t=0 a source is turned on smoothly and specified to run continuously, as described
by its effect on the expectation value áF(t)ñ. That is, F(t) is given throughout the changing interval [0,t] and is specified to continue to change in a known way until further notice. ^5 Although
any complete theory of nonequilibrium must be a continuum field theory, we shall omit spatial dependence explicitly here in the interest of clarity and return to address that point later. For
convenience we consider only a single driven operator; multiple operators, both driven and constrained, are readily included. Based on the probability model of I, the PME then provides the density
matrix for thermal driving:
é ó t ù
= 1 Z[t] exp ë -bH- õ l(t¢)F(t¢) dt¢ û ,
é ó t ù
=Tr exp ë -bH- õ l(t¢)F(t¢) dt¢ û ,
and the Lagrange-multiplier function is formally obtained from
áF(t)ñ[t]=- d dl(t) lnZ[t] , (22)
for t in the driving interval. Reference to the equilibrium state is made explicit not only because it provides a measure of how far the system is removed from equilibrium, but also because it
removes all uncertainty as to the previous history of the system prior to introduction of the external source; clearly, these are not essential features of the construction.
Since r[t] can now be considered an explicit function of t, we can employ the operator identity
to compute the time derivative:
¶[t]r[t]=r[t]l(t) é áF(t)ñ[t]- F(t) ù , (23)
ë û
where the overline denotes a generalized Kubo transform with respect to the operator lnr[t]:
ó 1
F(t) º õ e^-ulnr[t] F(t)e^ulnr[t] du , (24)
which arises here from the possible noncommutativity of F(t) with itself at different times.
The expression (23) has the form of what is often called a `master equation', but it has an entirely different origin and is exact; it is, in fact, the ¶[t]r term in the equation of motion (I-40).
Because l(t) is defined only on the information-gathering interval [0,t], Eq.(23) just specifies the rate at which r[t] is changing in that interval. Although r[t] does not evolve by unitary
transformation under time-independent H in the Heisenberg picture, in this case it does evolve explicitly, and in the Schrödinger picture this time variation will be in addition to the canonical time
evolution. In turn, an unambiguous time dependence for the entropy is implied, as follows.
The theoretical maximum entropy S[t]=-kTr[r[t]lnr[t]] is obtained explicitly by substitution from (21),
ó t
1 k S[t]=lnZ[t]+báH ñ[t] + õ l(t¢)áF(t¢)ñ[t] dt¢ ; (25)
it is the continuously re-maximized information entropy. Equation (25) indicates explicitly that áH ñ[t] changes only as a result of changes in, and correlation with F. The constraint that H is
explicitly not driven implies that áH ñ[t] and áF(t¢)ñ[t] are no longer independent, and that means that l(t) cannot be determined directly from S[t] by functional differentiation in (25); this has
important consequences.
The expectation value of another operator at time t is áC ñ[t]=Tr[r[t] C], and direct differentiation yields
é × ù
=Tr ë C(t)¶[t]r[t] +r[t] (t) û
=á (t)ñ[t] -l(t)K[CF]^t(t,t) ,
where the superposed dot denotes a total time derivative. We have here introduced the covariance function
K[CF]^t(t¢,t) º á F(t¢) C(t)ñ[t]-áF(t¢)ñ[t]áC(t)ñ[t] = - dáC(t)ñ[t] dl(t) , (27)
which is a quantum mechanical generalization of the static covariance (15). Note that all of the preceding entities are completely nonlinear, in that expectation values, Kubo transforms, and
covariance functions are all written in terms of the density matrix r[t], which is the meaning of the superscript t on K[CF]^t. Although time-translation invariance is not a property of the general
nonequilibrium system, it is not difficult to show that the reciprocity relation K^t[CF](t¢,t)=K^t[FC](t,t¢) is valid.
Let us introduce a new notation into (26), which at first appears to be only a convenience:
s[C](t) º d dt áC(t)ñ[t]-á (t)ñ[t] = -l(t)K[CF]^t(t,t) . (28)
For a number of choices of C and F the equal-time covariance function vanishes, but if C=F an illuminating interpretation first noticed by Mitchell (1967) emerges:
º d dt áF(t)ñ[t]-á (t)ñ[t]
Owing to the specification of thermal driving, dáF(t)ñ[t]/dt is the total time rate-of-change of áF(t)ñ[t] in the system at time t, whereas
is the rate of change produced by internal relaxation. Hence, s[F](t) must be the rate at which F is driven or transferred by the external source, and is often what is measured or controlled
experimentally. One need know nothing else about the details of the source, because its total effect on the system is expressed by the second equality in (29), which is similar to the first line of
(17). If the source strength is given, then (29) is a nonlinear transcendental equation determining the Lagrange multiplier function l(t).
An important reason for eventually including spatial dependence is that we can now derive the macroscopic equations of motion. For example, if F(t) is one of the conserved densities e(x,t) in a
simple fluid and J(x,t) the corresponding current density, then the local microscopic continuity equation
(x,t)+Ñ·J(x,t)=0 (30)
is satisfied irrespective of the the state of the system. When this is substituted into (29) we obtain the macroscopic conservation law
d dt áe(x,t)ñ[t] +Ñ·áJ(x,t)ñ[t] = s[e](x,t) , (31)
which is completely nonlinear. Specification of sources therefore provides automatically the thermokinetic equations of motion; for example, if e is the momentum density mj(x,t), so that J is the
stress tensor T[ik], then a series of transformations turns (31) into the Navier-Stokes equations of fluid dynamics.
The notion of thermal driving provides a basis for nonequilibrium thermodynamics, which can be developed in much the same way as is done for the equilibrium theory (e.g., Grandy, 1987). As with that
case, the operator F can also depend on an external variable a, so that at time t the entropy is S[t]=S[t][áHñ[t], áF(t)ñ[t]; a]; of course, we could also include a number of other measured variables
{F[i]}, though only H and F will be employed here. But now S[t] is also a function of time and, from (25), its total time derivative is
æ ö × ó t
1 k dS[t] dt = è ¶lnZ[t] ¶a ø +b dáHñ[t] dt -l(t) õ l(t¢)K^t[FF](t,t¢) dt¢ . (32)
a 0
Although ¶[t] Z[t] contributes to
, its contribution is cancelled because
¶[t]lnZ[t] = -l(t)áF(t)ñ[t] , (33)
which also provides a novel representation for Z[t] upon integration. In principle, then, one can follow the increase (or decrease) of entropy in the presence of external sources (or sinks).
The most common type of external variable a is the system volume V, so that in the equilibrium theory (¶áHñ/¶V)dV=-P dV is an element of work. This suggests a general interpretation of the first term
on the right-hand side of (32). As an example, in the present scenario consider the simple process of an adiabatic free expansion of a gas, wherein only the work term is involved in (32). We can now
model this by specifying a form for a = V; for example,
would, for b very large, rapidly inflate the volume to double its size over an interval from t=0 to some later time t. The coefficient of
in (32) is proportional to the pressure, so that one also needs an equation of state for the gas; but usually the pressure is proportional to V^-1 and therefore decreases exponentially as well. In
the case of an ideal gas, integration of this form for
over (0,t) yields the expected change S[t]-S[0]=kNln2. This result is almost independent of the model as long as V(t) @ 2V[0].
. In this case we can also explicitly evaluate the term containing the Hamiltonian and rewrite (31) as
ó t
=-bl(t)K^t[HF](t,0)-l(t) õ l(t¢)K^t[FF](t,t¢) dt¢
where we have employed (29) and defined a new parameter
ó t
g[F](t) º b K^t[HF](t,0) K^t[FF](t,t) + õ l(t¢) K^t[FF](t,t¢) K^t[FF](t,t) dt¢ . (35)
Although this expression for g at first glance seems only a bookkeeping convenience, it is actually of some physical significance, as suggested by (20). As noted above, the thermal driving constraint
on H prevents áHñ[t] and áF(t)ñ[t] from being completely independent; indeed, neither of them is independent of áF(t¢)ñ[t]. In turn, and unlike the equilibrium case, ¶áf[m]ñ/¶l[n] and ¶l[n]/ ¶áf[m]ñ
are no longer the respective elements of a pair of mutually inverse matrices. Thus, dS[t]/dáF(t)ñ[t] does not determine l(t); rather, from (25),
ó t
dS[t] dáF(t)ñ[t] = dáHñ[t] dl(t) dl(t) dáF(t)ñ[t] + õ l(t¢) dáF(t¢)ñ[t] dl(t) dl(t) dáF(t)ñ[t] dt¢ . (36)
Owing to interdependencies we can now write dl(t)/dáF(t)ñ[t]=1/K^t[FF](t,t), and hence the right-hand side of (36) is just g[F](t), which now has the general definition
æ ö
g[F](t) º è dS[t] dáF(t)ñ[t] ø . (37)
[(thermal) || (driving)]
The subscript ``thermal driving" reminds us that this derivative is evaluated somewhat differently than in the equilibrium formalism. When the source strength s[F](t) is specified the Lagrange
multiplier itself is determined from (29).
Physically, g[F] is a transfer potential in the same sense that the ls in Eq.(16) are thought of as potentials. Just as products of potentials and expectation values appear in the structure of the
equilibrium entropy, in thermal driving the entropy production (34) is always a sum of products of transfer potentials and source terms measuring the rate of transfer. So, the entropy production is
not in general given by products of `fluxes' and `forces', and S[t] and
are not simple generalizations of equilibrium quantities. But the ordinary potentials also play another role in equilibrium: if two systems in contact can exchange energy and particles, then they are
in equilibrium if the temperatures and chemical potentials of the two are equal. Similarly, if two systems can exchange quantities F[i] under thermal driving, then the conditions for migrational
equilibrium at time t are
Migrational equilibrium in stationary processes is discussed, for example, by Tykodi (1967).
What is the physical interpretation to be given to S[t]? Clearly it refers only to the information encoded in the distribution of (21) and cannot refer to the internal entropy of the system. In
equilibrium the maximum of this information entropy is the same as the experimental entropy, but that is not necessarily the case here. For example, if the driving is removed at time t=t[1], then S[t
[1]] in (25) can only provide the entropy of that nonequilibrium state at t=t[1]; its value will remain the same during subsequent relaxation, owing to unitary time evolution. Although the maximum
information (or theoretical) entropy provides a complete description of the system based on all known physical constraints on that system, it cannot describe the ensuing relaxation, for it contains
no new information about that process. Nevertheless, S[t] does have a definite physical interpretation.
The form of s[F] in (29) suggests a natural separation of the entropy if that expression is substituted into the second line of (34):
× æ × ö
1 k t =g[F](t) è d dt áF(t)ñ[t] -á (t)ñ[t] ø . (39)
S F
has the qualitative form
, as intuition might have suggested. The first term on the right-hand side of (39) must represent the total time rate-of-change of entropy
arising from the thermal driving of F(t), whereas the second term is the rate-of-change of internal entropy
owing to relaxation. Thus, the total rate of entropy production can be written
× × ×
tot (t)= t + int (t) , (40)
S S S
where the entropy production of transfer owing to the external source,
, is given by (34). This latter quantity is a function only of the driven variable F(t), whereas the internal entropy depends on all variables, driven or not, necessary to describe the nonequilibrium
state and is determined by the various relaxation processes taking place in the system. Calculation of
, of course, depends on a rather detailed model of the system; we'll have more to say on this below. ^6
In an equilibrium system the major role of S is associated with the Second Law, and this law in its traditional form has little to say about nonequilibrium processes. In these latter processes,
however, it is
, rather than S[t] itself that plays the major role, as is seen in (34)-(37). That is,
governs the transfer process in terms of the rate of driving and the transfer potential, in much the same way that S governs the direction of changes between equilibrium states through dQ/T. In
nonequilibrium processes
also governs the rate; this is true even in the steady state when one takes into account sources and sinks.
The distinction between theoretical entropy in equilibrium scenarios and in nonequilibrium processes cannot be emphasized enough. If external forces are removed, it is a mathematical theorem that
neither r[t] nor S[t] can evolve into their equilibrium counterparts. This is a singular limit, as discussed earlier, and unless these distinctions are clearly recognized few real advances can be
made in nonequilibrium statistical mechanics.
Constant Driving Rate and Spatial Variation
To complete the general development, logical consistency requires an examination of thermal driving at a constant rate. For this purpose it will first be useful to record the generalizations of the
primary equations of thermal driving to include spatial coordinates:
¶[t]r[t] = r[t] ó l(x¢,t) é áF(x¢,t)ñ[t]- F(x¢,t) ù d^3x¢ , (41)
õ ë û
ó ó t
1 k S[t] = lnZ[t] +báHñ[t] + õ d^3x¢ õ dt¢ l(x¢,t¢)áF(x¢,t¢)ñ[t] , (42)
=-b ó l(x¢,t)K[HF]^t(x¢,t) d^3x¢
ó ó ó t
- õ d^3x^¢¢l(x^¢¢,t) õ d^3x¢ õ dt¢l(x^¢¢,t¢)K[FF]^t(x^¢¢,t;x¢,t¢) ,
s[F](x,t) = - ó l(x¢,t)K[FF]^t(x¢,t;x,t) d^3x¢ . (44)
This last expression can be inverted by introducing an inverse integral operator:
l(x,t) = - ó é K^t[FF](x¢,t;x,t) ù -1 s[F](x¢,t) d^3x¢ , (45)
õ ë û
which is a nonlinear integral equation for l(x,t). Thus, the right-hand side of (45) is really only a shorthand notation for the iterated solution. Upon substitution of (45) into (43) we find that
× ó
1 k t = õ g[F](x,t)s[F](x,t) d^3x , (46)
º b ó d^3x¢ é K^t[FF](x,t;x¢,t) ù -1 K^t[HF](x¢,t)
õ ë û
ó ó ó t é ù -1
+ õ d^3x¢ õ d^3x^¢¢ õ dt¢l(x^¢¢,t) ë K^t[FF](x,t;x¢,t) û K[FF]^t(x¢,t;x^¢¢,t¢) .
We can verify this expression for g[F] from the more general definition
æ ö
g[F](x,t) º è dS[t] dáF(x,t)ñ[t] ø , (48)
[(thermal) || (driving)]
if we note two properties of functional differentiation. First, the ordinary chain rule for partial differentiation of F[x(s),y(s)] with respect to s,
¶F ¶s = ¶F ¶x ¶x ¶s + ¶F ¶y ¶y ¶s ,
generalizes to
= ó dáG(x,t)ñ dl[F](x¢,t) dál[F](x¢,t) dáF(x,t)ñ d^3x¢
= ó K^t[GF](x¢,t;x,t) é K^t[FF](x,t;x¢,t) ù -1 d^3x¢ ,
õ ë û
for example. Second, in Eq.(42) for S[t] the upper limit t on the time integral, and the subscript on áHñ[t], prevent the functional derivative from yielding merely l(x,t), which is determined by s[F
](x,t) at any rate. Rather, we obtain (47) for g[F](x,t).
Specification of constant driving means that s[F] is constant in time, and from (29) or (44) this in turn implies that l(x,t) must actually be independent of time in the steady state. This last
assertion follows because the covariance function in these equations is time independent, owing to the re-emergence of unitary time evolution in the absence of internal time variation. That is, the
integrals in (21), generalized to include spatial variables, can now be rewritten in the form
ó ó t
õ d^3x¢l(x¢) õ F(x¢,t¢) dt¢ . (50)
But now the form of the time integral no longer makes sense in the context of time-independent driving.
If a constant rate of driving is specified as a constraint on the initial probability distribution we take this to mean that the initial data were constant in the distant past, and at least up to the
time of observation. In requiring this one faces the possibility of a divergent integral, so that it is necessary to regularize the integral, along the lines of methods often employed in quantum
field theory. In the present case we rewrite the time integral in (50) as a time average over the past:
ó 0
lim 1 t õ F(x¢,t¢) dt¢ . (51)
t®¥ -t
This, however, is just the diagonal part of the operator F(x¢) as given by Eq.(11), and hence constant driving corresponds with our definition of the steady state. In this scenario we can then
replace all the time integrations over operators by the diagonal parts of those operators and omit all time dependence. We see that, in the sense of this procedure, the steady state is also a
singular limit of the general nonequilibrium state, in that the latter does not reduce in a completely straightforward mathematical way to the former.
In the steady state we expect time derivatives of all expectation values to vanish; hence from (29) we have the further implication that the constant rate of driving is exactly balanced by the rate
of internal relaxation. This is how the system responds to steady currents.
Although there exist stationary currents within the system, the steady driving takes place in the terminal parts, or boundaries of the system, and such currents imply irreversible dissipation. There
must then be an overall rate of dissipation or entropy production generated by the external sources. This rate is provided by Eq.(46), now rewritten in the form
× ó
1 k t = õ g[F](x)s[F](x) d^3x . (52)
The general definition of g[F](x) still applies, but the explicit form is now
º b ó d^3x¢ é K^ss[F^dF^d](x;x¢) ù -1 K^ss[HF^d](x¢)
õ ë û
+ ó d^3x¢ ó d^3x^¢¢l(x^¢¢) é K^ss[F^dF^d](x;x¢) ù -1 K[F^dF^d]^ss(x¢,;x^¢¢) .
õ õ ë û
4. The Linear Approximation
Much, though not all, of the work on macroscopic nonequilibrium phenomena has of necessity centered on small departures from equilibrium, or the linear approximation, so it is of some value to
outline that reduction of the present theory and discuss briefly some applications. We envision situations in which the system has been in thermal equilibrium in the remote past and later found to
produce data of the form considered above. By considering both classes of data we obtain a measure of the departure from equilibrium. In describing the general method of linearization the character
of the perturbing term and the scenario under consideration are immaterial; hence, we can take the distribution (21) with integration limits replaced by the space-time region R as our generic model
and, for brevity, temporarily omit space dependences. ^7 Thus, we consider the model
ì ó ü
= 1 Z exp í -bH- õ l(t)F(t) dt ý ,
î R þ
ì ó ü
=Trexp í -bH- õ l(t)F(t) dt ý ,
î R þ
where b refers to the temperature of the previous equilibrium state - no other value of b makes sense until the system returns to equilibrium.
By linear approximation we mean ``linear in the departure from equilibrium." In the present case that means that the entire integral in (54) and (55) is in some sense small. An expansion of the
exponential operator follows from repeated application of the identity
é ó 1 ù
e^A+B=e^A ë 1+ õ e^-xA Be^x(A+B) dx û , (56)
where B is the small perturbation. The first-order, or linear approximation to the expectation value of another operator C is (Heims and Jaynes, 1962; Jaynes, 1979; Grandy, 1988)
ó 1
áCñ @ áCñ[0] - õ e^-xABe^xA C 0 dx+ áBñ[0]áCñ[0] , (57)
. In (57) we again encounter the Kubo transform of the operator B with respect to A, the nonlinear form of which was introduced in (24).
Application of this approximation scheme to (54) and (55) reveals that the leading-order departure of the expectation value of C at time t from its equilibrium value is
áC(t)ñ-áCñ[0]=- õ K[CF](t,t¢)l(t¢) dt¢ , (58)
where K[CF] º K^0[CF] is the linearized version of the covariance function defined in (27):
º á F(t¢) C(t)ñ[0]-áFñ[0]áCñ[0]
and á¼ñ[0] is an expectation value in terms of the equilibrium distribution r[0]. Time independence of the Hamiltonian confers the same property upon the single-operator expectations, and also
guarantees time-translation invariance: K[CF](t,t¢)=K[CF](t-t¢). One verifies the reciprocity relation
from a change of variables and cyclic invariance of the trace. Note that it is always the second variable that carries the Kubo transform. If C and F are Hermitian, K[CF] is real and K[FF] ³ 0. In
this case K[CF] has all the properties of a scalar product on a linear vector space, and thus satisfies the Schwarz inequality: K[CC]K[FF]-K[CF]^2 ³ 0, with equality if and only if C=cF, with c a
real constant.
The covariance function (59) clearly depends only on equilibrium properties of the system. Quite generally, then, small departures from equilibrium caused by anything are described principally by
equilibrium fluctuations. While this provides some useful physical insight, the other side of the coin is that covariance functions are exceedingly difficult to calculate for interacting particles,
other than in some kind of perturbation theory. The linear approximation represents considerable progress, but formidable mathematical barriers remain. In practice, however, it is usually the
relations among these and other quantities that interest us; after all, we seldom evaluate from first principles the derivatives in the Maxwell relations, yet they provide us with important insights.
Linear hydrodynamics provides one area in which various approximation schemes for correlation functions have proved fruitful.
In the absence of external driving the Lagrange multiplier function l(t) is determined formally by (7), but one suspects that if we set C=F and restrict t to the region R, then (58) becomes a
Fredholm integral equation determining l(t) in the only interval in which it is defined. This indeed turns out to be the case, though the demonstration that the two procedures are equivalent requires
a little effort (Grandy, 1988). This is, in fact, a very rich result, and to discuss it in slightly more detail it will be convenient to specify R more definitely, as [-t,0], say. Thus, the
ó 0
áF(t)ñ-áFñ[0]=- õ K[FF](t-t¢)l(t¢) dt¢ (61)
is now seen to have several interpretations as t ranges over (-¥,¥). When t > 0 it gives the predicted future of F(t); with -t £ t £ 0 it provides a linear integral equation determining l(t); and
when t < -t it yields the retrodicted past of F(t). This last observation underscores the facts that K[FF](t) is not necessarily a causal function unless required to be so, and that these expressions
are based on probable inference; in physical applications the dynamics enters into computation of the covariance function, but does not dictate its interpretation in various time domains. Although
physical influences must propagate forward in time, logical inferences about the present can affect our knowledge of the past as well as the future. Retrodiction, of course, is at the heart of fields
such as archeology cosmology, geology, and paleontology.
When the perturbed system is spatially nonuniform we find that (58) and (59) are replaced by
=- õ K[CF](x,t;x^¢,t^¢)l(x^¢,t^¢) d^3x^¢ dt^¢ ,
=á F(x^¢,t^¢) C(x,t)ñ[0] -áF(x^¢)ñ[0]áC(x)ñ[0] ,
so that in its causal domain K[CF](x,t;x^¢,t^¢) takes the form of a Green function. Note that the single-operator expectation values are also independent of x in an initially homogeneous system, and
that the generalization to include a number of operators F[k](x,t) is straightforward.
If the equilibrium system is also space-translation invariant it is useful to employ the notation r º x-x¢. Generally, the operators encountered in covariance functions possess definite
transformation properties under space inversion (parity) and time reversal. Under the former A(r,t) becomes P[A]A(-r,t), P[A]=±1, and under the latter T[A]A(r,-t), T[A]=±1. For operators describing a
simple fluid, say , PT=+1 and one verifies that the full reciprocity relation holds:
The efficacy of these equations of the linear approximation will become apparent as we present some sample applications.
The generic model for a macroscopic fluid is most readily described as a continuum in terms of various densities, and representations in terms of quantum-mechanical operators are defined in terms of
field operators in a Fock representation (e.g., Fetter and Walecka, 1971). The three basic density operators in the fluid are the number density n, momentum density mj, and energy density h, where j
is the particle current-density operator. Unless so specified, these generally have no explicit time dependence, so that their equations of motion in the Heisenberg picture are
But the left-hand sides of these equations are also involved in statements of the local microscopic conservation laws in the continuum, which usually relate time derivatives of densities to
divergences of the corresponding currents. The differential conservation laws are thus obtained by evaluating the commutators on the right-hand sides in the forms
The superposed dot in these equations indicates a total time derivative. In the absence of external forces and sources (65) are equivalent to unitary transformations, and the Hamiltonian and
total-number operator, respectively, are given by
H= ó h(x,t) d^3x , N= ó n(x,t) d^3x , (67)
õ õ
both independent of time.
The current density j is just the usual quantum-mechanical probability current density, so that (66a) is easily verified. Identification of the energy current density q and stress tensor T, however,
is far from straightforward; in fact, they may not be uniquely defined for arbitrary particle-particle interactions. But if the Hamiltonian is rotationally invariant we can restrict the discussion to
spherically-symmetric two-body potentials. Two further symmetry properties arise from time independence and spatial uniformity in the equilibrium system: time-translation and space-translation
invariance, respectively. These latter two invariances are expressed in terms of volume-integrated, or total energy, number, and momentum operators, so that the commutators [H,P], [H,N], [P,N] all
vanish. Specification of these symmetry properties defines a simple fluid, and the operators q and T can be identified uniquely by evaluation of the commutators in Eqs.(65b,c). The algebra is tedious
and the results are given, for example, by Puff and Gillis (1968), and Grandy (1988). Thus, the five local microscopic conservation laws (66) completely characterize the simple fluid and lead to five
long-lived hydrodynamic modes. Local disturbances of these quantities cannot be dissipated locally, but must spread out over the entire system.
As a first application of the linear theory we return to the steady-state scenario of Eqs.(12) and (13) and also incorporate a term -bH in the exponentials to characterize an earlier equilibrium
reference state. Denoting the deviation from equilibrium as DF(x) = F(x)-áF(x)ñ[0], we find that in linear approximation another operator C will have expectation value
=- õ l(x¢)K[CF](x-x¢) d^3x¢
ó ó 0
+ lim õ d^3x¢ õ e^et l(x¢)K[C[(F)\dot]](x-x¢,t) dt ,
e® 0^+ R -¥
where we have employed the expression (11) for the diagonal part of an operator, and subscripts ss refer to the steady-state distribution. Specify F(x) to be one of the fluid densities d(x), so that
the continuity equations (66) lead to the identity
d dt K[dB](x,t)=-Ñ·K[jB](x,t) , (69)
and thus K[C[(d)\dot]] in (68) can be replaced by -Ñ¢·K[CJ]. Let R(x) be the system volume V, and presume K[CJ] to vanish at large distances. An integration by parts then reduces (68) to
=- õ l(x¢)K[Cd](x-x¢) d^3x¢
ó ó 0
+ lim õ d^3x¢ õ e^et Ñ¢l(x¢)·K[CJ](x-x¢,t) dt ,
e® 0^+ V -¥
in which we have dropped the surface term.
Classical hydrodynamics corresponds to a long-wavelength approximation by presuming that Ѣl varies so slowly that it is effectively constant over the range for which K[CJ] is appreciable. ^8 With
this in mind we can extract the gradient from the integral and write
@ - õ l(x¢)K[Cd](x-x¢) d^3x¢
ó ó 0
+Ñl· lim õ d^3x¢ õ e^et K[CJ](x-x¢,t) dt ,
e® 0^+ v -¥
which is the fundamental equation describing linear transport processes in the steady state. The integration region v is the correlation volume, outside of which the correlations vanish; it is
introduced here simply as a reminder that the spatial correlations are presumed to be of short range.
As an example, let d be the number density n with gradient characterized by the deviation Dn(x)=n(x)-ánñ[0]. The specified density gradient and the predicted current density, respectively, are then
=- õ l(x¢)K[nn](x-x¢) d^3x¢
ó ó 0
+Ñl· õ d^3x¢ õ e^et K[nj](x-x¢,t) dt
v -¥
=- õ l(x¢)K[nn](x-x¢) d^3x¢ ,
=- õ l(x¢)K[jn](x-x¢) d^3x¢
ó ó 0
+Ñl· õ d^3x¢ õ e^et K[jj](x-x¢,t) dt
v -¥
ó ó 0
= Ñl· õ d^3x¢ õ e^et K[jj](x-x¢,t) dt ,
v -¥
where the limit e® 0^+ is understood. We have noted that the second term of the first line in (72) and the first term of the first line in (73) vanish by symmetry.
Now take the gradient in (72), make the long-wavelength approximation, and eliminate Ñl between this result and (73), which leads to the relation
ó ¥ ó
=- õ e^-et dt õ K[jj](x-x¢,t) d^3x¢ ·Ñán(x)ñ[ss]
0 v
with the proviso that e® 0^+. This is Fick's law of diffusion, in which we have identified the diffusion tensor D that can now be calculated in principle from microscopic dynamics; owing to spatial
uniformity in the equilibrium system D(x) is actually independent of x. For more general nonequilibrium states the same type of calculation produces a quantity D(x,t) having the same form as that in
(74), and the long-wavelength approximation also involves one of short memory. (By `short memory' we mean that recent information is the most relevant, not that the system somehow forgets.)
It is remarkable that linear constitutive equations such as Fick's law arise from almost nothing more than having some kind of data available over a space-time region. These relations have long been
characterized as phenomenological, since they are not derived from dynamical laws. We now see why this is so, for the derivation here shows that they are actually laws of of inference. Indeed, what
we usually mean by `phenomenological' is `inferred from experience', a notion here put on a sound footing through probability theory. When they are coupled with the corresponding conservation laws,
however, one does obtain macroscopic dynamical laws, such as the diffusion equation.
Because it involves a slightly different procedure, and will provide a further example below, let us consider thermal conductivity (which need not be restricted to fluids). A steady gradient in
energy density is specified in the form of a deviation Dh(x)=h(x)-áhñ[0]. By a calculation similar to the above we find for the expected steady-state heat current
ó ó ¥
áq(x)ñ[ss]= õ d^3x¢ õ e^-et Ñl(x¢)·K[qq](x-x¢, t) dt , (75)
v 0
where the limit e® 0^+ is understood, and we have not yet invoked the long-wavelength limit. In this case we do not eliminate Ñl, for it contains the gradient of interest. Both dimensionally, and as
dictated by the physical scenario, l must be b(x)=[kT(x)]^-1, a space-dependent temperature function. Although such a quantity may be difficult to measure in general, it is well-defined in the steady
state. With this substitution the long-wavelength approximation of constant temperature gradient in (75) yields
ó ó ¥
@ -ÑT· õ d^3x¢ õ e^-et K[qq](x-x¢, t) kT^2(x¢) dt
v 0
in which we identify the thermal conductivity tensor k, which again is independent of x. This is Fourier's law of thermal conductivity; it applies to solids as well as fluids, but calculation of the
covariance function remains a challenge. It is left to the reader to verify that k, as well as D in (74), are positive.
A common model employing (76) is that of a uniform conducting rod of length L and thermal conductivity k. We can calculate the constant rate of transfer of entropy from the source to the sink by
means of (52), in which the transfer potential g(x) is simply the spatial temperature distribution b(x), and s(x) is the (constant) rate of driving on the end boundaries of the rod. In this case the
driving rate is given by the heat current áqñ[ss] itself, inserting thermal energy at one end and taking it out at the other. Hence,
ó L é ù
= õ 1 kT(x) (-kÑT) ë d[x,0]-d[x,L] û dx
which is identical to the more intuitively obtained result (e.g., Palffy-Muhoray, 2001). Although (52) itself is completely nonlinear, one notes that we have employed the linear form of Fourier's law
(76) for the current. This calculation illustrates the importance of boundary conditions in describing stationary processes; Tykodi (1967) has also emphasized the role of terminal parts in describing
the steady state. Indeed, the entropy generated in this process is entirely that of the external world. It is also of some interest to note that
is by no means a minimum in this state (Palffy-Muhoray, 2001).
An important feature of the thermal driving mechanism is that the actual details of the thermal driving source are irrelevant, and only the rates and strengths at which system variables are driven
enter the equations. It should make no difference in many situations whether the driving is thermal or mechanical; we examine the latter context here.
The theory of dynamical response was described very briefly in Eqs.(I-5)-(I-8), and the linear version follows as described there. The underlying scenario is that a well-defined external field is
imposed on a system that has been in thermal equilibrium in the remote past, as described by the Hamiltonian H[0]. It is then presumed that the response to this disturbance can be derived by adding a
time-dependent term to the Hamiltonian, so that effectively H=H[0]-Fv(t), t > 0, where v(t) describes the external field and F is a system operator to which it couples. Some of the difficulties with
this approach were sketched in I, including the observation that r(t) can only evolve unitarily. We now see that these problems can be resolved by noting that dynamical response is just a special
case of thermal driving.
For eventual comparison with the results of linear response theory we shall need an identity for the time derivative of the covariance function. Direct calculation in the definition (59) yields
where f[CF] is the linear response function. Clearly, the covariance function contains a good deal more information than does the dynamic response function.
The derivation of the generic maximum-entropy distribution in (I-14) disguises a subtle point regarding that procedure. We note from (I-16) that the Lagrange multiplier l can also be determined from
the maximum entropy:
Together with (I-15) this reveals a reciprocity implying that the probability distribution can be obtained by specifying either áfñ or l. An example of this choice is illustrated in the canonical
distribution (I-10), which could be obtained by specifying either the energy or the temperature; this option was also exercised in the model of spatial inhomogeneity of Eq.(10). Thus, we return to
Eqs.(21), replacing H with H[0], and let l(t¢) be the independent variable. In linear approximation (29) expresses l(t) directly in terms of the source strength, or driving rate, and dimensional
considerations suggest that we write this variable in the form
=b d dt¢ é q(t-t¢)v(t¢) ù
ë û
=b é -d(t-t¢)v(t¢)+q(t-t¢) d dt¢ v(t¢) ù ,
ë û
with the condition that v(0)=0. The step-function q(t-t¢) is included in (80) because l is defined only on the interval [0,t].
Substitution of (80) into (21) yields the distribution relevant to a well-defined external field,
é ó t é ù ù
= 1 Z[t] exp ë -bH[0]+b õ ë d(t-t¢)-q(t-t¢) d dt¢ û v(t¢) F(t¢) dt¢ û
é ó t × ù
= 1 Z[t] exp ë -bH[0] +b õ v(t¢) (t¢) dt¢ û ,
0 F
and Z[t], as usual, is the trace of the numerator. Although the exponential contains what appears to be an effective Hamiltonian, we do not assert that
is an addition to the equilibrium Hamiltonian H[0]; there is no rationalé of any kind for such an assertion. The Lagrange multiplier function l(t) is a macroscopic quantity, as is its expression as
an independent variable in (80). The linear approximation (58), along with the identity (78), yields the departure from equilibrium of the expected value of another operator C at any future time t
under driving by the external field:
ó t
=b õ v(t¢)K[C[(F)\dot]](t-t¢) dt¢
ó t
= b õ v(t¢) d dt¢ K[CF](t-t¢) dt¢
ó t
= õ v(t¢)f[CF](t-t¢) dt¢ ,
which is precisely the result obtained in linear response theory. But now we also have the time-evolved probability distribution (81) from which we can develop the associated thermodynamics. Equation
(82) confirms that, at least linearly, both r[t] and a unitarily evolved r(t) will predict the same expectation values. But, as suggested following (78), r(t) contains no more macroscopic information
than it had to begin with.
As an example of an external source producing a time-varying field, suppose a component of electric polarization M[i](t) is specified, leading to the density matrix
é ó t ù
r[t]= 1 Z[t] exp ë -bH[0] + õ l[i](t¢)M[i](t¢ dt¢ û . (83)
We presume no spontaneous polarization, so that in linear approximation the expectation of another component at time t is
ó t
áM[j](t)ñ = õ l(t¢)á M[i](t¢) M[j](t)ñ[0] dt¢ . (84)
Now, with the additional knowledge that (84) is the result of turning on an external field one might be led to think that the Lagrange multiplier is simply a field component, say E[i](t). But (80)
shows that, even when the effect is to add a time-dependent term to the Hamiltonian, the actual source term is somewhat more complicated; only the d-function term in (80) corresponds to that
possibility, and the actual source term also describes the rate of change of the field. This again illustrates the earlier observation that the covariance function contains much more information than
the dynamic response function.
With (80) we can rewrite (84) explicitly as
ó t
-b õ á M[i](t¢) M[j](t)ñ[0] dE[i](t¢) dt¢ dt¢ ,
which is just the result obtained from the theory of dynamic response. But we've uncovered much more, because now one can do thermodynamics. In the present scenario we have specified thermal driving
of the polarization and incorporated that into a density matrix; additionally, the Lagrange multiplier has been chosen to be the independent variable corresponding to an external field, which allows
us to identify the source strength. Thus, from (25) we have a definite expression for the time-dependent entropy of the ensuing nonequilibrium state:
ó t
=lnZ[t] +báHñ[t]- õ l(t¢)áM(t¢)ñ[t] dt¢ ,
ó t
@ 1 k S[0] +b õ l(t¢)K[H[0]M](t¢) dt¢+O(l) ,
where the second line is the linear approximation and we have identified the entropy of the equilibrium system as S[0]=klnZ[0]+kbáH[0]ñ[0]. In the case of dynamic response, if one makes the linear
approximation to r(t) in (I-6) and computes the entropy similarly, it is found that S(t)-S[0] vanishes identically, as expected. With (78), (80), and (86), however, the entropy difference can also be
written in terms of the linear response function:
æ ö ó t
1 k è S[t]-S[0] ø @ b õ v(t¢)f[H[0]M](t¢) dt¢ , (87)
once again exhibiting the canonical form DQ(t)/T. These remarks strongly suggest that the proper theory of response to a dynamical perturbation is to be found as a special case of thermal driving.
When external sources are removed we expect the system to relax to a (possibly new) state of thermal equilibrium. If the driving ceases at time t=t[1], say, then from that point on the system is
described by (21) with the replacement t® t[1] everywhere, barring any further external influence. These equations define the nonequilibrium state at t=t[1], from which the subsequent behavior can be
As discussed earlier, S[t[1]] as given by (25) cannot evolve to the entropy of some equilibrium state, for the same reason that r[t[1]] cannot evolve to a canonical equilibrium distribution; both
evolve from t=t[1] under unitary transformation. It should be sufficient, however, to show that the macrovariables describing the thermodynamic system may relax to a set of equilibrium values. Then,
with those predicted values, we can construct a new canonical density matrix via entropy maximization that will describe the new equilibrium state. The value S[t[1]] remains the entropy of the
nonequilibrium state at the time the driving was removed.
If we add energy and matter, say, to the system, then the total energy and particle number of the ensuing equilibrium state are fixed at the cutoff t=t[1]. The energy and number densities, however,
continue to evolve to uniform values over the relaxation period, and these processes define that period. Often the densities themselves are the driven variables; for example, the pot of water is
heated over the area of the bottom of the pot. But in the example of electric polarization the total moment is not fixed at cutoff, and its decay in zero field defines the relaxation process. In view
of these various possibilities, we shall consider a generic variable f(t) whose expectation value describes the relaxation.
Calculation of the exact expectation values is essentially intractable, of course, so we again employ the linear approximation. For example, at time t ³ t[1] the expectation of interest is
ó t[1]
@ õ l(t¢)K[ff](t-t¢) dt¢
ó t[1]
= õ s[f](t¢) K[ff](t-t¢) K[ff](0) dt¢ ,
where we've utilized (29). Although everything on the right-hand side of (88) is presumably known, the actual details of the relaxation process depend crucially on the behavior of K[ff](t-t¢) for t >
t[1]. But if f is the driven energy operator, say, then Df will be independent of time and (88) provides the new total energy of the equilibrium system.
In the discussion following Eq.(60) it was noted that the covariance functions satisfy the Schwarz inequality. From this we can see that the ratio r(t-t¢)=|K[ff](t-t¢)/K[ff](0)| in (88), and
therefore the integrand, reach their maxima at t¢=t where r(0)=1. Further, r(t-t¢) is less than unity for t¢ < t, and again for all t > t[1]; the exact magnitude of r depends on the decay properties
of K[ff](t-t¢). In any event, the major contribution to the integral arises from the region around the cutoff t=t[1]. The relaxation time t can be estimated by studying the asymptotic properties of D
f(t), which in turn requires an examination of K[ff](t-t¢). We seek a time t[2] for which K[ff] no longer contributes appreciably to the integral:
by some criterion. Then, t @ t[2]-t[1].
For many covariance functions the ratio r(t-t¢) in (88) will tend to some constant value as t/t[1] becomes large, while others may tend to zero. For example, if we turn off the burner under a pot of
water at t=t[1], the total energy of the equilibrium system will be áE(t[1])ñ[t[1]], so that K[EE] would be expected to reach its nonzero asymptotic form very quickly. But in the polarization example
of (84) we expect the correlations to decay to zero as the system relaxes back to the unpolarized state; this may, or may not, be rapid. One can only uncover the particular behavior from a detailed
study of the covariance functions as determined by the relaxation mechanisms specific to the system, which are generally governed by particle interactions.
There is also a connection here with the rate-of-change of internal entropy,
. When the source is removed the entropy production of transfer immediately vanishes for t > t[1]. Equation (40) then implies that the total rate of entropy production is entirely that of relaxation,
and the compelling conclusion is that
is actually the relaxation rate itself and may be observable. This interpretation is reinforced by comparing the time derivative of áf(t)ñ[t[1]] in (88) with the definition of
in (39).
Finally, the preceding discussion suggests that one can actually construct an explicit expression for the relaxation entropy S[int], and hence for
. Equation (88) provides a continuous set of data áf(t)ñ[t[1]] from the cutoff at t=t[1] to the effective equilibrium point at t=t[2]. These predicted values are as good as any other data for
constructing a `relaxation distribution' via the PME, and hence a maximum entropy of relaxation. Note carefully, though, that this entropy production should disappear at equilibrium while S[t]
possibly approaches the thermodynamic entropy of that state; the latter is determined by values of variables characterizing the equilibrium state and which most often have already been set at t=t[1].
In a simple fluid, say, the approach to equilibrium is described by taking f(t) in (88) to be a density that may approach a constant value denoting a homogeneous state. Further details of this
construction will be discussed elsewhere.
5. Summary
The aim of this discussion has been to expand the concept of theoretical entropy in equilibrium thermodynamics to encompass macroscopic systems evolving in time. In doing so we find that the maximum
information entropy S[t], while providing a complete description of the nonequilibrium state at any instant, does not assume the dominant role it does in an equilibrium context. Rather, the rates and
directions of processes are the most important features of nonequilibrium systems, and the rate of entropy production
takes the form of a transfer potential times a rate of transfer, or a generalized intuitive form of
. This suggests that in nonequilibrium thermodynamics it is
that governs the ongoing macroscopic processes and can be expressed as a measurable quantity via Eq.(34). In the absence of external sources (or sinks) the rate of entropy production simply describes
the relaxation rate; the theoretical maximum entropy itself characterizes only the nonequilibrium state from which the system is relaxing to the singular equilibrium limit. Further thought leads us
to conclude that these interpretations can also be applied to ongoing processes.
Many writers have expressed a belief in the existence of an additional variational principle characterizing nonequilibrium states and processes in much the same way that the Gibbs algorithm governs
equilibrium states. Although various candidates have been put forth in special contexts, none have achieved that same lofty position. This is perhaps not surprising in light of the foregoing
discussion, for it would seem that the Gibbs variational principle has all along been the rule governing all thermodynamic states, possibly because it has its roots in a fundamental rule of
probability theory: the principle of maximum information entropy. The difficulty has not been our inability to find a compelling definition of physical entropy for nonequilibrium states; rather, it
was a failure to understand the specific role of entropy over the entire spectrum of thermodynamic states. It's only the nature of the constraints that changes, from constants of the motion to steady
state to time dependent, while the principle remains the same. This is a satisfying result in that it provides a certain economy of principles.
The formalism presented here applies to macroscopic systems arbitrarily far from equilibrium, although the nonlinear equations provide formidable mathematical barriers to any detailed calculations.
While the linear approximation is the most fruitful approach, even here the covariance functions remain somewhat complicated and resistant to exact computation; various attacks have produced some
progress, however, in the context of linear hydrodynamics. At present it is the formal relations containing covariance functions that can prove most useful, in which carefully chosen models of these
nonequilibrium correlations can play a role similar to that of potential models in equilibrium statistical mechanics. Although the present results may lay some groundwork for a complete theory of
nonequilibrium thermodynamics, there is a great deal of room for expansion and further development. Perhaps a third paper in this series will describe additional steps along these lines.
Berry, M. (2002), ``Singular Limits," Physics Today 55 (5), 10.
Boltzmann, L. (1895), ``On certain questions of the theory of gases," Nature 51, 413.
Clausius, R. (1865), ``Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanische Wärmetheorie," Ann. d. Phys.[2] 125, 390.
de Groot, S.R. and P. Mazur (1962), Nonequilibrium Thermodynamics, North-Holland, Amsterdam.
Evans, R. (1979), ``The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids," Adv. Phys. 28, 143.
Fano, U. (1957), ``Description of States in Quantum Mechanics by Density Matrix and Operator Techniques," Rev. Mod. Phys. 29, 74.
Fetter, A.L. and J.D. Walecka (1971), Quantum Theory of Many-Particle Systems, McGraw-Hill, New York.
Gibbs, J.W. (1902), Elementary Principles in Statistical Mechanics, Yale University Press, New Haven, Conn.
Grandy, W.T., Jr.(1987), Foundations of Statistical Mechanics, Vol.I: Equilibrium Theory, Reidel, Dordrecht.
(1988), Foundations of Statistical Mechanics, Vol.II: Nonequilibrium Phenomena, Reidel, Dordrecht.
(2004), ``Time Evolution in Macroscopic Systems. I: Equations of Motion," Found. Phys. 34, 1.
Jaynes, E.T. (1963), ``Information Theory and Statistical Mechanics," in K.W. Ford (ed.), Statistical Physics, Benjamin, New York.
(1967), ``Foundations of Probability Theory and Statistical Mechanics," in M. Bunge (ed.), Delaware Seminar in the Foundations of Physics, Springer-Verlag, New York.
(1979), ``Where Do We Stand On Maximum Entropy?," in R.D.Levine and M. Tribus (eds.), The Maximum Entropy Formalism, M.I.T. Press, Cambridge, MA.
Kubo, R., M. Toda, and N. Hashitsume (1985), Statistical Physics II, Springer-Verlag, Berlin.
Mitchell, W.C. (1967), ``Statistical Mechanics of Thermally Driven Systems," Ph.D. thesis, Washington University, St. Louis (unpublished).
Nakajima, S. (1958), ``On Quantum Theory of Transport Phenomena," Prog. Theor. Phys. 20, 948.
Onsager, L. (1931), ``Reciprocal Relations in Irreversible Processes. I," Phys. Rev. 37, 405.
Palffy-Muhory, P. (2001), ``Comment on `A check of Prigogine's theorem of minimum entropy production in a rod in a nonequilibrium stationary state' by Irena Danielewicz-Ferchmin and A. Ryszard
Ferchmin [Am. J. Phys. 68 (10), 962-965 (2000)]," Am. J. Phys. 69, 825.
Puff, R.D. and N.S. Gillis (1968), ``Fluctuations and Transport Properties of Many-Particle Systems," Ann. Phys. (N.Y.) 6, 364.
Tykodi, R.J. (1967), Thermodynamics of Steady States, Macmillan, New York.
Equations of Motion
^1Equation (n) of that paper will be denoted here by (I-n).
^2Because there are numerous `entropies' defined in different contexts, we shall denote the experimental equilibrium entropy of Clausius as S without further embellishments, such as subscripts.
^3Requiring the equilibrium system to have no `memory' of its past precludes `mysterious' effects such as those caused by spin echos.
^4This prescription for stationarity was advocated earlier by Fano (1957), and has also been employed by Nakajima (1958) and by Kubo, et al (1985).
^5The lower limit of the driving interval is chosen as 0 only for convenience.
^6Equation (40) is reminiscent of, but not equivalent to, similar expressions for entropy changes, such as dS = dS[ext] + dS[int], that can be found in works on phenomenological nonequilibrium
thermodynamics (e.g., de Groot and Mazur, 1962).
^7The equilibrium distribution is taken as canonical only for convenience; for example, one could just as well use the grand canonical form, as well as include different types of particle.
^8We presume that the fluctuations are not correlated over the entire volume.
File translated from T[E]X by T[T]H, version 3.10.
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Functional and Logic Programming
Title: Functional and Logic Programming
Code: FLP
Ac.Year: 2012/2013
Term: Summer
Study plans:
Language: Czech
Public info: http://www.fit.vutbr.cz/study/courses/FLP/public/
Private info: http://www.fit.vutbr.cz/study/courses/FLP/private/
Credits: 5
Completion: accreditation+exam (written)
Type of ┌──────────┬─────────────┬────────────────┬────────────────┬──────────────────┬───────┐
instruction: │ Hour/sem │ Lectures │ Sem. Exercises │ Lab. exercises │ Comp. exercises │ Other │
│ Hours: │ 26 │ 0 │ 0 │ 12 │ 14 │
│ │ Examination │ Tests │ Exercises │ Laboratories │ Other │
│ Points: │ 60 │ 20 │ 0 │ 0 │ 20 │
Guarantee: Kolář Dušan, doc. Dr. Ing., DIFS
Lecturer: Kolář Dušan, doc. Dr. Ing., DIFS
Instructor: Křivka Zbyněk, Ing., Ph.D., DIFS
Židek Stanislav, Ing., DIFS
Faculty: Faculty of Information Technology BUT
Department: Department of Information Systems FIT BUT
Substitute for:
Learning objectives:
Obtaining a basic knowledge and practical experience in functional and logic programming. Introduction into formal concepts used as a theoretical basis for both paradigms.
Practical applications and broader introduction into lambda calculus and predicate logic within the context of functional and logic programming languages. Within functional programming, abstract data
types are discussed, as well as the use of recursion and induction, manipulation of lists and infinite data structures in language Haskell. Experience in logic programming is gained in programming
languages Prolog (cut operator, state space search, database modification), and Goedel. Moreover, principles of their implementation are mentioned too.
Knowledge and skills required for the course:
Processing (analysis, evaluation/interpretion/compilation) of programming languages, predicate logic.
Subject specific learning outcomes and competences:
Students will get basic knowledge and practical experience in functional and logic programming (two important representatives of declarative programming). Moreover, they will get basic information
about theoretical basis of both paradigms and implementation techniques.
Generic learning outcomes and competences:
Use and understanding of recursion for expression of algorithms.
Syllabus of lectures:
1. Introduction to functional programming, lambda calculus
2. Programming language Haskell, introduction, lists
3. User-defined data types, type classes, and arrays in Haskell
4. Input/Ouput in Haskell - type classes IO and Monad
5. Simple applications/programs in Haskell
6. Proofs in functional programming
7. Denotational semantics, implementation of functional languages
8. Introduction to logic programming, Prolog
9. Lists, cut operator, and sorting in Prolog
10. Data structures, text strings, operators - extensions of SWI Prolog
11. Searching state space, clause management, and parsing in Prolog
12. Goedel - logic programming language not using Horn clauses
13. Implementation of logic languages, CLP, conclusion
Syllabus of computer exercises:
1. Introduction to Haskell environment (Hugs), simple functions, recursion, lists
2. Infinite data structures
3. User defined data types, input/output
4. Practical problem
5. Introduction to Prolog environment (Hugs), lists
6. Practical problem
Syllabus - others, projects and individual work of students:
1. A simple program in Haskell programming language (Hugs, GHC, GHCi).
2. A simple program in Prolog/Gödel/CLP(R) (SWIPL, Gödel, CiaoProlog).
Fundamental literature:
1. Thompson, S.: Haskell, The Craft of Functional Programming, ADDISON-WESLEY, 1999, ISBN 0-201-34275-8
2. Nilsson, U., Maluszynski, J.: Logic, Programming and Prolog (2ed), John Wiley & Sons Ltd., 1995
3. Hill, P., Lloyd, J.: The Gödel Programming Language, MIT Press, 1994, ISBN 0-262-08229-2
4. Jones, S.P.: Haskell 98 Language and Libraries, Cambridge University Press, 2003, p. 272, ISBN 0521826144
Study literature:
1. Thompson, S.: Haskell, The Craft of Functional Programming, ADDISON-WESLEY, 1999, ISBN 0-201-34275-8
2. Nilsson, U., Maluszynski, J.: Logic, Programming and Prolog (2ed), John Wiley & Sons Ltd., 1995
3. Hill, P., Lloyd, J.: The Gödel Programming Language, MIT Press, 1994, ISBN 0-262-08229-2
Controlled instruction:
• Mid-term exam - written form, questions and exersises to be answered and solved (there are even questions with selection of one from several predefined answers), no possibility to have a second/
alternative trial - 20 points.
• Projects realization - 2 projects, implementation of a simple program according to the given specification - one in a functional programming language the other in a logic programming language -
20 points all projects together.
• Final exam - written form, questions and exersises to be answered and solved (there are even questions with selection of one from several predefined answers), 2 another corrections trials
possible (60 points - the minimal number of points which can be obtained from the final exam is 25, otherwise, no points will be assigned to a student).
Progress assessment:
• Mid-term exam, for which there is only one schedule and, thus, there is no possibility to have another trial.
• Two projects should be solved and delivered in a given date during a term.
Exam prerequisites:
At the end of a term, a student should have at least 50% of points that he or she could obtain during the term; that means at least 20 points out of 40. Plagiarism and not allowed cooperation will
cause that involved students are not classified and disciplinary action can be initiated. | {"url":"http://www.fit.vutbr.cz/study/course-l.php?id=8608","timestamp":"2014-04-18T00:13:57Z","content_type":null,"content_length":"14644","record_id":"<urn:uuid:6c347822-803b-4075-afa7-3731cc4e6359>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00085-ip-10-147-4-33.ec2.internal.warc.gz"} |