2010 May 29: Searching this blog.
The command-F search is limited to a page of 10 posts at a time… and because the posts have been split, it does not search the non-dislayed portions of posts, or other pages.
I have added a “Search this blog” widget, over on the right. It really seems to search the entire blog. It will pull up multiple pages of posts containing the search string — but it won’t show you where the string occurs.
But now you can use command-F on the pages the widget selected!
I tested this by searching for “orbit” with the widget… it pulled up two pages of displayed posts, and the bibliography, too — and command-F worked within each page.
Odd… I wasn’t paying close enough attention. The blog search does select every post or other page containing the search string, but it does not expand the posts. So we have to do that ourselves. OTOH, at least we are guaranteed that each post selected does in fact contain the search string. That is an improvement.
In summary, searching the blog is a 2-step 3-step process. First, use the box on the right to select all posts containing the search string; second, expand a selected post; third, use command-F to actually locate the search string within that post.
(Of course, command-F will show locations in the opening portions of each post, but there may be more in the hidden portions.)
6 aug 2008: Finding books by date of entry or latest edit. If you want to find, for example, the books I entered or edited in June 2008, do a search on “jun 2008”. I have made sure that all the months are 3 letters, and all the years are 4 digits, and if there’s a day, it precedes the month. If you want to find all the entries I touched in 2008, search on “2008]”, where the “]” will skip over dates of publication. I am, in fact, planning a mass edit of the entries, and you should be able to find all the changed entries by searching on DD MMM.
older posts
if you click on the “math PCA” category, you get a page on which the earliest post is dated Feb 11; it is one of the Jolliffe posts, but not the first one. where are the rest? there is a link at the bottom of that page, “older posts”, and it will give you all the rest of the posts in that category. I suspect that I have no more than 10 posts per page, and that eventually there will more multiple pages of earler posts. I’ll let you know.
blech
if you see any of those yellow and red signs (excluding that one, of course), please try again in a little while. the chances are very high that wordpress is being flaky. “it’s not my fault.”
i do make mistakes, and there are typos in these posts, but that image is real hard to miss; it’s exactly what i see when i preview an equation that isn’t acceptable latex to wordpress. i promise you: there aren’t any of those left when i publish a post. that much i do get right.
please keep coming back despite some technical glitches.
searching
on a mac, apple-F opens up a search box. it is limited to the current page, but you can change the page. on windows, control-F opens a search box. the exact form and location depends on the browser.
bibliography
the bibliography is now current.
note that you can search through it.
Rip's Applied Mathematics Blog 
January 9, 2011 at 9:26 am
This is a very nice site. I have been using Mathematica’s NonlinearModelFit function and ordering the results according to their AIC & BIC scores. I wanted to calculate my own AIC/BIC values AND compare them to those produced by Mathematica.
I didn’t know what they had done. Your page was a big help in clearing that up for me. I might reference it in a paper unless you know if those formulae are published somewhere within Wolfram’s pages?
January 9, 2011 at 1:38 pm
Hi JEP,
I’m glad to have been of help.
My source for the AIC formula was McQuarrie & Tsai, “Regression and Time Series Model Selection”, 981-02-3242-X, p. 21. I modified their SIC (BIC) formula on p. 23 by using (k+1) where they used k.
Then i confirmed that my two equations matched Mathematica’s answers. Mathematica’s brief discussion of AIC and BIC can be found in the tutorial for statistical model analysis, but I do not recall seeing the actual equations anywhere on Wolfram.
BTW, I’m going to copy this comment to the “selection criteria” post.
March 4, 2014 at 9:27 am
If you’ve done office parties or corporate work, make sure to mention that as well.
If I knew how to bake cookies, I would do that and send
over a care package but Im much better at telling jokes so I go there and entertain, shake hands and thank the men and women who serve
our country. Brad Garret is a stand-up comedian but also well known for his acting career.
March 16, 2019 at 11:18 am
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March 27, 2019 at 10:53 am
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May 7, 2019 at 10:15 am
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May 9, 2019 at 6:49 pm
indeed can be considered and stated as prof dr mircea orasanu and drd horia orasanu and followed with other as forms for suggestions For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. For historical reasons, the term Jordan measure is now well-established, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets in each have a Jordan measure of 0, while , a countable union of them, is not Jordan-measurable.[1] For this reason, some authors[2] prefer to use the term Jordan content (see the article on content).
The Peano-Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.[3
Molnár Antal Zeneiskola-Charpentier-TeDeum-20091218
II. DEFINITIONS AND DESIGNATIONS
Consider the discrete optimization problem (which we refer to as Problem A)
,
where – is a non-decreasing -order-convex function on a partially set .
Let be an optimal solution of Problem A, and let be the point obtained by the following iterative procedure [4]:
which halts on the step if either or is the maximal element of the set (the set contains the zero , as we have stipulated). This point is called the gradient maximum os the function on the set [4].
By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number
.
By perturbations of problem A by means problem
May 10, 2019 at 9:53 am
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and if there are m different constraints then, we have
Let’s look at an example,
The constraints:
Let’s use the following generalized coordinates , from which we get:
and .
As for the constraint forces we get,
By applying the D’Alembert principle (making note that the virtual displacement is no longer compatible with the constraints) we get,
Note that
From which we may call the term:
constraint generalized forces.
The constrained Lagrange equations thus become:
with
The result we get is differential equations and algebraic equations with unknowns and .
While this approach simplifies the derivation procedure and allows for detection of constraint forces, it is a rather difficult numerical problem to solve.
Let’s look at an example:
Here we show a disk that rolls without slipping as it is pulled by a force F. Let’s use and as our generalized coordinates.
As seen in the image, there is a horizontal constraint between the roll of the wheel due to its attachment to the yoke. Namely, the velocity of the pin in the direction must match the velocity of the yoke.
May 12, 2019 at 7:10 am
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the 1970’s have accurate estimates become available. In
a very important series of six articles published in 1974
and 1975, Allan Sandage and G. A. Tammann estimate
that the big bang occurred about 15 billion years ago.12
Therefore, according to the big bang model, the universe
began to exist with a great explosion from a state of in-
finite density about 15 billion years ago. Four of the
world’s most prominent astronomers describe that event
in these words.
June 19, 2019 at 6:58 am
there exist a so called by a problem of prof dr mircea orasanu observed by prof dr mircea orasanu and prof drd horia orasanu and must extended specially for LAGRANGIAN and other and followed with By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number
.
By perturbations of problem A by means problem B
,
where is a non-decreasing -order-convex function on a partially set and .
Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .
Theorem. Let and be guaranteed error estimates for the gradient algorithm in Problems A and B, respectively. Then .
To prove Theorem, we need the following lemma.
Lemma. The gradient maximum and the global maximum of any -ordered-convex non-decreasing function on are connected by the following relations:
, (1)
July 7, 2019 at 7:21 am
these can be extended in more ways observed prof dr mircea orassanu and prof drd horia orasanu and followed as is seen that si have In this model, the real numbers are defined to be the set of Dedekind cuts. Order is easy to define. We declare E < F if E is a subset of F. Addition is also easy to define. The sum of the cuts E and F is the set of all sums x + y, where x is in E and y is in F. The product is a little more complicated to define. Once defined, it is straightforward to verify the real number axioms. The main disadvantage of this method is the level of sophistication required to organize and execute these “straightforward” verifications.
In any event, the Dedekind cuts form a complete ordered field. The additive identity in the Dedekind model is the open interval from minus infinity to 0. The multiplicative identity is the open interval from minus infinity to 1. More generally, each rational number r corresponds to the cut (– ∞, r), and this correspondence allows us to identify the rational numbers with a subfield of the Dedekind model for the real numbers.
COMPLETION by cauchy sequences
The most far-reaching method for constructing the real numbers is due independently to Charles Meray (1869, 1872) and Georg Cantor (1872, 1883). Again one begins with the rational numbers. One considers the set of all sequences {xn} of rational numbers such that xn-xm tends to zero as n and m tend to infinity. Such sequences are called Cauchy sequences. We introduce an equivalence relation in the set of Cauchy sequences by declaring two Cauchy sequences {xn} and {yn} to be equivalent if xn – yn tends to zero as n tends to infinity. The real numbers are then defined to be the set of equivalence classes of Cauchy sequences. Addition and multiplication are easy to define. The sum of the equivalence classes represented by two such sequences {xn} and {yn} is defined to be the equivalence class of {xn + yn}, and similarly for the product. It is straightforward to verify the axioms of an ordered field, and a little more complicated to verify the completion axiom. The main disadvantage of the method is the excess labor and the level of sophistication required for working with equivalence classes rather than just sequences. The advantage of the method is that it can be used in a fairly general context to embed metric spaces in “complete” spaces. (A metric space can be embedded as a dense subset of a complete metric space, which is essentially unique.)
July 8, 2019 at 7:52 am
these are very important considered prof dr mircea orasanu and prof drd horia orasanu and followed so that to appeared many posters for us This passage should be considered as decisive in the history of sets: it is extremely rich, incorporating as it does all crucial ideas related to the notions of set and mapping as used in algebra (cf. DUGAC 1976, 29). There is no doubt that it had to be difficult to understand for any reader of the time, since they were completely unaccustomed to any use of the notion of set in algebra. But for this very reason, it had to attract strongly the attention of readers to this new vision─whether to accept it or refuse it.
DEDEKIND’s terminology might seem strange, but from his definitions it is clear that ‘divisor’ and ‘multiple’ mean the two sides of an inclusion; ‘greatest common divisor’ designs the intersection; and ‘least common multiple’ refers to the union, in the sense of the smallest structure that contains both. The selection of these expressions is easy to understand if we consider that DEDEKIND was dealing with algebraic number theory. In the case of Z, the inclusion of principal ideals corresponds strictly to the divisibility of their generators; this led him to employ the analogy between inclusion and division throughout the supplement on ideal theory. Moreover, his intention was to establish terminology in such a way that theorems could be formulated with exactly the same wording as in elementary number theory.
There exists evidence that DEDEKIND’s terminology was understood by at least one outstanding mathematician as related to the general notion of set. CANTOR used DEDEKIND’s terminology for inclusion, union and intersection of fields in his decisive series of papers ‘Über unendliche, lineare Punktmannigfaltigkeiten’, from 1880 to 1884. This is particularly striking because that terminology is rather inappropriate in a general set-theoretical setting such as the one to which CANTOR applied it─DEDEKIND himself replaced it in such a context (see below).
In the text on fields, DEDEKIND’s treatment of ‘substitutions’ or mappings, and particularly those which transfer structure, was equally a model. The fact that he is considering morphisms which may not be injective, is made clear by the comment on zero images: when he says that the numbers b constitute a field “if not all of them are zero”, it means that his notion of mapping allows for the trivial case in which all elements of the original have 0 as their image. (We have seen that non-injective maps were already present in his 1856-58 ‘Gruppen-Studien’.) Moreover, the clarity with which the reflexive, symmetric and transitive properties of field ‘conjugation’ are set forth in the text deserves attention.
The example we have considered here is only that of fields, discussed by DEDEKIND in the first paragraph of his 1871 ideal theory, but the whole exposition insisted on the importance of set-structures for algebraic number theory. The very viewpoint adopted involved a constant exercise in the translation of problems stated in terms of numbers to new and more abstract set-formulations: DEDEKIND replaced KUMMER’s ideal numbers with the kind of sets of numbers that he called ‘ideals’ (cf. section 4.2).
4.1.2. Sets and maps in a general setting. The year after the publication of this first version of ideal theory, DEDEKIND began to write the draft for his later book Was sind und was sollen die Zahlen?. The 1872 draft begins with definitions of some set operations (DUGAC 1976, 293-294); he presented them in a general setting, abstracted from the structural restrictions that are necessary in an algebraic context. The word ‘system’ was defined in a way that undeniably pointed to the extensional notion of set, and the most characteristic part of DEDEKIND’s algebraic terminology changed in accordance with the new abstract framework: the word ‘divisor’ was replaced by ‘part’ [Theil], meaning subset. Meanwhile, he maintained the expression ‘least common multiple’ for the union set, although he also wrote ‘compound system’, which was to become his final choice.
Nevertheless, set operations were by no means his primary interest. They were completely clear in themselves, and DEDEKIND had long been accustomed to them. The whole draft was primarily devoted to a study of the notions of mapping and chain, the second being based on the first. The term ‘mapping’ [Abbildung] appears also at the very beginning of the draft (DUGAC 1976, 294), and is defined in its general sense. We will return to this essential part of Was sind und was sollen die Zahlen? later on.
The translation of set-structural operations to a general set-theoretical framework, which DEDEKIND sketched in his 1872 draft, reveals clearly what I have already stated: that DEDEKIND’s mature set theory has its roots in his algebraic ideas, that emerged during the late 1850s.
It would be interesting to know about DEDEKIND’s motivation for writing the 1872-78 draft. It probably was one or two of the following factors. In 1872, he published his theory of the real number system after learning for the first time about two other theories, those of WEIERSTRASS and CANTOR. The knowledge that some mathematicians had developed ideas similar to his own might have motivated him to publish his more general reflections on the foundations of the natural numbers, and arithmetic as a whole, since they established the general framework for his theory of the real numbers. CANTOR had also developed the notion of derived set (see CANTOR 1872), which perhaps was even closer to DEDEKIND’s viewpoint, since it used the notion of set. On the other hand, it was