Boundaries and the james theorem

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Boundaries and the james theorem

In mathematicsparticularly functional analysisJames' theoremnamed for Robert C. Jamesstates that a Banach space B is reflexive if and only if every continuous linear functional on B attains its supremum on the closed unit ball in B.

A stronger version of the theorem states that a weakly closed subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C. The hypothesis of completeness in the theorem cannot be dropped James The space X considered can be a real or complex Banach space.

Its topological dual is denoted by X '. A Banach space being reflexive if and only if its closed unit ball is weakly compact one deduces from this, since the norm of a continuous linear form is the upper bound of its module on this ball:. Historically, these sentences were proved in reverse order. InJames had proved the reflexivity criterion for separable Banach spaces and for general Banach spaces.

Since the reflexivity is equivalent to the weak compactness of the unit sphere, Victor L.

Bayes' theorem

Klee reformulated this as a compactness criterion for the unit sphere in and assumes that this criterion characterizes any weakly compact quantities. This was then actually proved by RC James in From Wikipedia, the free encyclopedia. Redirected from James' theorem. Hahn—Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein—Milman theorem min-max theorem Gelfand—Naimark theorem Banach—Alaoglu theorem.

Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis. Categories : Theorems in functional analysis. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version.In probability theory and statisticsBayes's theorem alternatively Bayes's law or Bayes's rulenamed after Reverend Thomas Bayesdescribes the probability of an eventbased on prior knowledge of conditions that might be related to the event.

One of the many applications of Bayes's theorem is Bayesian inferencea particular approach to statistical inference. When applied, the probabilities involved in Bayes's theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence.

Bayesian inference is fundamental to Bayesian statistics. Bayes's theorem is stated mathematically as the following equation: [2]. Assuming 0. The Positive predictive value PPV of a test is proportion of persons who are actually positive out of all those testing positive, and can be calculated from a sample as:. If sensitivity, specificity, and prevalence are known, PPV can be calculated using Bayes theorem.

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We can write:. In this case, it says that the probability that someone tests positive is the probability that a user tests positive, times the probability of being a user, plus the probability that a non-user tests positive, times the probability of being a non-user.

This is true because the classifications user and non-user form a partition of a setnamely the set of people who take the drug test. This combined with the definition of conditional probability results in the above statement.

See Figure 1 for an illustration using a frequency box, and note how small the pink area of true positives is compared to the blue area of false positives. Based on incidence rate, the following table presents the corresponding numbers perpeople. If a randomly selected item is defective, what is the probability it was produced by machine C? Once again, the answer can be reached without using the formula by applying the conditions to a hypothetical number of cases.

For example, if the factory produces 1, items, will be produced by Machine A, by Machine B, and by Machine C. Let Y denote the event that a randomly chosen item is defective.

Then, we are given the following information:. If the item was made by the first machine, then the probability that it is defective is 0. Overall, we have.

To answer the original question, we first find P Y. That can be done in the following way:. We are given that Y has occurred, and we want to calculate the conditional probability of X C. By Bayes's theorem. Although machine C produces half of the total output, it produces a much smaller fraction of the defective items.

The interpretation of Bayes's rule depends on the interpretation of probability ascribed to the terms. The two main interpretations are described below. Figure 2 shows a geometric visualization similar to Figure 1.

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Gerd Gigerenzer and co-authors have pushed hard for teaching Bayes Rule this way, with special emphasis on teaching it to physicians. In the Bayesian or epistemological interpretationprobability measures a "degree of belief". Bayes's theorem links the degree of belief in a proposition before and after accounting for evidence. If the coin is flipped a number of times and the outcomes observed, that degree of belief will probably rise or fall, but might even remain the same, depending on the results.

For proposition A and evidence B. For more on the application of Bayes's theorem under the Bayesian interpretation of probability, see Bayesian inference.

In the frequentist interpretationprobability measures a "proportion of outcomes".Furthermore, the theorem has applications in fluid mechanics and electromagnetism. Conversely, we can calculate the line integral of vector field F along the boundary of surface S by translating to a double integral of the curl of F over S.

James's theorem

Let S be an oriented smooth surface with unit normal vector N. Furthermore, suppose the boundary of S is a simple closed curve C. The orientation of S induces the positive orientation of C if, as you walk in the positive direction around C with your head pointing in the direction of Nthe surface is always on your left. Let S be a piecewise smooth oriented surface with a boundary that is a simple closed curve C with positive orientation Figure 6.

If F is a vector field with component functions that have continuous partial derivatives on an open region containing Sthen. Suppose surface S is a flat region in the xy -plane with upward orientation. We look at an intuitive explanation for the truth of the theorem and then see proof of the theorem in the special case that surface S is a portion of a graph of a function, and Sthe boundary of S, and F are all fairly tame.

First, we look at an informal proof of the theorem. This proof is not rigorous, but it is meant to give a general feeling for why the theorem is true. Let S be a surface and let D be a small piece of the surface so that D does not share any points with the boundary of S. We choose D to be small enough so that it can be approximated by an oriented square E.

Let D inherit its orientation from Sand give E the same orientation. To approximate the flux over the entire surface, we add the values of the flux on the small squares approximating small pieces of the surface Figure 6.

Let F be an approximating square with an orientation inherited from S and with a right side E l E l so F is to the left of E.

In other words, the right side of F F is the same curve as the left side of Ejust oriented in the opposite direction. The same goes for the line integrals over the other three sides of E.

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These three line integrals cancel out with the line integral of the lower side of the square above Ethe line integral over the left side of the square to the right of Eand the line integral over the upper side of the square below E Figure 6.

After all this cancelation occurs over all the approximating squares, the only line integrals that survive are the line integrals over sides approximating the boundary of S. In the limit, as the areas of the approximating squares go to zero, this approximation gets arbitrarily close to the flux.

Furthermore, assume that f f has continuous second-order partial derivatives. Suppose that S is oriented upward. By Equation 6. Therefore, four of the terms disappear from this double integral, and we are left with. We can quickly confirm this theorem for another important case: when vector field F is conservative.

Therefore, the theorem allows us to compute surface integrals or line integrals that would ordinarily be quite difficult by translating the line integral into a surface integral or vice versa.It allowed us to benefit from every hour we were in Iceland. The small car was in good condition (Ford Focus). Great service, great suggestions. The map outlined and prepared for us was great, along with ideas and suggestions for each day.

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In fact, the National Institute on Aging estimate that 8 in 10 people with MCI are diagnosed with Alzheimer's disease within 7 years of their MCI diagnosis.

boundaries and the james theorem

But what goes on in the brain between being diagnosed with MCI and being diagnosed with Alzheimer's. Researchers from the University of Birmingham, the University of Kent both in the United Kingdom and the University of California, Davis set out to investigate this in their new study. Lead study author Dr. Ali Mazaheri, of the University of Birmingham, explains the rationale for the investigation. Mazaheri continues, "if there were anomalies in brain activity during language processing in MCI patients which could provide insight into their likelihood of developing Alzheimer's.

Previous studies have shown that it takes the brain of an average person 250 milliseconds to process a written word. The brain activity associated with word processing can be seen on an electroencephalogram (EEG), which is a procedure that measures the electrical activity of one's brain by placing tiny electrodes on the scalp. For the current research, Dr.

boundaries and the james theorem

Mazaheri and his colleagues used an EEG to study the brain activity of 25 participants while they were shown words on a computer screen.

Participants comprised healthy seniors and elders diagnosed with MCI, as well as MCI patients who had received an Alzheimer's diagnosis within 3 years of being diagnosed with MCI.

There are days when she can make a sentence then others when it is nothing more than gibberish. I feel that dementia is a hard disease to actually diagnose as there are so many others that can cause the same symptoms. So I am wondering how can you really use this as a tester. Am so lucky I found this link that provides knowledge about good health.

boundaries and the james theorem

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