Statistical clicking

Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

In Excel 97 and later versions you would click on Help, then select Contents and Index. This will give you three tabbed options first select Index, and type Analysis ToolPak (without the quotes). A number of choices appear, from which you can select general information . You are now offered three options Install and use Analysis ToolPak, Supplemental information about statistical methods and algorithms, and Ways to analyze statistics. Click on the first choice, and find out how to install the Analysis ToolPak. The same information would be available from Ways to analyze statistics. 

Step 4. Click on Statistical on the left scroll menu and click on L1NEST on the right scroll menu then hit OK. 

Visit www.aaamath.com and, under Math Topics on the right-hand side of the screen, scroll down to click on Statistics. There are brief descriptions, followed by interactive practice problems dealing with mean, median, mode, and range. 

The website www.mathgoodies.com/lessons is another good resource for practice with both statistics and probability. Once on this page, scroll down to Introduction to Statistics, or alternately, Probability. From this topic, click on the X under Description, then scroll down to Challenge Exercises. Apart from the exercises, there are selected lessons you can view. 

A third Internet resource is www.mathleague.com/help/help.htm. From this page, click on either using data and statistics, or percent and probability. 

In a spreadsheet, enter the mean in cell A2 and the standard deviation in cell B2. To find the area under the Gaussian curve from — oo to 900 h in cell C4, we select cell C4 and go to the INSERT menu and choose FUNCTION. In the window that appears, select the Statistical functions and find NORMDIST from the list of possibilities. Double click on NORMDIST and another window appears asking for four values that will be used by NORMDIST. (If you click on help, you will find a cryptic explanation of how to use NORMDIST.)... 

Returning to Figure 4-8, we want to know whether the mean values of the two sets of data are statistically the same or not. In the tools menu, select DATA analysis. In the window that appears, select t-Test Two-Sample Assuming Equal Variances. Click OK. The next window asks you in which cells the data are located. Write B5 B12 for Variable 1 and... 

Summary. Two principal methods for removal of low frequency noise transients are currently available. The model-based separation approach has shown more flexibility and generality, but is computationally rather intensive. It is felt that future work in the area should consider the problem from a realistic physical modelling perspective, which takes into account linear and non-linear characteristics of gramophone and film sound playback systems, in order to detect and correct these artifacts more effectively. Such an approach could involve both experimental work with playback systems and sophisticated non-linear modelling techniques. Statistical approaches related to those outlined in the click removal work (section 4.3.4) may be applicable to this latter task. 

Statistical functions are selected from two menus within Excel. The first approach is through the Function Wizard. Click the Function Wizard, fx to activate the Function Wizard dialog box select Statistical under the Function Category list and... 

Click the Function Wizard icon and select Statistical and LINEST. 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

The 3D-PSSM (Kelley et ah, 2000) server at http //www.bmm.icnet.uk/ 3dpssm/ offers online protein fold recognition. On the submission form, enter your e-mail address and a one-line description of the query protein, then paste the query sequence into the sequence box and click the Submit button. The query sequence is used to search the Fold library for homologues. You will be informed of the URL where the result is located for 4 days. The output includes a summary table (hits with statistics models that can be viewed with RasMol classifications and links) and fold recognition by 3D-PSSM with a printout as exemplified in Figure 12.13. The alignment displays consensus sequence, secondary structures (C for coil, E for extended, and H for helix), and core score (0 for exterior to 9 for interior core). 

The amine function served also as the starting point for the first covalent linkage of Pcs to single-walled carbon nanotubes (SWNTs) [94], The pipes with open-end and surface-bound acyl chloride moieties were used to prepare the Pc-SWNTs system by amide-bond formation (Fig. 14). Accordingly, statistical reaction of 4-aminophthalonitrile with 4-tcr/-bu(yIph111alonitrile in the presence of zinc ions delivered the monoamino Pc that was then employed in the conjugation with the acid chloride modified carbon nanotubes (CNTs). Here, it should also be mentioned that other functions have been applied to the covalent modification of CNTs, i.e., amide [95], ester [96,97], or click chemistry [98],... 

You can check the system resources in Windows 95/98 by using the Resource Meter (select Start > Programs > Accessories > System Tools, if you have installed the Microsoft Plus pack). The resource meter will show up as a small bar graph on your Taskbar at the bottom of the screen. If you double-click the bar graph, it will bring up a screen that shows the available system resources (similar to the Windows 3.1 statistics). Figure 10.2 shows this window. 

When you double-click the printer icon, you will see the screen shown in Figure 16.10. From here you can see any pending print jobs listed as well as their statistics. Notice that there is one print job currently being printed. If you want to stop the printer, you can choose Printer > Pause Printing and Windows will... 

When you have finished setting up the backup, you can click Start Backup. During the backup, MSBACKUP will display how long the backup will take, how many disks it will take, and how much data is being backed up. When the backup is complete, it will display a screen telling you all the statistics about the backup that was performed, including any files that were skipped and the speed (in KB per minute) at which the backup took place. 

We will complete and document the spreadsheet in Chapter 6 after we have explored some of the important calculations of statistical analysis. For now, click on File/Save As. .. in the menu bar, enter a file name such as grav chloride, and save the Excel spreadsheet on a floppy disk or other medium for retrieval and... 

We have now constructed a general-purpose spreadsheet that you may use to make basic statistical calculations. To complete this part of the exercise, select a convenient location, construct a formula to display the number of degrees of freedom, and then add a label in an adjacent cell to identify this important variable. Save the file for future use in problems and laboratory calculations. Use the spreadsheet now to check the calculations in Example 6-1. To clear the data from your worksheet, just click and drag to highlight cells B2 B7, and. strike [Delete]. Alternatively, you may simply click on B2, and begin typing the data. Terminate each piece of data with [ J]. Be sure to delete the data in cells B7 D7. 

Point your Web browser to http //chemistry.brookscole.com/skoogfac/. From the Chapter Resources menu, choose Web Works, and locate the Chapter 7 section. Click on the link to the statistics on-line textbook. Click on the ANOVA/MANOVA button. Read about the partitioning of the sum of squares in ANOVA procedures. Click on the F-distribution link in this section. Look at the tail areas for an F-dis-tribution with both degrees of freedom equal to 10. Determine the value of F for a significance level of 0.10 with both degrees of freedom equal to 10. 

Now, click on cell B8, and then click on the Insert (Paste) Function icon shown in the margin so that the Insert Function window appears, and then click on Statistical. The window appears as follows. 

Note that numerous statistical functions appear in the window labeled Select a function . Use the mouse to scroll down the list of functions until you come to the SLOPE function, and then click on it. The function appears in bold under the left-hand window, and a description of the function appears below it. Read the description of the slope function, and then click OK. The following window appears just below the formula bar. 

Now we see how the LINEST function can accomplish many important functions in a single procedure. Begin by using the mouse to select an array of cells two cells wide and five cells high, such as E2 F6. Then click on the Insert Function icon, select STATISTICAL and LINEST in the left and right windows, respectively, and click on OK. Select the Known y s and Known x s as before, then click on the box labeled Const and type true. Also type true in the box labeled Stats. When you click on each of the latter two boxes, notice that a description of the meaning of these logical variables appears below the box. To activate the LINEST function, you must now type the rather unusual keystroke combination Ctrl-1-Shift+ [J]. This keystroke combination must be used whenever you perform a function on an array of cells. The worksheet should now appear as follows. 

The atom pairs button enables a special mode for the display of a list of atom pairs if the mouse is clicked on a certain descriptor point. The corresponding distance is calculated, and a list of atom pairs together with their original values from the distance matrix is displayed. The statistics button calculates some statistical parameters for the actual descriptor and for a superimposed descriptor if available. The peak areas button separates the individual peaks and calculates the peak areas. The transform buttons enable the display of an additional transformed descriptor. 

Also in the Reference list and threshold menu, optionally select a Background List such as a particular microarray, a custom gene or a network object list, to define the universe of possible network nodes for the most accurate statistical calculations. Click OK. 

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