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http://stdatu.stsci.edu/iue/manual/newsips/node87.html	Next: 8.2.3 1995 Wavelength Calibration Up: 8.2 Low-Dispersion Wavelength Calibration Previous: 8.2.1 Parameterization of the ## 8.2.2 Application of the Dispersion Relations The dispersion solutions derived from individual WAVECAL images display variations from the averaged dispersion solutions discussed above. These variations arise from two sources, the first of which is simply the measurement uncertainties involved in locating the Pt-Ne emission features in the WAVECAL images and the corresponding uncertainties in the coefficients of the derived dispersion solutions. Second, global shifts in the location of the spectral format within individual images are known to occur as a function of camera temperature (THDA) and time (see e.g., Thompson 1988 and Garhart 1993). These shifts occur both parallel and perpendicular to the dispersion direction and consequently result in changes to the wavelength zeropoint (A1) as well as the spatial location (B1) of spectra within the low-dispersion SI. These format shifts appear to be a translational shift only, with no change in image scale. As a result, the higher order dispersion term (A2) remains constant. The dispersion coefficients as actually applied to science images are determined by adding to the mean zeropoint terms corrections appropriate for the observation date and THDA of the particular image. The correction terms and Ws, which represent the offset that is added to the mean wavelength and spatial zeropoints, are defined by the following general expressions: Ws = W1s + W2st + W3sT + W4st2 + W5st3 where and Ws are the corrections to be added to the A1 and B1 terms of the mean dispersion relation, respectively, T is the THDA at the end of exposure, and t is time expressed as the total number of elapsed days since 1 January 1978. A first-order (i.e., linear) fit is sufficient to characterize the correlation between THDA and zeropoint for all cameras. The correlation between time and zeropoint, however, has second- and third-order dependencies for some cameras. The and Ws coefficients in use for the low-dispersion mode for pre-1990 LWP and SWP images are listed in Table 8.2. Coefficients appropriate for more recent LWP and SWP images are discussed in Chapter 8.2.3. The database of exposures used to generate the coefficients in Table 8.2 include WAVECAL images obtained through mid-1993 for the LWP and mid-1991 for the SWP. The respective coefficients for the LWR camera are listed in Table 8.3. In the case of the LWR, a separate set of coefficients are tabulated for the ITF A and ITF B calibrations. In addition, each ITF has an early and late epoch. This is due to the fact that the wavelength zeropoint shift was dramatically different between these two epochs. The cutoff date for the early and late epochs is 1980.1 for ITF A and 1979.9 for ITF B. The shift of the spectrum center does not exhibit this trend, so the spatial coefficients are identical for each epoch. The zeropoint shifts and corresponding polynomial fits are illustrated graphically for each camera in Figures 8.2-8.5. Next: 8.2.3 1995 Wavelength Calibration Up: 8.2 Low-Dispersion Wavelength Calibration Previous: 8.2.1 Parameterization of the Karen Levay 12/4/1997	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8281353116035461, "perplexity": 2232.6562576705674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591150.71/warc/CC-MAIN-20180719164439-20180719184439-00412.warc.gz"}
http://math.stackexchange.com/questions/266193/geometry-in-a-circle?answertab=oldest	# Geometry in a circle Can somebody explain to me why the line $DB$ is equal to $\sin(\alpha + \beta)$. I haven't been able to figure it out. My image (which I would post if I could) is located here: For example, if I try to use the law of cosines then, $$(\cos \alpha)^2 + (\cos \beta)^2 - 2(\cos \alpha) (\cos \beta) (\cos(\alpha + \beta)) = DB$$ which does not obviously equal $\sin(\alpha + \beta)$. - Do you mean the line DB? – Amzoti Dec 28 '12 at 0:12 For any point $E$ on the same arc as $A$, the angle $DEB$ is $\alpha+\beta$. – Andres Caicedo Dec 28 '12 at 6:21 The circle is a circle with unit diameter. Recall the sine rule which goes $$\dfrac{a}{\sin(A)} = \dfrac{b}{\sin(B)} = \dfrac{c}{\sin(C)} = 2R$$ In you case, $2R = 1$, and $\angle A = \alpha + \beta$. And you use the fact that for a cyclic quadrilateral the product of the diagonals is the sum of the product of opposite sides, which is called Ptolemy's theorem, i.e. $$BD \cdot AC = BC \cdot AD + AB \cdot CD$$ to conclude that $$\sin(\alpha + \beta) \times 1 = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$$ - The reason why that line has length $\sin(\alpha+\beta)$ is the same as the reason why the line of length $\sin\alpha$ has that length: In a circle of unit diameter, the length of the chord connecting the endpoints of an arc is the sine of an angle whose vertex is on the circle and whose rays pass through the endpoints of the arc. - (I use $a$ and $b$ instead of $\alpha$ and $\beta$ because I am lazy.) $(\cos a)^2 + (\cos b)^2 - 2(\cos a) (\cos b) (\cos(a + b))$ $= (\cos a)^2 + (\cos b)^2 - 2(\cos a)(\cos b)(\cos a \cos b - \sin a \sin b)$ $=(\cos a)^2 + (\cos b)^2 - 2(\cos a)^2(\cos b)^2 + 2 (\cos a)(\cos b)(\sin a)(\sin b) $$= (\cos a)^2(1-(\cos b)^2) + (\cos b)^2(1-(\cos a)^2) + 2(\cos a)(\cos b)(\sin a \sin b)$$= (\cos a)^2 (\sin b)^2 + (\cos b)^2 (\sin a)^2 + 2(\cos a)(\cos b)(\sin a \sin b)$ $= (\cos a \sin b + \cos b \sin a)^2$ $= (\sin(a+b))^2$. So I think the expression you have is for $DB^2$, not $DB$, but, other than that, your expression was correct - just not simplified. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9348791837692261, "perplexity": 263.7294717664242}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830834.3/warc/CC-MAIN-20140820021350-00329-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/physics-histogram-help.244772/	Physics histogram help 1. Jul 13, 2008 Niki4444 Hello! I have a question. When I am provided with a histogram, with % of molecules along the y-axis and speed range (m/s) along the x-axis, how do I use this to determine the average speed (v avg) and v-rms (root mean square) of the molecules in the gas? I am very puzzled. Any suggestions or thoughts to point me in the right direction would be hugely appreciated. Thank you so much. 2. Jul 13, 2008 dynamicsolo For the average speed, you will perform what is called a "weighted average" on the histogram. You have some number of bins for velocity and some percentage of the total number of molecules in each speed bin. Convert the percentages into decimal fractions (e.g., 6% = .06). Find the center of the velocity range for each bin. You will then add up the terms: (fraction in first bin) · (speed midpoint for first bin) + (fraction in second bin) · (speed midpoint for second bin) + ... + (fraction in last bin) · (speed midpoint for last bin). The sum will be the weighted average for molecular speed. You do something similar for root-mean-square speed, but now the sum is: (fraction in first bin) · (speed midpoint for first bin)^2 + (fraction in second bin) · (speed midpoint for second bin)^2 + ... + (fraction in last bin) · (speed midpoint for last bin)^2 [that is to say, square each speed midpoint first, multiply that by its corresponding fraction, then sum all the terms]. Finally, take the square root of the sum you've found; this is your root-mean-squared speed. It will be a different result from the weighted average. Last edited: Jul 13, 2008 3. Jul 13, 2008 Niki4444 Thank you! That makes much more sense than the confusion I was having over what is v-avg, what is v-rms, and what is v-squared.... from a picture. Thank you, thank you, thank you! I get it now. Have something to add? Similar Discussions: Physics histogram help	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8841590285301208, "perplexity": 1939.019000284553}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280504.74/warc/CC-MAIN-20170116095120-00471-ip-10-171-10-70.ec2.internal.warc.gz"}
http://mathoverflow.net/users/9871/diverietti?tab=activity	# diverietti less info reputation 927 bio website math.jussieu.fr/~diverio location Paris age 34 member for 3 years, 5 months seen yesterday profile views 2,558 I am a researcher in complex analytic and algebraic geometry at the Institut de Mathématiques de Jussieu. # 469 Actions Feb23 comment Rational or elliptic curves on Calabi-Yau threefolds @LiYutong, this is because trivial canonical class implies existence of Ricci flat Kähler metrics (this is Yau), then Kobayashi-Lübke inequalities give you that $c_2(X)\ge 0$ and you have equality if and only if the Ricci flat metric is indeed flat itself. Now you conclude by the classical Bieberbach theorem. Nov13 awarded Enlightened Nov13 awarded Nice Answer Oct7 awarded Yearling Sep30 revised When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$? fixed a typo in the name of Angelo Vistoli. Sep30 comment Euler Sequence on Homogeneous Spaces Your proof is more or less the proof I knew... Sep23 comment motivation for multiplier ideal sheaves I don't think I have much to add to this very detailed answer. It is anyhow amazing how many others applications there are, completely independent of the original motivation! For an excellent reference, see Lazarsfeld's book "Positivity in Algebraic Geometry". Sep23 comment motivation for multiplier ideal sheaves Since you wrote my name explicitly, I'll try to write something later :) Sep11 comment Variety $X$ such that $TX$ is ample on any curve in $X$ yes, there is a minus sign before the canonical. Sep5 awarded complex-geometry Sep4 awarded Self-Learner Sep4 revised holomorphic sectional curvature and total scalar curvature added tag Sep4 answered holomorphic sectional curvature and total scalar curvature Sep2 comment Yau's conjecture for positive Chern class Yes, maybe for the sake of completeness, and since moreover your answer has been accepted, you could add also Tian's contribution to the subject. Best, Aug16 comment Rational or elliptic curves on Calabi-Yau threefolds Arbitrary I don't know, generic yes. But probably, this you already know! Aug13 revised If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$ added 2 characters in body Aug13 comment If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$ Problem solved, I guess :) Aug13 revised If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$ Clarified and improved last argument Aug13 revised Example of Calabi-Yau 3-fold fibered by both K3 surface and abelian surface? fixed some typos Aug12 revised If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$ nota bene added	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8887534141540527, "perplexity": 1411.7300476393964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999639602/warc/CC-MAIN-20140305060719-00049-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/singularity-in-laplacian-operator.627767/	# Singularity in Laplacian operator 1. Aug 12, 2012 ### MrHigh I am trying to understand the following basic problem, $\partial_{xx} f^\alpha (x) = \alpha (\alpha-1) \frac{1}{f^{2-\alpha}} \partial_x f + \alpha \frac{1}{f^{1-\alpha}} \partial_{xx} f$ So it is not hard to see that if $f$ tends to zero the laplacian becomes undefined (im not sure if i am using proper terminology). that should happen for any $\alpha < 2$. I don't understand why the singulary happens and how the trivial f(x) = 0 for all x can be applied. thanks for the attention. Last edited: Aug 12, 2012	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9777933955192566, "perplexity": 566.3184958973902}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160641.81/warc/CC-MAIN-20180924185233-20180924205633-00004.warc.gz"}
http://rrmh.org/ionescu-s-irna/e32066-what-is-the-following-quotient-3-sqrt-8	Using the positive square roots: sqrt(16) / sqrt(8) = 4 / 2sqrt(2) which reduces to 2 / sqrt(2). What is the domain of the function given below? Find the following quotient and express the answer in standard form of a complex number. a) f(x) = â¦ 3 â¦ Answer Save. See the answer. notthejake. NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. Expert Answer . The square root of 8 is the square root of 4 times the square root of 2. What is the simplified form of ? Simplify the following quotient of complex numbers into the form a + bi. When -17 is divided by 3. HELLLPPPP! What is the simplified form of ? The following is called the quotient rule: "The derivative of the quotient of two functions is equal to . The slope is important in calculus â¦ For example, accepting for the moment that the derivative of sin x â¦ Assume x â 0. Three multiplied by three equals nine, but negative three multiplied by negative three also equals nine, so. Determine the difference quotient of p(x)=2+3(sqrt(9x-7))? PLAY. $4\sqrt x + 9\sqrt x = \left( {4 + 9} \right)\sqrt x = 13\sqrt x \hspace{0.25in}3\,\,\sqrt[{10}]{5} - 11\,\,\sqrt[{10}]{5} = \left( {3 - 11} \right)\sqrt[{10}]{5} = - 8\,\,\sqrt[{10}]{5}$ Weâve already seen some multiplication of radicals in the last part of the previous example. Download PDF's . The only answer that's correct (but not in its simplest form) is A. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the â¦ The quotient of 12 apples by 3 apples is 4. Biology. Learn. When -73 is divided by 7. â¦ Math-Help asap please~ For each function, the point given is the maximum or minimum. For each of the following, find the quotient and remainder (guaranteed by the Division Algorithm) and then summarize the results by writing an equation of the form $$a = bq + r$$, where $$0 \le r < b$$. Surds fraction calculator (square root quotient) The online square root calculator can symplify surds root quotients in exact form. 2-sqrt 8/4+sqrt12; Question: What Is The Following Quotient? NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. Calculate the average rate of change for the given graph from x = 3 to x = 5 and select the correct answer below.Courtesy of Texas Instruments a.5 b.3 c.1 d.0 Asked By adminstaff @ 25/12/2019 08:24 AM Graphing a function or exploring a table of values to determine a limit can â¦ The velocity v of a freefalling skydiver is â¦ The difference quotient allows you to find the derivative, which â¦ View Answer. Test. Every positive real number has two square roots, one positive and one negative. Maths. i know the difference quotient formula is f(x)=f(x+h)-f(x)/h . It can be helpful to separate the numerator and denominator of a â¦ 4 1/3 4 1/2 4 3/4 4 7/8 Please explain how to do the problems 2 and 3 :(. When 17 is divided by 3. Differential Equations. Flashcards. Simplify the radical expression sqrt 56x^2 28x 2x sqrt 14*** 2x sqrt 7 sqrt 14x2 3. Match. Here we use the limit definition of the derivative to compute d/dx sqrt(x+3). the 2+3 before the squareroot is getting me anxious!! Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. Working a difference quotient involving a square root Suppose f(x) = p x and suppose we want to simplify the differnce quotient f(x+h) f(x) h as much as possible (say, to eliminate the h in the denominator). 8 â¦ The remainder is $$(33 + 19 \sqrt{3}) \cdot\frac{\sqrt{3}}{3}=19 + 11 \sqrt{3}$$ Notice that the last remainder $19 + 11 \sqrt{3}$ divides the last divider $33 + 19 \sqrt{3}$. What is the following quotient 1/1+ sqrt 3 What is the following quotient 9+ sqrt 2/4 - sqrt7 What is the following profuct? The square root of 4 is either 2 or -2. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange Section 1-8 Radical Rules PRODUCT RULE: QUOTIENT RULE: a √ xâ a √ y = a √ xy a √ x a √ y = a √ x y Example: Example: √ 10 â √ x = √ 10 x √ 10 √ 2 = √ 10 2 = √ 5 More directly, when determining a product or quotient of radicals and the indices (the small â¦ i have to make a 100 on this project and this is the only problem that is causing me to rethink myself. What is the following quotient 3 sqrt 8/ 4 sqrt 6 . Jan 3â¦ Evaluate the expression 1 + 1 4 + 9 and write the result in the form a+bi . Therefore, we stop here, the $\gcd$ is $19 + 11 \sqrt{3}$. When 73 is divided by 7. STUDY. Remember that every number actually has two square roots. NCERT RD Sharma Cengage KC Sinha. Answer to: Use the Product Rule or the Quotient Rule to find the derivative of the given function. Step 2: Now click the button âCalculate Quotientâ to get the result Step 3: Finally, the difference quotient will be displayed in the new window. What is Meant by Difference Quotient? 1+3/5 1+3x/5 1-3x/5 * 25-15x * is my . You perform this exercise because the difference quotient is the basis of the definition of the derivative. )Ben's cookie recipe calls for 3 1/4 cups of flour. Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. Integration by trigonometric substitution Calculator online with solution and steps. 2-sqrt 8/4+sqrt12. Using the Quotient Rule to Simplify Square Roots. Since $$(â4)^{2}=16$$, we can say that â4 is a square root of 16 as well. _____ 3. sqrt 490y^5w^6 2 sqrt . Physics. As next, we will â¦ For example, three squared is nine (3 2 = 9), so the square root of nine is three. The square root of a number is that number that when multiplied by itself yields the original number. 0 0 379; Katy. 1 Answer. Thus, for calculating the product of the following square roots sqrt(33)sqrt(6), enter simplify_surd(sqrt(33)sqrt(6)), the result 3sqrt(22) is returned. :) First of all, let me simplify the expression using the following power rule: sqrt(x) = x^(1/2) and the following logarithmic rule: ln(a^r) = r ln(a) Thus, we can simplify as follows: ln sqrt((2x-8)/(3x+3)) = ln [((2x-8)/(3x+3)) ^(1/2)] = 1/2 * ln ((2x-8)/(3x+3)) This is easier already. Books. 10x18 10x6 50x18 50x6. 3/4 divided by using 2/3 3/4/2/3 This seems a liitle daunting until now each and every thing, yet merely think of of the case of two/2. what is the following quotient? 5-3x/5? Answer to: Let f be a differentiable function such that f(9) = 18 and f (9) = 7 . The square root of 16 is either 4 or -4. a. Simplify the radical expression. f'(x) = (15)/((3x+3)(2x-8)) Let's do this. Previous question Next question Get more help from Chegg. It can be helpful to separate the numerator and denominator of a â¦ This problem has been solved! Get 1:1 help now from â¦ Created by. Gravity. NCERT P Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan. The final solution is {eq}\boxed{\color{blue}{ \sqrt{3\sqrt{3} + 3 + 8\sqrt{7}}}} {/eq} Become a member and unlock all Study Answers Try it risk-free for 30 days No, Because one has a quotient of 0.83 and the other has a quotient of 0.916 No, because one ratio reduce to5/6 and the other reduces to 3/4. Finding the Limit of a Sum, a Difference, and a Product. Write. noelle_lee6. you could convert this to 2a million/2 So, 3/4/2/3 might properly be converted to 3/43/2 this is 9/8 Relevance. In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced / Ë k w oÊ Ê Én t /) is a quantity produced by the division of two numbers. Substituting the deï¬nition of f into the quotient, we have f(x+h) f(x) h = p x+h x h at which point we are stuck, as far as â¦ The difference quotient shows up in most high school Algebra II classes as an exercise you do after your instructor shows you the composition of functions. If we are looking at the product of two radicals â¦ 8/x-m * 8-x/m x-m/8 8-x/m 2.Which sum of difference is equivalent to the following expression? Spell. sqrt -11 sqrt -55 I am having trouble with these 3 problems, Please be detailed so i know how to get the answer â¦ Lv 7. Terms in this set (10) What is the simplified form of â100x36? Using the Quotient Rule to Simplify Square Roots. Square root of -44 Also Simplify the following expression. the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the square of the denominator." Note: the division process is very similar to â¦ To evaluate expressions with exponents, refer to the rules of exponents in the table below. The Square Root Function. Solved exercises of Integration by trigonometric substitution. A useful mathematical differentiation calculator to simplify the functions. What Is The Following Quotient? 2-sqrt 8/4+sqrt12. â¦ Find three consecutive integers such that the sum of the first and twice the second is equal to the third plus four. if the recipe was increase by a factr of 1 1/2, how much flour would ben need? Use the difference quotient to verify that the slope of the tangent at this point is zero. For example, 4 is a square root of 16, because $$4^{2}=16$$. In symbols, this is \sqrt{9} = 3. Use differentials to approximate (25)^1/3? What is the Algebraic expression for the following word phrase: the quotient of 8 and the difference of x and m? a. ? Assume 36x3 â¦ 6/5x10 6/5x2 6/5x10 6/5x2. -8-5i/6-3i Also Simplify the following expression; express in terms of i. Therefore the integral quotient is $3 - \sqrt{3}$. Chemistry. In Calculus, the difference quotient is used to determine the slope of the secant line between two points. The âââ symbol tells you to take the square root of a number, and you can find this on most calculators. An online derivative calculator that differentiates a given function with respect to a given variable by using analytical differentiation. Free simplify calculator - simplify algebraic expressions step-by-step Class 12 Class 11 Class 10 Class 9 Class 8 â¦ Detailed step by step solutions to your Integration by trigonometric substitution problems online with our math solver and calculator. Root calculator can symplify surds root quotients in exact form solver and calculator 8 is the following quotient by analytical. Number actually has two square roots is either 2 or -2 by trigonometric substitution calculator online solution. Question: What is the maximum or minimum factr of 1 1/2, how much flour would need... Which â¦ Integration by trigonometric substitution problems online with our math solver and calculator if the recipe was by... Would Ben need by step solutions to your Integration by trigonometric substitution problems online with our math solver calculator... Slope of the derivative to compute d/dx sqrt ( 9x-7 ) ) detailed step step. Use the Product Rule or the quotient Rule: the derivative of the quotient Rule find! =F ( x+h ) -f ( x ) =2+3 ( sqrt ( 9x-7 ) ) 19... That every number actually has two square roots, one what is the following quotient 3 sqrt 8 and negative... A factr of 1 1/2, how much flour would Ben need root quotients in exact form of. 2 } =16\ ) ; express in terms of i on this project and this \sqrt! 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Derivative calculator that differentiates a given variable by Using analytical differentiation important in Calculus â¦ square!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9363487362861633, "perplexity": 991.9694250797356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488259200.84/warc/CC-MAIN-20210620235118-20210621025118-00475.warc.gz"}
http://worldwidescience.org/topicpages/r/reflexive+banach+space.html	Sample records for reflexive banach space 1. The cofinal property of the Reflexive Indecomposable Banach spaces OpenAIRE Argyros, Spiros A; Raikoftsalis, Theocharis 2010-01-01 It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\\ell_p$ saturated space with $1 2. Rigidity of commuting affine actions on reflexive Banach spaces OpenAIRE Rosendal, Christian 2012-01-01 We give a simple argument to show that if {\\alpha} is an affine isometric action of a product G x H of topological groups on a reflexive Banach space X with linear part {\\pi}, then either {\\pi}(H) fixes a unit vector or {\\alpha}\|G almost fixes a point on X. It follows that any affine isometric action of an abelian group on a reflexive Banach space X, whose linear part fixes no unit vectors, almost fixes points on X. 3. On J. Borwein's concept of sequentially reflexive Banach spaces OpenAIRE Ørno, Peter 1991-01-01 A Banach space$X$is reflexive if the Mackey topology$\\tau(X^,X)$on$X^$agrees with the norm topology on$X^$. Borwein [B] calls a Banach space$X${\\it sequentially reflexive\\/} provided that every$\\tau(X^,X)$convergent {\\it sequence\\/} in$X^$is norm convergent. The main result in [B] is that$X$is sequentially reflexive if every separable subspace of$X$has separable dual, and Borwein asks for a characterization of sequentially reflexive spaces. Here we answer that question b... 4. Metrical characterization of super-reflexivity and linear type of Banach spaces OpenAIRE Baudier, Florent 2007-01-01 We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric. 5. First and Second Order Convex Sweeping Processes in Reflexive Smooth Banach Spaces International Nuclear Information System (INIS) In this paper we establish new characterizations of the normal cone of closed convex sets in reflexive smooth Banach spaces and then we use those results to prove the existence of solutions for first order convex sweeping processes and their variants in reflexive smooth Banach spaces. The case of second order convex sweeping processes is also studied. (author) 6. Some fixed point theorems in fuzzy reflexive Banach spaces International Nuclear Information System (INIS) In this paper, we first show that there are some gaps in the fixed point theorems for fuzzy non-expansive mappings which are proved by Bag and Samanta, in [Bag T, Samanta SK. Fixed point theorems on fuzzy normed linear spaces. Inf Sci 2006;176:2910-31; Bag T, Samanta SK. Some fixed point theorems in fuzzy normed linear spaces. Inform Sci 2007;177(3):3271-89]. By introducing the notion of fuzzy and α- fuzzy reflexive Banach spaces, we obtain some results which help us to establish the correct version of fuzzy fixed point theorems. Second, by applying Theorem 3.3 of Sadeqi and Solati kia [Sadeqi I, Solati kia F. Fuzzy normed linear space and it's topological structure. Chaos, Solitons and Fractals, in press] which says that any fuzzy normed linear space is also a topological vector space, we show that all topological version of fixed point theorems do hold in fuzzy normed linear spaces. 7. Composite iterative schemes for maximal monotone operators in reflexive Banach spaces Directory of Open Access Journals (Sweden) Cho Yeol 2011-01-01 Full Text Available Abstract In this article, we introduce composite iterative schemes for finding a zero point of a finite family of maximal monotone operators in a reflexive Banach space. Then, we prove strong convergence theorems by using a shrinking projection method. Moreover, we also apply our results to a system of convex minimization problems in reflexive Banach spaces. AMS Subject Classification: 47H09, 47H10 8. On solvability of general nonlinear variational-like inequalities in reflexive Banach spaces Directory of Open Access Journals (Sweden) Soo Hak Shim 2005-07-01 Full Text Available We introduce and study a new class of general nonlinear variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality, we establish two existence and uniqueness theorems of solutions for the general nonlinear variational-like inequality. 9. Convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive Banach spaces International Nuclear Information System (INIS) Let E be a real reflexive Banach space with uniformly Gateaux differentiable norm. Let K be a nonempty closed convex subset of E. Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive mappings. Let T1, T2, ..., TN be a family of nonexpansive self-mappings of K, with F := intersectioni=1N Fix(Ti) ≠ 0, F = Fix(TNTN-1... T1) = Fix(T1TN ... T2) = ... Fix(TN-1TN-2... T1TN). Let { λn} be a sequence in (0, 1) satisfying the following conditions: C1 : lim λn 0; C2 : Σ λn = ∞ . For a fixed δ element of (0, 1), define Sn : K → K by Snx := (1 - δ )x + δTnx for all x element of K where Tn = Tn mod N. For arbitrary fixed u, x0 element of K, let B := { x element of K : TNTN-1... T1x γx+(1- γ)u, for some γ > 1} be bounded and let the sequence {xn} be defined iteratively by xn+1 λn+1u + (1 - λn+1 )Sn+1xn, for n ≥ 0. Assume that lim n→∞ vertical bar vertical bar Tnxn - Tn+1 xn vertical bar vertical bar = 0. Then, {xn} converges strongly to a common fixed point of the family T1, T2, ..., TN. Convergence theorem is also proved for non-self maps. (author) 10. Genus$n$Banach spaces OpenAIRE Casazza, Peter G.; Lammers, Mark C. 1999-01-01 We show that the classification problem for genus$n$Banach spaces can be reduced to the unconditionally primary case and that the critical case there is$n=2$. It is further shown that a genus$n$Banach space is unconditionally primary if and only if it contains a complemented subspace of genus$(n-1)$. We begin the process of classifying the genus 2 spaces by showing they have a strong decomposition property. 11. Topological duals of Banach spaces as Banach algebras and applications International Nuclear Information System (INIS) Banach algebra structures on the topological dual X of a Banach space X are defined, and some algebraic and topological properties of such Banach algebras are studied. In this paper, on the second dual space X of a Banach space X, different products, each of which turns X into an associative Banach algebra which satisfies similar properties are also defined. 13 refs 12. Multipliers of Banach valued weighted function spaces Directory of Open Access Journals (Sweden) Serap Öztop 2000-10-01 Full Text Available We generalize Banach valued spaces to Banach valued weighted function spaces and study the multipliers space of these spaces. We also show the relationship between multipliers and tensor product of Banach valued weighted function spaces. 13. Separably injective Banach spaces CERN Document Server Avilés, Antonio; Castillo, Jesús M F; González, Manuel; Moreno, Yolanda 2016-01-01 This monograph contains a detailed exposition of the up-to-date theory of separably injective spaces: new and old results are put into perspective with concrete examples (such as l∞/c0 and C(K) spaces, where K is a finite height compact space or an F-space, ultrapowers of L∞ spaces and spaces of universal disposition). It is no exaggeration to say that the theory of separably injective Banach spaces is strikingly different from that of injective spaces. For instance, separably injective Banach spaces are not necessarily isometric to, or complemented subspaces of, spaces of continuous functions on a compact space. Moreover, in contrast to the scarcity of examples and general results concerning injective spaces, we know of many different types of separably injective spaces and there is a rich theory around them. The monograph is completed with a preparatory chapter on injective spaces, a chapter on higher cardinal versions of separable injectivity and a lively discussion of open problems and further lines o... 14. On asymptotic models in Banach spaces OpenAIRE Halbeisen, Lorenz; Odell, Edward 2001-01-01 A well known application of Ramsey's Theorem to Banach Space Theory is the notion of a spreading model (e'_i) of a normalized basic sequence (x_i) in a Banach space X. We show how to generalize the construction to define a new creature (e_i), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the... 15. Adjoint Operators on Banach Spaces CERN Document Server Gill, Tepper L; Zachary, Woodford W 2010-01-01 In this paper, we report on new results related to the existence of an adjoint for operators on separable Banach spaces and discuss a few interesting applications. (Some results are new even for Hilbert spaces.) Our first two applications provide an extension of the Poincar\\'{e} inequality and the Stone-von Neumann version of the spectral theorem for a large class of$C_0$-generators of contraction semigroups on separable Banach spaces. Our third application provides a natural extension of the Schatten-class of operators to all separable Banach spaces. As a part of this program, we introduce a new class of separable Banach spaces. As a side benefit, these spaces also provide a natural framework for the (rigorous) construction of the path integral as envisioned by Feynman. 16. A pythagorean approach in Banach spaces OpenAIRE Gao Ji 2006-01-01 Let be a Banach space and let be the unit sphere of . Parameters , , , and , where and are introduced and studied. The values of these parameters in the spaces and function spaces are estimated. Among the other results, we proved that a Banach space with , or is uniform nonsquare; and a Banach space with , or has uniform normal structure. 17. Fusion Frames and -Frames in Banach Spaces Indian Academy of Sciences (India) Amir Khosravi; Behrooz Khosravi 2011-05-01 Fusion frames and -frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, -frames, Banach -frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that -frames, fusion frames and Banach -frames are stable under small perturbations and invertible operators. 18. Smooth analysis in Banach spaces CERN Document Server Hjek, Petr 2014-01-01 This bookis aboutthe subject of higher smoothness in separable real Banach spaces.It brings together several angles of view on polynomials, both in finite and infinite setting.Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into in 19. Characterizing R-duality in Banach spaces DEFF Research Database (Denmark) Christensen, Ole; Xiao, Xiang Chun; Zhu, Yu Can 2013-01-01 R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces.......R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces.... 20. Existence of zeros for operators taking their values in the dual of a Banach space Directory of Open Access Journals (Sweden) Ricceri Biagio 2004-01-01 Full Text Available Using continuous selections, we establish some existence results about the zeros of weakly continuous operators from a paracompact topological space into the dual of a reflexive real Banach space. 1. On factorization of operators between Banach spaces OpenAIRE Tokarev, Eugene 2002-01-01 A finite-dimensional analogue of the known Gordon-Lewis constant of a Banach space X is introduced; in its definition are used only finite rank operators. It is shown that there exist Banach spaces such that the standard Gordon-Lewis constant of X is finite and its finite-dimensional analogue GLfin(X) is infinite. Moreover, for any Banach space X of cotype 2 its finite-dimensional constant GLfin(X) is finite too. 2. Isometries on Banach spaces function spaces CERN Document Server Fleming, Richard J 2002-01-01 Fundamental to the study of any mathematical structure is an understanding of its symmetries. In the class of Banach spaces, this leads naturally to a study of isometries-the linear transformations that preserve distances. In his foundational treatise, Banach showed that every linear isometry on the space of continuous functions on a compact metric space must transform a continuous function x into a continuous function y satisfying y(t) = h(t)x(p(t)), where p is a homeomorphism and \|h\| is identically one.Isometries on Banach Spaces: Function Spaces is the first of two planned volumes that survey investigations of Banach-space isometries. This volume emphasizes the characterization of isometries and focuses on establishing the type of explicit, canonical form given above in a variety of settings. After an introductory discussion of isometries in general, four chapters are devoted to describing the isometries on classical function spaces. The final chapter explores isometries on Banach algebras.This treatment p... 3. Regularization methods in Banach spaces CERN Document Server Schuster, Thomas; Hofmann, Bernd; Kazimierski, Kamil S 2012-01-01 Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Usually the mathematical model of an inverse problem consists of an operator equation of the first kind and often the associated forward operator acts between Hilbert spaces. However, for numerous problems the reasons for using a Hilbert space setting seem to be based rather on conventions than on an approprimate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, sparsity constraints using general Lp-norms or the B 4. Compactness in Banach space theory - selected problems OpenAIRE Avilés, Antonio; Kalenda, Ondřej F. K. 2010-01-01 We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia compacta and Radon-Nikod\\'{y}m compacta. 5. Banach spaces of analytic functions CERN Document Server Hoffman, Kenneth 2007-01-01 A classic of pure mathematics, this advanced graduate-level text explores the intersection of functional analysis and analytic function theory. Close in spirit to abstract harmonic analysis, it is confined to Banach spaces of analytic functions in the unit disc.The author devotes the first four chapters to proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the f 6. Coefficient Quantization for Frames in Banach Spaces OpenAIRE Casazza, P. G.; Dilworth, S. J.; Odell, E.; Schlumprecht, Th.; Zsak, Andras 2007-01-01 Let$(e_i)$be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces. 7. Geometry of Banach spaces and biorthogonal systems OpenAIRE Dilworth, S. J.; Girardi, Maria; Johnson, W. B. 1999-01-01 A separable Banach space X contains$\\ell_1$isomorphically if and only if X has a bounded wc_0^-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded wc_0^-biorthogonal system. 8. Simultaneous approximation in scales of Banach spaces International Nuclear Information System (INIS) The problem of verifying optimal approximation simultaneously in different norms in a Banach scale is reduced to verification of optimal approximation in the highest order norm. The basic tool used is the Banach space interpolation method developed by Lions and Peetre. Applications are given to several problems arising in the theory of finite element methods 9. On Uniformly finitely extensible Banach spaces OpenAIRE Castillo, Jesús M. F.; Ferenczi, Valentin; Moreno, Yolanda 2013-01-01 We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, \\emph{On automorphic Banach spaces}, Israel J. Math. 169 (2009) 29--45 and Castillo-Plichko, \\emph{Banach spaces in various positions.} J. Funct. Anal. 259 (2010) 2098-2138. We show that they have the Uniform Approximation Property of Pe\\l czy\\'nski and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstraus... 10. Condensation rank of injective Banach spaces OpenAIRE Gazor, Majid 2011-01-01 The condensation rank associates any topological space with a unique ordinal number. In this paper we prove that the condensation rank of any infinite dimensional injective Banach space is equal to or greater than the first uncountable ordinal number. 11. Isometric embeddings of compact spaces into Banach spaces OpenAIRE Dutrieux, Yves; Lancien, Gilles 2008-01-01 We show the existence of a compact metric space$K$such that whenever$K$embeds isometrically into a Banach space$Y$, then any separable Banach space is linearly isometric to a subspace of$Y$. We also address the following related question: if a Banach space$Y$contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space$X$, does it necessarily contain a subspace isometric to$X$? We answer positively this question when$X$is a polyhedral fi... 12. On Λ-Type Duality of Frames in Banach Spaces Directory of Open Access Journals (Sweden) Renu Chugh 2013-11-01 Full Text Available Frames are redundant system which are useful in the reconstruction of certain classes of spaces. The dual of a frame (Hilbert always exists and can be obtained in a natural way. Due to the presence of three Banach spaces in the definition of retro Banach frames (or Banach frames duality of frames in Banach spaces is not similar to frames for Hilbert spaces. In this paper we introduce the notion of Λ-type duality of retro Banach frames. This can be generalized to Banach frames in Banach spaces. Necessary and sufficient conditions for the existence of the dual of retro Banach frames are obtained. A special class of retro Banach frames which always admit a dual frame is discussed. 13. A Gaussian Average Property for Banach Spaces OpenAIRE Casazza, Peter G.; Nielsen, Niels Jorgen 1996-01-01 In this paper we investigate a Gaussian average property of Banach spaces. This property is weaker than the Gordon Lewis property but closely related to this and other unconditional structures. It is also shown that this property implies that certain Hilbert space valued operators defined on subspaces of the given space can be extended. 14. On Sharp Triangle Inequalities in Banach Spaces II Directory of Open Access Journals (Sweden) Kichi-Suke Saito 2010-01-01 Full Text Available Sharp triangle inequality and its reverse in Banach spaces were recently showed by Mitani et al. (2007. In this paper, we present equality attainedness for these inequalities in strictly convex Banach spaces. 15. On Banach spaces without the approximation property OpenAIRE Reinov, Oleg I. 2002-01-01 A. Szankowski's example is used to construct a Banach space similar to that of "An example of an asymptotically Hilbertian space which fails the approximation property", P.G. Casazza, C.L. Garc\\'{\\i}a, W.B. Johnson [math.FA/0006134 ()]. 16. Completely positive linear operators for Banach spaces Directory of Open Access Journals (Sweden) Mingze Yang 1994-03-01 Full Text Available Using ideas of Pisier, the concept of complete positivity is generalized in a different direction in this paper, where the Hilbert space Ã¢Â„Â‹ is replaced with a Banach space and its conjugate linear dual. The extreme point results of Arveson are reformulated in this more general setting. 17. Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces OpenAIRE Sergey Ajiev 2013-01-01 We introduce and study Clarkson, Dolnikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hlder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide... 18. Boundary Controllability of Integrodifferential Systems in Banach Spaces Indian Academy of Sciences (India) K Balachandran; E R Anandhi 2001-02-01 Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory. 19. Discretization of variational regularization in Banach spaces CERN Document Server Poeschl, C; Scherzer, O 2010-01-01 Consider a nonlinear ill-posed operator equation$F(u)=y$where$F$is defined on a Banach space$X$. In general, for solving this equation numerically, a finite dimensional approximation of$X$and an approximation of$F$are required. Moreover, in general the given data$\\yd$of$y$are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the$L^\\infty$--space. 20. Nonlinear evolution equations on Banach space OpenAIRE Ahmed, N.U. 1991-01-01 In this paper we consider the questions of existence and uniqueness of solutions of certain semilinear and quasilinear evolution equations on Banach space. We consider both deterministic and stochastic systems. The approach is based on semigroup theory and fixed point theorems. Our results allow the nonlinear perturbations in all the semilinear problems to be bounded or unbounded with reference to the base space, thereby increasing the scope for applications to partial differential equations.... 1. Fixed point theorems for generalized Lipschitzian semigroups in Banach spaces Directory of Open Access Journals (Sweden) Jong Soo Jung 1999-03-01 Full Text Available Fixed point theorems for generalized Lipschitzian semigroups are proved in p-uniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, in Lp spaces, in Hardy space Hp, and in Sobolev spaces Hk,p, for 1 2. p-representable operators in Banach spaces Directory of Open Access Journals (Sweden) Roshdi Khalil 1986-12-01 Full Text Available Let E and F be Banach spaces. An operator TâL(E,F is called p-representable if there exists a finite measure Î¼ on the unit ball, B(E, of E and a function gâLq(Î¼,F, 1p+1q=1, such thatTx=â«B(Eâ©x,xâªg(xdÎ¼(xfor all xâE. The object of this paper is to investigate the class of all p-representable operators. In particular, it is shown that p-representable operators form a Banach ideal which is stable under injective tensor product. A characterization via factorization through Lp-spaces is given. 3. Boundedness of biorthogonal systems in Banach spaces Czech Academy of Sciences Publication Activity Database Hájek, Petr Pavel; Montesinos, V. 2010-01-01 Roč. 177, č. 1 (2010), s. 145-154. ISSN 0021-2172 R&D Projects: GA ČR GA201/07/0394 Institutional research plan: CEZ:AV0Z10190503 Keywords : M-basis * Banach spaces Subject RIV: BA - General Mathematics Impact factor: 0.630, year: 2010 http://link.springer.com/article/10.1007%2Fs11856-010-0041-x 4. Topological Vector-Space Valued Cone Banach Spaces OpenAIRE Nayyar Mehmood; Akbar Azam; Suzana Aleksic 2014-01-01 In this paper we introduce the notion of tvs-cone normed spaces, discuss related topological concepts and characterize the tvs-cone norm in various directions. We construct generalize locally convex tvs generated by a family of tvs-cone seminorms. The class of weak contractions properly includes large classes of highly applicable contractions like Banach, Kannan, Chatterjea and quasi etc. We prove fixed point results in tvs-cone Banach spaces for nonexpansive self mappings and self/non-self w... 5. Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions CERN Document Server Betancor, Jorge J; Rodríguez-Mesa, Lourdes 2011-01-01 In this paper we consider the space$BMO_o(\\mathbb{R},X)$of bounded mean oscillations and odd functions on$\\mathbb{R}$taking values in a UMD Banach space$X$. The functions in$BMO_o(\\mathbb{R},X)$are characterized by Carleson type conditions involving Bessel convolutions and$\\gamma$-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain$\\gamma$-radonifying Carleson inequalities for Bessel-Poisson integrals of$BMO_o(\\mathbb{R},X)$functions hold. 6. The Diameter of the Isomorphism Class of a Banach Space OpenAIRE Johnson, W. B.; Odell, E. 2003-01-01 We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c_0. We call X elastic if for some K < infinity for every Banach space Y which embeds into X, the space Y is K-isomorphic to a subspace of X. We also prove that if X is a separable Banach space such that for some K < infinity every isomorph of X is K-e... 7. Introduction to Banach spaces and algebras CERN Document Server Allan, Graham 2010-01-01 Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complexanalysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration.The text be 8. Some results concerning frames in Banach spaces OpenAIRE S. K. Kaushik 2007-01-01 A necessary and sufficient condition for the associated sequence of functionals to a complete minimal sequence to be a Banach frame has been given. We give the definition of a weak-exact Banach frame, and observe that an exact Banach frame is weak-exact. An example of a weak-exact Banach frame which is not exact has been given. A necessary and sufficient condition for a Banach frame to be a weak-exact Banach frame has been obtained. Finally, a necessary condition for the perturbation of a ret... 9. Frame Wavelet Sets and Wavelets in Banach Spaces Directory of Open Access Journals (Sweden) Devendra Kumar 2014-06-01 Full Text Available In this paper, we try to study a special class of frame wavelets in Banach spaces whose Fourier transforms are supported by frame wavelet sets. Our results generalize the various results of [2] to Banach spaces other than Hilbert spaces using the Feichtinger and Grochenig theory. 10. Discretization of variational regularization in Banach spaces International Nuclear Information System (INIS) Consider a nonlinear ill-posed operator equation F(u) = y, where F is defined on a Banach space X. In this paper we analyze finite-dimensional variational regularization, which takes into account operator approximations and noisy data. As shown in the literature, depending on the setting, convergence of the regularized solutions of the finite-dimensional problems can be with respect to the strong or just a weak topology. In this paper our contribution is twofold. First, we derive convergence rates in terms of Bregman distances in the convex regularization setting under appropriate sourcewise representation of a solution of the equation. Secondly, for particular regularization realizations in nonseparable Banach spaces, we discuss the finite-dimensional approximations of the spaces and the type of convergence, which is needed for the convergence analysis. These considerations lay the fundament for efficient numerical implementation. In particular, we emphasize on the space X of finite total variation functions and analyze in detail the cases when X is the space of the functions of finite bounded deformation and the L∞-space. The latter two settings are of interest in numerous problems arising in optimal control, machine learning and engineering 11. Discretization of variational regularization in Banach spaces Science.gov (United States) Pschl, Christiane; Resmerita, Elena; Scherzer, Otmar 2010-10-01 Consider a nonlinear ill-posed operator equation F(u) = y, where F is defined on a Banach space X. In this paper we analyze finite-dimensional variational regularization, which takes into account operator approximations and noisy data. As shown in the literature, depending on the setting, convergence of the regularized solutions of the finite-dimensional problems can be with respect to the strong or just a weak topology. In this paper our contribution is twofold. First, we derive convergence rates in terms of Bregman distances in the convex regularization setting under appropriate sourcewise representation of a solution of the equation. Secondly, for particular regularization realizations in nonseparable Banach spaces, we discuss the finite-dimensional approximations of the spaces and the type of convergence, which is needed for the convergence analysis. These considerations lay the fundament for efficient numerical implementation. In particular, we emphasize on the space X of finite total variation functions and analyze in detail the cases when X is the space of the functions of finite bounded deformation and the L?-space. The latter two settings are of interest in numerous problems arising in optimal control, machine learning and engineering. 12. Impulsive integral equations in Banach spaces and applications Directory of Open Access Journals (Sweden) Dajun Guo 1992-01-01 Full Text Available In this paper, we first use the fixed point theory to prove two existence theorems of positive solutions for the impulsive Fredholm integral Equations in Banach spaces. And then, we offer some applications to the two-point boundary value problems for the second order impulsive differential equations in Banach spaces. 13. Unbounded operators on Banach spaces over the quaternion field OpenAIRE Ludkovsky, S. V. 2004-01-01 Resolvents of quasi-linear operators and operator algebras in Banach spaces over the quaternion field are investigated. Spectral theory of unbounded nonlinear operators in quaternion Banach spaces is studied. Strongly continuous semigroups of quaternion quasi-linear operators are investigated and the polar decomposition theorem for them is proved. 14. Nonlinear impulsive Volterra integral equations in Banach spaces and applications OpenAIRE Dajun Guo 1993-01-01 In this paper, we first extend results on the existence of maximal solutions for nonlinear Volterra integral equations in Banach spaces to impulsive Volterra integral equations. Then, we give some applications to initial value problems for first order impulsive differential equations in Banach spaces. The results are demonstrated by means of an example of an infinite system for impulsive differential equations. 15. Minimal and Maximal Operator Space Structures on Banach Spaces OpenAIRE P., Vinod Kumar; Balasubramani, M. S. 2014-01-01 Given a Banach space$X$, there are many operator space structures possible on$X$, which all have$X$as their first matrix level. Blecher and Paulsen identified two extreme operator space structures on$X$, namely$Min(X)$and$Max(X)$which represents respectively, the smallest and the largest operator space structures admissible on$X$. In this note, we consider the subspace and the quotient space structure of minimal and maximal operator spaces. 16. Amenability for dual Banach algebras OpenAIRE Runde, Volker 2002-01-01 We define a Banach algebra A to be dual if$A = (A_\\ast)^\\ast$for a closed submodule$A_\\ast$of$A^\\ast$. The class of dual Banach algebras includes all$W^\\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certa... 17. BANACH CONTRACTION PRINCIPLE ON CONE RECTANGULAR METRIC SPACES Directory of Open Access Journals (Sweden) Ismat Beg 2009-08-01 Full Text Available We introduce the notion of cone rectangular metric space and prove {sc Banach} contraction mapping principle in cone rectangular metric space setting. Our result extends recent known results. 18. On Shrinking and Boundedly Complete Schauder Frames of Banach spaces OpenAIRE Liu, Rui 2009-01-01 This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will ... 19. Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces OpenAIRE Jong Soo Jung 2008-01-01 Abstract Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of , a contractive mapping (or a weakly contractive mapping), and nonexpansive mapping with the fixed point set . Let be generated by a new composite iterative scheme: , , . It is proved that converges strongly to a point in , which is a s... 20. On nested sequences of convex sets in a Banach space OpenAIRE Castillo, Jesús M. F.; González, Manuel; Papini, Pier Luigi 2014-01-01 In this paper we study different aspects of the representation of weak-compact convex sets of the bidual$X^{}$of a separable Banach space$X$via a nested sequence of closed convex bounded sets of$X$. 1. Some Questions Arising from the Homogeneous Banach Space Problem OpenAIRE Casazza, Peter G. 1992-01-01 We review the current state of the homogeneous Banach space problem. We then formulate several questions which arise naturally from this problem, some of which seem to be fundamental but new. We give many examples defining the bounds on the problem. We end with a simple construction showing that every infinite dimensional Banach space contains a subspace on which weak properties have become stable (under passing to further subspaces). Implications of this construction are considered. 2. Banach spaces with many boundedly complete basic sequences failing PCP CERN Document Server Perez, Gines Lopez 2009-01-01 We prove that there exist Banach spaces not containing$\\ell_1$, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the problem of the Remark 2 in H. P. Rosenthal. "Boundedly complete weak-Cauchy sequences in Banach spaces with PCP." J. Funct. Anal. 253 (2007) 772-781. 3. Quantitative Hahn-Banach Theorems and Isometric Extensions forWavelet and Other Banach Spaces Directory of Open Access Journals (Sweden) Sergey Ajiev 2013-05-01 Full Text Available We introduce and study Clarkson, Dolnikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hlder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. 4. On the Bounded Approximation Property in Banach spaces OpenAIRE Castillo, Jesús M. F.; Moreno, Yolanda 2013-01-01 We prove that the kernel of a quotient operator from an$\\mathcal L_1$-space onto a Banach space$X$with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case$\\ell_1$-- and Figiel, Johnson and Pe\\l czy\\'nski --case$X^$separable. Given a Banach space$X$, we show that if the kernel of a quotient map from some$\\mathcal L_1$-space onto$X$has the BAP then every kernel of every quotient map from any$\\mathcal L_1$-space onto$X$has the BAP. T... 5. Banach frames for multivariate alpha-modulation spaces DEFF Research Database (Denmark) Borup, Lasse; Nielsen, Morten 2006-01-01 The ?-modulation spaces [$Mathematical Term$], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for ?-modulation spaces in the multivariate setting. The frames constructed are unions of independent... 6. Construction of generalized atomic decompositions in Banach spaces Directory of Open Access Journals (Sweden) Raj Kumar 2014-06-01 Full Text Available G-atomic decompositions for Banach spaces with respect to a model space of sequences have been introduced and studied as a generalization of atomic decompositions. Examples and counter example have been provided to show its existence. It has been proved that an associated Banach space for G-atomic decomposition always has a complemented subspace. The notion of a representation system is introduced and exhibits its relation with G-atomic decomposition. Also It has been observed that G-atomic decompositions are exactly compressions of Schauder decompositions for a larger Banach space. We give a characterization for finite G-atomic decomposition in terms of finite-dimensional expansion of identity. Keywords: complemented coefficient spaces, finite-dimensional expansion of identity, G-atomic decomposition, representation system. 7. Tauberian theorems for generalized functions with values in Banach spaces International Nuclear Information System (INIS) We state and prove Tauberian theorems of a new type. In these theorems we give sufficient conditions under which the values of a generalized function (distribution) that are assumed to lie in a locally convex topological space actually belong to some narrower (Banach) space. These conditions are stated in terms of 'general class estimates' for the standard average of this generalized function with a fixed kernel belonging to a space of test functions. The applications of these theorems are based, in particular, on the fact that asymptotical (and some other) properties of the generalized functions under investigation can be described in terms of membership of certain Banach spaces. We apply these theorems to the study of asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of generalized functions of small growth (tempered distributions), and to the study of Banach spaces of Besov-Nikol'skii type 8. Tauberian theorems for generalized functions with values in Banach spaces Energy Technology Data Exchange (ETDEWEB) Drozhzhinov, Yu N; Zav' yalov, B I [Steklov Mathematical Institute, Russian Academy of Sciences (Russian Federation) 2002-08-31 We state and prove Tauberian theorems of a new type. In these theorems we give sufficient conditions under which the values of a generalized function (distribution) that are assumed to lie in a locally convex topological space actually belong to some narrower (Banach) space. These conditions are stated in terms of 'general class estimates' for the standard average of this generalized function with a fixed kernel belonging to a space of test functions. The applications of these theorems are based, in particular, on the fact that asymptotical (and some other) properties of the generalized functions under investigation can be described in terms of membership of certain Banach spaces. We apply these theorems to the study of asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of generalized functions of small growth (tempered distributions), and to the study of Banach spaces of Besov-Nikol'skii type. 9. Reproducing Kernel Banach Spaces with the l1 Norm CERN Document Server Song, Guohui; Hickernell, Fred J 2011-01-01 Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given. 10. Metric embeddings bilipschitz and coarse embeddings into Banach spaces CERN Document Server Ostrovskii, Mikhail I 2013-01-01 Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The book will help readers to enter and to work in this very rapidly developing area having many important connections with different parts of mathematics and computer science. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include embeddability of locally finite metric spaces into Banach spaces is finitely determined, constructions of embeddings, distortion in terms of Poinc 11. Semigroups preserving a convex set in a Banach space OpenAIRE Shigekawa, Ichiro 2011-01-01 We discuss semigroups that preserve a convex set in a Banach space or a Hilbert space. We give sufficient conditions for which a semigroup preserves a convex set. Using this, we show that various issues can be treated in a unified way. We also discuss the problem in the Hilbert space setting, in which we use the sesquilinear form associated with a semigroup. 12. Banach's space: Lviv and the Scottish cafe OpenAIRE Henderson, D. 2004-01-01 In a sense Stefan Banach is the patron saint of this workshop. He was one of the most prominent former residents of Lviv. His hangout'', the Scottish Cafe, is shown in the street scene that is the cover of the book of abstracts. 13. Banach's space: Lviv and the Scottish cafe Directory of Open Access Journals (Sweden) D.Henderson 2004-01-01 Full Text Available In a sense Stefan Banach is the patron saint of this workshop. He was one of the most prominent former residents of Lviv. His hangout'', the Scottish Cafe, is shown in the street scene that is the cover of the book of abstracts. 14. A GLIDING HUMP PROPERTY AND BANACH-MACKEY SPACES Directory of Open Access Journals (Sweden) CHARLES SWARTZ 2001-08-01 Full Text Available We consider the Banach-Mackey property for pairs of vector spaces E and E' which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and anoter measure theoretic property are Banach-Mackey pairs, i. e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given 15. Banach spaces and groups - order properties and universal models CERN Document Server Shelah, S; Shelah, Saharon; Usvyatsov, Alex 2003-01-01 We deal with two natural examples of almost-elementary classes: the class of all Banach spaces (over R or C) and the class of all groups. We show both of these classes do not have the strict order property, and find the exact place of each one of them in Shelah's SOP_n (strong order property of order n) hierarchy. Remembering the connection between this hierarchy and the existence of universal models, we conclude, for example, that there are few'' universal Banach spaces (under isometry) of regular cardinalities. 16. Riesz Isomorphisms of Tensor Products of Order Unit Banach Spaces Indian Academy of Sciences (India) T S S R K Rao 2009-06-01 In this paper we formulate and prove an order unit Banach space version of a Banach–Stone theorem type theorem for Riesz isomorphisms of the space of vector-valued continuous functions. Similar results were obtained recently for the case of lattice-valued continuous functions in [5] and [6]. 17. Total subspaces in dual Banach spaces which are not norming OpenAIRE Ostrovskii, Mikhail I. 1993-01-01 The main result: the dual of separable Banach space$X$contains a total subspace which is not norming over any infinite dimensional subspace of$X$if and only if$X$has a nonquasireflexive quotient space with the strictly singular quotient mapping. 18. The reconstruction property in Banach spaces and a perturbation theorem DEFF Research Database (Denmark) Casazza, P.G.; Christensen, Ole 2008-01-01 Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop a more general perturbation theory that does not force equiva... 19. Smooth approximations of norms in separable Banach spaces Czech Academy of Sciences Publication Activity Database Hájek, Petr Pavel; Talponen, J. 2014-01-01 Roč. 65, č. 3 (2014), s. 957-969. ISSN 0033-5606 R&D Projects: GA ČR(CZ) GAP201/11/0345 Institutional support: RVO:67985840 Keywords : Banach space * approximation Subject RIV: BA - General Mathematics Impact factor: 0.640, year: 2014 http://qjmath.oxfordjournals.org/content/65/3/957 20. On polynomial equations in Banach space, perturbation techniques and applications Directory of Open Access Journals (Sweden) Ioannis K. Argyros 1987-03-01 Full Text Available We use perturbation techniques to solve the polynomial equation in Banach space. Our techniques provide more accurate information on the location of solutions and yield existence and uniqueness in cases not covered before. An example is given to justify our method. 1. A characterization of Banach space with non expansive map International Nuclear Information System (INIS) In this paper, we characterize a Banach space X with a certain non expansive map T: X → X. We shall prove that T is non expansive if and only if each three dimensional subspace of X has the property that, after rotation by π/2, its unit ball coincides with its dual unit ball. (authors). 3 refs 2. The M-basis Problem for Separable Banach Spaces OpenAIRE Gill, Tepper L. 2013-01-01 In this note we show that, if$\\mcB$is separable Banach space, then there is a biorthogonal system$\\{x_n, x_n^\\}$such that, the closed linear span of$\\{x_n\\},\\bar{\\left\\langle {\\{x_n\\}}\\right\\rangle}=\\mcB$and$\\left\\\| {x_n} \\right\\\|\\left\\\| {x_n^} \\right\\\| = 1$for all$n$. 3. On some impulsive fractional differential equations in Banach spaces Directory of Open Access Journals (Sweden) JinRong Wang 2010-01-01 Full Text Available This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \\(PC\\-mild solutions for some fractional differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration. 4. On the Weak Relatively Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Yongfu Su 2010-01-01 Full Text Available In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive mappings and give two examples which are weak relatively nonexpansive mappings but not relatively nonexpansive mappings in Banach space l2 and Lp[0,1]? ?(1 5. On the Weak Relatively Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Su Yongfu 2010-01-01 Full Text Available Abstract In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive mappings and give two examples which are weak relatively nonexpansive mappings but not relatively nonexpansive mappings in Banach space and . 6. Fast regularizing sequential subspace optimization in Banach spaces International Nuclear Information System (INIS) We are concerned with fast computations of regularized solutions of linear operator equations in Banach spaces in case only noisy data are available. To this end we modify recently developed sequential subspace optimization methods in such a way that the therein employed Bregman projections onto hyperplanes are replaced by Bregman projections onto stripes whose width is in the order of the noise level 7. Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Jung JongSoo 2008-01-01 Full Text Available Abstract Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of , a contractive mapping (or a weakly contractive mapping, and nonexpansive mapping with the fixed point set . Let be generated by a new composite iterative scheme: , , . It is proved that converges strongly to a point in , which is a solution of certain variational inequality provided that the sequence satisfies and , for some and the sequence is asymptotically regular. 8. Iterative Approximation to Convex Feasibility Problems in Banach Space Directory of Open Access Journals (Sweden) Shih-Sen Chang 2007-04-01 Full Text Available The convex feasibility problem (CFP of finding a point in the nonempty intersection Ã¢ÂˆÂ©i=1NCi is considered, where NÃ¢Â‰Â¥1 is an integer and each Ci is assumed to be the fixed point set of a nonexpansive mapping Ti:EÃ¢Â†Â’E, where E is a reflexive Banach space with a weakly sequentially continuous duality mapping. By using viscosity approximation methods for a finite family of nonexpansive mappings, it is shown that for any given contractive mapping f:CÃ¢Â†Â’C, where C is a nonempty closed convex subset of E and for any given x0Ã¢ÂˆÂˆC the iterative scheme xn+1=P[ÃŽÂ±n+1f(xn+(1Ã¢ÂˆÂ’ÃŽÂ±n+1Tn+1xn] is strongly convergent to a solution of (CFP, if and only if {ÃŽÂ±n} and {xn} satisfy certain conditions, where ÃŽÂ±nÃ¢ÂˆÂˆ(0,1,Tn=Tn(modN and P is a sunny nonexpansive retraction of E onto C. The results presented in the paper extend and improve some recent results in Xu (2004, O'Hara et al. (2003, Song and Chen (2006, Bauschke (1996, Browder (1967, Halpern (1967, Jung (2005, Lions (1977, Moudafi (2000, Reich (1980, Wittmann (1992, Reich (1994. 9. Quantitative coarse embeddings of quasi-Banach spaces into a Hilbert space OpenAIRE Kraus, Michal 2015-01-01 We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space. 10. Embeddings of groups into Banach spaces OpenAIRE Jolissaint, Pierre-Nicolas; Valette, Alain 2015-01-01 L'objectif de cette thèse est de faire des liens entre les propriétés algébriques et géométriques des groupes. Une façon d'étudier un espace métrique quelconque est de le représenter comme un sous-ensemble d'un espace de Banach dont la géométrie est bien comprise. En premier lieu, nous étudions les plongements bi-Lipschitz de graphes de Cayley de groupes finis dans les espaces Lp [0; 1]. En particulier, nous donnons une borne inferieure pour la distortion de tels plongements à l'aide d'invari... 11. Integral operators on the section space of a Banach bundle Directory of Open Access Journals (Sweden) D. A. Robbins 1993-09-01 Full Text Available Let Ï:EâX and Ï:FâX be bundles of Banach spaces, where X is a compact Hausdorff space, and let V be a Banach space. Let Î(Ï denote the space of sections of the bundle Ï. We obtain two representations of integral operators T:Î(ÏâV in terms of measures. The first generalizes a recent result of P. Saab, the second generalizes a theorem of Grothendieck. We also study integral operators T:Î(ÏâÎ(Ï which are C(X-linear. 12. Integrals and Banach spaces for finite order distributions CERN Document Server Talvila, Erik 2011-01-01 Let$\\Bc$denote the real-valued functions continuous on the extended real line and vanishing at$-\\infty$. Let$\\Br$denote the functions that are left continuous, have a right limit at each point and vanish at$-\\infty$. Define$\\acn$to be the space of tempered distributions that are the$n$th distributional derivative of a unique function in$\\Bc$. Similarly with$\\arn$from$\\Br$. A type of integral is defined on distributions in$\\acn$and$\\arn$. The multipliers are iterated integrals of functions of bounded variation. For each$n\\in\\N$, the spaces$\\acn$and$\\arn$are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to$\\Bc$and$\\Br$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space$\\ac^1$is the completion of the$L^1$functions in the Alexiewicz norm. The space$\\ar^1$contains all finite signed Borel measure... 13. On best proximity points in metric and Banach spaces CERN Document Server Espinola, Rafa 2009-01-01 In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to different results which complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with. 14. Some problems on ordinary differential equations in Banach spaces Czech Academy of Sciences Publication Activity Database Hájek, Petr Pavel; Vivi, P. 2010-01-01 Roč. 104, č. 2 (2010), s. 245-255. ISSN 1578-7303 R&D Projects: GA AV ČR IAA100190801; GA ČR GA201/07/0394 Institutional research plan: CEZ:AV0Z10190503 Keywords : Banach space * ODE * Peano's theorem Subject RIV: BA - General Mathematics Impact factor: 0.400, year: 2010 http://link.springer.com/article/10.5052%2FRACSAM.2010.16 15. Polynomial algebras and smooth functions in Banach spaces Czech Academy of Sciences Publication Activity Database D'Alessandro, Stefania; Hájek, Petr Pavel 2014-01-01 Roč. 266, č. 3 (2014), s. 1627-1646. ISSN 0022-1236 R&D Projects: GA ČR(CZ) GAP201/11/0345; GA MŠk(CZ) 7AMB12FR003 Institutional support: RVO:67985840 Keywords : polynomials in Banach space Subject RIV: BA - General Mathematics Impact factor: 1.322, year: 2014 http://www.sciencedirect.com/science/article/pii/S0022123613004588 16. Weak compactness and sigma-Asplund generated Banach spaces Czech Academy of Sciences Publication Activity Database Fabian, Marián; Montesinos, V.; Zizler, Václav 2007-01-01 Roč. 181, č. 2 (2007), s. 125-152. ISSN 0039-3223 R&D Projects: GA AV ČR IAA1019301; GA AV ČR(CZ) IAA100190610 Institutional research plan: CEZ:AV0Z10190503 Keywords : epsilon-Asplund set * epsilon-weakly compact set * weakly compactly generated Banach space Subject RIV: BA - General Mathematics Impact factor: 0.568, year: 2007 17. Elements of mathematical theory of evolutionary equations in Banach spaces CERN Document Server Samoilenko, Anatoly M 2013-01-01 Evolutionary equations are studied in abstract Banach spaces and in spaces of bounded number sequences. For linear and nonlinear difference equations, which are defined on finite-dimensional and infinite-dimensional tori, the problem of reducibility is solved, in particular, in neighborhoods of their invariant sets, and the basics for a theory of invariant tori and bounded semi-invariant manifolds are established. Also considered are the questions on existence and approximate construction of periodic solutions for difference equations in infinite-dimensional spaces and the problem of extendibi 18. A proximal point method for nonsmooth convex optimization problems in Banach spaces Directory of Open Access Journals (Sweden) Y. I. Alber 1997-01-01 Full Text Available In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces. 19. Greedy Algorithms for Reduced Bases in Banach Spaces KAUST Repository DeVore, Ronald 2013-02-26 Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n-dimensional space X n⊂X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width dn(F)X. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) in the case X=H is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457-1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces. © 2013 Springer Science+Business Media New York. 20. A remark on contraction semigroups on Banach spaces OpenAIRE Lin, P K 1994-01-01 Let$X$be a complex Banach space and let$J:X \\to X^$be a duality section on$X$(i.e.$\\langle x,J(x)\\rangle=\\\|J(x)\\\|\\\|x\\\|=\\\|J(x)\\\|^2=\\\|x\\\|^2$). For any unit vector$x$and any ($C_0$) contraction semigroup$T=\\{e^{tA}:t \\geq 0\\}$, Goldstein proved that if$X$is a Hilbert space and if$\|\\langle T(t) x,J(x)\\rangle\| \\to 1 $as$t \\to \\infty$, then$x$is an eigenvector of$A$corresponding to a purely imaginary eigenvalue. In this article, we prove the similar result holds if$X$is a stri... 1. On moduli of convexity in Banach spaces OpenAIRE Reif Jiří 2005-01-01 Let be a normed linear space, an element of norm one, and and the local modulus of convexity of . We denote by the greatest such that for each closed linear subspace of the quotient mapping maps the open -neighbourhood of in onto a set containing the open -neighbourhood of in . It is known that . We prove that there is no universal constant such that , however, such a constant exists within the class of Hilbert spaces . If is a Hilbert space with , then . 2. On moduli of convexity in Banach spaces Directory of Open Access Journals (Sweden) Reif Jiří 2005-01-01 Full Text Available Let be a normed linear space, an element of norm one, and and the local modulus of convexity of . We denote by the greatest such that for each closed linear subspace of the quotient mapping maps the open -neighbourhood of in onto a set containing the open -neighbourhood of in . It is known that . We prove that there is no universal constant such that , however, such a constant exists within the class of Hilbert spaces . If is a Hilbert space with , then . 3. p-adic Banach space operators and adelic Banach space operators Directory of Open Access Journals (Sweden) Ilwoo Cho 2014-01-01 Full Text Available In this paper, we study non-Archimedean Banach \\(\\-algebras \\(\\frak{M}_{p}\\ over the \\(p\\-adic number fields \\(\\mathbb{Q}_{p}\\, and \\(\\frak{M}_{\\mathbb{Q}}\\ over the adele ring \\(\\mathbb{A}_{\\mathbb{Q}}\\. We call elements of \\(\\frak{M}_{p}\\, \\(p\\-adic operators, for all primes \\(p\\, respectively, call those of \\(\\frak{M}_{\\mathbb{Q}}\\, adelic operators. We characterize \\(\\frak{M}_{ \\mathbb{Q}}\\ in terms of \\(\\frak{M}_{p}\\'s. Based on such a structure theorem of \\(\\frak{M}_{\\mathbb{Q}}\\, we introduce some interesting \\(p\\-adic operators and adelic operators. 4. Geometry and Gâteaux smoothness in separable Banach spaces Czech Academy of Sciences Publication Activity Database Hájek, Petr Pavel; Montesinos, V.; Zizler, Václav 2012-01-01 Roč. 6, č. 2 (2012), s. 201-232. ISSN 1846-3886 R&D Projects: GA ČR(CZ) GAP201/11/0345; GA AV ČR IAA100190901 Institutional research plan: CEZ:AV0Z10190503 Keywords : Gâteaux differentiable norms * extreme points * Radon-Nikodým property Subject RIV: BA - General Math ematics Impact factor: 0.529, year: 2012 http://oam.ele- math .com/06-15/Geometry-and-Gateaux-smoothness-in-separable-Banach-spaces 5. Weakly convex and proximally smooth sets in Banach spaces International Nuclear Information System (INIS) We establish interconnections between the conditions of weak convexity in the sense of Vial, weak convexity in the sense of Efimov-Stechkin, and proximal smoothness of sets in Banach spaces. We prove a theorem on the separation by a sphere of two disjoint sets, one of which is weakly convex in the sense of Vial and the other is strongly convex. We also prove that weakly convex and proximally smooth sets are locally connected, and study questions related to the preservation of the conditions of weak convexity and proximal smoothness under passage to the limit 6. On nonautonomous second-order differential equations on Banach space Directory of Open Access Journals (Sweden) Nguyen Thanh Lan 2001-04-01 Full Text Available We show the existence and uniqueness of classical solutions of the nonautonomous second-order equation: uÃ¢Â€Â³(t=A(tuÃ¢Â€Â²(t+B(tu(t+f(t, 0Ã¢Â‰Â¤tÃ¢Â‰Â¤T; u(0=x0, uÃ¢Â€Â²(0=x1 on a Banach space by means of operator matrix method and apply to Volterra integrodifferential equations. 7. Smoothness in Banach spaces. Selected problems Czech Academy of Sciences Publication Activity Database Fabian, Marián; Montesinos, V.; Zizler, Václav 2006-01-01 Roč. 100, č. 2 (2006), s. 101-125. ISSN 1578-7303 R&D Projects: GA ČR(CZ) GA201/04/0090; GA AV ČR(CZ) IAA100190610 Institutional research plan: CEZ:AV0Z10190503 Keywords : smooth norm * renorming * weakly compactly generated space Subject RIV: BA - General Mathematics 8. Chaotification of a class of discrete systems based on heteroclinic cycles connecting repellers in Banach spaces International Nuclear Information System (INIS) This paper is concerned with chaotification of a class of discrete dynamical systems in Banach spaces via the feedback control technique. A chaotification theorem based on heteroclinic cycles connecting repellers for maps in Banach spaces is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations. 9. Numerical analysis of semilinear stochastic evolution equations in Banach spaces Science.gov (United States) Hausenblas, Erika 2002-10-01 The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in Lp-spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank-Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences. 10. Homomorphisms and functional calculus on algebras on entire functions on Banach spaces Directory of Open Access Journals (Sweden) H. M. Pryimak 2015-07-01 Full Text Available The paper is devoted to study homomorphisms of algebras of entire functionson Banach spaces to a commutative Banach algebra. In particular, it is proposed amethod to construct homomorphisms vanishing on homogeneouspolynomials of degree less or equal that a fixed number$n.$11. Perturbation of n dimensional AQ - mixed type functional equation via Banach spaces and Banach algebra : Hyers direct and alternative fixed point methods OpenAIRE Arunkumar, M 2013-01-01 In this paper, the authors obtain the general solution and generalized Ulam - Hyers stability of n dimensional additive quadratic functional equation in Banach spaces using direct and fixed point methods. We also investigate the stability of the above equation in Banach algebra using direct and fixed point approach. 12. Stochastic integration in Banach spaces theory and applications CERN Document Server Mandrekar, Vidyadhar 2015-01-01 Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integrati... 13. Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces CERN Document Server Lindenstrauss, Joram; Tiaer, Jaroslav 2012-01-01 This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysi 14. An Inversion Formula for the Gaussian Radon Transform for Banach Spaces OpenAIRE Holmes, Irina 2013-01-01 We provide a disintegration theorem for the Gaussian Radon transform Gf on Banach spaces and use the Segal-Bargmann transform on abstract Wiener spaces to find a procedure to obtain f from its Gaussian Radon transform Gf. 15. Viscosity Approximation of Common Fixed Points for -Lipschitzian Semigroup of Pseudocontractive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Kim JongKyu 2009-01-01 Full Text Available We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007 and the pseudocontractive mapping in (Zegeye et al., 2007 to the pseudocontractive semigroup in Banach spaces under different conditions. 16. Parameter choice in Banach space regularization under variational inequalities International Nuclear Information System (INIS) The authors study parameter choice strategies for the Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depends on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader, the authors review in an appendix a few instances where the validity of a variational inequality can be established. (paper) 17. Porous sets for mutually nearest points in Banach spaces Directory of Open Access Journals (Sweden) Chong Li 2008-01-01 Full Text Available Let \\(\\mathfrak{B}(X\\ denote the family of all nonempty closed bounded subsets of a real Banach space \\(X\\, endowed with the Hausdorff metric. For \\(E, F \\in \\mathfrak{B}(X\\ we set \\(\\lambda_{EF} = \\inf \\{\\\|z - x\\\| : x \\in E, z \\in F \\}\\. Let \\(\\mathfrak{D}\\ denote the closure (under the maximum distance of the set of all \\((E, F \\in \\mathfrak{B}(X \\times \\mathfrak{B}(X\\ such that \\(\\lambda_{EF} \\gt 0\\. It is proved that the set of all \\((E, F \\in \\mathfrak{D}\\ for which the minimization problem \\(\\min_{x \\in E, z\\in F}\\\|x - z\\\|\\ fails to be well posed in a \\(\\sigma\\-porous subset of \\(\\mathfrak{D}\\. 18. Controllability of impulsive neutral functional differential inclusions with infinite delay in Banach spaces International Nuclear Information System (INIS) In this work, we establish a sufficient condition for the controllability of the first-order impulsive neutral functional differential inclusions with infinite delay in Banach spaces. The results are obtained by using the Dhage's fixed point theorem. 19. On Property () in Banach Lattices, Calderón–Lozanowskiĭ and Orlicz–Lorentz Spaces Indian Academy of Sciences (India) Paweł Kolwicz 2001-08-01 The geometry of Calderón–Lozanowskiĭ spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property () in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property () in Banach function lattices. Namely a criterion for property () in Banach function lattice is presented. In particular we get that in Banach function lattice property () implies uniform monotonicity. Moreover, property () in generalized Calderón–Lozanowskiĭ function spaces is studied. Finally, it is shown that in Orlicz–Lorentz function spaces property () and uniform convexity coincide. 20. Initial value problems for integro-differential equations of Volterra type in Banach spaces Directory of Open Access Journals (Sweden) Dajun Guo 1994-01-01 Full Text Available This paper investigates the extremal solutions of initial value problems for first order integro-differential equations of Volterra type in Banach spaces by means of establishing a comparison result. 1. Approximating zero points of accretive operators with compact domains in general Banach spaces Directory of Open Access Journals (Sweden) Miyake Hiromichi 2005-01-01 Full Text Available We prove strong convergence theorems of Mann's type and Halpern's type for resolvents of accretive operators with compact domains and apply these results to find fixed points of nonexpansive mappings in Banach spaces. 2. Ishikawa iteration process with errors for nonexpansive mappings in uniformly convex Banach spaces Directory of Open Access Journals (Sweden) Li Shenghong 2000-07-01 Full Text Available We shall consider the behaviour of Ishikawa iteration with errors in a uniformly convex Banach space. Then we generalize the two theorems of Tan and Xu without the restrictions that C is bounded and limsupnsn<1. 3. Fourier transform of function on locally compact Abelian groups taking value in Banach spaces OpenAIRE Radyna, Yauhen; Sidorik, Anna 2008-01-01 We consider Fourier transform of vector-valued functions on a locally compact group$G$, which take value in a Banach space$X$, and are square-integrable in Bochner sense. If$G$is a finite group then Fourier transform is a bounded operator. If$G$is an infinite group then Fourier transform$F: L_2(G,X)\\to L_2(\\widehat G,X)$is a bounded operator if and only if Banach space$X$is isomorphic to a Hilbert one. 4. Approximation of Common Fixed Points for Families of Mappings in Banach Space Directory of Open Access Journals (Sweden) Zhiming Cheng 2012-06-01 Full Text Available In this paper. We introduce a general iterative method for the family of mappings and prove the strong convergence of the new iterative scheme in Banach space. The new iterative method includes the iterative scheme of Khan and Domlo and Fukhar-ud-din [Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 341 (2008 111]. The results generalize the corresponding results. 5. Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces OpenAIRE Azagra Rueda, Daniel 1997-01-01 We prove that for every infinite-dimensional Banach space X with a Frechet differentiable norm, the sphere S-X is diffeomorphic to each closed hyperplane in X. We also prove that every infinite-dimensional Banach space Y having a (not necessarily equivalent) C-p norm (with p is an element of N boolean OR {infinity}) is C-p diffeomorphic to Y \\ {0}. 6. Strong convergence theorems for uniformly L-Lipschitzian mappings in Banach spaces International Nuclear Information System (INIS) Let E be a real reflexive Banach space with uniform Gateaux differentiable norm, K be a nonempty bounded closed and convex subset of E , T : K → K be a uniformly L-Lipschitzian mapping such that F (T) := {x element of K : Tx = x} ≠ 0, u element of K be fixed and let {αn}n≥0 and {γn}n≥0 subset of (0, 1) be such that limn→∞ αn = 0 = limn→ ∞ γn and limn→ ∞(βn - 1)/ αn = 0, where βn Σj=0n λj and λj = 1 + αjγjL. Let Sn := (1 - αnγn)I + αnγnTn. It is proved that there exists some integer N0 > 1, such that for each n ≥ N0, there exists unique xn element of K such that xn = αnu+(1 -αn) 1/ (n + 1) Σj=0n Sjxn. If φ : E → R is defined by φ (y) := LIMn vertical bar vertical bar xn -y vertical bar vertical bar2 for all y element of E here LIM denotes a Banach limit, vertical bar vertical bar xn - Txn vertical bar vertical bar → 0 as n → ∞ and Kmin intersection F (T) ≠ 0, where Kmin := {x element of E : φ (x) = min (u element of K) φ (u) }, then it is proved that {xn} converges strongly to a fixed point of T. As an application, it is proved that the iterative process, z0 element of K, zn+1 alpha#nu + (1 - αn) 1/ (n + 1) Σj=0n Sjzn , n ≥ 0, under suitable conditions on the iteration parameters, converges strongly to a fixed point of T. (author) 7. On the fixed point property in direct sums of Banach spaces with strictly monotone norms OpenAIRE Prus, Stanis?aw; Wi?nicki, Andrzej 2007-01-01 It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis. 8. Isometric uniqueness of a complementably universal Banach space for Schauder decompositions OpenAIRE Garbulińska, Joanna 2014-01-01 We present an isometric version of the complementably universal Banach space$\\mathbb{P}$with a Schauder decomposition. The space$\\mathbb{P}$is isomorphic to Pe{\\l}czy\\'nski's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property. 9. Skipped Blocking and other Decompositions in Banach spaces CERN Document Server Bellenot, S F 2004-01-01 Necessary and sufficient conditions are given for when a sequence of finite dimensional subspaces (X_n) can be blocked to be a skipped blocking decompositon (SBD). The condition is order independent, so permutations of conditional basis, for example can be so blocked. This condition is implied if (X_n) is shrinking, or (X_n) is a permutation of a FDD, or if X is reflexive and (X_n) is separating. A separable space X has PCP, if and only if, every norming decomposition (X_n) can be blocked to be a boundedly complete SBD. Every boundedly complete SBD is a JT-decomposition. 10. Arens Regularity Of Bilinear Forms And Unital Banach Module Spaces CERN Document Server Azar, Kazem Haghnejad 2010-01-01 Assume that$A$,$B$are Banach algebras and$m:A\\times B\\to B$,$m^\\prime:A\\times A\\to B$are bounded bilinear mappings. We will study the relation between Arens regularities of$m$,$m^\\prime$and the Banach algebras$A$,$B$. For Banach$A-bimoduleB$, we show that$B$factors with respect to$A$if and only if$B^{}$is an unital$A^{}-module$, and we define locally topological center for elements of$A^{}$and will show that when locally topological center of mixed unit of$A^{}$is$B^{*}$, then$B^$factors on both sides with respect to$A$if and only if$B^{}$has a unit as$A^{}-module$. 11. Accretive operators and Banach Alaoglu theorem in Linear 2-normed spaces OpenAIRE P. K Harikrishnan; Bernardo La Fuerza Guilln; K. T Ravindran 2011-01-01 In this paper we introduce the concept of accretive operator in linear 2-normed spaces, focusing on the relationships and the various aspects of accretive, m-accretive and maximal accretive operators. We prove the analogous of Banach-Alaoglu theorem in linear 2- normed spaces, obtaining an equivalent definition for accretive operators in linear 2-normed spaces. 12. Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces OpenAIRE Hiroko Manaka 2014-01-01 Let$E$be a smooth Banach space with a norm$‖·‖$. Let$V(x,y)={‖x‖}^{2}+{‖y‖}^{2}-2{\\langle}x,Jy{\\rangle}$for any$x,y\\in E$, where${\\langle}·,·{\\rangle}$stands for the duality pair and$J$is the normalized duality mapping. We define a$V$-strongly nonexpansive mapping by$V(·,·)$. This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a$V$-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this pa... 13. Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces Science.gov (United States) Jin, Qinian; Stals, Linda 2012-10-01 Nonstationary iterated Tikhonov regularization is an efficient method for solving ill-posed problems in Hilbert spaces. However, this method may not produce good results in some situations since it tends to oversmooth solutions and hence destroy special features such as sparsity and discontinuity. By making use of duality mappings and Bregman distance, we propose an extension of this method to the Banach space setting and establish its convergence. We also present numerical simulations which indicate that the method in Banach space setting can produce better results. 14. Periodic and almost periodic solutions for multi-valued differential equations in Banach spaces Directory of Open Access Journals (Sweden) E. Hanebaly 2000-03-01 Full Text Available It is known that for$omega$-periodic differential equations of monotonous type, in uniformly convex Banach spaces, the existence of a bounded solution on${Bbb R}^+$is equivalent to the existence of an omega-periodic solution (see Haraux [5] and Hanebaly [7, 10]. It is also known that if the Banach space is strictly convex and the equation is almost periodic and of monotonous type, then the existence of a continuous solution with a precompact range is equivalent to the existence of an almost periodic solution (see Hanebaly [8]. In this note we want to generalize the results above for multi-valued differential equations. 15. Iterative approximation of a solution of a general variational-like inclusion in Banach spaces International Nuclear Information System (INIS) In this paper, we introduce a class of η-accretive mappings in a real Banach space, and show that the η-proximal point mapping for η-m-accretive mapping is Lipschitz continuous. Further we develop an iterative algorithm for a class of general variational-like inclusions involving η-accretive mappings in real Banach space, and discuss its convergence criteria. The class of η-accretive mappings includes several important classes of operators that have been studied by various authors. (author) 16. A New Two-Step Iterative Process with Errors for Common Fixed Points in Banach Spaces Directory of Open Access Journals (Sweden) Esref Turkmen 2011-12-01 Full Text Available In this paper, a new two-step iterative scheme with errors is introduced for two asymptotically nonexpansive nonself-mappings. Several convergence theorems are established in real Banach spaces and real uniformly convex Banach spaces. Our theorems improve and extend the results due to Agarwal-O'Regan-Sahu [R.P. Agarwal, Donal O'Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J.Nonliear Convex. Anal.8(1(2007 61--79] and many other papers. 17. Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Takahashi Wataru 2007-01-01 Full Text Available Using the convex combination based on Bregman distances due to Censor and Reich, we define an operator from a given family of relatively nonexpansive mappings in a Banach space. We first prove that the fixed-point set of this operator is identical to the set of all common fixed points of the mappings. Next, using this operator, we construct an iterative sequence to approximate common fixed points of the family. We finally apply our results to a convex feasibility problem in Banach spaces. 18. FOREWORD: Tackling inverse problems in a Banach space environment: from theory to applications Tackling inverse problems in a Banach space environment: from theory to applications Science.gov (United States) Schuster, Thomas; Hofmann, Bernd; Kaltenbacher, Barbara 2012-10-01 Inverse problems can usually be modelled as operator equations in infinite-dimensional spaces with a forward operator acting between Hilbert or Banach spaces—a formulation which quite often also serves as the basis for defining and analyzing solution methods. The additional amount of structure and geometric interpretability provided by the concept of an inner product has rendered these methods amenable to a convergence analysis, a fact which has led to a rigorous and comprehensive study of regularization methods in Hilbert spaces over the last three decades. However, for numerous problems such as x-ray diffractometry, certain inverse scattering problems and a number of parameter identification problems in PDEs, the reasons for using a Hilbert space setting seem to be based on conventions rather than an appropriate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, non-Hilbertian regularization and data fidelity terms incorporating a priori information on solution and noise, such as general Lp-norms, TV-type norms, or the Kullback-Leibler divergence, have recently become very popular. These facts have motivated intensive investigations on regularization methods in Banach spaces, a topic which has emerged as a highly active research field within the area of inverse problems. Meanwhile some of the most well-known regularization approaches, such as Tikhonov-type methods requiring the solution of extremal problems, and iterative ones like the Landweber method, the Gauss-Newton method, as well as the approximate inverse method, have been investigated for linear and nonlinear operator equations in Banach spaces. Convergence with rates has been proven and conditions on the solution smoothness and on the structure of nonlinearity have been formulated. Still, beyond the existing results a large number of challenging open questions have arisen, due to the more involved handling of general Banach spaces and the larger variety of concrete instances with special properties. The aim of this special section is to provide a forum for highly topical ongoing work in the area of regularization in Banach spaces, its numerics and its applications. Indeed, we have been lucky enough to obtain a number of excellent papers both from colleagues who have previously been contributing to this topic and from researchers entering the field due to its relevance in practical inverse problems. We would like to thank all contributers for enabling us to present a high quality collection of papers on topics ranging from various aspects of regularization via efficient numerical solution to applications in PDE models. We give a brief overview of the contributions included in this issue (here ordered alphabetically by first author). In their paper, Iterative regularization with general penalty term—theory and application to L1 and TV regularization, Radu Bot and Torsten Hein provide an extension of the Landweber iteration for linear operator equations in Banach space to general operators in place of the inverse duality mapping, which corresponds to the use of general regularization functionals in variational regularization. The L∞ topology in data space corresponds to the frequently occuring situation of uniformly distributed data noise. A numerically efficient solution of the resulting Tikhonov regularization problem via a Moreau-Yosida appriximation and a semismooth Newton method, along with a δ-free regularization parameter choice rule, is the topic of the paper L∞ fitting for inverse problems with uniform noise by Christian Clason. Extension of convergence rates results from classical source conditions to their generalization via variational inequalities with a priori and a posteriori stopping rules is the main contribution of the paper Regularization of linear ill-posed problems by the augmented Lagrangian method and variational inequalities by Klaus Frick and Markus Grasmair, again in the context of some iterative method. A powerful tool for proving convergence rates of Tikhonov type but also othe 19. Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces CERN Document Server Sukochev, F 2010-01-01 {\\it We study the class of all rearrangement-invariant (=r.i.) function spaces$E$on$[0,1]$such that there exists$00$does not depend on$n$. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces$exp(L_p)$,$p\\ge 1$. We further apply our results to the study of Banach-Saks index sets in r.i. spaces. 20. Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group OpenAIRE Abreu, Luis Daniel; Grchenig, Karlheinz 2010-01-01 Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L^2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, w... 1. Psi-exponential dichotomy for linear differential equations in a Banach space Directory of Open Access Journals (Sweden) Atanaska Georgieva 2013-07-01 Full Text Available In this article we extend the concept psi-exponential and psi-ordinary dichotomies for homogeneous linear differential equations in a Banach space. With these two concepts we prove the existence of psi-bounded solutions of the appropriate inhomogeneous equation. A roughness of the psi-dichotomy is also considered. 2. Nonuniform Exponential Stability and Instability of Evolution Operators in Banach Space OpenAIRE Mihaela Tomescu; Andrea Minda 2006-01-01 In this paper is presenting a parallel between nonuniform exponential stability and nonuniform exponential instability of evolution operators in Banach spaces, beginning to present the concept of the evolution operator with nonuniform exponential decay, respectively growth, next with the concept of the nonuniform stability, respectively instability, nonuniform exponential stability, respectively instability, nonuniform integrable stability, respectively instability and... 3. Nonuniform Exponential Stability and Instability of Evolution Operators in Banach Space Directory of Open Access Journals (Sweden) Mihaela Tomescu 2006-10-01 Full Text Available In this paper is presenting a parallel between nonuniform exponential stability and nonuniform exponential instability of evolution operators in Banach spaces, beginning to present the concept of the evolution operator with nonuniform exponential decay, respectively growth, next with the concept of the nonuniform stability, respectively instability, nonuniform exponential stability, respectively instability, nonuniform integrable stability, respectively instability and relationship between this concepts. 4. N-th order impulsive integro-differential equations in Banach spaces Directory of Open Access Journals (Sweden) Manfeng Hu 2004-03-01 Full Text Available We investigate the maximal and minimal solutions of initial value problem for N-th order nonlinear impulsive integro-differential equation in Banach space by establishing a comparison result and using the upper and lower solutions methods. 5. Known results and open problems on C1 linearization in Banach spaces OpenAIRE Munhoz Rodrigues, Hildebrando; Solà-Morales Rubió, Joan 2012-01-01 The purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimen- sional Banach spaces, especially in the C 1 case, and also to present some open problems that we believe that are still important for the understanding of the theory. 6. Bound and periodic solutions of the Riccati equation in Banach space Directory of Open Access Journals (Sweden) A. Ya. Dorogovtsev 1995-01-01 Full Text Available An abstract, nonlinear, differential equation in Banach space is considered. Conditions are presented for the existence of bounded solutions of this equation with a bounded right side, and also for the existence of stationary (periodic solutions of this equation with a stationary (periodic process in the right side. 7. The generalized hardy operator with kernel and variable integral limits in banach function spaces OpenAIRE Lang J.; Gogatishvill A 1999-01-01 Let we have an integral operator where and are nondecreasing functions, and are non-negative and finite functions, and is nondecreasing in , nonincreasing in and for and . We show that the integral operator where and are Banach functions spaces with -condition is bounded if and only if . Where and 8. Convergence rates for an iteratively regularized Newton–Landweber iteration in Banach space International Nuclear Information System (INIS) In this paper, we provide convergence and convergence rate results for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting. Numerical experiments illustrate the performance of the method. (paper) 9. On the fixed points of nonexpansive mappings in direct sums of Banach spaces OpenAIRE Wi?nicki, Andrzej 2011-01-01 We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional. 10. Existence Results for a Coupled System of Nonlinear Fractional Differential Equations in Banach Spaces OpenAIRE Yuping Cao; Chuanzhi Bai 2014-01-01 We investigate boundary value problems for a coupled system of nonlinear fractional differential equations involving Caputo derivative in Banach spaces. A generalized singular type coupled Gronwall inequality system is given to obtain an important a priori bound. Existence results are obtained by using fixed point theorems and an example is given to illustrate the results. 11. On common fixed points of compatible mappings in metric and Banach spaces Directory of Open Access Journals (Sweden) M. S. Khan 1988-06-01 Full Text Available We prove a number of results concerning the existence of common fixed points of a family of maps satisfying certain contractive conditions in metric and Banach spaces. Results dealing with the stucture of the set of common fixed points of such maps are also given. Our work is an improvement upon the previously known results. 12. Existence and convergence theorems for a class of multi-valued variational inclusions in Banach spaces International Nuclear Information System (INIS) An existence theorem for a new class of multi-valued variational inclusion problems is established in smooth Banach spaces. Further, it is shown that a sequence of a Mann-type iteration algorithm is strongly convergent to the solutions in this class of variational inclusion problems. (author) 13. Integral equations of fractional order with multiple time delays in Banach spaces Directory of Open Access Journals (Sweden) Mouffak Benchohra 2012-04-01 Full Text Available In this article, we give sufficient conditions for the existence of solutions for an integral equation of fractional order with multiple time delays in Banach spaces. Our main tool is a fixed point theorem of Monch type associated with measures of noncompactness. Our results are illustrated by an example. 14. Approximate Fixed Point Theorems in Banach Spaces with Applications in Game Theory OpenAIRE 2002-01-01 In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games. 15. Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces Directory of Open Access Journals (Sweden) Zhaohui Gu 2008-02-01 Full Text Available Let X be a uniformly convex Banach space, and let S,Â T be a pair of mean nonexpansive mappings. In this paper, it is proved that the sequence of Ishikawa iterations associated with S and T converges to the common fixed point of S and T. 16. Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Isa Yildirim 2011-03-01 Full Text Available In this paper, we consider a composite iterative algorithm for approximating common fixed points of two nonself asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces. 17. Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces Directory of Open Access Journals (Sweden) 2009-02-01 Full Text Available We study the influence of computational errors on the convergence to compact sets of orbits of nonexpansive mappings in Banach and metric spaces. We first establish a convergence theorem assuming that the computational errors are summable and then provide examples which show that the summability of errors is necessary for convergence. 18. A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Gu Feng 2006-01-01 Full Text Available The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003, Xu and Ori (2001, and Zhou and Chang (2002. 19. A Bourgain-like property of Banach spaces with no copies of$c_0$OpenAIRE Pérez, A.; Raja, M 2016-01-01 We give a characterization of the existence of copies of$c_{0}$in Banach spaces in terms of indexes. As an application, we deduce new proofs of James Distortion theorem and Bessaga-Pe{\\l}czynski theorem about weakly unconditionally Cauchy series. 20. Asymptotic behaviour of the solutions of Schroedinger equation with impulse effect in a Banach space International Nuclear Information System (INIS) The present paper studies the asymptotic behaviour of the solutions of linear homogeneous differential Schroedinger equation with impulse effect in a Banach space and finds a dependence between their asymptotic behaviour and the spectrum of the linear Hamiltonian operator. 6 refs 1. Wide and tight spherical hulls of bounded sets in Banach spaces OpenAIRE He, Chan; Wu, Senlin; Zhang, Xinling 2016-01-01 Let$A$be a bounded closed convex set in a Banach space. The boundaries of the wide spherical hull$\\eta (A)$and the tight spherical hull$\\theta (A)$are characterized, the existence of diametral points of these three sets are discussed, and a further relation between these three sets is clarified. Moreover, a new characterization of balls is presented. 2. Viscosity Approximation of Common Fixed Points for L-Lipschitzian Semigroup of Pseudocontractive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Xue-song Li 2009-01-01 Full Text Available We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007 and the pseudocontractive mapping in (Zegeye et al., 2007 to the pseudocontractive semigroup in Banach spaces under different conditions. 3. Functional analysis an introduction to metric spaces, Hilbert spaces, and Banach algebras CERN Document Server Muscat, Joseph 2014-01-01 This textbook is an introduction to functional analysis suited to final year undergraduates or beginning graduates. Its various applications of Hilbert spaces, including least squares approximation, inverse problems, and Tikhonov regularization, should appeal not only to mathematicians interested in applications, but also to researchers in related fields. Functional Analysis adopts a self-contained approach to Banach spaces and operator theory that covers the main topics, based upon the classical sequence and function spaces and their operators. It assumes only a minimum of knowledge in elementary linear algebra and real analysis; the latter is redone in the light of metric spaces. It contains more than a thousand worked examples and exercises, which make up the main body of the book. 4. Characterization of the absolutely summing operators in a Banach space using μ-approximate l_1 sequences Directory of Open Access Journals (Sweden) N. L. Braha 2005-05-01 Full Text Available In this paper we will give a characterization of 1-absolutely summing operators using μ-approximate l_1 sequences. Exactly if (x_n _1^∞ is μ-approximate l_1 , basic and normalized sequence in Banach space X then every bounded linear operator T from X into Banach space Y is 1-absolutely summing if and only if Y is isomorphic to Hilbert space. 5. Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces International Nuclear Information System (INIS) This paper treats the embedding of finite-dimensional subsets of a Banach space B into finite-dimensional Euclidean spaces. When the Hausdorff dimension of X ? X is finite, dH(X ? X) k are injective on X. The proof motivates the definition of the 'dual thickness exponent', which is the key to proving that a prevalent set of such linear maps have Hlder continuous inverse when the box-counting dimension of X is finite and k > 2dB(X). A related argument shows that if the Assouad dimension of X ? X is finite and k > dA(X ? X), a prevalent set of such maps are bi-Lipschitz with logarithmic corrections. This provides a new result for compact homogeneous metric spaces via the Kuratowksi embedding of (X, d) into L?(X) 6. Convergence theorems of common elements for mixed equilibrium problems, variational inequality problems and fixed point problems in Banach spaces Directory of Open Access Journals (Sweden) Kriengsak Wattanawitoon 2011-01-01 Full Text Available In this paper, a hybrid iterative scheme is introduced for theapproximation method for finding a common element of the set of mixed equilibrium problems, variational inequality problems and fixed point problems of two quasi-$phi$-nonexpansive mappings ina real uniformly convex and uniformly smooth Banach space. Then, we obtain a strong convergence theorem for the sequence generated by those process in Banach spaces. Moreover, we obtain new resultfor finding a zero point of maximal monotone operators in a Banach space. Our results improve and extend the corresponding results announced by Takahashi and Zembayashi [Strong and weak convergencetheorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009 45--57.] Qin, Cho and Kang [Convergence theorems of common elements forequilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009, 20--30.] Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithmfor fixed point problems and equilibrium problems of tworelatively quasi-nonexpansive mappings, Nonlinear Anal: Hybrid Systems. 3 (2009 11--20.] Cholamjiak [A hybrid iterative scheme for equilibrium problems, variational inequality problems and fixed point problems in Banach spaces, Fixed Point Theory Appl., (2009, Article ID 719360, 18 pages] and many authors. 7. Inexact Newton–Landweber iteration for solving nonlinear inverse problems in Banach spaces International Nuclear Information System (INIS) By making use of duality mappings, we formulate an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme providing the increments by applying the Landweber iteration in Banach spaces to the local linearized equations. It has the advantage of reducing computational work by computing more cheap steps in each inner scheme. We first prove a convergence result for the exact data case. When the data are given approximately, we terminate the method by a discrepancy principle and obtain a weak convergence result. Finally, we test the method by reporting some numerical simulations concerning the sparsity recovery and the noisy data containing outliers. (paper) 8. Buried object detection by means of a Lp Banach-space inversion procedure Science.gov (United States) Estatico, Claudio; Fedeli, Alessandro; Pastorino, Matteo; Randazzo, Andrea 2015-01-01 Electromagnetic inspection techniques are becoming powerful tools for buried object detection and subsurface prospection in several applicative fields, such as civil engineering and archeology. However, the nonlinearity and ill-posedness of the underlying inverse problem make the development of efficient imaging techniques a very challenging task. In the present paper, an algorithm based on a regularizing approach in Lp Banach spaces is proposed for tackling such problems. The effectiveness of the approach is verified by means of numerical simulations in a noisy environment aimed at evaluating the reconstruction capabilities with respect to the choice of several model parameters. The reported results show that, for small targets, the use of Lp Banach spaces with 1 < p < 2 allows to obtain a better localization of different buried scatterers. 9. Maximal regular boundary value problems in Banach-valued weighted space Directory of Open Access Journals (Sweden) Veli B. Shakhmurov 2005-02-01 Full Text Available This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted Lp spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties. 10. Some s-numbers of an integral operator of Hardy type in Banach function spaces Czech Academy of Sciences Publication Activity Database Edmunds, D.; Gogatishvili, Amiran; Kopaliani, T.; Samashvili, N. 2016-01-01 Roč. 207, July (2016), s. 79-97. ISSN 0021-9045 R&D Projects: GA ČR GA13-14743S Institutional support: RVO:67985840 Keywords : Hardy type operators * Banach function space s * s-numbers * compact linear operators Subject RIV: BA - General Mathematics Impact factor: 0.951, year: 2014 http://www.sciencedirect.com/science/article/pii/S0021904516000265 11. Some s-numbers of an integral operator of Hardy type in Banach function spaces Czech Academy of Sciences Publication Activity Database Edmunds, D.; Gogatishvili, Amiran; Kopaliani, T.; Samashvili, N. 2016-01-01 Roč. 207, July (2016), s. 79-97. ISSN 0021-9045 R&D Projects: GA ČR GA13-14743S Institutional support: RVO:67985840 Keywords : Hardy type operators * Banach function spaces * s-numbers * compact linear operators Subject RIV: BA - General Mathematics Impact factor: 0.951, year: 2014 http://www. science direct.com/ science /article/pii/S0021904516000265 12. A reinterpretation of set differential equations as differential equations in a Banach space OpenAIRE Rasmussen, Martin; Rieger, Janosch; Webster, Kevin 2015-01-01 Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of$\\R^d$with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example,... 13. Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces OpenAIRE Saeidi Shahram 2008-01-01 Abstract Let be a left amenable semigroup, let be a representation of as Lipschitzian mappings from a nonempty compact convex subset of a smooth Banach space into with a uniform Lipschitzian condition, let be a strongly left regular sequence of means defined on an -stable subspace of , let be a contraction on , and let , , and be sequences in (0, 1) such that , for all . Let , for all . Then, under suitable hypotheses on the constants, we show that converges stron... 14. Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces OpenAIRE Tomonari Suzuki 2006-01-01 We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let be a bounded closed convex subset of a uniformly smooth Banach space . Let be an infinite family of commuting nonexpansive mappings on . Let and be sequences in satisfying for . Fix and define a sequence in by for . Then converges strongly to , where is the unique sunny nonexpansive retraction from onto . 15. On the range of the derivative of Gâteaux-Smooth functions on separable Banach spaces Czech Academy of Sciences Publication Activity Database Deville, R.; Hájek, Petr Pavel 2005-01-01 Roč. 145, č. 2 (2005), s. 257-269. ISSN 0021-2172 R&D Projects: GA AV ČR(CZ) IAA1019003; GA AV ČR(CZ) IAA1019205; GA ČR(CZ) GA201/01/1198 Institutional research plan: CEZ:AV0Z10190503 Keywords : Gâteaux-Smooth functions * Banach space * Lipschitz function Subject RIV: BA - General Mathematics Impact factor: 0.448, year: 2005 16. Approximating fixed points of non-self asymptotically nonexpansive mappings in Banach spaces OpenAIRE Yongfu Su; Xiaolong Qin 2006-01-01 Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T:K?E be an asymptotically nonexpansive mapping with {kn}?[1,?) such that ?n=1?(kn?1) 17. Some Results for a Finite Family of Uniformly L-Lipschitzian Mappings in Banach Spaces OpenAIRE Shih-Sen Chang; Jia Lin Huang; Xiong Rui Wang 2007-01-01 The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper not only correct some mistakes appeared in the paper by Ofoedu (2006) but also improve and extend some recent results by Chang (2001), Cho et al. (2005), Ofoedu (2006), Schu (1991), and Zeng (2003, 2005). 18. Fixed point iteration for asymptotically quasi-nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Somyot Plubtieng 2005-07-01 Full Text Available Suppose that C is a nonempty closed convex subset of a real uniformly convex Banach space X. Let T:CÃ¢Â†Â’C be an asymptotically quasi-nonexpansive mapping. In this paper, we introduce the three-step iterative scheme for such map with error members. Moreover, we prove that if T is uniformly L-Lipschitzian and completely continuous, then the iterative scheme converges strongly to some fixed point of T. 19. On some Banach space constants arising in nonlinear fixed point and eigenvalue theory Directory of Open Access Journals (Sweden) Erzakova Nina A 2004-01-01 Full Text Available As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps. 20. On some Banach space constants arising in nonlinear fixed point and eigenvalue theory Directory of Open Access Journals (Sweden) Martin Väth 2004-12-01 Full Text Available As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps. 1. Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Suzuki Tomonari 2006-01-01 Full Text Available We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let be a bounded closed convex subset of a uniformly smooth Banach space . Let be an infinite family of commuting nonexpansive mappings on . Let and be sequences in satisfying for . Fix and define a sequence in by for . Then converges strongly to , where is the unique sunny nonexpansive retraction from onto . 2. Weak and Strong Convergence Theorems for Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Jing Zhao 2008-03-01 Full Text Available The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping T and a finite family of nonexpansive mappings {Ti}i=1N, respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others. 3. Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Huimin He 2010-01-01 Full Text Available Let K be a nonempty compact convex subset of a uniformly convex Banach space E and let T be a multivalued nonexpansive mapping. For the implicit iterates x0?K, xn=?nxn-1+(1-?nyn, yn?Txn, n?1. We proved that {xn} converges strongly to a fixed point of T under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007. 4. Hyers-Ulam stability of linear second-order differential equations in complex Banach spaces Directory of Open Access Journals (Sweden) Yongjin Li 2013-08-01 Full Text Available We prove the Hyers-Ulam stability of linear second-order differential equations in complex Banach spaces. That is, if y is an approximate solution of the differential equation$y''+ alpha y'(t +eta y = 0$or$y''+ alpha y'(t +eta y = f(t$, then there exists an exact solution of the differential equation near to y. 5. Weak and Strong Convergence Theorems for Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Su Yongfu 2008-01-01 Full Text Available Abstract The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping and a finite family of nonexpansive mappings , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others. 6. Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group CERN Document Server Abreu, Luis Daniel 2010-01-01 Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L^2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lattice sampling theorem discussed by J. M. Whittaker in Chapter 5 of his book "Interpolatory Function Theory". 7. Embedding theorems in Banach-valued -spaces and maximal -regular differential-operator equations Directory of Open Access Journals (Sweden) Shakhmurov Veli B 2006-01-01 Full Text Available The embedding theorems in anisotropic Besov-Lions type spaces are studied; here and are two Banach spaces. The most regular spaces are found such that the mixed differential operators are bounded from to , where are interpolation spaces between and depending on and . By using these results the separability of anisotropic differential-operator equations with dependent coefficients in principal part and the maximal -regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the quasielliptic partial differential equations and the parabolic Cauchy problems are studied. 8. On Linear Isometries of Banach Lattices in$\\mathcal{C}_0()$-Spaces Indian Academy of Sciences (India) José M Isidro 2009-11-01 Consider the space$\\mathcal{C}_0()$endowed with a Banach lattice-norm$\\\|\\cdot p\\\|$that is not assumed to be the usual spectral norm$\\\|\\cdot p\\\|_∞$of the supremum over . A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry of the Banach lattice$(\\mathcal{C}_0(),\\\|\\cdot p\\\|)$induces a partition of into a family of finite subsets$S\\subset$along with a bijection$T:→$which preserves cardinality, and a family$[u(S):S\\in]$of surjective linear maps$u(S):\\mathcal{C}(T(S))→\\mathcal{C}(S)$of the finite-dimensional -algebras$\\mathcal{C}(S)$such that $$(U f)\|_{T(S)}=u(S)(f\|_S) \\quad \\forall f\\in\\mathcal{C}_0() \\quad \\forall S\\in.$$ Here we endow the space of finite sets$S\\subset$with a topology for which the bijection and the map are continuous, thus completing the analogy with the classical result. 9. Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Jong Soo Jung 2008-05-01 Full Text Available Let E be a reflexive Banach space with a uniformly GÃƒÂ¢teaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f:CÃ¢Â€Â‰Ã¢Â€Â‰Ã¢Â†Â’Ã¢Â€Â‰Ã¢Â€Â‰C a contractive mapping (or a weakly contractive mapping, and T:CÃ¢Â€Â‰Ã¢Â€Â‰Ã¢Â†Â’Ã¢Â€Â‰Ã¢Â€Â‰C nonexpansive mapping with the fixed point set F(TÃ¢Â€Â‰Ã¢Â€Â‰Ã¢Â‰Â Ã¢Â€Â‰Ã¢Â€Â‰Ã¢ÂˆÂ…. Let {xn} be generated by a new composite iterative scheme: yn=ÃŽÂ»nf(xn+(1Ã¢ÂˆÂ’ÃŽÂ»nTxn, xn+1=(1Ã¢ÂˆÂ’ÃŽÂ²nyn+ÃŽÂ²nTyn, (nÃ¢Â‰Â¥0. It is proved that {xn} converges strongly to a point in F(T, which is a solution of certain variational inequality provided that the sequence {ÃŽÂ»n}Ã¢ÂŠÂ‚(0,1 satisfies limnÃ¢Â†Â’Ã¢ÂˆÂžÃŽÂ»n=0 and Ã¢ÂˆÂ‘n=1Ã¢ÂˆÂžÃŽÂ»n=Ã¢ÂˆÂž, {ÃŽÂ²n}Ã¢ÂŠÂ‚[0,a for some 0 10. Hilbert space frames containing a Riesz basis and Banach spaces which have no subspace isomorphic to$c_0$OpenAIRE Casazza, Peter G.; Christensen, Ole 1995-01-01 We prove that a Hilbert space frame$\\fti$contains a Riesz basis if every subfamily$\\ftj , J \\subseteq I ,$is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to$c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements. 11. Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces International Nuclear Information System (INIS) In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in a Banach space setting and prove optimal convergence rates under approximate source conditions. These are related to the classical concept of source conditions that is available only in Hilbert space. We provide results in the framework of general index functions, which include, e.g. Hölder and logarithmic rates. Concerning the regularization parameters in each Newton step as well as the stopping index, we provide both a priori and a posteriori strategies, the latter being based on the discrepancy principle 12. Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces Directory of Open Access Journals (Sweden) A. Abkar 2010-01-01 Full Text Available In the first part of this paper, we prove the existence of common fixed points for a commuting pair consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition in a uniformly convex Banach space. In this way, we generalize the result of Dhompongsa et al. (2006. In the second part of this paper, we prove a fixed point theorem for upper semicontinuous mappings satisfying the Suzuki condition in strictly L(? spaces; our result generalizes a recent result of Domnguez-Benavides et al. (2009. 13. Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Abkar A 2010-01-01 Full Text Available In the first part of this paper, we prove the existence of common fixed points for a commuting pair consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition in a uniformly convex Banach space. In this way, we generalize the result of Dhompongsa et al. (2006. In the second part of this paper, we prove a fixed point theorem for upper semicontinuous mappings satisfying the Suzuki condition in strictly spaces; our result generalizes a recent result of Domnguez-Benavides et al. (2009. 14. On Borel structures in the Banach space C(\\beta\\omega) OpenAIRE Marciszewski, Witold; Plebanek, Grzegorz 2013-01-01 M. Talagrand showed that, for the Cech-Stone compactification \\beta\\omega\\ of the space of natural numbers, the norm and the weak topology generate different Borel structures in the Banach space C(\\beta\\omega). We prove that the Borel structures in C(\\beta\\omega) generated by the weak and the pointwise topology are also different. We also show that in C(\\omega), where \\omega=\\beta\\omega - \\omega, there is no countable family of pointwise Borel sets separating functions from C(\\omega). 15. Variational Inequalities and Improved Convergence Rates for Tikhonov Regularisation on Banach Spaces CERN Document Server Grasmair, Markus 2011-01-01 In this paper we derive higher order convergence rates in terms of the Bregman distance for Tikhonov like convex regularisation for linear operator equations on Banach spaces. The approach is based on the idea of variational inequalities, which are, however, not imposed on the original Tikhonov functional, but rather on a dual functional. Because of that, the approach is not limited to convergence rates of lower order, but yields the same range of rates that is well known for quadratic regularisation on Hilbert spaces. 16. Evolutionary problems in non-reflexive spaces Czech Academy of Sciences Publication Activity Database Kružík, Martin; Zimmer, J. 2010-01-01 Roč. 16, č. 1 (2010), s. 1-22. ISSN 1262-3377 R&D Projects: GA AV ČR IAA1075402 Institutional research plan: CEZ:AV0Z10750506 Keywords : concentrations * energetic solution * energies with linear growth * oscillations * relaxation Subject RIV: BA - General Mathematics Impact factor: 1.084, year: 2009 http://library.utia.cas.cz/separaty/2008/MTR/kruzik-evolutionary problems in non-reflexive spaces.pdf 17. A forward backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space Science.gov (United States) Bredies, Kristian 2009-01-01 We consider the task of computing an approximate minimizer of the sum of a smooth and a non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman-Taylor-distance estimates, rates of convergence for the forward-backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation-based image restoration in higher dimensions are presented. 18. Density of a semigroup in a Banach space International Nuclear Information System (INIS) We study conditions on a set M in a Banach space X which are necessary or sufficient for the set R(M) of all sums x1+⋯+xn, xk∈M, to be dense in X. We distinguish conditions under which the closure R(M)-bar is an additive subgroup of X, and conditions under which this additive subgroup is dense in X. In particular, we prove that if M is a closed rectifiable curve in a uniformly convex and uniformly smooth Banach space X, and does not lie in a closed half-space {x∈X:f(x)≥0}, f∈X∗, and is minimal in the sense that every proper subarc of M lies in an open half-space {x∈X:f(x)>0}, then R(M)-bar =X. We apply our results to questions of approximation in various function spaces 19. Differences of weighted composition operators between weighted Banach spaces of holomorphic functions and weighted Bloch type spaces Directory of Open Access Journals (Sweden) Elke Wolf 2010-01-01 Full Text Available We consider analytic self-maps ø‑1,ø ‑2 of the open unit disk as well as analytic mapsψ 1, ψ2. These maps induce differences of weighted composition operators acting between weighted Banach spaces of holomorphic functions and weighted Bloch type spaces. In this article we give necessary and sufficient conditions for such a difference to be bounded resp. compact.Nosotros consideramos auto aplicaciones ø‑1,ø ‑2 del disco unitario abierto bien como aplicaciones analíticas ψ1, ψ2. Estas aplicaciones inducen diferencias de compición de operadores con peso actuando entre espacios de Banach pesados de funciones holomorfas y espacios de tipo Bloch con peso. En este artículo damos condiciones necesarias y suficientes para que tal diferencia sea acotada, respectivamente, compacta. 20. Some compactness tests in Banach spaces by Cesaro means of Fourier coefficients Science.gov (United States) Öztürk, Seda 2015-09-01 Let H be a complex Banach space, T be the topological group of the unit circle with respect to the Euclidean topology, α be a strongly continuous isometric linear representation of T in H, {Fkα}k ∈ℤ be the family of Fourier coefficients with respect to α, and {σkα}k ∈ℤ be Cesaro means of the family {Fkα}k ∈ℤ . In this work, we give some compactness tests for closed subsets of H. 1. Oscillation and the mean ergodic theorem for uniformly convex Banach spaces OpenAIRE Avigad, Jeremy; Rute, Jason 2012-01-01 Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k \|\| A_{t_{k+1}} x - A_{t_k} x \|\|^p 2. On the unbounded behaviour for some non-autonomous systems in Banach spaces International Nuclear Information System (INIS) By modifying our previous methods and by using the notion of integral solution introduced by Benilan, we study the asymptotic behaviour of unbounded trajectories for the quasi-autonomous dissipative system: du/dt + Au is not an element of f where X is a real Banach space, A an accretive (possibly multivalued) operator in X x X, and f - f∞ is an element of Lp((0, +∞);X) for some f∞ is an element of X and 1 ≤ p < ∞. (author). 24 refs 3. C0-semigroups of linear operators on some ultrametric Banach spaces Directory of Open Access Journals (Sweden) Toka Diagana 2006-06-01 Full Text Available C0-semigroups of linear operators play a crucial role in the solvability of evolution equations in the classical context. This paper is concerned with a brief conceptualization of C0-semigroups on (ultrametric free Banach spaces E. In contrast with the classical setting, the parameter of a given C0-semigroup belongs to a clopen ball ÃŽÂ©r of the ground field K. As an illustration, we will discuss the solvability of some homogeneous p-adic differential equations. 4. Gabor multipliers for weighted Banach spaces on locally compact abelian groups OpenAIRE Pandey, S. S. 2009-01-01 We use a projective groups representation$\\rho$of the unimodular group$\\mathcal{G} \\times \\hat{\\mathcal{G}}$on$L^2(\\mathcal{G}$) to define Gabor wavelet transform of a function$f$with respect to a window function$g$, where$\\mathcal{G}$is a locally compact abelian group and$\\hat{\\mathcal{G}}$its dual group. Using these transforms, we define a weighted Banach$\\mathcal{H}^{1, \\rho}_w(\\mathcal{G})$and its antidual space$\\mathcal{H}^{{1}^{\\sim}, \\rho}_w(\\math... 5. On the regularity of mild solutions to complete higher order differential equations on Banach spaces Directory of Open Access Journals (Sweden) Nezam Iraniparast 2015-09-01 Full Text Available For the complete higher order differential equation u(n(t=Σk=0n-1Aku(k(t+f(t, t∈ R (* on a Banach space E, we give a new definition of mild solutions of (. We then characterize the regular admissibility of a translation invariant subspace al M of BUC(R, E with respect to ( in terms of solvability of the operator equation Σj=0n-1AjXal Dj-Xal Dn = C. As application, almost periodicity of mild solutions of (* is proved. 6. Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers OpenAIRE Talponen, Jarno 2014-01-01 In this note various geometric properties of a Banach space $X$ are characterized by means of weaker corresponding geometric properties involving an ultrapower $X^\\mathcal{U}$. The characterizations do not depend on the particular choice of the free ultrafilter $\\mathcal{U}$. For example, a point $x\\in S_X$ is an MLUR point if and only if $j(x)$ (given by the canonical inclusion $j\\colon X \\to X^\\mathcal{U}$) in $B_{X^\\mathcal{U}}$ is an extreme point; a point $x\\in S_X$ is LUR if and only if... 7. Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Udomene Aniefiok 2006-01-01 Full Text Available Suppose is a nonempty closed convex subset of a real Banach space . Let be two asymptotically quasi-nonexpansive maps with sequences such that and , and . Suppose is generated iteratively by where and are real sequences in . It is proved that (a converges strongly to some if and only if ; (b if is uniformly convex and if either or is compact, then converges strongly to some . Furthermore, if is uniformly convex, either or is compact and is generated by , where , are bounded, are real sequences in such that and , are summable; it is established that the sequence (with error member terms converges strongly to some . 8. Toward a General Law of the Iterated Logarithm in Banach Space OpenAIRE Einmahl, Uwe 1993-01-01 A general bounded law of the iterated logarithm for Banach space valued random variables is established. Our results implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables in the domain of attraction of a Gaussian law and (c) new LIL results for random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence $\\{\\sqrt{nLLn}\\}$ does not work. Basic ingredients of our proof are an infinite-dimensional Fuk-Nag... 9. Maximal regular boundary value problems in Banach-valued function spaces and applications Directory of Open Access Journals (Sweden) Veli B. Shakhmurov 2006-07-01 Full Text Available The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valued Lp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied. 10. Approximation of fixed points of Lipschitz pseudo-contractive mapping in Banach spaces International Nuclear Information System (INIS) Let K be a subset of a real Banach space X. A mapping T:K → X is called pseudo-contractive if the inequality \|\|x-y\|\| ≤ \|\|(1+r)(x-y)-r(Tx-Ty)\|\| holds for all x,y in K and r > 0. Fixed points of Lipschitz pseudo-contractive maps are approximated under appropriate conditions, by an iteration process of the type introduced by W.R. Mann. This gives an affirmative answer to the problem stated by T.L. Hicks and J.R. Rubicek (J. Math. Anal. Appl. 59 (1977) 504). (author). 28 refs 11. An analytic Koszul complex in a Banach space OpenAIRE Patyi, Imre 2005-01-01 We show that the holomorphic ideal sheaf of a linear section of a pseudoconvex open subset $\\Omega$ of, say, a Hilbert space $X=\\ell_2$ is acyclic. We also prove an analog of Hefer's lemma, i.e., if $f:\\Omega\\times\\Omega\\to\\CC$ is holomorphic and $f(x,x)=0$ for $x\\in\\Omega$, then there is a holomorphic $g:\\Omega\\times\\Omega\\to X^$ with values in the dual space $X^$ of $X$ such that $f(x,y)=g(x,y)(x-y)$ 12. The Banach space $H^1(X,d,\\mu)$, II OpenAIRE Müller, Paul F. X. 1994-01-01 In this paper we give the isomorphic classification of atomic $H^1(X,d,\\mu)$, where $(X,d,\\mu)$ is a space of homogeneous type, hereby completing a line of investigation opened by the work of Bernard Maurey [Ma1], [Ma2], [Ma3] and continued by Lennard Carleson [C] and Przemyslaw Wojtaszczyk [Woj1], [Wpj2]. 13. A characterization of subspaces of weakly compactly generated Banach spaces Czech Academy of Sciences Publication Activity Database Fabian, Marián; Montesinos, V.; Zizler, V. 2004-01-01 Roč. 69, č. 2 (2004), s. 457-464. ISSN 0024-6107 R&D Projects: GA ČR GA201/01/1198 Institutional research plan: CEZ:AV0Z1019905 Keywords : subspace of a weakly compactly generated space * Eberlein compact * epsilon-weakly compact set Subject RIV: BA - General Mathematics Impact factor: 0.663, year: 2004 14. Convergence rates and finite-dimensional approximations for nonlinear ill-posed problems involving monotone operators in Banach spaces International Nuclear Information System (INIS) The purpose of this paper is to investigate convergence rates for an operator version of Tikhonov regularization constructed by dual mapping for nonlinear ill-posed problems involving monotone operators in real reflective Banach spaces. The obtained results are considered in combination with finite-dimensional approximations for the space. An example is considered for illustration. (author). 15 refs 15. Ã¢Â„Â‚-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation Directory of Open Access Journals (Sweden) Lars Filipsson 2006-07-01 Full Text Available We investigate the concepts of linear convexity and Ã¢Â„Â‚-convexity in complex Banach spaces. The main result is that any Ã¢Â„Â‚-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a Ã¢Â„Â‚-convex domain ÃŽÂ© in the Banach space X and a point pÃ¢ÂˆÂ‰ÃŽÂ©, there is a complex hyperplane through p that does not intersect ÃŽÂ©. We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined in Ã¢Â„Â‚-convex domains. 16. A convergence rates result for an iteratively regularized Gauss–Newton–Halley method in Banach space International Nuclear Information System (INIS) The use of second order information on the forward operator often comes at a very moderate additional computational price in the context of parameter identification problems for differential equation models. On the other hand the use of general (non-Hilbert) Banach spaces has recently found much interest due to its usefulness in many applications. This motivates us to extend the second order method from Kaltenbacher (2014 Numer. Math. at press), (see also Hettlich and Rundell 2000 SIAM J. Numer. Anal. 37 587620) to a Banach space setting and analyze its convergence. We here show rates results for a particular source condition and different exponents in the formulation of Tikhonov regularization in each step. This includes a complementary result on the (first order) iteratively regularized Gauss–Newton method in case of a one-homogeneous data misfit term, which corresponds to exact penalization. The results clearly show the possible advantages of using second order information, which get most pronounced in this exact penalization case. Numerical simulations for an inverse source problem for a nonlinear elliptic PDE illustrate the theoretical findings. (paper) 17. Newton-Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Method in Banach Spaces OpenAIRE Rongfei Lin; Yueqing Zhao; Zdeněk Šmarda; Yasir Khan; Qingbiao Wu 2013-01-01 Newton-Kantorovich and Smale uniform type of convergence theorem of a deformed Newton method having the third-order convergence is established in a Banach space for solving nonlinear equations. The error estimate is determined to demonstrate the efficiency of our approach. The obtained results are illustrated with three examples. 18. Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi-I-Nonexpansive Mapping in Banach Space Directory of Open Access Journals (Sweden) Farrukh Mukhamedov 2010-01-01 Full Text Available We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mapping T and an asymptotically quasi-nonexpansive mapping I, defined on a nonempty closed convex subset of a Banach space. 19. On the Krasnoselskii-type fixed point theorems for the sum of continuous and asymptotically nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Arunchai Areerat 2011-01-01 Full Text Available Abstract In this article, we prove some results concerning the Krasnoselskii theorem on fixed points for the sum A + B of a weakly-strongly continuous mapping and an asymptotically nonexpansive mapping in Banach spaces. Our results encompass a number of previously known generalizations of the theorem. 20. Almost Stable Iteration Schemes for Local Strongly Pseudocontractive and Local Strongly Accretive Operators in Real Uniformly Smooth Banach Spaces Directory of Open Access Journals (Sweden) Shin Min Kang 2008-08-01 Full Text Available In this paper we establish the strong convergence and almost stability of the Ishikawa iteration methods with errors for the iterative approximations of either fixed points of local strongly pseudocontractive operators or solutions of nonlinear operator equations with local strongly accretive type in real uniformly smooth Banach spaces. Our convergence results extend some known results in the literature. 1. Perturbation analysis for the Moore–Penrose metric generalized inverse of closed linear operators in Banach spaces OpenAIRE Du, Fapeng; Chen, Jianlong 2016-01-01 In this paper, we characterize the perturbations of the Moore–Penrose metric generalized inverse of closed operator in Banach spaces. Under the condition $R(\\delta T)\\subset R(T)$ , $N(T)\\subset N(\\delta T)$ , respectively, we get some new results about upper-bound estimates of $\\\|\\bar{T}^{M}\\\|$ and $\\\|\\bar{T}^{M}-T^{M}\\\|$ . 2. Picard iterations for nonlinear Lipschitz strong pseudo-contractions in uniformly smooth Banach spaces International Nuclear Information System (INIS) Suppose E is a real uniformly smooth Banach space and K is a nonempty closed convex and bounded subset of E, T:K ? K is a Lipschitz pseudo-contraction. It is proved that the Picard iterates of a suitably defined operator converges strongly to the unique fixed point of T. Furthermore, this result also holds for the slightly larger class of Lipschitz strong hemi-contractions. Related results deal with strong convergence of the Picard iterates to the unique solution of operator equations involving Lipschitz strongly accretive maps. Apart from establishing strong convergence, our theorems give existence, uniqueness and convergence-rate which is at least as fast as a geometric progression. (author). 51 refs 3. Convergence of hybrid steepest descent method for variational inequalities in Banach spaces International Nuclear Information System (INIS) Let E be a real q-uniformly smooth Banach space with constant dq, q ? 2. Let T : E ? E and G : E ? E be a nonexpansive map and an ?-strongly accretive map which is also ? - Lipschitzian, respectively. Let {?n} be a real sequence in [0, 1] satisfying some appropriate conditions. For ? element of (0, ( q ?/dq ?q )q-1 ), define a sequence { xn} iteratively in E by x0 element of E, xn+1 = T?n+1 xn = Txn - ? ?n+1 G(Txn), n ? 0. Then, {xn} converges strongly to the unique solution x* of the variational inequality problem VI(G,K) (search for x* element of K such that q(y - x)> ? 0 for all y element of K), where K := Fix(T) { x element of E : Tx = x} ? 0. A convergence theorem related to fi nite family of nonexpansive maps is also proved. (author) 4. Models of CT dose profiles in Banach space; with applications to CT Dosimetry CERN Document Server Weir, Victor J 2015-01-01 This paper consists of two parts.In the first part, the scatter components of computed tomograpahy dose profiles are modeled using various functions including the solution to Riccati's differential equation. These scatter functions are combined with primary components such as a trapezoidal function and a constructed function that uses the analytic continuation of Heaviside step function. A mathematical theory is developed in Banach space. The modeled function, which is the product of the scatter and primary functions, is used to accurately fit data from the O-arm cone beam imaging system. In a second part of the paper, an approach to dosimtery is developed that shows that the results obtained from the use of a pencil shaped ion chamber is equivalent to that from a farmer chamber. This result is verified by presenting some preliminary experimental data measured in a 64 slice Siemens Sensation scanner. 5. Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus Directory of Open Access Journals (Sweden) George A. Anastassiou 2015-10-01 Full Text Available We present a semilocal convergence study of Newton-type methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies require that the operator involved is Frchet differentiable. In the present study we assume that the operator is only continuous. This way we extend the applicability of Newton-type methods to include fractional calculus and problems from other areas. Moreover, under the same or weaker conditions, we obtain weaker sufficient convergence criteria, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. Special cases are provided where the old convergence criteria cannot apply but the new criteria can apply to locate zeros of operators. Some applications include fractional calculus involving the Riemann-Liouville fractional integral and the Caputo fractional derivative. Fractional calculus is very important for its applications in many applied sciences. 6. Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces Directory of Open Access Journals (Sweden) Chen Rudong 2011-01-01 Full Text Available Abstract We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities. Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011 and Lu et al. (Nonlinear Anal, 71(3-4, 1032-1041, 2009. 2000 Mathematics subject classification 47H05; 47H09; 65J15 7. Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces Directory of Open Access Journals (Sweden) Yao JC 2009-01-01 Full Text Available Let be a nonempty closed convex subset of a Banach space with the dual , let be a continuous mapping, and let be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI( : find such that for all , where is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings, respectively, we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao (2008 for finding approximate solutions of the VI( . Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI( , which is also a fixed point of . 8. An iterative regularization method for the solution of the split feasibility problem in Banach spaces International Nuclear Information System (INIS) The split feasibility problem (SFP) consists of finding a common point in the intersection of finitely many convex sets, where some of the sets arise by imposing convex constraints in the range of linear operators. We are concerned with its solution in Banach spaces. To this end we generalize the CQ algorithm of Byrne with Bregman and metric projections to obtain an iterative solution method. In case the sets projected onto are contaminated with noise we show that a discrepancy principle renders this algorithm a regularization method. We measure the distance between convex sets by local versions of the Hausdorff distance, which in contrast to the standard Hausdorff distance allow us to measure the distance between unbounded sets. Hereby we prove a uniform continuity result for both kind of projections. The performance of the algorithm is demonstrated with some numerical experiments 9. The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space OpenAIRE Aneke, S. J. 2010-01-01 The equation = , where = + , with being a K-positive definite operator and being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point , where is Frchet differentiable and a... 10. Quasi-geostrophic equations with initial data in Banach spaces of local measures Directory of Open Access Journals (Sweden) 2005-06-01 Full Text Available This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations $$displaylines{ partial_{t}heta+u ablaheta+( -Delta ^{gamma}heta =0 quad hbox{on }mathbb{R}^{d}imes] 0,+infty[cr heta( x,0 =heta_{0}(x, quad xinmathbb{R}^{d} }$$ where 0 less than $gammaleq 1$ is a fixed parameter and the velocity field $u=(u_{1},u_{2},dots,u_{d}$ is divergence free; i.e., $abla u=0$. The initial data $heta_{0}$ is taken in Banach spaces of local measures (see text for the definition, such as Multipliers, Lorentz and Morrey-Campanato spaces. We will focus on the subcritical case 1/2 less than $gammaleq1$ and we analyse the well-posedness of the system in three basic spaces: $L^{d/r,infty}$, $dot {X}_{r}$ and $dot {M}^{p,d/r}$, when the solution is global for sufficiently small initial data. Furtheremore, we prove that the solution is actually smooth. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. 11. Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces OpenAIRE Sokhuma, K.; A. Kaewkhao 2010-01-01 Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that and for all . We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary by , , where and , are sequences of positive numbers satisfying , , converges strongly to a common fixed point of and ; that is, there exists such tha... 12. Comparison of fastness of the convergence among Krasnoselskij, Mann, and Ishikawa iterations in arbitrary real Banach spaces Directory of Open Access Journals (Sweden) 2006-01-01 Full Text Available Let be an arbitrary real Banach space and a nonempty, closed, convex (not necessarily bounded subset of . If is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant , then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of . 13. On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces Directory of Open Access Journals (Sweden) M. Eshaghi Gordji 2009-01-01 Full Text Available We establish the general solution of the functional equation f(nx+y+f(nx−y=n2f(x+y+n2f(x−y+2(f(nx−n2f(x−2(n2−1f(y for fixed integers n with n≠0,±1 and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces. 14. Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces International Nuclear Information System (INIS) Let E be a real uniformly convex Banach space whose dual space E satisfies the Kadec- Klee property, K be a closed convex nonempty subset of E . Let T1, T2, . . . , Tm : K → K be asymptotically nonexpansive mappings of K into E with sequences (respectively) {kin}n=1∞ satisfying kin → 1 as n → ∞, i = 1, 2 , ...,m and Σn=1∞(kin - 1) in}n=1∞ be a sequence in [ε, 1 - ε ], for each i element of { 1, 2 , . . . ,m} (respectively). Let {xn} be a sequence generated for m ≥ 2 by, x1 element of K, xn+1 = (1 - α1n)xn + α1nT1nyn+m-2, yn+m-2 = (1 - α2n)xn + α2nT2nyn+m-3, ..., yn = (1 - αmn)xn + αmnTmnxn , n ≥ 1. Let Intersectioni=1m F (Ti) ≠ 0 . Then, {xn} converges weakly to a common fixed point of the family {Ti}i=1m. Under some appropriate condition on the family {Ti}i=1m, a strong convergence theorem is also roved. (author) 15. Quadratic equations in Banach space, perturbation techniques and applications to Chandrasekhar's and related equations International Nuclear Information System (INIS) In this dissertation perturbation techniques are developed, based on the contraction mapping principle which can be used to prove existence and uniqueness for the quadratic equation x = y + lambdaB(x,x) (1) in a Banach space X; here B: XxX→X is a bounded, symmetric bilinear operator, lambda is a positive parameter and y as a subset of X is fixed. The following is the main result. Theorem. Suppose F: XxX→X is a bounded, symmetric bilinear operator and that the equation z = y + lambdaF(z,z) has a solution z/sup / of sufficiently small norm. Then equation (1) has a unique solution in a certain closed ball centered at z/sup /. Applications. The theorem is applied to the famous Chandrasekhar equation and to the Anselone-Moore system which are of the form (1) above and yields existence and uniqueness for a solution of (1) for larger values of lambda than previously known, as well as more accurate information on the location of solutions 16. Approximating fixed points of non-self asymptotically nonexpansive mappings in Banach spaces Directory of Open Access Journals (Sweden) Yongfu Su 2006-01-01 Full Text Available Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T:K?E be an asymptotically nonexpansive mapping with {kn}?[1,? such that ?n=1?(kn?10. Starting from arbitrary x1?K, define the sequence {xn} by x1?K, zn=P(?n''T(PTn?1xn+(1??n''xn, yn=P(?n'T(PTn?1zn+(1??n'xn, xn+1=P(?nT(PTn?1yn+(1??nxn. (i If the dual E* of E has the Kadec-Klee property, then { xn} converges weakly to a fixed point p?F(T; (ii if T satisfies condition (A, then {xn} converges strongly to a fixed point p?F(T. 17. Reproducing Kernel Banach Spaces with the l1 Norm II: Error Analysis for Regularized Least Square Regression CERN Document Server Song, Guohui 2011-01-01 A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learning rate estimate of l1-regularization in machine learning. 18. Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces CERN Document Server Ramm, A G 2010-01-01 Let $F(u)=h$ be an operator equation in a Banach space $X$, $\\\|F'(u)-F'(v)\\\|\\leq \\omega(\\\|u-v\\\|)$, where $\\omega\\in C([0,\\infty))$, $\\omega(0)=0$, $\\omega(r)>0$ if $r>0$, $\\omega(r)$ is strictly growing on $[0,\\infty)$. Denote $A(u):=F'(u)$, where $F'(u)$ is the Fr\\'{e}chet derivative of $F$, and $A_a:=A+aI.$ Assume that () $\\\|A^{-1}_a(u)\\\|\\leq \\frac{c_1}{\|a\|^b}$, $\|a\|>0$, $b>0$, $a\\in L$. Here $a$ may be a complex number, and $L$ is a smooth path on the complex $a$-plane, joining the origin and some point on the complex $a-$plane, $00$ is a small fixed number, such that for any $a\\in L$ estimate () holds. It is proved that the DSM (Dynamical Systems Method) \\bee \\dot{u}(t)=-A^{-1}_{a(t)}(u(t))[F(u(t))+a(t)u(t)-f],\\quad u(0)=u_0,\\ \\dot{u}=\\frac{d u}{dt}, \\eee converges to $y$ as $t\\to +\\infty$, where $a(t)\\in L,$ $F(y)=f$, $r(t):=\|a(t)\|$, and $r(t)=c_4(t+c_2)^{-c_3}$, where $c_j>0$ are some suitably chosen constants, $j=2,3,4.$ Existence of a solution $y$ to the equation $F(u)=f$ is assumed. It is also assu... 19. Chaotic Banach algebras CERN Document Server Shkarin, Stanislav 2010-01-01 We construct an infinite dimensional non-unital Banach algebra $A$ and $a\\in A$ such that the sets $\\{za^n:z\\in\\C,\\ n\\in\\N\\}$ and $\\{({\\bf 1}+a)^na:n\\in\\N\\}$ are both dense in $A$, where $\\bf 1$ is the unity in the unitalization $A^{\\#}=A\\oplus \\spann\\{{\\bf 1}\\}$ of $A$. As a byproduct, we get a hypercyclic operator $T$ on a Banach space such that $T\\oplus T$ is non-cyclic and $\\sigma(T)=\\{1\\}$. 20. Comparison of fastness of the convergence among Krasnoselskij, Mann, and Ishikawa iterations in arbitrary real Banach spaces Directory of Open Access Journals (Sweden) K. N. V. V. Vara Prasad 2007-01-01 Full Text Available Let E be an arbitrary real Banach space and K a nonempty, closed, convex (not necessarily bounded subset of E. If T is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant LÃ¢Â‰Â¥1, then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of T. 1. Convergence Theorems for a Maximal Monotone Operator and a $V$ -Strongly Nonexpansive Mapping in a Banach Space OpenAIRE Hiroko Manaka 2010-01-01 Let E be a smooth Banach space with a norm $\\Vert \\cdot\\,\\!\\Vert$ . Let $V(x,y)={\\Vert x\\Vert}^{2}+{\\Vert y\\Vert }^{2}-2\\langle x,Jy\\rangle$ for any $x,y\\in E$ , where $\\langle \\cdot\\,\\!,\\cdot\\,\\!\\rangle$ stands for the duality pair and J is the normalized duality mapping. With respect to this bifunction $V(\\cdot\\,\\!,\\cdot\\,\\!)$ , a generalized nonexpansive mapping and a $V$ -strongly nonexpansive mapping are defined in $E$ . In this paper, using the properties of generalized nonexpansi... 2. The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces OpenAIRE Du, Fapeng; Xue, Yifeng 2012-01-01 In this paper, we investigate the invertibility of $I_Y+\\delta TT^+$ when $T$ is a closed operator from $X$ to $Y$ with a generalized inverse $T^+$ and $\\delta T$ is a linear operator whose domain contains $D(T)$ and range is contained in $D(T^+)$. The characterizations of the stable perturbation $T+\\delta T$ of $T$ by $\\delta T$ in Banach spaces are obtained. The results extend the recent main results of Huang's in Linear Algebra and its Applications. 3. On the Banach Problem on Surjections OpenAIRE Tokarev, Eugene 2002-01-01 Is shown that any separable superreflexive Banach space X may be isometrically embedded in a separable superreflexive Banach space Z=Z(X) (which, in addition, is of the same type and cotype as X) such that its conjugate admits a continuous surjection on each its subspace. This gives an affirmative answer on S. Banach problem: Whether there exists a Banach space X, non isomorphic to a Hilbert space, which admits a continuous linear surjection on each its subspace and is essentially different f... 4. Metatext Phenomenon: Mode of Irony in Reflexive-Interpretative Space OpenAIRE Kuznetsova, Anna V. 2013-01-01 The article presents metatext and correlative interaction of irony in the reflexive-interpretative space of the literary text, clarifies the status of metatext space as linguo-cognitive activity of text producer. Irony serves as a system factor, forming metatext within the linguo-rhetoric script of text. Context plays special role in singling out of linguistic and discursive irony; multilevel character of irony is determined by context breadth, required for its decoding. The integrated nature... 5. Double-dual n-types over Banach spaces not containing Ã¢Â„Â“1 Directory of Open Access Journals (Sweden) Markus Pomper 2004-07-01 Full Text Available Let E be a Banach space. The concept of n-type overE is introduced here, generalizing the concept of type overE introduced by Krivine and Maurey. Let EÃ¢Â€Â³ be the second dual of E and fix gÃ¢Â€Â³1,Ã¢Â€Â¦gÃ¢Â€Â³nÃ¢ÂˆÂˆEÃ¢Â€Â³. The function ÃÂ„:EÃƒÂ—Ã¢Â„ÂnÃ¢Â†Â’Ã¢Â„Â, defined by letting ÃÂ„(x,a1,Ã¢Â€Â¦,an=Ã¢Â€Â–x+Ã¢ÂˆÂ‘i=1naigÃ¢Â€Â³iÃ¢Â€Â– for all xÃ¢ÂˆÂˆE and all a1,Ã¢Â€Â¦,anÃ¢ÂˆÂˆÃ¢Â„Â, defines an n-type over E. Types that can be represented in this way are called double-dual n-types; we say that (gÃ¢Â€Â³1,Ã¢Â€Â¦gÃ¢Â€Â³nÃ¢ÂˆÂˆ(EÃ¢Â€Â³n realizes ÃÂ„. Let E be a (not necessarily separable Banach space that does not contain Ã¢Â„Â“1. We study the set of elements of (EÃ¢Â€Â³n that realize a given double-dual n-type over E. We show that the set of realizations of this n-type is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given 1-type over a separable Banach space E is convex. The proof makes use of Henson's language for normed space structures and uses ideas from mathematical logic, most notably the LÃƒÂ¶wenheim-Skolem theorem. 6. Ring homomorphisms on real Banach algebras Directory of Open Access Journals (Sweden) Sin-Ei Takahasi 2003-08-01 Full Text Available Let B be a strictly real commutative real Banach algebra with the carrier space ÃŽÂ¦B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ÃÂ:AÃ¢Â†Â’B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ÃÂ(A is contained in the radical of B. 7. Positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations in ordered Banach spaces Directory of Open Access Journals (Sweden) Liang Yue 2011-01-01 Full Text Available Abstract The paper deals with the existence of positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations - u ? ( t + M u ( t = f ( t , u ( t , ( S u ( t , 0 < t < 1 , u ? ( 0 = u ? ( 1 = ? and u ? ( t + M u ( t = f ( t , u ( t , ( S u ( t , 0 < t < 1 , u ? ( 0 = u ? ( 1 = ? in an ordered Banach space E with positive cone K, where M > 0 is a constant, f : [0, 1] K K ? K is continuous, S : C([0, 1], K ? C([0, 1], K is a Fredholm integral operator with positive kernel. Under more general order conditions and measure of noncompactness conditions on the nonlinear term f, criteria on existence of positive solutions are obtained. The argument is based on the fixed point index theory of condensing mapping in cones. Mathematics Subject Classification (2000: 34B15; 34G20. 8. Modification of Otolith Reflex Asymmetries Following Space Flight Science.gov (United States) Clarke, Andrew H.; Schoenfeld, Uwe; Wood, Scott J. 2011-01-01 We hypothesize that changes in otolith-mediated reflexes adapted for microgravity contribute to perceptual, gaze and postural disturbances upon return to Earth s gravity. Our goal was to determine pre- versus post-fight differences in unilateral otolith reflexes that reflect these adaptive changes. This study represents the first comprehensive examination of unilateral otolith function following space flight. Ten astronauts participated in unilateral otolith function tests three times pre-flight and up to four times after Shuttle flights from landing day through the subsequent 10 days. During unilateral centrifugation (UC, +/- 3.5cm at 400deg/s), utricular function was examined by the perceptual changes reflected by the subjective visual vertical (SVV) and by video-oculographic measurement of the otolith-mediated ocular counter-roll (OOR). Unilateral saccular reflexes were recorded by measurement of collic Vestibular Evoked Myogenic Potential (cVEMP). Although data from a few subjects were not obtained early post-flight, a general increase in asymmetry of otolith responses was observed on landing day relative to pre-flight baseline, with a subsequent reversal in asymmetry within 2-3 days. Recovery to baseline levels was achieved within 10 days. This fluctuation in the asymmetry measures appeared strongest for SVV, in a consistent direction for OOR, and in an opposite direction for cVEMP. These results are consistent with our hypothesis that space flight results in adaptive changes in central nervous system processing of otolith input. Adaptation to microgravity may reveal asymmetries in otolith function upon to return to Earth that were not detected prior to the flight due to compensatory mechanisms. 9. Some Extensions of Banach's Contraction Principle in Complete Cone Metric Spaces Directory of Open Access Journals (Sweden) Raja P 2008-01-01 Full Text Available Abstract In this paper we consider complete cone metric spaces. We generalize some definitions such as -nonexpansive and -uniformly locally contractive functions -closure, -isometric in cone metric spaces, and certain fixed point theorems will be proved in those spaces. Among other results, we prove some interesting applications for the fixed point theorems in cone metric spaces. 10. Some results on nearest points problems in Banach lattices Directory of Open Access Journals (Sweden) 2015-04-01 Full Text Available In this paper we prove that for two equivalent norms such that $X$ becomes an STM and LLUM space the dominated best approximation problem have the same solution. We give some conditions such that under these conditions the Frchet differentiability of the nearest point map is equivalent to the continuity of metric projection in the dominated best approximation problem. Also we prove that these conditions are equivalent to strong solvability of the dominated best approximation problem. We prove these results in an STM, reflexive STM and UM spaces. In continuation we give some weaker conditions in the problem of best approximation. We delete the dominated condition and examine these results for a closed solid subset in Banach lattice. We prove that for two equivalent norms such that $X$ becomes an STM and LLUM space the best approximation problem have the same solution. 11. Networks for the weak topology of Banach and Fréchet spaces Czech Academy of Sciences Publication Activity Database Gabriyelyan, S.; Kąkol, Jerzy; Kubiś, Wieslaw; Marciszewski, W. 2015-01-01 Roč. 432, č. 2 (2015), s. 1183-1199. ISSN 0022-247X R&D Projects: GA ČR(CZ) GA14-07880S Institutional support: RVO:67985840 Keywords : Fréchet space * space of continuous functions * locally convex space Subject RIV: BA - General Mathematics Impact factor: 1.120, year: 2014 http://www.sciencedirect.com/science/article/pii/S0022247X15006836 12. Infinite-dimensional Grassmann-Banach algebras CERN Document Server Ivashchuk, V D 2000-01-01 A short review on infinite-dimensional Grassmann-Banach algebras (IDGBA) is presented. Starting with the simplest IDGBA over $K = {\\bf R}$ with $l_1$-norm (suggested by A. Rogers), we define a more general IDGBA over complete normed field $K$ with $l_1$-norm and set of generators of arbitrary power. Any $l_1$-type IDGBA may be obtained by action of Grassmann-Banach functor of projective type on certain $l_1$-space. In non-Archimedean case there exists another possibility for constructing of IDGBA using the Grassmann-Banach functor of injective type. 13. Reflexivity: The Creation of Liminal Spaces-Researchers, Participants, and Research Encounters. Science.gov (United States) 2016-03-01 Reflexivity is defined as the constant movement between being in the phenomenon and stepping outside of it. In this article, we specify three foci of reflexivity-the researcher, the participant, and the encounter-for exploring the interview process as a dialogic liminal space of mutual reflection between researcher and participant. Whereas researchers' reflexivity has been discussed extensively in the professional discourse, participants' reflexivity has not received adequate scholarly attention, nor has the promise inherent in reflective processes occurring within the encounter. PMID:25987582 14. Nonseparability of Banach spaces of bounded harmonic functions on Riemann surfaces OpenAIRE Nakai, Mitsuru 2011-01-01 The separability of certain seminormed spaces of harmonic functions on Riemann surfaces will be considered. An application of the result obtained in the above to some inverse inclusion problem in the classification theory of Riemann surfaces will be appended. 15. Group actions and ergodic theory on Banach function spaces / Richard John de Beer OpenAIRE De Beer, Richard John 2014-01-01 This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measure... 16. Iterative approximation of the solution of a monotone operator equation in certain Banach spaces International Nuclear Information System (INIS) Let X=Lp (or lp), p ≥ 2. The solution of the equation Ax=f, f is an element of X is approximated in X by an iteration process in each of the following two cases: (i) A is a bounded linear mapping of X into itself which is also bounded below; and, (ii) A is a nonlinear Lipschitz mapping of X into itself and satisfies ≥ m \|x-y\|2, for some constant m > 0 and for all x, y in X, where j is the single-valued normalized duality mapping of X into X* (the dual space of X). A related result deals with the iterative approximation of the fixed point of a Lipschitz strictly pseudocontractive mapping in X. (author). 12 refs 17. Functional analysis and applied optimization in Banach spaces applications to non-convex variational models CERN Document Server Botelho, Fabio 2014-01-01 This book introduces the basic concepts of real and functional analysis. It presents the fundamentals of the calculus of variations, convex analysis, duality, and optimization that are necessary to develop applications to physics and engineering problems. The book includes introductory and advanced concepts in measure and integration, as well as an introduction to Sobolev spaces. The problems presented are nonlinear, with non-convex variational formulation. Notably, the primal global minima may not be attained in some situations, in which cases the solution of the dual problem corresponds to an appropriate weak cluster point of minimizing sequences for the primal one. Indeed, the dual approach more readily facilitates numerical computations for some of the selected models. While intended primarily for applied mathematicians, the text will also be of interest to engineers, physicists, and other researchers in related fields. 18. On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces OpenAIRE Castro Gonzlez, Nieves; Vlez Cerrada, Jos Ygnacio 2008-01-01 En este trabajo se estudia la perturbacin de la inversa generalizada grupo en el mbito de los operadores lineales y acotados sobre un espacio de Banach complejo. Se establecen, en primer lugar, caracterizaciones de los {1,2}-inversos generalizados de operadores perturbados que verifican una condicin de no singularidad. Posteriormente se caracteriza la clase de operadores perturbados para los cuales existe el operador inverso grupo y verifican ciertas condiciones geomtricas. Se... 19. ANALYSIS OF AN OPERATOR EQUATION WITH THE DIFFERENTIABLE IN BANACH SPACE OPERATOR AND ITS APPLICATION TO THE INVESTIGATION OF REACTOR MODEL DYNAMICS Directory of Open Access Journals (Sweden) Michał Podowski 1977-01-01 Full Text Available An operator equation of the following formλx = z + F(x,where z - an element of Banach space X and F(x∈X for x∈D⊂X, is investigated. In the cas of m-order differentiable operator F (m>3, conditions of existence and uniqueness of a solution of considered equation are formulated and an estimation of the norm of this solution as a function of the norm z is obtained.The results of theoretical analysis are applied to the point reactor model with nonlinear, temperature-type reactivity feedback. The conditions of the asymptotic stability for the autonomous system as well as for the system including the external reactivity oscilations, are formulated. 20. Analysis of an operator equation with the differentiable in Banach space operator and its application to the investigation of reactor model dynamics International Nuclear Information System (INIS) An operator equation of the following form lambda x = z + F(x), where z - an element of Banach space X and F(x) epsilonX for xepsilon DcX, is investigated. In the case of m-order differentiable operator F (m>= 3), conditions of existence and uniqueness of a solution of considered equation are formulated and an estimation of the norm of this solution as a function of the norm z is obtained. The results of theoretical analysis are applied to the point reactor model with nonlinear, temperature-type reactivity feedback. The conditions of the asymptotic stability for the autonomous system as well as for the system including the external reactivity oscillations, are formulated. (author) 1. Infinite-dimensional Grassmann-Banach algebras OpenAIRE Ivashchuk, V. D. 2000-01-01 A short review on infinite-dimensional Grassmann-Banach algebras (IDGBA) is presented. Starting with the simplest IDGBA over $K = {\\bf R}$ with $l_1$-norm (suggested by A. Rogers), we define a more general IDGBA over complete normed field $K$ with $l_1$-norm and set of generators of arbitrary power. Any $l_1$-type IDGBA may be obtained by action of Grassmann-Banach functor of projective type on certain $l_1$-space. In non-Archimedean case there exists another possibility for constructing of I... 2. Ultrametric Banach algebras CERN Document Server Escassut, Alain 2003-01-01 In this book, ultrametric Banach algebras are studied with the help of topological considerations, properties from affinoid algebras, and circular filters which characterize absolute values on polynomials and make a nice tree structure. The Shilov boundary does exist for normed ultrametric algebras. In uniform Banach algebras, the spectral norm is equal to the supremum of all continuous multiplicative seminorms whose kernel is a maximal ideal. Two different such seminorms can have the same kernel. Krasner-Tate algebras are characterized among Krasner algebras, affinoid algebras, and ultrametri 3. Existence and Global Asymptotical Stability of Periodic Solution for the T-Periodic Logistic System with Time-Varying Generating Operators and T0-Periodic Impulsive Perturbations on Banach Spaces OpenAIRE Qian Chen; Wei, W.; Xiang, X.(Department of Physics, Princeton University, Princeton, NJ, 08544, USA); JinRong Wang 2008-01-01 This paper studies the existence and global asymptotical stability of periodic PC-mild solution for the T-periodic Logistic system with time-varying generating operators and T0-periodic impulsive perturbations on Banach spaces. Two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator corresponding to homogenous well-posed T-periodic system with time-varying generating operators and T0-periodic impulsive perturbations are given. It is shown that th... 4. Reduction of LieJordan Banach algebras and quantum states International Nuclear Information System (INIS) In this paper, it is shown that the concept of dynamical correspondence for Jordan Banach algebras is equivalent to a Lie structure compatible with the Jordan one. Then a theory of reduction of LieJordan Banach algebras in the presence of quantum constraints is presented and compared to the standard reduction of C-algebras of observables of a quantum system. The space of states of the reduced LieJordan Banach algebra is characterized in terms of Dirac states on the physical algebra of observables and its GNS representations described in terms of states on the unreduced algebra. (paper) 5. An inner product for a Banach-algebra International Nuclear Information System (INIS) An inner product is defined on a commutative Banach algebra with an essential involution and the resultant inner product space is shown to be a topological algebra. Several conditions for its completeness are established and moreover, a decomposition theorem is proved. It is shown that every commutative Banach algebra with an essential involution has an auxiliary norm which turns it into an A-algebra. (author). 6 refs 6. Spectral synthesis for Banach Algebras II OpenAIRE Feinstein, J. F.; Somerset, D. W. B. 1999-01-01 This paper continues the study of spectral synthesis and the topologies $\\tau_{\\infty}$ and $\\tau_r$ on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C$^$-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of $\\tau_{\\infty}$. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of $\\tau_r$. 7. Impulsive moving mirror model and the stability of linear homogeneous differential equations with impulse effect in a Banach space International Nuclear Information System (INIS) From a special class of systems has been used the linear homogeneous differential equations with impulse effect in Minkowski space field theory with time dependent boundary conditions, i.e. those of moving mirrors. The field theoretical approach for studing the properties of the vacuum starts from an analysis of the behaviour of local field quantities in Minkowski space with uniformly moving mirrors. For the impulsive moving mirror model is the real process of interaction between the quantum field and the external mirror a subject to disturbances in its evolution acting in time very short compared with the entire duration of the process. The stability of the process in the stability of the vacuum state energy. 7 refs 8. Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces Directory of Open Access Journals (Sweden) Zhang Peiguo 2011-01-01 Full Text Available Abstract By obtaining intervals of the parameter λ, this article investigates the existence of a positive solution for a class of nonlinear boundary value problems of second-order differential equations with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator. MSC: 34B15; 34B16. 9. Q.H.I. spaces OpenAIRE Ferenczi, Valentin 1996-01-01 A Banach space $X$ is said to be Q.H.I. if eve\\-ry infinite dimensional quo\\-tient spa\\-ce of $X$ is H.I.: that is, a space is Q.H.I. if the H.I. property is not only stable passing to subspaces, but also passing to quotients and to the dual. We show that Gowers-Maurey's space is Q.H.I.; then we provide an example of a reflexive H.I. space ${\\cal X}$ whose dual is not H.I., from which it follows that $\\cal X$ is not Q.H.I. 10. Moduli and Characteristics of Monotonicity in Some Banach Lattices Czech Academy of Sciences Publication Activity Database Foralewski, P.; Hudzik, H.; Kaczmarek, R.; Krbec, Miroslav -, - (2010), s. 852346. ISSN 1687-1812 R&D Projects: GA AV ČR IAA100190804; GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : Banach lattice characteristics of monotonicity * Orlicz function space * Orlicz sequence space Subject RIV: BA - General Mathematics http://www. fixed pointtheoryandapplications.com/content/2010/1/852346 11. Derivations into Duals of Ideals of Banach Algebras M E Gorgi; T Yazdanpanah 2004-11-01 We introduce two notions of amenability for a Banach algebra $\\mathcal{A}$. Let be a closed two-sided ideal in $\\mathcal{A}$, we say $\\mathcal{A}$ is -weakly amenable if the first cohomology group of $\\mathcal{A}$ with coefficients in the dual space * is zero; i.e., $H^1(\\mathcal{A},I^) =\\{0\\}$, and, $\\mathcal{A}$ is ideally amenable if $\\mathcal{A}$ is -weakly amenable for every closed two-sided ideal in $\\mathcal{A}$. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study the -weak amenability of a Banach algebra $\\mathcal{A}$ for some special closed two-sided ideal . 12. Additive Functional Inequalities in Banach Modules Directory of Open Access Journals (Sweden) An JongSu 2008-01-01 Full Text Available Abstract We investigate the following functional inequality in Banach modules over a -algebra and prove the generalized Hyers-Ulam stability of linear mappings in Banach modules over a -algebra in the spirit of the Th. M. Rassias stability approach. Moreover, these results are applied to investigate homomorphisms in complex Banach algebras and prove the generalized Hyers-Ulam stability of homomorphisms in complex Banach algebras. 13. Wave-front sets of Banach function types CERN Document Server Coriasco, Sandro; Toft, Joachim 2009-01-01 We introduce the wave-front set for distributions with respect to Fourier images of weighted translation invariant Banach function spaces. We prove that usual mapping properties for pseudo-differential operators hold in the context of such wave-front sets. 14. Endomorphic Elements in Banach Algebras CERN Document Server Babalola, V A 2004-01-01 The use of the properties of actions on an algebra to enrich the study of the algebra is well-trodden and still fashionable. Here, the notion and study of endomorphic elements of (Banach) algebras are introduced. This study is initiated, in the hope that it will open up, further, the structure of (Banach) algebras in general, enrich the study of endomorphisms and provide examples. In particular, here, we use it to classify algebras for the convenience of our study. We also present results on the structure of some classes of endomorphic elements and bring out the contrast with idempotents. 15. Strong convergence theorems for a common fixed point of a finite family of Bregman weak relativity nonexpansive mappings in reflexive Banach spaces. Science.gov (United States) 2014-01-01 We introduce an iterative process for finding an element of a common fixed point of a finite family of Bregman weak relatively nonexpansive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. PMID:24757423 CERN Document Server Wagon, Stan 1985-01-01 The Banach-Tarski paradox is a most striking mathematical construction: it asserts that a solid ball may be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large as the original. This volume explore 17. Tensor products of abstract Besov spaces OpenAIRE Dmytryshyn M.I. 2012-01-01 We prove the interpolation theorem for tensor products of abstract Besov spaces, associated with closed operators in Banach spaces and show its application to the problems of approximations of elements of tensor products of Banach spaces. 18. Tensor products of abstract Besov spaces Directory of Open Access Journals (Sweden) Dmytryshyn M.I. 2012-12-01 Full Text Available We prove the interpolation theorem for tensor products of abstract Besov spaces, associated with closed operators in Banach spaces and show its application to the problems of approximations of elements of tensor products of Banach spaces. 19. Reflexive Aero Structures for Enhanced Survivability Project Data.gov (United States) National Aeronautics and Space Administration — Cornerstone Research Group Inc. (CRG) proposes to develop an advanced reflexive structure system to increase the survivability of aerostructures. This reflexive... 20. Several complex variables and Banach algebras International Nuclear Information System (INIS) This paper aims to present certain applications of the theory of holomorphic functions of several complex variables to the study of commutative Banach algebras. The material falls into the following sections: (A) Introcution to Banach algebras (this will not presuppose any knowledge of the subject); (B) Groups of differential forms (mainly concerned with setting up a useful language); (C) Polynomially convex domains. (D) Holomorphic functional calculus for Banach algebras; (E) Some applications of the functional calculus. (author) 1. Projectivity of Banach and $C^$-algebras of continuous fields CERN Document Server Cushing, David 2011-01-01 We give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective $C^$-algebras $\\A$ defined by locally trivial continuous fields $\\mathcal{U} = \\{\\Omega,(A_t)_{t \\in \\Omega},\\Theta\\}$ such that each $C^$-algebra $A_{t}$ has a strictly positive element. For a commutative $C^$-algebra $\\D$ contained in ${\\cal B}(H)$, where $H$ is a separable Hilbert space, we show that the condition of left projectivity of $\\D$ is equivalent to the existence of a strictly positive element in $\\D$ and so to the spectrum of $\\D$ being a Lindel$\\ddot{\\rm o}$f space. 2. Erratum to: “Polynomial algebras on classical Banach spaces” Czech Academy of Sciences Publication Activity Database D'Alessandro, Stefania; Hájek, Petr Pavel; Johanis, M. 2015-01-01 Roč. 207, č. 2 (2015), s. 1003-1012. ISSN 0021-2172 R&D Projects: GA ČR(CZ) GAP201/11/0345; GA MŠk(CZ) 7AMB12FR003 Institutional support: RVO:67985840 Keywords : Banach space polynomial algebra Subject RIV: BA - General Mathematics Impact factor: 0.787, year: 2014 http://link.springer.com/article/10.1007%2Fs11856-015-1155-y 3. The Giesy--James theorem for general index $p$, with an application to operator ideals on the $p$th James space CERN Document Server Bird, Alistair; Laustsen, Niels Jakob 2011-01-01 A theorem of Giesy and James states that $c_0$ is finitely representable in James' quasi-reflexive Banach space $J_2$. We extend this theorem to the $p$th quasi-reflexive James space $J_p$ for each $p \\in (1,\\infty)$. As an application, we obtain a new closed ideal of operators on $J_p$, namely the closure of the set of operators that factor through the complemented subspace $(\\ell_\\infty^1 \\oplus \\ell_\\infty^2 \\oplus...\\oplus \\ell_\\infty^n \\oplus...)_{\\ell_p}$ of $J_p$. 4. Rotund renormings in spaces of bochner integrable functions OpenAIRE Fabian, M. 2015-01-01 We show that if μ is a probability measure and X is a Banach space, then the Lebesgue-Bochner space L1(μ,X) admits an equivalent norm which is rotund (uniformly rotund in every direction, locally uniformly rotund, or midpoint locally uniformly rotund) if X does. We also prove that if X admits a uniformly rotund norm, then the space L1(μ,X) has an equivalent norm whose restriction to every reflexive subspace is uniformly rotund. This is done via the Luxemburg norm associated to a suitable Orli... 5. The normed and Banach envelopes of Weak L^1 OpenAIRE Leung, Denny H. 1998-01-01 The space Weak L^1 consists of all measurable functions on [0,1] such that q(f) = sup_{c>0} c \\lambda{t : \|f(t)\| > c} is finite, where \\lambda denotes Lebesgue measure. Let \\rho be the gauge functional of the unit ball {f : q(f) \\leq 1} of the quasi- norm q, and let N be the null space of \\rho. The normed envelope of Weak L^1, which we denote by W, is the space (Weak L^1/N, \\rho). The Banach envelope of Weak L^1, \\overline{W}, is the completion of W. We show that \\overline{W} is isometrically... 6. Torsional vestibulo-ocular reflex measurements for identifying otolith asymmetries possibly related to space motion sickness susceptibility Science.gov (United States) Peterka, Robert J. 1993-01-01 Recent studies have identified significant correlations between space motion sickness susceptibility and measures of disconjugate torsional eye movements recorded during parabolic flights. These results support an earlier proposal which hypothesized that an asymmetry of otolith function between the two ears is the cause of space motion sickness. It may be possible to devise experiments that can be performed in the 1 g environment on earth that could identify and quantify the presence of asymmetric otolith function. This paper summarizes the known physiological and anatomical properties of the otolith organs and the properties of the torsional vestibulo-ocular reflex which are relevant to the design of a stimulus to identify otolith asymmetries. A specific stimulus which takes advantage of these properties is proposed. 7. Modification of Otolith-Ocular Reflexes, Motion Perception and Manual Control During Variable Radius Centrifugation Following Space Flight Science.gov (United States) Wood, Scott J.; Clarke, A. H.; Rupert, A. H.; Harm, D. L.; Clement, G. R. 2009-01-01 8. BanachMazur Distance Between Convex Quadrangles Directory of Open Access Journals (Sweden) Lassak Marek 2014-12-01 Full Text Available It is proved that the Banach-Mazur distance between arbitrary two convex quadrangles is at most 2. The distance equals 2 if and only if the pair of these quadrangles is a parallelogram and a triangle. 9. Bounded Hochschild cohomology of Banach algebras with a matrix-like structure CERN Document Server Gronbaek, N 2003-01-01 Let B be a unital Banach algebra. A projection in B which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal A in B. In this set-up we prove a theorem to the effect that the bounded Hochschild cohomology H^n(A,A^) vanishes for all n>=1. The hypothesis of this theorem involve (i) strong H-unitality of A, (ii) a growth condition on diagonal matrices in A, (iii) an extension of A in B with trivial bounded simplicial homology. As a corollary we show that if X is an infinite dimensional Banach space with the bounded approximation property, L_1(\\mu,\\Omega) is an infinite dimensional L_1-space, and A is the Banach algebra of approximable operators on L_p(X,\\mu,\\Omega), (1==0. 10. The Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game OpenAIRE Fishman, Lior; Reams, Vanessa; Simmons, David 2015-01-01 We introduce two new mathematical games, the Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game, merging well-known games. We investigate the properties of the games, as well as providing an application to Diophantine approximation theory, analyzing the geometric structure of certain Diophantine sets. 11. On the Cauchy Functional Inequality in Banach Modules Directory of Open Access Journals (Sweden) Park Choonkil 2008-01-01 Full Text Available Abstract We investigate the following functional inequality: in Banach modules over a -algebra, and prove the generalized Hyers-Ulam stability of linear mappings in Banach modules over a -algebra. 12. Banach-Mazur Games with Simple Winning Strategies OpenAIRE Grädel, Erich; Leßenich, Simon 2012-01-01 We discuss several notions of "simple" winning strategies for Banach-Mazur games on graphs, such as positional strategies, move-counting or length-counting strategies, and strategies with a memory based on finite appearance records (FAR). We investigate classes of Banach-Mazur games that are determined via these kinds of winning strategies. Banach-Mazur games admit stronger determinacy results than classical graph games. For instance, all Banach-Mazur games with omega-r... 13. Moduli and Characteristics of Monotonicity in Some Banach Lattices Directory of Open Access Journals (Sweden) Miroslav Krbec 2010-01-01 Full Text Available First the characteristic of monotonicity of any Banach lattice X is expressed in terms of the left limit of the modulus of monotonicity of X at the point 1. It is also shown that for Kthe spaces the classical characteristic of monotonicity is the same as the characteristic of monotonicity corresponding to another modulus of monotonicity ?^m,E. The characteristic of monotonicity of Orlicz function spaces and Orlicz sequence spaces equipped with the Luxemburg norm are calculated. In the first case the characteristic is expressed in terms of the generating Orlicz function only, but in the sequence case the formula is not so direct. Three examples show why in the sequence case so direct formula is rather impossible. Some other auxiliary and complemented results are also presented. By the results of Betiuk-Pilarska and Prus (2008 which establish that Banach lattices X with ?0,m(X<1 and weak orthogonality property have the weak fixed point property, our results are related to the fixed point theory (Kirk and Sims (2001. 14. Moduli and Characteristics of Monotonicity in Some Banach Lattices Directory of Open Access Journals (Sweden) Krbec Miroslav 2010-01-01 Full Text Available Abstract First the characteristic of monotonicity of any Banach lattice is expressed in terms of the left limit of the modulus of monotonicity of at the point . It is also shown that for Kthe spaces the classical characteristic of monotonicity is the same as the characteristic of monotonicity corresponding to another modulus of monotonicity . The characteristic of monotonicity of Orlicz function spaces and Orlicz sequence spaces equipped with the Luxemburg norm are calculated. In the first case the characteristic is expressed in terms of the generating Orlicz function only, but in the sequence case the formula is not so direct. Three examples show why in the sequence case so direct formula is rather impossible. Some other auxiliary and complemented results are also presented. By the results of Betiuk-Pilarska and Prus (2008 which establish that Banach lattices with and weak orthogonality property have the weak fixed point property, our results are related to the fixed point theory (Kirk and Sims (2001. 15. Connes-amenability of multiplier Banach algebras OpenAIRE Hayati, Bahman; Amini, Massoud 2010-01-01 Let $B$ be a Banach algebra with bounded approximate identity, and let $M(B)$ be its multiplier algebra. If there exists a continuous linear injection $B^{}\\rightarrow M(B)$ such that, for every $b\\in B$ and every $u,v\\in B^{}$ , $\\langle u,vb\\rangle_{B}=\\langle v,bu\\rangle_{B}$ , then $M(B)$ is a dual Banach algebra and the following are equivalent: ¶ (i) $B$ is amenable; ¶ (ii) $M(B)$ is Connes amenable; ¶ (iii) $M(B)$ has a normal, virtual diagonal. 16. Smooth renormings of the Lebesgue-Bochner function space L-1(mu, X) OpenAIRE Fabian, M.; Lajara, S. 2012-01-01 We show that, if μ is a probability measure and X is a Banach space, then the space L 1 (μ,X) of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that X has such a norm, and that if X admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then L 1 (μ,X) has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth. 17. On dominated polynomials between Banach spaces CERN Document Server Botelho, Geraldo; Rueda, Pilar 2008-01-01 In this paper, among other results, we prove a conjecture concerning coincidence theorems for dominated polynomials. We also obtain an abstract version of Pietsch Domination Theorem (PDT) which unifies and generalizes several different nonlinear approaches; our result recovers, as a particular case, the well-known PDT for dominated multilinear mappings. 18. Spectral theory of linear operators and spectral systems in Banach algebras CERN Document Server 2003-01-01 This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg... 19. Banach ultrapowers and multivalued nonexpansive mappings Science.gov (United States) Wisnicki, Andrzej; Wosko, Jacek 2007-02-01 The Banach ultrapower construction is applied in fixed point theory for multivalued mappings. We introduce the notion of ultra-asymptotic centers and use it to remove the separability assumption from the results of Dominguez Benavides, Lorenzo Ramirez (2004) and Dhompongsa, Kaewcharoen, Kaewkhao (2006). 20. Derivations And Cohomological Groups Of Banach Algebras CERN Document Server 2010-01-01 Let $B$ be a Banach $A-bimodule$ and let $n\\geq 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action $\\pi_\\ell:A\\times B\\rightarrow B$ of $A^{(2n)}$ on $B^{(2n)}$ is $B^{(2n)}$ and $H^1(A^{(2n+2)},B^{(2n+2)})=0$, then we have $H^1(A,B^{(2n)})=0$, and we find the relationships between cohomological groups such as $H^1(A,B^{(n+2)})$ and $H^1(A,B^{(n)})$, spacial $H^1(A,B^)$ and $H^1(A,B^{(2n+1)})$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H^1_{w^}(L^\\infty(G)^,L^\\infty(G)^{*})=0$. Let $G$ be an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\\infty(G),.)$ such that $Z^1(L^1(G),L^\\infty(G))=\\{L_{f}:~f\\in L^\\infty(G)\\}.$ We also obtain some conclusions in the Arens regularity of module actions and weak amenability of Banach algebras. We introduce some new concepts as $left-weak^-to-weak$ convergence property [... 1. Natural examples of Valdivia compact spaces Science.gov (United States) Kalenda, Ondrej F. K. 2008-04-01 We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L1 spaces, Banach lattices and noncommutative L1 spaces. 2. (2-1)-Ideal amenability of triangular banach algebras 2015-05-01 Let $\\mathcal{A}$ and $\\mathcal{B}$ be two unital Banach algebras and let $\\mathcal{M}$ be an unital Banach $\\mathcal{A}$, $\\mathcal{B}$-module. Also, let $\\mathcal{T}=\\left[\\begin{smallmatrix} \\mathcal{A} & \\mathcal{M}\\\\ & \\mathcal{B}\\end{smallmatrix}\\right]$ be the corresponding triangular Banach algebra. Forrest and Marcoux (Trans. Amer. Math. Soc. 354 (2002) 1435–1452) have studied the -weak amenability of triangular Banach algebras. In this paper, we investigate (2-1)-ideal amenability of $\\mathcal{T}$ for all ≥ 1. We introduce the structure of ideals of these Banach algebras and then, we show that (2-1)-ideal amenability of $\\mathcal{T}$ depends on (2-1)-ideal amenability of Banach algebras $\\mathcal{A}$ and $\\mathcal{B}$. 3. Reflexive Aero Structures for Enhanced Survivability Project Data.gov (United States) National Aeronautics and Space Administration — Cornerstone Research Group Inc. (CRG) will develop an advanced reflexive structure technology system to increase the survivability of future systems constructed of... 4. Hereditary properties of Amenability modulo an ideal of Banach algebras Directory of Open Access Journals (Sweden) Hamidreza Rahimi 2014-10-01 Full Text Available In this paper we investigate some hereditary properties of amenability modulo an ideal of Banach algebras. We show thatif $(e_{\\alpha}_{\\alpha}$ is a bounded approximate identity modulo $I$ of a Banach algebra $A$ and $X$ is a neo-unital modulo $I$, then $(e_{\\alpha}_{\\alpha}$ is a bounded approximate identity for $X$. Moreover we show that amenability modulo an ideal of a Banach algebra $A$ can be only considered by the neo-unital modulo $I$ Banach algebra over $A$ 5. ZAG-Otolith: Modification of Otolith-Ocular Reflexes, Motion Perception and Manual Control during Variable Radius Centrifugation Following Space Flight Science.gov (United States) Wood, S. J.; Clarke, A. H.; Rupert, A. H.; Harm, D. L.; Clement, G. R. 2009-01-01 Two joint ESA-NASA studies are examining changes in otolith-ocular reflexes and motion perception following short duration space flights, and the operational implications of post-flight tilt-translation ambiguity for manual control performance. Vibrotactile feedback of tilt orientation is also being evaluated as a countermeasure to improve performance during a closed-loop nulling task. METHODS. Data is currently being collected on astronaut subjects during 3 preflight sessions and during the first 8 days after Shuttle landings. Variable radius centrifugation is utilized to elicit otolith reflexes in the lateral plane without concordant roll canal cues. Unilateral centrifugation (400 deg/s, 3.5 cm radius) stimulates one otolith positioned off-axis while the opposite side is centered over the axis of rotation. During this paradigm, roll-tilt perception is measured using a subjective visual vertical task and ocular counter-rolling is obtained using binocular video-oculography. During a second paradigm (216 deg/s, centrifugation, and measure the time course of postflight recovery. This study will also address how adaptive changes in otolith-mediated reflexes correspond to one's ability to perform closed-loop nulling tasks following G-transitions, and whether manual control performance can be improved with vibrotactile feedback of orientation. 6. On sequence spaces for Fr\\'echet frames OpenAIRE Pilipović, Stevan; Stoeva, Diana T. 2008-01-01 We analyze the construction of a sequence space $\\widetilde{\\Theta}$, resp. a sequence of sequence spaces, in order to have $\\{g_i\\}$ as a $\\widetilde{\\Theta}$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fr\\'echet space $X_F=\\cap_{s\\in {\\mathbb N}_0} X_s$, where $\\{X_s\\}_{s\\in {\\mathbb N}_0}$ is a sequence of Banach spaces. 7. Multipliers of Weighted Semigroups and Associated Beurling Banach Algebras S J Bhatt; P A Dabhi; H V Dedania 2011-11-01 Given a weighted discrete abelian semigroup $(S,)$, the semigroup $M_(S)$ of -bounded multipliers as well as the Rees quotient $M_(S)/S$ together with their respective weights $\\overline{}$ and $\\overline{}_q$ induced by are studied; for a large class of weights , the quotient $\\ell^1(M_(S),\\overline{})/\\ell^1(S,)$ is realized as a Beurling algebra on the quotient semigroup $M_(S)/S$; the Gel’fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples. 8. A G.N.S.-type theorem for a non-involutive Banach algebra International Nuclear Information System (INIS) Given an involutive Banach algebra A with a bounded approximate identity, the elegant construction by Gelfand, Naimark and Segal, associates to each positive linear functional f on A, a triple (Hf, πf, ξf), where Hf is a Hilbert space with scalar product denoted by: , πf is a representation of A into L(Hf), the space of bounded linear operators on Hf, ξf is a cyclic vector in Hf, such that: f(x)=f(x)ξf, ξf >, for all x is an element of A. In this result, the existence of a (multiplicative) involution on A, is central. The aim of this paper is to show that such a construction may be performed for a non-involutive Banach algebra, to obtain a similar triple (H, ω, ξ) as above. Moreover, this procedure indeed enables us to associate to the topological dual of any Banach space, a liminary C-subalgebra of L(H). A notion of compact Hilbert algebras will be introduced, together with a method of construction of a large collection of such spaces. (author). 14 refs 9. An inner product for a Banach-algebra International Nuclear Information System (INIS) An intrinsic inner product for a commutative Banach-algebra is defined. Several conditions for its completeness are established. It is shown that any Banach-algebra with proper and continuous involution has an auxiliary norm that turns it into an A-algebra. (author). 7 refs 10. Normas tensoriales construidas mediante espacios de sucesiones de Banach Directory of Open Access Journals (Sweden) Patricia Gmez Palacio 2003-01-01 Full Text Available En este art culo se de ne una norma tensorial gc a partir de un espacio de sucesiones de Banach . Para cada par de espacios de Banach E y F, se caracterizan los elementos de la complecci n del espacio E g F, Eb g F, y se caracteriza su espacio dual (E g F0 11. Compact multipliers on spaces of analytic functions OpenAIRE Mleczko, Paweł 2008-01-01 In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\\infty\\hookrightarrow X\\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multiplier... 12. Left introverted subspaces of duals of Banach algebras and $WEAK^-$continuous derivations on dual Banach algebras CERN Document Server Gordji, M E 2006-01-01 Let $X$ be a left introverted subspace of dual of a Banach algebra. We study $Z_t(X^),$ the topological center of Banach algebra $X^$. We fined the topological center of $(X\\cA)^$, when $\\cA$ has a bounded right approximate identity and $\\cA\\subseteq X^.$ So we introduce a new notation of amenability for a dual Banach algebra $\\cal A$. A dual Banach algebra $\\cal A$ is weakly Connes-amenable if the first $weak^-$continuous cohomology group of $\\cal A$ with coefficients in $\\cal A$ is zero; i.e., $H^1_{w^}(\\cal A, \\cal A)=\\{o\\}$. We study the weak Connes-amenability of some dual Banach algebras. 13. ZAG-Otolith: Modification of Otolith-Ocular Reflexes, Motion Perception and Manual Control during Variable Radius Centrifugation Following Space Flight Science.gov (United States) Wood, S. J.; Clarke, A. H.; Rupert, A. H.; Harm, D. L.; Clement, G. R. 2009-01-01 14. Very Smooth Points of Spaces of Operators T S S R K Rao 2003-02-01 In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is an -ideal in the space of bounded operators, a very smooth operator attains its norm at a unique vector (up to a constant multiple) and ( ) is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon–Nikodým property. We give an example of a smooth Banach space without any very smooth points. 15. Spaces of small metric cotype CERN Document Server Veomett, Ellen 2010-01-01 Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result. 16. Ideal Amenability of Banach Algebras on Locally Compact Groups M Eshaghi Gordji; S A R Hosseiniun 2005-08-01 In this paper we study the ideal amenability of Banach algebras. Let $\\mathcal{A}$ be a Banach algebra and let be a closed two-sided ideal in $\\mathcal{A}, \\mathcal{A}$ is -weakly amenable if $H^1(\\mathcal{A},I^∗)=\\{0\\}$. Further, $\\mathcal{A}$ is ideally amenable if $\\mathcal{A}$ is -weakly amenable for every closed two-sided ideal in $\\mathcal{A}$. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenability of Banach algebras on locally compact groups. 17. On Reflection: Is Reflexivity Necessarily Beneficial in Intercultural Education? Science.gov (United States) Blasco, Maribel 2012-01-01 This article explores how the concept of reflexivity is used in intercultural education. Reflexivity is often presented as a key learning goal in acquiring intercultural competence (ICC). Yet, reflexivity can be defined in different ways, and take different forms across time and space, depending on the concepts of selfhood that prevail and how 18. General Hormander and Mikhlin conditions for multipliers of Besov spaces OpenAIRE 2008-01-01 Here a new condition for the geometry of Banach spaces is introduced and the operator--valued Fourier multiplier theorems in weighted Besov spaces are obtained. Particularly, connections between the geometry of Banach spaces and Hormander-Mikhlin conditions are established. As an application of main results the regularity properties of degenerate elliptic differential operator equations are investigated. 19. Countable linear combinations of characters on commutative Banach algebras OpenAIRE Feinstein, J. F. 2014-01-01 An elegant but elementary result of Wolff from 1921, when interpreted in terms of Banach algebras, shows that it is possible to find a sequence of distinct characters $\\phi_n$ on the disc algebra and an $\\ell_1$ sequence of complex numbers $\\lambda_n$, not all zero, such that $\\sum_{n=1}^\\infty \\lambda_n \\phi_n =0.$ We observe that, even for general commutative, unital Banach algebras, this is not possible if the closure of the countable set of characters has no perfect subsets. 20. On the Cauchy Functional Inequality in Banach Modules Directory of Open Access Journals (Sweden) Choonkil Park 2008-05-01 Full Text Available We investigate the following functional inequality: Ã¢Â€Â–f(x+f(y+f(zÃ¢Â€Â–Ã¢Â‰Â¤Ã¢Â€Â–f(x+y+zÃ¢Â€Â– in Banach modules over a CÃ¢ÂˆÂ—-algebra, and prove the generalized Hyers-Ulam stability of linear mappings in Banach modules over a CÃ¢ÂˆÂ—-algebra. 1. Representations of -Algebras, Locally Compact Groups, and Banach -Algebraic Bundles Banach -Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis CERN Document Server Fell, James M G 1988-01-01 This is an all-encompassing and exhaustive exposition of the theory of infinite-dimensional Unitary Representations of Locally Compact Groups and its generalization to representations of Banach algebras. The presentation is detailed, accessible, and self-contained (except for some elementary knowledge in algebra, topology, and abstract measure theory). In the later chapters the reader is brought to the frontiers of present-day knowledge in the area of Mackey normal subgroup analysisand its generalization to the context of Banach -Algebraic Bundles. 2. Almost Lie structures on an anchored Banach bundle CERN Document Server Cabau, Patrick 2011-01-01 Under appropriate assumptions, we generalize the concept of linear almost Poisson struc- tures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the "Hilbert snake" 3. Emotionally Colorful Reflexive Games CERN Document Server Tarasenko, Sergey 2011-01-01 This study addresses the matter of reflexive control of the emotional states by means of Reflexive Game Theory (RGT). It is shown how to build a bridge between RGT and emotions. For this purpose the Pleasure-Arousal-Dominance (PAD) model is adopted. The major advantages of RGT are its ability to predict human behavior and unfold the entire spectra of reflexion in the human mind. On the other hand, PAD provides ultimate approach to model emotions. It is illustrated that emotions are reflexive processes and, consequently, RGT fused with PAD model is natural solution to model emotional interactions between people. The fusion of RGT and PAD, called Emotional Reflexive Games (ERG), inherits the key features of both components. Using ERG, we show how reflexive control can be successfully applied to model human emotional states. Up to date, EGR is a unique methodology capable of modeling human reflexive processes and emotional aspects simultaneously. 4. Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes CERN Document Server Di Girolami, Cristina 2011-01-01 This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of M\\'etivier and Pellaumail which is quite restrictive. We make use of the notion of $\\chi$-covariation which is a generalized notion of covariation for processes with values in two Banach spaces $B_{1}$ and $B_{2}$. $\\chi$ refers to a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$. We investigate some $C^{1}$ type transformations for various classes of stochastic processes admitting a $\\chi$-quadratic variation and related properties. If $\\X^1$ and $\\X^2$ admit a $\\chi$-covariation, $F^i: B_i \\rightarrow \\R$, $i = 1, 2$ are of class $C^1$ with some supplementary assumptions then the covariation of the real processes $F^1(\\X^1)$ and $F^2(\\X^2)$ exist. A detailed analysis will be devoted to the so-called window processes. Let $X$ be a real continuous process; the $C([-\\tau,0])$-valued process $X(\\cdot)$ defined by $X_t(y)... 5. Reflexive intensification in Spanish: Toward a complex reflexive? DEFF Research Database (Denmark) Pedersen, Johan Spanish, intensifier, intensification, reflexive pronouns, anaphor, reanalysis, grammaticalization, sí, mismo......Spanish, intensifier, intensification, reflexive pronouns, anaphor, reanalysis, grammaticalization, sí, mismo... 6. Cruciate ligament reflexes DEFF Research Database (Denmark) Krogsgaard, Michael R; Dyhre-Poulsen, Poul; Fischer-Rasmussen, Torsten 2002-01-01 The idea of muscular reflexes elicited from sensory nerves of the cruciate ligaments is more than 100 years old, but the existence of such reflexes has not been proven until the recent two decades. First in animal experiments, a muscular excitation could be elicited in the hamstrings when the ant... 7. A generalization of the weak amenability of some Banach algebra OpenAIRE Azar, Kazem Haghnejad 2010-01-01 Let$A$be a Banach algebra and$A^{}$be the second dual of it. We show that by some new conditions,$A$is weakly amenable whenever$A^{}$is weakly amenable. We will study this problem under generalization, that is, if$(n+2)-th$dual of$A$,$A^{(n+2)}$, is$T-S-$weakly amenable, then$A^{(n)}$is$T-S-$weakly amenable where$T$and$S$are continuous linear mappings from$A^{(n)}$into$A^{(n)}$. 8. The Socle and Finite Dimensionality of some Banach Algebras Indian Academy of Sciences (India) Ali Ghaffari; Ali Reza Medghalchi 2005-08-01 The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact group , is compact if there exists a measure in$\\mathrm{Soc} (L^1(G))$such that () ≠ 0. We also prove that is finite if$\\mathrm{Soc}(M(G))$is closed and every nonzero left ideal in () contains a minimal left ideal. 9. On Reflexive Data Models Energy Technology Data Exchange (ETDEWEB) Petrov, S. 2000-08-20 An information system is reflexive if it stores a description of its current structure in the body of stored information and is acting on the base of this information. A data model is reflexive, if its language is meta-closed and can be used to build such a system. The need for reflexive data models in new areas of information technology applications is argued. An attempt to express basic notions related to information systems is made in the case when the system supports and uses meta-closed representation of the data. 10. Convexity and w-compactness in Banach spaces Czech Academy of Sciences Publication Activity Database Granero, A. S.; Hájek, Petr Pavel; Santalucía, V. M. 2004-01-01 Roč. 328, č. 4 (2004), s. 625-631. ISSN 0025-5831 R&D Projects: GA ČR GA201/01/1198; GA AV ČR IAA1019205 Institutional research plan: CEZ:AV0Z1019905 Keywords : krein theorem * weak compactness Subject RIV: BA - General Mathematics Impact factor: 0.790, year: 2004 11. Stability of stationary and periodic solutions equations in Banach space Directory of Open Access Journals (Sweden) A. Ya. Dorogovtsev 1997-01-01 Full Text Available Linear difference and differential equations with operator coefficients and random stationary (periodic input are considered. Conditions are presented for the mean stability of stationary (periodic solutions under small perturbation of the coefficients. 12. Stochastic processes on non-Archimedean Banach spaces OpenAIRE Ludkovsky, S. V. 2003-01-01 Non-Archimedean analogs of Markov quasimeasures and stochastic processes are investigated. They are used for the development of stochastic antiderivations. The non-Archimedean analog of the ItÃƒÂ´ formula is proved. 13. Smooth linearization for a saddle on Banach spaces OpenAIRE Munhoz Rodrigues, Hildebrando; Solà-Morales Rubió, Joan 2003-01-01 As a continuation of a previous work on linearization of class C1 of diﬀeomorphisms and ﬂows in inﬁnite dimensions near a ﬁxed point, in this work we deal with the case of a saddle point with some non-resonance restrictions for the linear part. Our result can be seen as an extension of results by P. Hartman [2] and Aronson, Belitskii and Zhuzhoma [1] in dimension two. We also present an application to a system of nonlinear wave equations. 14. Existence theorems for functional differential equations in Banach spaces OpenAIRE Sghir, A 2009-01-01 This paper concernes with the study of existence theorems for a general class of functional differential equations of the form u'(t)=f(t,u circ g(t,cdot)).The obtained results generalize the retarded functional differential equations [5], [6], [8] and cover singular functional differential equations [1], [3], [4], [7], [9], [12]. 15. Lectures given at the Banach Center and C.I.M.E. Joint Summer School CERN Document Server Lachowicz, Mirosław 2008-01-01 The aim of this volume that presents Lectures given at a joint CIME and Banach Center Summer School, is to offer a broad presentation of a class of updated methods providing a mathematical framework for the development of a hierarchy of models of complex systems in the natural sciences, with a special attention to Biology and Medicine. Mastering complexity implies sharing different tools requiring much higher level of communication between different mathematical and scientific schools, for solving classes of problems of the same nature. Today more than ever, one of the most important challenges derives from the need to bridge parts of a system evolving at different time and space scales, especially with respect to computational affordability. As a result the content has a rather general character; the main role is played by stochastic processes, positive semigroups, asymptotic analysis, kinetic theory, continuum theory and game theory. 16. Functional calculus extensions on dual spaces CERN Document Server Terauds, Venta 2008-01-01 In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one. 17. Identification of Spinal Reflexes: OpenAIRE Vlugt, E. de 2004-01-01 Visco-elasticity of joints is important for the maintenance of the human body posture and can in two manners be regulated. By means of cocontraction of antagonistic muscle groups and by neural reflexive feedback of muscle length and muscle strength, measured both by means of sensors in the muscles. The impact of the reflexive regulation is considerable and infestation of this system can give serious motor deviations, like at the Parkinson's Disease. Except anatomical knowledge is there still ... 18. Reflex sympathetic dystrophy International Nuclear Information System (INIS) Reflex sympathetic dystrophy is characterized by intense and disproportionate local pain, associated with vasomotor and trophic changes. Extremities are commonly involved, especially when a trauma or surgery, even minor, has occurred. Likewise, spontaneous or idiopathic presentation of this syndrome is much less frequent. Here we describe the clinical picture of a young woman presenting with idiopathic reflex sympathetic dystrophy. Then we present a brief review, emphasizing on diagnosis and treatment of this disease. 19. Orthogonalities, transitivity of norms and characterizations of Hilbert spaces OpenAIRE Martini, Horst; Wu, Senlin 2015-01-01 We introduce three concepts, called$I$-vector,$IP$-vector, and$P$-vector, which are related to isosceles orthogonality and Pythagorean orthogonality in normed linear spaces. Having the Banach-Mazur rotation problem in mind, we prove that an almost transitive real Banach space, whose dimension is at least three and which contains an$I$-vector (an$IP$-vector, a$P$-vector, or a unit vector whose pointwise James constant is$\\sqrt2$, respectively) is a Hilbert space. 20. Too Busy for Reflexivity? DEFF Research Database (Denmark) Ratner, Helene What Danish school managers can teach STS researchers about epistemological ideals and pragmatic morals. Reflexivity has an ambivalent status in both anthropology and Science and Technology Studies. On the one hand, the critique of representation at the heart of the reflexivity debates of the 1980s...... highlighted non-symmetric relationships between observer and observed and accused the academic text of enacting a realist genre, concealing the relativism entailed in textual production (Clifford and Marcus 1986, Woolgar 1988, Ashmore 1989). On the other hand, the reflexivity program produced fears of a...... “corrosive relativism in which everything is but a more or less clever expression of opinion” (Geertz 1988:2, 3) and it has suffered the little flattering accusations of piling "layer upon layer of self-consciousness to no avail" (Latour 1988:170) with little “interest [for] … theoretically ambitious... 1. Corneomandibular reflex: Anatomical basis Directory of Open Access Journals (Sweden) Michele Pistacchi 2015-01-01 Full Text Available Corneomandibular reflex is a pathological phenomenon evident in cases of severe brainstem damage. It is considered to be a pathological exteroceptive reflex, associated with precentro bulbar tract lesions. The sign is useful in distinguishing central neurological injuries to metabolic disorders in acutely comatose patients, localizing lesions to the upper brainstem area, determining the depth of coma and its evolution, providing evidence of uncal or transtentorial herniation in acute cerebral hemisphere lesions, and it is a marker of supraspinal level impairment in amyotrophic lateral sclerosis and multiple sclerosis. This sign was evident in a patient with severe brain damage. We discuss the literature findings and its relevance in prognosis establishment. 2. Geometry of Mntz spaces and related questions CERN Document Server Gurariy, Vladimir 2005-01-01 Starting point and motivation for this volume is the classical Muentz theorem which states that the space of all polynomials on the unit interval, whose exponents have too many gaps, is no longer dense in the space of all continuous functions. The resulting spaces of Muentz polynomials are largely unexplored as far as the Banach space geometry is concerned and deserve the attention that the authors arouse. They present the known theorems and prove new results concerning, for example, the isomorphic and isometric classification and the existence of bases in these spaces. Moreover they state many open problems. Although the viewpoint is that of the geometry of Banach spaces they only assume that the reader is familiar with basic functional analysis. In the first part of the book the Banach spaces notions are systematically introduced and are later on applied for Muentz spaces. They include the opening and inclination of subspaces, bases and bounded approximation properties and versions of universality. 3. Sharp form of the Sobolev trace theorems. [Proof of trace theorems for Besov spaces Energy Technology Data Exchange (ETDEWEB) Scott, R. 1977-05-01 Trace theorems for Besov spaces are proved by use of piecewise polynomial approximation theory and the K-method of interpolating Banach spaces. These theorems are limiting cases of standard embedding results. 4. Filtrations of Totally Reflexive Modules OpenAIRE Tracy, Denise A. Rangel 2014-01-01 In this paper, we will introduce a subcategory of totally reflexive modules that have a saturated filtration by other totally reflexive modules. We will prove these are precisely the totally reflexive modules with an upper-triangular presentation matrix. We conclude with an investigation of the ranks of$\\operatorname{Ext}^1$of two such modules over a specific ring. 5. Reflexivity in Pigeons Science.gov (United States) Sweeney, Mary M.; Urcuioli, Peter J. 2010-01-01 A recent theory of pigeons' equivalence-class formation (Urcuioli, 2008) predicts that reflexivity, an untrained ability to match a stimulus to itself, should be observed after training on two "mirror-image" symbolic successive matching tasks plus identity successive matching using some of the symbolic matching stimuli. One group of pigeons was… 6. Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I Directory of Open Access Journals (Sweden) Rassias JM 2009-01-01 Full Text Available We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let be a Banach algebra and a Banach -module, and . The mappings , and are defined and it is proved that if (resp., is dominated by then is a generalized (resp., linear module- left derivation and is a (resp., linear module- left derivation. It is also shown that if (resp., is dominated by then is a generalized (resp., linear module- derivation and is a (resp., linear module- derivation. 7. Characterizing Hilbert spaces using Fourier transform over the field of p-adic numbers OpenAIRE Radyna, Yauhen; Radyno, Yakov; Sidorik, Anna 2008-01-01 We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field$Q_p$of$p$-adic numbers. Precisely, Banach space$X$is isomorphic to a Hilbert one if and only if Fourier transform$F: L_2(Q_p,X)\\to L_2(Q_p,X)$in space of functions, which are square-integrable in Bochner sense and take value in$X$, is a bounded operator. 8. Existence of solutions for discontinuous hyperbolic partial differential equations in Banach algebras Directory of Open Access Journals (Sweden) Bapurao C. Dhage 2006-03-01 Full Text Available In this paper, we prove an existence theorem for hyperbolic differential equations in Banach algebras under Lipschitz and Caratheodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions. 9. Simplified and Equivalent Characterizations of Banach Limit Functional and Strong Almost Convergence OpenAIRE You, Chao 2009-01-01 In this paper, we give simplified and equivalent characterizations of Banach limit functional, which is the minimum requirement to characterize strong almost convergence. With this machinery, we show that Hajdukovic's quasi-almost convergence is equivalent to strong almost convergence. 10. Impulsive discontinuous hyperbolic partial differential equationsof fractional order on Banach algebras Directory of Open Access Journals (Sweden) Said Abbas 2010-07-01 Full Text Available This article studies the existence of solutions and extremal solutions to partial hyperbolic differential equations of fractional order with impulses in Banach algebras under Lipschitz and Caratheodory conditions and certain monotonicity conditions. 11. Mentalis muscle related reflexes. Science.gov (United States) Gündüz, Ayşegül; Uyanık, Özlem; Ertürk, Özdem; Sohtaoğlu, Melis; Kızıltan, Meral Erdemir 2016-05-01 The mentalis muscle (MM) arises from the incisive fossa of the mandible, raises and protrudes the lower lip. Here, we aim to characterize responses obtained from MM by supraorbital and median electrical as well as auditory stimuli in a group of 16 healthy volunteers who did not have clinical palmomental reflex. Reflex activities were recorded from the MM and orbicularis oculi (O.oc) after supraorbital and median electrical as well as auditory stimuli. Response rates over MM were consistent after each stimulus, however, mean latencies of MM response were longer than O.oc responses by all stimulation modalities. Shapes and amplitudes of responses from O.oc and MM were similar. Based on our findings, we may say that MM motoneurons have connections with trigeminal, vestibulocochlear and lemniscal pathways similar to other facial muscles and electrophysiological recording of MM responses after electrical and auditory stimulation is possible in healthy subjects. PMID:26721248 12. Reflex Sympathetic Dystrophy Syndrome OpenAIRE Annil Mahajan, Pawan Suri, Ghulam Hussain Bardi, J.B. Singh, Dheeraj Gandotra, Vijay Gupta 2004-01-01 Reflex Sympathetic Dystrophy (RSD) or Complex Regional Pain Syndome Type-I (CRPS-I), adisease of unknown prevalance, complicates any minor trauma, stroke, myocardial infection, colle’sfracture, peripheral nerve injury and in one-fourth of cases without any precipitant factor. Anawareness of RSD and the injuries, illnesses and drugs that can provoke it is the first step to learnfor an early treatment and better outcome. Here we present a neglected case of RSD followingminor trauma who presente... 13. Corporeal reflexivity and autism. Science.gov (United States) Ochs, Elinor 2015-06-01 Ethnographic video recordings of high functioning children with autism or Aspergers Syndrome in everyday social encounters evidence their first person perspectives. High quality visual and audio data allow detailed analysis of children's bodies and talk as loci of reflexivity. Corporeal reflexivity involves displays of awareness of one's body as an experiencing subject and a physical object accessible to the gaze of others. Gaze, demeanor, actions, and sotto voce commentaries on unfolding situations indicate a range of moment-by-moment reflexive responses to social situations. Autism is associated with neurologically based motor problems (e.g. delayed action-goal coordination, clumsiness) and highly repetitive movements to self-soothe. These behaviors can provoke derision among classmates at school. Focusing on a 9-year-old girl's encounters with peers on the playground, this study documents precisely how autistic children can become enmeshed as unwitting objects of stigma and how they reflect upon their social rejection as it transpires. Children with autism spectrum disorders in laboratory settings manifest diminished understandings of social emotions such as embarrassment, as part of a more general impairment in social perspective-taking. Video ethnography, however, takes us further, into discovering autistic children's subjective sense of vulnerability to the gaze of classmates. PMID:25939529 14. Composition operators on vector-valued analytic function spaces: a survey OpenAIRE Laitila, Jussi; Tylli, Hans-Olav 2015-01-01 We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic ... 15. Soleus stretch reflex during cycling DEFF Research Database (Denmark) Grey, Michael James; Pierce, C. W.; Milner, T. E.; Sinkjær, Thomas 2001-01-01 The modulation and strength of the human soleus short latency stretch reflex was investigated by mechanically perturbing the ankle during an unconstrained pedaling task. Eight subjects pedaled at 60 rpm against a preload of 10 Nm. A torque pulse was applied to the crank at various positions durin...... depressed during active cycling as has been shown with the H-reflex. This lack of depression may reflect a decreased susceptibility of the stretch reflex to inhibition, possibly originating from presynaptic mechanisms.... 16. Best approximation in Orlicz spaces Directory of Open Access Journals (Sweden) S. Ayesh 1991-06-01 Full Text Available Let X be a real Banach space and (ÃŽÂ©,ÃŽÂ¼ be a finite measure space and ÃÂ• be a strictly icreasing convex continuous function on [0,Ã¢ÂˆÂž with ÃÂ•(0=0. The space LÃÂ•(ÃŽÂ¼,X is the set of all measurable functions f with values in X such that Ã¢ÂˆÂ«ÃŽÂ©ÃÂ•(Ã¢Â€Â–cÃ¢ÂˆÂ’1f(tÃ¢Â€Â–dÃŽÂ¼(t0. One of the main results of this paper is: Ã‚Â“For a closed subspace Y of X, LÃÂ•(ÃŽÂ¼,Y is proximinal in LÃÂ•(ÃŽÂ¼,X if and only if L1(ÃŽÂ¼,Y is proximinal in L1(ÃŽÂ¼,XÃ¢Â€Â²Ã¢Â€Â‹Ã¢Â€Â². As a result if Y is reflexive subspace of X, then LÃÂ•(ÃÂ•,Y is proximinal in LÃÂ•(ÃŽÂ¼,X. Other results on proximinality of subspaces of LÃÂ•(ÃŽÂ¼,X are proved. 17. Reflex Sympathetic Dystrophy Syndrome Directory of Open Access Journals (Sweden) Annil Mahajan, Pawan Suri, Ghulam Hussain Bardi, J.B. Singh, Dheeraj Gandotra, Vijay Gupta 2004-07-01 Full Text Available Reflex Sympathetic Dystrophy (RSD or Complex Regional Pain Syndome Type-I (CRPS-I, adisease of unknown prevalance, complicates any minor trauma, stroke, myocardial infection, colle’sfracture, peripheral nerve injury and in one-fourth of cases without any precipitant factor. Anawareness of RSD and the injuries, illnesses and drugs that can provoke it is the first step to learnfor an early treatment and better outcome. Here we present a neglected case of RSD followingminor trauma who presented to us after 6-7 months of onset of disease. Delay in treatment resultedin partial recovery of the patient. 18. Approximately -Jordan Homomorphisms on Banach Algebras OpenAIRE Karimi T; Kaboli Gharetapeh S; Gordji MEshaghi 2009-01-01 Let , and let be two rings. An additive map is called -Jordan homomorphism if for all . In this paper, we establish the Hyers-Ulam-Rassias stability of -Jordan homomorphisms on Banach algebras. Also we show that (a) to each approximate 3-Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique 3-ring homomorphism near to , (b) to each approximate -Jordan homomorphism between two commutative Banach algebras there corresp... 19. Acoustic reflex and general anaesthesia. Science.gov (United States) Farkas, Z 1983-01-01 Infant and small children are not always able to cooperate in impedance measurements. For this reason it was decided, -in special cases, -to perform acoustic reflex examination under general anaesthesia. The first report on stapedius reflex and general anaesthesia was published by Mink et al. in 1981. Under the effect of Tiobutabarbital, Propanidid and Diazepam there is no reflex response. Acoustic reflex can be elicited with Ketamin-hydrochlorid and Alphaxalone-alphadolone acetate narcosis. The reflex threshold remains unchanged and the amplitude of muscle contraction is somewhat increased. The method was used: 1. to assess the type and degree of hearing loss in children with cleft palate and/or lip prior to surgery. 2. to exclude neuromuscular disorders with indication of pharyngoplasties. 3. to quantify hearing level in children--mostly multiply handicapped--with retarded speech development. The results of Behavioral Observation and Impedance Audiometry are discussed and evaluated. PMID:6577558 20. [Reflex seizures, cinema and television]. Science.gov (United States) Olivares-Romero, Jesús 2015-12-16 In movies and television series are few references to seizures or reflex epilepsy even though in real life are an important subgroup of total epileptic syndromes. It has performed a search on the topic, identified 25 films in which they appear reflex seizures. Most seizures observed are tonic-clonic and visual stimuli are the most numerous, corresponding all with flashing lights. The emotions are the main stimuli in higher level processes. In most cases it is not possible to know if a character suffers a reflex epilepsy or suffer reflex seizures in the context of another epileptic syndrome. The main conclusion is that, in the movies, the reflex seizures are merely a visual reinforcing and anecdotal element without significant influence on the plot. PMID:26662874 1. Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory OpenAIRE Mierczyński, Janusz; Shen, Wenxian 2012-01-01 This is the first of a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. It focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of... 2. Finiteness of the space of n-cycles for a n-concave complex space OpenAIRE 2015-01-01 We show that for n$\\ge$2 the space of closed n--cycles in a strongly n--concave complex space has a natural structure of reduced complex space locally of finite dimension and represents the functor "analytic family of n--cycles" parametrized by banach analytic sets. 3. Simple strategies for Banach-Mazur games and fairly correct systems Directory of Open Access Journals (Sweden) Thomas Brihaye 2013-07-01 Full Text Available In 2006, Varacca and Völzer proved that on finite graphs, omega-regular large sets coincide with omega-regular sets of probability 1, by using the existence of positional strategies in the related Banach-Mazur games. Motivated by this result, we try to understand relations between sets of probability 1 and various notions of simple strategies (including those introduced in a recent paper of Grädel and Lessenich. Then, we introduce a generalisation of the classical Banach-Mazur game and in particular, a probabilistic version whose goal is to characterise sets of probability 1 (as classical Banach-Mazur games characterise large sets. We obtain a determinacy result for these games, when the winning set is a countable intersection of open sets. 4. Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras CERN Document Server Azar, Kazem Haghnejad 2010-01-01 In this note, we study the Arens regularity of projective tensor product$A\\hat{\\otimes}B$whenever$A$and$B$are Arens regular. We establish some new conditions for showing that the Banach algebras$A$and$B$are Arens regular if and only if$A\\hat{\\otimes}B$is Arens regular. We also introduce some new concepts as left-weak$^$-weak convergence property [$Lw^wc-$property] and right-weak$^$-weak convergence property [$Rw^wc-$property] and for Banach algebra$A$, suppose that$A^$and$A^{}$, respectively, have$Rw^wc-$property and$Lw^wc-$property. Then if$A^{}$is weakly amenable, it follows that$A$is weakly amenable. We also offer some results concerning the relation between these properties with some special derivation$D:A\\rightarrow A^$. We obtain some conclusions in the Arens regularity of Banach algebras. 5. Isometries of a function space Directory of Open Access Journals (Sweden) U. D. Vyas 1987-09-01 Full Text Available It is proved here that an isometry on the subset of all positive functions of L1âLp(â can be characterized by means of a function h together with a Borel measurable mapping Ï of â, thus generalizing the Banach-Lamparti theorem of Lp spaces. 6. Pelczynski's property (V) on spaces of vector valued functions CERN Document Server Randrianantoanina, N 1994-01-01 Let$E$be a separable Banach space and$\\Omega$be a compact Hausdorff space. It is shown that the space$C(\\Omega,E)$has property (V) if and only if$E$does. Similar result is also given for Bochner spaces$L^p(\\mu,E)$if$1 7. The Sociology of Lesbian and Gay Reflexivity or Reflexive Sociology? OpenAIRE Brian Heaphy 2008-01-01 This article is concerned with sociological conceptualisations of lesbian and gay sexualities as reflexive forms of existence, and identifies core problems with these. Our sociological narratives about lesbian and gay reflexivity tend to be partial in two senses. First, they talk about and envision only very particular - and relatively exclusive â€“ experience, and fail to adequately account for the significance of difference and power in shaping diverse lesbian and gay experiences. Second, t... 8. Some properties of space of compact operators CERN Document Server Randrianantoanina, N 1994-01-01 Let $X$ be a separable Banach space, $Y$ be a Banach space and $\\Lambda$ be a subset of the dual group of a given compact metrizable abelian group. We prove that if $X^$ and $Y$ have the type I-$\\Lambda$-RNP (resp. type II-$\\Lambda$-RNP) then $K(X,Y)$ has the type I-$\\Lambda$-RNP (resp. type II-$\\Lambda$-RNP) provided $L(X,Y)=K(X,Y)$. Some corollaries are then presented as well as results conserning the separability assumption on $X$. Similar results for the NearRNP and the WeakRNP are also presented. 9. Management of Reflex Anoxic Seizures Directory of Open Access Journals (Sweden) J Gordon Millichap 2013-10-01 Full Text Available Investigators at the Roald Dahl EEG Unit, Alder Hey Children’s NHS Foundation, Liverpool, UK, review the definition, pathophysiology, clinical presentation, and management of reflex anoxic seizures (RAS in children. 10. Hoffmann reflex in idiopathic scoliosis. Science.gov (United States) Czernicki, Krzysztof; Dobosiewicz, Krystyna; Jedrzejewska, Anna; Durmala, Jacek 2006-01-01 This study was carried out in order to determine the dependence of selected Hoffmann reflex parameters on type, progression and morphology of idiopathic scoliosis (IS). Data collected from 129 girls with IS (59 progressive and 70 non-progressive cases) aged 7-16 years and 24 healthy subjects were analysed. H-reflex index (IH) and H/M amplitude index (IH/M) were calculated. Progressive left lumbar scoliosis expressed a significant decrease of IH values and a distinct tendency to IH/M depletion compared to non-progressives and controls. Progressive right thoracic scoliosis expressed marked tendency to IH decrease compared to nonprogressive scoliosis. No significant differences in H-reflex parameters were observed between the convex and concave side of the curvature or between types of scoliosis. H-reflex analysis in idiopathic scoliosis supports the hypothesis of a primary neurological disorder in progressive IS. PMID:17108418 11. Oculocardiac reflex during strabismus surgery Directory of Open Access Journals (Sweden) Mehryar Taghavi Gilani 2016-12-01 Full Text Available The activation of oculucardiac reflex (OCR is common during the strabismus surgeries. OCR is known as a trigemino-vagal reflex, which leads to the various side effects including bradycardia, tachycardia, arrhythmia, or in some cases cardiac arrest. This reflex could be activated during intraorbital injections, hematomas, and mechanical stimulation of eyeball and extraocular muscles surgeries. The incidence of OCR varies in a wide range, from 14% to 90%, that depends on anesthetic strategy and drug used for the surgery. The efficacy of various anticholinergic and anesthetic agents on declining the OCR reflex has been evaluated in different studies, especially in children. Although the detection of OCR goes back to 1908, its exact effect is not well recognized during strabismus surgery. In this review, we aimed to summarize the studies investigated the efficacy and potential of various anesthetic medications on inhibiting the OCR in children undergoing strabismus surgery. 12. Management of Reflex Anoxic Seizures OpenAIRE J. Gordon Millichap 2013-01-01 Investigators at the Roald Dahl EEG Unit, Alder Hey Children’s NHS Foundation, Liverpool, UK, review the definition, pathophysiology, clinical presentation, and management of reflex anoxic seizures (RAS) in children. 13. Weakly countably determined spaces of high complexity CERN Document Server Avilés, Antonio 2009-01-01 We prove that there exist weakly countably determined spaces of complexity higher than coanalytic. On the other hand, we also show that coanalytic sets can be characterized by the existence of a cofinal adequate family of closed sets. Therefore the Banach spaces constructed by means of these families have at most coanalytic complexity. 14. Spaces of continuous functions over Dugundji compacta OpenAIRE Banakh, Taras; Kubis, Wieslaw 2006-01-01 We show that for every Dugundji compact $K$ of weight aleph one the Banach space $C(K)$ is 1-Plichko and the space $P(K)$ of probability measures on $K$ is Valdivia compact. Combining this result with the existence of a non-Valdivia compact group, we answer a question of Kalenda. 15. The minimal displacement and extremal spaces OpenAIRE Bolibok, Krzysztof; Wi?nicki, Andrzej; Wo?ko, Jacek 2013-01-01 We show that both separable preduals of $L_{1}$ and non-type I $C^$-algebras are strictly extremal with respect to the minimal displacement of $k$-Lipschitz mappings acting on the unit ball of a Banach space. In particular, every separable $C(K)$ space is strictly extremal. 16. SYSTEM REFLEXIVE STRATEGIC MARKETING MANAGEMENT Directory of Open Access Journals (Sweden) Andrii A. DLIGACH 2012-07-01 Full Text Available This article reviews the System Reflexive paradigm of strategic marketing management, being based on the alignment of strategic economic interests of stakeholders, specifically, enterprise owners and hired managers, and consumers. The essence of marketing concept of management comes under review, along with the strategic management approaches to business, buildup and alignment of economic interests of business stakeholders. A roadmap for resolving the problems of modern marketing is proposed through the adoption of System Reflexive marketing theory. 17. Reflexive fatherhood in everyday life DEFF Research Database (Denmark) Westerling, Allan 2015-01-01 This article looks at fathering practices in Denmark, using the findings from a research project on everyday family life in Denmark. It takes a social psychological perspective and employs discursive psychology and theories about reflexive modernisation. It shows how fathers orient towards intimacy...... in their relationships with their children. Moreover, it discusses how fathers’ relatedness reflects individualisation and detraditionalisation. It is argued that reflexive modernisation entails subjective orientations that enable novel pathways to intimacy in contemporary father–child relationships... 18. Polynomial Bridgeland Stable Objects and Reflexive Sheaves CERN Document Server Lo, Jason 2011-01-01 On a smooth projective threefold, we show that there are only two isomorphism types for the moduli of stable objects with respect to Bayer's standard polynomial Bridgeland stability - the moduli of Gieseker-stable sheaves and the moduli of PT-stable objects - under the following assumptions: no two of the stability vectors are collinear, and the degree and rank of the objects are relatively prime. We also describe a close relation between the intersection of the moduli spaces of PT-stable and dual-PT-stable objects, and the moduli of reflexive sheaves. 19. Dentocardiac Reflex: an Allegedly New Subform of the Trigeminocardiac Reflex Directory of Open Access Journals (Sweden) Amr Abdulazim 2011-05-01 Full Text Available Trigeminocardiac reflex (TCR is currently defined as a sudden bradycardia and decrease in mean arterial blood pressure by 20% during the manipulation of the branches of trigeminal nerve. TCR, especially during the last decade has been mostly studied in the course of neurosurgical procedures which are supposed to elicit the central subtype of TCR. Previously the well-known oculocardiac reflex was also considered as a subtype of TCR. Recently, surgeons dealing with the other branches of the fifth cranial nerve have become more interested in this reflex. Some noteworthy points have been published discussing new aspects of the trigeminocardiac reflex (TCR in simple oral surgical procedures. Arakeri et al. have reviewed the similarities and differences between TCR, vasovagal response (VVR, and syncope. They have also explained a new possible pathway for the reflex during the simple extraction of upper first molars. The present paper aims to briefly discuss these recently presented points. Although the discussed concepts are noteworthy and consistent our preliminary results of our yet to be published studies, it seemed timely for us to discuss some possible shortcomings that may affect the results of such assessments. 20. A Banach Algebra Approach To Amalgamated R- and S-transforms OpenAIRE Aagaard, L 2004-01-01 We give a Banach algebra approach to the additiveness property of Voiculescu's amalgamated R-transform. We also define an amalgamated S-transform and prove that it is multiplicative on products of amalgamated free random variables when the algebra of amalgamation is commutative. 1. Central limit theorem for the Banach-valued weakly dependent random variables International Nuclear Information System (INIS) The central limit theorem (CLT) for the Banach-valued weakly dependent random variables is proved. In proving CLT convergence of finite-measured (i.e. cylindrical) distributions is established. A weak compactness of the family of measures generated by a certain sequence is confirmed. The continuity of the limiting field is checked 2. On solvability of some quadratic functional-integral equation in Banach algebra International Nuclear Information System (INIS) Using the technique of a suitable measure of non-compactness in Banach algebra, we prove an existence theorem for some functional-integral equations which contain, as particular cases, a lot of integral and functional-integral equations that arise in many branches of nonlinear analysis and its applications. Also, the famous Chandrasekhar's integral equation is considered as a special case. (author) 3. Subspaces of almost Daugavet spaces OpenAIRE Lücking, Simon 2010-01-01 We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\\ell_1$, then $Z$ has the almost Daugavet property, too. 4. Stochastic processes on non-Archimedean spaces. III. Stochastic processes on totally disconnected topological groups OpenAIRE Ludkovsky, S. V. 2001-01-01 Stochastic processes on totally disconnected topological groups are investigated. In particular they are considered for diffeomorphism groups and loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a quasi-invariance and a pseudo-differentiability of transition measures are proved. Transition measures are used for the construction of strongly continuous representations including irreducible of these groups. In addition stochastic processes on general Banach-Lie groups, l... 5. Composition Operators on Laguerre-Cauchy Type Spaces Directory of Open Access Journals (Sweden) El-Bachir Yallaoui 2014-01-01 Full Text Available We study the action of the composition operators on the analytic function spaces whose kernels are special cases of Laguerre polynomials. These function spaces become Banach spaces when the kernels are integrated with respect to the complex Borel measures of the unit circle. Necessary and sufficient conditions for the composition operators to be compact are found. 6. On some interpolation properties in locally convex spaces International Nuclear Information System (INIS) The aim of this paper is to introduce the notion of interpolation between locally convex spaces, the real method, and to present some elementary results in this setting. This represents a generalization from the Banach spaces framework to the locally convex spaces sequentially complete one, where the operators acting on them are locally bounded 7. On some interpolation properties in locally convex spaces Energy Technology Data Exchange (ETDEWEB) Pater, Flavius [Department of Mathematics, Politehnica University of Timi?oara, 300004 Timi?oara (Romania) 2015-03-10 The aim of this paper is to introduce the notion of interpolation between locally convex spaces, the real method, and to present some elementary results in this setting. This represents a generalization from the Banach spaces framework to the locally convex spaces sequentially complete one, where the operators acting on them are locally bounded. 8. Fixed Points of Non-expansive Operators on Weakly Cauchy Normed Spaces Directory of Open Access Journals (Sweden) Sahar M. Ali 2007-01-01 Full Text Available We proved the existence of fixed points of non-expansive operators defined on weakly Cauchy spaces in which parallelogram law holds, the given normed space is not necessarily be uniformly convex Banach space or Hilbert space, we reduced the completeness and the uniform convexity assumptions which imposed on the given normed space. 9. The Intuitionistic Fuzzy Normed Space of Coefficients OpenAIRE Bilalov, B.T.; Farahani, S. M.; F. A. Guliyeva 2012-01-01 Intuitionistic fuzzy normed space is defined using concepts of $t$ -norm and $t$ -conorm. The concepts of fuzzy completeness, fuzzy minimality, fuzzy biorthogonality, fuzzy basicity, and fuzzy space of coefficients are introduced. Strong completeness of fuzzy space of coefficients with regard to fuzzy norm and strong basicity of canonical system in this space are proved. Strong basicity criterion in fuzzy Banach space is presented in terms of coefficient operator. 10. [The oculocardiac reflex in blepharoplasties]. Science.gov (United States) Rippmann, V; Scholz, T; Hellmann, S; Amini, P; Spilker, G 2008-08-01 The oculocardiac reflex (OCR) is a well-known phenomenon in ophthalmic surgery, but is rarely described in aesthetic blepharoplasty surgery. It was first mentioned in 1908 by Ascher and Dagnini. Since then, ophthalmologists and anaesthesiologists have regarded the onset of the oculocardiac reflex as a significant intraoperative problem, which is undermined by several case reports that describe dysrhythmias which have haved caused morbidity and death. Per definition the OCR is caused by ocular manipulation and involves intraoperative bradycardia by a change of 20 beats/minute compared to the preoperative heart rate or any dysrhythmia during the manipulation via a trigeminal-vagal-mediated reflex arc. Having operated on a 48-year-old, healthy woman in our clinic, who underwent a cardiac arrest during the blepharoplasty procedure, followed by a successful resuscitation, we investigated the onset of the OCR in our blepharoplasty patients within the last 3 years. The onset of the OCR was noted in 22 of 110 (20 %) blepharoplasty patients, mainly affecting younger, low-weighted patients operated under local anaesthesia. Awareness and treatment of this potentially life-threatening oculocardiac reflex are necessary. In most cases the onset of the reflex may be avoided by a gentle operation technique and by refraining from severe traction to the muscle or fat pad. The best treatment of a profound bradycardia caused by the OCR is to release tension to the muscle or fat pad in order to permit the heart rate to return to normal. Intraoperative monitoring is of utmost importance. PMID:18716987 11. Stochastic processes on non-Archimedean spaces. I. Stochastic processes on Banach spaces OpenAIRE Ludkovsky, S. V. 2001-01-01 Non-Archimedean analogs of Markov quasimeasures and stochastic processes are investigated. Thery are used for the development of stochastic antiderivations. The non-Archimedean analog of the It$\\hat o$ formula is proved. 12. A contribution to group representations in locally convex spaces International Nuclear Information System (INIS) Let U be a continuous representation of a (connected) locally compact group G in a separated locally convex space E. It is shown that the study of U is equivalent to the study of a family Usub(i) of continuous representations of G in Frechet spaces Fsub(i). If U is equicontinuous, the Fsub(i) are Banach spaces, and the Usub(i) are realized by isomeric operators. When U is topologically irreducible, it is Naemark equivalent to a Frechet (or isomeric Banach, in the equicontinuous case) continuous representation. Similar results hold for semi-groups. (Auth.) 13. Survey on nonlocal games and operator space theory Science.gov (United States) Palazuelos, Carlos; Vidick, Thomas 2016-01-01 This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states. 14. Survey on nonlocal games and operator space theory International Nuclear Information System (INIS) This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states 15. Survey on nonlocal games and operator space theory Energy Technology Data Exchange (ETDEWEB) Palazuelos, Carlos, E-mail: [email protected] [Instituto de Ciencias Matemáticas (ICMAT), Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid (Spain); Vidick, Thomas, E-mail: [email protected] [Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California 91125 (United States) 2016-01-15 This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states. 16. Complemented copies of $\\ell_{1}$ in spaces of vector valued measures and applications CERN Document Server Randrianantoanina, N 1995-01-01 Let $X$ be a Banach space and $(\\Omega,\\Sigma)$ be a measure space. We provide a characterization of sequences in the space of $X$-valued countably additive measures on $\\Omega,\\Sigma)$ of bounded variation that generate complemented copies of $\\ell_1$. As application, we prove that if a dual Banach space $E^$ has Pe\\l czy\\'nski's property (V) then so does the space of $E^$-valued countably additive measures with bounded variation. Another application, we show that for a Banach space $X$, the space $\\ell_\\infty(X)$ contains a complemented copy of $\\ell_1$ if and only if $X$ contains all $\\ell_1^n$ uniformly complemented. 17. Two ways to support reflexivity DEFF Research Database (Denmark) 2013-01-01 A current challenge to public managers is the lack of a well-defined role. How can master’s programmes prepare managers to live up to an undefined function? In this paper we argue that the lack of role description enhances the need for reflexivity and show how it is done at Master in Educational......’. This requires participants to conduct experiments in their own organization, to reflect on and analyse their experiences with concepts from the curriculum. While the new language and the experimental teaching format are difficult, the participants learn a reflexive practice that can enable them to life...... up to an undefined role.... 18. 2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW-algebras Science.gov (United States) Ayupov, Shavkat; Kudaybergenov, Karimbergen 2016-03-01 The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW-algebras and show that any 2-local derivation on an arbitrary AW-algebra is a derivation. 19. Sur la continuité automatique des épimorphismes dans les ☆-algèbres de Banach OpenAIRE Y. Tidli; Tajmouati, A.; Oukhtite, L. 2004-01-01 Nous étudions les problèmes de continuité automatique dans des algèbres de Banach avec involutions. Nous obtenons aussi des nouveoux résultats concernant ☆-idéals des ☆-algèbres.We study the automatic continuity problems for Banach algebras with involutions. We also obtain some new results concerning ☆-ideals of ☆-algebras. 20. Reflexivity and technology in adult learning OpenAIRE Neil Selwyn 2005-01-01 It is argued by influential commentators such as Ulrich Beck and Scott Lash that we now live in a ‘reflexively modern' age. People are seen to now be free of the structures of modern society and driven instead by individualised opportunities to reflexively engage with their fast-changing social worlds and identities. Taking the notion of reflexive modernisation as its starting point, this paper explores the roles that information technologies (ITs) may play in supporting adults' reflexive jud... 1. Educating the Reflexive Practitioner Directory of Open Access Journals (Sweden) Marc J. Neveu 2012-09-01 2. Tonic vibration reflexes and background force level Science.gov (United States) Lackner, James R.; Dizio, Paul; Fisk, John 1992-01-01 On earth, the functional stretch reflex is an important component in the maintenance of posture and muscle tone. In parabolic flight experiments, it is evaluated whether the functional stretch reflex, as reflected in the tonic vibration reflex, adjusts appropriately for changes in background gravitoinertial force level. Virtually immediate alterations of appropriate sign occurred. 3. Stereotype locally convex spaces Science.gov (United States) Akbarov, S. S. 2000-08-01 We give complete proofs of some previously announced results in the theory of stereotype (that is, reflexive in the sense of Pontryagin duality) locally convex spaces. These spaces have important applications in topological algebra and functional analysis. 4. Stereotype locally convex spaces International Nuclear Information System (INIS) We give complete proofs of some previously announced results in the theory of stereotype (that is, reflexive in the sense of Pontryagin duality) locally convex spaces. These spaces have important applications in topological algebra and functional analysis 5. Stereotype locally convex spaces Energy Technology Data Exchange (ETDEWEB) Akbarov, S S 2000-08-31 We give complete proofs of some previously announced results in the theory of stereotype (that is, reflexive in the sense of Pontryagin duality) locally convex spaces. These spaces have important applications in topological algebra and functional analysis. 6. The reflexive case study method DEFF Research Database (Denmark) Rittenhofer, Iris 2015-01-01 This paper extends the international business research on small to medium-sized enterprises (SME) at the nexus of globalization. Based on a conceptual synthesis across disciplines and theoretical perspectives, it offers management research a reflexive method for case study research of postnational... 7. Some properties of the noncommutative Hp(r ,s ) (𝒜; ??) and Hp(𝒜; ?1) spaces Science.gov (United States) Tulenov, Kanat 2015-09-01 In this paper, we introduce the noncommutative Hp(r ,s ) (𝒜; ??) and Hp (𝒜; ?1) spaces. Then, it is shown that both spaces are Banach spaces for r, s ? 2 (and resp. p ? 1) and the analogue of Saito's theorem for the Hp(r ,s ) (𝒜; ??) and Hp(𝒜; ?1) spaces are proved. 8. On the Banach algebra Ã¢Â„Â¬(lp(ÃŽÂ± Directory of Open Access Journals (Sweden) Bruno de Malafosse 2004-11-01 Full Text Available We give some properties of the Banach algebra of bounded operators Ã¢Â„Â¬(lp(ÃŽÂ± for 1Ã¢Â‰Â¤pÃ¢Â‰Â¤Ã¢ÂˆÂž, where lp(ÃŽÂ±=(1/ÃŽÂ±Ã¢ÂˆÂ’1Ã¢ÂˆÂ—lp. Then we deal with the continued fractions and give some properties of the operator ÃŽÂ”h for h>0 or integer greater than or equal to one mapping lp(ÃŽÂ± into itself for pÃ¢Â‰Â¥1 real. These results extend, among other things, those concerning the Banach algebra SÃŽÂ± and some results on the continued fractions. 9. Bessel and Grüss Type Inequalities in Inner Product Modules over Banach -Algebras Directory of Open Access Journals (Sweden) Dragomir SS 2011-01-01 Full Text Available We give an analogue of the Bessel inequality and we state a simple formulation of the Grüss type inequality in inner product -modules, which is a refinement of it. We obtain some further generalization of the Grüss type inequalities in inner product modules over proper -algebras and unital Banach -algebras for -seminorms and positive linear functionals. 10. Star product realizations of kappa-Minkowski space DEFF Research Database (Denmark) Durhuus, Bergfinnur; Sitarz, Andrzej 2013-01-01 We define a family of star products and involutions associated with κ -Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is... 11. Weighted norms and Volterra integral equations in LP spaces Directory of Open Access Journals (Sweden) Jaroslaw Kwapisz 1991-01-01 Full Text Available A new simple proof of existence and uniqueness of solutions of the Volterra integral equation in Lebesque spaces is given. It is shown that the weighted norm technique and the Banach contraction mapping principle can be applied (as in the case of continuous functions space. 12. ON STRONG AND WEAK CONVERGENCE IN n-HILBERT SPACES Directory of Open Access Journals (Sweden) Agus L. Soenjaya 2014-03-01 Full Text Available We discuss the concepts of strong and weak convergence in n-Hilbert spaces and study their properties. Some examples are given to illustrate the concepts. In particular, we prove an analogue of Banach-Saks-Mazur theorem and Radon-Riesz property in the case of n-Hilbert space. 13. Strong and Convergence Theorems for Multivalued Mappings in Spaces Directory of Open Access Journals (Sweden) Laowang W 2009-01-01 Full Text Available We show strong and convergence for Mann iteration of a multivalued nonexpansive mapping whose domain is a nonempty closed convex subset of a CAT(0 space. The results we obtain are analogs of Banach space results by Song and Wang [2009, 2008]. Strong convergence of Ishikawa iteration are also included. 14. Tsirelson's space CERN Document Server Casazza, Peter G 1989-01-01 This monograph provides a structure theory for the increasingly important Banach space discovered by B.S. Tsirelson. The basic construction should be accessible to graduate students of functional analysis with a knowledge of the theory of Schauder bases, while topics of a more advanced nature are presented for the specialist. Bounded linear operators are studied through the use of finite-dimensional decompositions, and complemented subspaces are studied at length. A myriad of variant constructions are presented and explored, while open questions are broached in almost every chapter. Two appendices are attached: one dealing with a computer program which computes norms of finitely-supported vectors, while the other surveys recent work on weak Hilbert spaces (where a Tsirelson-type space provides an example). 15. Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I Directory of Open Access Journals (Sweden) Huai-Xin Cao 2009-01-01 Full Text Available We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let 𝒜 be a Banach algebra and X a Banach 𝒜-module, f:X→X and g:𝒜→𝒜. The mappings Δf,g1, Δf,g2, Δf,g3, and Δf,g4 are defined and it is proved that if ∥Δf,g1(x,y,z,w∥ (resp., ∥Δf,g3(x,y,z,w,α,β∥ is dominated by φ(x,y,z,w, then f is a generalized (resp., linear module-𝒜 left derivation and g is a (resp., linear module-X left derivation. It is also shown that if ∥Δf,g2(x,y,z,w∥ (resp., ∥Δf,g4(x,y,z,w,α,β∥ is dominated by φ(x,y,z,w, then f is a generalized (resp., linear module-𝒜 derivation and g is a (resp., linear module-X derivation. 16. On the L-characteristic of nonlinear superposition operators in lp-spaces International Nuclear Information System (INIS) In this paper we describe the L-characteristic of the nonlinear superposition operator F(x) f(s,x(s)) between two Banach spaces of functions x from N to R. It was shown that L-characteristic of the nonlinear superposition operator which acts between two Lebesgue spaces has so-called Σ-convexity property. In this paper we show that L-characteristic of the operator F (between two Banach spaces) has the convexity property. It means that the classical interpolation theorem of Reisz-Thorin for a linear operator holds for the nonlinear operator superposition which acts between two Banach spaces of sequences. Moreover, we consider the growth function of the operator superposition in mentioned spaces and we show that one has the logarithmically convexity property. (author). 7 refs 17. A note on mutiplication operators on Köthe-Bochner spaces Directory of Open Access Journals (Sweden) S. S. Khurana 2012-02-01 Full Text Available Let (Ω, A, μ is a finite measure space, E an order continuous Banach function space over μ, X a Banach space and E(X the Köthe-Bochner space. A new simple proof is given of the result that a continuous linear operator T: E(X ® E(X is a multiplication operator (by a function in L¥ iff T(g f, x* > x =g T(f, x* > x for everyg Î L¥, f Î E(X, x Î X, x* Î X. 18. Integrating Reflexivity in Livelihoods Research DEFF Research Database (Denmark) Prowse, Martin 2010-01-01 Much poverty and development research is not explicit about its methodology or philosophical foundations. Based on the extended case method of Burawoy and the epistemological standpoint of critical realism, this paper discusses a methodological approach for reflexive inductive livelihoods researc...... that overcomes the unproductive social science dualism of positivism and social constructivism. The approach is linked to a conceptual framework and a menu of research methods that can be sequenced and iterated in light of research questions.......Much poverty and development research is not explicit about its methodology or philosophical foundations. Based on the extended case method of Burawoy and the epistemological standpoint of critical realism, this paper discusses a methodological approach for reflexive inductive livelihoods research... 19. Integrating Reflexivity in Livelihoods Research OpenAIRE Prowse, Martin 2010-01-01 Much poverty and development research is not explicit about its methodology or philosophical foundations. Based on the extended case method of Burawoy and the epistemological standpoint of critical realism, this paper discusses a methodological approach for reflexive inductive livelihoods research that overcomes the unproductive social science dualism of positivism and social constructivism. The approach is linked to a conceptual framework and a menu of research methods that can be sequenced ... 20. Reflexivity in Narratives on Practice DEFF Research Database (Denmark) Jakobsen, Helle Nordentoft; Olesen, Lektor Birgitte Ravn Previous research has shown how reflexivity is a precondition for knowledge co-production through productive dialogue in organisational contexts because it entails a re-ordering, re-arranging and re-designing of what one knows and therefore creates new angles of vision. In this paper, we draw on ...... have the potential to bring about productive learning in organizational contexts since they appear to stimulate participants’ relational engagement and reflexivity.......Previous research has shown how reflexivity is a precondition for knowledge co-production through productive dialogue in organisational contexts because it entails a re-ordering, re-arranging and re-designing of what one knows and therefore creates new angles of vision. In this paper, we draw on...... dialogic conception of practice as they entail a conceptual reframing of key elements in practice. In addition, the narratives expose a situational and relational, rather than normative, focus which allows for reflections on emotional and bodily experiences. In conclusion, we argue that practice narratives... 1. On the Carleman ultradifferentiable vectors of a scalar type spectral operator OpenAIRE Markin, Marat V. 2016-01-01 A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space. 2. Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation OpenAIRE Ghenciu, Ioana 2015-01-01 For a Banach space X and a measure space (Ω,Σ), let M(Ω,X) be the space of all X-valued countably additive measures on (Ω,Σ) of bounded variation, with the total variation norm. In this paper we give a characterization of weakly precompact subsets of M(Ω,X). 3. The near Radon-Nikodym property in Lebesgue-Bochner function spaces CERN Document Server Randrianantoanina, N; Randrianantoanina, Narcisse; Saab, Elias 1997-01-01 Let $X$ be a Banach space $E$ a K\\"othe function space that does not contain $c_0$. It is shown that the vector valued function space $E(X)$ has the Near Radon Nikodym property if and only if $X$ does. 4. Duality of Variable Exponent Triebel-Lizorkin and Besov Spaces OpenAIRE Noi, Takahiro 2012-01-01 We will prove the duality and reflexivity of variable exponent Triebel-Lizorkin and Besov spaces. It was shown by many authors that variable exponent Triebel-Lizorkin spaces coincide with variable exponent Bessel potential spaces, Sobolev spaces, and Lebesgue spaces when appropriate indices are chosen. In consequence of the results, these variable exponent function spaces are shown to be reflexive. 5. Avaliao dos reflexos espinhais em bezerros Spinal reflexes in calves Directory of Open Access Journals (Sweden) Alexandre Secorun Borges 1997-12-01 Full Text Available Este trabalho objetivou avaliar, quantificar e padronizar a ocorrncia dos reflexos espinhais em bezerros da raa holandesa de 15 a 90 dias de idade, os quais foram submetidos a avaliao nos membros torcicos (reflexo carpo radial, reflexo bicipital, reflexo tricipital e reflexo flexor e nos membros plvicos (reflexo patelar, reflexo tibial cranial, reflexo gastrocnmio, reflexo citico e reflexo flexor. Para quantificao da resposta involuntria frente ao reflexo realizado, padronizou-se a ausncia do reflexo como sendo o algarismo 0; resposta discreta do reflexo como sendo l e a presena evidente da resposta como sendo 2. Os reflexos mais evidentes e constantes foram os reflexos flexor, carpo radial, patelar e tricipital. Os reflexos menos evidentes e menos freqentes foram os reflexos tibial cranial, bicipital, gastrocnmio e citico.A study was carried out to evaluate the spinal reflexes of Holstein calves (15 to 90 days. Spinal reflexes were tested and graded (0 = absence, I = mild reflex, 2 = evident in thoracic (extensor carpi reflex, biceps reflex, triceps reflex and flexor reflex and pelvic (patellar reflex, cranial tibial reflex, gastrocnemius reflex, ciatic reflex and flexor reflex limbs. The flexor reflex, extensor carpi reflex, patellar and triceps reflex were elicited in most animals. While the least evident reflexes were, cranial tibial reflex, biceps reflex, gastrocnemius reflex and ciatic reflex. 6. The emergence of reflexive global labour law OpenAIRE Rogowski, Ralf 2015-01-01 The article introduces the main tenets of reflexive labour law and uses this perspective to interpret core trends in global labour law. It suggests a conceptual distinction between international and global labour law and identifies a transformation in the global labour law regime related to processes of reflexivity and constitutionalisation. The first part of the article analyses reflexivity within the International Labour Organization (ILO) in relation to its policy of defining labour standa... 7. Arakeri’s Reflex: an Alternative Pathway for Dento-Cardiac Reflex Mediated Syncope OpenAIRE Veena Arali; Shailaja Reddy; C.G Raghuram; Gururaj Arakeri 2010-01-01 Introduction: Dentocardiac reflex, a variant of trigeminocardiac reflex elicited specifically during tooth extraction procedures in den-tal/maxillofacial surgery and is believed to cause syncope with an afferent link mediated by posterior superior alveolar nerve. Another variant of trigeminocardiac reflex which is also of interest to the oral and maxillofacial surgeon is oculocardiac reflex which can be triggered by direct or indirect manipulation of eye globe or muscles around it.The hypothe... 8. On the Carleman classes of vectors of a scalar type spectral operator OpenAIRE Markin, Marat V. 2004-01-01 The Carleman classes of a scalar type spectral operator in a reflexive Banach space are characterized in terms of the operator's resolution of the identity. A theorem of the Paley-Wiener type is considered as an application. 9. On Presic Type Generalization of the Banach Contraction Mapping Principle Directory of Open Access Journals (Sweden) S. Presic 2007-10-01 Full Text Available Let (X, d be a metric space, k a positive integer and T a mapping of Xk into X. In this paper we proved that if T satisfies conditions (2.1 and (2.2 below, then there exists a unique x in X such that T(x, x,, x = x, This result generalizes the corresponding theorems of the second author [4], [5] and the theorem of Dhage [3]. 10. Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces OpenAIRE Bargetz, Christian; Dymond, Michael; Reich, Simeon 2016-01-01 We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\\mathrm{CAT}(\\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\\sigma$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical poi... 11. Existence Results for Some Nonlinear Functional-Integral Equations in Banach Algebra with Applications Directory of Open Access Journals (Sweden) Lakshmi Narayan Mishra 2016-04-01 Full Text Available In the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. By utilizing the techniques of noncompactness measures, we operate the fixed point theorems such as Darbo's theorem in Banach algebra concerning the estimate on the solutions. The results obtained in this paper extend and improve essentially some known results in the recent literature. We also provide an example of nonlinear functional-integral equation to show the ability of our main result. 12. Lattices generated by skeletons of reflexive polytopes OpenAIRE Haase, Christian; Nill, Benjamin 2005-01-01 Lattices generated by lattice points in skeletons of reflexive polytopes are essential in determining the fundamental group and integral cohomology of Calabi-Yau hypersurfaces. Here we prove that the lattice generated by all lattice points in a reflexive polytope is already generated by lattice points in codimension two faces. This answers a question of J. Morgan. 13. A Model of the Hoffmann Reflex OpenAIRE Ward, T. 2000-01-01 The Hoffman reflex which has applications in assessment of nervous system damage in stroke, spinal injury, Parkinson's disease and other conditions is studied here both practically and theoretically. Using recorded Hoffman reflex data a model is proposed that captures the qualitative behaviour of the phenomenon and allows investigation of how different waveform morphologies and patterns of recruitment can be related to underlying physiological variables. 14. Reflexivity: Towards a Theory of Lifelong Learning. Science.gov (United States) Edwards, Richard; Ranson, Stewart; Strain, Michael 2002-01-01 The current notion of lifelong learning in policy and practice is dominated by behaviorist, adaptive accumulation of skills and qualifications. An alternative is reflexive lifelong learning, developed through social learning networks within the context of dislocation and uncertainty. It involves the reflexive practices of metacognitive analysis 15. $L^p$-Spaces as Quasi -Algebras OpenAIRE Bagarello, F.; C. Trapani 1994-01-01 The Banach space $L^p(X,\\mu)$, for $X$ a compact Hausdorff measure space, is considered as a special kind of quasi -algebra (called CQ-algebra) over the C-algebra $C(X)$ of continuous functions on $X$. It is shown that, for $p \\geq 2$, $(L^p(X,\\mu), C(X))$ is -semisimple (in a generalized sense). Some consequences of this fact are derived. 16. Widths of embeddings of 2-microlocal Besov spaces CERN Document Server Zhang, Shun 2011-01-01 We consider the asymptotic behaviour of the approximation, Gelfand and Kolmogorov numbers of compact embeddings between 2-microlocal Besov spaces with weights defined in terms of the distance to a $d$-set $U\\subset \\mathbb{R}^n$. The sharp estimates are shown in most cases, where the quasi-Banach setting is included. 17. A topological obstruction for small-distortion embeddability into spaces of continuous functions on countable compact metric spaces OpenAIRE Baudier, Florent Pierre 2013-01-01 We give the first lower bound on the $\\scr C(K)$-distortion of the class of separable Banach spaces, for $K$ a countable compact in the family $\\{ [0,\\omega],[0,\\omega\\cdot2],\\cdots, [0,\\omega^2], \\cdots, [0,\\omega^k\\cdot n],\\cdots,[0,\\omega^\\omega]\\}$. 18. Topological centers of module actions and cohomological groups of Banach Algebras CERN Document Server 2010-01-01 In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\\ell_{B^{}}(A^{})$ and ${Z}^\\ell_{A^{}}(B^{})$ be the topological centers of the left module action $\\pi_\\ell:~A\\times B\\rightarrow B$ and the right module action $\\pi_r:~B\\times A\\rightarrow B$, respectively. We investigate some relationships between topological center of $A^{}$, ${Z}_1({A^{}})$ with respect to the first Arens product and topological centers of module actions ${Z}^\\ell_{B^{}}(A^{})$ and ${Z}^\\ell_{A^{}}(B^{})$. On the other hand, if $A$ has Mazure property and $B^{}$ has the left $A^{}-factorization$, then $Z^\\ell_{A^{}}(B^{})=B$, and so for a locally compact non-compact group $G$ with compact covering number $card(G)$, we have $Z^\\ell_{M(G)^{}}{(L^1(G)^{})}= {L^1(G)}$ and $Z^\\ell_{L^1(G)^{}}{(M(G)^{})}= {M(G)}$. By using the Arens regularity of module actions, we study some cohomological groups properties of Banach algebra and we... 19. Common fixed points in best approximation for Banach operator pairs with Ciric type I-contractions Science.gov (United States) Hussain, N. 2008-02-01 The common fixed point theorems, similar to those of Ciric [Lj.B. Ciric, On a common fixed point theorem of a Gregus type, Publ. Inst. Math. (Beograd) (N.S.) 49 (1991) 174-178; Lj.B. Ciric, On Diviccaro, Fisher and Sessa open questions, Arch. Math. (Brno) 29 (1993) 145-152; Lj.B. Ciric, On a generalization of Gregus fixed point theorem, Czechoslovak Math. J. 50 (2000) 449-458], Fisher and Sessa [B. Fisher, S. Sessa, On a fixed point theorem of Gregus, Internat. J. Math. Math. Sci. 9 (1986) 23-28], Jungck [G. Jungck, On a fixed point theorem of Fisher and Sessa, Internat. J. Math. Math. Sci. 13 (1990) 497-500] and Mukherjee and Verma [R.N. Mukherjee, V. Verma, A note on fixed point theorem of Gregus, Math. Japon. 33 (1988) 745-749], are proved for a Banach operator pair. As applications, common fixed point and approximation results for Banach operator pair satisfying Ciric type contractive conditions are obtained without the assumption of linearity or affinity of either T or I. Our results unify and generalize various known results to a more general class of noncommuting mappings. 20. In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements Czech Academy of Sciences Publication Activity Database Hudzik, H.; Kaczmarek, R.; Krbec, Miroslav 2016-01-01 Roč. 90, č. 1 (2016), s. 249-261. ISSN 0001-9054 Institutional support: RVO:67985840 Keywords : symmetric space s * K-monotone symmetric Banach space s * strict monotonicity * lower local uniform monotonicity Subject RIV: BA - General Mathematics Impact factor: 0.918, year: 2014 http://link.springer.com/article/10.1007%2Fs00010-015-0379-6 1. A Valdivia compact space with no $G_\\delta$ points and few nontrivial convergent sequences OpenAIRE Correa, Claudia; Daniel V. Tausk 2015-01-01 We give an example of a Valdivia compact space with no $G_\\delta$ points and no nontrivial convergent sequences in the complement of a dense $\\Sigma$-subset. The example is related to a problem concerning twisted sums of Banach spaces. 2. Fuzzy optimization for portfolio selection based on Embedding Theorem in Fuzzy Normed Linear Spaces Directory of Open Access Journals (Sweden) Solatikia Farnaz 2014-05-01 Full Text Available Background: This paper generalizes the results of Embedding problem of Fuzzy Number Space and its extension into a Fuzzy Banach Space C(? C(?, where C(? is the set of all real-valued continuous functions on an open set ?. 3. Description of the cohomology of Banach algebras and locally convex algebras in the language of A∞-structures International Nuclear Information System (INIS) We consider the following problem: how can one apply the methods employed for describing the cohomology of algebras and based on the use of the A∞-structures of Stasheff to describe the cohomology of Banach algebras and locally convex algebras 4. The oculocardiac reflex in blepharoplasty surgery. Science.gov (United States) Matarasso, A 1989-02-01 The oculocardiac reflex (OCR), a previously undescribed phenomenon in aesthetic blepharoplasty surgery, involves intraoperative bradycardia exceeding 10 percent of the preoperative heart rate or any dysrhythmia during ocular manipulation. It is a trigeminal-vagal-mediated reflex arc. The oculocardiac reflex was noted to occur in 25 of 100 patients (25 percent) undergoing blepharoplasty. A data sheet designed and distributed for use in the operating room identified a reflex-prone patient (RPP) as a young, anxious female, with a cardiac history, operated on under light anesthesia with aggressive fat pad resection. The oculocardiac reflex was more likely to occur in a reflex-prone patient during traction on the medial fat pads and in the left eye. Despite anticipating the fatigue phenomenon in those patients who exhibited a profound bradycardia (35 to 40 beats per minute), it was necessary to release traction in order to permit the heart rate to return to normal. Awareness and treatment of this potentially life-threatening oculocardiac reflex are necessary. Careful patient surveillance and monitoring are mandatory. PMID:2911623 5. Reflexivity and technology in adult learning Directory of Open Access Journals (Sweden) Neil Selwyn 2005-03-01 Full Text Available It is argued by influential commentators such as Ulrich Beck and Scott Lash that we now live in a ‘reflexively modern' age. People are seen to now be free of the structures of modern society and driven instead by individualised opportunities to reflexively engage with their fast-changing social worlds and identities. Taking the notion of reflexive modernisation as its starting point, this paper explores the roles that information technologies (ITs may play in supporting adults' reflexive judgements about, and reflexive engagements with, education and learning. Through an analysis of interview data with 100 adults in the UK the paper finds that whilst a minority of interviewees were using ITs to support and inform reflexive engagementwith learning, the majority of individuals relayed little sign of technology-supported reflexivity when it came to their (nonengagement with education. For most people ITs were found, at best, to reinforce pre-established tendencies to ‘drift' through the formal education system. The paper concludes by considering the implications of these findings for ongoing efforts in developed countries to establish technology-supported ‘learning societies'. 6. On H–Dichotomy for Skew-Evolution Semiflows in Banach Spaces Directory of Open Access Journals (Sweden) Stoica Codruţa 2012-04-01 Full Text Available The aim of this paper is to define and characterize a particular case of dichotomy for skew-evolution semiflows, called the H–dichotomy, as a useful tool in describing the behaviors for the solutions of evolution equations that describe phenomena from engineering or economics. The paper emphasizes also other asymptotic properties, as ω–growth and ω–decay, H–stability and H–instability, as well as the classic concept of exponential dichotomy. 7. Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces OpenAIRE Zajicek, Ludek 2013-01-01 The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \\R \\to \\R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \\in X$) which holds almost everywhere for an arbitrary function $f:\\R^2\\to \\R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Bana... 8. On Milmanâs moduli for Banach spaces Directory of Open Access Journals (Sweden) Mariusz Szczepanik 2001-08-01 Full Text Available We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given. 9. Compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions Directory of Open Access Journals (Sweden) 2014-10-01 Full Text Available We characterize compact composition operators on real Banachspaces of complex-valued bounded Lipschitz functions on metricspaces, not necessarily compact, with Lipschitz involutions anddetermine their spectra. 10. Complex analysis in Banach spaces holomorphic functions and domains of holomorphy in finite and infinite dimensions CERN Document Server Mujica, J 1985-01-01 Problems arising from the study of holomorphic continuation and holomorphic approximation have been central in the development of complex analysis in finitely many variables, and constitute one of the most promising lines of research in infinite dimensional complex analysis. This book presents a unified view of these topics in both finite and infinite dimensions. 11. Perturbed optimization in Banach spaces II : a theory based on a strong directional constraint qualification OpenAIRE Bonnans, J. Frederic; Cominetti, Roberto 1994-01-01 Nous tudions la sensibilit du cot optimal et des solutions de problmes d'optimisation dans deux cas. Le premier est quand des multiplicateurs existent mais seule la condition suffisante d'optimalit faible est satisfaite. Le second cas est lorsque l'ensemble des multiplicateurs est vide. 12. Existence of Solutions of Nonlinear Integrodifferential Equations of Sobolev Type with Nonlocal Condition in Banach Spaces K Balachandran; K Uchiyama 2000-05-01 In this paper we prove the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using semigroup theory and the Schauder fixed point theorem. 13. On the mild solutions of higher-order differential equations in Banach spaces Directory of Open Access Journals (Sweden) Nguyen Thanh Lan 2003-08-01 Full Text Available For the higher-order abstract differential equation u(n(t=Au(t+f(t, tÃ¢ÂˆÂˆÃ¢Â„Â, we give a new definition of mild solutions. We then characterize the regular admissibility of a translation-invariant subspace Ã¢Â„Â³ of BUC(Ã¢Â„Â,E with respect to the above-mentioned equation in terms of solvability of the operator equation AXÃ¢ÂˆÂ’XÃ°ÂÂ’ÂŸn=C. As applications, periodicity and almost periodicity of mild solutions are also proved. 14. Existence and analyticity of a parabolic evolution operator for nonautonomous linear equations in Banach spaces Directory of Open Access Journals (Sweden) 2009-02-01 Full Text Available We give conditions for the parabolic evolution operator to be analytic with respect to a coefficient operator. We also show that the solution of a homogeneous parabolic evolution equation is analytic with respect to the coefficient operator and to the initial data. We apply our results to example that can not be studied by the standard methods. 15. The fixed point theorems of 1-set-contractive operators in Banach space Directory of Open Access Journals (Sweden) Wang Shuang 2011-01-01 Full Text Available Abstract In this paper, we obtain some new fixed point theorems and existence theorems of solutions for the equation Ax = μx using properties of strictly convex (concave function and theories of topological degree. Our results and methods are different from the corresponding ones announced by many others. MSC: 47H09, 47H10 16. Implicit Iteration Process for Common Fixed Points of Strictly Asymptotically Pseudocontractive Mappings in Banach Spaces Directory of Open Access Journals (Sweden) Chang Shih-sen 2008-01-01 Full Text Available Abstract In this paper, a new implicit iteration process with errors for finite families of strictly asymptotically pseudocontractive mappings and nonexpansive mappings is introduced. By using the iterative process, some strong convergence theorems to approximating a common fixed point of strictly asymptotically pseudocontractive mappings and nonexpansive mappings are proved. The results presented in the paper are new which extend and improve some recent results of Osilike et al. (2007, Liu (1996, Osilike (2004, Su and Li (2006, Gu (2007, Xu and Ori (2001. 17. Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces Directory of Open Access Journals (Sweden) Mouffak Benchohra 2012-01-01 Full Text Available The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness. 18. Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces Directory of Open Access Journals (Sweden) 2006-01-01 Full Text Available The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations. 19. Monotone iterative method for semilinear impulsive evolution equations of mixed type in Banach spaces OpenAIRE Pengyu Chen; Jia Mu 2010-01-01 We use a monotone iterative method in the presence of lower and upper solutions to discuss the existence and uniqueness of mild solutions for the initial value problem $$displaylines{ u'(t)+Au(t)= f(t,u(t),Tu(t)),quad tin J,; t eq t_k,cr Delta u \|_{t=t_k}=I_k(u(t_k)) ,quad k=1,2,dots ,m,cr u(0)=x_0, }$$ where $A:D(A)subset Eo E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)(tgeq 0)$ in $E$. Under wide monotonicity conditions and the non-compac... 20. [Research progress of rectoanal inhibitory reflex]. Science.gov (United States) Yin, Shuhui; Zhao, Ke 2015-12-01 The understanding of rectoanal inhibitory reflex (RAIR) is progressing for the latest 100 years. From the discovery of its important role in diagnosis of Hirschsprung's disease to all aspects of its development, reflex pathways, neural regulation and physiological functions, there have been more in-depth explorations. It is now recognized that a number of other diseases also have a more specific performance of RAIR. It has become an important and indispensable part to anorectal manometry. Research progress of rectoanal inhibitory reflex is reviewed in this article. PMID:26704013 1. New Molecular Knowledge Towards the Trigemino-Cardiac Reflex as a Cerebral Oxygen-Conserving Reflex OpenAIRE Sandu, N.; Spiriev, T.; Lemaitre, F; Filis, A.; Schaller, B. 2010-01-01 The trigemino-cardiac reflex (TCR) represents the most powerful of the autonomous reflexes and is a subphenomenon in the group of the so-called “oxygen-conserving reflexes”. Within seconds after the initiation of such a reflex, there is a powerful and differentiated activation of the sympathetic system with subsequent elevation in regional cerebral blood flow (CBF), with no changes in the cerebral metabolic rate of oxygen (CMRO2) or in the cerebral metabolic rate of glucose (CMRglc). Such an ... 2. Star product realizations of kappa-Minkowski space OpenAIRE Durhuus, Bergfinnur; Sitarz, Andrzej 2011-01-01 We define a family of star products and involutions associated with $\\kappa$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $\\kappa$-Minkowski space. Furthermore, we ... 3. Entrepreneurship Teaching Conducted as Strategic Reflexive Conversation DEFF Research Database (Denmark) Kristiansson, Michael The paper intends exploring and ascertaining whether the concept of strategic reflexive conversation can profitably be applied to entrepreneurship. As a start, a process of conceptualisation is undertaken, which is instrumental in placing the notion of strategic reflexive conversation into a...... knowledge management perspective. Strategic reflexive conversation is presented in an enhanced and updated version, which is contrasted to entrepreneurship through reflection. The findings indicate and it can be concluded that, with some important reservations, strategic reflexive conversation can...... advantageously, and on a pilot basis, be applied to entrepreneurship in practical environments and within the framework of entrepreneurship-centred teaching. The present theoretical investigation is solely of an introductory nature and steps are considered that can lead to the planning of additional exploratory... 4. The Multiple Faces of Reflexive Research Designs OpenAIRE Müller, Karl H. 2015-01-01 Reflexive research can be grouped into five clusters with circular relations between two elements x ↔ x, namely circular relations between observers, between scientific building blocks like concepts, theories or models, between systemic levels, between rules and rule systems or as circular relations or x ↔ y between these four components. By far the most important cluster is the second cluster which becomes reflexive through a re-entry operation RE into a scientific element x and which establ... 5. Weighted composition operators between weak spaces of vector-valued analytic functions OpenAIRE Hassanlou, Mostafa; Laitila, Jussi; VAEZI, Hamid 2014-01-01 We consider weighted composition operators on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of weighted composition operators on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of these operators between two different vector-valued Bloch-type spaces. This result appears to be ... 6. Infinitesimally Lipschitz functions on metric spaces OpenAIRE Durand-Cartagena, Estibalitz; Jaramillo Aguado, Jesús Ángel 2013-01-01 For a metric space $X$, we study the space $D^{\\infty}(X)$ of bounded functions on $X$ whose infinitesimal Lipschitz constant is uniformly bounded. $D^{\\infty}(X)$ is compared with the space $\\LIP^{\\infty}(X)$ of bounded Lipschitz functions on $X$, in terms of different properties regarding the geometry of $X$. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare $D^{\\infty}(X)$ with the Newtonian-Sobolev space $N^{1, \\infty}(X)$. In pa... 7. The passive of reflexive verbs in Icelandic Directory of Open Access Journals (Sweden) 2011-10-01 Full Text Available The Reflexive Passive in Icelandic is reminiscent of the so-called New Passive (or New Impersonal in that the oblique case of a passivized object NP is preserved. As is shown by recent surveys, however, speakers who accept the Reflexive Passive do not necessarily accept the New Passive, whereas conversely, speakers who accept the New Passive do also accept the Reflexive Passive. Based on these results we suggest that there is a hierarchy in the acceptance of passive sentences in Icelandic, termed the Passive Acceptability Hierarchy. The validity of this hierarchy is confirmed by our diachronic corpus study of open access digital library texts from Icelandic journals and newspapers dating from the 19th and 20th centuries (tímarit.is. Finally, we sketch an analysis of the Reflexive Passive, proposing that the different acceptability rates of the Reflexive and New Passives lie in the argument status of the object. Simplex reflexive pronouns are semantically dependent on the verbs which select them, and should therefore be analyzed as syntactic arguments only, and not as semantic arguments of these verbs. 8. Next generation control system for reflexive aerostructures Science.gov (United States) Maddux, Michael R.; Meents, Elizabeth P.; Barnell, Thomas J.; Cable, Kristin M.; Hemmelgarn, Christopher; Margraf, Thomas W.; Havens, Ernie 2010-04-01 Cornerstone Research Group Inc. (CRG) has developed and demonstrated a composite structural solution called reflexive composites for aerospace applications featuring CRG's healable shape memory polymer (SMP) matrix. In reflexive composites, an integrated structural health monitoring (SHM) system autonomously monitors the structural health of composite aerospace structures, while integrated intelligent controls monitor data from the SHM system to characterize damage and initiate healing when damage is detected. Development of next generation intelligent controls for reflexive composites were initiated for the purpose of integrating prognostic health monitoring capabilities into the reflexive composite structural solution. Initial efforts involved data generation through physical inspections and mechanical testing. Compression after impact (CAI) testing was conducted on composite-reinforced shape memory polymer samples to induce damage and investigate the effectiveness of matrix healing on mechanical performance. Non-destructive evaluation (NDE) techniques were employed to observe and characterize material damage. Restoration of mechanical performance was demonstrated through healing, while NDE data showed location and size of damage and verified mitigation of damage post-healing. Data generated was used in the development of next generation reflexive controls software. Data output from the intelligent controls could serve as input to Integrated Vehicle Health Management (IVHM) systems and Integrated Resilient Aircraft Controls (IRAC). Reflexive composite technology has the ability to reduce maintenance required on composite structures through healing, offering potential to significantly extend service life of aerospace vehicles and reduce operating and lifecycle costs. 9. Symmetries and constraints in classical and quantum mechanics: Lie-Jordan Banach algebras and their applications OpenAIRE Ferro, Leonardo 2013-01-01 El objetivo principal de esta disertación es el estudio de la teoría de las álgebras de Lie–Jordan Banach y su papel en el marco de la mecánica clásica y cuántica y sus aplicaciones en diferentes ámbitos de las matemáticas. La investigación algebraica de los fundamentos de la física surgió de la búsqueda de una formulación “excepcional” para la mecánica cuántica, que finalmente culminó en la formulación de las álgebras de Jordan y las posteriores C∗–álgebras............. 10. Intuitionistic fuzzy {\\psi}-{\\phi}-contractive mappings and fixed point theorems in non-Archimedean intuitionistic fuzzy metric spaces OpenAIRE Dinda, Bivas; T. K. SAMANTA; Jebril, Iqbal H. 2011-01-01 In this paper intuitionistic fuzzy {\\psi}-{\\phi}-contractive mappings are introduced. Intuitionistic fuzzy Banach contraction theorem for M-complete non-Archimedean intuitionistic fuzzy metric spaces and intuitionistic fuzzy Elelstein contraction theorem for non-Archimedean intuitionistic fuzzy metric spaces by intuitionistic fuzzy {\\psi}-{\\phi}-contractive mappings are proved. 11. Moduli of convexity and smoothness of reflexive subspaces of L^1 OpenAIRE Lajara, S.; Pallares, A.; Troyanski, S. 2011-01-01 We show that for any probability measure \\mu there exists an equivalent norm on the space L^1(\\mu) whose restriction to each reflexive subspace is uniformly smooth and uniformly convex, with modulus of convexity of power type 2. This renorming provides also an estimate for the corresponding modulus of smoothness of such subspaces. 12. The trigeminocardiac reflex – a comparison with the diving reflex in humans Science.gov (United States) Lemaitre, Frederic; Schaller, Bernhard 2015-01-01 The trigeminocardiac reflex (TCR) has previously been described in the literature as a reflexive response of bradycardia, hypotension, and gastric hypermotility seen upon mechanical stimulation in the distribution of the trigeminal nerve. The diving reflex (DR) in humans is characterized by breath-holding, slowing of the heart rate, reduction of limb blood flow and a gradual rise in the mean arterial blood pressure. Although the two reflexes share many similarities, their relationship and especially their functional purpose in humans have yet to be fully elucidated. In the present review, we have tried to integrate and elaborate these two phenomena into a unified physiological concept. Assuming that the TCR and the DR are closely linked functionally and phylogenetically, we have also highlighted the significance of these reflexes in humans. PMID:25995761 13. Common subspaces of $L_{p}$-spaces CERN Document Server Koldobsky, A B 1992-01-01 For $n\\geq 2, p2,$ does there exist an $n$-dimensional Banach space different from Hilbert spaces which is isometric to subspaces of both $L_{p}$ and $L_{q}$? Generalizing the construction from the paper "Zonoids whose polars are zonoids" by R.Schneider we give examples of such spaces. Moreover, for any compact subset $Q$ of $(0,\\infty)\\setminus \\{2k, k\\in N\\},$ we can construct a space isometric to subspaces of $L_{q}$ for all $q\\in Q$ simultaneously. This paper requires vanilla.sty 14. Generalized Kӧthe $p$-dual spaces OpenAIRE Galdames Bravo, O. 2014-01-01 Let us consider a Banach function space $X$. The Kӧthe dual space can be characterized as the space of multipliers from $X$ to $L^1$. We extend this characterization to the space of multipliers from $X$ to $L^p$ in order to define the Kӧthe $p$-dual space of $X$. We analyze the properties of this space so as to use it as a tool for studying $p$-th power factorable operators. In particular, we compute $q$-concavity for these spaces and type and cotype when $X$ is an AM-spa... 15. Automorphisms in spaces of continuous functions on Valdivia compacta CERN Document Server Avilés, Antonio 2009-01-01 We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(I). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(I), all isomorphism between subspaces of C(K) of size less than aleph_omega extend to automorphisms of C(K). 16. Snakes and articulated arms in an Hilbert space CERN Document Server Pelletier, Fernand 2011-01-01 The purpose of this paper is to give an illustration of results on integrability of distributions and orbits of vector fields on Banach manifolds obtained in [Pe] and [LaPe]. Using arguments and results of these papers, in the context of a separable Hilbert space, we give a generalization of a Theorem of accessibility contained in [Ha], [Ro] and proved for a finite dimensional Hilbert space 17. Stochastic processes on non-Archimedean spaces. II. Stochastic antiderivational equations OpenAIRE Ludkovsky, S. V. 2001-01-01 Stochastic antiderivational equations on Banach spaces over local non-Archimedean fields are investigated. Theorems about existence and uniqiuness of the solutions are proved under definite conditions. In particular Wiener processes are considered in relation with the non-Archimedean analog of the Gaussian measure. 18. Boundedness and invertibility of multidimensional integral operators with anisotropically homogeneous kernels in weighted Lp-spaces Science.gov (United States) Elena, Miroshnikova 2014-12-01 In weighted space Lp (Rn,?), 1 < p < ?, a new broad class of integral operators with anisotropically homogeneous kernels is investigated. For such operators boundedness theorem is proved. Also the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type is considered. For elements of the algebra symbolic calculation is constructed, the invertibility criterion is obtained in terms of operator symbol. 19. An improvement of dimension-free Sobolev imbeddings in r spaces Czech Academy of Sciences Publication Activity Database Fiorenza, A.; Krbec, Miroslav; Schmeisser, H.-J. 2014-01-01 Roč. 267, č. 1 (2014), s. 243-261. ISSN 0022-1236 R&D Projects: GA ČR GAP201/10/1920 Institutional support: RVO:67985840 Keywords : imbedding theorem * small Lebesgue space * rearrangement-invariant Banach Subject RIV: BA - General Mathematics Impact factor: 1.322, year: 2014 http://www.sciencedirect.com/science/article/pii/S0022123614001724 20. Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces II CERN Document Server Zhang, Shun 2011-01-01 We determine the sharp asymptotic degree of the Gelfand and Kolmogorov numbers of compact embeddings between weighted function spaces of Besov and Triebel-Lizorkin type with polynomial weights in the non-limiting case. This complements our previous results in the context of the quasi-Banach setting, $0 < p, q \\le \\infty$. In addition the comparisons with corresponding approximation numbers are made. 1. Reflexive structures an introduction to computability theory CERN Document Server Sanchis, Luis E 1988-01-01 Reflexive Structures: An Introduction to Computability Theory is concerned with the foundations of the theory of recursive functions. The approach taken presents the fundamental structures in a fairly general setting, but avoiding the introduction of abstract axiomatic domains. Natural numbers and numerical functions are considered exclusively, which results in a concrete theory conceptually organized around Church's thesis. The book develops the important structures in recursive function theory: closure properties, reflexivity, enumeration, and hyperenumeration. Of particular interest is the treatment of recursion, which is considered from two different points of view: via the minimal fixed point theory of continuous transformations, and via the well known stack algorithm. Reflexive Structures is intended as an introduction to the general theory of computability. It can be used as a text or reference in senior undergraduate and first year graduate level classes in computer science or mathematics. 2. Intrathecal baclofen and the H-reflex. OpenAIRE Macdonell, R A; Talalla, A; SWASH, M.; D.Grundy 1989-01-01 Baclofen was given intrathecally to six patients with severe lower limb spasticity due to traumatic spinal cord injury. The effects of the drug on spasticity and the ratio between the maximum amplitude of the H reflex and the M response from the soleus (Hmax/Mmax ratio) were assessed. In each patient, spasticity was reduced following intrathecal baclofen and in four patients there was a reduction in the amplitude of the H reflex and Hmax/Mmax ratio. These results suggest that the Hmax/Mmax ra... 3. Reflex control for safe autonomous robot operation International Nuclear Information System (INIS) This paper describes the design of an autonomous, sonar-based world mapping system for collision prevention in robotic systems. Obstacle detection and mapping is performed as a task that competes with higher-level tasks for the robot's attention. All tasks are integrated within a hierarchy, organized and co-ordinated by schemes analogous to biological reflexes and fixed action patterns. It is illustrated how the existence of low-level reflex behaviours can enhance the survivability and autonomy of complex systems and simplify the design of complex higher-level controls like our autonomous sonar-based world mapping system 4. The Reflexive Teacher Educator in TESOL Roots and Wings CERN Document Server Edge, Julian 2010-01-01 Edge explores the construct of reflexivity in teacher education, differentiating it from, while locating it in, reflective practice, and introduces a framework (Copying, Applying, Theorising, Reflecting, Acting) to help teacher educators become reflexive professionals. 5. Unconditionally convergent series in the space C(Q) International Nuclear Information System (INIS) Let B be a Banach space and B* its dual Banach space. B contains csub(0) (B does not contain csub(0)) if B contains (does not contain) a subspace isomorphic to the space csub(0) of sequences of numbers tending to zero. The series ?sub(n=1)sup(infinity) xsub(n) of elements of B is weakly unconditionally convergent (w.u.c.) iff ?sub(n=1)sup(infinity)\|x*(xsub(n))\| 0. Series of elements of C(Q) are considered here. Subspaces of C(Q) isomorphic to c0 are constructed, and criteria for a series of elements of C(Q) to be w.u.c. or u.c. are given. Finally, an improved theorem of giving characterizations of the elements of subalgebras of C(Q) not containing c0 is presented 6. Nasal Reflexes: Implications for Exercise, Breathing, and Sex OpenAIRE James N. BARANIUK; Merck, Samantha J. 2008-01-01 Nasal patency, with both congestion and decongestion, is affected in a wide variety of reflexes. Stimuli that lead to nasal reflexes include exercise, alterations of body position, pressure, and temperature, neurological syndromes, and dentists. As anticipated, the vagal and trigeminal systems are closely integrated through nasobronchial and bronchonasal reflexes. However, perhaps of greater pathophysiological importance are the naso-hypopharyngea-laryngeal reflexes that become aggravated dur... 7. Anatomy and neuro-pathophysiology of the cough reflex arc OpenAIRE Polverino Mario; Polverino Francesca; Fasolino Marco; Andò Filippo; Alfieri Antonio; De Blasio Francesco 2012-01-01 Abstract Coughing is an important defensive reflex that occurs through the stimulation of a complex reflex arc. It accounts for a significant number of consultations both at the level of general practitioner and of respiratory specialists. In this review we first analyze the cough reflex under normal conditions; then we analyze the anatomy and the neuro-pathophysiology of the cough reflex arc. The aim of this review is to provide the anatomic and pathophysiologic elements of evaluation of the... 8. Coorbit description and atomic decomposition of Besov spaces CERN Document Server Christensen, Jens Gerlach; Olafsson, Gestur 2011-01-01 Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gr\\"ochenig and then later extended in [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of [3] and use this to derive atomic decompositions for the Besov spaces. 9. Reflexive Planning as Design and Work DEFF Research Database (Denmark) Lissandrello, Enza; Grin, John 2011-01-01 In recent years, planning theorists have advanced various interpretations of the notion of reflexivity, inspired by American pragmatism, complexity theory, hermeneutics, discursive and collaborative planning. Scholars agree that “reflexivity” has a strong temporal dimension: it not only aims to... 10. Dilemmas and Deliberations in Reflexive Ethnographic Research Science.gov (United States) Robinson, Janean Valerie 2014-01-01 This paper traces insights into the challenges and dilemmas experienced whilst researching students' interpretations and understandings of the Behaviour Management in Schools policy in Western Australia. Journal records, supported by student transcripts, are woven together in a reflexive ethnographic journey--from the beginning phase of… 11. Biological Motion Cues Trigger Reflexive Attentional Orienting Science.gov (United States) Shi, Jinfu; Weng, Xuchu; He, Sheng; Jiang, Yi 2010-01-01 The human visual system is extremely sensitive to biological signals around us. In the current study, we demonstrate that biological motion walking direction can induce robust reflexive attentional orienting. Following a brief presentation of a central point-light walker walking towards either the left or right direction, observers' performance… 12. Atropine treatment of reflex anoxic seizures. OpenAIRE McWilliam, R C; Stephenson, J B 1984-01-01 In 7 children with unusually severe or frequent reflex anoxic seizures atropine treatment, which was well tolerated, reduced seizure frequency by a mean value of 98%. Treatment withdrawal (five patients) was followed by an increase in seizure frequency and reintroduction (three patients) by restoration of control. 13. Self hypnotherapeutic treatment of habitual reflex vomiting. OpenAIRE Sokel, B S; Devane, S P; Bentovim, A; Milla, P. J. 1990-01-01 A 9 year old boy with intractable postprandial reflex vomiting was taught a self hypnotherapy technique incorporating relaxation exercises, mental imagery, and suggestions of symptom relief. The sequence was recorded on a personal stereo cassette tape. Vomiting was completely eliminated within four weeks. At 12 month review vomiting had not recurred. 14. A reflexive perspective in problem solving Directory of Open Access Journals (Sweden) Chio, José Angel 2013-01-01 Full Text Available The objective of this paper is to favour the methodological process of reflexive analysis in problem solving in the general teaching methods that concentrates in strengthening the dimensional analysis, to gain a greater preparation of the students for the solution of mathematical problems. 15. Vestibulospinal control of reflex and voluntary head movement Science.gov (United States) Boyle, R.; Peterson, B. W. (Principal Investigator) 2001-01-01 Secondary canal-related vestibulospinal neurons respond to an externally applied movement of the head in the form of a firing rate modulation that encodes the angular velocity of the movement, and reflects in large part the input "head velocity in space" signal carried by the semicircular canal afferents. In addition to the head velocity signal, the vestibulospinal neurons can carry a more processed signal that includes eye position or eye velocity, or both (see Boyle on ref. list). To understand the control signals used by the central vestibular pathways in the generation of reflex head stabilization, such as the vestibulocollic reflex (VCR), and the maintenance of head posture, it is essential to record directly from identified vestibulospinal neurons projecting to the cervical spinal segments in the alert animal. The present report discusses two key features of the primate vestibulospinal system. First, the termination morphology of vestibulospinal axons in the cervical segments of the spinal cord is described to lay the structural basis of vestibulospinal control of head/neck posture and movement. And second, the head movement signal content carried by the same class of secondary vestibulospinal neurons during the actual execution of the VCR and during self-generated, or active, rapid head movements is presented. 16. Feminist Politics as Reflexive Citizenship Directory of Open Access Journals (Sweden) Nikos Katrivesis 2008-11-01 Full Text Available This article examines the potential feminist contribution to a radically renovated citizenship, according to which we all ought to undertake our responsibilities towards family, society and world. This contribution mainly consists in the critical transgression of the traditional dualism between private and public space, which arises from the politicization of everyday experience and results to the emergence of an essentially participial civil society. 17. Functional calculus on real interpolation spaces for generators of $C_{0}$-groups OpenAIRE Haase, Markus; Rozendaal, Jan 2014-01-01 We study functional calculus properties of $C_{0}$-groups on real interpolation spaces, using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded $H^{\\infty}_{1}$-calculus on real interpolation spaces. Additional results are derived from this. 18. Multiple Solutions of Boundary Value Problems for th-Order Singular Nonlinear Integrodifferential Equations in Abstract Spaces OpenAIRE Yanlai Chen; Tingqiu Cao; Baoxia Qin 2015-01-01 The authors discuss multiple solutions for the nth-order singular boundary value problems of nonlinear integrodifferential equations in Banach spaces by means of the fixed point theorem of cone expansion and compression. An example for infinite system of scalar third-order singular nonlinear integrodifferential equations is offered. 19. Fixed point theory in distance spaces CERN Document Server Kirk, William 2014-01-01 This is a monograph on fixed point theory, covering the purely metric aspects of the theoryparticularly results that do not depend on any algebraic structure of the underlying space. Traditionally, a large body of metric fixed point theory has been couched in a functional analytic framework. This aspect of the theory has been written about extensively. There are four classical fixed point theorems against which metric extensions are usually checked. These are, respectively, the Banach contraction mapping principal, Nadlers well known set-valued extension of that theorem, the extension of Banachs theorem to nonexpansive mappings, and Caristis theorem. These comparisons form a significant component of this book. This book is divided into three parts. Part I contains some aspects of the purely metric theory, especially Caristis theorem and a few of its many extensions. There is also a discussion of nonexpansive mappings, viewed in the context of logical foundations. Part I also contains certain re... 20. Coorbit description and atomic decomposition of Besov spaces OpenAIRE Christensen, Jens Gerlach; Mayeli, Azita; Olafsson, Gestur 2011-01-01 Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gr\\"ochenig and then later extended in [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article we identify the homogeneous Besov spaces on stratified Lie groups introduced in [13] as coorbit spaces in the sense of [3] and use this to deri...	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8836637735366821, "perplexity": 1598.4670173994753}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471983019893.83/warc/CC-MAIN-20160823201019-00154-ip-10-153-172-175.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/43222/convex-hull-problem-with-a-twist/147851	# Convex hull problem with a twist I have a 2D set and would like to determine from them the subset of points which, if joined together with lines, would result in an edge below which none of the points in the set exist. This problem resembles the convex hull problem, but is fundamentally different in its definition. One approach to determine these points might be to evaluate the cross-product of only x_1, x_2 and x_3, where x_1 is on the 'hull', x_2's 'hull'-ness is being evaluated and x_3 is another point on the set (all other points in the set should yield positive cross products if x_2 is indeed on the hull), with the additional constraint that x_1 < x_2 in one dimension. I realize that this algorithm is not entirely perfect; the plot below shows that some valid points would be missed as a result of the convex hull constraint. How else can I define this edge? Hope the question is clear. - You want an edge or set of edges? – Aryabhata Jun 4 '11 at 17:40 I think you have not clearly stated your problem. So there is given a finite set $A$ of points in the plane. You want a subset which is somehow lying on the bottom (with respect to $y$). Such a subset will consist of of $>2$ points, so how can it define "a line"? Maybe this "line" is in fact a polygonal arc - why should all this be "fundamentally different" from searching for the convex hull of $A$? – Christian Blatter Jun 4 '11 at 21:06 This is called the "lower convex hull" (do a Google search). Many convex hull algorithms separately compute it and the upper convex hull to create the convex hull. – whuber Jun 4 '11 at 21:52 @Cristian : I am after a polygonal arc here. – Zaid Jun 5 '11 at 5:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8347459435462952, "perplexity": 562.9303734801285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396949.33/warc/CC-MAIN-20160624154956-00015-ip-10-164-35-72.ec2.internal.warc.gz"}
http://harvard.voxcharta.org/2012/04/26/a-new-mhd-code-with-adaptive-mesh-refinement-and-parallelization-for-astrophysics/	A new code, named MAP, is written in Fortran language for magnetohydrodynamics (MHD) calculation with the adaptive mesh refinement (AMR) and Message Passing Interface (MPI) parallelization. There are several optional numerical schemes for computing the MHD part, namely, modified Mac Cormack Scheme (MMC), Lax-Friedrichs scheme (LF) and weighted essentially non-oscillatory (WENO) scheme. All of them are second order, two-step, component-wise schemes for hyperbolic conservative equations. The total variation diminishing (TVD) limiters and approximate Riemann solvers are also equipped. A high resolution can be achieved by the hierarchical block-structured AMR mesh. We use the extended generalized Lagrange multiplier (EGLM) MHD equations to reduce the non-divergence free error produced by the scheme in the magnetic induction equation. The numerical algorithms for the non-ideal terms, e.g., the resistivity and the thermal conduction, are also equipped in the MAP code. The details of the AMR and MPI algorithms are described in the paper.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9434159398078918, "perplexity": 880.9203906549911}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510270313.12/warc/CC-MAIN-20140728011750-00404-ip-10-146-231-18.ec2.internal.warc.gz"}
http://jdh.hamkins.org/tag/od/	# When does every definable set have a definable member? CUNY Set Theory Seminar, October 2014 This will be a talk for the CUNY set theory seminar, October 10, 2014, 12pm GC 6417. Abstract. Although the concept of `being definable’ is not generally expressible in the language of set theory, it turns out that the models of ZF in which every definable nonempty set has a definable element are precisely the models of V=HOD. Indeed, V=HOD is equivalent to the assertion merely that every $\Pi_2$-definable set has an ordinal-definable element. Meanwhile, this is not true in the case of $\Sigma_2$-definability, because every model of ZFC has a forcing extension satisfying $V\neq\text{HOD}$ in which every $\Sigma_2$-definable set has an ordinal-definable element. This is joint work with François G. Dorais and Emil Jeřábek, growing out of some questions and answers on MathOverflow, namely, Definable collections without definable members A question asked by Ashutosh five years ago, in which François and I gradually came upon the answer together. Is it consistent that every definable set has a definable member? A similar question asked last week by (anonymous) user38200 Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable member? A question I had regarding the limits of an issue in my answer to the previous question. In this talk, I shall present the answers to all these questions and place the results in the context of classical results on definability, including a review of basic concepts for graduate students. # Algebraicity and implicit definability in set theory, CUNY, May 2013 This is a talk May 10, 2013 for the CUNY Set Theory Seminar. Abstract. An element a is definable in a model M if it is the unique object in M satisfying some first-order property. It is algebraic, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b \| M satisfies φ[b] } is a finite set containing a. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interesting new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open. This is joint work with Cole Leahy (MIT).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8646356463432312, "perplexity": 472.61380677166477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256812.65/warc/CC-MAIN-20190522123236-20190522145236-00306.warc.gz"}
http://math.stackexchange.com/questions/258239/probability-of-a-large-sub-sequence-within-a-huge-sequence	# probability of a large sub-sequence within a huge sequence You toss a fair coin one million times. What is the probability of getting at least one sequence of six heads followed by six tails? - Here's a way to get a ballpark estimate. You know how to get the probability that tosses $1$ through $12$ don't give you six heads followed by six tails. Let's call it $q$. The probability that tosses $13$ through $24$ don't give that result is also $q$. So is the probability for tosses $25$ through $36$. And so on. So the probability that none of these sequences of $12$ gives a $6,6$ is $q^{[100000/12]}$. That's a bit of an over estimate of the chances of not getting a $6,6$, since it doesn't account for, say, tosses $2$ through $13$. For practical purposes, it might be good enough to divide by $12$ to get an estimate of the probability that you never get your $6,6$. Here's another idea. The probability you get your $6,6$ in $1000000$ tosses is the probability you get it in $999999$ tosses, plus the product of the probability you don't get it in $999999$ tosses and the probability that you do get it in the last $12$ tosses. This lets you set up a recurrence equation, which maybe you can solve. - thanks for ideas – alexandreC Dec 14 '12 at 13:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9134290814399719, "perplexity": 124.82280717149766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657137906.42/warc/CC-MAIN-20140914011217-00282-ip-10-234-18-248.ec2.internal.warc.gz"}
http://mathhelpforum.com/algebra/96920-pri-5-maths-percentage-question-1-a.html	# Math Help - Pri 5 maths percentage question 1 1. ## Pri 5 maths percentage question 1 Mrs Wong bought some flour.She used 3/8 of it to bake some cakes and kept the rest.What percentage of the flour did she keep? 2. If Mrs. Wong had a whole package of flour (1, or 8/8), and she used 3/8 of it, how much is left over? Find that answer, and then convert it to a percent. I think that you should memorize the conversion between certain fractions and percents, like fractions with an 8 as the denominator. But if don't have them memorized at the moment, then set up a proportion. For example, if I want to convert 1/8 to percent, I would write $\frac{1}{8} = \frac{x}{100}$ and solve for x by cross-multiplying: $8x = 100$ $x = 12.5$ So 1/8 = 12.5%. 01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9810068607330322, "perplexity": 2376.0914022199527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999676149/warc/CC-MAIN-20140305060756-00015-ip-10-183-142-35.ec2.internal.warc.gz"}
http://tex.stackexchange.com/questions/20767/adding-full-page-illustrations-before-chapters-in-classic-thesis	# Adding full-page illustrations before chapters in classic-thesis I use classic-thesis, and I would like to do the following: each chapter should begin on an odd page (right on a dual-page view), and on the left page should be displayed a picture I choose. would that be possible? thank you EDIT: with classicthesis, this will put title pages on the right pages instead of the left : \documentclass[10pt, a4paper,twoside,openright,titlepage]{scrreprt} \makeatletter \newcommand\ChapImage{\huge image page -- should be on the left!} \newcommand\Invcleardoublepage{ \clearpage\if@twoside \ifodd\c@page \null\newpage\if@twocolumn\null% \newpage\fi\fi\fi } \renewcommand\chapter{% \if@openright \ifodd\c@page{\clearpage}\else{\Invcleardoublepage}\fi { \pagestyle{empty} \ChapImage \clearpage } \else \clearpage \fi \thispagestyle{\chapterpagestyle}% \global\@topnum\z@ \@afterindentfalse \secdef\@chapter\@schapter } \makeatother \usepackage[T1]{fontenc} % la codifica dei font \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{changepage,calc} % per impostare i margini del frontespizio \usepackage{lipsum} % genera testo fittizio \usepackage{classicthesis-ldpkg} % carica molti pacchetti utili a ClassicThesis \usepackage[eulerchapternumbers,% % numeri dei capitoli in Euler subfig,% % compatibilità con subfig beramono,% % Bera Mono come font a spaziatura fissa eulermath,% % AMS Euler come font per la matematica pdfspacing% % migliora il riempimento di riga con PDFLaTeX ]{classicthesis} % lo stile ClassicThesis %\usepackage[english]{arsclassica} % modifica alcuni aspetti di ClassicThesis \begin{document} \pagestyle{plain} \begin{titlepage} \changetext{}{}{}{((\paperwidth - \textwidth) / 2) - \oddsidemargin - \hoffset - 1in}{} \null\vfill \large \sffamily {Title page} \vfill \end{titlepage} \pagenumbering{roman} \clearpage \lipsum[1-4] \clearpage \setcounter{tocdepth}{2} \thispagestyle{empty} \tableofcontents \markboth{\spacedlowsmallcaps{\contentsname}}{\spacedlowsmallcaps{\contentsname}} \pagenumbering{arabic} \chapter{First chapter} \lipsum[1-10] \chapter{Second chapter} \lipsum[1-8] \chapter{Third chapter} \lipsum[1-10] \chapter{Fourth chapter} \lipsum[1-8] \end{document} - maybe you could try to redefine \chapter to, instead of doing a \cleardoublepage, do a \clearpage\thispagestyle{empty}\includegraphics[<options>]{path/to/image}, but this is just a long shot – henrique Jun 15 '11 at 1:00 There is a problem: if the previous chapter ends in an even page there's no place for the picture! In this case one should add a blank page, the picture and finally the chapter! – Spike Jun 15 '11 at 7:17 I think there's a confusion here: what exactly do you mean with odd page? I assumed you meant an odd-numbered page. – Gonzalo Medina Jun 16 '11 at 20:41 The basic idea is to redefine \chapter as defined in the corresponding .cls file. I used scrbook.cls, but the modifications needed are clear mutatis mutandis for other classes). To guarantee that all chapters will begin in an odd numbered page and that the images get included in the (even numbered) page before the beginning of the chapter, I used an auxiliary command \Invcleardoublepage that acts as an "inverse" to \cleardoublepage: it flushes all material and starts a new page, but starts in a new even numbered page. Finally, with the help of the xparse package I defined a \MyChapter command with one optional arguments (the entry for the ToC) and two mandatory arguments (the title of the chapter, and the name of the image to be used). \documentclass[11pt,a5paper,footinclude=true,headinclude=true]{scrbook} \usepackage{xparse} \usepackage{lipsum} \usepackage{graphicx} \newcommand\ChapImage{} \makeatletter % A command that acts as an "inverse" cleardoublepage: % flush all material and start a new page, start new even numbered page \newcommand\Invcleardoublepage{\clearpage\if@twoside \ifodd\c@page \null\newpage\if@twocolumn\null% \newpage\fi\fi\fi } \renewcommand\chapter{% \if@openright \ifodd\c@page\clearpage\else\Invcleardoublepage\fi% NEW \thispagestyle{empty}\ChapImage\clearpage \else\clearpage \fi \thispagestyle{\chapterpagestyle}% \global\@topnum\z@ \@afterindentfalse \secdef\@chapter\@schapter } \makeatother \DeclareDocumentCommand\MyChapter{omm}{% \renewcommand\ChapImage{\includegraphics[width=.95\textwidth,height=.95\textheight]{#3}} \IfNoValueTF{#1} {\chapter{#2}}{\chapter[#1]{#2}} } \begin{document} \tableofcontents \MyChapter[Entry in ToC]{Test Chapter with an Image}{image1} \lipsum[1-3] \MyChapter{Another Test Chapter with another Image}{image3} \lipsum[1-3] \end{document} I used the demo option for graphicx to make my example compilable for everyone, do not use that option in your actual code. If the document uses \part, then the following redefinition (or a similar one if the class is not scrbook) should also be added to the preamble: \renewcommand\part{% \if@openright \ifodd\c@page\clearpage\else\Invcleardoublepage\fi% NEW \else\clearpage \fi \thispagestyle{\partpagestyle}% \if@twocolumn \onecolumn \@tempswatrue \else \@tempswafalse \fi \vbox to\z@{\vss\use@preamble{part@o}\strut\par}% \vskip-\baselineskip\nobreak% \secdef\@part\@spart } - Thank you! But when I do this with classic thesis, it does not work properly: if ChapImage is not defined, it correctly displays a blank page at the left and the chapter page at the right, but if I define ChapImage, it shows a blank page at the left, the image at the right, then it starts the chapter on the next left page (perhaps because the chapter command has been redefined in classic-thesis). Do you have some idea what the problem might be? – oulipo Jun 15 '11 at 6:51 And why was the redefinition of part needed? (I don't use parts in my manuscript) – oulipo Jun 15 '11 at 9:04 @oulipo: Please add to your question a minimal version of your document showing that my solution doesn't work as expected: as you can see, my example code do works and it uses the classicthesis package. And I redefined \part because I couldn't possible know if you would use it or not. – Gonzalo Medina Jun 15 '11 at 11:36 @oulipo: I improved the code. There's no need to redefine commands anymore, and the new definition is more flexible. – Gonzalo Medina Jun 16 '11 at 20:12 Thank you! – oulipo Jun 17 '11 at 8:34 I would define a command that would call both the image as well as the chapter, \documentclass[oneside]{book} \usepackage[demo]{graphicx} \usepackage{lipsum} \pagestyle{empty} \makeatletter \def\chapter{\clearpage\thispagestyle{plain}\global\@topnum \z@\@afterindentfalse \secdef\@chapter\@schapter } \makeatother \title{My Picture Chapters} \begin{document} \maketitle \newcommand{\chapterwithpic}[3][]{% \includegraphics[width=1.2\textwidth]{./graphics/#3} \chapter[#1]{#2} } \chapterwithpic[option]{Amato}{pic} \lipsum \end{document} - +1 but I think it is better to make the optional argument as #1 as in \newcommand{\MyChapterPic}[3][]{...}. – xport Jun 15 '11 at 3:21 @xport ... agreed, won't you please edit my answer? I am off to work. – Yiannis Lazarides Jun 15 '11 at 3:47 @Yiannis: well done. – xport Jun 15 '11 at 4:13 Thank you, but actually this won't show the picture on the left page and the chapter on the right page since the chapter command will do a cleardoublepage before – oulipo Jun 15 '11 at 6:42 @Yiannis: but, using your code, there's no guarantee that every chapter will begin in an odd-numbered page. – Gonzalo Medina Jun 16 '11 at 19:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8964542746543884, "perplexity": 3299.14413294088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657132025.91/warc/CC-MAIN-20140914011212-00033-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://mathhelpforum.com/calculus/10556-series-converge-absolutely-conditionally.html	for the series, converge absolutely, conditionally.. Determin for the series if it converge absolutely, converge conditionally, orit diverges? equation: http://img219.imageshack.us/img219/8494/untitled3xa.jpg Need plenty of help for these type of problems. Thanks.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9867389798164368, "perplexity": 1470.5830817182234}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00577-ip-10-147-4-33.ec2.internal.warc.gz"}
http://slideplayer.com/slide/3501516/	Stoichiometry of Precipitation Reactions Presentation on theme: "Stoichiometry of Precipitation Reactions"— Presentation transcript: Stoichiometry of Precipitation Reactions 1. Identify the species present in the combined solution, and determine what reaction occurs. 2. Write the balanced net ionic equation for the reaction. 3. Calculate the moles of reactants. 4. Determine which reactant is limiting. 5. Calculate the moles of reactants or products, as required. 6. Convert to grams or other units, as required. Determining the Mass of Products Formed Calculate the mass of solid NaCl that must be added to 1.50 L of a M AgNO3 solution to precipitate all the Ag+ ions in the form of AgCl. Acid-Base Reactions The Bronsted-Lowry Definitions: An acid is a proton donor. A base is a proton acceptor. Arrhenius definitions: Acid generate protons (hydronium) Bases generate hydroxide Lewis definitions Acids are electron pair acceptors Bases are electron pair donors Performing Calculations for Acid-Base Reactions 1.List the species present in the combined solution before any reaction occurs, and decide what reaction will occur. 2. Write the balance net ionic equation for this reaction. 3. Calculate the moles of reactants. For reactions in solution, use the volumes of the original solutions and their molarities. 4. Determine the limiting reactant. 5. Calculate the moles of the required reactant and product. 6. Convert to grams or volume (of solution), as required. Example Problem What volume of a M HCl solution is needed to neutralize 25.0 mL of M NaOH? In a certain experiment, 28.0 mL of M HNO3 and 53.0 mL of M KOH are mixed. Calculate the amount of water formed in the resulting reaction. What is the concentration of H+ or OH- ions in excess after the reaction goes to completion? Acid-Base Titrations Volumetric analysis is a technique for determining the amount of a certain substance by doing a titration. A titration is a process in which a solution of known concentration is used to determine the concentration of another solution through a monitored reaction. The equivalence point is reached when all of the moles of H+ ions present in the original volume of acid solution have reacted with an equivalent number of moles of OH- ions added. This is also known as the stoichiometric point. Titration An acid-base indicator is a chemical that changes color once the equivalence point is reached. The endpoint of a titration is the point in a titration where the indicator actually changes color. Three requirements for a titration 1. The exact reaction between titrant (known solution) and analyte (unknown solution or one you are analyzing) must be known (and rapid). Three requirement (cont.) 2. The stoichiometric (equivalence) point must be marked accurately. 3. The volume of titrant required to reach the stoichiometric point must be known accurately. Problem solving The first step in the analysis of a complex solution is to write down the components and focus on the chemistry of each one. Take a big problem and look at the small problems within. Neutralization Titration A student carries out an experiment to standardize (determine the exact concentration of) a sodium hydroxide solution. To do this, the student weighs out a g sample of potassium hydrogen phthalate (KHC8H4O4, often abbreviated KHP. KHP (molar mass g/mol) has one acidic hydrogen. The student dissolves the KHP in distilled water, adds phenolphthalein as an indicator, and titrates the resulting solution with the sodium hydroxide solution to the phenolphthalein endpoint. The difference between the final and initial buret readings indicates that mL of the sodium hydroxide solution is required to react exactly with the g KHP. Calculate the concentration of the sodium hydroxide solution. Neutralization Analysis An environmental chemist analyzed the effluent (the released waste material) from an industrial process known to produce the compounds carbon tetrachloride (CCl4) and benzoic acid (HC7H5O2), a weak acid that has one acidic hydrogen atom per molecule. A sample of this effluent weighing g was shaken with water, and the resulting aqueous solution required mL of M NaOH for neutralization. Calculate the mass percent of HC7H5O2 in the original sample.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8946701288223267, "perplexity": 3541.4543139321913}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888302.37/warc/CC-MAIN-20180119224212-20180120004212-00241.warc.gz"}
https://www.groundai.com/project/dizzyrnn-reparameterizing-recurrent-neural-networks-for-norm-preserving-backpropagation/	DizzyRNN: Reparameterizing Recurrent Neural Networks for Norm-Preserving Backpropagation DizzyRNN: Reparameterizing Recurrent Neural Networks for Norm-Preserving Backpropagation Victor Dorobantu* Per Andre Stromhaug* Jess Renteria* Abstract The vanishing and exploding gradient problems are well-studied obstacles that make it difficult for recurrent neural networks to learn long-term time dependencies. We propose a reparameterization of standard recurrent neural networks to update linear transformations in a provably norm-preserving way through Givens rotations. Additionally, we use the absolute value function as an element-wise non-linearity to preserve the norm of backpropagated signals over the entire network. We show that this reparameterization reduces the number of parameters and maintains the same algorithmic complexity as a standard recurrent neural network, while outperforming standard recurrent neural networks with orthogonal initializations and Long Short-Term Memory networks on the copy problem. Cornell University, Ithaca, NY 1 Defining the problem Recurrent neural networks (RNNs) are trained by updating model parameters through gradient descent with backpropagation to minimize a loss function. However, RNNs in general will not prevent the loss derivative signal from decreasing in magnitude as it propagates through the network. This results in the vanishing gradient problem, where the loss derivative signal becomes too small to update model parameters (Bengio et al., 1994). This hampers training of RNNs, especially for learning long-term dependencies in data. 2 Signal scaling analysis The prediction of an RNN is the result of a composition of linear transformations, element-wise non-linearities, and bias additions. To observe the sources of vanishing and exploding gradient problems in such a network, one can observe the minimum and maximum scaling properties of each transformation independently, and compose the resulting scaling factors. 2.1 Linear transformations Let be an arbitrary linear transformation, where is a matrix of rank . Theorem 1. The singular value decomposition (SVD) of is , for orthogonal and , and diagonal with diagonal elements , the singular values of . From the SVD, Corollaries 1 and 2 follow. Corollary 1. Let and be the minimum and maximum singular values of , respectively. Then . Corollary 2. Let and be the minimum and maximum singular values of , respectively. Then and are also the minimum and maximum singular values of . Proofs for these corollaries are deferred to the appendix. Let be a scalar function of . Then ∂L∂x=∂L∂y∂y∂x=AT∂L∂y In an RNN, this relation describes the scaling effect of a linear transformation on the backpropagated signal. By Corollary 2, each linear transformation scales the loss derivative signal by at least the minimum singular value of the corresponding weight matrix and at most by the maximum singular value. Theorem 2. All singular values of an orthogonal matrix are . By Corollary 2, if the linear transformation is orthogonal, then the linear transformation will not scale the loss derivative signal. 2.2 Non-linear functions Let be an arbitrary element-wise non-linear transformation. Let be a scalar function of . Then ∂L∂x=∂L∂y∂y∂x=f′(x)⊙∂L∂y where denotes the first derivative of and denotes the element-wise product. The -th element of is scaled at least by and at most by . 2.3 Bias Let be an arbitrary addition of bias to . Let be a scalar function of . Then ∂L∂x=∂L∂y∂y∂x=∂L∂y Though additive bias does not preserve the norm during a forward pass over the network, it does preserve the norm of the backpropagated signal during a backward pass. 3 Previous Work In general, the singular values of weight matrices in RNNs are allowed to vary unbounded, leaving the network susceptible to the vanishing and exploding gradient problems. A popular approach to mitigating this problem is through orthogonal weight initialization, first proposed by Saxe et al. (Saxe et al., 2013). Later, identity matrix initialization was introduced for RNNs with ReLU non-linearities, and was shown to help networks learn longer time dependencies (Le et al., 2015). Arjovsky et al. (Arjovsky et al., 2015) introduced the idea of an orthogonal reparametrization of weight matrices. Their approach involves composing several simple complex-valued unitary matrices, where each simple unitary matrix is parametrized such that updates during gradient descent happen on the manifold of unitary matrices. The authors prove that their network cannot have an exploding gradient, and believe that this is the first time a non-linear network has been proven to have this property. Wisdom et al. (Wisdom et al., 2016) note that Arjovsky’s approach does not parametrize all orthogonal matrices, and propose a method of computing the gradients of a weight matrix such that the update maintains orthogonality, but also allows the matrix to express the full set of orthogonal matrices. Jia et al. (Jia, 2016) propose a method of regularizing the singular values during training by periodically computing the full SVD of the the weight matrices, and clipping the singular values to have some maximum allowed distance from . The authors show this has comparable performance to batch normalization in convolutional neural networks. As computing the SVD is an expensive operation, this approach may not translate well to RNNs with large weight matrices. 4 DizzyRNN We propose a simple method of updating orthogonal linear transformations in an RNN in a way that maintains orthogonality. We combine this approach with the use of the absolute value function as the non-linearity, thus constructing an RNN that provably has no vanishing or exploding gradient. We term an RNN using this approach a Dizzy Recurrent Neural Network (DizzyRNN). The reparameterization maintains the same algorithmic space and time complexity as a standard RNN. 4.1 Givens rotation An orthogonal matrix may be constructed as a product of Givens rotations (Golub & Van Loan, 2012). Each rotation is a sparse matrix multiplication, depending on only two elements and modifying only two elements, meaning each rotation can be performed in time. Additionally, each rotation is represented by one parameter: a rotation angle. These rotation angles can be updated directly using gradient descent through backpropagation. Let and denote the indices of the fixed dimensions in one rotation, with . Let express this rotation by an angle . The rotation matrix is sparse and orthogonal with the following form: each diagonal element is except for the -th and -th diagonal elements, which are . Additionally, two off-diagonal elements are non-zero; the element is and the element is . All remaining off-diagonal elements are . Let be a scalar function of . Then ∂L∂x=∂L∂y∂y∂x=RTa,b(θ)∂L∂y Recall that since the matrix is orthogonal with minimum and maximum singular values of , the transpose also has minimum and maximum singular values of (by Corollary 2). To update the rotation angles, note that the only elements of that differ from the corresponding element of are and . Each can be expressed as and . The derivative of with respect to the parameter is thus ∂L∂θ =∂L∂ya∂ya∂θ+∂L∂yb∂yb∂θ =[∂L∂ya∂L∂yb][−sinθcosθ−cosθ−sinθ][xaxb] To simplify this expression, define the matrix as Ea,b=[eTaeTb] where is a column vector of zeros with a in the -th index. The matrix selects only the -th and -th indices of a vector. Additionally, define the matrix as R∂(θ)=[−sinθcosθ−cosθ−sinθ] Note that always has this form; it does not depend on indices and . Now the derivative of with respect to the parameter can be represented as ∂L∂θ=(Ea,b∂L∂y)TR∂(θ)Ea,bx This multiplication can be implemented in time. 4.2 Parallelization Through Packed Rotations While the DizzyRNNs maintain the same algorithmic complexity as standard RNNs, it is important to perform as many Givens rotations in parallel as possible in order to get good performance on GPU hardware. Since each Givens rotation only affects two values in the input vector, we can perform Givens rotations in parallel. We therefore only need sequential operations, each of which has computational and space complexity. We refer to each of these operations as a packed rotation, representable by a sparse matrix multiplication. 4.3 Norm preserving non-linearity Typically used non-linearities like tanh and sigmoid strictly reduce the norm of a loss derivative signal during backpropagation. ReLU only preserves the norm in the case that each input element is non-negative. We propose the use of an element-wise absolute value non-linearity (denoted as abs). Let be the element-wise absolute value of , and let be a scalar function of . Then ∂L∂x=∂L∂y∂y∂x=sign(x)⊙∂L∂y The use of this non-linearity preserves the norm of the backpropagated signal. 4.4 Eliminating Vanishing and Exploding Gradients Let represent packed rotations, be a hidden state at time step , be an input vector, and be a bias vector. Define the hidden state update equation as ht=abs(P1⋯Pn−1ht−1+Wxxt+b) If is square, it can also be represented as packed rotations , resulting in the hidden state update equation ht=abs(P1⋯Pn−1ht−1+Q1⋯Qn−1xt+b) 4.5 Eliminating Vanishing and Exploding Gradients Arvosky et al. claim to provide the first proof of a network having no exploding gradient (through their uRNN) (Arjovsky et al., 2015). We show that DizzyRNN has no exploding gradient and, more importantly, no vanishing gradient. Let a state update equation for an RNN be defined as ht=f(Whht−1+Wxxt+b) Let be a loss function over the RNN. Theorem 3. In a DizzyRNN cell, Theorem 4. If is square, then Therefore, the network can propagate loss derivative signals through an arbitrarily large number of state updates and stacked cells. The proofs for Theorems and are deferred to the Appendix. 5 Incorporating Singular Value Regularization 5.1 Exposing singular values Let be an arbitrary linear transformation, where is a matrix of rank . Such a matrix can be represented by the DizzyRNN reparameterization through a construction , where and are orthogonal matrix and is a diagonal matrix. This construction represents a singular value decomposition of a linear transformation; however, the diagonal elements of (the singular values) can be updated directly along with the rotation angles of and . Additionally, the distribution of singular values can be penalized easily, regularizing the network while allowing full expressivity of linear transformations. 5.2 Diagonal matrix A matrix-vector product where is a diagonal matrix can be represented as the element-wise vector product , where is the vector of the diagonal elements of . Let be a scalar function of . Then ∂L∂x =∂L∂y∂y∂x=σ⊙∂L∂y ∂L∂σ =∂L∂y∂y∂σ=x⊙∂L∂y Each computation can be performed in time. 5.3 Singular value regularization For a DizzyRNN, an additional term can be added to the loss function to penalize the distance of the singular values of each linear transformation from . For each cell in the stack, let denote the vector of all singular values of all linear transformations in the cell; the regularization term is then , where is a penalty factor and is the vector of all ones. The loss function can now be rewritten as L′=L+12λM∑i=1∥σ(i)−e∥22 for a DizzyRNN with a stack height of where represents the vector of all singular values associated with the -th cell in the stack. Note that setting the hyperparameter to allows the singular values to grow or decay unbounded, and setting to constrains each linear transformation to be orthogonal. Additionally, note that initializing the singular values of each linear transformation to is equivalent to an orthogonal initialization of the DizzyRNN. 6 Experimental results We implemented DizzyRNN in Tensorflow and compared the performance of DizzyRNN with standard RNNs, Identity RNNs (Le et al., 2015), and Long Short-Term Memory networks (LSTM) (Hochreiter & Schmidhuber, 1997). We evaluated each network on the copy problem described in (Arjovsky et al., 2015). We modify the loss function in this problem to only quantify error on the copied portion of the output, making our baseline accuracy (guessing at random). The copy problem for our experiments consisted of memorizing a sequence of 10 one-hot vectors of length 10 and outputting the same sequence (via softmax) upon seeing a delimiter after a time lag of 90 steps. We use a stack size of and use only a subset of the total possible packed rotations for every orthogonal matrix. All experiments consist of epochs with 10 batches of size 100, sampled directly from the underlying distribution. DizzyRNN manages to reach near perfect accuracy in under 100 epochs while other models either fail to break past the baseline or plateau at a low test accuracy. Note that 100 epochs corresponds to 100000 sampled training sequences. 7 Conclusion DizzyRNNs prove to be a promising method of eliminating the vanishing and exploding gradient problems. The key is using pure rotations in combination with norm-preserving non-linearities to force the norm of the backpropagated gradient at each timestep to remain fixed. Surprisingly, at least for the copy problem, restricting weight matrices to pure rotations actually improves model accuracy. This suggests that gradient information is more valuable than model expressiveness in this domain. Further experimentation with sampling packed rotations will be a topic of future work. Additionally, we would like to augment other state-of-the-art networks with Dizzy reparameterizations, such as Recurrent Highway Networks (Zilly et al., 2016). References • Arjovsky et al. (2015) Arjovsky, Martín, Shah, Amar, and Bengio, Yoshua. Unitary evolution recurrent neural networks. CoRR, abs/1511.06464, 2015. • Bengio et al. (1994) Bengio, Yoshua, Simard, Patrice, and Frasconi, Paolo. Learning long-term dependencies with gradient descent is difficult. IEEE transactions on neural networks, 5(2):157–166, 1994. • Golub & Van Loan (2012) Golub, Gene H and Van Loan, Charles F. Matrix computations, volume 3. JHU Press, 2012. • Hochreiter & Schmidhuber (1997) Hochreiter, Sepp and Schmidhuber, Jürgen. Long short-term memory. Neural computation, 9(8):1735–1780, 1997. • Jia (2016) Jia, Kui. Improving training of deep neural networks via singular value bounding. arXiv preprint arXiv:1611.06013, 2016. • Le et al. (2015) Le, Quoc V., Jaitly, Navdeep, and Hinton, Geoffrey E. A simple way to initialize recurrent networks of rectified linear units. CoRR, abs/1504.00941, 2015. • Saxe et al. (2013) Saxe, Andrew M., McClelland, James L., and Ganguli, Surya. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. CoRR, abs/1312.6120, 2013. • Wisdom et al. (2016) Wisdom, Scott, Powers, Thomas, Hershey, John, Le Roux, Jonathan, and Atlas, Les. Full-capacity unitary recurrent neural networks. In Advances In Neural Information Processing Systems, pp. 4880–4888, 2016. • Zilly et al. (2016) Zilly, Julian Georg, Srivastava, Rupesh Kumar, Koutník, Jan, and Schmidhuber, Jürgen. Recurrent highway networks. arXiv preprint arXiv:1607.03474, 2016. Appendix Let be an arbitrary linear transformation, where is a matrix of rank . Theorem 1. The singular value decomposition of is expressed as , for orthogonal and , and diagonal . and are the -th columns of and , respectively, and is the -th diagonal element of . The columns of are the left singular vectors, the columns of are the right singular vectors, and the diagonal elements of are the singular values. Corollary 1. Let and be the minimum and maximum singular values of , respectively. Then . Proof. Express as , i.e. a linear combination of the orthonormal set of left singular vectors. The -th coefficient of the combination is the scalar . Since the set of left singular vectors is orthonormal, the -norm of is the Pythagorean sum of the coefficients of the linear combination, i.e., . Note that the term is the magnitude of the projection of onto . Without loss of generality, fix the norm of to . Let and be the right singular vectors corresponding to and , respectively. Then, the norm of is minimized for parallel to , and maximized for parallel to . The corresponding norms of are and . Since the set of right singular vectors is orthonormal, , and the corresponding norms of are and . Corollary 2. Let and be the minimum and maximum singular values of , respectively. Then and are also the minimum and maximum singular values of . Proof. Since where and are orthogonal, . By the same construction as in the previous corollary, if , then . For all such that , the quantity is minimized and maximized for and , respectively, where and are the left singular vectors corresponding to and . The corresponding minimum and maximum norms are and . Let a state update equation for an RNN be defined as ht=f(Whht−1+Wxxt+b) Let be a loss function over the RNN. Theorem 3. In a DizzyRNN cell, Proof. Let and express ∂L∂ht−1=∂L∂ht∂ht∂y∂y∂ht−1=WTh(f′(y)⊙∂L∂ht) The -norms of each side of this equation are ∥∥∥∂L∂ht−1∥∥∥2=∥∥∥WTh(f′(y)⊙∂L∂ht)∥∥∥2 In a DizzyRNN, is the absolute value function, thus the elements of are or . is orthogonal since it is defined by a composition of Givens rotations. Neither and scale the norm of the vector , thus ∥∥∥∂L∂ht−1∥∥∥2=∥∥∥∂L∂ht∥∥∥2 If instead is the ReLU non-linearity, then is only equal to in the case where all values in are non-negative, resulting in a diminishing gradient in all other cases. Theorem 4. If is square, then Proof. By symmetry with the proof of Theorem 3, the norms are shown to be equal. ∎ You are adding the first comment! How to quickly get a good reply: • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made. • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements. • Your comment should inspire ideas to flow and help the author improves the paper. The better we are at sharing our knowledge with each other, the faster we move forward. The feedback must be of minimum 40 characters and the title a minimum of 5 characters	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9451889991760254, "perplexity": 1023.5193297485977}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704821381.83/warc/CC-MAIN-20210127090152-20210127120152-00065.warc.gz"}
http://slideplayer.com/slide/3913248/	# James T. Shipman Jerry D. Wilson Charles A. Higgins, Jr. Electricity and Magnetism Chapter 8. ## Presentation on theme: "James T. Shipman Jerry D. Wilson Charles A. Higgins, Jr. Electricity and Magnetism Chapter 8."— Presentation transcript: James T. Shipman Jerry D. Wilson Charles A. Higgins, Jr. Electricity and Magnetism Chapter 8 Electric Charge and Electric Force Electric charge is a fundamental quantity – we don’t really know what it is –But we can describe it, use it, control it –Electricity runs motors, lights, heaters, A/C, stereos, TV’s, computers, etc. Electric Forces – at the microscopic level they hold atoms and molecules together –Electric Forces hold matter together Gravitational Forces hold the universe together Magnetism is also closely associated with electricity Intro Electric Charge and Electric Force Experimental evidence leads us to conclude that there are two types charges –Positive (+) –Negative (-) All matter is composed of atoms, which in turn are composed of subatomic particles –Electrons (-) –Protons (+) –Neutron (neutral) Section 8.1 Electric Force An electric force exists between any two charged particles – either attractive or repulsive Section 8.1 Law of Charges Law of Charges – like charges repel, and unlike charges attract –Two positives repel each other –Two negatives repel each other –Positive and negative attract each other Section 8.1 Negative/Positive An object with an excess of electrons is said to be negatively charged An object with a deficiently of electrons is said to be positively charged Static electricity is the study of charges at rest. Section 8.1 Repulsive and Attractive Electrical Forces Two negative charges repel Two positive charges repel One negative and one positive attract Repulse Attract Section 8.1 Charging by Induction Section 8.1 Conductors/Insulators Electrical conductor – materials in which an electric charge flows readily (most metals, due to the outer, loosely bound electrons) Electrical insulator – materials that do not conduct electricity very well due to very tight electron bonding (wood, plastic, glass) Semiconductor – not good as a conductor or insulator (graphite) Section 8.2 Simple Electrical Circuit Electrons flow from negative terminal to positive terminal (provided by the chemical energy of the battery) -- negative to positive Open switch – not a complete circuit and no flow of current (electrons) Closed switch – a complete circuit and flow of current (electrons) exists Closed Circuit Required – to have a sustained electrical current Section 8.2 Simple Electrical Circuit The light bulb offers resistance. The kinetic energy of the electric energy is converted to heat and radiant energy. Section 8.2 Forms of Electric Current Direct Current (DC) – the electron flow is always in one direction, from (-) to (+) –Used in batteries and automobiles Alternating Current (AC) – constantly changing the voltage from positive to negative and back –Used in homes. –60 Hz (cycles/sec) and Voltage of 110-120 V Section 8.2 Electrical Safety Wires can become hot as more and more current is used on numerous appliances. Fuses are placed in the circuit to prevent wires from becoming too hot and catching fire. The fuse filament is designed to melt (and thereby break the electrical circuit) when the current gets too high. Two types of fuses: Edison and S-type Circuit Breakers are generally now used. Section 8.3 Electrical Safety with Dedicated Grounding A dangerous shock can occur if an internal ’hot’ wire comes in contact with the metal casing of a tool. This danger can be minimized by grounding the case with a dedicated wire through the third wire on the plug. Section 8.3 Fuses Section 8.3 Thermal type Circuit Breaker – as the current through the bimetallic strip increases, it becomes warmer (joule heat) and bends – “tripping” the circuit breaker. Magnetism Closely associated with electricity is magnetism. In fact electromagnetic waves consists of both vibrating electric and magnetic fields. These phenomena are basically inseparable. A bar magnet has two regions of magnetic strength, called the poles. One pole is designated “north,” one “south.” Section 8.4 Magnetic Poles The N pole of a magnet is “north-seeking” – it points north. The S pole of a magnet is “south-seeking” – it points south. Magnets also have repulsive forces, specific to their poles, called … Law of Poles – Like poles repel and unlike poles attract –N-S attract –S-S & N-N repel Section 8.4 Law of Poles All magnets have two poles – they are dipoles Section 8.4 Source of Magnetism The source of magnetism is moving and spinning electrons. Hans Oersted, a Danish physicist, first discovered that a compass needle was deflected by a current-carrying wire. –Current open  deflection of compass needle –Current closed  no deflection of compass needle A current-carrying wire produces a magnetic field: stronger current  stronger field Electromagnet – can be switched on & off Section 8.4 Magnetic Materials Most materials have many electrons going in many directions, therefore their magnetic effect cancels each other out  non-magnetic A few materials are ferromagnetic (iron, nickel, cobalt) – in which many atoms combine to create magnetic domains (local regions of magnetic alignment within a single piece of iron) A piece of iron with randomly oriented magnetic domains is not magnetic. Section 8.4 Magnetization The magnetic domains are generally random, but when the iron is placed in a magnetic field the domains line up (usually temporarily). Section 8.4 Earth’s Magnetic Field This planet’s magnetic field exists within the earth and extends many hundreds of miles into space. The aurora borealis (northern lights) and aurora australis (southern lights) are associated with the earth’s magnetic field. Although this field is weak compared to magnets used in the laboratory, it is thought that certain animals use it for navigation. Section 8.4 Earth’s Magnetic Field Similar to the pattern from a giant bar magnet being present within the earth (but one is not present!) Section 8.4 Download ppt "James T. Shipman Jerry D. Wilson Charles A. Higgins, Jr. Electricity and Magnetism Chapter 8." 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https://cs.stackexchange.com/questions/102673/mathematical-physicss-equivalent-for-computer-science/102696	# “Mathematical Physics”'s equivalent for Computer Science Mathematical Physics is a well defined scientific fields that deals with the application of mathematics , mathematical tools , and mathematical methods in the theories and the problems of physics. Is there a well defined scientific field that deals with the application of mathematics , mathematical tools , and mathematical methods in the theories and the problems of computer science ? • Theoretical computer science? – Yuval Filmus Jan 10 at 15:01 • Computer science? – David Richerby Jan 10 at 15:27	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8671901226043701, "perplexity": 1066.6579867980588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540528490.48/warc/CC-MAIN-20191210180555-20191210204555-00273.warc.gz"}
https://www.taylorfrancis.com/chapters/edit/10.1201/b13680-15/diffuse-optical-imaging-application-brain-imaging-jeremy-hebden?context=ubx&refId=bd1299cb-67f7-4816-873a-c7690793928f	## ABSTRACT Although the relatively low attenuation of near-infrared light in most tissues facilitates transmittance measurements over distances of several centimeters, the profound scatter of light renders those measurements sensitive to a much larger volume of tissue than that occupying a direct line between the source and the detector. e light traveling between two points on the surface of a uniform diusing medium separated by a few centimeters has typically explored a banana-shaped volume of tissue, oen referred to as the photon measurement density function (PMDF; see Figure 10.1) [2]. e breadth of the PMDF limits the spatial resolution achievable using the diuse optical imaging (DOI) methods, although as discussed later, gains can be achieved by measuring more than just the intensity of the transmitted light, and by a judicious combination of multiple measurements with overlapping PMDFs. Note that due to the dominance of scatter, the terms “transmitted” and “reected” are somewhat arbitrary when applied to optical measurements on the head, although the latter term is generally used when the separation between the source and the detector is less than about one-quarter of the circumference.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9255186915397644, "perplexity": 1047.1233692471583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499758.83/warc/CC-MAIN-20230129180008-20230129210008-00041.warc.gz"}
http://en.wikipedia.org/wiki/Talk:Limit_(mathematics)	# Talk:Limit (mathematics) WikiProject Mathematics (Rated B-class, Top-importance) This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. Mathematics rating: B Class Top Importance Field: Analysis One of the 500 most frequently viewed mathematics articles. Wikipedia Version 1.0 Editorial Team / Vital ## Copy of removed paragraph Removed this: ====A Brief Note Regarding Division by Zero==== In general, but not in all cases, should u directly substitute c for x (into f(x)) and obtain an illegal fraction with division by zero, check to see whether the numerator equals zero. In cases where such substitution results in 0 / 0, a limit probably exists; in other cases (such as 17 / 0) a limit is less likely. For instance; if f(x) = x³ + 1 / x - 1; then, if one substitutes 1 for x, one will obtain 2 / 0; the limit of f(x) (as x approaches 1) does not exist. I can't be bothered to do the graph offhand, but there will be a limit: either + or - inf. User:Tarquin oops Pizza Puzzle Plus and minus infinity are not limits according to the definition in the article. Please make sure that you have some understanding of the article before you go removing bits. -- Oliver P. 15:42 8 Jun 2003 (UTC) I'm not aware that infinity is a limit; because, infinity is not a real number and my understanding is that limits must be real numbers. Pizza Puzzle Yes, that's what I just said. I said it in reply to your statement that "there will be a limit: either + or - inf". If you have changed your mind, and are retracting your previous statement, please replace what you removed from the article. -- Oliver P. 16:02 8 Jun 2003 (UTC) No sir! I did not state that there will be a limit either + or - inf. The user who does not sign his messages stated that. I have added: which I believe is what u are referring to above. There is now the question of, if the above user was wrong, does that mean I can reinsert my text: • For instance; if f(x) = x³ + 1 / x - 1; then, if one substitutes 1 for x, one will obtain 2 / 0; the limit of f(x) (as x approaches 1) does not exist. or would that be a hostile revert? He had initially removed the entire paragraph, which I put most of it back in, but I didnt put the final line back since there was a debate of sorts regarding it. ## Infinite limit • As x approaches 0, F(x) = 1 / x² is not approaching a limit as it is unbounded; a function which approaches infinity is not approaching a limit. Note that as x approaches infinity, F(x) = 1 / x² does approach a limit of 0. Pizza Puzzle Oh, I see! In that case, I apologise unreservedly for having accused you. I'll blame Tarquin for my error, though, since he was the phantom non-signer. ;) There is a problem in that there are different ways of defining what a limit is. I'll give the article some thought, and come back to it later. I wouldn't object to you putting that example back in, although you should leave out the idea of substitution; a limit only depends on the behaviour as you appraoch the point, not at the point itself. -- Oliver P. 16:15 8 Jun 2003 (UTC) The subsitution point is, IF you substitute, and you get division by zero, if you get 0 / 0, then there is probably a limit, otherwise there probably isn't. Pizza Puzzle Oh, I'll think about it later. I should be doing work... -- Oliver P. 16:29 8 Jun 2003 (UTC) Now here, this text says (in so many words): "The limit, L of f(x), as f(x) increases (or decreases) without bound is an infinite limit. Be sure that you see that the equal sign in "L = infinity" does not mean that the limit exists. Rather, this tells you that the limit fails to exist by being boundless." It would appear, that it is correct to refer to "infinite limits" but one should understand that an "infinite limit" is not a limit. See also: "unbounded limit" Pizza Puzzle Would it be too much to expect User: AxelBoldt to explain some of his more "major" changes? It appears that a great deal of information was deleted. If he had a problem with it, it would have been more appropriate to discuss it or improve it; rather than merely deleting it. Pizza Puzzle Too many subsections before the formal definition. I don't think an encyclopedia article should go that way. I will try to rewrite this later. Wshun I see limits in this way. If the function is continous for all R then at the limit the function will have a definte value. It doesn't matter if you are trying to find the limit at + or - infinity, or the limit of a function as it approaches a certain value c. In both cases you are dealing with an infinte number of values. If there was no definte value at the limit then limits would'nt be of much use in calculus. ## Inconsistent graphic? At Limit (mathematics) § Limit of a function, the prose discusses a single scenario, and the right side of the graphic purporting to show it almost shows a zoomed-out view of the left side, but not quite. If the two sides are meant to represent the same thing, the left side needs the vertical line intersection with the x-axis at c - δ to be labeled "S". On the right side, f(x) needs to be equal to L + ε at x = c + δ (i.e. the second hump needs to be above the green-highlighted area). —[AlanM1(talk)]— 23:17, 14 June 2013 (UTC) ## Questionable example The article states that f(x) = x²-1/x-1 is undefined at x=0. I would disagree, since it can easily be simplified to f(x) = x+1. It seems the same as arguing that x²/x would be undefined at 0 (or actually any g(x)*x/x). I can see how the example is convenient in other ways, because the formula is simple, but I would propose to either replace it by sin(x)/x, or at least note that the statement "f(x) is not defined for x=0" is debatable and that the example was chosen for its simplicity. - Jay 84.171.79.63 (talk) 19:11, 28 June 2014 (UTC) I don't see your point. The function is not defined at x=0. and sin(x)/x = sinc(x), which usually has sinc(0)=1. — Arthur Rubin (talk) 03:40, 2 July 2014 (UTC) To clarify... the function in the article is f(x) = (x²-1)/(x-1) (rather than f(x) = x²-1/x-1 ) and the article states that it is not defined at x=1 (rather than at x=0). Meters (talk) 22:45, 7 July 2014 (UTC) I still don't see the IP's point. Just because $\frac {x^2-1} {x-1}$ can be simplified to x+1, doesn't mean that it is simplified. And our article "sinc" does specify that sinc(x) = sin(x)/x when x ≠ 0. One could make a better case that $e^{x^{-2}}$ isn't defined at x = 0, but that isn't quite correct either, when we work on the extended real line. — Arthur Rubin (talk) 16:08, 9 July 2014 (UTC) I think the IP's point might be that in practice, other than to come up (in a textbook section about limits) with a function with a specific value excluded, no-one would ever define a function like $\frac {x^2-1} {x-1}$. A less trivial and perhaps somehow "better" example would be, for instance, $f(x) = \frac {\ln{x}} {x-1}$ with a hole at x=1, because it cannot be trivially simplified. - DVdm (talk) 16:53, 9 July 2014 (UTC)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8768183588981628, "perplexity": 695.4802146511983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500825567.38/warc/CC-MAIN-20140820021345-00168-ip-10-180-136-8.ec2.internal.warc.gz"}
http://mathoverflow.net/users/3384/stankewicz	# stankewicz less info reputation 1024 bio website stankewicz.net location age member for 5 years, 3 months seen Apr 21 at 10:32 profile views 1,380 Postdoc at the University of Bristol 16 Thinking and Explaining 15 Isogeny classes of elliptic curves 10 roadmap for studying arithmetic geometry 9 Deducing BSD from Gross-Zagier and Kolyvagin 9 The significance of modularity for all Galois representations # 2,634 Reputation +5 Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup? +10 Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q +10 Is the number of twists of a curve with a section in a given field finite +10 is there any way to bound the number of CM points by height functions? # 4 Questions 24 Biographic Data/Stories about André Néron 15 Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup? 9 Sequences of Squares with all square differences 4 Cubic forms and Hasse Principle # 50 Tags 124 nt.number-theory × 23 12 algebraic-number-theory × 3 86 elliptic-curves × 14 11 riemann-surfaces × 4 85 ag.algebraic-geometry × 21 10 moduli-spaces × 3 79 arithmetic-geometry × 14 10 quaternions × 2 17 algebraic-curves × 4 10 modular-forms × 2 # 4 Accounts MathOverflow 2,634 rep 1024 TeX - LaTeX 108 rep 5 Stack Overflow 101 rep 1 Ask Ubuntu 101 rep 2	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8537133932113647, "perplexity": 2673.3196535746006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246654292.99/warc/CC-MAIN-20150417045734-00149-ip-10-235-10-82.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/380705/if-n-j-p-1-cdot-ldots-cdot-p-t-fracp-1-cdot-ldots-cdot-p-tp-j-th	# If $n_j = p_1\cdot \ldots \cdot p_t - \frac{p_1\cdot \ldots \cdot p_t}{p_j}$, then $\phi(n_j)=\phi(n_k)$ for $1 \leq j,k \leq t$ Show that if $p_1,\ldots p_t$ are the first $t$ prime numbers, and $n_j = p_1\cdot \ldots \cdot p_t - \frac{p_1\cdot \ldots \cdot p_t}{p_j}$, then $\phi(n_j)=\phi(n_k)$ for $1 \leq j,k \leq t$ and conclude that the equation $\phi(x)=m$ has infinitely many solutions. Here $\phi(\cdot)$ is the Euler Totient function. I am really stuck on this one. First of all $p_j \nmid n_j$ because even though $p_j$ divides the first term in $n_j$ it does not divide the second. Therefore $gcd(n_j,p_j)=1$. However, can we use this to prove the theorem? Thanks for any help! EDIT At first the whole expression for $n_j$ seemed really confusing to me. I think it can be rewritten as: $$n_j = \left(1-\frac{1}{p_j}\right) \prod\limits_{i=1}^{t} p_i = (p_j-1)\left(\prod\limits_{i=1}^{j-1}p_i \right)\cdot \left(\prod\limits_{i=j+1}^{t}p_i \right)$$ Maybe this helps! Let $\phi$ denote the Euler totient function. Let $d$, $n$ be positive integers with $d$ dividing some power of $n$, that is, every prime divisor of $d$ also divides $n$. Then $\phi(dn) = d\,\phi(n)$. As $N = p_1\cdots p_t$ is the product of the first $t$ prime numbers, every prime factor of $p_j-1$ still divides $N/p_j$. Therefore, $$\phi(n_j) = \phi\left((p_j - 1)\,\frac{N}{p_j}\right) = (p_j - 1)\,\phi\left(\frac{N}{p_j}\right) = \phi(p_j)\,\phi\left(\frac{N}{p_j}\right)$$ As $\phi$ is multiplicative and $N/p_j$ is coprime to $p_j$, we get $$\phi(n_j) = \phi(N)$$ which implies the statement $\phi(n_j) = \phi(n_k)$. Nevertheless, $\phi(x) = m$ with given $m$ has only finitely many solutions for $x$. Too see this, consider the number of prime factors of $x$, counted with multiplicity. Let us denote this $\Omega(x)$. You can deduce from the product formula for $\phi(x)$ that $\Omega(\phi(x)) \geq \Omega(x) - 1$ (the $-1$ is for the possible prime divisor 2). Therefore $$\Omega(x) \leq \Omega(m) + 1$$ and since the greatest prime divisor of $x$ cannot exceed $m+1$, we can easily estimate an upper bound for $x$ as $$x \leq (m+1)^{\Omega(m)+1}$$ We conclude that there can only be finitely many $x$ with $\phi(x) = m$. Call $N = p_1 p_2 \ldots p_t$ for compactness. Then: $$n_j = N - N / p_j = (p_j - 1) N / p_j$$ As the totient is multiplicative and never zero: $$\phi(n_j) = \phi(p_j - 1) \phi(N) / \phi(p_j) = \phi(p_j - 1) \phi(N) / (p_j - 1)$$ If it was $\phi(n_j) = \phi(n_k)$, then it would be: $$\phi(p_j - 1) / (p_j - 1) = \phi(p_k - 1) / (p_k - 1)$$ But: \begin{align} \phi(5 - 1) / (5 - 1) &= 2 / 4 = 1/2 \\ \phi(7 - 1) / (7 - 1) &= 2 / 6 = 1/3 \end{align} Too nice to be true :-( • Hey thanks for your answer. I think the question is meant to mean that there exists $j$ and $k$ smaller than $t$ such that $\phi(n_j)=\phi(n_k)$. For instance in the example you gave $\phi(3)/(3-1) = 1/2$. So it does not need to hold for all $j$ and $k$ smaller than $t$. – Slugger May 3 '13 at 23:18 • @TeunVerstraaten OK, then $p_j = 3$ and $p_k = 5$ will work for OP's needs, for any $t$ – vonbrand May 3 '13 at 23:55 • Yeah but $j$ needs to specified beforehand. Thats why in the question I used $n_j$. And then you show that some $k$ exists to satisfy the condition. – Slugger May 4 '13 at 0:04 • @vonbrand: $\phi$ is multiplicative, but not completely multiplicative. – ccorn May 4 '13 at 1:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.995789647102356, "perplexity": 88.40821154235194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357935.29/warc/CC-MAIN-20210226175238-20210226205238-00422.warc.gz"}
https://math.stackexchange.com/questions/2107742/solving-a-cauchy-problem	# Solving a Cauchy problem Given the equation $$y' = e^t \sqrt[3]{y^2}$$ (a) consider the related Cauchy problem with $y(t_0)=y_0$. What $P(t_0,y_0)$ ensures the problem has a unic solution? (b) find the general integral of the given equation. I'm not able to answer the first question. I know that if $f(t,y(t))=e^t \sqrt[3]{y^2}$ is lipschizt with respect to $t$ and continuos with respect to $t$, then there exists an interval $I$ centered in $t_0$ where the solution is unic. But how can I apply this theorem? Also, do I just need this theorem? Oss: I shall suppose $y(t)$ of class $C^1$ for regularity of $f$. Plus, how can I solve the differential equation? Do I need some other hypothesis on $y(t)$? Thanks • for b) write the equation as $$\frac{dy}{(y^2)^{1/3}}=e^tdt$$ – Dr. Sonnhard Graubner Jan 21 '17 at 19:18	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9841060638427734, "perplexity": 229.90544461381538}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256763.42/warc/CC-MAIN-20190522043027-20190522065027-00533.warc.gz"}
http://mathoverflow.net/questions/105696/solving-in-x-involving-both-exponential-and-logarithmic-function	# solving in x involving both exponential and logarithmic function Is it possible to solve a function with both exponential and logarithm such as $a x^2 - b.\log(x) = c$ in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$? - Too easy for MO ; try Lambert w-function – Feldmann Denis Aug 28 '12 at 9:10 Good question, wrong website. We're about research. You'll get a better reception at math.stackexchange.com – Gerry Myerson Aug 28 '12 at 13:08 @Tom and Gerry, thanks for the positive response. Well it did come up in my research and I thought this would be a good place to ask. I got a closed form solution on Wolfram Alpha but no clue how it was reached. – pratikag Aug 28 '12 at 14:34 I will honestly admit that I laughed at "Tom and Gerry". – Vidit Nanda Aug 28 '12 at 22:00 Short explanation. If you substitute $x=\sqrt{t}$ in the equation $ax^{2}-b\log x=c$ you can rewrite it as $-\frac{2a}{b}e^{-2c/b}=\left( -\frac{2a}{b}t\right) e^{-2at/b}$. Since by definition of the Lambert $W$ function $Y=Xe^{X}$ iff $X=W(Y)$ this means that $W\left( -\frac{2a}{b}e^{-2c/b}\right) =-\frac{2a}{b}t=-\frac{2a}{b}x^{2}$. And solving for $x$ you get $x=\left( -\frac{b}{2a}W\left( -\frac{2a}{b}e^{-2c/b}\right) \right) ^{1/2}$. – Américo Tavares Aug 29 '12 at 0:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9035119414329529, "perplexity": 662.4238627150934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860125857.44/warc/CC-MAIN-20160428161525-00118-ip-10-239-7-51.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/centrifugal-force-units.412154/	# Centrifugal force units. 1. Jun 24, 2010 I found this on Wikipedia for the equation relating to centrifugal force. F = d/dt (mv) So I have my mass and velocity but I just multiplied the two, that can't be correct can it. You have to derive something I imagine. Also would the units just be in lbs or Newtons. I'm using feet/s and lbs. Help I'm pretty rusty on my calculus. 2. Jun 24, 2010 ### Staff: Mentor If your velocity is constant, there is no net force (no acceleration). A better way to write it (assuming constant mass) is F = m dv/dt So the units in mks are Newtons = kg m / s^2 3. Jun 24, 2010 The odd thing I found is was this equation for centripetal force. where v^2 = F R/M Couldn't I just use this equation for centrifugal force since all centrifugal and centripetal are is the name for the direction which the force is going on the wheel? 4. Jun 24, 2010 what is mks. how do you perform that equation? 5. Jun 24, 2010 ### Staff: Mentor mks = meters, kilograms, seconds. It's one of the two metric systems of units. The other is cgs = centimeters, grams, seconds. If you mean the F = m dv/dt equation, the dv/dt is a timie derivative of velocity. That is, how much the velocity chages over time, which is called the acceleration. v = dx/dt (velocity is change in position with time, in mks units of m/s) a = dv/dt (acceleration is change of velocity with time, in mks units of m/s^2) 6. Jun 24, 2010 ### K^2 SI is the more common name for MKS. Centrifugal force is given by F = mv²/R. Thats kg(m/s)²/m = kgm/s² = N. So the units are exactly the same as these of F = ma. 7. Jun 24, 2010 Sorry I'm a little lost on figuring out the derivative. How do you set that up? 8. Jun 24, 2010 ### Staff: Mentor What exactly are you trying to calculate? 9. Jun 25, 2010 The centrifugal force of an object based upon the radius, weight or mass of the object. In the long run I would like to calculate the force exerted by the object at any point in the radius of the circle while rotating around at a constant velocity. In this case would we now be talking about acceleration? 10. Jun 25, 2010 ### Staff: Mentor Realize that 'centrifugal force', at least in standard physics usage, is a fictitious force that only exists when analyzing motion from a rotating frame. You don't need centrifugal force to understand rotation. OK. You can easily calculate the centripetal force acting on the object. And for each force acting on the object there will be a corresponding force exerted by the object. Anything moving in a circle will have a centripetal acceleration. 11. Jun 25, 2010 I understand the rotation, I would just like to know what the force of that object would be at different points from the center as it rotates. Like when you twirl a rock on a string around your finger, as you bring it in I'm sure the force changes and so does the speed. Am I making any sense? I figured that the centripetal force would be ficticious if any, how do you even get an object to go to the inside of a rotation. Don't want to get off topic, but maybe this is a goodtime for a description of the two. 12. Jun 25, 2010 ### K^2 It's just a choice of the coordinate system. If your coordinate system rotates, you have centrifugal force. If it doesn't, there isn't one. If you are standing on a rotating disk, it's convenient for you to define your position relative to disk. That's your choice of coordinate system, and you then end up observing outwards pull you call Centrifugal Force. That being canceled by the friction with the ground, you remain stationary in that system. If, however, you decide to take your bearings relative to an object that's outside the disk, you note that you aren't just standing still. You are moving. In fact, you are constantly accelerating towards the center of the disk. The force that provides that acceleration is still the same friction with the ground that was countering Centrifugal Force in the rotating frame. So as long as you choose coordinate system without rotation, you really don't need centrifugal force. 13. Jun 28, 2010 Simply put I just would like to find the force of a single object rotating in a circle relating to weight, RPM or any omega, and distance from center. Would the equation below work for what I am trying to do. Or do I need to derive something, If this is the case, I could use some help. 14. Jun 28, 2010 ### uart Yes the equation $F= m v^2/r$ will work just fine if you use SI units of kg, meters and seconds (which of course means the force is in Newtons). Feel free to ask if you need to know how that equation is derived. 15. Jun 28, 2010 ### Staff: Mentor For a mass m moving in a circle of radius r at constant speed v (or angular speed ω), the net force on it (also known as the centripetal force) is given by: Fc = mv²/r = mω²r 16. Jun 28, 2010 What units would that be, I would initially think lbs. but it how does the ft/s cancel. lbs (ft/s)^2/ft = lbsft/s^2 that can't be right. what am I forgetting? 17. Jun 28, 2010 ### Staff: Mentor You are forgetting that a pound is a unit of force (for example, weight) not mass. In the English system the unit of mass is the slug. slug ft/s^2 ≡ lbs At the earth's surface, an object's weight in pounds is related to its mass in slugs via: Weight = massg = mass (32 ft/s^2) In the SI system: kg * m/s^2 ≡ Newtons 18. Jun 28, 2010 So for example: you have an item which ways 5lbs which would equal 5/32.2 = .15528 slugs. So according to this equation. Fc = mv²/r = mω²r Fc = (.15528 slugs50ft/s ^2)/2 ft = 194 (lbf*s^2/ft)(ft^2/s^2)/(ft) = 194 lb right. And this force would be towards the center of the cirlce? 19. Jun 28, 2010 ### Staff: Mentor Right. Yes. This is the net force towards the center required to make it go in a circle at that speed and radius. 20. Jun 28, 2010 To determine the centrifugal force, would it be the same as the centripetal? equal opposing reacting force? Similar Discussions: Centrifugal force units.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8845060467720032, "perplexity": 997.289123177832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948567042.50/warc/CC-MAIN-20171215060102-20171215080102-00186.warc.gz"}
https://rbspgway.jhuapl.edu/biblio?keyword=refractive%20index	# Bibliography Van Allen Probes Bibliography is from August 2012 through September 2021 Notice: Clicking on the title will open a new window with all details of the bibliographic entry. Clicking on the DOI link will open a new window with the original bibliographic entry from the publisher. Clicking on a single author will show all publications by the selected author. Clicking on a single keyword, will show all publications by the selected keyword. ## Found 1 entries in the Bibliography. ### Showing entries from 1 through 1 2018 Highly Oblique Lower-Band Chorus Statistics: Dependencies of Wave Power on Refractive Index and Geomagnetic Activity We use 3 years of Van Allen Probes observations of highly oblique lower-band chorus waves at low latitudes over L = 4\textendash6 to provide a comprehensive statistics of the distribution of their magnetic and electric powers and full energy density as a function of wave refractive index N, L shell, and geomagnetic activity AE. We use the refractive index calculated either in the cold plasma approximation or in the quasi-electrostatic (hot plasma) approximation and either observed wave electric fields or corrected wave elect ... Shi, R.; Mourenas, D.; Artemyev, A.; Li, W.; Ma, Q.; Published by: Journal of Geophysical Research: Space Physics Published on: 06/2018 YEAR: 2018 DOI: 10.1029/2018JA025337 1	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8583817481994629, "perplexity": 4209.343361573102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571472.69/warc/CC-MAIN-20220811133823-20220811163823-00280.warc.gz"}
http://mathoverflow.net/questions/163775/bound-for-the-frattini-subgroup-of-a-p-group	# Bound for the Frattini subgroup of a $p$-group Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to get a bound for $\dim \Phi(G)$ as a (polynomial) function of $\dim G / \Phi(G)$? Edit: I should probably write a little more. First of all, the exponent of such a group is $p$ or $p^2$ as the $p^{\text{th}}$ power of any element falls in the Frattini subgroup. If the exponent of the group is $p$ then $\Phi(G) = G'$, since $\Phi(G)=G^pG'$ in general. Now suppose that $g_1,\dots,g_n$ is a basis for $G / \Phi(G)$. Then it is easy to see that the elements $[g_i,g_j]$, $0 \leq i,j \leq n$ generate $G'$ because, in this particular case, the $p^{\text{th}}$ power map and the commutator map (with fixed first or second coordinate) are homomorphisms. So if the exponent of $G$ is $p$ then $\dim \Phi(G)$ is at most $\dim G / \Phi(G) \choose 2$. My guess is that there are $p$-groups which achieve this upper bound. What happens when the exponent of the group is $p^2$? I guess what I'm really asking is whether the upper bound remains quadratic in that case too. The answer to that question is $\textbf{no}$, as Geoff's post establishes. Is there an upper bound that takes into account $p$ as well? - Yes, there are $p$-groups that achieve the bound for exponent $p$; namely, the relatively free groups of rank $n$, class $2$, and exponent $p$ have commutator subgroup that is free abelian of rank $\binom{n}{2}$; this group can be realized as $F_n/F_n^p(F_n)_3$, where $F_n$ is the absolutely free group of rank $n$, and $(F_n)_3$ is the third term of the lower central series of $F_n$. – Arturo Magidin Apr 19 '14 at 0:18 There can be no polynomial bound which is independent of $p$. Consider the group $G = C_{p} \wr C_{p},$ where $C_{p}$ denotes the cyclic group of order $p.$ Then $\Phi(G) = G^{\prime}$ has order $p^{p-1},$ (and is elementary Abelian) yet $[G: \Phi(G)] = p^{2}.$ For your second question the answer is yes. The bound follows from Schreier's inequality: if $\Phi(G)$ has index $p^d$, then it follows that $\Phi(G)$ can be generated by $p^d(d-1)+1$ elements. Note that this is true without assuming that the Frattini subgroup is elementary abelian. Your first claim is true for the exponent $p^2$ case, if one requires that $\Phi(G)$ is central. Once you fix the number of generators $d$, the largest such group can be defined to be the quotient of the free group $F$ on $d$ generators by the subgroup $[F,F^p[F,F]] (F^p[F,F])^p$. In that case the Frattini subgroup can be generated by $d(d+1)/2$ elements.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833948612213135, "perplexity": 48.46492879582243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375099755.63/warc/CC-MAIN-20150627031819-00026-ip-10-179-60-89.ec2.internal.warc.gz"}
https://collaborate.princeton.edu/en/publications/weak-lensing-of-type-ia-supernovae-from-the-dark-energy-survey	# Weak lensing of Type Ia Supernovae from the Dark Energy Survey DES Collaboration Research output: Contribution to journalReview articlepeer-review 3 Scopus citations ## Abstract We consider the effects of weak gravitational lensing on observations of 196 spectroscopically confirmed Type Ia Supernovae (SNe Ia) from years 1 to 3 of the Dark Energy Survey (DES).We simultaneously measure both the angular correlation function and the non-Gaussian skewness caused by weak lensing. This approach has the advantage of being insensitive to the intrinsic dispersion of SNe Ia magnitudes. We model the amplitude of both effects as a function of σ8, and find σ8 =1.2+0.9-0.8. We also apply our method to a subsample of 488 SNe from the Joint Light-curve Analysis (JLA; chosen to match the redshift range we use for this work), and find σ8 =0.8+1.1-0.7. The comparable uncertainty in σ8 between DES-SN and the larger number of SNe from JLA highlights the benefits of homogeneity of the DES-SN sample, and improvements in the calibration and data analysis. Original language English (US) 4051-4059 9 Monthly Notices of the Royal Astronomical Society 496 3 https://doi.org/10.1093/MNRAS/STAA1852 Published - 2020 ## All Science Journal Classification (ASJC) codes • Astronomy and Astrophysics • Space and Planetary Science ## Keywords • Cosmological parameters • Cosmology: Observations • Large-scale structure of the universe ## Fingerprint Dive into the research topics of 'Weak lensing of Type Ia Supernovae from the Dark Energy Survey'. Together they form a unique fingerprint.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8282003998756409, "perplexity": 3730.187700528742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710734.75/warc/CC-MAIN-20221130092453-20221130122453-00253.warc.gz"}
http://mathhelpforum.com/calculus/169864-derivative-fourier-transform.html	# Math Help - derivative of the fourier transform 1. ## derivative of the fourier transform my book defines the fourier transform as F(ξ) = integral (-infinite to infinite) f(x)e^iξx dx. my book takes the derivative of the fourier transform and does: DF(ξ) = integral (D_ξ (f(x) e^iξx)) dx = integral (ixf(x)e^iξx) dx = ixF(ξ), where D_ξ is the partial derivative with respect to ξ my book signifies the fourier transform by f with a small hat on top. i put F instead since i don't know how to type f with a small hat on top. so the result of the differentiation is the following: ixf(ξ) and the whole thing has a small hat on top of it. what i don't understand is why isn't the derivative iξf(ξ) all with a hat on top? from the definition of the fourier transform i gathered that you take whatever is in front of the e^iξx and you replace the x with ξ and you put a hat on it. in the derivative, the function in front of e^iξx is ixf(x), but couldn't it be written as g(x) = ixf(x)? then by the definition the integral would be g(ξ) with a hat on top which would be iξf(ξ) with a hat on top wouldn't it? my apologies if this is messy to read. thanks. 2. Originally Posted by oblixps my book defines the fourier transform as F(ξ) = integral (-infinite to infinite) f(x)e^iξx dx. my book takes the derivative of the fourier transform and does: DF(ξ) = integral (D_ξ (f(x) e^iξx)) dx = integral (ixf(x)e^iξx) dx = ixF(ξ), where D_ξ is the partial derivative with respect to ξ my book signifies the fourier transform by f with a small hat on top. i put F instead since i don't know how to type f with a small hat on top. so the result of the differentiation is the following: ixf(ξ) and the whole thing has a small hat on top of it. what i don't understand is why isn't the derivative iξf(ξ) all with a hat on top? from the definition of the fourier transform i gathered that you take whatever is in front of the e^iξx and you replace the x with ξ and you put a hat on it. in the derivative, the function in front of e^iξx is ixf(x), but couldn't it be written as g(x) = ixf(x)? then by the definition the integral would be g(ξ) with a hat on top which would be iξf(ξ) with a hat on top wouldn't it? my apologies if this is messy to read. thanks. Look at the derivation of the FT of a derivative, you should have something like: $\displaystyle \frac{d}{d\xi}\left( \mathfrak{F}(f(x))(\xi) \right)=-i \widehat{xf(x)}(\xi)=-i \mathfrak{F}(xf(x))(\xi)$ (You should not have the minus sign as you appear to have a definition of the FT at variance with what I regard as the usual definition) CB 3. thanks for clearing this up. I was just a little confused the first time I encountered this in my textbook.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9866310358047485, "perplexity": 567.6217499063556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657131304.74/warc/CC-MAIN-20140914011211-00121-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://www.physicsforums.com/threads/find-a-force.396302/	# Homework Help: Find A Force 1. Apr 18, 2010 ### nmegabyte While two forces act on it, a particle is to move at the constant velocity v = (3 m/s) - (4 m/s) . One of the forces is F1 = (4 N) + (- 7 N) . What is the other force? i know the formula Fnet=ma and i cant use it to find mass in this case how can i find mass first so and i know that velocity is my acceleration here 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 2. Apr 18, 2010 I assume your notation "F1 = (4 N) + (- 7 N)" is a vector component notation? I.e., 4 in the x-direction, and -7 in the y-direction? Whatever it is, what does Newton's 2nd law tell you about the not force, if the velocity is constant? 3. Apr 18, 2010 ### nmegabyte if it is moving, it continues to move at constant velocity i believe so 4. Apr 18, 2010 Yes, but we know it is moving with a constant velocity. So? What does the sum of the forces equal? 5. Apr 18, 2010 ### nmegabyte i guess zero they cancel each other? 6. Apr 18, 2010 ### nmegabyte but i need to find a mass in order to find F2, F2 = ma - F1 7. Apr 18, 2010 Why do you think you need the mass? This is only about the forces. Yes, they cancel out. You know one of them. So you can find the other one. 8. Apr 18, 2010 ### nmegabyte F2 = ma - F1 using this formula i can find the other force how i am not that great at physics. 9. Apr 18, 2010 Look, F1 + F2 = 0, so F2 = -F1, right? And you are given F1. 10. Apr 18, 2010 ### nmegabyte now i understand why cuz when constant velocity acc is 0 and when acc is zero the net force is 0 so F1+F2 =0 is this correct conclusion? 11. Apr 18, 2010	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9254558086395264, "perplexity": 1090.5010517487276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826856.55/warc/CC-MAIN-20181215105142-20181215131142-00055.warc.gz"}
http://mathoverflow.net/questions/90918/sufficient-conditions-for-gradient-descent-convergence?sort=newest	# Sufficient conditions for gradient descent convergence I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps treating $\hat{\nabla}$ as the true gradient: $$x^{t+1} = x^{t} - \lambda \hat{\nabla}$$ What are sufficient conditions on $g$ such that this converges to the optima? In particular, are there results of the form "if $\\|\hat{\nabla}-\nabla\\|<\epsilon$ and some-property-of-$f$ then gradient descent treating $\hat{\nabla}$ as the gradient converges to the optima"? - Are there further restrictions on $g$? Do you know the form of $g$? Is it linear? Nonlinear? It seems to me that it's difficult to develop general sufficiency conditions on an arbitrary $g$. For instance, we may say that a necessary condition might be that $g$ ought to have the same sign as $\nabla f$, but if $g$ is a nonlinear function that changes signs depending on the region, this statement may be problematic. A trivial sufficiency condition would be that $g(x) = x$. – Gilead Mar 11 '12 at 18:16 Oh, it seems that I just took $g$ to be the identity map. More generally, we can have $g(\nabla f(x)) = D\nabla f(x)$, where $D$ is a strictly positive definite matrix. That would ensure that $g(\nabla f(x))$ is a descent direction. Given that, and some minor technical assumptions, should ensure sufficiency. However, if $g$ is allowed to be a nonlinear transformation, then things can be trickier. However, maybe you have a more specific $g$ is mind? – Suvrit Mar 11 '12 at 18:26 There are certainly convergence theorems that work as long as the step direction is a descent direction for the function being minimized and the step length is selected so as to satisfy some special conditions (e.g. the Armijo conditions.) I don't think it's possible to say much more without knowing exactly what's being done to the gradient. – Brian Borchers Mar 11 '12 at 18:29 Based on Brian's comment, perhaps broad sufficiency conditions would be that $g$ (1) maps to a descent direction; and (2) satisfies Armijo conditions in the domain of interest. Those are pretty general conditions, but it's harder to get more specific without additional restrictions on $g$. – Gilead Mar 12 '12 at 0:35 Ah, there may be another difficulty here. Hinge loss functions are non-smooth, and many standard convergence proofs generally stipulate that the function of interest is at least once-differentiable everywhere. The standard gradient descent method is undefined for non-differentiable functions. One may need to look at subgradient methods or bundle methods which may have completely different convergence criteria. (I'm not familiar with those) – Gilead Mar 12 '12 at 15:04 Ok, after reading your comments, and some thinking, here is one way to tackle what seems to be going on: 1. You have a nondifferentiable loss function. 2. You wish to compute a subgradient of the loss, but the subgradient is too expensive to compute 3. So you compute only a small part of some subgradient. This is, the classic setting of an inexact subgradient projection method, where essentially you are iterating as follows: $$x^{k+1} = \Pi_X(x^k - \alpha_k(g^k+e^k)),$$ where $g^k$ is a subgradient of your loss function and $e^k$ is an error in the subgradient computation, which can be used to model the fact that you are not using all the components of the loss function to compute a subgradient. Depending on what you are doing, this type of method might be cast as an online, stochastic, or incremental subgradient method. I recommend that you have a look at the recent survey, your inexact computations will probably fit the general frameworks discussed therein. D. P. Bertsekas, "Incremental Gradient, Subgradient, and Proximal Methods for Convex Optimization: A Survey", Lab. for Information and Decision Systems Report LIDS-P-2848, MIT, August 2010; this is an extended version of a chapter in the edited volume Optimization for Machine Learning, by S. Sra, S. Nowozin, and S. J. Wright, MIT Press, Cambridge, MA, 2012, pp. 85-119. - I'm confused- the original poster referred to the gradient as if this was a smooth problem. How did we jump to considering nonsmooth problems? – Brian Borchers Mar 12 '12 at 19:44 @Brian, in a comment, he mentioned that the objective $f$ was a sum of hinge losses. en.wikipedia.org/wiki/Hinge_loss – Gilead Mar 12 '12 at 22:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9045230746269226, "perplexity": 225.8471282632254}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860127496.74/warc/CC-MAIN-20160428161527-00076-ip-10-239-7-51.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/pressure-gradient-term-in-navier-stokes.803212/	# Pressure gradient term in Navier-Stokes 1. Mar 14, 2015 ### Urmi Roy Hi, I've been thinking about the Navier-Stokes equations and trying to build skill in implementing it in various situations. In a particular situation, if I have a fluid flowing down an inclined surface such that it forms a film of finite height which is smaller than the length of flow, there is no applied pressure. So am I allowed to cross off the pressure term in the momentum conservation equation along the direction of flow? In a detailed solution to this problem, our instructor non-dimensionalizes the equation to show that the pressure term goes away. However, I'm wondering, since there is no applied pressure, can't I just cross it off (like Couette flow)? Also, in another problem (the situation is that oil is flowing up an inclined surface submerged in water) the instructor replaces the pressure gradient by the buoyancy force . So in the inclined flow case, am I not allowed to simply replace the pressure term by the hysdrostatic pressure? I think my question boils down to whether the pressure term in the NS equation is an external force or internal property of the fluid. 2. Mar 15, 2015 ### Staff: Mentor It doesn't have to be non-dimensionalized to do this. The force balance in the direction perpendicular to the surface at any location shows that the pressure is varying hydrostatically in that direction. So it's equal to the atmospheric pressure plus ρgcosθy, where y is the distance measured downward normal to the air interface. If the film thickness is constant, this means that the pressure is not varying with distance along the incline. I don't follow the description of the oil problem. Also, I don't understand what you mean about external and internal. Pressure is part of the stress tensor, so it has to be internal, except that it has to match the adjacent medium at an interface. Chet 3. Mar 15, 2015 ### Urmi Roy So in the problem the film thickness is actually varying (increasing) along the flow direction.So if I follow your method and draw a force balance on it, it means the pressure near the end of the film where its thickness is highest is greater? The confusion relating to the 'internal'/'external'-ness of the pressure arises because of the couette flow with no pressure applied i.e. shear flow example. Here fluid flows from one side to another due to the external force of the plate. So there is no external pressure gradient, but the fluid should develop a pressure gradient within itself so that it does indeed flow from one side to the other. Does this sound correct? 4. Mar 15, 2015 ### Staff: Mentor I think you forgot to include the word "if" somewhere in this text. In your problem, is the film thickness changing as the fluid flows down the incline? Even so, if the rate of change of the thickness with distance along the incline is very small, you can still approximate the pressure gradient along the incline as negligible. I'm having trouble understanding your description. Are you talking about shear flow between concentric rotating cylinders, or are you talking about shear flow between parallel plates? Chet 5. Mar 17, 2015 ### Urmi Roy Ok, so if the rate of change of thickness along the plane is small, the film is of practically uniform thickness along the incline...and the only net force on a small slice of fluid is gravitational. If the film were changing thickness (say increasing along the incline), the net force on a slice of fluid is the gravitational force (pointing down the incline) minus some net hydrostatic force (up the incline)...then there is a pressure gradient in the fluid such that pressure is greater as you go down the incline? Let's say shear flow between parallel plates. Doing a similar force balance above, gravity acts downward (perpendicular to plates) so it doesn't cause acceleration of fluid. There is no applied pressure gradient. The fluid moves because of momentum transfer from the moving plate. If we had shear flow between rotating cylinders, it's a similar situation except that if the cylinders are vertical, gravity will come into play and a pressure gradient will come into play. Is this all okay? 6. Mar 17, 2015 ### Staff: Mentor Yes. If the film thickness is varying rapidly, there will be a pressure gradient along the direction of flow. Just think of a horizontal trough (open channel) in which you are adding fluid rapidly at the left end of the trough so that there is a depth gradient along the direction of flow from left to right. This depth gradient causes a pressure gradient along the trough, which results in the flow. Yes. The pressure gradient will be in the vertical direction (hydrostatic), perpendicular to the shear flow. So, the vertical pressure gradient will not affect the shear flow. There will also be a radial pressure gradient to provide the centripetal force. However, this also will be perpendicular to the velocity gradient, so it won't affect the shear flow either. To affect the shear flow, you would need to have a circumferential pressure gradient, but this is prevented by the axial symmetry. Chet 7. Mar 18, 2015 ### Urmi Roy Ok thanks Chet, I'm feeling much better about this now. So to summarize, the pressure in a fluid can be from two main sources: 1. hydrostatic pressure 2. applied pressure . Also, just because a fluid is moving, it doesn't mean that the fluid 'creates its own pressure gradient within itslef' (what I was previously calling 'internal pressure'), (like in the shear flow between parallel plates). 8. Mar 18, 2015 ### Staff: Mentor Well, if you look at the terms in the Navier Stokes equations, you will see that it can be from three sources: 1. viscous stresses (e.g., flow in a pipe) 2. gravitational forces (hydrostatic) Applied pressure is what you have to impose on the boundaries of the fluid to overcome the viscous forces, gravitational forces, and inertial forces. There is a close connection between the pressure and the boundary conditions. This is because, for an incompressible fluid, the pressure is determined only up to an arbitrary constant. It is necessary for the boundary conditions to establish the value of that constant. For example, in the case of the flow down an inclined plane, the pressure at the free surface was atmospheric. But, please, don't try too hard to categorize it. After you have solved some problems, you well get a much better feel for how all this works. You just need to get some experience and have patience. Yes. As I said, you will soon have a feel for all this. Chet 9. Mar 19, 2015 ### Urmi Roy Ok, thanks I think I get it :-) 10. Apr 27, 2015 ### Urmi Roy So I have a related question to what we were discussing earlier. I usually don't need to non-dimensionalize the momentum balance in the y-direction, but this one time I did and I realized I was a bit confused. Please take a look at the description of my non-dimensionalization procedure in the attached. It turns out that after proper rearrangement, the non-dimensionalization in the y direction is very similar to the x-direction one (when I rearrange to bring in the Re_L i.e. the Re taking characteristic length in x direction), except with (L/h) everywhere. Now in the situation L>>h, it looks like the convective term foes away entirely and we're left with the pressure term., the gravity term and the viscous terms. So now without the Re being >>1, we can't eliminate the viscous terms. However in most situations we are only supposed to be left with the pressure and gravitational term so that it is just hydrostatics. My Prof got to that conclusion by not going to the extent of rearranging everything to get the Re_L and just said all terms other than the pressure term and gravity term are multiplied by (h/L)^2 so they're small. However isn't there a way to get to that conclusion from where I ended up? #### Attached Files: File size: 37 KB Views: 149 • ###### (2).JPG File size: 31.7 KB Views: 149 11. Apr 27, 2015 ### Staff: Mentor I'm pretty sure I can help you, but I'm unable to read the attachments you sent. It's just too small on my computer screen, and, unlike ordinary web pages, I'm unable to zoom on the attachment. Any chance you can Latex it? Chet 12. Apr 28, 2015 ### Urmi Roy So unfortunately I've never used Latex before :-( ...would probably be a useful thing to learn. I've taken a different set of pictures, more zoomed in, so pleasetake a look if this is better. If not, I'll try to learn Latex :-) Thanks a lot! 13. Apr 28, 2015 ### Staff: Mentor OK. Here's what I got, assuming y is the vertical direction: I set $P_c=ρu^2$. So, for the x direction: $$\left[\frac{\partial v_x^}{\partial t^}+v_x^\frac{\partial v_x^}{\partial x^}+v_y^\frac{\partial v_x^}{\partial y^}\right]=-\frac{\partial p^}{\partial x^}+\frac{ηL}{ρuh^2}\left[\left(\frac{h}{L}\right)^2\frac{\partial ^2 v_x^}{\partial x^{2}}+\frac{\partial ^2 v_x^}{\partial y^{2}}\right]$$ For the y direction, I get: $$\left(\frac{h}{L}\right)^2\left[\frac{\partial v_y^}{\partial t^}+v_x^\frac{\partial v_y^}{\partial x^}+v_y^\frac{\partial v_y^}{\partial y^}\right]=-\frac{\partial p^}{\partial y^}+\frac{ηL}{ρuh^2}\left[\left(\frac{h}{L}\right)^4\frac{\partial ^2 v_y^}{\partial x^{2}}+\left(\frac{h}{L}\right)^2\frac{\partial ^2 v_y^}{\partial y^{2}}\right]+\frac{gh}{u^2}$$ Chet 14. Apr 28, 2015 ### Urmi Roy Yup, that's what I get too,but if you rearrange a little more (y direction), you will get Reynolds number (=rhouL/mu) multiplied by (L/h)^2 in the viscous terms...except your rho has gone away. So then how would you prove that the y direction equation boils down to hydrostatics? Currently in your equation, you have (h/L)^2 multiplied into all but the pressure and gravity terms, so they're small....but after the rearrangement to get the Re in there, the situation looks a but different. 15. Apr 28, 2015 ### Staff: Mentor Well, for this particular situation, in which the length scale in the y direction is much smaller than the length scale in the x direction, the more appropriate definition of the Reynolds number would be $Re = \frac{ρuh^2}{ηL}$. So presumably, with this definition, Re would be held finite while h/L becomes small. This would be accomplished with adjustment to u and/or η. Chet 16. Apr 29, 2015 ### Urmi Roy Ok, so is it okay to set Pc=rhou^2 without knowing if the flow is viscous dominated? Also, in relation to what you were saying, if the h is not much smaller than L, then it seems that the y direction equation doesn't boil down to only hydrostatics...then we'd have a problem. Anyway, I think I'm getting it! 17. Apr 29, 2015 ### Staff: Mentor Sure. Don't forget that the shear stress is equal to the friction factor times rhou^2 even in laminar flow. Another possible choice is ηu/h. If you have time, why don't you try that and see if it leads to anything interesting. I wouldn't characterize this as having a problem. The problem is just more 2D in character, and you just can't use the same simplifying approximation to solve the equations. That doesn't mean that they can't be solved. Chet 18. May 2, 2015 ### Urmi Roy This is a very interesting point, about how even in laminar flow the shear stress is proportional to rho*u^2! I have tried out the other non-dimensionalization with the viscous dominated flows and I just get a 1/Re with the pressure term. Hopefully none of my problems on the exam should end up with that :-) 19. May 2, 2015 ### Staff: Mentor Actually, no. Don't forget that the friction factor is inversely proportional to the Reynolds number, so the dimensional pressure in rectilinear laminar flow is directly proportional to u. Chet Similar Discussions: Pressure gradient term in Navier-Stokes	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8633514046669006, "perplexity": 609.9467595318649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948581053.56/warc/CC-MAIN-20171216030243-20171216052243-00098.warc.gz"}
https://www.physicsforums.com/threads/velocity-of-transverse-waves-problem.609636/	# Velocity of Transverse Waves problem 1. May 28, 2012 ### rusty65 1. The problem statement, all variables and given/known data Two children are sending signals along a cord of total mass 0.54 kg tied between tin cans with a tension of 37 N. It takes the vibrations in the string 0.53 s to go from one child to the other. How far apart are the children? 2. Relevant equations Velocity of transverse wave on a cord = sqrt(F_t/$\mu$) F_t = Tension Force $\mu$ = mass per unit length -> m/l 3. The attempt at a solution I attempted plugging the given values into the formula for velocity of a transverse wave on a cord, and came up with a distance of 4.387 meters. However, after getting the problem wrong (on masterphysics) I realized that the mass given for the cord is its total mass rather than mass per unit length. Seeing as what I am asked to find is the distance between the children (length of the cord) I dont see any way of solving this problem. Am i simply missing the proper formula? Any help would be greatly appreciated. 2. May 28, 2012 ### Infinitum Hello rusty65!! The mass per unit length, as you've written in your relevant equations is m/l. Putting this into the velocity equation and multiplying by time, $vt = t\sqrt{\frac{lF}{{m}}}$ What is "vt" in that above equation? 3. May 29, 2012 ### rusty65 vt is equal to the distance, but the trouble im having is that the distance, d, that i am attempting to find is equal to the length of the string, l. So i must either be using the wrong formula, or some key piece of information is escaping me. This is where im at right now, using the information given: d = t√((F_t * l)/m) ---plugged in----> d = 0.53√(37l/0.54) So ive still got two unknowns, d and l, which, according to the wording of the problem, seem to me to be equal to one another. 4. May 29, 2012 ### rusty65 Scratch that, I figured it out! Since d = l, I replaced l with d in the equation. d = 0.53sqrt(37l/0.54) ---> d = 0.53sqrt(37d/0.54) d/sqrt(d) = 0.53sqrt(37/0.54) ---> d/sqrt(d) = 4.387 d/sqrt(d) = d^(1/2) ---> sqrt(d) = 4.387 d = (4.387)^2 d = 19.246! Took me a while to get it through my thick head, but I got it now. And thanks for the help! 5. May 29, 2012 ### Infinitum Yep! That is what I was suggesting. Good to see you figured it out Similar Discussions: Velocity of Transverse Waves problem	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.94696044921875, "perplexity": 1007.415575929759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106358.80/warc/CC-MAIN-20170820073631-20170820093631-00505.warc.gz"}
https://economy.spbstu.ru/en/article/2010.19.29/	# The economic estimation efficiency of introduction bank cards in payment system at the enterprise Authors: Abstract: Being based on that over 80 % of the market of bank payment cards is presented by the cards which have been given out within the limits of "salary" projects, in the given research the technique is offered and the economic estimation of efficiency of introduction of bank payment cards is spent to payment system at the enterprise.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8424115180969238, "perplexity": 1848.9987607217934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370497042.33/warc/CC-MAIN-20200330120036-20200330150036-00193.warc.gz"}
https://open.library.ubc.ca/cIRcle/collections/48630/items/1.0044035	# Open Collections ## BIRS Workshop Lecture Videos ### Skew t-Copula and its Estimation: For Application to Risk Aggregation Yoshiba, Toshinao #### Description Correlation structure of risk factors matters to financial portfolio risk management. When the risk factors are specified by some assets’ return, risk managers have to care about lower tail dependence of those factors rather than upper. On the other hand, it is practical to specify overall nontail dependence by sample linear or rank correlation matrix. Based on this background, we focus on the application of skew t-copula. Skew t-copula is defined by a multivariate skew t-distribution and its marginal. As indicated in Kotz and Nadarajah (2004), various types of multivariate skew t-distribution have been proposed. That implies various types of skew t-copula can exist. Three types of skew t-copula are known so far. The first type was mentioned in Demarta and McNeil (2005), which is based on multivariate version of generalized hyperbolic (GH) skew t-distribution proposed in Barndorff-Nielsen (1977) (See also Aas and Haff (2006)). The second type was constructed in Smith et al. (2012) based on Sahu et al. (2003). Kollo and Pettere (2010) tried to construct the third type based on Azzalini and Capitanio (2003). This paper is constructed as follows. First, we improve the skew t-copula approach proposed in Kollo and Pettere (2010) and derive log-likelihood function to estimate the parameters by maximum likelihood estimation (MLE). Second, we indicate that MLE for the skew t-copula requires fast and accurate quantile calculation for univariate skew t-distribution. Third, we show the difference between upper and lower tail dependence of the skew t-copula citing Fung and Seneta (2010) and Bortot (2010). Finally, we introduce a method of moment approach for the parameter estimation of the skew t-copula.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9366476535797119, "perplexity": 1738.7568606256018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335504.37/warc/CC-MAIN-20220930212504-20221001002504-00351.warc.gz"}
https://www.physicsforums.com/threads/determining-the-number-of-solutions-using-generating-functions.224296/	# Determining the Number of Solutions Using Generating Functions 1. Mar 25, 2008 ### alec_tronn 1. The problem statement, all variables and given/known data Determine the number of solutions in nonegative intergers to the equation: a + 2b + 4c = 10$$^{30}$$ 2. Relevant equations The generating function I've found is f(x) = 1/[(1-x$$^{4}$$)(1-x$$^{2}$$)(1-x)] 3. The attempt at a solution I'm pretty sure I need to get from here to an explicit formula, but I'm not sure how to start. Any hints to get me started on this one? 2. Mar 26, 2008 ### alec_tronn Alright, using the Apart function in Mathematica, I separated the generating function into: [-1/(8(-1+x)^3)] + [1/(4(-1+x)^2)]- [9/(32(-1+x))]+ [1/(16(1+x)^2)]+ [5/(32(1+x))]+ [(1+x)/(8(1+x^2))] Then, I turned those each into the followings infinite series: [(-1/2)$$\Sigma$$(n+1)x^n][(1\2)$$\Sigma$$x^n], so the coefficient for 10^30 is -1\4[((10^30)/2) +1], (-1\2)$$\Sigma$$(n+1)x^n, so the coefficient is -1\2(10^30+1) (9\30)$$\Sigma$$x^n, so the coefficient is 9\32 (1\4)$$\Sigma$$(n+1)(-x)^n, so the coefficient is (1\4)(10^30+1) (5\32)$$\Sigma$$(-x)^n, so the coefficient is 5\32 (1\8)$$\Sigma$$x^(2n+1), so the coefficent is 1\8, I add up all the coefficients is (-1\2)((10^30)+1)+(1\2) Thats negative, so it can't be the answer. I'm awfully rusty in the Calc 2 skills of making functions like that into series... so if anybody could catch any of my mistakes I'd be forever grateful!!!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9267874956130981, "perplexity": 1414.4978092669292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279368.44/warc/CC-MAIN-20170116095119-00147-ip-10-171-10-70.ec2.internal.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/diagram-shows-block-mass-frictionless-horizontal-surface-seen--three-forces-magnitudes-app-q1269972	# Newtons laws problem 0 pts ended The diagram below shows a block of mass on a frictionless horizontal surface, as seen from above. Three forces of magnitudes , , and are applied to the block, initially at rest on the surface, at angles shown on the diagram. In this problem, you will determine the resultant (total) force vector from the combination of the three individual force vectors. All angles should be measured counterclockwise from the positive x axis (i.e., all angles are positive). A)Calculate the magnitude of the total resultant force acting on the mass. Express your answer in Newtons to three significant figures. B)What is the magnitude of the mass's acceleration vector, ? 2 sig figs C)What angle does make with the positive x axis? 2 sig figs D)What is the direction of ? In other words, what angle does this vector make with respect to the positive x axis? 2 sig figs E)How far (in meters) will the mass move in 5.0 s? 2 sig figs F)What is the magnitude of the velocity vector of the block at ? 2 sig figs G)In what direction is the mass moving at time ? That is, what angle does the velocity vector make with respect to the positive x axis? 2 sig figs	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9415518045425415, "perplexity": 413.95703600754405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00556-ip-10-147-4-33.ec2.internal.warc.gz"}
http://www.freedomandpower.ws/index.php/al-lessons/46-lesson-011-what-is-science-and-how-it-works-part-3-scientific-truth-or-fact	User Rating: 0 / 5 You may have heard expressions that have the word “scientific” behind them. Because of this, most people assume that they are the same as saying “truth” or “fact”. Nothing could be farther from the truth. As we have seen before, science creates models that describe portions of our reality. Then, we attempt to verify those models using measurements. If the measurements coincide with the forecast from the model, then we say that the model is true. Now, allow me to throw a wrench in the process. What happens if scientists John, Indi, Muller, Xi and Santiago repeat the same test independently and obtain results that seem to verify the model. But, scientist Umu does not. Is the model then true or false? Here is where the definition of scientific truth or fact comes into play: A scientific model is true or a fact when the majority of the scientific tests (or experiments) confirm it. Notice that the definition does not say that all tests must confirm, but the majority. Why is this important? Because as science creates imperfect models, we will also obtain imperfect results. However, we can have sufficient reassurance that the model is sufficiently close to reality to be useful if the majority of the experiment confirm it. And this is the best that science can do! Because of this, science is not a religion nor a certainty nor a truth. Note: please see the Glossary if you are unfamiliar with certain words. Continue to What is science and how it works – Part 4	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8759771585464478, "perplexity": 722.4186766351382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202872.8/warc/CC-MAIN-20190323141433-20190323163433-00454.warc.gz"}
http://www.ams.org/joursearch/servlet/DoSearch?f1=msc&v1=76D05&jrnl=one&onejrnl=tran	# American Mathematical Society My Account · My Cart · Customer Services · FAQ Publications Meetings The Profession Membership Programs Math Samplings Policy and Advocacy In the News About the AMS You are here: Home > Publications AMS eContent Search Results Matches for: msc=(76D05) AND publication=(tran) Sort order: Date Format: Standard display Results: 1 to 10 of 10 found Go to page: 1 [1] Lorenzo Brandolese and Maria E. Schonbek. Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Amer. Math. Soc. 364 (2012) 5057-5090. Abstract, references, and article information View Article: PDF [2] Jean-Yves Chemin and Isabelle Gallagher. Large, global solutions to the Navier-Stokes equations, slowly varying in one direction. Trans. Amer. Math. Soc. 362 (2010) 2859-2873. MR 2592939. Abstract, references, and article information View Article: PDF This article is available free of charge [3] Alexey Cheskidov. Blow-up in finite time for the dyadic model of the Navier-Stokes equations. Trans. Amer. Math. Soc. 360 (2008) 5101-5120. MR 2415066. Abstract, references, and article information View Article: PDF This article is available free of charge [4] Rabi N. Bhattacharya, Larry Chen, Scott Dobson, Ronald B. Guenther, Chris Orum, Mina Ossiander, Enrique Thomann and Edward C. Waymire. Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations. Trans. Amer. Math. Soc. 355 (2003) 5003-5040. MR 1997593. Abstract, references, and article information View Article: PDF This article is available free of charge [5] Franco Flandoli and Marco Romito. Partial regularity for the stochastic Navier-Stokes equations. Trans. Amer. Math. Soc. 354 (2002) 2207-2241. MR 1885650. Abstract, references, and article information View Article: PDF This article is available free of charge [6] A. Fursikov, M. Gunzburger and L. Hou. Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications. Trans. Amer. Math. Soc. 354 (2002) 1079-1116. MR 1867373. Abstract, references, and article information View Article: PDF This article is available free of charge [7] Avner Friedman and Juan J. L. Velázquez. Time-dependent coating flows in a strip, Part I: The linearized problem. Trans. Amer. Math. Soc. 349 (1997) 2981-3074. MR 1422605. Abstract, references, and article information View Article: PDF This article is available free of charge [8] Calixto P. Calderón. Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$ . Trans. Amer. Math. Soc. 318 (1990) 179-200. MR 968416. Abstract, references, and article information View Article: PDF This article is available free of charge [9] Calixto P. Calderón. Addendum to the paper: Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$'' [Trans.\ Amer.\ Math.\ Soc.\ {\bf 318} (1990), no.\ 1, 179--200; MR0968416 (90k:35199)] . Trans. Amer. Math. Soc. 318 (1990) 201-207. MR 1018571. Abstract, references, and article information View Article: PDF This article is available free of charge [10] David Hoff. Global existence for $1$D, compressible, isentropic Navier-Stokes equations with large initial data . Trans. Amer. Math. Soc. 303 (1987) 169-181. MR 896014. Abstract, references, and article information View Article: PDF This article is available free of charge Results: 1 to 10 of 10 found Go to page: 1	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8144943714141846, "perplexity": 2299.275484115292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701163729.14/warc/CC-MAIN-20160205193923-00306-ip-10-236-182-209.ec2.internal.warc.gz"}
https://www.investopedia.com/terms/r/random-walk-index.asp	## What Is the Random Walk Index? The random walk index (RWI) is a technical indicator that compares a security's price movements to random movements in an effort to determine if it's in a statistically significant trend. It can be used to generate trade signals based on the strength of the underlying price trend. ### Key Takeaways • The random walk index has two lines, a RWI High and RWI Low, which measure uptrend and downtrend strength. • When the RWI High is above the RWI Low, it means there is more upward strength than downward strength, and vice versa. • When either the RWI High or RWI Low is above one, it indicates a strong, non-random, trend is present. Readings below one mean movement could be random because there is not enough strength to indicate otherwise. ## Understanding the Random Walk Index The random walk index was created by Michael Poulos in order to determine if a security's current price action is exhibiting "random walk" or is the result of a statistically significant trend, higher or lower. Random walk refers to market or security movements that are within the realm of statistical "noise" levels and not consistent with a confirmed or definable trend. The technical indicator was originally published in Technical Analysis of Stocks and Commodities magazine in a 1990's article entitled, "Of Trends And Random Walks." Market trend and random walk studies go back for decades, highlighted by R.A. Stevenson's publication of "Commodity Futures: Trends or Random Walks?" in the March 1970 issue of The Journal Of Finance. ## Calculating the Random Walk Index William Feller, a mathematician who specialized in probability theory, proved that the bounds of randomness, also known as displacement distances, could be calculated by taking the square foot of the number of binary events, which refer to two-sided outcomes with equal probability (like a coin toss). Logically speaking, any movement outside of these bounds suggests the movement is not inherently random in nature. The RWI applies these mathematical principles when measuring an uptrend and downtrend to determine if it's random or statistically meaningful. Because the indicator measures both uptrend and downtrend strength, the indicator has two lines and requires separate calculations for both. The calculation for high periods, or RWI High, is: \begin{aligned} &\text{RWI High} = \frac { \text{High} - \text{Low}_n}{ \text{ATR} \times \sqrt{n} } \\ &\textbf{where:} \\ &n = \text{number of days in the period} \\ &\text{ATR} = \text{average true range} \\ \end{aligned} In other words, if you're calculating the RWI High of the last five days, take the high from today minus the low from the prior period and calculate RWI High. Then calculate using today's high minus the low two days ago. Do this for each day going back five trading sessions. Your RWI High value is the highest value of the last five days or for however many periods (n) were chosen. RWI Low is calculated as follows: \begin{aligned} &\text{RWI Low} = \frac { \text{High}_n - \text{Low} }{ \text{ATR} \times \sqrt{n} } \\ \end{aligned} The method is similar to the approach above, except now we will use today's low and the high from the prior period to create the first calculation. Then use the high from two days ago. Do this for each of the n periods. The RWI Low value is the lowest number of the n calculations completed. Each day (or period) the calculations are completed again. ## Trading the Random Walk Index The random walk index is typically used over two to seven periods for short-term trading and scalping and eight to 64 periods for long-term trading and investments. Market players may wish to experiment with these settings to determine what works best for their overall strategies. Readings above 1.0 indicate that a security is trending higher or lower, while readings below 1.0 suggest that a security may be moving randomly. If RWI Low is above one, it indicates a strong downtrend; if RWI High is above one, it indicates a strong uptrend. Often times, traders and market timers will enter long positions when a long-term RWI High is greater than 1.0 and the short-term RWI Low is also above 1.0. This means the trader tracks two RWI calculation, a longer-term one, say 64-periods, and a short-term one, say seven-periods. A trader buys when the long-term RWI High is above 1.0, which indicates a long-term strong uptrend, but the short-term RWI Low is also above 1.0, showing that in the short term the price has dropped, providing a favorable entry into the long-term uptrend. Short positions may be entered when the long-term RWI Low is greater than 1.0 and the short-term RWI High peaks above one as well. Some traders may look to use crossovers of the two lines to indicate potential trades. This will work well when strong trends develop, but it will result in lots of losing trades if the price doesn't trend well since crossovers could occur without a strong trend resulting. That said, some traders may wish to utilize this approach, potentially in conjunction with other forms of technical analysis. ## Example of How to Use the Random Walk Index The daily chart of Apple Inc. (AAPL) has a 30-period RWI indicator applied to it. When the price is falling the red line, or RWI Low, is on top. When the price is rising the green line, or RWI High, is on top. When either one of these lines is above one, the black horizontal line, it indicates a strong trend. On the left, there is a strong uptrend. The RWI High moves above 1.0, and the RWI Low is below 1.0. Then a strong downtrend begins. The RWI Low moves above 1.0, and the RWI High is well below 1.0. This is followed by another uptrend with similar conditions to the prior uptrend. Then the stock enters a weak trending period. Neither the RWI Low or High maintains its position above 1.0 for long. For a brief period, the two lines even become tangled around the zero mark, signaling a very weak trend, or choppy trading, in both directions. ## The Difference Between the Random Walk Index and the Average Directional Index (ADX) These two indicators look quite similar and, in fact, are quite similar. The average directional index (ADX) is composed of directional movement lines (DI+ and DI-) which move in a very similar ways to the RWI Low and High. The ADX is a third line on the ADX indicator and shows the strength of the trend. Readings above 25 indicate a strong trend. ## Limitations of the Random Walk Index The RWI is a lagging indicator. It uses past data in its calculation and there is nothing inherently predictive about it. While the indicator can move above one to signal a strong trend, it can easily slip back below one very quickly. It can also go from a weak trend to a strong trend with little forewarning from the indicator. Waiting for the indicator to move above one before taking a trade in that direction can sometimes result in a poor entry. The price has already been moving in that direction for some time and may be ready to reverse or enter a pullback. The random walk index is best used in conjunction with price action analysis or other forms of technical analysis.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8240063190460205, "perplexity": 2421.254452329407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143963.79/warc/CC-MAIN-20200219000604-20200219030604-00075.warc.gz"}
https://www.goldbook.iupac.org/terms/view/S06219	## synchronous https://doi.org/10.1351/goldbook.S06219 A @C01234@ in which the @P04845@ concerned (generally bond rupture and bond formation) have progressed to the same extent at the @T06468@ is said to be synchronous. The term figuratively implies a more or less synchronized progress of the changes. However, the progress of the bonding change (or other @P04845@) has not been defined quantitatively in terms of a single parameter applicable to different bonds or different bonding changes. The concept is therefore in general only qualitatively descriptive and does not admit an exact definition except in the case of concerted processes involving changes in two identical bonds.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8793854713439941, "perplexity": 1423.3953833257701}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330968.54/warc/CC-MAIN-20190826042816-20190826064816-00370.warc.gz"}
http://qchu.wordpress.com/2011/01/02/the-schrodinger-equation-on-a-finite-graph/	Feeds: Posts ## The Schrödinger equation on a finite graph One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation by hand, but it is conceptually cleaner to use the symmetries of the situation and representation theory. Deducing these results using representation theory has the added benefit that it identifies which parts of the situation depend only on symmetry and which parts depend on the particular form of the Hamiltonian. This is nicely explained in Singer’s Linearity, symmetry, and prediction in the hydrogen atom. For awhile now I’ve been interested in finding a toy model to study the basic structure of the arguments involved, as well as more generally to get a hang for quantum mechanics, while avoiding some of the mathematical difficulties. Today I’d like to describe one such model involving finite graphs, which replaces the infinite-dimensional Hilbert spaces and Lie groups occurring in the analysis of the hydrogen atom with finite-dimensional Hilbert spaces and finite groups. This model will, among other things, allow us to think of representations of finite groups as particles moving around on graphs. Physics I am going to be vague about the mathematics here because it’s all about to get a lot simpler anyway. I am also not going to justify the physics because I am not particularly knowledgeable in this area. In quantum mechanics, the states of a quantum system in a configuration space $X$ are described by unit vectors in the Hilbert space $L^2(X)$ up to phase, that is, up to multiplication by a complex number of norm $1$. The system does not have a definite state in $X$; instead, if the state is $\psi \in L^2(X)$, the probability that the state of the system lies in a region $U \subset X$ is the integral of $\|\psi(x)\|^2$ over $U$. Observables in a quantum system are given by self-adjoint operators $A : L^2(X) \to L^2(X)$, and their values can be predicted as follows: if $A$ is sufficiently nice, it has an orthonormal basis of eigenvectors $\psi_1, \psi_2, ...$ with corresponding real eigenvalues $\lambda_1, \lambda_2, ...$. If our quantum system is in a state $\psi \in L^2(X)$, then $A$ takes the value $\lambda_k$ with probability $\|\langle \psi, \psi_k \rangle\|^2$, and the state $\psi$ is then projected to the $\lambda_k$-eigenspace (wave function collapse). In particular, the expected value of $A$ is given by $\langle v, Av \rangle^2$. (If $A$ has continuous spectrum, as is the case with the position operator, then one must do more work. Discrete spectrum, however, occurs in many applications and is responsible for the discreteness after which quantum mechanics is named, and we will only be dealing with discrete spectrum anyway.) Time evolution $\psi \mapsto U(t) \psi$ of a quantum system is described by a strongly continuous one-parameter unitary group, that is, a collection of unitary operators $U(t) : L^2(X) \to L^2(X), t \in \mathbb{R}$ such that $U(t+s) = U(t) U(s)$ and such that $\lim_{t \to t_0} U(t) = U(t_0)$ in the strong operator topology. By Stone’s theorem, there exists a self-adjoint operator $A$ such that $U(t) = e^{it A}$. (Since $U(t)$ can stand for a more general one-parameter group of symmetries of our quantum system, this is a form of Noether’s theorem in quantum mechanics.) An appropriate multiple of $A$ is then the observable corresponding to energy; it is called the Hamiltonian $H$, we label its eigenvalues $E_k$, and the dynamics of our quantum system are then described by the Schrödinger equation $\displaystyle i \hbar \frac{d}{dt} \psi(t) = H \psi(t)$ where $\psi(t) = U(t) \psi(0) : \mathbb{R} \to L^2(X)$ describes the time evolution of our system (and should perhaps be written $\psi(t, x)$) and $\hbar$ is the reduced Planck constant. Note that this is equivalent to $\psi(t)$ having the form $\displaystyle \psi(t) = e^{- i \frac{H}{\hbar} t} \psi(0)$. Given the eigenvalues and eigenvectors of $H$, we can write down the family of solutions $\displaystyle e^{-i \frac{H}{\hbar} t} \psi_k = e^{-i \frac{E_k}{\hbar} t} \psi_k$ to the Schrödinger equation, and since the $\psi_k$ are an orthonormal basis, these solutions span the space of solutions (in the Hilbert space sense). These are the solutions for which the energy $H$ takes the definite value $E_k$, so they are called the base states relative to $H$. A generic state in $L^2(X)$ is a linear combination of base states, so energy does not take a definite value for such a state. However, time evolution preserves the probability distribution on the set of possible energies of a state, which is a form of conservation of energy. More generally, an observable (a self-adjoint operator) $A : L^2(X) \to L^2(X)$ is preserved by time evolution if and only if it commutes with the Hamiltonian $H$. If the eigenspaces of $H$ are all one-dimensional, as one would expect to happen for generic choices of $H$, then knowing the energy of a base state uniquely determines it up to phase. However, in many quantum systems of interest, $H$ is far from generic, and the eigenspaces are larger; this means that more than one base state will have the same energy in general, which physicists refer to as degeneration. In the degenerate case, one must use additional observables other than energy to name states. A general mechanism which produces degeneration is the existence of a sufficiently large group $G$ of symmetries of the Hamiltonian, that is, a group $G$ of unitary operators on $L^2(X)$ which commute with $H$. The reason is that $G$ acts on the eigenspaces of $H$, so if the representation of $G$ on $X$ is nontrivial then the eigenspaces of $H$ must carry nontrivial irreducible subrepresentations (and in particular must degenerate if $G$ is non-abelian). This is certainly the case for an important example, the case of a single electron in a hydrogen atom, where $X = \mathbb{R}^3$ and $G$ includes the group $\text{SO}(3)$ (but, as it turns out, actually includes the larger group $\text{SO}(4)$). Here, the decomposition of the eigenspaces of $H$ into irreducible representations of $G$ is responsible for the list of possible states of the electron. The idea here is that as long as we are going to attempt to label different parts of each energy eigenspace, we might as well do it in a $G$-invariant way. In any case, the decomposition of $L^2(X)$ into irreducible representations of $G$, which exists independent of the form of the Hamiltonian, strongly constrains the energy eigenspaces of any Hamiltonian with $G$-symmetry. The Laplacian Solutions to the Schrödinger equation are called wave functions; the historical reason is related to wave-particle duality. Informally, one can think of an eigenvector $\psi_k$ of the Hamiltonian with eigenvalue $E_k$ as vibrating at angular frequency $\omega = \frac{E_k}{\hbar}$, a manifestation of the de Broglie relations connecting energy and frequency. This vibration is not noticeable if the state of the system is $\psi_k$, since time evolution will only change its phase, but if the state of the system is a superposition of states then they will interfere with one another roughly as if they were waves with the corresponding angular frequencies, as one can readily verify. (It’s not healthy to take this intuition too seriously, though. Quantum mechanics is quantum mechanics, regardless of one’s intuitions about either waves or particles.) Part of wave-particle duality is a mathematical similarity between the eigenfunctions of Hamiltonians which occur in nature and solutions to the wave equation. A particle in $\mathbb{R}^n$ moving under the influence of a potential $V(x) : \mathbb{R}^n \to \mathbb{R}$ has quantum Hamiltonian $\displaystyle H = - \frac{\hbar^2}{2m} \Delta + V$ where $\Delta$ is the Laplacian, $m$ is the mass of the particle, and $V$ denotes the operator $L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ corresponding to multiplication by $V$. This is true more generally for a particle in a Riemannian manifold, where $\Delta$ is the Laplace-Beltrami operator. Here the first term is the kinetic energy of the particle and the second term is the potential energy. It follows that solutions to the Schrödinger equation in this case involve eigenfunctions of the Laplacian, which are the same functions which correspond to the standing wave solutions of the wave equation. This suggests that in order to find finite toy models of quantum systems, we should find a finite analogue of the Laplacian. The usual choice is as follows. Our configuration space $X$ will be the vertices of an undirected finite graph with no self-loops, but possibly with multiple edges. We denote by $V$ and $E$ its edge and vertex sets, respectively, and by $L^2(V)$ the Hilbert space of functions $V \to \mathbb{C}$ with inner product $\displaystyle \langle \psi, \phi \rangle = \sum_{v \in V} \overline{\psi(v)} \phi(v)$. The state of our particle is then described by a unit vector in $L^2(V)$. To construct a Hamiltonian we define the Laplacian (or combinatorial Laplacian) $\displaystyle \Delta \psi(v) = \sum_{(w, v) \in E} (\psi(w) - \psi(v))).$ (There is a sign convention to choose here, and I am choosing the one which gives the better analogy to the usual Laplacian.) In other words, if we define $D : L^2(V) \to L^2(V)$ by $D \psi(v) = d_v \psi(v)$ (the degree operator) and $A : L^2(V) \to L^2(V)$ by $A \psi(v) = \sum_{(v,w) \in E} \psi(w)$ (the adjacency operator), then $\Delta = A - D$. (In particular, if $X$ is regular then the Laplacian differs from the adjacency matrix by an identity matrix, so talking about the eigenvalues of one is equivalent to talking about the eigenvalues of the other.) The Laplacian naturally arises in the study of random walks and electrical networks (see, for example, Doyle and Snell). Informally it measures the extent to which the value of $\psi$ at $v$ differs from the average of the value at its neighbors, and if we take $X = \mathbb{Z}^n$ with its natural graph structure then $\Delta$ can be written as a sum of discrete second finite differences which approximates the Laplacian on $\mathbb{R}^n$. For example, the combinatorial Laplacian on $\mathbb{Z}$ is just $\displaystyle \Delta f(n) = f(n+1) - 2f(n) + f(n-1)$. Before looking at the Schrödinger equation, it is perhaps a good idea to look at two other natural differential equations one can define using the Laplacian on a finite graph which generalize to $\mathbb{Z}^n$ to give, in the limit, well-known partial differential equations. First, the heat equation $\displaystyle \frac{d}{dt} \psi(t) = \Delta \psi(t)$ can be motivated as follows: we want a function $\psi : \mathbb{R} \to L^2(V)$ describing the evolution of a heat distribution on the vertices over time. Heat should tend to propagate from hot regions to cool regions, so a given vertex should lose heat proportional to how much hotter it is than its neighbors. (There should be a constant in there describing how quickly heat propagates, but it is not particularly important for the discussion that follows.) Since heat is the result of Brownian motion of particles, we expect that heat equation on a finite graph is related to the behavior of random walks on the graph. What are the possible stable heat distributions on a finite graph? These are precisely the (real-valued) solutions to $\Delta \psi(t) = 0$, or the harmonic functions. Proposition: A (real-valued) function on a finite graph is harmonic if and only if it is constant on connected components. Proof. Verify that $\displaystyle -\langle \psi, \Delta \psi \rangle = \sum_{(v, w) \in E} \|\psi(v) - \psi(w)\|^2$ hence that $\Delta \psi = 0$ implies $\psi(v) = \psi(w)$ for every $(v, w) \in E$. On the other hand any such function is harmonic. One can also argue by a finite form of the maximum modulus principle: $\psi$ is harmonic if and only if the value of $\psi$ at any vertex is the average of its neighbors, so since there are only finitely many vertices in each connected component, there must be some vertex for which $\psi(v)$ is maximal. Since this maximal value is the average over all its neighbors, the neighbors of $v$ must share that value. In physical terms, there cannot be a hottest point on any connected component, since heat would have flowed away from it to some other point. The argument above establishes more than we asked for: it shows that, not only is the Laplacian self-adjoint (hence has all real eigenvalues), but it is also negative-semidefinite (since its negative represents a positive-semidefinite quadratic form), and the multiplicity of the eigenvalue $0$ is the number of connected components of the graph, and the remaining eigenvalues are all negative. The general solution to the heat equation is a linear combination of the solutions $\displaystyle e^{\Delta t} \psi_k = e^{\lambda_k t} \psi_k$ corresponding to each (real) eigenvector of the Laplacian $\psi_k$ with eigenvalue $\lambda_k$; since the nonzero eigenvalues are negative, these solutions have temperature decaying to zero as is physically reasonable. In fact these solutions of the heat equation correspond to modes of “pure decay,” where the temperature decays exponentially in the simplest possible way, and the general solution is a linear combination of these modes. The rate of decay from generic initial conditions is controlled by the eigenvalue closest to zero, and if the graph is connected, this is the eigenvalue second smallest in absolute value $\lambda_1$, a fundamental invariant called the algebraic connectivity in graph theory. It is related to several other measures of connectivity in graphs and is important in the theory of expander graphs. Similarly, the wave equation $\displaystyle \frac{d^2}{dt^2} \psi(t) = \Delta \psi(t)$ can be motivated as follows: imagine that our graph $X$ describes a system of point masses connected by springs, one for each edge. Hooke’s law then tells us that, if we displace these point masses very slightly, the restoring force that pulls a given vertex back to equilibrium is proportional to the sum of the differences in the displacements between that vertex and its neighbors. The general solution to the wave equation is a linear combination of the solutions $\cos (\sqrt{\|\lambda_k\|} t) \psi_k, \sin (\sqrt{\|\lambda_k\|} t) \psi_k$ (keeping in mind that the eigenvalues $\lambda_k$ are negative), and it is because of this that the eigenvectors $\psi_k$ of the Laplacian on anything are called its harmonics. The solutions above are the standing waves, and so we know we can express any solution to the wave equation as a superposition of standing waves. I think the following would be a very fun project for a graph theory class: for various interesting graphs, plot using mathematical software the solutions to the heat and wave equations from various initial conditions. I would do this myself if I had more time. The Schrödinger equation (without potential) Replacing the continuous Laplacian by a discrete Laplacian, and in the absence of a potential, the Schrödinger equation on a finite graph $X$ (which we will assume to be connected) reads $\displaystyle i \hbar \frac{d}{dt} \psi(t) = - \frac{\hbar^2}{2m} \Delta \psi(t)$. The energy eigenvalues $E_k$ of the Hamiltonian are related to the eigenvalues $\lambda_k$ of the Laplacian by $E_k = - \frac{\hbar^2}{2m} \lambda_k$; in particular, they are non-negative, and the vacuum state, which has energy zero, has multiplicity $1$. The corresponding eigenvector, the all-ones vector, corresponds to a state with position smeared out evenly over all of $X$. The general solution is a superposition of the solutions $\displaystyle e^{-i \frac{H}{\hbar} t} \psi_k = e^{i \frac{\hbar}{2m} \lambda_k} \psi_k$ which are the states in which the energy has a definite value $E_k$. And for a general state $\psi$, the probability that a particle moving around $X$ will be measured at location $v \in X$ is given by $\frac{\|\psi(v)\|^2}{\sum_v \|\psi(v)\|^2}$. If the eigenvalues of the Laplacian all have multiplicity $1$, then as mentioned above, the energy constitutes a completely satisfying label for base states. Consider, therefore, what happens when the graph $X$ has a nontrivial automorphism group $G$. In that case, each energy eigenspace decomposes into a direct sum of representations of $G$, at least one of which will be nontrivial, so if these irreducible representations have dimension greater than $1$ some eigenspace must degenerate. (There is a purely graph-theoretic consequence of this argument: if the eigenvalues of the Laplacian of a graph have multiplicity $1$, then the automorphism group of the graph is abelian.) One of these eigenstates, the vacuum state, always corresponds to a copy of the trivial representation. In general the character of the representation of $G$ on $L^2(V)$ is $\text{Fix}(g)$, so the number of copies of the trivial representation is $\displaystyle \frac{1}{\|G\|} \sum_{g \in G} \text{Fix}(g)$ which is also, by Burnside’s lemma, the number of orbits of the action of $G$ on $X$. (The subspace spanned by copies of the trivial representation is precisely the subspace spanned by functions constant on orbits.) In particular, if $G$ acts transitively then there are no other copies of the trivial representation. In any case, this suggests that we should restrict our attention to the positive subspace, the orthogonal complement of the all-ones vector (where the Hamiltonian has positive eigenvalues). Example. Let $X$ be the complete graph $K_n$. Then $G = S_n$ acts, not only transitively, but double transitively, on $X$. This implies (this is a nice exercise) that the positive subspace is an irreducible representation of $S_n$, which means that it must correspond to a single energy eigenspace of dimension $n-1$. By computing the trace of the Laplacian, or otherwise, we see that the corresponding eigenvalue is $\lambda = -n$, hence corresponds to energy $E = \frac{n \hbar^2}{2m}$. We can further slice up this eigenspace by picking out permutations in $S_n$, but I don’t see a particularly good reason to do this in this case. Time evolution starting from the state $\psi(0) = \langle 1, 0, 0, ... \rangle$ (where the particle is at some vertex with probability $1$; we can prepare this state by observing the position of the particle) is as follows. We write $\displaystyle \psi(0) = \frac{1}{n} \left( \langle 1, 1, 1, ... \rangle + \langle n-1, -1, -1, ... \rangle \right)$ where the first vector is in the zero eigenspace and the second is in the positive subspace, and the Schrödinger equation then gives $\displaystyle \psi(t) = \frac{1}{n} \left( \langle 1, 1, 1, ... \rangle + e^{ - i \frac{n \hbar}{2m} t} \langle n-1, -1, -1, ... \rangle \right)$. Thus at time $t$, the probability that the particle is at its starting vertex is $\displaystyle \frac{1}{n^2} \left( n^2 - 2(n-1) + 2(n-1) \cos \frac{n \hbar}{2m} t \right)$ and the probability that it is at any particular other vertex is $\displaystyle \frac{1}{n^2} \left( 2 - 2 \cos \frac{n \hbar}{2m} t \right)$. This particle is in a superposition of two base states: the vacuum state where it is uniformly distributed over all other vertices, and a positive-energy state where it is found at its original vertex with high probability and at the other vertices with low probability. The observed behavior is “interference” between these two states. One can think of these results as a very simple manifestation of the uncertainty principle: the particle resists being located entirely at its starting vertex, so it distributes itself a little across all the other vertices. (We will give a slightly better manifestation of the uncertainty principle later.) The complete graph is somewhat anomalous in how degenerate its positive eigenspace is: the only way the group of automorphisms of a graph can act double transitively on it in general is if it is either the complete graph or the empty graph, so in any other case the positive subspace decomposes into two or more irreducible representations. We can get a grip on how many of these there are as follows. By character theory, if $L^2(V)$ decomposes into a direct sum of $n_k$ copies of the irreducible representation $V_k$ of $G$, then $\displaystyle \sum n_k^2 = \frac{1}{\|G\|} \sum_{g \in G} \text{Fix}(g)^2$. By Burnside’s lemma, this is just the number of orbits of $G$ acting on $V \times V$. There is always one orbit consisting of pairs $(v, v)$ of identical vertices, and the number of remaining orbits constrains how many representations can occur in the positive subspace. In particular, since $n_k^2 \ge 1$, it follows that the number of orbits of $G$ on pairs of distinct vertices is an upper bound on the number of representations the positive subspace decomposes into (hence on the number of distinct positive eigenvalues the Laplacian can have). One can think of the number of orbits as “the number of ways two distinct vertices can stand in relation to each other up to automorphism,” and for structured graphs that makes it relatively easy to count directly. Example. Consider the Kneser graph $KG_{n,2}$ of $2$-element subsets of $[n] = \{ 1, 2, ... n \}$, where two subsets are joined by an edge if they are disjoint. The graph has automorphism group $G = S_n$ acting in the obvious way, two distinct vertices can only be related to each other in two ways up to the action of $G$: they can be connected or not connected. It follows that the positive eigenspace decomposes into two irreducible representations of $S_n$ of total dimension ${n \choose 2} - 1$. Given an element $a \in [n]$, consider the vector $v_a$ in the positive subspace equal to $n-2$ on subsets containing $a$ and equal to $-2$ on subsets not containing $a$. The vectors $v_1, ... v_n$ add to zero and $S_n$ acts on them via its standard permutation reprsentation, hence they span a $n-1$-dimensional irreducible representation corresponding to the positive subspace of the representation of $S_n$ on $K_n$. The orthogonal complement of this representation is a ${n \choose 2} - n$-dimensional irreducible representation whose character is now easy to write down. I will leave the computation of the corresponding energy eigenvalues as an exercise. Momentum In ordinary quantum mechanics, linear momentum comes from the self-adjoint operators which generate linear translations via Stone’s theorem, and angular momentum comes from the self-adjoint operators which generate rotations. So momentum is generally related to spatial symmetries. In the finite graph model, spatial symmetries – the automorphisms $G$ of the graph – are discrete, so Stone’s theorem doesn’t apply. But we can still give a definition of a quantity which behaves something like momentum. Definition: Let $g \in G$ and suppose that a state $\psi \in L^2(V)$ is an eigenvector for $g$ with eigenvalue $e^{ix}, x \in S^1 \simeq \mathbb{R}/2\pi\mathbb{Z}$. Then $x$ is the $g$-momentum of $\psi$. In other words, it’s what the eigenvalue of a self-adjoint operator generating $g$ would be if it existed. The $g$-momentum does not come from a self-adjoint operator and so is not an observable as we have narrowly defined it, but nevertheless it is still a number which behaves like ordinary momentum; one might think of it as the momentum the particle has in the “direction” given by $g$. And it is still possible, for an arbitrary $\psi \in L^2(V)$, to write down the probability that the $g$-momentum is one of its several possible values, and to decompose the energy eigenspaces of the Hamiltonian by the eigenspaces of $g$, each one corresponding to a particular value for the $g$-momentum. And since $g$ commutes with the Laplacian, $g$-momentum is conserved by time evolution. Example. Let $X$ be the cycle graph $C_n$, with vertices $V = \{ 0, 1, 2, ... n-1 \}$. Then $G$ is the dihedral group $D_n$. Letting $g$ be the rotation $v \mapsto v+1 \bmod n$, the decomposition of $L^2(V)$ into eigenspaces of $g$ is straightforward, since it is the same as the decomposition of the regular representation of $\mathbb{Z}/n\mathbb{Z}$: the eigenspace spanned by the eigenvector $\psi_k = \langle 1, \zeta_n^k, \zeta_n^{2k}, ... \zeta_n^{(n-1)k} \rangle$ where $\zeta_n = e^{ \frac{2 \pi i}{n} }$ is the one where the $g$-momentum has value $\frac{2 \pi k}{n}$. In this particular case the Laplacian can be written in terms of $g$: it is precisely equal to $g + g^{-1} - 2$, hence $\psi_k$ is an eigenvector of the Laplacian with eigenvalue $\lambda_k = 2 \cos \frac{2 \pi k}{n} - 2$, hence energy $E_k = \frac{\hbar^2}{m} \left( 1 - \cos \frac{2 \pi k}{n} \right)$. Since $E_k = E_{-k}$ it follows that eigenvectors $\psi_k$ pair up into energy eigenspaces (and in fact, pair up into irreducible representations of $D_n$), and we can slice up each pair according to whether the $g$-momentum is directed clockwise or counterclockwise (as measured by whether it is closer to $0$ in the clockwise or counterclockwise direction). This situation is the discrete analogue of the particle in a ring, and indeed note that the eigenvectors are very similar and that if $k$ is very small compared to $n$ we have $2 - 2 \cos x \approx x^2$, hence $\displaystyle E_k \approx \frac{\hbar^2}{2m} \left( \frac{2 \pi k}{n} \right)^2$ which is very close to the lower energy levels of the spectrum in the continuous case. (One should think of $\frac{2\pi}{n}$ as a conversion factor between the discrete and continuous Laplacians.) The cycle graph is also not a bad place to demonstrate another form of the uncertainty principle: position and $g$-momentum operators do not commute. Here by a position operator we mean a self-adjoint operator $x_v$ which projects $\psi \in L^2(V)$ onto the subspace of functions nonzero except at a particular $v \in V$. In particular, as is already clear from our computations above, there is no state in $L^2(V)$ in which both position and $g$-momentum are uniquely determined (a common eigenvector). Furthermore, a state in which position is completely determined is a state with maximum ambiguity in $g$-momentum: $\displaystyle \langle 1, 0, 0, ... \rangle = \sum_{k=0}^{n-1} \frac{1}{n} \langle 1, \zeta_n^k, \zeta_n^{2k}, ... \zeta_n^{(n-1)k} \rangle$. Conversely, a state in which $g$-momentum is completely determined is a state with maximum ambiguity in position. One should think of this in terms of the discrete Fourier transform, since the story parallels perfectly the story of the Fourier transform on $\mathbb{R}^d$ which describes the relationship between position and momentum for free particles in $\mathbb{R}^d$. Representations, duals, and tensor products There is a philosophy going back at least as far as Wigner that irreducible representations of the symmetry group $G$ of a quantum system should be identified with the possible types of particles in that system. This seems pretty reasonable: if $\psi$ is a wave function describing some particle then $g \psi$ for $g \in G$ should be thought of as the same particle, just moving in a different direction, so it is natural to collect all of these wave functions into the same representation of $G$. Thus an arbitrary wave function is a superposition of different types of particles corresponding to the different irreducible representations of $G$ occurring in the decomposition of $L^2(V)$. This gives us a physical language for making sense of basic operations on representations. For example, if $W$ is an irreducible subrepresentation of $L^2(V)$, then since the character of $L^2(V)$ is real, it follows that the dual representation $W^{\ast}$ is also an irreducible subrepresentation. (Concretely, it is obtained from $W$ by taking the complex conjugate of the corresponding wave functions. If $W$ has a basis of wave functions with real coordinates, then $W \simeq W^{\ast}$.) In physical language, one might call $W^{\ast}$ the antiparticle of $W$. Note that if $\psi_k \in W$ has energy eigenvalue $E_k$, then $\psi_k$ evolves as $\displaystyle e^{- i \frac{E_k}{\hbar} t} \psi_k$ whereas its antiparticle $\overline{\psi_k}$ evolves as $\displaystyle e^{-i \frac{E_k}{\hbar} t} \overline{\psi_k}$. The probabilities one computes from this evolution are the same as the probabilities one computes from its complex conjugate $\displaystyle e^{i \frac{E_k}{\hbar} t} \psi_k$ and so one might say that an antiparticle is “the same thing” as the original particle going backwards in time. But in the finite graph model one should be a little more careful about this, since the above object is not, strictly speaking, a solution to the Schrödinger equation when $E_k$ is positive. Particles and antiparticles are probably best known for colliding. To give this operation some kind of meaning in the finite graph model, we must first discuss the following general construction. If one quantum system is given by a Hilbert space $K_1$ and another is given by a Hilbert space $K_2$, then to study them together, we work in the Hilbert space tensor product $K = K_1 \otimes K_2$, and we call the corresponding quantum system the composite system. This is the natural thing to do if, for example, $K_1$ and $K_2$ are the Hilbert spaces of states of two particles and you want to study the particles together. (But it is also the natural thing to do if $K_1$ and $K_2$ describe two aspects of the same object, e.g. the $x$ and $y$-coordinates of a particle in the plane.) Given any self-adjoint operator $A$ on $K_1$ we get a self-adjoint operator $A \otimes 1$ on $K$, and similarly for $K_2$, so one can make all of the same observations on each subsystem as before. In addition, for every symmetry $g$ of $K_1$ there is a corresponding symmetry $g \otimes 1$ of $K$, and similarly for $K_2$. If $K_1, K_2$ are both equipped with Hamiltonians $H_1, H_2$, then the operators giving the energy of each subsystem are $H_1 \otimes 1, 1 \otimes H_2$, so the total energy is a new Hamiltonian $H = H_1 \otimes 1 + 1 \otimes H_2$ (assuming there is no interaction between the subsystems; otherwise we need a third term to describe that interaction.) The time evolution operator given by the Schrödinger equation for $K$ is then $U(t) = e^{-i \frac{H}{\hbar} t} = e^{-i \frac{H_1}{\hbar} t} \otimes e^{-i \frac{H_2}{\hbar} t}$ since $H_1 \otimes 1$ and $1 \otimes H_2$ commute. It follows that if $\psi, \psi'$ are states in $H_1, H_2$, then the time evolution of the state $\psi \otimes \psi'$ in $H$ is given by the individual evolution of each state in each subsystem. Of course, in general a state in $H$ is a superposition of states of the form $\psi \otimes \psi'$; this is the mechanism behind quantum entanglement. If $K_1, K_2$ are finite graph models $L^2(V_1), L^2(V_2)$ on graphs $X_1, X_2$, then their tensor product can be identified with $L^2(V_1 \times V_2)$, and the composite Hamiltonian is the Hamiltonian of a new finite graph model given by a new graph structure on $V_1 \times V_2$, the Cartesian product (or box product) of graphs $X_1 \Box X_2$. The box product has Laplacian $\Delta = \Delta_1 \otimes 1 + 1 \otimes \Delta_2$ so it defines the correct composite Hamiltonian. There is even a Wikipedia article about this construction, in the special case of the usual discrete Laplacian on $\mathbb{Z}$. (Note that the Cartesian product of $n$ copies of $\mathbb{Z}$ is $\mathbb{Z}^n$ with the obvious graph structure, a nice property not shared by the tensor product of graphs. More generally, it does the obvious thing to Cayley graphs.) Thus one can identify a pair of free particles traveling on $X_1$ and $X_2$ with a single free particle traveling on $X_1 \Box X_2$. Remark: Philosophically, it seems to me that the distinction between the box product and the tensor product comes from whether one wants to treat the Laplacian as a one-dimensional Lie algebra acting on the graph or whether one wants to treat, say, the adjacency matrix as a monoid acting on the graph. For every pair of eigenvectors $v, w$ of the Laplacians $\Delta_1, \Delta_2$ with eigenvalues $\lambda, \mu$, there is an eigenvector $v \otimes w$ of the composite Laplacian $\Delta$ with eigenvalue $\lambda + \mu$, and this is a complete orthonormal basis of eigenvectors. The energy eigenspaces of $\Delta$ decompose into irreducible representations of $G_1 \times G_2$ where $G_i$ is the symmetry group of $X_i$, and these irreducible representations are exactly what you would expect: tensor products of irreducible representations of the groups $G_1, G_2$. (Note that $X_1 \Box X_2$ may have additional symmetries which allow us to group some of these representations together into larger irreducible representations.) Now suppose that $X_1 = X_2 = X$. Then one can think of a pair of free particles on $X_1, X_2$ as traveling on the same copy of $X$ (although not interacting), and instead of allowing $G_1 \times G_2 = G \times G$ as the symmetry group we should only consider symmetries we can perform on both particles at once, that is, the diagonal copy $(g, g), g \in G$ of $G$ inside $G \times G$. Then $L^2(V \times V)$ is the tensor square of the representation of $G$ given by $L^2(V)$, so one can think of a tensor product $W_1 \otimes W_2$ of two subrepresentations $W_1, W_2$ of $L^2(V)$ as describing the behavior of a pair of particles on $X$, one of which is of type $W_1$ and one of which is of type $W_2$. Since $W_1 \otimes W_2$ is not usually itself irreducible, this representation breaks up into irreducible components, which correspond to the different types of ways (up to symmetry) that a particle of type $W_1$ and a particle of type $W_2$ can behave together on $X$. A collision between a particle of type $W_1$ and a particle of type $W_2$ can then be thought of as a morphism $W_1 \otimes W_2 \to W$ of representations of $G$, where $W$ is some subrepresentation of $L^2(V)$. That is, it is a linear and $G$-invariant process for obtaining a new particle. And if $W_2 = W_1^{\ast}$, there is a canonical $G$-invariant morphism $W_1 \otimes W_1^{\ast} \to 1$ (where $1$ denotes the trivial representation) given by the dual pairing. Physicists call this morphism particle-antiparticle annihilation, since its result is the vacuum state. The dual morphism $1 \to W_1 \otimes W_1^{\ast}$ (whose image corresponds to the identity operator $W_1 \to W_1$) is particle-antiparticle creation. If one wants to think of all these morphisms as taking place in the same Hilbert space, then the natural thing to do is to set up a Hilbert space which is large enough to accompany any finite number of particles. The operation on Hilbert spaces correspond to studying a subsystem $K_1$ or a subsystem $K_2$ (in contrast to the tensor product, where we want to study the first and the second) is the direct sum $K_1 \oplus K_2$, so we construct the tensor algebra $\displaystyle T(L^2(V)) = \bigoplus_{k \ge 0} L^2(V^k)$. Since the tensor products of a faithful representation of a finite group contain all irreducible representations of that group, it follows that we can get all irreducible representations of $G$ by considering enough different particles moving around on $X$. In practice, people don’t work with the full tensor algebra. This is because particles in physics of the same type are actually identical (for example, every pair of electrons is identical, as is every pair of photons, etc.), so switching the identities of two particles is a physical symmetry. The two simplest cases are bosons, where switching the two particles gives exactly the same composite wave function, and fermions, where switching the two particles gives the negative of the original wave function. This means that instead of working in the full tensor square $L^2(V) \otimes L^2(V)$, we work instead in either the symmetric square (for bosons) or the exterior square (for fermions). Similarly, for $n$ particles, instead of working in the full tensor power we work in either the symmetric power or the exterior power. So to work with a variable number of particles, instead of working in the tensor algebra, we work in either the symmetric algebra or the exterior algebra; in either case the corresponding construction is called Fock space. Note that the need to work in exterior powers for fermions is responsible for the Pauli exclusion principle. Fermions and bosons are not easy to explain in the context of the finite graph model itself; they are a consequence of the spin-statistics theorem, which is relativistic. In any case, we now have a physical interpretation of the symmetric and exterior powers, and of the symmetric and exterior algebras. (Regardless of whether one cares about fermions and bosons in the finite graph model, the Hilbert space $L^2(V^n)$ describing $n$ identical particles on $X$ still has an $S_n$-symmetry relative to which it decomposes in terms of irreducible representations of $S_n$, so one still needs to understand the corresponding Schur functors to understand the ways in which $n$ particles can behave up to symmetry.) ### 13 Responses 1. Very interesting! Why is the usual choice to define X on graph with no loops? Doesn’t that rule out all the interesting symmetry cases? • I don’t know if that’s the usual choice; I’m guessing here. The reason I said to ignore loops is that, as far as I can tell, the Laplacian ignores loops. What are the interesting cases that this rules out? • Loops act as a “potential energy” for a particle located at that vertex. • Yes, that is a consistent interpretation, but it seems more natural to me to regard that as separate from the “kinetic energy” term. There is no reasonable sense I can see in which the Laplacian, abstractly defined, notices loops. • Ah, I think this might be a terminological issue: when I said “loops” I meant “edges from a vertex to itself.” 2. Ah ok, that makes sense, since you talk about cycle graph later. I’m used to them being called “self-loops” • Yes, I usually call them that as well but I was sloppy this time. My mistake. 3. I have not finished reading this post, but I’ve been reading your blog for awhile and being more physics-minded, I haven’t really ever had much to say. But I found what I’ve read so far to be very fascinating and insightful. Taking the exponential of a Hamiltonian is much less mysterious now… If you’re interested in further generalization that connects quantum mechanics with graph theory, there is a recent paper by Pearson and Bellissard that interprets the Cantor set as the space of paths of an infinite tree, turns it into a non-commutative Riemannian manifold, and then defines an analogue of the Laplace operator on it. The paper is mostly self-contained, which is really nice: http://people.math.gatech.edu/~jeanbel/Publi/cantor08.pdf • Thanks for the link! I’m glad I was able to reach out to the physicists. I’ve been trying to teach myself some physics lately and this post helped me collect my thoughts. 4. How quantum spin systems on graphs fit into this? Take quantum Ising system on a graph G. There’s a Laplacian associated with G, but that’s different from the “quantum laplacian” you talk about, right? • I don’t think it’s the same. As far as I know, the quantum Ising model is still about a collection of particles embedded in a graph, whereas this is about a single particle “moving” (in some generalized sense) freely on the graph. (One shouldn’t take this model too seriously; for example it does not seem to have a corresponding path integral model.) 5. [...] (that is, a Hilbert space and a Hamiltonian ) has a path-connected Lie group of symmetries . The previous post may have misled you into thinking that this implies that has a unitary representation on . [...] 6. [...] an earlier post we introduced the Schrödinger picture of quantum mechanics, which can be summarized as follows: [...]	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 383, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9609180688858032, "perplexity": 121.42100990874208}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507443598.37/warc/CC-MAIN-20141017005723-00206-ip-10-16-133-185.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/as-maths-exam-mechanics.57626/	# AS maths exam mechanics 1. Dec 23, 2004 ### NinaL I have a mechanics exam coming up, and I'm going through a textbook answering questions. I'm really stuck on the following: A lift ascends from rest with an accceleration of 0.5ms before slowing with an acceleration of -0.75ms for the next stop. If the total journey time is 10 secs, what is the distance between the two stops? The thing is, I know that you have to use the SUVAT equations (displacement, initial velocity, final velocity, time) but I have no idea what to do with this. Why are they giving me two accelerations? I've tried splitting the question into two parts, but i don't know the time travelled at each acceleration. Please help! This is for my AS maths exam, on Jan the 12th, and with my luck, this kind of thing will come up, just because i'm not prepared for it!!. :) Thank you! Last edited: Dec 23, 2004 2. Dec 23, 2004 ### Staff: Mentor Treat the motion in two parts. In part one, the lift goes from a speed of 0 to V in $T_1$ seconds; in part two, it goes from V to 0 in $T_2$ seconds. Figure out $T_1$ and $T_2$. (Hint: use $V_f = V_i + aT$.) Then use the times to figure out the distance traveled. 3. Dec 23, 2004 ### NinaL Thank you. I'm sorry, but I'm not feeling particularly intelligent today, or perhaps i've just called it something different, but could you explain the above formula? Thank you ever so much! 4. Dec 23, 2004 ### Beretta Since it started from rest V initial = 0m/s So first Velocity V= 0m/s + (0.5m/s^2)(t1) Second final velocity since it stoped 0m/s = V(initial) + (-0.75m/s^2)(t2 = 10s) and then you plug first Velocity in the second equation. 0m/s = [0m/s + (0.5m/s^2)(t1)] + (-0.75m/s^2)(t2 = 10s) Last edited: Dec 23, 2004 5. Dec 23, 2004 ### Staff: Mentor $V_f = V_i + aT$ is one of the basic kinematic formulas describing uniformly accelerated motion. It tells you how to calculate the final speed ($V_f$) a uniformly accelerated object will attain after T seconds given the initial speed ($V_i$). Hints: In part one, the initial speed is 0, call the final speed V. In part two, the initial speed is V, the final speed is zero. Now apply that equation for each part, and make use of the fact that $T_1 + T_2 = 10$ seconds. You should be able to solve for the two times. 6. Dec 24, 2004 ### NinaL Thanks a lot, I finally figured out how to do it! :) On the subject of Mechanics though, does anyone know any good websites that feature revision material, explanations etc? Thank you! Similar Discussions: AS maths exam mechanics	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9635199904441833, "perplexity": 1049.0902318551944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825264.94/warc/CC-MAIN-20171022132026-20171022152026-00767.warc.gz"}
http://softmatterlab.org/2016/03/01/jstatphys_2016_163_659/	# The Small-mass Limit for Langevin Dynamics published in J. Stat. Phys. The small-mass limit for Langevin dynamics with unbounded coefficients and positive friction David P. Herzog, Scott Hottovy & Giovanni Volpe Journal of Statistical Physics 163(3), 659—673 (2016) DOI: 10.1007/s10955-016-1498-8 arXiv: 1510.04187 A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to three physically realizable examples where the coefficients defining the Langevin equation for these examples grow unboundedly either at a boundary, such as a wall, and/or at the point at infinity. This unboundedness violates the assumptions of previous limit theorems in the literature. The main result of this paper proves convergence for such examples. This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9530767798423767, "perplexity": 1312.288811656206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104054564.59/warc/CC-MAIN-20220702101738-20220702131738-00557.warc.gz"}
http://cnp.phys.vt.edu/chandler/	The CHANDLER Detector Project Reactor Neutrino Detection Nuclear reactors are a very intense source of electron antineutrinos which come from the β-decay of the neutron-rich fission fragments. Neutrinos are extraordinarily weakly interacting such that they can pass all the way through the Earth unhindered, and so the shielding around a nuclear reactor provides no impediment to their passage. The neutrinos produced in the core arrive at the detector just as they were produced, and as such they carry information about their point of origin and the processes inside the reactor which created them. Most of these neutrinos pass through the detector without leaving a trace, but a tiny fraction interact in the detector, giving us the possibility of learning about the inner workings of the core. For each gigawatt of thermal power a reactor produces about 2×1020 electron antineutrinos per second. A one-ton detector placed 15 meters from the core of a typical commercial reactor should see about 5000 neutrino interactions per day. With such a detector we could track the plutonium content in a reactor core by measuring slight differences in the energies of the neutrinos produced in the fission of uranium compared to plutonium. Reactor neutrino are detected through a process known as inverse β-decay, in which an electron antineutrino interacts with a proton (a hydrogen nucleus in the plastic scintillator) to produce a positron (or positive electron) and a neutron. $\bar{\nu}_e+p\to e^++n$ In this process, the positron acquires the energy of the neutrino, minus the mass difference between the proton and the neutron plus positron. Within the detector the positron energy is deposit promptly in the scintillator, producing light proportional to that energy deposition. In this way the energy of each neutrino can measured on an event-by-event basis. The neutron, which carries just a tiny fraction of the available energy, bounces around for a while (about 100 μs) and eventually captures on a nucleus. In the capture process the excess binding energy is release into the scintillator creating a delayed light pulse. The CHANDLER Technology The CHANDLER detector technology is comprised of cubes of wavelength shifting plastic scintillator cubes and thin sheets of lithium-6 (6Li) loaded zinc sulfide (ZnS) scintillator. The 6 cm cubes are arraigned in layers of up to 20×20 cubes which are separated by the 6Li-loaded ZnS sheets. The cubes and sheets are well suited for detecting electron antineutrinos from nuclear reactors, which produce a positron and a neutron when they interact in the plastic cubes. The positron produces a prompt flash of light in the cube, while the neutron bounces around for a while before capturing on the 6Li in the sheet producing a delayed flash of light. The correlation between these two distinct events provides a clean indication of a neutrino interaction. The light from both the sheets and cubes is transported by total-internal-reflection along the rows and columns of cubes to the surface of the detector where it is read out by light detectors known as photomultiplier tubes (PMTs). This unique method for reading out the light, known as a Raghavan optical lattice, was invented by the late Center for Neutrino Physics professor Raju Raghavan. It provides precise spatial information for the neutrino interaction and neutron capture, which can be used to separate the true neutrino events, which must be close together in both time and space, from the random correlation of unassociated neutron captures and positron-like events that would otherwise form fake neutrino events. The following brief video illustrates how this technology works: The CHANDLER R&D Program Stages in the R&D Program: Cube String, MicroCHANDLER and MiniCHANDLER In the early stages of the program we tested the optical properties of the cubes in strings of up to 10 cubes. This allowed us to create an optical model for the light transportation in the cubes and to determine the energy response of the scintillator. With a combination of measurements and simulation we have shown that the full-scale detector should measure the positron energy with a resolution of better than 6.5% at 1 MeV. Next we constructed a prototype detector of 3×3×3 cubes and 4 sheets. This detector, known as MicroCHANDLER has been used to study the neutron response and to measure the background rate from false coincidences. With MicroCHANDLER we obtained a neutron event purity of better than 98%. We have recently completed construction of the next-level prototype, known as MiniCHANDLER, which is constructed of 320 cubes (8×8×5) and 6 sheets. MiniCHANDLER was designed to be large enough to see neutrinos from a nuclear reactor. Once we finish commissioning MiniCHANDLER in the lab it will be transported in our Mobile Neutrino Lab to the North Anna Nuclear Power Plant near Richmond, Virginia for a full system demonstration. The MiniCHANDLER Detector at different stages in its assembly Neutron Identification in CHANDLER In CHANDLER the scintillation light produced in the 6Li-loaded ZnS is released much more slowly than the scintillation light in the plastic cubes (200 ns compared to 10 ns). This means that the signal pulse observed in the PMT that comes from neutron capture is much wider than from positron-like events in formed in the cubes. This fact can be used to form a simple neutron identification (or "Neutron ID") variable by taking the ration of the signal pulse integral to the signal pulse amplitude. Due to their longer light release time, neutron capture pulses produced in the ZnS have larger values of this ratio than signals formed in the plastic scintillating cubes. The above figure shows this Neutron ID parameter for signals readout simultaneously on the two sides of the detector (the x and y views). There are two clear populations of events: those with high Neutron ID values in both views, these are neutron-like events, and those with low Neutron ID values in both views, which are positron-like events. The MiniCHANDLER Deployment at the North Anna Nuclear Power Plant In June of 2017 the MiniCHANDLER Detector was loaded into the Mobile Neutrino Lab and deployed to Dominion Energy's North Anna Nuclear Power Plant, where it took data for four and a half months. The Mobile Neutrino Lab with detector sat just outside the secondary containment of reactor 2, which is located about 25 meters from the center of the reactor core. The MiniCHANDLER Detector Mobile Neutrino Lab (left) and the Mobile Neutrino Lab deployed at North Anna (right) At this distance we expect that MiniCHANDLER should see about 100 neutrino interactions per day. We are still analyzing the data, and hope to have a result soon. If we succeed in seeing neutrinos with MiniCHANDLER, it will be the world's smallest reactor neutrino detector and the first ever mobile neutrino detector. It will also demonstrate the potential for CHANDLER technology as a reactor neutrino detector both for studying the properties of neutrinos and for applications like nuclear non-proliferation and characterization of new reactor designs. The following brief video highlights the North Anna Deployment in cartoon form:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 1, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.802257239818573, "perplexity": 1436.735817327827}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376828018.77/warc/CC-MAIN-20181216234902-20181217020902-00257.warc.gz"}
http://mathhelpforum.com/calculus/20608-partial-fraction-integrate.html	Math Help - Partial Fraction and integrate 1. Partial Fraction and integrate ok it says to integrate. .... Integral (5x^4 - 2x^3 +2x^2 +4) / (x^3 - x^2) dx so since the numerator's degree is higher we have to do long divison right? i got 5x + 3 + (x^2+4)/(x^3-x^2) but now... what do we use to integrate? is it just the fraction portion of the answer to long division? im stuck... 2. Re: Re: Attached Thumbnails 3. thanks but how did you get a 9? 4. Originally Posted by runner07 ok it says to integrate. .... Integral (5x^4 - 2x^3 +2x^2 +4) / (x^3 - x^2) dx so since the numerator's degree is higher we have to do long divison right? i got 5x + 3 + (x^2+4)/(x^3-x^2) but now... what do we use to integrate? is it just the fraction portion of the answer to long division? im stuck... In your long division you should have got 5x +3 +[(5x^2 +4) /(x^3 -x^2) You forgot the coefficient 5 of the x^2 term. Now you need to decompose the rational or fraction portion of the integrand, because the degree of the numerator now is lower than that of the degree of the denominator. You cannot use long division anymore. (5x^2 +4) /(x^3 -x^2) (5x^2 +4) / (x^2)(x -1) (5x^2 +4) / (x^2)(x-1) = A/x +B/(x^2) +C/(x-1) Multiply both sides by (x^2)(x-1), 5x^2 +4 = A(x)(x-1) +B(x-1) +C(x^2) -------------(i) When x = 0, in (i), 0 +4 = 0 +B(-1) +0 So, B = -4 --------------------- When x = 1, in (i), 5(1^2) +4 = 0 +0 +C(1^2) So, C = 9 ------------------------- When x = 2, and B = -4, and C = 9, in (i), 5(2^2) +4 = A(2)(2-1) +(-4)(2-1) +9(2^2) 24 = 2A -4 +36 24 +4 -36 = 2A A = -8/2 = -4 -------------------** Therefore, INT.[(5x^4 - 2x^3 +2x^2 +4) / (x^3 - x^2)]dx = INT.[5x +3 -4/x -4/(x^2) +9/(x-1)]dx And so on, and so forth.....	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8936548233032227, "perplexity": 3010.1801794137205}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637897717.20/warc/CC-MAIN-20141030025817-00158-ip-10-16-133-185.ec2.internal.warc.gz"}
http://ahay.org/blog/2007/03/12/madagascar-programs-guide-sfscale/	A short section on sfscale has been added to the Madagascar programs guide. Thanks to Nick Vlad for filling other missing entries in the guide!	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8914476037025452, "perplexity": 4758.738319504411}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347396300.22/warc/CC-MAIN-20200527235451-20200528025451-00214.warc.gz"}
https://socratic.org/questions/how-do-you-simplify-sqrt18-15sqrt2#608031	Algebra Topics # How do you simplify sqrt18+15sqrt2? ##### 1 Answer May 5, 2018 $18 \sqrt{2}$ #### Explanation: $15 \sqrt{2}$ is already simplified, but to simplify $\sqrt{18}$ , we need to figure out its factors. the factors of $18$ are $3 , 3 , \mathmr{and} 2$ $\sqrt{18}$ simplified is $3 \sqrt{2}$ since $3 \sqrt{2}$ and $15 \sqrt{2}$ both have the number $2$ in the root you can add these two radicals to get $18 \sqrt{2}$ ##### Impact of this question 956 views around the world You can reuse this answer Creative Commons License	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9728834629058838, "perplexity": 1356.678786841202}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710869.86/warc/CC-MAIN-20221201185801-20221201215801-00087.warc.gz"}
https://proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset/Proof_1	# Infinite Set has Countably Infinite Subset/Proof 1 ## Theorem Every infinite set has a countably infinite subset. ## Proof Let $S$ be an infinite set. Suppose that there exists an injection $\psi: \N \to S$. Let $T$ be the image of $\psi$. From Injection to Image is Bijection, it follows that $\psi^{-1}: T \to \N$ is a bijection. Hence, $T$ is a countably infinite subset of $S$. Now, suppose that that there exists a surjection $\phi: \N \to S$. From Surjection from Natural Numbers iff Countable, it follows that $S$ is countably infinite. So, from Set is Subset of Itself, we have that $S$ is a countably infinite subset of $S$. $\blacksquare$ #### Axiom of Choice This proof depends on the Axiom of Choice, by way of Between Two Sets Exists Injection or Surjection. Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial. Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted. However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9957159161567688, "perplexity": 409.0551305773268}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999261.43/warc/CC-MAIN-20190620145650-20190620171650-00463.warc.gz"}
https://quantumcomputing.stackexchange.com/questions/2466/expressing-square-root-of-swap-gate-in-terms-of-cnot	# Expressing “Square root of Swap” gate in terms of CNOT How could a $\sqrt{SWAP}$ circuit be expressed in terms of CNOT gates & single qubit rotations? • CNOT & $\sqrt{SWAP}$ Gates Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations. Both CNOT and $\sqrt{SWAP}$ are universal two-qubit gates and can be transformed into each other. Edit The questions are not identical, and are likely to attract different audiences. -DaftWullie Can the difference be regarded as applied vs theoretic? • quantumcomputing.stackexchange.com/questions/2228/… – Nelimee Jun 26 '18 at 5:53 • Possible duplicate of How to implement the "Square root of Swap gate" on the IBM Q (composer)? – Sanchayan Dutta Jun 26 '18 at 6:55 • @Blue I think that while the answers are essentially the same, the questions are not identical, and are likely to attract different audiences. – DaftWullie Jun 26 '18 at 7:01 • May have been better to post edit as comment? Also, to be clear, the circuit that @DaftWullie posted at the end is what I was after (which I did not see in the other post). – meowzz Jun 26 '18 at 7:55 • @Blue It is indeed a candidate for closure. But given there are slight differences and willing answerers, I'll let it live. – James Wootton Jun 26 '18 at 8:42 As pointed out by @Nelimee, this question is essentially answered in this question, even if that question seems more specific. However, for the sake of completeness... (Note that I make no claims about minimality of construction with respect to, for example, number of controlled-not gates.) Let's start with a unitary matrix for the square root of SWAP: $$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{2}+\frac{i}{2} & \frac{1}{2}-\frac{i}{2} & 0 \\ 0 & \frac{1}{2}-\frac{i}{2} & \frac{1}{2}+\frac{i}{2} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$$ Note that if we pre- and post-multiply by a controlled NOT, controlled off qubit 2, this transforms into the form of a controlled-$U$ gate: $$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{2}+\frac{i}{2} & \frac{1}{2}-\frac{i}{2} \\ 0 & 0 & \frac{1}{2}-\frac{i}{2} & \frac{1}{2}+\frac{i}{2} \\ \end{array} \right)$$ There are standard constructions for this controlled-$U$ gate. So, overall, we have a circuit that looks like where $ABC=\mathbb{I}$ and $AXBXC=e^{i\pi/4}\sqrt{X}$. Again, there are standard routes towards finding $A$, $B$ and $C$. For instance, we can define $R_Y(\theta)$ to be a rotation about the $Y$ axis by an angle $\theta$, i.e. $R_Y(\theta)=e^{i\theta Y}$ (there may be a factor of $\frac12$ here compared to some definitions). Then, if $$A=R_Y(\theta)R_Z(\phi)\qquad B=R_Z(-2\phi)\qquad C=R_Z(\phi)R_Y(-\theta),$$ then it's easy to verify that $ABC=\mathbb{I}$. Furthermore, $AXBXC=R_Y(\theta)R_Z(4\phi)R_Y(-\theta)$ since $XR_Z(\phi)X=R_Z(-\phi)$. You can think of this as $R_Z(4\phi)$ specifying the eigenvalues that we want, and the $R_Y(\theta)$ is changing the basis from the computational basis into something else. In the present case, we have $\theta=\pi/2$ and $4\phi=\pi/4$. The one little thing that we haven't got right yet is a phase factor. Our $AXBXC$ is creating the correct rotation up to an overall phase of $e^{i\pi/4}$: $$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{i}{\sqrt{2}} \\ 0 & 0 & -\frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \end{array} \right)$$ The trick to get this right is to apply an $R_Z(\pi/8)$ on the first qubit and remove a global phase of $e^{i\pi/8}$. Thus, we have I think that you want answers to explicitly use full cnots, rather than partial versions. But since you already have an answer for that, I'll contribute a different perspective. A $$\mathrm{SWAP}$$ can be thought of as a cnot that has been conjugated by oppositely oriented cnots. $$\mathrm{SWAP} = {\rm cx}(k,j) \,\, {\rm cx}(j,k) \,\, {\rm cx}(k,j)$$ To make a $$\sqrt{\mathrm{SWAP}}$$, we can instead use a square root of cnot conjugated by oppositely oriented cnots. $$\sqrt{\mathrm{SWAP}} = {\rm cx}(k,j) \,\, \sqrt{{\rm cx}(j,k)} \,\, {\rm cx}(k,j)$$ To verify that this is indeed a $$\sqrt{\mathrm{SWAP}}$$, you can simply square it and verify that it ends up in the way we expect (using the fact that cnots square to identity). • Awesome! Your $\mathrm{SWAP}$ approach makes me think of an unanswered question.. – meowzz Jun 26 '18 at 9:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8832916021347046, "perplexity": 412.331624918279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256997.79/warc/CC-MAIN-20190523003453-20190523025453-00325.warc.gz"}
https://encyclopediaofmath.org/wiki/Infinite_induction	Infinite induction Carnap's rule, $\omega$-rule A non-elementary derivation rule with an infinite number of premises. More exactly, let the variable $x$ in some logico-mathematical language be regarded as ranging over the natural numbers, and let $\phi(x)$ be a formula of this language. If each formula in the infinite list $$\phi(0),\phi(1),\ldots,\phi(n),\ldots,$$ can be derived, the rule of infinite induction permits to conclude that the formula $\forall x\phi(x)$ is derivable as well. The use of the rule of infinite induction in deriving formulas usually renders the problem of existence of a derivation undecidable. An axiomatic system containing an $\omega$-rule is called a semi-formal theory (semi-formal axiomatic system). They play an important role in proof theory. In order to render the concept of a derivation in the theory effective, additional restrictions must be imposed to ensure that the premises can be effectively derived. It may be required, for example, that derivations of the premises are enumerated by some general recursive function (the so-called constructive rule of infinite induction). It is known that formal arithmetic (cf. Arithmetic, formal), completed by the constructive rule of infinite induction, is complete with respect to classical truth. The rule of infinite induction has also found use in constructing semantics of constructive mathematics by the method of stepwise semantic systems. Still another term for the $\omega$-rule is rule of complete induction.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9872283935546875, "perplexity": 540.3456919807951}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662520817.27/warc/CC-MAIN-20220517194243-20220517224243-00043.warc.gz"}
https://socratic.org/questions/if-a-spring-has-a-constant-of-4-kg-s-2-how-much-work-will-it-take-to-extend-the--11	Physics Topics # If a spring has a constant of 4 (kg)/s^2, how much work will it take to extend the spring by 87 cm ? Feb 3, 2018 Here,work done is equals to the energy stored in the spring as elastic potential energy = $\frac{1}{2} k {x}^{2}$ Given, $k = 4 , x = \frac{87}{100}$ So,work done = $1.5138 J$ ALTERNATIVELY Force required to stretch the spring is $F = k x$ So,work done here is $\mathrm{dW} = F . \mathrm{ds} = k x \mathrm{dx}$ So,integrating from $W = 0 \to W$ and $x = 0 \to \frac{87}{100}$ $W = \frac{1}{2} k {x}^{2} = 1.5138 J$ ##### Impact of this question 219 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8579000234603882, "perplexity": 1524.721795153514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575751.84/warc/CC-MAIN-20190922221623-20190923003623-00420.warc.gz"}
https://katalog.uu.se/profile/?id=N17-1304	# Chao Zhang associate senior lecturer at Department of Chemistry - Ångström Laboratory, Structural Chemistry Email: chao.zhang[AT-sign]kemi.uu.se Telephone: +4618-471 3721 Lägerhyddsvägen 1 Box 538 751 21 Uppsala Short presentation According to the two-volume "Modern Electrochemistry" written by Bockris and Reddy, there are two kinds of electrochemistry. The first one is "The physical chemistry of ionically conducting solutions" and the second one is "The physical chemistry of electrically charged interfaces". I am a computational physical chemist working on these topics in energy storage/conversion and nanochemistry. Drop me a line if you are interested in a thesis work (Master/PhD) or a research project (Postdoc). Keywords: density functional theory molecular dynamics dielectrics electrolyte solid-electrolyte interfaces solid-state batteries Also available at My courses Research ### Density functional theory based molecular dynamics (aka Car-Parrinello MD) A respectable senior colleague once told me that a person doing computational electrochemistry needs to know four things (five to include Machine Learning): Electronic structure theory, statistical mechanics, classical electrodynamics and chemical thermodynamics. This puts the density functional theory based molecular dynamics (DFTMD) as the method of choice. To myself, DFTMD is not just a method but a spirit (or a bridge) which connects the "hard" world (solid state/surface science community) and "soft" world (soft matter/liquid state theory community). ### Finite-field molecular dynamics simulations My recent practice following DFTMD spirit is to explore the constant electric displacement D Hamiltonian in modelling charged solid-liquid interfaces. Constant D Hamiltonian was designed by Stengel, Spaldin and Vanderbilt (SSV) for treating spontaneous polarization in groundstate ferroelectric systems [1]. The macroscopic electric field E and electric displacement D are conjugate variables in macroscopic Maxwell theory. Either E or D can be used as fixed thermodynamic boundary conditions. From classical electrodynamics, we know that the electric displacement D is continuous at a dielectric interface but not the electric field E. That is the reason why it is convenient to use the electric displacement D as the fundamental variable to simulate an interface. According to the classical Debye theory, switching the electric boundary condition from constant potential (E) to constant charge (D) leads to a speed-up of the relaxation time of the macroscopic polarization by a factor comparable to the dielectric constant of the medium. This would be two orders of magnitude difference for aqueous solutions and makes dielectric properties of charged solid-electrolyte interfaces accessible to DFTMD (in theory). ### Dielectric properties at charged solid-electrolyte interfaces We showed that the simulation of finite temperature polarization fluctuations and dielectric constant in polar liquids are now doable in DFTMD [2,3]. The advantage of constant D simulations is not only to speed up simulations but also to eliminate the finite size effect for modelling the electric double layer due to the periodic boundary condition [4]. Now the Helmholtz capacitance of charged solid-liquid interface can be calculated from the supercell polarization [5, 6]. This methodology was further extended to treat the charge compensation between polar surfaces and the electrolyte solution [7, 8]. Its DFTMD implementation is available in one of our community codes CP2K (www.cp2k.org). ### The dead-layer at water interfaces A recent spin-off of constant D simulations is to study dielectric properties under confinement. We showed that the low dielectric constant of nanoconfined water found in molecular dynamics simulations can be largely explained by the so-called dielectric dead-layer effect using a simple capacitor model [9]. Later, this proposed dead-layer effect at water interfaces is confirmed by experiments [10]. [1] Stengel, M., Spaldin, N. A. Vanderbilt, D. Electric Displacement as the Fundamental Variable in Electronic Structure Calculations. Nat. Phys., 2009, 5: 304. [2] Zhang, C. and Sprik, M. Computing the Dielectric Constant of Liquid Water at Constant Dielectric Displacement. Phys. Rev. B, 2016, 93: 144201. [3] Zhang, C., Hutter, J. and Sprik, M. Computing the Kirkwood g-factor by Combining Constant Maxwell Electric Field and Electric Displacement Simulations: Application to the Dielectric Constant of Liquid Water, J. Phys. Chem. Lett., 2016, 7: 2696. [4] Zhang, C. and Sprik, M. Finite Field Methods for the Supercell Modeling of Charged Insulator-Electrolyte Interfaces, Phys. Rev. B, 2016, 94: 245309 (Editors’ Suggestion). [5] Zhang, C. Computing the Helmholtz Capacitance of Charged Insulator-Electrolyte Interfaces from the Supercell Polarization, J. Chem. Phys., 2018, 149: 031103 (Communication). [6] Zhang, C., Hutter, J. and Sprik, M. Coupling of Surface Chemistry and Electric Double Layer at TiO2 Electrochemical Interfaces, J. Phys. Chem. Lett., 2019, 10: 3871. [7] Sayer, T., Zhang, C. and Sprik, M. Charge Compensation at the Interface between the Polar NaCl(111) Surface and a NaCl Aqueous Solution, J. Chem. Phys., 2017, 147: 104702. [8] Sayer, T., Sprik, M. and Zhang, C. Finite electric displacement simulations of polar ionic solid-electrolyte interfaces: Application to NaCl(111)/aqueous NaCl solution, J. Chem. Phys., 2019, 150: 041716 (Editor's pick). [9] Zhang, C. Note: On the Dielectric Constant of Nanoconfined Water. J. Chem. Phys., 2018, 148: 156101. [10] Fumagalli, L. et al., Anomalously low Dielectric Constant of Confined Water. Science, 2018, 360: 1339. 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https://bird.bcamath.org/handle/20.500.11824/4/browse?authority=89c45ba8-6dfd-401d-9d8a-52ace40b904d&type=author	Now showing items 1-1 of 1 • #### Bayesian approach to inverse scattering with topological priors (2020) We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite ...	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9313167929649353, "perplexity": 800.0744272166269}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585183.47/warc/CC-MAIN-20211017210244-20211018000244-00059.warc.gz"}
http://en.wikipedia.org/wiki/Johnson_scheme	# Johnson scheme In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors X of length and weight n, such that $v=\left\|X\right\|=\binom{\ell}{n}$.[1][2][3] Two vectors xy ∈ X are called ith associates if dist(xy) = 2i for i = 0, 1, ..., n. The eigenvalues are given by $p_{i}\left(k\right)=E_{i}\left(k\right),$ $q_{k}\left(i\right)=\frac{\mu_{k}}{v_{i}}E_{i}\left(k\right),$ where $\mu_{i}=\frac{\ell-2i+1}{\ell-i+1}\binom{\ell}{i},$ and Ek(x) is an Eberlein polynomial defined by $E_{k}\left(x\right)=\sum_{j=0}^{k}(-1)^{j}\binom{x}{j} \binom{n-x}{k-j}\binom{\ell-n-x}{k-j},\qquad k=0,\ldots,n.$ ## References 1. ^ P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,“ IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2477–2504, 1998. 2. ^ P. Camion, "Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding," in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands, 1998. 3. ^ F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, New York, 1978.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9764109253883362, "perplexity": 2368.728696933106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00494-ip-10-147-4-33.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/588824/satisfying-equality-between-logarithmic-expressions	# Satisfying equality between logarithmic expressions Apologies in advance for any misused terminology, or if this is the wrong place for the question (I think it's okay though). I am given a group of logarithmic expressions such as: • $- (a \log(a) + (1 - a)\log(\frac{1-a}{2}))$ • $- (b \log(b) + (1 - b)\log(\frac{1-b}{4}))$ • $- (c \log(c) + (1 - c)\log(\frac{1-c}{5}))$ My goal is to find values of $a$, $b$, and $c$ that make these expressions equal. The above is just an example of the sort of expressions I get, but they are always of the form $-(x\log(x) + (1-x) log(\frac{1-x}{n}))$, where $n$ is a an integer $\ge 1$, and $x$ is some symbol that doesn't occur in more than one expression in the group. Ultimately I'll be using a computer to solve these problems. But my question is: Can I solve this by hand? And if so, how? Alternatively, can I distill the problem down to something simpler to code up? I imagine I'd start by creating a set of equations that expresses equality between the expressions. So between expression 1 and 2 I made: $- (a \log(a) + (1 - a)\log(\frac{1-a}{2})) = - (b \log(b) + (1 - b)\log(\frac{1-b}{4}))$ But I'm not sure how to go about solving for the values of the variables. I found a bunch of info on solving systems of linear equations but I'm not sure what to do with the logs. • Just to clarify before I attempt to answer - $\log$ refers to the natural logarithm, yes? – Zubin Mukerjee Dec 2 '13 at 0:32 • @ZubinMukerjee: I can assure you that it makes no difference at all since $\log_a(x)=k\cdot\ln(x)$ for some constant $k$. In fact this constant is $k=\log_a(\mbox{e})=1/\ln(a)$. This is known as 'change of base for logarithms'. – String Dec 2 '13 at 0:38 • I think he was just asking for clarification before writing up his solution. – Doc Dec 2 '13 at 0:40 • I'm assuming base 2. The expressions were derived from some information entropy equations. Each expression is the entropy of some distribution. (This has made me realize some errors, and I've changed the expressions accordingly.) – oadams Dec 2 '13 at 0:53 • I see that the expressions only make sence for $a,b,c,x,...\in(0,1)$. Also if $x+y=1$ the general form of the expressions is $$x\log(x)-y\log(y)+y\log(n)$$ – String Dec 2 '13 at 1:02 I wrote this when the expressions in the question didn't have leading negatives. However, I think the method still works in the same way. This solution requires computation that cannot be done (or at least would be very tedious) by hand. Let's simplify the general form of the expressions you're setting equal to each other: $$x \log(x) - \left(1-x\right))\log\left(\frac{1-x}{n}\right)$$ $$x \log(x) - \left(1-x\right)\left(\log\left(1-x\right)-\log(n)\right)$$ Note that for both $\log(x)$ and $\log(1-x)$ to be real numbers, $x$ must satisfy $0 < x < 1$. We'll continue with this assumption. $$x\log(x) - \left(1-x\right)\log\left(1-x\right) + \left(1-x\right)\log\left(n\right)$$ Set this equal to a constant $k$: $$x\log(x) - \left(1-x\right)\log\left(1-x\right) + \left(1-x\right)\log\left(n\right) = k$$ $$x\log(x) - \left(1-x\right)\log\left(1-x\right) = \left(x-1\right)\log\left(n\right)+k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ This is a graph (from Wolfram Alpha) of the left side of the above equation, strictly between $x=0$ and $x=1$: We want to find a fixed value of $k$ for which there exist values of $x$, call them $x_n$ (in your question, these were the $a$, $b$, $c$, etc.) for every integer $n \geq 1$, so that equation $(1)$ holds. In particular, this value of $k$ must yield a solution when $n=1$. When $n=1$, $\log(n) = \log(1) = 0$, so $\left(x-1\right)\log\left(n\right)+k$ is just the horizontal line given by $y=k$. This means $k$ must be between approximately $-0.14$ and $0.14$ in order for there to be a solution to equation $(1)$. When $n>1$, $y=(x-1)\log(n) + k$ is a line with positive slope that intersects the $y$-axis at $y=k-\log(n)$. I propose that $k=0.1$ yields a solution $x_n$ for all integers $n \geq 1$. This is a graph of the right side of equation $(1)$, superimposed on the graph of the left side, when $n = 2$ and $k = 0.1$: Note that the line $y=(x-1)\log(n) + 0.1$ intersects the vertical line $x=1$ at $y=0.1$, regardless of what we choose $n$ to be. My graphical skills are poor, so you'll have to imagine the set of lines $y=(x-1)\log(n) + 0.1$ over all integers $n \geq 1$ as a set of lines that all have non-negative slope and all pass through the point $(x,y) = (1,0.1)$. It should be clear that there must be an $x_n$ that solves equation $(1)$ for each $n$. The value we chose for $k$, $0.1$, was certainly not the only one. Any $k$ in some neighborhood of $k=0.1$ would yield a set of $x_i$ that work with equation $(1)$. This implies that the solutions to your infinite system of equations are nowhere close to unique. When setting all of the equations equal to each other, I would recommend choosing a value of $k$ to set them all equal to, e.g. $k=0.1$. Finding the values of $x_n$, once you've fixed a $k$ shouldn't be too difficult to code. For each $n$, finding $x_n$ is finding the intersection of a line that goes through $(x,y) = (1, k)$ with slope $\log(n)$, with the curve $y=x\log(x) - \left(1-x\right)\log\left(1-x\right)$. I hope that helps. • Thank you for this thorough and instructional answer. Although I unfairly changed the sign not just at the start of the expression, but in the middle as well, the principles here showed me what to do nonetheless. – oadams Dec 2 '13 at 6:11 $$x \log(x) - (1-x) \log\left(\frac{1-x}{n}\right) = \log \left(x^x \left(\frac{n}{1-x}\right)^{1-x}\right)$$ but that's about all the simplification you'll get. In particular, you're unlikely to get closed-form solutions when equating these for different $x$'s and different $n$'s. Disclaimer: This is nothing near a solution - just an observation: Here is a dynamic graph (for changing values of $n$) of $$f_n(x)=f_1(x)+(1-x)\log(n)$$ where $f_1(x)=x\log(x)-(1-x)\log(1-x)$: As can be seen above the graph of $f_1(x)$ is simply added to a line through $(0,\log(n))$ and $(1,0)$. • I think this puts much more clearly the part in my answer where I say "imagine the set of lines ..." :) – Zubin Mukerjee Dec 2 '13 at 1:37 • @ZubinMukerjee: I did not add in much brain-work, though :) Great observations on your part! I just programmed it into GeoGebra to consider what happens when $n$ changes... – String Dec 2 '13 at 1:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9472126960754395, "perplexity": 210.19364886270202}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524111.50/warc/CC-MAIN-20190715195204-20190715221204-00450.warc.gz"}
https://community.jmp.com/t5/Discussions/Response-surface-final-equation-in-terms-of-actual-factors/td-p/45651	## Response surface final equation in terms of actual factors New Contributor Joined: Oct 8, 2017 Hi I am very new to JMP and recently im using DOE for response surface methodology. My question is; the equation in the softwear is given in terms of coded factors, how i can use this softwear to convert the equation with actual factors? Many thanks Yang 1 ACCEPTED SOLUTION Accepted Solutions Super User Joined: Jun 22, 2012 Solution What you are displaying are the Parameter Estimates for the model that is specified for your DOE. The definition of it can be found in the DOE Documentation Help==>Books==>Design of Experiments Guide If what you displayed is an attempt to follow what Dan indicated to do, let me clarify that for you. What was specified was to "save the prediction formula from the model fitting results". To do this, click on the red triangle and select Save Columns==>Prediction Formula This saves the Prediction Formula to the data table. Then, "Open up the formula from the data table.". To do this, go to the data table, right click on the header for the new column that has been added to the data table, and select "Formula" The formula window will be opened up. Then you can "From the red triangle pop up menu of the equation editor, choose simplify" This is the prediction formula for your Model Jim 6 REPLIES Joined: Apr 3, 2013 One way is to save the prediction formula from the model fitting results. Open up the formula from the data table. From the red triangle pop up menu of the equation editor, choose simplify. Dan Obermiller New Contributor Joined: Oct 8, 2017 Hi dan Parameter Estimates Term Estimate Std Error t Ratio Prob>\|t\| Intercept 19,956667 0,320239 62,32 <,0001 Temp(45,75) -1,40125 0,196105 -7,15 0,0008 time(30,120) 1,27 0,196105 6,48 0,0013 E(0,1,1) 3,82625 0,196105 19,51 <,0001 Temptime -1,31 0,277335 -4,72 0,0052 TempE -1,3075 0,277335 -4,71 0,0053 timeE 0,515 0,277335 1,86 0,1224 TempTemp -2,127083 0,288659 -7,37 0,0007 timetime -0,809583 0,288659 -2,80 0,0378 EE -1,747083 0,288659 -6,05 0,0018 This is what i got for the formular, i have no idea how to do with it Super User Joined: Jun 22, 2012 Solution What you are displaying are the Parameter Estimates for the model that is specified for your DOE. The definition of it can be found in the DOE Documentation Help==>Books==>Design of Experiments Guide If what you displayed is an attempt to follow what Dan indicated to do, let me clarify that for you. What was specified was to "save the prediction formula from the model fitting results". To do this, click on the red triangle and select Save Columns==>Prediction Formula This saves the Prediction Formula to the data table. Then, "Open up the formula from the data table.". To do this, go to the data table, right click on the header for the new column that has been added to the data table, and select "Formula" The formula window will be opened up. Then you can "From the red triangle pop up menu of the equation editor, choose simplify" This is the prediction formula for your Model Jim New Contributor Joined: Oct 8, 2017 Thank you very much! New Contributor Joined: Oct 8, 2017 Hi Jim I have another question about RSM result. how can i obtain the anova table or the "Parameter Estimates" like this? i mean with the quadric and linear factor which give SS and MS values in the result? Many thanks Super User Joined: Jun 22, 2012 The source table you are referring to is found in the "EffectsTests" paragraph in the Anova(RSM) output. By default the Mean Square is not displayed, since it is really a redundant statistic, however, it can be displayed if you right click on the table and select "Columns" and then select Mean Square. Jim	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8480021357536316, "perplexity": 3749.5741135207354}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590559.95/warc/CC-MAIN-20180719051224-20180719071224-00203.warc.gz"}
http://math.stackexchange.com/questions/150385/graph-of-a-rel-morphism	# Graph of a Rel-morphism Let $F=(f;A;B)$ is a morphism of the category $\mathbf{Rel}$ (the category whose objects are sets and morphisms are defined as binary relations). How to name and how to denote $f$ when we know $F$? I propose to call $f$ the graph of $F$. Right name? But how to denote it? Are there a standard notation? I propose the following (non-standard) notation: $\mathrm{GR}\, F= f$. - In practice you won't have to distinguish between $f$ and $F$. And I would call $f$ just a relation from $A$ to $B$. – Martin Brandenburg May 27 '12 at 14:22 Graph of F is perfectly good. I would write: $Graph(F)=f$, since GR in itself is a bit opaque as a name. Edited after taking comments into account: $\mathbf{Rel}$ is a bit atypical in the sense that it is named after its morphisms, while most categories are named after their objects (see MacLane's CWM chapter 1 notes). We can however form the category of arrows of $\mathbf{Rel}$, ie. $\mathbf{Rel}^\mathbf{2}$ (see CWM again) where the relations are the objects. So we could then write: $U(F)=f$ Where $U:\mathbf{Rel}^\mathbf{2}\to \mathbf{Set}$ is the forgetful functor from $\mathbf{Rel}^\mathbf{2}$ to $\mathbf{Set}$ (since the graph of F is a set). $\mathbf{2}$ is the category with 2 objects and just one morphism between them Correction: On second thought: graph of F is not a good name. That is because graph is normally meant to be an ordered pair (Wikipedia), while your $f$ is just a subset of $A \times B$. So I would simply call $f$ "the underlying set of $F$" and use the $U(F)=f$ notation to derive it - I'm tempted to apply a -1: the category $\textbf{Rel}$ is not a category whose objects are relations. – Zhen Lin May 28 '12 at 12:40 There is no functor U here ... – Martin Brandenburg May 28 '12 at 13:41 @ZhenLin Ops...you are very right. My answer was not very well thought out. I would prefer to delete it if possible or otherwise see if there is a possible category where relations are objects and morphisms are...what? – magma May 28 '12 at 13:45 There is a 2-dimensional category $\mathfrak{Rel}$, and the hom-category $\mathfrak{Rel}(X, Y)$ is an ordinary category whose objects are relations from $X$ to $Y$ and morphisms are, well, maps of sets. But this is just a really fancy way of talking about $\mathscr{P}(X \times Y)$. – Zhen Lin May 28 '12 at 13:51 @zhenLin yes or we could consider this: $\mathbf{Rel'}$: objects are relations (ordered triples), morphisms from <r,A,B> to <r',C,D> are <f,f'> with $r' \circ f =r \circ f'$ where composition in $\mathbf{Rel}$ is meant. This is called the category of arrows of $\mathbf{Rel}$ (MacLane's CWM page 40) – magma May 28 '12 at 14:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9347860217094421, "perplexity": 328.64416042620627}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929956.54/warc/CC-MAIN-20150521113209-00328-ip-10-180-206-219.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/1714067/problem-about-biking-and-car-driving-from-one-point-to-another/1714086	# Problem about biking and car driving from one point to another A cyclist begins to bike from point A at $6$:$00$ am to point B. At $8$:$00$ am a car begins to drive from point A also to point B and drove at a velocity $60$ km/h faster than the bike. The car met the bike after they had driven/biked $\frac{5}{16}$ from the the distance between A and B. Immediately the car continued driving. He got to point B and waited $2$ hours there. Then, drove back to point B. After the car had driven $\frac14$ of the distance between A and B he met the bike that was still on his way to point B. Find the velocity of the cyclist and the distance between A and B. In this type of problems I like to do charts. I put $x$ as the velocity of the cyclist, $t$ the time of the cyclist to meet the car and $y$ the distance between A and B. In addition, $t_2$ is the time that takes the bike to get from A to B. So what I did is the following: $$\begin{array}{\|c\|c\|c\|} \hline \text{Until they met}& \text{v velocity} & \text{t time} &\text{d distance} \\ \hline \text{Cyclist} & x & t & \frac{5}{16}y\\ \hline \text{Car} & x + 60 & t-2 & \frac{5}{16}y\\ \hline \text{Cyclist} & x & t_2 & y \\\hline \end{array}$$ I'm stuck here. I don't know how to continue. I don't know how to use the other data like he waited $2$ hours, in the way back he met again the bike, and so far. Let the biker's velocity be $x$. Then $x+60$ is the speed of the car. Let the distance between $A$ and $B$ be $d$. Now, according to the first statement, to cover $\frac{5}{16}$ of the distance from $A$ to $B$, the biker took $2$ hours more than the car. So we write :$$\frac{5d}{16x} - \frac{5d}{16(x+60)} = 2$$. For the second part, let's calculate the time between the two meetings using the bike. So the bike first met at $\frac{5d}{16}$, and next he meets him at $\frac{3d}{4}$, so the distance travelled by the biker was $\frac{7d}{16}$, and the time it took was $\frac{7d}{16x}$. Now use the car. The car traveled the remaining $\frac{11d}{16}$ of the distance, waited for $2$ hours, and came back $\frac{d}{4}$ distance, therefore he traveled $\frac{15d}{16}$ distance and waited two hours before the next meeting, so the times taken was $\frac{15d}{16(x+60)} + 2$. Now the time waited by the car and the bike were equal hence we write down the second equation: $$\frac{15d}{16(x+60)} + 2=\frac{7d}{16x}$$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8289716839790344, "perplexity": 164.47411867881883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00168.warc.gz"}
http://www.thespectrumofriemannium.com/tag/quantum-physics/	## LOG#150. Bohr and Doctor Who: A=mc³. The year 2013 is coming to its end…And I have a final gift for you. An impossible post! This year was the Bohr model 100th anniversary. I have talked about this subject already, here, here and here. The hydrogen spectrum is very important in Astronomy, … Continue reading ## LOG#148. Path integral (III). Round 3! Fight… with path integrals! XD Introduction: generic aspects Consider a particle moving in one dimension (the extension to ND is trivial), the hamiltonian being of the usual form: The fundamental question and problem in the path integral (PI) … Continue reading ## LOG#136. Flavor ν mixtures. Neutrino oscillations are one of the most surprising “sounds” in the whole Universe. Since neutrinos do oscillate/mix, they are massive. And due to mass, they can experiment “mixing” or “changes” of flavor (mass and flavor basis are different!). Even more, … Continue reading ## LOG#114. Bohr’s legacy (II). Dedicated to Niels Bohr and his atomic model (1913-2013) 2nd part: Electron shells, Quantum Mechanics and The Periodic Table Niels Bohr (1923) was the first to propose that the periodicity in the properties of the chemical elements might be explained … Continue reading ## LOG#113. Bohr’s legacy (I). Dedicated to Niels Bohr and his atomic model (1913-2013) 1st part: A centenary model This is a blog entry devoted to the memory of a great scientist, N. Bohr, one of the greatest master minds during the 20th century, one … Continue reading	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8596962690353394, "perplexity": 3082.6969905609917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189462.47/warc/CC-MAIN-20170322212949-00104-ip-10-233-31-227.ec2.internal.warc.gz"}
https://proofwiki.org/wiki/Space_with_Open_Point_is_Non-Meager	# Space with Open Point is Non-Meager Jump to navigation Jump to search ## Theorem Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$ be an open point. Then $T$ is a non-meager space. ## Proof Let $x \in S$ be an open point of $T$. That is: $\left\{{x}\right\} \in \tau$ Recall that: a topological space is non-meager if it is not meager and: a topological space is meager if and only if it is a countable union of subsets of $S$ which are nowhere dense in $S$. Aiming for a contradiction, suppose that $T$ is meager. Let: $\displaystyle T = \bigcup \mathcal S$ where $\mathcal S$ is a countable set of subsets of $S$ which are nowhere dense in $S$. Then: $\exists H \in \mathcal S: x \in H$ and so: $\left\{{x}\right\} \subseteq H$ We have that $H$ is nowhere dense in $T$. By definition, its closure $H^-$ contains no open set of $T$ which is non-empty. But from Set is Subset of its Topological Closure we have that: $H \subseteq H^-$ $\left\{{x}\right\} \subseteq H^-$ So $H$ is not nowhere dense. Therefore $T$ cannot be a countable union of subsets of $S$ which are nowhere dense in $S$. That is, $T$ is not meager. Hence the result by definition of non-meager. $\blacksquare$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9026771187782288, "perplexity": 201.37300031332302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670255.18/warc/CC-MAIN-20191119195450-20191119223450-00002.warc.gz"}
https://www.physicsoverflow.org/12572/is-there-a-physical-intuition-for-diamagnetic-inequality	Is there a physical intuition for diamagnetic inequality? + 4 like - 0 dislike 754 views Diamagnetic inequality implies, quantum mechanically, for a charged particle without intrinsic magnetic moment(or to say ignoring spin-magnetic field interaction) in some potential $V(x)$, when subjected to some arbitrary external magnetic field $A(x)$ (vector potential), the ground state energy(more generally the infimum of all energy expectation values) is always bigger than the one without magnetic field$^1$. The statement seems simple enough to make it tempting to find a physical intuition. A classical picture just does not help, since the situation becomes trivial: the ground state is always such that the particle is sitting still at the minimum of $V(x)$, when applying a magnetic field, this is still the the ground state. Does anyone has an idea about what the intuition should be, if any? What I want is something more physically pictorial, like "magnetic field curves the electron's trajectory and hence...". But I admit this might possibly be too much to ask. Update: The Landau level seems to be a good exactly-solvable example, without the magnetic field, we are dealing with a free particle of which the ground state energy (again the infimum of all energy expectation values, if we wish to stay in Hilbert space)is 0, while with a uniform magnetic field turned on the ground state energy becomes $\frac{1}{2}\hbar\omega$. Crossposted from my own post at stackexchange: Is there a physical intuition for diamagnetic inequality? 1.Stablity of Matter in Quantum Mechanics, E.Lieb and R. Seiringer, Cambridge University Press. Lemma 4.1, and parts of Theorem 3.3, discussing when ground state wavefunction can be chosen to be real and positive. + 3 like - 0 dislike I) The diamagnetic inequality $$\tag{1} \left\|\vec{\nabla}\|\psi(\vec{r})\|\right\| ~\leq~ \left\|(\vec{\nabla}+i\vec{A}(\vec{r}))\psi(\vec{r})\right\|$$ is proven rigorously in Lieb & Loss, Analysis, sections 7.19 - 7.22, as user Willie Wong explains in his mathoverflow answer. II) However, it seems that OP is not asking for rigor, but rather intuition. Here is a heuristic proof. As good physicists let us assume that all involved functions are smooth/differentiable. It turns out that the absolute value $\|\psi(\vec{r})\|$ of the wave function $\psi(\vec{r})$ cannot be differentiable and have a zero $\|\psi(\vec{r})\|=0$ unless the zero is of at least second order, i.e, a stationary point $\vec{\nabla}\|\psi(\vec{r})\|=\vec{0}$. In that case the inequality (1) is trivially satisfied. III) Let us therefore assume from now on that the wave function $\psi(\vec{r})\neq 0$ does not have a zero. Then we may locally polar decompose the wave function $$\tag{2}\psi(\vec{r})~=~R(\vec{r})e^{i\theta(\vec{r})}, \qquad R(\vec{r})~>~0 , \qquad \theta(\vec{r})~\in~\mathbb{R}.$$ The square of the inequality (1) becomes a triviality: $$\left\|\vec{\nabla}\|\psi(\vec{r})\|\right\|^2 ~=~\left\|\vec{\nabla}R(\vec{r})\right\|^2 ~\leq~ \left\|\vec{\nabla}R(\vec{r})\right\|^2+ R(\vec{r})^2\left\|\vec{\nabla}\theta(\vec{r})+\vec{A}(\vec{r})\right\|^2$$ $$~=~ \left\|e^{i\theta(\vec{r})}\left\{\vec{\nabla}R(\vec{r})+iR(\vec{r})(\vec{\nabla}\theta(\vec{r})+\vec{A}(\vec{r}))\right\}\right\|^2~=~$$ $$\tag{3} ~=~ \left\|(\vec{\nabla}+i\vec{A}(\vec{r}))R(\vec{r})e^{i\theta(\vec{r})}\right\|^2~=~\left\|(\vec{\nabla}+i\vec{A}(\vec{r}))\psi(\vec{r})\right\|^2.$$ This post imported from StackExchange Physics at 2014-04-04 05:13 (UCT), posted by SE-user Qmechanic answered Mar 12, 2014 by (3,110 points) Thanks for the reply, and +1 for putting a proof. I am aware of this kind of proof. What I really want is something more physically pictorial, like "magnetic field curves the electron's trajectory and hence...". But I admit this might possibly be too much to ask. This post imported from StackExchange Physics at 2014-04-04 05:13 (UCT), posted by SE-user Jia Yiyang + 3 like - 0 dislike Consider the path integral in imaginary time. Here, the wavefunction at time t is the sum over all paths of a weight. The weight coming from the kinetic energy term is the (real-valued) probability of a random walk taking this path, it's a product of Gaussians when you discretize. The potential energy at position x is an additional positive or negative decay rate in time for each path whenever it is going through x. If you consider the ground state wavefunction, you weight the starting points with the weight equal to the ground state, and you reproduce the ground state at time t, rescaled by exp(-Et) where E is the ground state energy. So the rate of shrinkage is the ground state energy. The ground state is real and positive without a magnetic field. Now when you turn on a magnetic field, you add a term which is a phase (it is complex in imaginary time too). It adds a complex phase on each path equal to the line-integral of the vector potential along the path. The only possible effect is to reduce the value of the integral at time T, since adding together paths with phase is always less than adding together paths without phases. That means all states decay faster in the presence of magnetic field, and that means the ground state energy went up. answered Apr 5, 2014 by (7,720 points) Thanks for the reply, it is good to have you answering my question again. I need a few clarifications: If I undestand you correctly, you are talking about the path integral representation of $\langle x\|e^{-tH}\|0\rangle= \langle x\|0\rangle e^{-tE}$, where $\|0\rangle$ is the ground state. (1)You claim "adding together paths with phase is always less than adding together paths without phases.", I don't understand why this is true, as a naive counterexample, if we have two positive real numbers $a$ and $b$, $a+b$ is not necessarily bigger than $\|ae^{i\alpha}+be^{i\beta}\|$(Edit: This is silly, it is necessarily bigger).(2)When magnetic field is turned on, the wavefunction of ground state would change, so we must consider a path integral with a different choice of initial weight, does it jeopardize some of your arguments？ Ah I see what you did now, you basically argued $\langle\psi\|e^{-tH_B}\|\psi\rangle\leq \langle\psi\|e^{-tH}\|\psi\rangle$ for all $\psi$ with positive wavefunction, where the subscript B means the presence of a magnetic field. Because it holds for all t, in first order we must have$\langle\psi\|-tH_B\|\psi\rangle\leq \langle\psi\|-tH\|\psi\rangle$, i.e. $\langle\psi\|H_B\|\psi\rangle\geq \langle\psi\|H\|\psi\rangle$, which is just diagmagnetic inequality, which of course indicates a increase in ground state energy! In essence you proved diagmagnetic inequality using path integral. This is cool and definitely adds something to my intuitition. +1. Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. 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http://nag.com/numeric/cl/nagdoc_cl23/html/F08/f08upc.html	f08 Chapter Contents f08 Chapter Introduction NAG C Library Manual NAG Library Function Documentnag_zhbgvx (f08upc) 1 Purpose nag_zhbgvx (f08upc) computes selected the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form $Az=λBz ,$ where $A$ and $B$ are Hermitian and banded, and $B$ is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues. 2 Specification #include #include void nag_zhbgvx (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Integer ka, Integer kb, Complex ab[], Integer pdab, Complex bb[], Integer pdbb, Complex q[], Integer pdq, double vl, double vu, Integer il, Integer iu, double abstol, Integer m, double w[], Complex z[], Integer pdz, Integer jfail[], NagError fail) 3 Description The generalized Hermitian-definite band problem $Az = λ Bz$ is first reduced to a standard band Hermitian problem $Cx = λx ,$ where $C$ is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem. The eigenvectors are normalized so that $zH A z = λ and zH B z = 1 .$ 4 References Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44 Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912 Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press 5 Arguments 1: orderNag_OrderTypeInput On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument. Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor. 2: jobNag_JobTypeInput On entry: indicates whether eigenvectors are computed. ${\mathbf{job}}=\mathrm{Nag_EigVals}$ Only eigenvalues are computed. ${\mathbf{job}}=\mathrm{Nag_DoBoth}$ Eigenvalues and eigenvectors are computed. Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$. 3: rangeNag_RangeTypeInput On entry: if ${\mathbf{range}}=\mathrm{Nag_AllValues}$, all eigenvalues will be found. If ${\mathbf{range}}=\mathrm{Nag_Interval}$, all eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found. If ${\mathbf{range}}=\mathrm{Nag_Indices}$, the ilth to iuth eigenvalues will be found. Constraint: ${\mathbf{range}}=\mathrm{Nag_AllValues}$, $\mathrm{Nag_Interval}$ or $\mathrm{Nag_Indices}$. 4: uploNag_UploTypeInput On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangles of $A$ and $B$ are stored. If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangles of $A$ and $B$ are stored. Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$. 5: nIntegerInput On entry: $n$, the order of the matrices $A$ and $B$. Constraint: ${\mathbf{n}}\ge 0$. 6: kaIntegerInput On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the number of superdiagonals, ${k}_{a}$, of the matrix $A$. If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the number of subdiagonals, ${k}_{a}$, of the matrix $A$. Constraint: ${\mathbf{ka}}\ge 0$. 7: kbIntegerInput On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the number of superdiagonals, ${k}_{b}$, of the matrix $B$. If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the number of subdiagonals, ${k}_{b}$, of the matrix $B$. Constraint: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$. 8: ab[$\mathit{dim}$]ComplexInput/Output Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$. On entry: the upper or lower triangle of the $n$ by $n$ Hermitian band matrix $A$. This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of ${A}_{ij}$, depends on the order and uplo arguments as follows: • if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, ${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{a}+i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{a}\right),\dots ,j$; • if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, ${A}_{ij}$ is stored in ${\mathbf{ab}}\left[i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{a}\right)$; • if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, ${A}_{ij}$ is stored in ${\mathbf{ab}}\left[j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{a}\right)$; • if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, ${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{a}+j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{a}\right),\dots ,i$. On exit: the contents of ab are overwritten. 9: pdabIntegerInput On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab. Constraint: ${\mathbf{pdab}}\ge {\mathbf{ka}}+1$. 10: bb[$\mathit{dim}$]ComplexInput/Output Note: the dimension, dim, of the array bb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdbb}}×{\mathbf{n}}\right)$. On entry: the upper or lower triangle of the $n$ by $n$ Hermitian positive definite band matrix $B$. This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of ${B}_{ij}$, depends on the order and uplo arguments as follows: • if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, ${B}_{ij}$ is stored in ${\mathbf{bb}}\left[{k}_{b}+i-j+\left(j-1\right)×{\mathbf{pdbb}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{b}\right),\dots ,j$; • if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, ${B}_{ij}$ is stored in ${\mathbf{bb}}\left[i-j+\left(j-1\right)×{\mathbf{pdbb}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{b}\right)$; • if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, ${B}_{ij}$ is stored in ${\mathbf{bb}}\left[j-i+\left(i-1\right)×{\mathbf{pdbb}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{b}\right)$; • if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, ${B}_{ij}$ is stored in ${\mathbf{bb}}\left[{k}_{b}+j-i+\left(i-1\right)×{\mathbf{pdbb}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{b}\right),\dots ,i$. On exit: the factor $S$ from the split Cholesky factorization $B={S}^{\mathrm{H}}S$, as returned by nag_zpbstf (f08utc). 11: pdbbIntegerInput On entry: the stride separating row or column elements (depending on the value of order) of the matrix $B$ in the array bb. Constraint: ${\mathbf{pdbb}}\ge {\mathbf{kb}}+1$. 12: q[$\mathit{dim}$]ComplexOutput Note: the dimension, dim, of the array q must be at least • $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoBoth}$; • $1$ otherwise. The $\left(i,j\right)$th element of the matrix $Q$ is stored in • ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$; • ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, the $n$ by $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form. If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, q is not referenced. 13: pdqIntegerInput On entry: the stride separating row or column elements (depending on the value of order) in the array q. Constraints: • if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$; • otherwise ${\mathbf{pdq}}\ge 1$. 14: vldoubleInput 15: vudoubleInput On entry: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, the lower and upper bounds of the interval to be searched for eigenvalues. If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Indices}$, vl and vu are not referenced. Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$. 16: ilIntegerInput 17: iuIntegerInput On entry: if ${\mathbf{range}}=\mathrm{Nag_Indices}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned. If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Interval}$, il and iu are not referenced. Constraints: • if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$; • if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$. 18: abstoldoubleInput On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to $abstol+ε maxa,b ,$ where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_{1}$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this function returns with NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990). 19: mInteger Output On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$. If ${\mathbf{range}}=\mathrm{Nag_AllValues}$, ${\mathbf{m}}={\mathbf{n}}$. If ${\mathbf{range}}=\mathrm{Nag_Indices}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$. 20: w[n]doubleOutput On exit: the eigenvalues in ascending order. 21: z[$\mathit{dim}$]ComplexOutput Note: the dimension, dim, of the array z must be at least • $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdz}}×{\mathbf{n}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoBoth}$; • $1$ otherwise. The $\left(i,j\right)$th element of the matrix $Z$ is stored in • ${\mathbf{z}}\left[\left(j-1\right)×{\mathbf{pdz}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$; • ${\mathbf{z}}\left[\left(i-1\right)×{\mathbf{pdz}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, z contains the matrix $Z$ of eigenvectors, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left[i-1\right]$. The eigenvectors are normalized so that ${Z}^{\mathrm{H}}BZ=I$. If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, z is not referenced. 22: pdzIntegerInput On entry: the stride separating row or column elements (depending on the value of order) in the array z. Constraints: • if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$; • otherwise ${\mathbf{pdz}}\ge 1$. 23: jfail[$\mathit{dim}$]IntegerOutput Note: the dimension, dim, of the array jfail must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$. On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, then • if NE_NOERROR, the first m elements of jfail are zero; • if NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge. If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, jfail is not referenced. 24: failNagError Input/Output The NAG error argument (see Section 3.6 in the Essential Introduction). 6 Error Indicators and Warnings NE_ALLOC_FAIL Dynamic memory allocation failed. On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value. NE_CONVERGENCE The algorithm failed to converge; $〈\mathit{\text{value}}〉$ eigenvectors did not converge. NE_ENUM_INT_2 On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$. Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$; otherwise ${\mathbf{pdq}}\ge 1$. On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$. Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$; otherwise ${\mathbf{pdz}}\ge 1$. NE_ENUM_INT_3 On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{il}}=〈\mathit{\text{value}}〉$, ${\mathbf{iu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$. Constraint: if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$; if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$. NE_ENUM_REAL_2 On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{vl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{vu}}=〈\mathit{\text{value}}〉$. Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$. NE_INT On entry, ${\mathbf{ka}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{ka}}\ge 0$. On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{n}}\ge 0$. On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{pdab}}>0$. On entry, ${\mathbf{pdbb}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{pdbb}}>0$. On entry, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{pdq}}>0$. On entry, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{pdz}}>0$. NE_INT_2 On entry, ${\mathbf{ka}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kb}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$. On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ka}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{pdab}}\ge {\mathbf{ka}}+1$. On entry, ${\mathbf{pdbb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kb}}=〈\mathit{\text{value}}〉$. Constraint: ${\mathbf{pdbb}}\ge {\mathbf{kb}}+1$. NE_INTERNAL_ERROR An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance. NE_MAT_NOT_POS_DEF If ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}={\mathbf{n}}+〈\mathit{\text{value}}〉$, for $1\le 〈\mathit{\text{value}}〉\le {\mathbf{n}}$, then nag_zpbstf (f08utc) returned ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}=〈\mathit{\text{value}}〉$: $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed. 7 Accuracy If $B$ is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of $B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of $B$ would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds. The total number of floating point operations is proportional to ${n}^{3}$ if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$ and ${\mathbf{range}}=\mathrm{Nag_AllValues}$, and assuming that $n\gg {k}_{a}$, is approximately proportional to ${n}^{2}{k}_{a}$ if ${\mathbf{job}}=\mathrm{Nag_EigVals}$. Otherwise the number of floating point operations depends upon the number of eigenvectors computed. The real analogue of this function is nag_dsbgvx (f08ubc). 9 Example This example finds the eigenvalues in the half-open interval $\left(0.0,2.0\right]$, and corresponding eigenvectors, of the generalized band Hermitian eigenproblem $Az=\lambda Bz$, where $A = -1.13i+0.00 1.94-2.10i -1.40+0.25i 0.00i+0.00 1.94+2.10i -1.91i+0.00 -0.82-0.89i -0.67+0.34i -1.40-0.25i -0.82+0.89i -1.87i+0.00 -1.10-0.16i 0.00i+0.00 -0.67-0.34i -1.10+0.16i 0.50i+0.00$ and $B = 9.89i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00 .$ 9.1 Program Text Program Text (f08upce.c) 9.2 Program Data Program Data (f08upce.d) 9.3 Program Results Program Results (f08upce.r)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 251, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.998764157295227, "perplexity": 3256.4391371117185}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929171.55/warc/CC-MAIN-20150521113209-00058-ip-10-180-206-219.ec2.internal.warc.gz"}
https://cs.stackexchange.com/questions/29996/given-a-non-deterministic-mealy-machine-m-if-l-is-regular-is-ml-regula	# Given a non-deterministic Mealy machine $M$, if $L$ is regular, is $M(L)$ regular? Consider a nondeterministic Mealy machine, $M$, defined as follows: $M = (Q, \Sigma, \Delta, \delta, \tau, q_0)$ where 1. $Q$ is a finite set of states 2. $\Sigma$ is an input alphabet 3. $\Delta$ is an output alphabet 4. $\delta : Q \times \Sigma_{\varepsilon} \to \mathcal{P}(Q)$ is the transition function 5. $\tau : Q \times \Sigma_{\varepsilon} \to \mathcal{P}(\Delta)$ is the output function 6. $q_0 \in Q$ is the start state Let $M(x)$ for $x \in \Sigma^$ denote the set of all strings that $M$ could output on input $x$. Note that $M$ could fail to process the entirety of its input, and thus a string is output only if $M$ processes all of its input. Also, we define, for $L \subset \Sigma^$, $M(L) = \bigcup\limits_{w \in L} M(w)$. Given this, if $L$ is regular, does it follow that $M(L)$ is regular? I've attempted to solve this by first showing that for every non-deterministic Mealy machine, there exists an equivalent deterministic Mealy machine. By "equivalent" I mean that the input-output behavior of the two is identical. The problem, however, is that the output function for a deterministic Mealy machine can only output a single character at a time. How, then, could I get the simulating deterministic Mealy machine to output a set of strings? • Where did you get the definition? For each state and input the transition and output functions give sets of states and output symbols. Probably the choice of state and output should be connected in some way? You cannot assume determinism, for the reason you state. You do not need it. Look for "product construction". – Hendrik Jan Sep 14 '14 at 21:19 • So you are suggesting that $\tau$ outputs some symbol in $\Delta$? But how does this account for the case in which 2 distinct transitions exiting 1 particular state are such that one has the label '$x/y$', while the other has the label '$x/z$', where $y \neq z$? – David Smith Sep 14 '14 at 21:27 • I may have misread what you wrote. Do you mean that "the transition and output functions give sets of states and [sets of] output symbols [, respectively]"? – David Smith Sep 14 '14 at 21:42 • Please don't post homework: public.asu.edu/~ccolbou/src/457hw1f14.pdf – Ryan Sep 15 '14 at 3:54 • 1. There seem to be some typos in your quesiton. Please proof-read it carefully. You define $M(L)=$ but there is nothing after the equality sign; I assume something got left out. 2. What research have you done? See, e.g., here: en.wikipedia.org/wiki/Finite_state_transducer. Your question should be answered in standard textbooks on automata theory. – D.W. Sep 15 '14 at 4:11 Instead, the idea is to construct an NFA which keeps track of both the Mealy machine $M$ and a machine $T$ accepting $L$. Denote by $w$ the (unknown) word in $L$ on which we run $M$. At each point in time we guess the next symbol in $w$ and transition in $T$, guess a corresponding transition in $M$, and "accept" the symbol output by the $M$-transition (if any). The accepting states of the NFA are those in which $T$ is at an accepting state.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8586500287055969, "perplexity": 436.49389737590855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703522150.18/warc/CC-MAIN-20210121004224-20210121034224-00697.warc.gz"}
https://ai.stackexchange.com/questions/7195/how-can-i-calculate-the-mean-best-fitness-measure-in-genetic-algorithms	# How can I calculate the “mean best fitness” measure in genetic algorithms? I've just started to learn genetic algorithms and I have found these measurements of runs that I don't understand: MBF: The mean best fitness measure (MBF) is the average of the best fitness values over all runs. AES: The average number of evaluation to solution. I have an initial random population. To evolve a population I do: 1. Tournament selection 2. One point crossover. 3. Random resetting. 4. Age based replacement with elitism (I replace the population with all offsprings generated). 5. If I have generated G generations (in other words, I have repeated these four points G times) or I have found the solution, the algorithm ends, otherwise, it comes back to point 1. Is the mean of the best fitness the mean fitness of all of each generations (G best fitness)? MBF = (BestFitness_0 + ... + BestFitness_G) / G I'm not English and I don't understand the meaning of "run" here. The typical way you'll see a GA measured is that an algorithm with a population size of $$N$$ is ran $$K$$ times from new random seeds each time. That gives you $$K$$ total runs of the algorithm, each of which, at the end, had a final population of $$N$$ individuals. If you take the best of those $$N$$ from each run, you get $$K$$ "best" solutions found. The average fitness value of those $$K$$ solutions is your MBF. For sudoku, you might have a fitness function that counted the number of rows, columns, or blocks that don't contain the correct digits of 1-9 and you minimize that function. You run your algorithm $$K$$ times from random seeds, and for each run, you record how many times you had to evaluate that fitness function before you found a $$0$$ (i.e., the puzzle was successfully solved). Average all $$K$$ of those counts, and that's your AES.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8300056457519531, "perplexity": 860.9865472217122}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991685.16/warc/CC-MAIN-20210512070028-20210512100028-00122.warc.gz"}
http://laneas.com/publication/distributed-energy-efficient-power-optimization-relay-aided-heterogeneous-networks	# Distributed Energy-Efficient Power Optimization for Relay-Aided Heterogeneous Networks Conference Paper ### Authors: I stupia; Luca Sanguinetti; G Bacci; L Vandendorpe ### Source: International Workshop on Wireless Networks (WiOpt - WCN), Hammamet, Tunisia, May (2014) ### Abstract: This paper presents an energy-efficient power allocation for relay-aided heterogeneous networks subject to coupling convex constraints, that make the problem at hand a generalized Nash equilibrium problem. The solution to the resource allocation problem is derived using a sequential penalty approach based on the advanced theory of quasi variational inequality, which allows the network to converge to its generalized Nash equilibrium in a distributed manner. The main feature of the proposed approach is its decomposability, which leads to a two-layer distributed algorithm with provable convergence. Full Text:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.970473051071167, "perplexity": 1441.841077269057}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864798.12/warc/CC-MAIN-20180522151159-20180522171159-00020.warc.gz"}
http://physics.stackexchange.com/questions/113643/do-particle-velocities-in-liquid-follow-the-maxwell-boltzmann-velocity-distribut/115005	# Do particle velocities in liquid follow the Maxwell-Boltzmann velocity distribution? The Maxwell-Boltzmann velocity distribution arises from non-reactive elastic collisions of particles and is usually discussed in the context of the kinetic theory (for gases). There are various offhand remarks, for example here (slide 5), that state without reference that particles observe a similar velocity distribution in liquid. Is that true? References? The main reason I'm curious is that it seems as though the mean free path would be extremely short in liquid vs. gas. I'm actually most curious about the nature of the collisions in liquid vs. gas, i.e., are collisions in liquid still (on average) elastic? EDIT: The linked post on particle velocity in liquids is definitely interesting and weighs in on this question, and I appreciate the distinction made between local position fluctuation vs. long range movement. Still, let me frame this question in a few different ways. 1. Gillespie proposed a stochastic framework for simulating chemical reactions which is formulated on the idea that non-reactive elastic collisions serve to 'uniformize' particle position so that the assumption of well-mixedness is always satisfied (see page 409 in the linked version). This is formulated from kinetic theory. A corollary to this is that a non-reactive collision between two molecules that are able to react does not induce local correlation, i.e., two particles able to react with each other that just collided, but didn't react are no more likely to react with each other in the next dt than any other particle pair in the volume. Gillespie's algorithm is commonly used in biology where biochemical species are modeled in the aqueous environment of cells. Is this valid, and if so why? 2. On a microscopic scale, suppose we are interested in two 'A' particles diffusing in one dimension in an aqueous environment. The two A particles collide but don't react. What is their behavior immediately after the collision? Is it a 'reflection' which conserves velocity and might correspond to a Neumann BC? In a gas that approach seems natural, but in liquid the short mean-free path makes me think that diffusive forces would rapidly dissipate any momentum from the A-A collision, which might imply the A particles collide and 'stop'. How should I be thinking about this? Just to bring it back to the original question, I think both (1) and (2) depend on the statistical velocity behavior of particles in liquid. - I'm curious about it too. This question arised recently in my work, and I haven't found a satisfatory answer yet. – user23873 May 20 at 21:46 I think your edits 1 and 2 would be better posted as a new question and possibly as two separate questions. – John Rennie May 21 at 15:59 What @JohnRennie said. – Kyle May 21 at 18:20 @vector07, I do not think that your question was well answered here or in the "duplicate" question suggested by the people below. I think Maxwell-Boltzmann distribution should be valid for molecules in liquid too, at least according to classical statistical physics, because the factor $e^{-\beta p^2/2m}$ in the Gibbs-Boltzmann probability density does not depend on potential energy and is the same whether the molecule is in gas, or a liquid. I do not know if there is a measurement supporting this theoretical result. – Ján Lalinský May 21 at 19:34 @JánLalinský I agree that this isn't a duplicate. Discussion continued here – vector07 May 27 at 14:48 In terms of the Hamiltonian formalism the MB distribution can be obtained for a set of particles without potential energy between them (it means free particles), under the assumption that in equilibrium they satisfy the MB statistics. It is worth noting that this is not an obvious fact, the MB statistics can be applied to any kind of object, in particular applied to an ideal gas gives the MB distribution for velocities. The free particle approximation is not valid for the liquid state, the mean free path in liquids is very short due to the intermolecular interactions and to derive the distribution of velocities is needed to take into account the potential term. - I think Maxwell-Boltzmann distribution should be valid for molecules in liquid too, at least according to classical statistical physics, because the factor $e ^{−\beta p^2/2m}$ in the Gibbs-Boltzmann probability density does not depend on potential energy and is the same whether the molecule is in gas, or a liquid. I do not know if there is a measurement supporting this theoretical result. are collisions in liquid still (on average) elastic? Elastic collision means that appreciable change in the kinetic and potential energy of two bodies happens to them only during short time interval and the energy long after that is the same as the energy long before that - the interaction of the two molecules is thought of as a scattering process. In liquids the interaction of the molecules may not be idealizable in this way, as the molecules are believed to be in incessant complicated motion constantly influencing each other (Brownian motion...) This does not seem to be a reason to abandon the Boltzmann statistics, however. - Generally speaking, it depends on the nature of the liquid. The assumptions behind the Maxwell-Boltzmann distribution are fairly simple: molecules can be approximated as point-masses and their only interactions are through collisions that exchange momentum and energy. So if you have liquids where this is true, then yes, the distribution will be correct. However, if you have liquids (or gases for that matter) that have large molecules (so that the point-mass assumption is invalid) or that have long-range interaction forces (like water for example), then the distribution will not be Maxwell-Boltzmann. - Another very important example of fluids with long-range interaction forces are plasmas. – Robin Ekman May 21 at 15:52 Classical particles must follow the Maxwell-Boltzmann velocity distribution, and this is a consequence of separability of momenta and position in the partition function, and that the Boltzmann factor weighing the probability of each state is $\propto \exp(-\beta\mathcal{H})$. This is not, however, to say that this is the distribution one would always obtain experimentally. I am not aware of an experimental procedure where one can directly sample velocities, but rather you observe displacements between times $0$ and $t$, say, and then you divide the size of the displacement by the time $t$ to get an effective velocity. Now, remember that the diffusion constant is defined as (in 3D) $$D = \lim_{t\to\infty}\frac{1}{6t}\langle (\mathbf{r}(t) - \mathbf{r}(0))^2\rangle$$ Note that here we assume that the displacements are taken an infinite time apart. Because the collisions cause random motion, and in stochastic Ito calculus the time is proportional to the square of displacements, this indeed evaluates as a finite constant in most cases. And in most cases the distribution of displacements (and therefore of effective velocities) at infinity is Gaussian. Now there are important exceptions to this rule. Fractional Brownian motion, for example, does not yield a regular diffusion constant, but undergoes something called anomalous diffusion (a hot topic in current physics research, anything dealing with anomalous diffusion, or diffusion seemingly anomalous often gets published in top venues). Anomalous diffusion looks like the following: $$D = \lim_{t\to\infty}\frac{1}{6t^\alpha}\langle (\mathbf{r}(t) - \mathbf{r}(0))^2\rangle$$ where the limit makes sense only for some $\alpha$. If $\alpha$ is larger than $1$, one calls the motion superdiffusive, and when it is smaller, subdiffusive. How is this related to velocity distributions? Well, to get an idea through the diffusion constant of the underlying velocity distribution, one wants the displacements to be a very short time apart. $$D = \lim_{t\to0}\frac{1}{6t^\alpha}\langle (\mathbf{r}(t) - \mathbf{r}(0))^2\rangle$$ Note the change of limit. Now what is $\alpha$ going to be? Obviously $2$, because there are no collisions in very short time scales and one is then dealing with ballistic motion, where displacements scale linearly with time. Shortly after (if the limit in $t$ were to be in the picosecond scale) one enters the subdiffusive regime caused by viscoelastic behaviour of the material, and finally, when the displacements are measured long enough apart, to the normal, diffusive regime (in most liquids, that is, but there are exceptions, of course, like fractional Brownian motion). Very few experimental methods can access the femtosecond, ballistic, regime, and it is only here that the displacement distribution should follow Maxwell-Boltzmann. For longer time scales, as you typically observe and are interested in, a Gaussian distribution might be a better approximation, but this does depend on the type of liquid. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.942609965801239, "perplexity": 347.6378107589637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009968.66/warc/CC-MAIN-20141125155649-00180-ip-10-235-23-156.ec2.internal.warc.gz"}
http://www.physicsforums.com/showthread.php?t=274902	# Conservation of angular momentum by jimgram Tags: flywheel, inertia, kinetic energy, momentum P: 92 I'm tasked with the design of a variable inertia flywheel. I've concentrated on the change in location of mass about an axis and varying 'r'. I'm trying to analyze what happens without any change in torque (I.E. the flywheel free spins). My HS physics tells me that the angular momentum must remain constant- Lo=Le. But also, kinetic energy must be conserved - Eko=Eke. As I run trials with re=ro/2, the results substantiate what I would expect, since L is a function of w and Ek is a function of w2 (I can calculate a new w to conserve one (e.g. L) but not both). I know that I can adjust r without adding torque to the axis of rotation and it seems that the work required should not enter into the conservation question (if I increase r, centrifugal force would make work a negative value). Where am I missing the conservation question? Can it be both angular momentum and kinetic energy and if not, which is conserved? And what happened to the other? I've searched this forum as well as other internet sites and have found references to this dichotomy, but each leaves the fundamental question unanswered. P: 191 I think your reasoning is Ok. It's just that Kinetic Energy is not conserved. As you indicated, if you increase r, that's negative work. It shows up as a reduction in Kinetic Energy. The angular velocity is less. If you can do this without applying a torque to the object, the Angular Momentum should remain the same. That's a challenging puzzle. P: 92 I've attached a sample of the basic calculations in a PDF form. I saw a form of this question in an earlier post regarding a physics class demonstration: a puck with a string attached is rotated on a table. The string is pulled from the axis such that the puck moves in toward the axis until the radius of rotation is 1/2 the original. Most of the responses suggested that the angualr velocity would double, however, in order to maintain a constant angular momentum, the velcocity must increase by a factor of 4. Essentially, attached spreadsheet indicates the same factor for an increase in kinetic energy, which is what you would expect. But why? Attached Files Mathcad - Change_R.pdf (16.0 KB, 26 views) P: 191 ## Conservation of angular momentum Quote by jimgram I've attached a sample of the basic calculations in a PDF form. I saw a form of this question in an earlier post regarding a physics class demonstration: a puck with a string attached is rotated on a table. The string is pulled from the axis such that the puck moves in toward the axis until the radius of rotation is 1/2 the original. Most of the responses suggested that the angular velocity would double, however, in order to maintain a constant angular momentum, the velcocity must increase by a factor of 4. Essentially, attached spreadsheet indicates the same factor for an increase in kinetic energy, which is what you would expect. But why? It appears to me that the other responses for the puck example are assuming Kinetic Energy is conserved. That would result in the conclusion that angular velocity that is only doubled. As you have identified, changing the location of the puck involves work, positive if r is reduced, negative if r is increased. Work being energy has to go someplace. It can go into heat or mechanical deformation. But in this case, it goes into rotational motion. That's why its not surprising to see the angular velocity in the puck example increases by a factor of 4, not 2. Total Energy is conserved. Kinetic Energy by itself is not. I am curious. Can you provide a link to the earlier post? P: 92 ML - Thanks for your reply. I've been off in the mountains with no internet- scary feeling. I suspect there's a more direct way to give you the link I referred to but I'm not that well versed in the forum features, so you might try this link: http://www.physicsforums.com/showthr...89#post1976989 Otherwise, search for "spinning puck". I've decided that the work involved in changing the radius not only must be included, it's also a major factor. I don't know yet where that might take me. For example, when you move a mass that is at radius=1m to radius=0.8m, did you move the mass 0.2m, or did you move it the distance that it actually moved (I.E. a spiral depending on the ratio of angular velocity to linear velocity)? I think probably not because that is factored in when the force required is due primary to centripetal force as opposed to the actual inertia of the mass being moved. But.....? P: 191 Welcome back from the mountains. With regard to the puck problem, I would say you have already solved it in your pdf link. Using the consept of conservation of momentum, reducing the radius by a half increases the angular velocity by a factor of 4. What else are you trying to calculate? The work done? The new Kenetic Energy? P: 92 The formula for kinetic energy is: K=1/2Iw^2. This yields a kinetic energy that is 4 times the the original. In the series of equations to get this result, the only gray area where additional work could be applied is in the simple assumption that r suddenly becomes r/2. But my problem is trying to understand how r to r/2 makes energy increase 4 times. I've attempted to use centripetal force to determine the work required to move the mass from r to r/2 (F+dt), but this work turns out substantially exceeding the kinetic energy gain. P: 191 The difference between the new and old Kinetic Energies must be exactly the work done. It would be hard to calculate the work directly. Remember, as you pull the puck in, the rotation increases but the centrifugal force goes down. So FORCE times the increment of DISTANCE gets smaller as the radius gets smaller. I could try to spend some time coming up with a formula if you really want corroboration. You really want to nail this down don't you. P: 92 My calculus is more than a little rusty. I calculated the centripetal force for r and for r/2 using the respective velocities and then used an average (Fe+Fi/2)+Fi times distance. The answer yields work that is about 1.6 times higher than the delta kinetic energy. I'm sure the average is not going to get the job done. I really appreciate your feedback. I can definitely use help. As it turns out, I actually need to produce this apparatus and predicting forces and work involved is pretty important. Thanks again P: 92 One thing I failed to mention: since velocity increases by 4 times (assuming r to r/2), and force is proportional to v^2, my numbers show centrifugal force increasing rather substantially. P: 191 Quote by jimgram One thing I failed to mention: since velocity increases by 4 times (assuming r to r/2), and force is proportional to v^2, my numbers show centrifugal force increasing rather substantially. You are right. I jumped too fast on that one. Here is a link to a web page that does the Work calculation using Integration. http://mysite.verizon.net/mikelizzi/...rMomentum.html P: 92 I believe you have it. Thank you so much. I shutter to think how long it woulkd have taken me to get there (or if). If I reduce this to pratice may I email you with a status? P: 191 I'm trying to keep my emails to a minimum. But I think you could post an update anytime on this thread. I would be interested in reading about your progress. P: 92 Mike - the result from your integration looks good but I have been trying to reconcile one aspect that has me baffled. You may have explained it in your statement regarding the refernce frame, but it still confuses me: Your equation for centrifugal force is: F=mw^2r but it is normally F = m*w^2/r It seems that force would in any case be inversely proportional to r. Can this be right? P: 191 You are thinking radial acceleration for a rotating object is "v^2/r". Yes but, when you use angular velocity its "omega squared times r". P: 92 Thanks - I got it. I see why Related Discussions General Engineering 0 Introductory Physics Homework 4 Introductory Physics Homework 4 Introductory Physics Homework 11 Introductory Physics Homework 9	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8775188326835632, "perplexity": 459.2420981527387}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011163856/warc/CC-MAIN-20140305091923-00036-ip-10-183-142-35.ec2.internal.warc.gz"}
https://voer.edu.vn/c/magnetic-force-on-a-current-carrying-conductor/0e60bfc6/b5a7048a	Giáo trình # College Physics Science and Technology ## Magnetic Force on a Current-Carrying Conductor Tác giả: OpenStaxCollege Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself. We can derive an expression for the magnetic force on a current by taking a sum of the magnetic forces on individual charges. (The forces add because they are in the same direction.) The force on an individual charge moving at the drift velocity ${v}_{d}$ is given by $F={\text{qv}}_{d}B\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$. Taking $B$ to be uniform over a length of wire $l$ and zero elsewhere, the total magnetic force on the wire is then $F=\left({\text{qv}}_{d}B\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \right)\left(N\right)$, where $N$ is the number of charge carriers in the section of wire of length $l$. Now, $N=\text{nV}$, where $n$ is the number of charge carriers per unit volume and $V$ is the volume of wire in the field. Noting that $V=\text{Al}$, where $A$ is the cross-sectional area of the wire, then the force on the wire is $F=\left({\text{qv}}_{d}B\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \right)\left(\text{nAl}\right)$. Gathering terms, $F=\left({\text{nqAv}}_{d}\right)\text{lB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta .$ Because ${\text{nqAv}}_{d}=I$ (see Current), $F=\text{IlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ is the equation for magnetic force on a length $l$ of wire carrying a current $I$ in a uniform magnetic field $B$, as shown in [link]. If we divide both sides of this expression by $l$, we find that the magnetic force per unit length of wire in a uniform field is $\frac{F}{l}=\text{IB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$. The direction of this force is given by RHR-1, with the thumb in the direction of the current $I$. Then, with the fingers in the direction of $B$, a perpendicular to the palm points in the direction of $F$, as in [link]. Calculating Magnetic Force on a Current-Carrying Wire: A Strong Magnetic Field Calculate the force on the wire shown in [link], given $B=1\text{.}\text{50 T}$, $l=5\text{.}\text{00 cm}$, and $I=\text{20}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{A}$. Strategy The force can be found with the given information by using $F=\text{IlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ and noting that the angle $\theta$ between $I$ and $B$ is $\text{90º}$, so that $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =1$. Solution Entering the given values into $F=\text{IlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ yields $F=\text{IlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\left(\text{20}\text{.0 A}\right)\left(0\text{.}\text{0500 m}\right)\left(1\text{.}\text{50 T}\right)\left(1\right)\text{.}$ The units for tesla are $\text{1 T}=\frac{N}{A\cdot m}$; thus, $F=1\text{.}\text{50 N.}$ Discussion This large magnetic field creates a significant force on a small length of wire. Magnetic force on current-carrying conductors is used to convert electric energy to work. (Motors are a prime example—they employ loops of wire and are considered in the next section.) Magnetohydrodynamics (MHD) is the technical name given to a clever application where magnetic force pumps fluids without moving mechanical parts. (See [link].) A strong magnetic field is applied across a tube and a current is passed through the fluid at right angles to the field, resulting in a force on the fluid parallel to the tube axis as shown. The absence of moving parts makes this attractive for moving a hot, chemically active substance, such as the liquid sodium employed in some nuclear reactors. Experimental artificial hearts are testing with this technique for pumping blood, perhaps circumventing the adverse effects of mechanical pumps. (Cell membranes, however, are affected by the large fields needed in MHD, delaying its practical application in humans.) MHD propulsion for nuclear submarines has been proposed, because it could be considerably quieter than conventional propeller drives. The deterrent value of nuclear submarines is based on their ability to hide and survive a first or second nuclear strike. As we slowly disassemble our nuclear weapons arsenals, the submarine branch will be the last to be decommissioned because of this ability (See [link].) Existing MHD drives are heavy and inefficient—much development work is needed. # Section Summary • The magnetic force on current-carrying conductors is given by $F=\text{IlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\mathrm{\theta ,}$ where $I$ is the current, $l$ is the length of a straight conductor in a uniform magnetic field $B$, and $\theta$ is the angle between $I$ and $B$. The force follows RHR-1 with the thumb in the direction of $I$. # Conceptual Questions Draw a sketch of the situation in [link] showing the direction of electrons carrying the current, and use RHR-1 to verify the direction of the force on the wire. Verify that the direction of the force in an MHD drive, such as that in [link], does not depend on the sign of the charges carrying the current across the fluid. Why would a magnetohydrodynamic drive work better in ocean water than in fresh water? Also, why would superconducting magnets be desirable? Which is more likely to interfere with compass readings, AC current in your refrigerator or DC current when you start your car? Explain. # Problems & Exercises What is the direction of the magnetic force on the current in each of the six cases in [link]? (a) west (left) (b) into page (c) north (up) (d) no force (e) east (right) (f) south (down) What is the direction of a current that experiences the magnetic force shown in each of the three cases in [link], assuming the current runs perpendicular to $B$? What is the direction of the magnetic field that produces the magnetic force shown on the currents in each of the three cases in [link], assuming $\mathbf{\text{B}}$ is perpendicular to $\mathbf{\text{I}}$? (a) into page (b) west (left) (c) out of page (a) What is the force per meter on a lightning bolt at the equator that carries 20,000 A perpendicular to the Earth’s $3\text{.}\text{00}×{\text{10}}^{-5}\text{-T}$ field? (b) What is the direction of the force if the current is straight up and the Earth’s field direction is due north, parallel to the ground? (a) A DC power line for a light-rail system carries 1000 A at an angle of $\text{30.0º}$ to the Earth’s $\text{5.00}×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}\text{-T}$ field. What is the force on a 100-m section of this line? (b) Discuss practical concerns this presents, if any. (a) 2.50 N (b) This is about half a pound of force per 100 m of wire, which is much less than the weight of the wire itself. Therefore, it does not cause any special concerns. What force is exerted on the water in an MHD drive utilizing a 25.0-cm-diameter tube, if 100-A current is passed across the tube that is perpendicular to a 2.00-T magnetic field? (The relatively small size of this force indicates the need for very large currents and magnetic fields to make practical MHD drives.) A wire carrying a 30.0-A current passes between the poles of a strong magnet that is perpendicular to its field and experiences a 2.16-N force on the 4.00 cm of wire in the field. What is the average field strength? 1.80 T (a) A 0.750-m-long section of cable carrying current to a car starter motor makes an angle of $\text{60º}$ with the Earth’s $5\text{.}\text{50}×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}\text{T}$ field. What is the current when the wire experiences a force of $\text{7.00}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}N$? (b) If you run the wire between the poles of a strong horseshoe magnet, subjecting 5.00 cm of it to a 1.75-T field, what force is exerted on this segment of wire? (a) What is the angle between a wire carrying an 8.00-A current and the 1.20-T field it is in if 50.0 cm of the wire experiences a magnetic force of 2.40 N? (b) What is the force on the wire if it is rotated to make an angle of $\text{90º}$ with the field? (a) $\text{30º}$ (b) 4.80 N The force on the rectangular loop of wire in the magnetic field in [link] can be used to measure field strength. The field is uniform, and the plane of the loop is perpendicular to the field. (a) What is the direction of the magnetic force on the loop? Justify the claim that the forces on the sides of the loop are equal and opposite, independent of how much of the loop is in the field and do not affect the net force on the loop. (b) If a current of 5.00 A is used, what is the force per tesla on the 20.0-cm-wide loop?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 57, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8607332110404968, "perplexity": 368.8941265230697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315258.34/warc/CC-MAIN-20190820070415-20190820092415-00535.warc.gz"}
https://or.stackexchange.com/questions/1316/representing-an-indicator-function-binary-variables-and-indicator-constraints/1317	# Representing an indicator function: binary variables and “indicator constraints” I want to represent the indicator function: $$\mathbb{1}_{(y=j)}$$ where $$y$$ is a non negative, integer variable. My attempt is as follows: define a binary variable: $$z_j =\begin{cases} 1 \qquad\text{if y=j} \\ 0 \qquad\text{otherwise} \end{cases}$$ the model of the indicator function would be: $$\sum_{j=0}^{n} z_j = 1$$ $$\sum_{j=0}^{n} j \cdot z_j = y$$ where $$n$$ is an upper bound for $$y$$. Actually, this can be conveniently modeled in OPL Cplex (for example) using indicator constraints such as follows: forall(j in 0..n){ (y == j) == (z[j] == 1); } QUESTIONS: 1. is my binary-variables-based formulation attempt correct? Do you know better (performance wise) formulations? 2. in a hypothetical academic journal paper, is the second formulation (based on indicator constraints) acceptable? If yes, how would you formally express it in the paper in a way that is implementation-independent (that is, in a way that does not depend upon how a specific solver models such a constraint)? Or it is better to provide the more general, binary-variables-based formulation? Thanks a lot. • This is correct, assuming $y \in \{0,\dots,n\}$. More generally, if $y$ takes (not necessarily integer) values in a finite set $V$, impose $\sum\limits_{j \in V} z_j=1$ and $\sum\limits_{j \in V} j z_j=y$. – RobPratt Aug 17 '19 at 16:39 • Something else to consider is to do a binary expansion of $y$, i.e., add the constraint $\sum\limits_{j=0}^m 2^j z_j=y$, where $2^m$ is an upper bound on $y$. Depending on the application you have in mind you might not be able to use this representation, but if you can I would imagine it's more efficient. – Ryan Cory-Wright Aug 17 '19 at 17:19 • Note that you will need to linearize a product of binary variables to recover the indicator function for a fixed $j$. This can be achieved via $y \le z_i, \forall i, y\ge\sum\limits_i z_i - (n-1)$. – Ryan Cory-Wright Aug 17 '19 at 17:27 • Related: or.stackexchange.com/q/76/38 – LarrySnyder610 Aug 18 '19 at 0:59 • @RyanCory-Wright considering the binary expansion, however, I should drop the $\sum\limits_{j=0}^{n} z_j = 1$ constraint, right? If so, I don't get the equivalence between the two formulations. – Libra Aug 21 '19 at 11:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8666791915893555, "perplexity": 520.0969933794348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141674082.61/warc/CC-MAIN-20201201104718-20201201134718-00359.warc.gz"}
http://www.statisticssolutions.com/logistic-regression-assumptions/	# Binary Logistic Regressions Binary logistic regressions, by design, overcome many of the restrictive assumptions of linear regressions. For example, linearity, normality and equal variances are not assumed, nor is it assumed that the error term variance is normally distributed. The major assumptions are: 1. That the outcome must be discrete, otherwise explained as, the dependent variable should be dichotomous in nature (e.g., presence vs. absent); 2. There should be no outliers in the data, which can be assessed by converting the continuous predictors to standardized, or z scores, and remove values below -3.29 or greater than 3.29. 3. There should be no high intercorrelations (multicollinearity) among the predictors. This can be assessed by a correlation matrix among the predictors. Tabachnick and Fidell (2012) suggest that as long correlation coefficients among independent variables are less than 0.90 the assumption is met. Also, there should be a linear relationship between the odds ratio, orEXP(B),and each independent variable. Linearity with an ordinal or interval independent variable and the odds ratio can be checked by creating a new variable that divides the existing independent variable into categories of equal intervals and running the same regression on these newly categorized versions as categorical variables. Linearity is demonstrated if the beta coefficients increase or decrease in linear steps (Garson, 2009). A larger sample is recommended in fitting with the maximum likelihood method; using discrete variables requires that there are enough responses in each category. References: Garson, G. D. (2009). Logistic Regression. Retrieved on August 12, 2009 from http://faculty.chass.ncsu.edu/garson/PA765/logistic.htm Tabachnick, B. G. & Fidell, L. S. (2012). Using multivariate statistics (6th ed.). Boston, MA: Pearson. Related Pages:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8181524276733398, "perplexity": 1807.1618169894502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864657.58/warc/CC-MAIN-20180522092655-20180522112655-00563.warc.gz"}
http://math.stackexchange.com/questions/207157/does-the-following-function-have-an-elementary-indefinite-integral	# Does the following function have an elementary indefinite integral? If $m$ and $n$ are integers greater than one, does the function $f(t)=[\frac{m-n}{m+1}-t^{n+1}]^{-\frac{1}{m+1}}$ have an elementary indefinite integral? I have tried to find the integral by trigonometric substitution but I could not find the answer, (supposing that the integral exists). - Already for certain simple cases like $m=1$ and $n=2$ or $3$, definitely no elementary antiderivative. – André Nicolas Oct 4 '12 at 13:59	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9323766231536865, "perplexity": 96.55093186081402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246644526.54/warc/CC-MAIN-20150417045724-00071-ip-10-235-10-82.ec2.internal.warc.gz"}
http://mathhelpforum.com/business-math/149308-marginal-profit-question.html	1. Marginal Profit Question Im having a lot of trouble trying to figure out how to solve this problem, any help is greatly appreciated! Thanks. The profit (measured in dollars) of a small company has been determined to be P(x)= 20000 (x / 100 + x^2), where x is the production level (in units). a) Compute the marginal profit b) Find the production level xmax that leads to the maximal profit within the range 0 ≤ x ≤ 30. c) How many dollars of profit does the company make at that production level? d) Does your answer change if x has to be within the range 0 ≤ x ≤ 10 instead? If yes, how? 2. a) the marginal profit is $\frac{dP}{dx}$ b) maximal profit occurs either at the point where marginal profit = 0, or x=0, or x=30. Find the profit at each of these points and see which one is higher hintThere is no point in the range 0 < x < 30 where marginal profit =0 in this case. c) this is equal to P(x), where x is the production level you found in part b d) if your existing answer is in the range 0 < x< 10, then it will not change. However if your existing answer is not in the range, you need to find a new one, which will be either: x=0 x=10 or somewhere in between where the marginal profit is zero (you can discount this possibility as if there was a point in the range with marginal profit =0, you would have found it in part b) 3. Thanks so much for your help! A.)20000 [(100-x^2)/(100+x^2)^2] B.)x=10 c)p=1000 d.) No, because in the previous problem, 10 had the highest amount of profit. 4. hmm, if i understood your function correctly i dont think you have the marginal profit right You wrote P(x)= 20000 (x / 100 + x^2) which is $P(x) = 20000 \left(0.01x + x^2 \right) = 200x + 20000x^2$ Did you mean $P(x) = 20000 \left(\frac{x}{100 + x^2} \right)$? I'll assume you meant $P(x) = 20000 \left(0.01x + x^2 \right) = 200x + 20000x^2$ a) $P(x) = 20000 \left(0.01x + x^2 \right) = 200x + 20000x^2$ $MP = \frac{dP}{dx} = 200 + 40000x$ b) MP = 0 has no solutions where x > 0 So max pforit occurs at either x=0 or x=30 p(0) = 0 p(30) = 1200200 so x=30 maximises profit c) p(30) = 1200200 d) Yes, the new maximum is at x=10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9151014089584351, "perplexity": 873.2286460854907}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123560.51/warc/CC-MAIN-20170423031203-00599-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/newtons-laws-pulleys.260730/	# Homework Help: Newton's laws- pulleys 1. Sep 30, 2008 ### crhscoog 1. The problem statement, all variables and given/known data http://img213.imageshack.us/img213/6969/physicsproblemhq3.png [Broken] A rope of negligible mass passes over a pulley of negligible mass attached to the ceiling as shown above. One end of the rope is held by Student A of mass 70 kg, who is at rest on the floor. The opposite end of the rope is held by Student B of mass 60 kg, who is suspended at rest above the floor. 1. Calculate the magnitude of the force exerted by the floor on Student A. 2. Student B now climbs up the rope at a constant acceleration of .25 m/s^2 with respect to the floor. Calculate the tension in the rope while Student B is accelerating. 3. As Student B is accelerating, is Student A pulled upward off the floor? Justify your answer. 4. With what minimum acceleration must Student B climb up the rope to lift Student A upward off the floor? 3. The attempt at a solution 1. Because it is at rest: T + N = Mg, or N = Mg - T = (70 - 60)g ~ 98 N I am currently doing 2-4 right now... I'll post what I get after I finish them. Last edited by a moderator: May 3, 2017	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8181967735290527, "perplexity": 844.5296627491527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267157203.39/warc/CC-MAIN-20180921131727-20180921152127-00228.warc.gz"}
http://mathhelpforum.com/advanced-applied-math/36020-simple-curvilinear-motion.html	## simple curvilinear motion heres the diagram for the problem.. www.i14.photobucket.com/albums/a322/guitaristx/asdf.jpg Im just trying to find the horizontal distance the water hits the building at....the answer should be in terms of horizontal distance from point B. Heres my work....any suggestions would be great....	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8808683753013611, "perplexity": 1844.8509331444627}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768397.103/warc/CC-MAIN-20141217075248-00006-ip-10-231-17-201.ec2.internal.warc.gz"}
https://flyingcoloursmaths.co.uk/student-asks-upper-bounds/	When you’ve got a value to the nearest whole number, why is the upper bound something $.5$ rather than $.4$? Doesn’t $.5$ round up? So I don’t have to keep writing something$.5$, let’s pick a number, and say we’ve got 12 to the nearest whole number. $12.5$ does indeed round up (at least in the GCSE maths convention that you break a tie by going up; in some sciences, the convention is that you round to the nearest even number, so you don’t introduce an upward bias in your data), but $12.4$ certainly isn’t the upper bound - for example, $12.49$ would still round down. So would $12.499$. And $12.4999$. And, for that matter, $12.4999999999$. In fact, you can carry this on forever and say the upper bound has to be $12.4\dot9$ - which is technically a correct answer. However, we already have a name for $12.4\dot9$ - it’s the same as $12.5$. (Aside: don’t believe me? If $x = 12.4\dot 9$, then $10x = 124.\dot9$. Take them away and you get $9x = 112.5$. Divide by 9… $x = 12.5$.) You should get the mark if you write $12.4\dot 9$, but why risk it? Saying something is 12 to the nearest whole number is the same as saying $11.5 \le x \lt 12.5$ - the upper bound (the supremum, if you want the technical term) is 12.5.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9445844888687134, "perplexity": 218.61681821485575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178359497.20/warc/CC-MAIN-20210227204637-20210227234637-00368.warc.gz"}
http://mathhelpforum.com/advanced-algebra/141557-prove-b-invariant-subspace.html	# Thread: Prove This Is A B-Invariant Subspace 1. ## Prove This Is A B-Invariant Subspace Let V be a vector space over a field K. Let $A, B:V \rightarrow V$ be two linear maps. Given any polynomial $f \in K[t]$ let $V_f(A) = \ker f(A)$. Prove that if AB=BA, then $V_f(A)$ is a B-invariant subspace of V. 2. Originally Posted by mathematicalbagpiper Let V be a vector space over a field K. Let $A, B:V \rightarrow V$ be two linear maps. Given any polynomial $f \in K[t]$ let $V_f(A) = \ker f(A)$. Prove that if AB=BA, then $V_f(A)$ is a B-invariant subspace of V. $\forall x\in \ker f(A)\,,\,f(A)x=0\Longrightarrow$ $0=B(0)=Bf(A)x=f(A)Bx\Longrightarrow Bx\in\ker f(A)=V_f(A)$ and we're done. The fact that $Bf(A)=f(A)B$ follows easily from $AB=BA$ and a little induction on $\deg f$. Tonio	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9971879720687866, "perplexity": 408.74861943922525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891807825.38/warc/CC-MAIN-20180217204928-20180217224928-00227.warc.gz"}
https://encyclopediaofmath.org/index.php?title=Darboux_theorem&oldid=30975	# Darboux theorem (diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff) Darboux theorem may may refer to one of the following assertions: • Darboux theorem on local canonical coordinates for symplectic structure; • Darboux theorem on intermediate values of the derivative of a function of one variable. For Darboux theorem on integrability of differential equations, see Darboux integral. ## Darboux theorems for symplectic structure 2010 Mathematics Subject Classification: Primary: 37Jxx,53Dxx [MSN][ZBL] Recall that a symplectic structure on an even-dimensional manifold $M^{2n}$ is a closed nondegenerate $C^\infty$-smooth differential 2-form $\omega$: $$\omega\in\varLambda^2(M),\qquad \rd \omega=0,\qquad \forall v\in T_p M\quad \exists w\in T_p M:\ \omega_p(v,w)\ne0.$$ The matrix $S(z)$ of a symplectic structure, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})=-S_{ji}(z)$ in any local coordinate system $(z_1,\dots,z_{2n})$ is antisymmetric and nondegenerate[1]: $\omega=\frac12\sum_{i,j=1}^{2n} S_{ij}(z)\,\rd z_i\land \rd z_j$. The standard symplectic structure on $\R^{2n}$ in the standard canonical coordinates $(x_1,\dots,x_n,p_1,\dots,p_n)$ is given by the form $$\omega=\sum_{i=1}^n \rd x_i\land \rd p_i.\tag 1$$ ### Local equivalence Theorem (Darboux theorem[2], sometimes also referred to as the Darboux-Weinstein theorem[3]). Any symplectic structure locally is $C^\infty$-equivalent to the standard to the standard syplectic structure (1): for any point $a\in M$ there exists a neighborhood $M\supseteq U\owns a$ and "canonical" coordinate functions $(x,p):(U,a)\to (\R^{2n},0)$, such that in these coordinates $\omega$ takes the form $\sum \rd x_i\land\rd p_i$. In particular, any two symplectic structures $\omega_1,\omega_2$ on $M$ are locally equivalent near each point: there exists the germ of a diffeomorphism $h:(M,a)\to(M,a)$ such that $h^\omega_1=\omega_2$. ### Relative versions Together with the "absolute" version, one has a "relative" version of the Darboux theorem[2][4]: if $M$ is a smooth manifold with two symplectic structures $\omega_1,\omega_2$, and $N$ is a submanifold on which the two 2-forms coincide[5], then near each point $a\in N\subseteq M$ one has a diffeomorphism $h:(M,a)\to(M,a)$ transforming $\omega_1$ to $\omega_2$ and identical on $N$: $$\omega_1=\omega_2\Big\|_{TN}\ \implies\ \exists h\in\operatorname{Diff}(M,a):\quad h^\omega_1=\omega_2,\quad h\|_N\equiv\operatorname{id}.$$ The assertion of the Darboux theorem on local normalization of antisymmetric 2-forms should be compared with a similar question about symmetric nondegenerate forms, which (if positive) define a Riemannian metric on $M$. It is well known that, although at a given point $a$ the Riemannian metric can be brought to the canonical form $\left<v,v\right>=\sum_{i=1}^n v_i^2$, such transformation is in general impossible in any open neighborhood of $a$: the obstruction, among other things, is represented by the curvature of the metric (which is zero for the "constant" standard Euclidean metric). In the same way the relative Darboux theorem means that submaniolds of the symplectic manifold have no "intrinsic" geometry: any two submanifolds $N,N'$ with equivalent (eventually, quite degenerate) restrictions of $\omega$ on $TN$, resp., $TN'$, can be transformed to each other by a diffeomorphism preserving the symplectic structure. ### Notes and references 1. Another way to formulate the nondegeneracy is to require that the highest wedge power $\omega\land\cdots\land\omega$ ($n$ times) is a nonvanishing volume form. 2. Arnold V. I., Givental A. B., Symplectic Geometry, Dynamical systems, IV, 1–138, Encyclopaedia Math. Sci., 4, Springer, Berlin, 2001. MR1866631. Chap. 2, Sect. 1 3. Guillemin V., Sternberg S., Geometric asymptotics, Mathematical Surveys, No. 14. American Mathematical Society, Providence, R.I., 1977. xviii+474 pp. MR0516965, Chap. IV, Sect. 1. 4. McDuff, D., Salamon, D., Introduction to symplectic topology (Second edition). Oxford Mathematical Monographs. Oxford University Press, New York, 1998. x+486 pp. MR1698616, Sect. 3.2. 5. This means that the 2-forms $\omega_i$ take the same value on any pair of vectors tangent to $N$. This condition is weaker than coincidence of the forms $\omega_i$ at all points of $N$. ## Darboux theorem for intermediate values of differentiable functions 2010 Mathematics Subject Classification: Primary: 26A06 [MSN][ZBL] If $f:[a,b]\to\R$ is a function which is differentiable at all points of the segment $[a,b]\subseteq\R$ (the right and left derivatives are assumed at the endpoints $a,b$ respectively), then its derivative assumes all intermediate values[1] (i.e., the range of the derivative $f'=\frac{\rd f}{\rd x}:[a,b]\to\R$ is a connected set). For functions $f\in C^1[a,b]$ whose derivative is continuous, this is a simple consequence of the intermediate value theorem for the derivative. For functions whose derivative exists at all points but is discontinuous, e.g., $f(x)=x^2\sin(1/x)$, $0\ne x\in[-1,1]$, $f(0)=0$, the assertion follows from the Fermat principle (the derivative of a differential function at an extremal point vanishes), applied to a suitable combination $f(x)-\alpha x$. See also the Darboux property. ### References 1. Darboux’s theorem (2012). In Encyclopædia Britannica. Retrieved from [the EB site]. How to Cite This Entry: Darboux theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_theorem&oldid=30975 This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9743571877479553, "perplexity": 493.0749928699155}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141743438.76/warc/CC-MAIN-20201204193220-20201204223220-00409.warc.gz"}
http://www.sfb45.de/publications-1/skeleta-of-affine-hypersurfaces	##### Personal tools You are here: Home Skeleta of Affine Hypersurfaces # Skeleta of Affine Hypersurfaces Helge Ruddat, Nicolò Sibilla, David Treumann, Eric Zaslow Number 18 Helge Ruddat 2012 A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8415643572807312, "perplexity": 1596.6522033965502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247514804.76/warc/CC-MAIN-20190222074154-20190222100154-00452.warc.gz"}
http://mathhelpforum.com/differential-geometry/131301-using-composition-continuity.html	# Thread: Using Composition With Continuity 1. ## Using Composition With Continuity Let $\displaystyle g$ be defined on $\displaystyle \mathbb{R}$ by $\displaystyle g(1) := 0$, and $\displaystyle g(x) := 2$ if $\displaystyle x \neq 1$, and let $\displaystyle f(x) := x +1$ for all $\displaystyle x \in \mathbb{R}$. How would you show that $\displaystyle \lim_{x\to0}g \circ f \neq (g \circ f)(0)$ ? 2. Originally Posted by CrazyCat87 Let $\displaystyle g$ be defined on $\displaystyle \mathbb{R}$ by $\displaystyle g(1) := 0$, and $\displaystyle g(x) := 2$ if $\displaystyle x \neq 1$, and let $\displaystyle f(x) := x +1$ for all $\displaystyle x \in \mathbb{R}$. How would you show that $\displaystyle \lim_{x\to0}g \circ f \neq (g \circ f)(0)$ ? $\displaystyle g(f(0))=g(0+1)=g(1)=0$ but $\displaystyle g(f(x))=g(x+1)=2,\text{ }x\ne 0$... 3. Originally Posted by Drexel28 $\displaystyle g(f(0))=g(0+1)=g(1)=0$ but $\displaystyle g(f(x))=g(x+1)=2,\text{ }x\ne 0$... So to show they're not equal, should I prove that the limit of $\displaystyle \lim_{x\to0}g \circ f =2$ ? 4. Originally Posted by CrazyCat87 So to show they're not equal, should I prove that the limit of $\displaystyle \lim_{x\to0}g \circ f =2$ ? Is there a need to prove it? 5. Originally Posted by Drexel28 Is there a need to prove it? Yea, I'm trying to prove that $\displaystyle \lim_{x \to 0}g \circ f = 2$ so that I can show that the two are different, and I'm a bit stuck... so $\displaystyle \lim_{x \to 0}g \circ f = \lim_{x \to 0}g(x+1)$ I wanna show $\displaystyle \|g(x+1)-0\|=\|g(x+1)\|<\epsilon$ ... 6. Originally Posted by CrazyCat87 Yea, I'm trying to prove that $\displaystyle \lim_{x \to 0}g \circ f = 2$ so that I can show that the two are different, and I'm a bit stuck... so $\displaystyle \lim_{x \to 0}g \circ f = \lim_{x \to 0}g(x+1)$ I wanna show $\displaystyle \|g(x+1)-0\|=\|g(x+1)\|<\epsilon$ ...	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9822912216186523, "perplexity": 203.93568867677513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863939.76/warc/CC-MAIN-20180620221657-20180621001657-00149.warc.gz"}
https://nbviewer.ipython.org/github/BMClab/BMC/blob/master/notebooks/ReferenceFrame.ipynb	# Frame of reference¶ Marcos Duarte Laboratory of Biomechanics and Motor Control (http://demotu.org/) Federal University of ABC, Brazil Motion (a change of position in space with respect to time) is not an absolute concept; a reference is needed to describe the motion of the object in relation to this reference. Likewise, the state of such reference cannot be absolute in space and so motion is relative. A frame of reference is the place with respect to we choose to describe the motion of an object. In this reference frame, we define a coordinate system (a set of axes) within which we measure the motion of an object (but frame of reference and coordinate system are often used interchangeably). Often, the choice of reference frame and coordinate system is made by convenience. However, there is an important distinction between reference frames when we deal with the dynamics of motion, where we are interested to understand the forces related to the motion of the object. In dynamics, we refer to inertial frame of reference (a.k.a., Galilean reference frame) when the Newton's laws of motion in their simple form are valid in this frame and to non-inertial frame of reference when the Newton's laws in their simple form are not valid (in such reference frame, fictitious accelerations/forces appear). An inertial reference frame is at rest or moves at constant speed (because there is no absolute rest!), whereas a non-inertial reference frame is under acceleration (with respect to an inertial reference frame). The concept of frame of reference has changed drastically since Aristotle, Galileo, Newton, and Einstein. To read more about that and its philosophical implications, see Space and Time: Inertial Frames. ## Frame of reference for human motion analysis¶ In anatomy, we use a simplified reference frame composed by perpendicular planes to provide a standard reference for qualitatively describing the structures and movements of the human body, as shown in the next figure. ## Cartesian coordinate system¶ As we perceive the surrounding space as three-dimensional, a convenient coordinate system is the Cartesian coordinate system in the Euclidean space with three orthogonal axes as shown below. The axes directions are commonly defined by the right-hand rule and attributed the letters X, Y, Z. The orthogonality of the Cartesian coordinate system is convenient for its use in classical mechanics, most of the times the structure of space is assumed having the Euclidean geometry and as consequence, the motion in different directions are independent of each other. ### Standardizations in movement analysis¶ The concept of reference frame in Biomechanics and motor control is very important and central to the understanding of human motion. For example, do we see, plan and control the movement of our hand with respect to reference frames within our body or in the environment we move? Or a combination of both? The figure below, although derived for a robotic system, illustrates well the concept that we might have to deal with multiple coordinate systems. For three-dimensional motion analysis in Biomechanics, we may use several different references frames for convenience and refer to them as global, laboratory, local, anatomical, or technical reference frames or coordinate systems (we will study this later). There has been proposed different standardizations on how to define frame of references for the main segments and joints of the human body. For instance, the International Society of Biomechanics has a page listing standardization proposals by its standardization committee and subcommittees: In [2]: from IPython.display import IFrame IFrame('https://isbweb.org/activities/standards', width='100%', height=400) Out[2]: Another initiative for the standardization of references frames is from the Virtual Animation of the Kinematics of the Human for Industrial, Educational and Research Purposes (VAKHUM) project. ## Determination of a coordinate system¶ In Biomechanics, we may use different coordinate systems for convenience and refer to them as global, laboratory, local, anatomical, or technical reference frames or coordinate systems. For example, in a standard gait analysis, we define a global or laboratory coordinate system and a different coordinate system for each segment of the body to be able to describe the motion of a segment in relation to anatomical axes of another segment. To define this anatomical coordinate system, we need to place markers on anatomical landmarks on each segment. We also may use other markers (technical markers) on the segment to improve the motion analysis and then we will also have to define a technical coordinate system for each segment. As we perceive the surrounding space as three-dimensional, a convenient coordinate system to use is the Cartesian coordinate system with three orthogonal axes in the Euclidean space. From linear algebra, a set of unit linearly independent vectors (orthogonal in the Euclidean space and each with norm (length) equals to one) that can represent any vector via linear combination is called a basis (or orthonormal basis). The figure below shows a point and its position vector in the Cartesian coordinate system and the corresponding versors (unit vectors) of the basis for this coordinate system. See the notebook Scalar and vector for a description on vectors. One can see that the versors of the basis shown in the figure above have the following coordinates in the Cartesian coordinate system: $$\hat{\mathbf{i}} = \begin{bmatrix}1\\0\\0 \end{bmatrix}, \quad \hat{\mathbf{j}} = \begin{bmatrix}0\\1\\0 \end{bmatrix}, \quad \hat{\mathbf{k}} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$ Using the notation described in the figure above, the position vector $\overrightarrow{\mathbf{r}}$ (or the point $\overrightarrow{\mathbf{P}}$) can be expressed as: $$\overrightarrow{\mathbf{r}} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$$ ### Definition of a basis¶ The mathematical problem of determination of a coordinate system is to find a basis and an origin for it (a basis is only a set of vectors, with no origin). There are different methods to calculate a basis given a set of points (coordinates), for example, one can use the scalar product or the cross product for this problem. A classical procedure in mathematics, employing the scalar product, is known as the Gram–Schmidt process. See the notebook Scalar and Vector for a demonstration of the Gram–Schmidt process and how to implement it in Python. Let's now define a basis using a common method in motion analysis (employing the cross product): Given the coordinates of three noncollinear points in 3D space (points that do not all lie on the same line), $\overrightarrow{\mathbf{m}}_1, \overrightarrow{\mathbf{m}}_2, \overrightarrow{\mathbf{m}}_3$, which would represent the positions of markers captured from a motion analysis session, a basis can be found following these steps: 1. First axis, $\overrightarrow{\mathbf{v}}_1$, the vector $\overrightarrow{\mathbf{m}}_2-\overrightarrow{\mathbf{m}}_1$ (or any other vector difference); 1. Second axis, $\overrightarrow{\mathbf{v}}_2$, the cross or vector product between the vectors $\overrightarrow{\mathbf{v}}_1$ and $\overrightarrow{\mathbf{m}}_3-\overrightarrow{\mathbf{m}}_1$ (or $\overrightarrow{\mathbf{m}}_3-\overrightarrow{\mathbf{m}}_2$); 1. Third axis, $\overrightarrow{\mathbf{v}}_3$, the cross product between the vectors $\overrightarrow{\mathbf{v}}_1$ and $\overrightarrow{\mathbf{v}}_2$. 1. Make all vectors to have norm 1 dividing each vector by its norm. The positions of the points used to construct a coordinate system have, by definition, to be specified in relation to an already existing coordinate system. In motion analysis, this coordinate system is the coordinate system from the motion capture system and it is established in the calibration phase. In this phase, the positions of markers placed on an object with perpendicular axes and known distances between the markers are captured and used as the reference (laboratory) coordinate system. For example, given the positions $\overrightarrow{\mathbf{m}}_1 = [1,2,5], \overrightarrow{\mathbf{m}}_2 = [2,3,3], \overrightarrow{\mathbf{m}}_3 = [4,0,2]$, a basis can be found with: In [2]: import numpy as np m1 = np.array([1, 2, 5]) m2 = np.array([2, 3, 3]) m3 = np.array([4, 0, 2]) v1 = m2 - m1 # first axis v2 = np.cross(v1, m3 - m1) # second axis v3 = np.cross(v1, v2) # third axis # Vector normalization e1 = v1/np.linalg.norm(v1) e2 = v2/np.linalg.norm(v2) e3 = v3/np.linalg.norm(v3) print('Versors:', '\ne1 =', e1, '\ne2 =', e2, '\ne3 =', e3) print('\nTest of orthogonality (cross product between versors):', '\ne1 x e2:', np.linalg.norm(np.cross(e1, e2)), '\ne1 x e3:', np.linalg.norm(np.cross(e1, e3)), '\ne2 x e3:', np.linalg.norm(np.cross(e2, e3))) print('\nNorm of each versor:', '\n\|\|e1\|\| =', np.linalg.norm(e1), '\n\|\|e2\|\| =', np.linalg.norm(e2), '\n\|\|e3\|\| =', np.linalg.norm(e3)) Versors: e1 = [ 0.40824829 0.40824829 -0.81649658] e2 = [-0.76834982 -0.32929278 -0.5488213 ] e3 = [-0.49292179 0.85141036 0.17924429] Test of orthogonality (cross product between versors): e1 x e2: 1.0 e1 x e3: 1.0000000000000002 e2 x e3: 0.9999999999999999 Norm of each versor: \|\|e1\|\| = 1.0 \|\|e2\|\| = 1.0 \|\|e3\|\| = 1.0 To define a coordinate system using the calculated basis, we also need to define an origin. In principle, we could use any point as origin, but if the calculated coordinate system should follow anatomical conventions, e.g., the coordinate system origin should be at a joint center, we will have to calculate the basis and origin according to standards used in motion analysis as discussed before. If the coordinate system is a technical basis and not anatomic-based, a common procedure in motion analysis is to define the origin for the coordinate system as the centroid (average) position among the markers at the reference frame. Using the average position across markers potentially reduces the effect of noise (for example, from soft tissue artifact) on the calculation. For the markers in the example above, the origin of the coordinate system will be: In [3]: origin = np.mean((m1, m2, m3), axis=0) print('Origin: ', origin) Origin: [2.33333333 1.66666667 3.33333333] Let's plot the coordinate system and the basis using the custom Python function CCS.py: In [4]: import sys sys.path.insert(1, r'./../functions') # add to pythonpath from CCS import CCS In [5]: markers = np.vstack((m1, m2, m3)) basis = np.vstack((e1, e2, e3)) In [6]: %matplotlib notebook markers = np.vstack((m1, m2, m3)) basis = np.vstack((e1, e2, e3)) CCS(xyz=[], Oijk=origin, ijk=basis, point=markers, vector=True); ### Gram–Schmidt process¶ The Gram–Schmidt process is a method for orthonormalizing (orthogonal unit versors) a set of vectors using the scalar product. The Gram–Schmidt process works for any number of vectors. For example, given three vectors, $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$, in the 3D space, a basis $\{\hat{e}_a, \hat{e}_b, \hat{e}_c\}$ can be found using the Gram–Schmidt process by: The first versor is in the $\overrightarrow{\mathbf{a}}$ direction (or in the direction of any of the other vectors): $$\hat{e}_a = \frac{\overrightarrow{\mathbf{a}}}{\|\|\overrightarrow{\mathbf{a}}\|\|}$$ The second versor, orthogonal to $\hat{e}_a$, can be found considering we can express vector $\overrightarrow{\mathbf{b}}$ in terms of the $\hat{e}_a$ direction as: $$\overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{b}}^\\| + \overrightarrow{\mathbf{b}}^\bot$$ Then: $$\overrightarrow{\mathbf{b}}^\bot = \overrightarrow{\mathbf{b}} - \overrightarrow{\mathbf{b}}^\\| = \overrightarrow{\mathbf{b}} - (\overrightarrow{\mathbf{b}} \cdot \hat{e}_a ) \hat{e}_a$$ Finally: $$\hat{e}_b = \frac{\overrightarrow{\mathbf{b}}^\bot}{\|\|\overrightarrow{\mathbf{b}}^\bot\|\|}$$ The third versor, orthogonal to $\{\hat{e}_a, \hat{e}_b\}$, can be found expressing the vector $\overrightarrow{\mathbf{C}}$ in terms of $\hat{e}_a$ and $\hat{e}_b$ directions as: $$\overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{c}}^\\| + \overrightarrow{\mathbf{c}}^\bot$$ Then: $$\overrightarrow{\mathbf{c}}^\bot = \overrightarrow{\mathbf{c}} - \overrightarrow{\mathbf{c}}^\\|$$ Where: $$\overrightarrow{\mathbf{c}}^\\| = (\overrightarrow{\mathbf{c}} \cdot \hat{e}_a ) \hat{e}_a + (\overrightarrow{\mathbf{c}} \cdot \hat{e}_b ) \hat{e}_b$$ Finally: $$\hat{e}_c = \frac{\overrightarrow{\mathbf{c}}^\bot}{\|\|\overrightarrow{\mathbf{c}}^\bot\|\|}$$ Let's implement the Gram–Schmidt process in Python. For example, consider the positions (vectors) $\overrightarrow{\mathbf{a}} = [1,2,0], \overrightarrow{\mathbf{b}} = [0,1,3], \overrightarrow{\mathbf{c}} = [1,0,1]$: In [3]: import numpy as np a = np.array([1, 2, 0]) b = np.array([0, 1, 3]) c = np.array([1, 0, 1]) The first versor is: In [4]: ea = a/np.linalg.norm(a) print(ea) [ 0.4472136 0.89442719 0. ] The second versor is: In [6]: eb = b - np.dot(b, ea)ea eb = eb/np.linalg.norm(eb) print(eb) [-0.13187609 0.06593805 0.98907071] And the third version is: In [8]: ec = c - np.dot(c, ea)ea - np.dot(c, eb)*eb ec = ec/np.linalg.norm(ec) print(ec) [ 0.88465174 -0.44232587 0.14744196] Let's check the orthonormality between these versors: In [13]: print(' Versors:', '\nea =', ea, '\neb =', eb, '\nec =', ec) print('\n Test of orthogonality (scalar product between versors):', '\n ea x eb:', np.dot(ea, eb), '\n eb x ec:', np.dot(eb, ec), '\n ec x ea:', np.dot(ec, ea)) print('\n Norm of each versor:', '\n \|\|ea\|\| =', np.linalg.norm(ea), '\n \|\|eb\|\| =', np.linalg.norm(eb), '\n \|\|ec\|\| =', np.linalg.norm(ec)) Versors: ea = [ 0.4472136 0.89442719 0. ] eb = [-0.13187609 0.06593805 0.98907071] ec = [ 0.88465174 -0.44232587 0.14744196] Test of orthogonality (scalar product between versors): ea x eb: 2.08166817117e-17 eb x ec: -2.77555756156e-17 ec x ea: 5.55111512313e-17 Norm of each versor: \|\|ea\|\| = 1.0 \|\|eb\|\| = 1.0 \|\|ec\|\| = 1.0 ## Polar and spherical coordinate systems¶ When studying circular motion in two or three dimensions, the use of a polar (for 2D) or spherical (for 3D) coordinate system can be more convenient than the Cartesian coordinate system. ### Polar coordinate system¶ In the polar coordinate system, a point in a plane is described by its distance $r$ to the origin (the ray from the origin to this point is the polar axis) and the angle $\theta$ (measured counterclockwise) between the polar axis and an axis of the coordinate system as shown next. The relation of the coordinates in the Cartesian and polar coordinate systems is: $$\begin{array}{l l} x = r\cos\theta \\ y = r\sin\theta \\ r = \sqrt{x^2 + y^2} \end{array}$$ ### Spherical coordinate system¶ The spherical coordinate system can be seen as an extension of the polar coordinate system to three dimensions where an orthogonal axis is added and a second angle is used to describe the point with respect to this third axis as shown next. The relation of the coordinates in the Cartesian and spherical coordinate systems is: $$\begin{array}{l l} x = r\sin\theta\cos\phi \\ y = r\sin\theta\sin\phi \\ z = r\cos\theta \\ r = \sqrt{x^2 + y^2 + z^2} \end{array}$$ ## Generalized coordinates¶ In mechanics, generalized coordinates are a set of coordinates that describes the configuration of a system. Generalized coordinates are usually selected for convenience (e.g., simplifies the resolution of the problem) or to provide the minimum number of coordinates to describe the configuration of a system. For instance, generalized coordinates are used to describe the motion of a system with multiple links where instead of using Cartesian coordinates, it's more convenient to use the angles between links as coordinates. ## Problems¶ 1. Right now, how fast are you moving? In your answer, consider your motion in relation to Earth and in relation to Sun. 2. Go to the website http://www.wisc-online.com/Objects/ViewObject.aspx?ID=AP15305 and complete the interactive lesson to learn about the anatomical terminology to describe relative position in the human body. 3. To learn more about Cartesian coordinate systems go to the website http://www.mathsisfun.com/data/cartesian-coordinates.html, study the material, and answer the 10 questions at the end. 4. Given the points in the 3D space, $m1 = [2,2,0], m2 = [0,1,1], m3 = [1,2,0]$, find an orthonormal basis. 5. Determine if the following points form a basis in the 3D space, $m1 = [2,2,0], m2 = [1,1,1], m3 = [1,1,0]$. 6. Derive expressions for the three axes of the pelvic basis considering the convention of the Virtual Animation of the Kinematics of the Human for Industrial, Educational and Research Purposes (VAKHUM) project (use RASIS, LASIS, RPSIS, and LPSIS as names for the pelvic anatomical landmarks and indicate the expression for each axis). 7. Determine the basis for the pelvis following the convention of the Virtual Animation of the Kinematics of the Human for Industrial, Educational and Research Purposes (VAKHUM) project for the following anatomical landmark positions (units in meters): $RASIS=[0.5,0.8,0.4], LASIS=[0.55,0.78,0.1], RPSIS=[0.3,0.85,0.2], LPSIS=[0.29,0.78,0.3]$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.848518431186676, "perplexity": 925.5423931122099}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684854.67/warc/CC-MAIN-20191018204336-20191018231836-00541.warc.gz"}
https://arxiv.org/abs/1708.02999	stat.ML (what is this?) # Title: Demixing Structured Superposition Signals from Periodic and Aperiodic Nonlinear Observations Abstract: We consider the demixing problem of two (or more) structured high-dimensional vectors from a limited number of nonlinear observations where this nonlinearity is due to either a periodic or an aperiodic function. We study certain families of structured superposition models, and propose a method which provably recovers the components given (nearly) $m = \mathcal{O}(s)$ samples where $s$ denotes the sparsity level of the underlying components. This strictly improves upon previous nonlinear demixing techniques and asymptotically matches the best possible sample complexity. We also provide a range of simulations to illustrate the performance of the proposed algorithms. Comments: arXiv admin note: substantial text overlap with arXiv:1701.06597 Subjects: Machine Learning (stat.ML) Cite as: arXiv:1708.02999 [stat.ML] (or arXiv:1708.02999v1 [stat.ML] for this version)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8574860095977783, "perplexity": 1362.8274756516266}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813712.73/warc/CC-MAIN-20180221182824-20180221202824-00137.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/dcdss.2012.5.559	# American Institute of Mathematical Sciences June 2012, 5(3): 559-566. doi: 10.3934/dcdss.2012.5.559 ## Exponential decay for solutions to semilinear damped wave equation 1 Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget du Lac, France 2 Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Received March 2010 Revised May 2010 Published October 2011 This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4]. Citation: Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559 ##### References: [1] J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations,, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473. Google Scholar [2] A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391. doi: 10.1007/s00030-005-0008-5. Google Scholar [3] J. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation,, Discrete. Contin. Dyn. Syst., 10 (2004), 787. doi: 10.3934/dcds.2004.10.787. Google Scholar [4] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, Ann. I. H. Poincaré, 23 (2006), 185. Google Scholar [5] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms,, J. Differential Equations, 109 (1994), 295. doi: 10.1006/jdeq.1994.1051. Google Scholar [6] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,, Arch. Rational Mech. Anal., 100 (1988), 191. doi: 10.1007/BF00282203. Google Scholar [7] R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms,, Nonlinear. Anal., 27 (1996), 1165. doi: 10.1016/0362-546X(95)00119-G. Google Scholar [8] R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type,, Hiroshima Math. J., 26 (1996), 475. Google Scholar [9] J. Esquivel-Avila, The dynamics of nonlinear wave equation,, J. Math. Anal. Appl., 279 (2003), 135. doi: 10.1016/S0022-247X(02)00701-1. Google Scholar [10] V. K. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equations of parabolic and hyperbolic type,, J. Soviet. Math., 10 (1978), 53. doi: 10.1007/BF01109723. Google Scholar [11] M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,, Comment. Math. Univ. Carolin., 30 (1989), 713. Google Scholar [12] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcal t}=-Au+\mathcal F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1. doi: 10.2307/1996814. Google Scholar [13] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,, SIAM J. Math. Anal., 5 (1974), 138. doi: 10.1137/0505015. Google Scholar [14] S. Messaoudi and B. Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms,, Math. Methods Appl. Sci., 27 (2004), 1687. doi: 10.1002/mma.522. Google Scholar [15] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,, Math. Methods Appl. Sci., 20 (1997), 151. doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S. Google Scholar [16] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273. doi: 10.1007/BF02761595. Google Scholar [17] G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,, C. R. Acad Sci. Paris Ser., 326 (1998), 191. Google Scholar [18] G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms,, J. Math. Anal. Appl., 239 (1999), 213. doi: 10.1006/jmaa.1999.6528. Google Scholar [19] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation,, Arch. Ration. Mech. Anal., 149 (1999), 155. doi: 10.1007/s002050050171. Google Scholar [20] Z. Yang, Existence and asymptotic behavior of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms,, Math. Meth. Appl. Sci., 25 (2002), 795. doi: 10.1002/mma.306. Google Scholar [21] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial. Diff. Eq., 15 (1990), 205. Google Scholar show all references ##### References: [1] J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations,, Quart. J. Math. Oxford. Ser. (2), 28 (1977), 473. Google Scholar [2] A. Benaissa and S. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 391. doi: 10.1007/s00030-005-0008-5. Google Scholar [3] J. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation,, Discrete. Contin. Dyn. Syst., 10 (2004), 787. doi: 10.3934/dcds.2004.10.787. Google Scholar [4] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations,, Ann. I. H. Poincaré, 23 (2006), 185. Google Scholar [5] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms,, J. Differential Equations, 109 (1994), 295. doi: 10.1006/jdeq.1994.1051. Google Scholar [6] A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,, Arch. Rational Mech. Anal., 100 (1988), 191. doi: 10.1007/BF00282203. Google Scholar [7] R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms,, Nonlinear. Anal., 27 (1996), 1165. doi: 10.1016/0362-546X(95)00119-G. Google Scholar [8] R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type,, Hiroshima Math. J., 26 (1996), 475. Google Scholar [9] J. Esquivel-Avila, The dynamics of nonlinear wave equation,, J. Math. Anal. Appl., 279 (2003), 135. doi: 10.1016/S0022-247X(02)00701-1. Google Scholar [10] V. K. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equations of parabolic and hyperbolic type,, J. Soviet. Math., 10 (1978), 53. doi: 10.1007/BF01109723. Google Scholar [11] M. Kopáčkova, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,, Comment. Math. Univ. Carolin., 30 (1989), 713. Google Scholar [12] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcal t}=-Au+\mathcal F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1. doi: 10.2307/1996814. Google Scholar [13] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,, SIAM J. Math. Anal., 5 (1974), 138. doi: 10.1137/0505015. Google Scholar [14] S. Messaoudi and B. Said Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms,, Math. Methods Appl. Sci., 27 (2004), 1687. doi: 10.1002/mma.522. Google Scholar [15] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,, Math. Methods Appl. Sci., 20 (1997), 151. doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S. Google Scholar [16] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273. doi: 10.1007/BF02761595. Google Scholar [17] G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,, C. R. Acad Sci. Paris Ser., 326 (1998), 191. Google Scholar [18] G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms,, J. Math. Anal. 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https://tex.stackexchange.com/questions/5036/how-to-prevent-latex-from-hyphenating-the-entire-document/5039#5039	# How to prevent LaTeX from hyphenating the entire document? I can't find where I can remove any kind of hyphenation and just have LaTeX just do a line break. edit: Honestly speaking I just don't like to read hyphenation anywhere and that's the only reason why I wanted to remove it. It is a matter of style, probably unexpected in LaTeX. The document has only a summary in a different language and I used the language packages to hyphenate properly. • If you are going to typeset your document without hyphenation I strongly recommend using \raggedright to avoid large spaces between the word. Nov 7 '10 at 11:07 • @WillRobertson But, the document looks too bad. In there a middle road? (for instance like \hyphenpenalty=5000) Aug 19 '19 at 8:31 • @cyriac I guess you want the ragged2e package. Aug 19 '19 at 12:13 • This isn’t precisely a way to prevent hyphenation, but unhyphenated documents will usually look much better if you turn font expansion on with microtype. (Not compatible with XeTeX, unfortunately.) This stretches the font slightly to reduce the amount of extra spacing. Jan 22 at 14:13 This is discussed in detail in the TeX FAQ. Summarising the information given there: 1. You can set \hyphenpenalty and \exhyphenpenalty to 10000, which will stop hyphenation, but as TeX will still try to hyphenate this is not hugely efficient. 2. As Joel says, you can use \usepackage[none]{hyphenat} to select a 'language' with no hyphenation at all. This works fine for a single language document, but not if you want to use babel or polyglossia for other language-specific effects. 3. Setting \righthyphenmin and \lefthyphenmin to very large values will prevent hyphenation as it tells TeX that it must have more characters in the word than are going to be available. The suggested value in the FAQ is 62. 4. You can set \hyphenchar\font=-1, which will prevent hyphenation for the current font: this is probably not the best way for an entire document but is how it is done for the tt font shape in LaTeX. Now, of those (2) is probably the best choice. However, what you did not say is why you want no hyphenation. TeX hyphenates when it cannot find a good line break without it, so you get few hyphens in most cases. The risk with no hyphenation at all is that the output looks bad. • @Splashy: see Will's comment about using \raggedright. Nov 9 '10 at 18:53 • @Splashy: The issue is that in order to do that you end up with unacceptably-long gaps between words (see what happens in a word processor). You can let TeX make bigger gaps using the \sloppy macro, which will hopefully avoid text running into the margins when there is no hyphenation. However, the effect may well be very bad looking. That's really the whole point here: TeX hyphenates to keep a good appearance only when acceptable fiddling with spacing has failed. Nov 9 '10 at 19:09 • Oh, also consider loading the microtype package, as this enables some other approaches to improving spacing and reducing the need for hyphenation in many cases. Nov 9 '10 at 19:10 • @Juan: One reason why one might want to avoid hyphenation is because the journal they're submitting to forbids its use: tandfonline.com/action/…. – Sara Apr 17 '13 at 9:13 • @JuanA.Navarro, another reason to prevent hyphenation, is that sometimes you want to copy/paste the text from LaTeX to another document. If you have hyphenation, you have to control the text to make sure the newly justified text is not hyphenated in the middle of sentences. – PLG Mar 14 '16 at 7:01 I use this and it works great for me in almost all documents: \tolerance=1 \emergencystretch=\maxdimen \hyphenpenalty=10000 • Good answer, the previous solutions mess up my document, this works for me!!! Apr 9 '15 at 21:46 • When I used hyphenpenalty=10000 alone some words pass the margin. This works great. Aug 21 '15 at 11:24 • This worked for me with Share Latex, and it avoids the problem of words going over the right-hand margin. Mar 6 '16 at 2:58 • This worked ultimately...Thank you so much..@Timharris Jul 9 '16 at 13:15 • Thank you! All of the other answers that I've seen make the words overflow into the margin rather than wrapping to the next line. Nov 2 '16 at 21:06 A quick google found \usepackage[none]{hyphenat} and more useful info here. • Note that this silently breaks the breaklines option of the listings package. See stackoverflow.com/a/8264050/1650137. Jan 3 '16 at 4:15 • Another reason for not using hyphenation is if you want a textually 'clean' way to cut-and-paste to other editing environments. For example I cut-and-paste text from the typeset PDF that LaTeX generates into other editing packages when I work with colleagues who do not use LaTeX (and aren't likely to either). Apr 15 '16 at 8:00 • I get badbox errors by using this. Oct 23 '20 at 11:15 If one uses babel, there's the hyphsubst package by Heiko Oberdiek: \documentclass[a4paper] ... \usepackage[german=nohyphenation,french=nohyphenation]{hyphsubst} \usepackage[german,french]{babel} provided the distribution knows about the virtual language nohyphenation that has no patterns (both TeX Live and MiKTeX should know it). If this is not the case, the following hack is equivalent \makeatletter\chardef\l@nohyphenation=255 \makeatother \usepackage[german=nohyphenation,french=nohyphenation]{hyphsubst} (at least if less than 256 languages are already defined in the format, which is quite likely). TeX will still possibly break lines at explicit hyphens, though. To explain it better: if you get an error about Unknown pattern nohyphenation then the document should be like \documentclass[a4paper] \makeatletter\chardef\l@nohyphenation=255 \makeatother \usepackage[german=nohyphenation,french=nohyphenation]{hyphsubst} \usepackage[german,french]{babel} • I am sorry, but I searched hyphsubst manual for the option nohyphenation, and I got nothing. I would be grateful if you could update your answer if it needs. – Diaa May 5 '17 at 17:03 • @DiaaAbidou There is no nohyphenation option: the options are of the form language1=language2 and nohyphenation chooses a language with no hyphenation pattern. May 5 '17 at 17:06 • I tried to compile a MWE with your code of babel, but it gives me an error Package hyphsubst Error: Unknown pattern nohyphenation.' \ProcessOptions*. Would you like me to post a new question about it? – Diaa May 5 '17 at 17:15 • @DiaaAbidou Then you have to follow the second strategy: “If this is not the case…” May 5 '17 at 17:17 • @DiaaAbidou You're welcome! May 5 '17 at 17:35 You can use the command: \raggedright or the environment: \begin{flushleft} \end{flushleft} § Paragraph alignment Based on the answers from another post, I found these settings to be perfect to prevent hyphenation without being ugly: \tolerance=9999 \emergencystretch=10pt \hyphenpenalty=10000 \exhyphenpenalty=100 \tolerance=9999 allows as much whitespace as possible. \emerencystretch=10pt allows some extra whitespace per line. \hyphenpenalty=10000 disables hyphens completly. \exhyphenpenalty=100 allows using hyphens which were already present. By increasing the width of the spaces between words, it is possible to give LaTeX more room to stretch or squeeze a line of text, thus reducing the frequency with which words have to be divided at line breaks. In many documents, the respacing achieved using the command below will virtually eliminate end-of-line hyphenation, without forcing text into the margin. \spaceskip=1.3\fontdimen2\font plus 1.3\fontdimen3\font minus 1.3\fontdimen4\font In this command, \fontdimen2 is the nominal or ideal distance between words, \fontdimen3 is the allowable extension of the inter-word space and \fontdimen4 is the allowable compression. It is perhaps worth noting also that hyphenation of a given word can be prevented manually by placing it in an \mbox{}. • This uses a different mechanism, but the result isn't greatly different from that achieved with \sloppy. May 9 '20 at 19:40 By using all the answer here, you might be noticed that space after period feels too wide, where you might be not noticed when there is hyphenation. This is due to latex use more than single space after period. See the discussion in SE answer here. Thus, you can add \frenchspacing to remove this behavior if you use \pretolerance=10000, and disable the french spacing using \nonfrenchspacing if you use hyphenation. Also, if your text contain hyphenation that you write manually, you can use \tolerance=9000 and \emergencystretch=0pt to allow change line between word using that hypen. Otherwise, you can set \tolerance=1 and \emergencystretch=\maxdimen to prohibit change line between words of your manual hypen. The MWE (based on this, and this) will be: \documentclass[12pt]{article} \tolerance=9000 \emergencystretch=0pt \hyphenpenalty=10000 \frenchspacing \begin{document} Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam eleifend tellus id ultrices feugiat. Sed a risus vitae nisi placerat posuere. Donec ullamcorper rhoncus purus, a ornare nunc. In tempus elementum tellus a dictum. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Phasellus pharetra mollis efficitur. Duis urna nunc, molestie vitae ante in, pharetra hendrerit arcu. Fusce varius lectus vitae leo facilisis, sed ultricies velit interdum. Nunc volutpat, neque iaculis tempor scelerisque, enim nunc posuere sapien, vitae tempor quam justo a odio. Suspendisse porta vel ante et sagittis. Sed sit amet malesuada ligula, id commodo diam. Donec posuere eros et orci dignissim tincidunt. Donec imperdiet, metus at lobortis rutrum, nisi felis pretium magna, quis lacinia magna erat eget quam. Duis eget dolor consequat, porttitor nunc vel, rutrum tellus. 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http://mathhelpforum.com/calculus/19573-rubber-band-stretch-problem.html	# Math Help - rubber band stretch problem 1. ## rubber band stretch problem The question is, is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? What I wrote: Yes. You may stretch it left and right, but the middle will remain in place assuming the two forces are equal. Pretend the rubber band is a graph. It may be stretched horizontally, but the y-intercept will not change unless the graph is shifted (the y-intercept is equivalent to the center of the rubber band) So the middle of the rubber band will remain in place as long as the rubber band itself is not moved or shifted other than the stretch, equal on both sides. It was graded and I was told the reasoning was not exactly correct, I need to connect it to the intermediate value theorum. Can anyone explain how I would do so? Thanks! 2. Originally Posted by mistykz The question is, is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? What I wrote: Yes. You may stretch it left and right, but the middle will remain in place assuming the two forces are equal. Pretend the rubber band is a graph. It may be stretched horizontally, but the y-intercept will not change unless the graph is shifted (the y-intercept is equivalent to the center of the rubber band) So the middle of the rubber band will remain in place as long as the rubber band itself is not moved or shifted other than the stretch, equal on both sides. It was graded and I was told the reasoning was not exactly correct, I need to connect it to the intermediate value theorum. Can anyone explain how I would do so? Thanks! The band originaly may be thought of as the interval [a,b], and after stretching [f(a), f(b)], f(a)<a, f(b)>b. Now consider the function g(x)=f(x)-x, This is negative at a, and positive at b, and continuous, so by the intermediate value theorem there is a point c in [a,b] such that g(c)=0 so f(c)=c, that is there is a fixed of invariant point on the band. (you could do this without constructing g, from f directly, but I prefer this way) RonL 3. ## CaptainBlack beat me to it... well well I post it anyway Originally Posted by mistykz The question is, is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? "..." It was graded and I was told the reasoning was not exactly correct, I need to connect it to the intermediate value theorum. Can anyone explain how I would do so? Thanks! There was no specification of how far left or right the rubber band ends where moved so I figure the teacher didn't like your special case explanation with equal lengths(or forces)to the quite general problem statement. The intermediate value theorem states that(from wolfram mathworld)... "If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x)=c." Suggestion: Let the rubber band itself constitute the x-axis. Then let f(x) be the distance point x have moved from it's original position when the rubber band is stretched. Now, the left endpoint a will move to the left and the right endpoint b to the right so that f(a)<0 and f(b)>0. As f is continuous and f(a)<0<f(b) "there is at least one number x in the closed interval such that" f(x)=0." 4. I realize this is beyond what the original poster intended, but wouldn't this problem fall under the "fixed point theorem?" -Dan 5. Originally Posted by F.A.P There was no specification of how far left or right the rubber band ends where moved so I figure the teacher didn't like your special case explanation with equal lengths(or forces)to the quite general problem statement. The intermediate value theorem states that(from wolfram mathworld)... "If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x)=c." Suggestion: Let the rubber band itself constitute the x-axis. Then let f(x) be the distance point x have moved from it's original position when the rubber band is stretched. Now, the left endpoint a will move to the left and the right endpoint b to the right so that f(a)<0 and f(b)>0. As f is continuous and f(a)<0<f(b) "there is at least one number x in the closed interval such that" f(x)=0." sir very nice proof but sir still i have a doubt in my mind .how can we prove it practically	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8457180261611938, "perplexity": 385.8368197414326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644063825.9/warc/CC-MAIN-20150827025423-00011-ip-10-171-96-226.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/108402/decomposition-of-matrices-in-semisimple-and-nilpotent-parts/108408	# Decomposition of Matrices in Semisimple and Nilpotent Parts For any matrix $A\in M_n(\mathbb F)$, where $\mathbb F$ is an algebraically closed field, there is a matrix $S\in M_n(\mathbb F)$ such that $$SAS^{-1}=D+N,$$ where $D$ is diagonal and $N$ nilpotent. Moreover, this decomposition is unique. Suppose now that $A\in M_n(\mathbb K)$, but $\mathbb K$ is not necessarily algebraically closed. It is also true that there is a matrix $L\in M_n(\mathbb K)$ such that $$LAL^{-1}=R+M,$$ where $M$ is nilpotent and $R$ is diagonalizable in the algebraic closure of $\mathbb K$? Moreover when we consider the decomposition in $\mathbb K$ and in the algebraic closure of $\mathbb K$ the nilpotent part is the same? - Also $L$ is redundant. You may take $L=E$ (the identity matrix). – Anton Klyachko Sep 29 '12 at 12:24 There is no uniqueness if you don't require that the two matrices in the decomposition commute. – BS. Sep 30 '12 at 10:00 1. You don't need to conjugate if you want $R$ to be diagonalizable (as opposed to diagonal). 2. I assume you want $R$ and $M$ to have coefficients in $K$, otherwise just work in the algebraic closure. 3. The statement is then true if $K$ is perfect and possibly false otherwise, as you can see by taking $A=[[0, 1], [t, 0]]$ in $K=F_2(t)$. - We should demand that $R$ and $M$ commute. – Anton Klyachko Sep 29 '12 at 15:58 The example in #3 (which readily adapts to any imperfect field $k$ using the $k$-linear multiplication by $a^{1/p}$ on $V = k(a^{1/p})$) isn't an entirely satisfactory counterexample because it is semisimple over $k$ (though not "geometrically semisimple"; i.e., not diagonalizable over $\overline{k}$). The Wikipedia entry has now been updated to give an example over any imperfect field $k$ in which the operator isn't a sum of two commuting $k$-linear operators that are respectively semisimple (just over $k$!) and nilpotent. – grp Oct 1 '12 at 11:23 @grp, what is the definition of semisimplicity that is distinct from geometric semisimplicity? (For those who learned the Jordan decomposition from Borel, they are the same by definition.) Is it that the minimal polynomial is irreducible over $k$? – L Spice Aug 1 '15 at 19:06 To amplify the points made by Laurent Berger, the literature I've seen (dating from around 1950) always specifies perfect fields; so I believe it was understood very early that Jordan-type decomposition breaks down for imperfect fields. I'm not sure whether this is discussed in the modern textbooks on theoretical linear algebra such as those by Curtis and Hoffman-Kunze. (In basic linear algebra it's rare to mention imperfect fields, though this becomes a real issue in Lie theory.) Beyond the Jordan normal form for a matrix (originally developed over a field of characteristic 0 containing all the eigenvalues), the work of Chevalley has been essential for the more flexible notion of "Jordan decomposition" and related matrix polynomials over a perfect field not containing the eigenvalues. Of course he was motivated especially by the theory of linear algebraic groups, but even for computational linear algebra his viewpoint is historically important and justifies the term Jordan-Chevalley decomposition Much of the history has been written down in a joint paper by Danielle Couty and colleagues: see the arXiv preprint here. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9340956211090088, "perplexity": 298.1723444854426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860118321.95/warc/CC-MAIN-20160428161518-00089-ip-10-239-7-51.ec2.internal.warc.gz"}
https://tw.voicetube.com/videos/23949	A1 初級 4701 It's been a long day without you my friend And I'll tell you all about it when I see you again We've come a long way from where we began Oh, I'll tell you all about it when I see you again When I see you again Why'd you have to leave so soon? Why'd you have to go? Why'd you have to leave me when I needed you the most? Cause I don't really know how to tell you Without feeling much worse I know you're in a better place But it's always gonna hurt Carry on ..... Give me all the strength I need to carry on It's been a long day without you my friend And I'll tell you all about it when I see you again We've come a long way from where we began Oh, I'll tell you all about it when I see you again When I see you again How do I breathe without you I'm feeling so cold I'll be waiting right here for you Till the day you're home Carry on ..... Give me all the strength I need to carry on So let the light guide your way Hold every memory as you go And every road you take will always lead you home It's been a long day without you my friend And I'll tell you all about it when I see you again We've come a long way from where we began Oh, I'll tell you all about it when I see you again When I see you again ...... When I see you again See you again ..... When I see you again ..... 玩命關頭7片尾曲 (Charlie Puth - See You Again (Paul Walker Tribute) (Piano Demo Version) (With Lyrics)) 4701 Sanny 發佈於 2015 年 4 月 25 日 1. 1. 單字查詢在字幕上選取單字即可即時查詢單字喔！ 2. 2. 單句重複播放可重複聽取一句單句，加強聽力！ 3. 3. 使用快速鍵使用影片快速鍵，讓學習更有效率！ 4. 4. 關閉語言字幕進階版練習可關閉字幕純聽英文哦！ 5. 5. 內嵌播放器可以將英文字幕學習播放器內嵌到部落格等地方喔 6. 6. 展開播放器可隱藏右方全文及字典欄位，觀看影片更舒適！ 1. 英文聽力測驗挑戰字幕英文聽力測驗！ 1. 點擊展開筆記本讓你看的更舒服 1. UrbanDictionary 俚語字典整合查詢。一般字典查詢不到你滿意的解譯，不妨使用「俚語字典」，或許會讓你有滿意的答案喔	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9401496052742004, "perplexity": 1239.9176394119693}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251672440.80/warc/CC-MAIN-20200125101544-20200125130544-00279.warc.gz"}
https://arxiv.org/abs/math/0206178	math (what is this?) (what is this?) # Title:A third-order Apery-like recursion for $ζ(5)$ Abstract: In 1978, Apery has given sequences of rational approximations to $ζ(2)$ and $ζ(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $ζ(4)$ has been discovered. The note contains a possible generalization of the above results for the number $ζ(5)$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.931850016117096, "perplexity": 506.7696824103437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742253.21/warc/CC-MAIN-20181114170648-20181114192648-00325.warc.gz"}
http://mathhelpforum.com/pre-calculus/192239-arithmetic-possibility-continuity.html	Math Help - The arithmetic of possibility of continuity 1. The arithmetic of possibility of continuity What is the possibility of continuity of combined function of f+g, f-g, fg, f/g at x=c, g(x) not equal to zero at x=c. For all the combination possible When f(x) Continuous & Discontinuous g(x) Continuous & Discontinuous 2. Re: The arithmetic of possibility of continuity You need to be aware that if f and g are both discontinuous at a point, it is possible for f+ g, f- g, fg, and even f/g to be continuous at that point. Can you see why?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9736495018005371, "perplexity": 1784.9407454414927}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049278385.58/warc/CC-MAIN-20160524002118-00036-ip-10-185-217-139.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/259596/type-of-place-versus-type-of-unitary-group	# Type of place versus type of unitary group Let $F$ be a totally real number field, $E$ a totally imaginary quadratic extension over $E$, and $V$ an hermitian $n$-dimensional vector space over $F$. I assume $n=2m$ is even. Let $U$ be a unitary group, i.e. the group au automorphism preserving a given hermitian form on $V$. The structure of local components of $U$ at the finite places of $F$ is among the following ones for instance, see those notes of Michael Harris: • $U_v \cong GL_n(V)$, this happens iff $v$ splits in $E$ • $U_v \cong U(n)$, the only one non-quasi-split unitary group of rank $n$ over $V_v$, and this happens a finite number of time • $U_v \cong U(m,m)$, the only one quasi-split unitary group of rank $n$ over $V_v$ I wonder if there is a characterization of those last two possibilities in terms of the behaviour of $v$ in $E$. Since there is only a finite number of places where $U_v$ is non-quasi-split, could we wait for it to happen iff the place ramifies? • @KevinBuzzard Yes thanks for having noticed it, I switched the "non", as usual $U(n)$ denotes the non-quasi split case ;) – Desiderius Severus Jan 14 '17 at 20:28 Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this question off the top of my head but I did want to know, so I just figured it out below; hopefully there are no errors (hopefully an expert will glance over it and let me know if there are). From the way you write (talking about "the only one non-quasi-split unitary group...") you seem to be assuming that $v$ is a finite place of $F$ (of course the case of infinite places is very well-known). As you know, if $v$ splits in $E$ then $E\otimes_F F_v$ is isomorphic to $F_v\oplus F_v$ and the unitary group becomes $GL_n$ at $v$. If $v$ does not split (i.e. we are in the inert or ramified case) then there is one prime $w$ above $v$ in $E$, and we have a local extension $E_w/F_v$ of degree 2. So we need to understand unitary groups over local fields in order to answer your question. So now let $L/K$ be a degree 2 extension of $p$-adic fields and say $V$ is a vector space over $L$ equipped with a Hermitian form (Hermitian for the action of $Gal(L/K)$ of course). If we choose an $L$-basis for $V$ then this form gives rise to a Hermitian matrix $\Phi$ (so $\overline{\Phi}^t=\Phi$). The determinant of $\Phi$ is an element of $L^\times$ which is equal to its Galois conjugate, so it's in $K^\times$. Let $c(\Phi)$ be the image of this determinant in the group $K^\times/N_{L/K}(L^\times)$, a group of order 2. It turns out that this invariant $c$ parametrises isomorphism classes of Hermitian matrices -- so in particular there are two isomorphism classes of Hermitian forms on $V$. For each isomorphism class we get a unitary group. The next step depends on whether $n$ is odd or even. If $n$ is odd then it turns out that even though the two forms are not isomorphic, the unitary groups are (this can be easily seen -- scaling the form by an element of $K^\times$ can change the isomorphism class of the form but clearly doesn't change the corresponding unitary group). But you are interested in the case $n$ even, and in this case it turns out we get two unitary groups, one quasi-split and one not quasi-split. In particular we see that it is easy to build a non-quasi-split unitary group over $K$ even if $L/K$ is unramified, and it is easy to build a quasi-split unitary group over $K$ even if $L/K$ is ramified -- we just need to get the determinants right. Moreover, we can do all these things over $E/F$ as well, and this is why life is not as simple as you think. To understand things better and to see what is true, we next need to understand when our local unitary group is quasi-split. You have some global Hermitian form giving rise to a global Hermitian matrix whose determinant is $d\in E$, and what we know so far is that if $v$ is a finite place of $F$ which is not split, and $w$ the unique place of $E$ above $v$, then whether or not $U$ is quasi-split at $v$ depends only on what the image of $d$ is in $F_v^\times/N_{E_w/F_v}(E_w^\times)$. So to see exactly what is going on, I need to tell you which element of $F_v^\times/N_{E_w/F_v}(E_w^\times)$ corresponds to the quasi-split case, and then we also need to check (for our own sanity) that in the global situation $d$ will give us a quasi-split local unitary group for all but finitely many $v$. Now here's the bad news. It turns out that the story locally (if I worked it out correctly) is the following. We can write $L=K(\sqrt{k})$ for some $k\in K$, and if my calculations are correct, the element of $K^\times/N_{L/K}(L^\times)$ corresponding to the quasi-split unitary group is $k^m$, where $n=2m$. This is because (if I got it right) if we let our Hermitian form be anti-diagonal with entries $+\sqrt{k},-\sqrt{k},+\sqrt{k},\ldots$ (this is Hermitian if I got it right) then this form gives us a quasi-split unitary group because the upper triangular matrices are a Borel. In particular, the naive guess that the quasi-split group corresponds to the identity element of $K^\times/N_{L/K}(L^\times)$ is not always true. The norm of the element $\sqrt{k}\in L$ is $-k$, so it seems to me that the element of $K^\times/N_{L/K}(L^\times)$ corresponding to the quasi-split unitary group is $(-1)^m$, which of course is the identity element iff $(-1)^m$ is the norm of an element of $L$. This happens if $m$ is even or if $L/K$ is unramified (because then any unit is a norm) but not in general. However, going back to the global situation, we have a global determinant $d\in F^\times$ and at all but finitely many places $d$ will be a local unit, and at all but finitely many places $E_w/F_v$ will be unramified, so it is true that the global unitary group is quasi-split at all but finitely many local places. In general then, to figure out which places your unitary group is not quasi-split you need to figure out what the determinant $d$ of a corresponding Hermitian form is, and then for each place of $F$ which is either ramified in $E$ or for which this determinant is not a unit (there are only finitely many of these), you need to figure out whether $(-1)^md$ is a local norm or not; the places for which this number is not a local norm are the places where the unitary group is ramified. • Dear Kevin, thank you for such a complete answer, even if I fall from the heaven I hoped for. This is not so harmful, but the parallel split/$\mathrm{GL}_n$ and finitely many non-quasi-split/finitely many ramified stoke me, so I wondered if... – Desiderius Severus Jan 14 '17 at 20:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9642574191093445, "perplexity": 113.10098253205271}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400213454.52/warc/CC-MAIN-20200924034208-20200924064208-00520.warc.gz"}
http://openstudy.com/updates/50bdd056e4b0de42629fe9c8	Got Homework? Connect with other students for help. It's a free community. • across Online now • laura* Helped 1,000 students Online now • Hero College Math Guru Online now Here's the question you clicked on: 55 members online • 0 viewing Inspired Group Title The blocks shown are released from rest with spring unstretched. Pulley & horizontal surface are frictionless. If k=400 N/m and M=4.5 kg, what is maximum extension of spring? http://s3.amazonaws.com/answer-board-image/20071222120396333222723984175004973.jpg one year ago one year ago Edit Question Delete Cancel Submit • This Question is Closed Best Response You've already chosen the best response. 1 where are you stuck? during max extension, they are not moving. for M, $$\Sigma F_y= T-F_g=0$$ $$T=F_g=4.59.8$$ for 2M, $$\Sigma F_x= T-F_s=0$$ $$F_s=T=4.59.8$$ $$kx=4.5*9.8$$ sub k and solve for x. • one year ago 2. Vincent-Lyon.Fr Best Response You've already chosen the best response. 0 Hi! Careful: this assumption is not correct: "during max extension, they are not moving. for M, $$\Sigma F_y= T-F_g=0$$" Velocity is zero, but they are accelerating, so $$\Sigma F_y$$ is not zero. Best way to solve this problem is to use conservation of mechanical energy. • one year ago Best Response You've already chosen the best response. 1 ah. missed that. sorry. i was thinking that it was not SHM • one year ago Best Response You've already chosen the best response. 1 by the conservation of ME, since there is no potential energy, or kinetic energy, and setting that height of M as 0 at the very beginning, and that at the max extension, the velocities are zero, i.e. still no kinetic energy, $$0=\frac{1}{2} kx^2 + MgX$$ sub it all in and this should do it. • one year ago • Attachments: See more questions >>> spraguer (Moderator) 5→ View Detailed Profile 23 • Teamwork 19 Teammate • Problem Solving 19 Hero • You have blocked this person. • ✔ You're a fan Checking fan status... Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9710671305656433, "perplexity": 4523.4444216581005}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507443883.29/warc/CC-MAIN-20141017005723-00007-ip-10-16-133-185.ec2.internal.warc.gz"}
https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers	# Integer Combination of Coprime Integers ## Theorem Two integers are coprime if and only if there exists an integer combination of them equal to $1$: $\forall a, b \in \Z: a \perp b \iff \exists m, n \in \Z: m a + n b = 1$ ### General Result Let $a_1, a_2, \ldots, a_n$ be integers. Then $\gcd \left\{ {a_1, a_2, \ldots, a_n}\right\} = 1$ if and only if there exists an integer combination of them equal to $1$: $\exists m_1, m_2, \ldots, m_n \in \Z: \displaystyle \sum_{k \mathop = 1}^n m_k a_k = 1$ ## Proof 1 $\displaystyle a$ $\perp$ $\displaystyle b$ $\displaystyle \leadsto \ \$ $\displaystyle \gcd \set {a, b}$ $=$ $\displaystyle 1$ Definition of Coprime Integers $\displaystyle \leadsto \ \$ $\displaystyle \exists m, n \in \Z: m a + n b$ $=$ $\displaystyle 1$ Bézout's Lemma Then we have: $\displaystyle \exists m, n \in \Z: m a + n b$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle \gcd \set {a, b}$ $\divides$ $\displaystyle 1$ Set of Integer Combinations equals Set of Multiples of GCD $\displaystyle \leadsto \ \$ $\displaystyle \gcd \set {a, b}$ $=$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle a$ $\perp$ $\displaystyle b$ Definition of Coprime Integers $\blacksquare$ ## Proof 2 ### Sufficient Condition Let $a, b \in \Z$ be such that $\exists m, n \in \Z: m a + n b = 1$. Let $d$ be a divisor of both $a$ and $b$. Then: $d \divides m a + n b$ and so: $d \divides 1$ Thus: $d = \pm 1$ and so: $\gcd \set {a, b} = 1$ Thus, by definition, $a$ and $b$ are coprime. $\Box$ ### Necessary Condition Let $a \perp b$. Thus they are not both $0$. Let $S$ be defined as: $S = \set {a m + b n: m, n \in \Z}$ $S$ contains at least one strictly positive integer, because for example $a^2 + b^2 \in S$. By Set of Integers Bounded Below has Smallest Element, let $d$ be the smallest element of $S$ which is strictly positive. Let $d = a x + b y$. It remains to be shown that $d = 1$. By the Division Theorem: $a = d q + r$ where $0 \le r < d$ Then: $\displaystyle r$ $=$ $\displaystyle a - d q$ $\displaystyle$ $=$ $\displaystyle a - \paren {a x + b y} q$ $\displaystyle$ $=$ $\displaystyle a \paren {1 - x q} + b \paren {- y q}$ $\displaystyle$ $\in$ $\displaystyle S$ But we have that $0 \le r < d$. We have defined $d$ as the smallest element of $S$ which is strictly positive Hence it follows that $r$ cannot therefore be strictly positive itself. Hence $r = 0$ and so $a = d q$. That is: $d \divides a$ By a similar argument: $d \divides b$ and so $d$ is a common divisor of both $a$ and $b$. But the GCD of $a$ and $b$ is $1$. Thus it follows that, as $d \in S$: $\exists m, n \in \Z: m a + n b = 1$ $\blacksquare$ ## Proof 3 ### Sufficient Condition Let $a, b \in \Z$ be such that $\exists m, n \in \Z: m a + n b = 1$. Let $d$ be a divisor of both $a$ and $b$. Then: $d \mathrel \backslash m a + n b$ and so: $d \mathrel \backslash 1$ Thus: $d = \pm 1$ and so: $\gcd \left\{ {a, b}\right\} = 1$ Thus, by definition, $a$ and $b$ are coprime. $\Box$ ### Necessary Condition Let $a \perp b$. Thus they are not both $0$. Let $S$ be defined as: $S = \left\{ {\lambda a + \mu b: \lambda, \mu \in \Z}\right\}$ $S$ contains at least one strictly positive integer, because for example: $a \in S$ (setting $\lambda = 1$ and $\mu = 0$) $b \in S$ (setting $\lambda = 0$ and $\mu = 1$) By Set of Integers Bounded Below has Smallest Element, let $d$ be the smallest element of $S$ which is strictly positive. Let $d = \alpha a + \beta b$. Let $c \in S$, such that $\lambda_0 a + \mu_0 b = c$ for some $\lambda_0, \mu_0 \in \Z$. By the Division Algorithm: $\exists \gamma, \delta \in \Z: c = \gamma d + \delta$ where $0 \le \delta < d$ Then: $\displaystyle \delta$ $=$ $\displaystyle c - \gamma d$ $\displaystyle$ $=$ $\displaystyle \left({\lambda_0 a + \mu_0 b}\right) - \gamma \left({\alpha a + \beta b}\right)$ $\displaystyle$ $=$ $\displaystyle \left({\lambda_0 - \gamma \alpha}\right) a + \left({\mu_0 - \gamma \beta}\right) b$ $\displaystyle$ $\in$ $\displaystyle S$ But we have that $0 \le \delta < d$. We have defined $d$ as the smallest element of $S$ which is strictly positive Hence it follows that $\delta$ cannot therefore be strictly positive itself. Hence $\delta = 0$ and so $c = \gamma d$. That is: $d \mathrel \backslash c$ and so the smallest element of $S$ which is strictly positive is a divisor of both $a$ and $b$. But $a$ and $b$ are coprime. Thus it follows that, as $d \mathrel \backslash 1$: $d = 1$ and so, by definition of $S$: $\exists m, n \in \Z: m a + n b = 1$ $\blacksquare$ Note that in the integer combination $m a + n b = 1$, the integers $m$ and $n$ are also coprime. ## Also known as This result is sometimes known as Bézout's Identity, as is the more general Bézout's Lemma. Some sources refer to this result as the Euclidean Algorithm, but the latter as generally understood is the procedure that can be used to establish the values of $m$ and $n$, and for any pair of integers, not necessarily coprime.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9855939149856567, "perplexity": 144.97033591818945}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525524.12/warc/CC-MAIN-20190718063305-20190718085305-00382.warc.gz"}
http://michaelnielsen.org/polymath1/index.php?title=Line-free_sets_correlate_locally_with_complexity-1_sets&oldid=486	# Line-free sets correlate locally with complexity-1 sets Warning: I think I can prove something rigorously, but will not be sure until it is completely written up. The writing up will continue when I have the time to do it. The aim of this page is to present a proof that if $\mathcal{A}$ is a dense subset of $[3]^n$ that contains no combinatorial line, then there is a combinatorial subspace X of $\mathcal{A}$ with dimension tending to infinity and a dense subset $\mathcal{B}$ of X of complexity 1. It is written in a slightly unconventional way, with first a short sketch, then a longer one that fleshes out a few details, and then a longer one still. That way, even while it is incomplete it should be understandable to some extent, and if I get stuck then it will be clearer where the problem lies. ## Short sketch of argument Throughout this sketch, $\mathcal{A}$ refers to a subset of $[3]^n$ of density $\delta$ in the uniform distribution on $[3]^n.$ We shall sometimes use letters such as x, y and z for elements of $[3]^n$ and we shall sometimes write them as triples (U,V,W) of sets that partition [n]. A triple of sets corresponds to the 1-set, the 2-set and the 3-set of a sequence. We shall pass freely between the two ways of thinking about $[3]^n,$ at each stage using whichever is more convenient. If (U,V,W) is an element of $[3]^n$ and (U',V',W') is an arbitrary triple of disjoint sets (not necessarily partitioning [n]), we shall write (U,V,W)++(U',V',W') for the sequence obtained from (U,V,W) by changing everything in U' to 1, everything in V' to 2, and everything in W' to 3. For example, writing § for an unspecified coordinate, we have 331322311++§§§1§22§3=331122213. (We think of (U',V',W') as "overwriting" (U,V,W).) If Z is a subset of [n], we shall also write $(U,V,W)++[3]^Z$ for the combinatorial subspace consisting of all $(U,V,W)++(U',V',W')$ with $(U',V',W')\in[3]^Z,$ and $(U,V,W)++[2]^Z$ for the subset of this combinatorial subspace consisting of all points with $W'=\emptyset.$ Step 1. If a, b and c are all within $C\sqrt n$ of n/3 and a+b+c=n, and if r, s and t are three integers that add up to 0 and are all at most $m=o(\sqrt{n})$ in modulus, then the size of the slice $\Gamma_{a,b,c}$ is 1+o(1) times the size of the slice $\Gamma_{a+r,b+s,c+t}.$ Step 2. If $\mu$ is some probability distribution on combinatorial subspaces of $[3]^n$ such that the distribution of a point x chosen uniformly at random from a subspace chosen randomly according to the distribution $\mu$ is approximately uniform, then we may assume that $\mu$-almost all subpaces $S\subset[3]^n$ contain at least $(\delta-\eta)\|S\|$ elements of $\mathcal{A}.$ Step 3. By 1,2 and an averaging argument, we find $(U,V,W)$ and $Z\subset U\cup V$ of size $o(\sqrt{n})$ (but not much smaller than $\sqrt{n}$) with two properties. First, out of all pairs $(U',V')\in[2]^Z,$ the proportion such that $(U,V,W)++(U',V',\emptyset)$ belongs to $\mathcal{A}$ is at least $\delta/2.$ Secondly, out of all triples $(U',V',W')\in[3]^Z,$ the proportion such that $(U,V,W)++(U',V',W')$ belongs to $\mathcal{A}$ is at least $\delta-\eta.$ Step 4. Fixing such (U,V,W) and Z, let us write (U',V',W') instead of (U,V,W)++(U',V',W'). Then if $U_1\subset U_2$ and $(U_1,Z\setminus U_1,\emptyset)$ and $(U_2,Z\setminus U_2,\emptyset)$ both belong to $\mathcal{A},$ then, writing $V_i$ for $Z\setminus U_i,$ we have that $(U_1,V_2,Z\setminus(U_1\cup V_2))$ does not belong to $\mathcal{A}.$ Step 5. Let $\mathcal{U}$ be the set of all U such that $(U,Z\setminus U,\emptyset)$ belongs to $\mathcal{A},$ and let $\mathcal{V}=\{Z\setminus U:U\in\mathcal{U}\}.$ Then, in an appropriate sense, the set of all pairs $(U_1,V_2)$ such that $U_1\in\mathcal{U}$ and $V_2\in\mathcal{V}$ is dense. It follows that $\mathcal{A}$ is disjoint from a dense set of complexity 1. Step 6. We can partition the set of all disjoint pairs $(U_1,V_2)$ according to which of the sets $\mathcal{U}\times\mathcal{V},$ $\mathcal{U}\times\mathcal{V}^c,$ $\mathcal{U}^c\times\mathcal{V}$ or $\mathcal{U}^c\times\mathcal{V}^c$ they belong to. There must be at least one of the three sets other than $\mathcal{U}\times\mathcal{V}$ in which $\mathcal{A}$ has a density increment. Thus, we have a local density increment on a set of complexity 1. ## Further details ### Step 1 This one is easy. First let us prove the comparable result in $[2]^n.$ That is, let us prove that if a is within $O(\sqrt{n})$ of n/2 and $r=o(\sqrt{n},$ then $\binom na=(1+o(1))\binom n{a+r}.$ This is because the ratio of $\binom nk$ to $\binom n{k+1}$ is (k+1)/(n-k), so if $k=n/2+O(\sqrt{n}),$ then the ratio is $1+O(n^{-1/2}).$ If we now multiply $r=o(\sqrt{n})$ such ratios together we get $1+o(1).$ To get from there to a comparable statement about the sizes of slices in $[3]^n,$ note that we can get from $(a,b,c)$ to $(a+r,b+s,c+t)$ by two operations where we add $o(\sqrt n)$ to one coordinate and subtract $o(\sqrt{n})$ from another. Each time we do so, we multiply by $1+o(1),$ by the result for $[2]^n$ (but applied to $[2]^p$ with p close to 2n/3). ### Step 2 First let us make the statement more precise. Let us say that a probability distribution $\nu$ on a finite set X is $\epsilon$-uniform if $\nu(A)$ never differs from $\|A\|/\|X\|$ by more than $\epsilon.$ (A probabilist would say that the total variation distance between $\nu$ and the uniform distribution is at most $\epsilon.$) Then the precise claim is the following. Let $\epsilon,\eta\gt0.$ Suppose that $\mu$ is a probability distribution on some collection $\Sigma$ of combinatorial subspaces of $[3]^n$ such that the distribution $\nu$ of a point x chosen uniformly at random from a subspace chosen randomly from $\Sigma$ according to the distribution $\mu$ is $\epsilon$-uniform. Then either we can find a combinatorial subspace $S\in\Sigma$ such that $\|\mathcal{A}\cap S\|/\|S\|\geq\delta+\epsilon$ or, when you choose S randomly according to the distribution $\mu,$ the probability that $\|\mathcal{A}\cap S\|/\|S\|\leq\delta-\eta$ is at most $2\epsilon/\eta.$ Proof. Let us first work out a lower bound for the expectation of $\delta(S):=\|\mathcal{A}\cap S\|/\|S\|.$ This expectation is $\sum_{S\in\Sigma}\mu(S)\delta(S),$ which is precisely the probability that you obtain a point in $\mathcal{A}$ if you first pick a random S and then pick a random point in S. In other words, it is $\nu(\mathcal{A}),$ which by hypothesis is within $\epsilon$ of $\delta,$ and is therefore at least $\delta-\epsilon.$ If the probability that $\delta(S)\lt\delta-\eta$ is p and $\delta(S)$ is bounded above by $\delta+\epsilon,$ then the expectation of $\delta(S)$ is at most $p(\delta-\eta)+(1-p)(\delta+\epsilon),$ which equals $\delta+\epsilon-p(\eta+\epsilon).$ If $p\gt2\epsilon/\eta,$ then this is less than $\delta+\epsilon-2\epsilon,$ which is a contradiction. $\Box$ ### Step 3 Now let us pick a random point $(U,V,W)$ and a random set $Z\subset[n]$ of size $m=o(\sqrt{n}).$ We claim first that the distribution of a random point in the combinatorial subspace $S=(U,V,W)++[3]^Z$ is approximately uniform, and also that the distribution of a random point in the set $T=(U,V,W)++[2]^Z$ is approximately uniform. To be continued tomorrow. ## Old stuff, probably to be junked For convenience we shall use equal-slices measure but this is not fundamental to the argument. The model of equal-slices measure we use is this. If p, q and r are non-negative real numbers with p+q+r=1, and $(X_1,\dots,X_n)$ are independent random variables with probabilities p, q and r of equalling 1, 2 and 3, respectively, then we define $\mu_{p,q,r}(\mathcal{A})$ to be the probability that $(X_1,\dots,X_n)$ lies in $\mathcal{A}.$ We then define the density of $\mathcal{A}$ to be the average of $\mu_{p,q,r}(\mathcal{A})$ over all possible triples p,q,r. Now let us do some averaging. Let us write $\delta_{p,q,r}$ for $\mu_{p,q,r}(\mathcal{A}).$ Let us also use the notation (U,V,W) for the $x\in[3]^n$ that has 1-set U, 2-set V and 3-set W. First, we prove two similar lemmas that are very simple, but also rather useful. Lemma 1. The probability distribution of (U,V,W) conditioned on W is the equal-slices measure of (U,V) with ground set $[n]\setminus W.$ Proof. We are asking for the distribution of the random variable $(X_1,\dots,X_n)$ when we condition on the event that $W_i=3$ for every $i\in W.$ Let us condition further on the value of r. Then for each fixed p, q such that p+q=1-r, and each $i\notin W,$ we have that $X_i=1$ with probability p/(1-r) and $X_i=2$ with probability q/(1-r). When we average over p and q, the numbers p/(1-r) and q/(1-r) are uniformly distributed over pairs of positive reals that add up to 1. For each r, we therefore obtain precisely the equal-slices probability distribution on the random variables $X_i$with $i\notin W,$ so the same is true when we average over r.$\Box$ It is obviously not the case that the set W in a random triple (U,V,W) is distributed according to equal-slices measure: rather, we choose r with density 2(1-r) and then choose elements of W independently with probability r. When we refer to a random set W or discuss probabilities of events associated with W, it will be this measure that we refer to. (In other words, we take the marginal distribution on W, just as we should.) To be continued, but possibly not for a while as I have a lot to do in the near future.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9784109592437744, "perplexity": 151.63058985931417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250591431.4/warc/CC-MAIN-20200117234621-20200118022621-00338.warc.gz"}
https://projecteuclid.org/euclid.ade/1384278130	### Penalization for non-linear hyperbolic system Thomas Auphan #### Abstract This paper proposes a volumetric penalty method to simulate the boundary conditions for a non-linear hyperbolic problem. The boundary conditions are assumed to be maximally strictly dissipative on a non-characteristic boundary. This penalization appears to be quite natural since, after a natural change of variable, the penalty matrix is an orthogonal projector. We prove the convergence towards the solution of the wished hyperbolic problem and that this convergence is sharp in the sense that it does not generate any boundary layer, at any order. The proof involves an approximation by asymptotic expansion and energy estimates in anisotropic Sobolev spaces. #### Article information Source Adv. Differential Equations Volume 19, Number 1/2 (2014), 1-29. Dates First available in Project Euclid: 12 November 2013	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9153544306755066, "perplexity": 618.0622924500537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689624.87/warc/CC-MAIN-20170923104407-20170923124407-00708.warc.gz"}
https://brilliant.org/problems/mathematics-in-nature-nautilius-shell/	# Mathematics in Nature: Nautilus shell Calculus Level 1 The Nautilus Spiral is given by the polar coordinates $r = e ^ \theta$. Its shape resembles that of the Nautilus Shell, hence its name. Which of the following statements about this curve is false? ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8454875946044922, "perplexity": 2692.2741572505015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540530452.95/warc/CC-MAIN-20191211074417-20191211102417-00536.warc.gz"}
https://deepai.org/publication/invariant-representations-from-adversarially-censored-autoencoders	# Invariant Representations from Adversarially Censored Autoencoders We combine conditional variational autoencoders (VAE) with adversarial censoring in order to learn invariant representations that are disentangled from nuisance/sensitive variations. In this method, an adversarial network attempts to recover the nuisance variable from the representation, which the VAE is trained to prevent. Conditioning the decoder on the nuisance variable enables clean separation of the representation, since they are recombined for model learning and data reconstruction. We show this natural approach is theoretically well-founded with information-theoretic arguments. Experiments demonstrate that this method achieves invariance while preserving model learning performance, and results in visually improved performance for style transfer and generative sampling tasks. Comments There are no comments yet. ## Authors • 26 publications • 16 publications • 22 publications • ### Understanding disentangling in β-VAE We present new intuitions and theoretical assessments of the emergence o... 04/10/2018 ∙ by Christopher P Burgess, et al. ∙ 0 read it • ### Wyner VAE: Joint and Conditional Generation with Succinct Common Representation Learning A new variational autoencoder (VAE) model is proposed that learns a succ... 05/27/2019 ∙ by J. Jon Ryu, et al. ∙ 0 read it • ### Learning Representations by Maximizing Mutual Information in Variational Autoencoder Variational autoencoders (VAE) have ushered in a new era of unsupervised... 12/21/2019 ∙ by Ali Lotfi Rezaabad, et al. ∙ 0 read it • ### Improving Disentangled Text Representation Learning with Information-Theoretic Guidance Learning disentangled representations of natural language is essential f... 06/01/2020 ∙ by Pengyu Cheng, et al. ∙ 5 read it • ### Variational Autoencoders Pursue PCA Directions (by Accident) The Variational Autoencoder (VAE) is a powerful architecture capable of ... 12/17/2018 ∙ by Michal Rolinek, et al. ∙ 0 read it • ### Auto-Encoding Total Correlation Explanation Advances in unsupervised learning enable reconstruction and generation o... 02/16/2018 ∙ by Shuyang Gao, et al. ∙ 0 read it • ### AutoBayes: Automated Inference via Bayesian Graph Exploration for Nuisance-Robust Biosignal Analysis Learning data representations that capture task-related features, but ar... 07/02/2020 ∙ by Andac Demir, et al. ∙ 74 read it ##### This week in AI Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. ## 1 Introduction We consider the problem of learning data representations that are invariant to nuisance variations and/or sensitive features. Such representations could be useful for fair/robust classification zemel2013-fairrep ; louppe2016-pivot ; xie2017controllable , domain adaptation tzeng2017-AdvDomAdapt ; shen2017-AdvRepLearn , privacy preservation hamm2016-minimax ; iwasawa2017privacy , and style transfer mathieu2016disentangling . We investigate how this problem can be addressed by extensions of the variational autoencoder (VAE) model introduced by kingma2013-VAE , where a generative model is learned as a pair of neural networks: an encoder that produces a representation from data , and a decoder that reconstructs the data from the representation . A conditional VAE sohn2015learning can be trained while conditioned on the nuisance/sensitive variable (i.e., the encoder and decoder each have as an additional input). In principle, this should yield an encoder that extracts representations that are invariant to , since the corresponding generative model (decoder) implicitly enforces independence between and . Intuitively, an efficient encoder should learn to exclude information about from , since is already provided directly to the decoder. However, as we demonstrate in our experiments, invariance is not sufficiently achieved in practice, possibly due to approximations arising from imperfect optimization and parametric models. The adversarial feature learning approach of edwards2015-censor proposes training an unconditioned autoencoder along with an adversarial network that attempts to recover a binary sensitive variable from the representation . However, this approach results in a challenging tradeoff between enforcing invariance and preserving enough information in the representation to allow decoder reconstruction and generative model learning. Our work proposes and investigates the natural combination of adversarial censoring with a conditional VAE, while also generalizing to allow categorical (non-binary) or continuous . Although an adversary is used to enforce invariance between and , the decoder is given both and as inputs enabling data reconstruction and model learning. This approach disentangles the representation from the nuisance variations , while still preserving enough information in to recover the data when recombined with . In Section 2.2, we present a theoretical interpretation for adversarial censoring as reinforcement of the representation invariance that is already implied by the generative model of a conditional VAE. Our experiments in Section 3 quantitatively and qualitatively show that adversarial censoring of a conditional VAE can achieve representation invariance while limiting degradation of model learning performance. Further, the performance in style transfer and generative sampling tasks appear visually improved by adversarial censoring (see Figures 3 and 4). ### 1.1 Further Discussion of Related Work The variational fair autoencoder of louizos2015-VFAE extends the conditional VAE by introducing an invariance-enforcing penalty term based on maximum mean discrepancy (MMD). However, this approach is not readily extensible to non-binary or continuous . Generative adversarial networks (GAN) and the broader concept of adversarial training were introduced by goodfellow2014GAN . The work of mathieu2016disentangling also combines adversarial training with VAEs to disentangle nuisance variations from the representation . However, their approach instead attaches the adversary to the output of the decoder, which requires a more complicated training procedure handling sample triplets and swapping representations, but also incorporates the learned similarity concept of larsen2016autoencoding . Our approach is much simpler to train since the adversary is attached to the encoder directly enforcing representation invariance. Addressing the problem of learning fair representations zemel2013-fairrep , further work on adversarial feature learning louppe2016-pivot ; xie2017controllable ; hamm2016-minimax ; iwasawa2017privacy ; tzeng2017-AdvDomAdapt ; shen2017-AdvRepLearn have used adversarial training to learn invariant representations tailored to classification tasks (i.e., in comparison to our work, they replace the decoder with a classifier). However, note that in louppe2016-pivot , the adversary is instead attached to the output of the classifier. Besides fairness/robustness, domain adaptation tzeng2017-AdvDomAdapt ; shen2017-AdvRepLearn and privacy hamm2016-minimax ; iwasawa2017privacy are also addressed. By considering invariance in the context of a VAE, our approach instead aims to produce general purpose representations and does not require additional class labels. GANs have also been combined with VAEs in many other ways, although not with the aim of producing invariant representations. However, the following concepts could be combined in parallel with adversarial censoring. As mentioned earlier, in larsen2016autoencoding , an adversary attached to the decoder learns a similarity metric to enhance VAE training. In makhzani2015-adversarial ; mescheder2017-adversarial , an adversary is used to approximate the Kullback–Leibler (KL)-divergence in the VAE training objective, allowing for more general encoder architectures and latent representation priors. Both donahue2017adversarial and dumoulin2017adversarially independently propose a method to train an autoencoder using an adversary that tries to distinguish between pairs of data samples and extracted representations versus synthetic samples and the latent representations from which they were generated. ## 2 Formulation In Section 2.1, we review the formulation of conditional VAEs as developed by kingma2013-VAE ; sohn2015learning . Sections 2.2 and 2.3 propose techniques to enforce invariant representations via adversarial censoring and increasing the KL-divergence regularization. ### 2.1 Conditional Variational Autoencoders The generative model for the data involves an observed variable and a latent variable . The nuisance (or sensitive) variations are modeled by , while captures other remaining information. Since the aim is to extract a latent representation that is free of the nuisance variations in , these variables are modeled by the joint distribution , where and are explicitly made independent. The generative model is from a parametric family of distributions that is appropriate for the data. The latent prior can be chosen to have a convenient form, such as the standard multivariate normal distribution . No knowledge or assumptions about the nuisance variable prior are needed since it is not directly used in the learning procedure. The method for learning this model involves maximizing the log-likelihood for a set of training samples with respect to the conditional distribution pθ(x\|s)=∫pθ(x\|z,s)p(z)dz. This objective is analogous to minimizing the KL-divergence between the true conditional distribution of the data and the model since argminθKL(p(x\|s)∥∥pθ(x\|s))=argmaxθE[logpθ(x\|s)], where the expectation is with respect to and can be approximated by . Using a variational posterior to approximate the actual posterior , the log-likelihood can be lower bounded by 1nn∑i=1logpθ(xi\|si)≥1nn∑i=1[E[logpθ(xi\|si,zi)]−KL(qϕ(z\|xi,si)∥∥p(z))]=:Ln(θ,ϕ), (1) where in each expectation. The quantity given by (1) is known as the variational or evidence lower bound (ELBO). The inequality in (1) follows since 1nn∑i=1logpθ(xi\|si)−Ln(θ,ϕ)=1nn∑i=1KL(qϕ(z\|xi,si)∥∥pθ(z\|xi,si))≥0. Thus, by optimizing both and to maximize the lower bound , while is trained toward the true conditional distribution of the data, is trained toward the corresponding posterior . In the VAE architecture, the generative model (decoder) and variational posterior (encoder) are realized as neural networks that take as input and , respectively, as illustrated in Figure 1, and output the parameters of their respective distributions. This architecture is specifically a conditional VAE, since the encoding and decoding are conditioned on the nuisance variable . When the encoder is realized as conditionally Gaussian: qϕ(z\|x,s)=N(z;μϕ(x,s),Σϕ(x,s)), (2) where the mean vector and diagonal covariance matrix are determined as a function of , and the latent variable distribution is set to the standard Gaussian , the KL-divergence term in (1) can be analytically derived and differentiated kingma2013-VAE . However, the expectations in (1 ) must be estimated by sampling. Hence, the learning procedure maximizes a sampled approximation of the ELBO , given by maxθ,ϕ 1nn∑i=1Li(θ,ϕ) (3) ≈maxθ,ϕLn(θ,ϕ) ⪅maxθ1nn∑i=1logpθ(xi\|si), where, for , Li(θ,ϕ):=−KL(qϕ(z\|xi,si)∥∥p(z))+1kk∑j=1logpθ(xi\|si,zi,j), (4) which approximates the expectations in (1) by sampling . ### 2.2 Representation Invariance via Adversarial Censoring In principle, optimal training with ideal parametric approximations should result in an encoder that accurately approximates the true posterior , for which and are independent by construction. Thus, the theoretically optimal encoder should produce a representation that is independent of the nuisance variable . In practice, however, since the encoder is realized as a parametric approximation and globally optimal convergence cannot be guaranteed, we often observe that the representation produced by the trained encoder is significantly correlated with the nuisance variable . Further, one may wish to train an encoder that does not use as an input, to allow the representation to be generated from the data alone. However, this additional restriction on the encoder may increase the challenge of extracting invariant representations. Invariance could be be enforced by minimizing the mutual information where is the latent representation generated by the encoder. Mutual information can be subtracted from the lower bound of (1), yielding 1nn∑i=1logpθ(xi\|si)≥Ln(θ,ϕ)−1nn∑i=1I(si;zi), where equality is still met for . Thus, incorporating a mutual information penalty term into the lower bound does not, in principle, change the theoretical maximum. However, since computing mutual information is generally intractable, we apply the approximation technique of barber2003-IMalgorithm , which utilizes a variational posterior and the lower bound I(s;z)≥h(s)+E[logqψ(s\|z)], (5) where equality is met for equal to the actual posterior for which the expectation and entropies are defined with respect to. Hence, maximizing over the variational posterior , which can also be similarly realized as a neural network, yields an approximation of . The entropy , although generally unknown, is constant with respect to the optimization variables. Incorporating this variational approximation of the mutual information penalty into (3), modulo dropping the constant , results in the adversarial training objective maxθ,ϕminψ1nn∑i=1[Li(θ,ϕ)−λkk∑j=1logqψ(si\|zi,j)], (6) where are the same samples used for as given by (4), and is a parameter that controls the emphasis on invariance. Note that when is a categorical variable (e.g., a class label), the additional, adversarial network to realize the variational posterior is essentially just a classifier trained (by minimizing cross-entropy loss) to recover from the representation generated by the encoder. In this approach, the VAE is adversarially trained to maximize the cross-entropy loss of this classifier combined with the original objective given by (3). Figure 1 illustrates the overall VAE training framework including adversarial censoring. ### 2.3 Invariance via KL-divergence Censoring Another approach to enforce invariance is to introduce a hyperparameter to increase the weight of the KL-divergence terms in (4), yielding the alternative objective terms (7) for which (4) is the special case when . The KL-divergence terms can be interpreted as regularizing the variational posterior toward the latent prior, which encourages the encoder to generate representations that are invariant to not only but also the data . While increasing further encourages invariant representations, it potentially disrupts model learning, since the overall dependence on the data is affected. ## 3 Experiments We evaluate the performance of various VAEs for learning invariant representations, under several scenarios for conditioning the encoder and/or decoder on the sensitive/nuisance variable : • Full: Both the encoder and decoder are conditioned on . In this case, the decoder is the generative model and the encoder is the variational posterior as described in Section 2.1. • Partial: Only the decoder is conditioned on . This case is similar to the previous, except that the encoder approximates the variational posterior without as an input. • Basic (unconditioned): Neither the encoder nor decoder are conditioned on . This baseline case is the standard, unconditioned VAE where is not used as an input. In combination with these VAE scenarios, we also examine several approaches for encouraging invariant representations: • Adversarial Censoring: This approach, as described in Section 2.2, introduces an additional network that attempts to recover from the representation . The VAE and this additional network are adversarially trained according to the objective given by (6). • KL Censoring: This approach, as described in Section 2.3, increases the weight on the KL-divergence terms, using the alternative objective terms given by (7). • Baseline (none): As a baseline, the VAE is trained according to the original objective given by (3) without any additional modifications to enforce invariance. ### 3.1 Dataset and Network Details We use the MNIST dataset, which consists of 70,000 grayscale, pixel images of handwritten digits and corresponding labels in . We treat the vectorized images in as the data , while the digit labels serve as the nuisance variable . Thus, our objective is to train VAE models that learn representations that capture features (i.e., handwriting style) invariant of the digit class . We use basic, multilayer perceptron architectures to realize the VAE (similar to the architecture used in kingma2013-VAE ) and the adversarial network. This allows us to illustrate how the performance of even very simple VAE architectures can be improved with adversarial censoring. We choose the latent representation to have 20 dimensions, with its prior set as the standard Gaussian, i.e., . The encoder, decoder, and adversarial networks each use a single hidden layer of 500 nodes with the activation function. In the scenarios where the encoder (or decoder) is conditioned on the nuisance variable, the one-hot encoding of is concatenated with (or , respectively) to form the input. The adversarial network uses a 10-dimensional softmax output layer to produce the variational posterior . We use the encoder to realize the conditionally Gaussian variational posterior given by (2). The encoder network produces a 40-dimensional vector (with no activation function applied) that represents the mean vector concatenated with the log of the diagonal of the covariance matrix . This allows us to compute the KL-divergence terms in (4) analytically as given by kingma2013-VAE . The output layer of the decoder network has 784 nodes and applies the sigmoid activation function, matching the size and scale of the images. We treat the decoder output, denoted by , as parameters of a generative model given by logpθ(x\|s,z)=784∑i=1xilogyi+(1−xi)log(1−yi), where and are the components of and , respectively. Although not strictly binary, the MNIST images are nearly black and white, allowing this Bernoulli generative model to be a reasonable approximation. We directly display to generate the example output images. We implemented these experiments with the Chainer deep learning framework chainer . The networks were trained over the 60,000 image training set for 100 epochs with 100 images per batch, while evaluation and example generation were performed with the 10,000 image test set. The adversarial and VAE networks were each updated alternatingly once per batch with Adam kingma2014adam . Relying on stochastic estimation over each batch, we set the sampling parameter in (4), (6), and (7). ### 3.2 Evaluation Methods We quantitatively evaluate the trained VAEs for how well they: • Learn the data model: We measure this with the ELBO score estimated by computing over the test data set (see (3) and (4)). • Produce invariant representations: We measure this via the adversarial approach described in Section 2.2. Even when not using adversarial censoring, we still train an adversarial network in parallel (i.e., its loss gradients are not fed back into the main VAE training) that attempts to recover the sensitive variable from the representation . The classification accuracy and cross-entropy loss of the adversarial network provide measures of invariance. Since the digit class is uniformly distributed over , the entropy is equal to and can be combined with the cross-entropy loss (see (5) and barber2003-IMalgorithm ) to yield an estimate of the mutual information , which we report instead. The VAEs are also qualitatively evaluated with the following visual tasks: • Style Transfer (Digit Change): An image from the test set is input to the encoder to produce a representation by sampling from . Then, the decoder is applied to produce the image , while changing the digit class to . • Generative Model Sampling: A synthetic image is generated by first sampling a latent variable from the prior , and then applying the decoder to produce the image for a selected digit class . ### 3.3 Results and Discussion Figure 2 presents the quantitative performance comparison for the various combinations of VAEs with full (), partial (), or no conditioning (), and with invariance encouraged by adversarial censoring ( red —), KL censoring ( blue —), or nothing (black). Each pair of red and blue curves represent varying emphasis on enforcing invariance (as the parameters and are respectively changed) and meet at a black point corresponding to the baseline (no censoring) case (where and ). Unsurprisingly, the baseline, unconditioned VAE produces a representation that readily reveals the digit class ( accuracy), since otherwise image reconstruction by the decoder would be difficult. However, even when partially or fully conditioned on , the baseline VAEs still significantly reveal (partial: , full: accuracies). Both adversarial and KL censoring are effective at enforcing invariance, with adversarial accuracy approaching chance and mutual information approaching zero as the parameters and are respectively increased. However, the adversarial approach has less of an impact on the model learning performance (as measured by the ELBO score). With conditional VAEs, adversarial censoring achieves invariance while having only a small impact on the ELBO score, and appears to visually improve performance (particularly for the partially conditioned case) in the style transfer and sampling tasks as shown in Figures 3 and 4. The worse model learning performance with KL censoring seems to result in blurrier (although seemingly cleaner) images, as also shown in Figures 3 and 4. Attempting to censor a basic (unconditioned) autoencoder (as proposed by edwards2015-censor ) rapidly degrades model learning performance, which manifests as severely degraded sampling performance as shown in Figure 5. The results in Figures 3 and 4 correspond to specific points in Figure 2 as follows: (a) baseline , (b) left-most (), (c) left-most (), (d) baseline , (e) left-most (), (f) left-most (). Figure 5 results correspond to points in Figure 2 as follows: (a) MNIST test examples, (b-c) two left-most (), (d) baseline , (e-f) two left-most (). Note that larger values for the and parameters were required for the unconditioned VAEs to achieve similar levels of invariance as the conditioned cases. ## 4 Conclusion The natural combination of conditional VAEs with adversarial censoring is a theoretically well-founded method to generate invariant representations that are disentangled from nuisance variations. Conditioning the decoder on the nuisance variable allows the representation to be cleanly separated and model learning performance to be preserved, since and are both used to reconstruct the data . Training VAEs with adversarial censoring visually improved performance in style transfer and generative sampling tasks. ## References • (1) D. Barber and F. Agakov. The IM algorithm: a variational approach to information maximization. In Advances in Neural Information Processing Systems (NIPS), pages 201–208, 2003. • (2) J. Donahue, P. Krähenbühl, and T. Darrell. Adversarial feature learning. In Proceedings of the International Conference on Machine Learning (ICML) , 2017. • (3) V. Dumoulin, I. Belghazi, B. Poole, O. Mastropietro, A. Lamb, M. Arjovsky, and A. Courville. Adversarially learned inference. In Proceedings of the International Conference on Machine Learning (ICML), 2017. • (4) H. Edwards and A. Storkey. Censoring representations with an adversary. In Proceedings of the International Conference on Learning Representations (ICLR), 2015. • (5) I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems (NIPS), pages 2672–2680, 2014. • (6) J. Hamm. Minimax filter: Learning to preserve privacy from inference attacks. Journal of Machine Learning Research, 18(129):1–31, 2017. • (7) Y. Iwasawa, K. Nakayama, I. E. Yairi, and Y. Matsuo. Privacy issues regarding the application of DNNs to activity-recognition using wearables and its countermeasures by use of adversarial training. In Proceedings of the International Joint Conferences on Artificial Intelligence (IJCAI) , pages 1930–1936, 2017. • (8) D. Kingma and J. Ba. Adam: A method for stochastic optimization. In Proceedings of the International Conference on Learning Representations (ICLR), 2015. • (9) D. P. Kingma and M. Welling. Auto-encoding variational bayes. In Proceedings of the International Conference on Learning Representations (ICLR), 2014. • (10) A. B. L. Larsen, S. K. Sønderby, H. Larochelle, and O. Winther. Autoencoding beyond pixels using a learned similarity metric. In Proceedings of the International Conference on Machine Learning (ICML), pages 1558–1566, 2016. • (11) C. Louizos, K. Swersky, Y. Li, M. Welling, and R. Zemel. The variational fair autoencoder. In Proceedings of the International Conference on Learning Representations (ICLR), 2016. • (12) G. Louppe, M. Kagan, and K. Cranmer. Learning to pivot with adversarial networks. In Advances in Neural Information Processing Systems (NIPS), pages 982–991, 2017. • (13) A. Makhzani, J. Shlens, N. Jaitly, I. Goodfellow, and B. Frey. Adversarial autoencoders. In Proceedings of the International Conference on Learning Representations (ICLR), 2016. • (14) M. F. Mathieu, J. J. Zhao, J. Zhao, A. Ramesh, P. Sprechmann, and Y. LeCun. Disentangling factors of variation in deep representation using adversarial training. In Advances in Neural Information Processing Systems (NIPS), pages 5040–5048, 2016. • (15) L. Mescheder, S. Nowozin, and A. Geiger. Adversarial variational bayes: Unifying variational autoencoders and generative adversarial networks. In Proceedings of the International Conference on Machine Learning (ICML), pages 2391–2400, 2017. • (16) J. Shen, Y. Qu, W. Zhang, and Y. Yu. Adversarial representation learning for domain adaptation. arXiv preprint arXiv:1707.01217, 2017. • (17) K. Sohn, H. Lee, and X. Yan. Learning structured output representation using deep conditional generative models. In Advances in Neural Information Processing Systems (NIPS), pages 3483–3491, 2015. • (18) S. Tokui, K. Oono, S. Hido, and J. Clayton. Chainer: a next-generation open source framework for deep learning. In Proceedings of Workshop on Machine Learning Systems (LearningSys) in The Twenty-ninth Annual Conference on Neural Information Processing Systems (NIPS), 2015. • (19) E. Tzeng, J. Hoffman, K. Saenko, and T. Darrell. Adversarial discriminative domain adaptation. In Computer Vision and Pattern Recognition (CVPR), 2017. • (20) Q. Xie, Z. Dai, Y. Du, E. Hovy, and G. Neubig. Controllable invariance through adversarial feature learning. arXiv preprint arXiv:1705.11122, 2017. • (21) R. Zemel, Y. Wu, K. Swersky, T. Pitassi, and C. Dwork. Learning fair representations. In Proceedings of the International Conference on Machine Learning (ICML), pages 325–333, 2013.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8262683749198914, "perplexity": 1736.928472777678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655902496.52/warc/CC-MAIN-20200710015901-20200710045901-00507.warc.gz"}
https://hal-univ-artois.archives-ouvertes.fr/hal-00442267	# Weighted composition operators as Daugavet centers Abstract : We investigate the norm identity $\\|uC_\varphi + T\\| = \\|u\\|_\infty + \\|T\\|$ for classes of operators on $C(S)$, where $S$ is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator $T : C(S)\rightarrow C(S)$. We also give a characterization of such weighted composition operator acting on the disk algebra $A(D).$ Keywords : Document type : Preprints, Working Papers, ... Cited literature [13 references] https://hal-univ-artois.archives-ouvertes.fr/hal-00442267 Contributor : Romain Demazeux Connect in order to contact the contributor Submitted on : Friday, December 18, 2009 - 9:22:39 PM Last modification on : Thursday, September 16, 2021 - 10:26:05 AM Long-term archiving on: : Thursday, June 17, 2010 - 10:03:37 PM ### Files Weighted_composition_operators... Files produced by the author(s) ### Identifiers • HAL Id : hal-00442267, version 1 • ARXIV : 0912.4032 ### Citation Romain Demazeux. Weighted composition operators as Daugavet centers. 2009. ⟨hal-00442267⟩ Record views	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8356850743293762, "perplexity": 4309.064881530796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060803.2/warc/CC-MAIN-20210928122846-20210928152846-00022.warc.gz"}
http://www.physicsforums.com/showthread.php?t=397858	# Introduction to electromagnetism - Maxwell's equations by krtica Tags: electromagnetism, wave Share this thread: P: 50 Q1) The amplitude of an electromagnetic wave is E0 = 471.0 V/m. Find the following values. (a) Erms 333.047 V/m (b) Brms __ nT (c) the intensity I __W/m2 (d) the radiation pressure Pr __nPa Q2)(a) For a given distance from a radiating electric dipole, at what angle (expressed as θ and measured from the dipole axis) is the intensity equal to 83 percent of the maximum intensity? (b) At what angle θ is the intensity equal to 3 percent of the maximum intensity? I'm not sure what the general formulas are. Mentor P: 41,140 Quote by krtica Q1) The amplitude of an electromagnetic wave is E0 = 471.0 V/m. Find the following values. (a) Erms 333.047 V/m (b) Brms __ nT (c) the intensity I __W/m2 (d) the radiation pressure Pr __nPa Q2)(a) For a given distance from a radiating electric dipole, at what angle (expressed as θ and measured from the dipole axis) is the intensity equal to 83 percent of the maximum intensity? (b) At what angle θ is the intensity equal to 3 percent of the maximum intensity? I'm not sure what the general formulas are. Hint for 1b -- the ratio of E/B is determined by the characteristic impedance of the medium that the EM wave is propagating through. And on question 2, you must have learned some things about dipole antenna radiation patterns, or they would not have asked you that question. HW Helper P: 2,322 Here are the general formulas: E=Bc Energy density=1/2epsilonE^2 Energy density=1/2*B^2/mu Dipole formulas: http://en.wikipedia.org/wiki/Dipole_antenna Related Discussions Special & General Relativity 26 Classical Physics 65 Advanced Physics Homework 4 General Physics 3 Introductory Physics Homework 11	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9664732217788696, "perplexity": 1843.8861307843717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535924131.19/warc/CC-MAIN-20140901014524-00234-ip-10-180-136-8.ec2.internal.warc.gz"}
https://zbmath.org/?q=an:0992.57024	× ## Detecting codimension one manifold factors with the disjoint homotopies property.(English)Zbl 0992.57024 A generalized $$n$$-manifold $$X$$ is a finite-dimensional, locally contractible metric space such that $$H_n(X,X \smallsetminus \{x\};\mathbb{Z}) \cong \mathbb{Z}$$ and otherwise $$H_i(X,X \smallsetminus\{x\}; \mathbb{Z})\cong 0$$; such a space is said to be resolvable if there exist an $$n$$-manifold $$M$$ and a cell-like mapping $$f$$ of $$M$$ onto $$X$$ $$(f$$ is cell-like provided for each $$x\in X$$ and neighborhood $$U$$ of $$f^{-1}(x)$$, the inclusion $$f^{-1}(x) \hookrightarrow U$$ is null-homotopic). An approximately 40 year old problem traceable to R. H. Bing asks whether, for every resolvable generalized $$n$$-manifold $$X$$, $$X\times \mathbb{R}$$ is a manifold. If so, then $$X$$ is said to be a codimension one manifold factor. It is known that in this setting $$X\times\mathbb{R}^2$$ always is an $$(n+2)$$-manifold. The reviewer proved [Pac. J. Math. 93, 277-298 (1981; Zbl 0415.57007)] that a generalized $$n$$-manifold $$X$$ $$(n>3)$$ is a codimension one manifold factor if each pair of maps $$f:I\to X$$, $$g:B^2\to X$$ can be approximated by maps $$F:I\to X$$, $$G:B^2\to X$$ with disjoint images. In what may be the first result on the topic since then, here the author introduces a Disjoint Homotopies Property and shows that all generalized $$n$$-manifolds $$X$$ $$(n>3)$$ having it are codimension one manifold factors; $$X$$ has the Disjoint Homotopies Property if any two homotopies $$f_t,g_t: I\to X$$ can be approximated by homotopies $$F_t,G_t:I\to X$$ such that $$F_t(I)\cap G_t(I) =\emptyset$$ for all $$t$$. The author introduces another property, the Plentiful 2-manifolds Property, which (for metric spaces $$X)$$ is characterized by the fact that each path in $$X$$ can be approximated by another path which lies in a 2-manifold $$N \subset X$$. Then in another key result the author establishes that all resolvable generalized $$n$$-manifolds having the Plentiful 2-manifolds Property are codimension one manifold factors. She concludes by constructing a new class of codimension one manifold factors, namely, the $$k$$-ghastly examples $$(2<k<n)$$, which contain no embedded $$k$$-cells but do contain embedded $$(k-1)$$-cells; these are constructed so as to have the Plentiful 2-manifolds Property. ### MSC: 57P05 Local properties of generalized manifolds 54B10 Product spaces in general topology 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) Zbl 0415.57007 Full Text: ### References: [1] Antoine, L., Sur l’homéomorphie de deux figures et de leurs voisinages, J. math. pures appl. (9), 86, 221-325, (1921) · JFM 48.0650.01 [2] F.D. Ancel, The locally flat approximation of cell-like embedding relations, Thesis, 1976-1977 [3] Bing, R.H., Upper semicontinuous decompositions of E3, Ann. of math. (2), 65, 363-374, (1957) · Zbl 0078.15201 [4] Bing, R.H., A decomposition of E3 into points and tame arcs such that the decomposition space is topologically different from E3, Ann. of math. (2), 65, 484-500, (1957) · Zbl 0079.38806 [5] Bing, R.H., The Cartesian product of a certain non-manifold and a line is E4, Ann. of math. (2), 70, 399-412, (1959) · Zbl 0089.39501 [6] Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S., Topology of homology manifolds, Ann. of math. (2), 143, 435-467, (1996) · Zbl 0867.57016 [7] Cannon, J.W., The recognition problem: what is a topological manifold?, Bull. amer. math. soc., 84, 832-866, (1978) · Zbl 0418.57005 [8] Cannon, J.W., Shrinking cell-like decomposition of manifolds. codimension three., Ann. of math. (2), 110, 83-112, (1979) · Zbl 0424.57007 [9] Cannon, J.W.; Bryant, J.L.; Lacher, R.C., The structure of generalized manifold having nonmanifold set of trivial dimension, (), 261-300 · Zbl 0476.57006 [10] Cannon, J.W.; Daverman, R.J., A totally wild flow, Indiana univ. math. J., 30, 371-387, (1981) · Zbl 0432.58018 [11] Cohen, M.M., Simplicial structures and transverse cellularity, Ann. of math. 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It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.913024365901947, "perplexity": 2488.039039051005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00191.warc.gz"}
https://physics.stackexchange.com/questions/428320/mass-moving-at-a-constant-velocity-vertically	# Mass moving at a constant velocity vertically Let's say I move a mass of 10kg up by 1m at a constant velocity. I would have done 10J of work on it, and gravity would have done -10J of work. However, wouldn't that imply that the net work done is 0J? How could the mass have increased its height? At first, I reasoned that this is because a greater than g acceleration was applied at the start to make the mass move up at a non-zero velocity, but if this is how it increases height, lifting the mass up to 2m would require the same amount of energy. Am I missing something? • "Am I missing something?", no you added the -10J work done by gravity. Why do you think this is? – Jasper Sep 12 '18 at 16:56 • Are we on a planet where g = $1 \space m/s^2$ in this scenario? – Aaron Stevens Sep 12 '18 at 20:37 The key is constant velocity By the relationship between total work done on an object and its change in kinetic energy. $$W_{tot}=\Delta K$$ Since the object is moving at a constant velocity, $\Delta K=0$, so we must have that $W_{tot}=0$, which is what you have expressed. So as you can see, a $0$ net work does not mean no movement. It just means no change in kinetic energy, which means no change in velocity. The confusion might come in with thinking about potential energy. The potential energy due to gravity is related to the work done by gravity: $$W_{grav}=-\Delta U_{grav}$$ So the potential energy has actually increased, and gravity does do negative work, but the net work is still $0$, so there is no change in velocity. At first, I reasoned that this is because a greater than g acceleration was applied at the start to make the mass move up at a non-zero velocity I think what you mean here is that you have to apply a force greater than gravity to get the object to start moving upwards to its eventual constant speed. This is correct. Based on the above discussion, we need a non-zero net work to change the speed of the object. This is achieved by lifting with a force larger than gravity. Then, once the object is moving, we decrease our applied force to be equal to the force of gravity. This is what allows for movement at a constant velocity (which you can argue is from $0$ net force or $0$ net work. These are the same thing in 1D motion). Once again, $0$ net work does not mean $0$ displacement, just like how $0$ net work does not mean $0$ velocity. We can even take this a step further and think about what happens when the object comes to rest again above our head. Then we know that the net work from start to finish is $0$, since we started and ended at rest. What gives? We did work lifting the object didn't we? We did! But gravity did the same amount of negative work the whole time. Another way to look at it is that we used energy to increase the potential energy of the object. But potential energy is not essential to understanding the work done in this system. Side note, I think you need to check your calculations on how much work is done by each force during the constant velocity movement you discuss in the question. In your question you didn’t state whether or not the mass is brought to rest at 1 meter, or if it continues past 1 meter with the same velocity. It makes a difference because if it still has velocity at 1 meter is has kinetic energy equal to $1/2mv^2$ at 1 meter in addition to the 100 J of potential energy (correcting for g = 10 and m = 10). If the external force remains then you will gain another 100 J of potential energy when you get to 2 meters. In the following, the mass will have constant velocity until just before it reaches 1 meter when it will decelerate to zero velocity at 1 meter. Recall that the gravitational force is a conservative force. This means if I start with a mass at rest at point 1, say with respect to the surface of the earth, and move the mass along any arbitrary path P up to point 2 and back to point 1, with any arbitrary velocity along the way, the net work done (me + gravity) will be zero. In the following scenario all motion is vertical, i.e., parallel to the force of gravity, since it should be obvious to all that no work is required (absent air friction) for horizontal motion. I start with moving the mass from point 1 to point 2. Clearly, to accomplish this, the mass has to move, i.e., it must attain some velocity along the way. To get things going I need to apply an external upward force, $F_{ext}$, slightly greater than $mg$, to give it small upward acceleration $a$. Let’s say I do this for a brief time $dt$ and therefore over a short distance $dh$. The mass thereby attains a small velocity $v=adt$ and a small increase in kinetic energy of $½ m(adt)^2$ and a small increase in the potential energy of the mass m of $(mg) dh$. By the work-energy principle: $$W = \Delta KE + \Delta PE$$ Since the initial KE and PE are zero, this becomes, for a differential time $dt$ and a differential increase in height of $dh$ $$dW = ½ m(adt)^2 + (mg) dh$$ We now immediately reduce our upward force so that it equals the downward gravitational force. The mass is now rising at constant velocity v and so there is no subsequent change in KE, however its potential energy keeps increasing. Re-applying the work-energy principle, realizing now that KE is constant and $\Delta KE=0$ $$dW = (mg)dh$$ We now ask ourselves, what is doing this work. Clearly, the only force that is responsible for raising the mass is the external upward force that I exert in order to perform work against gravity. But so far we are still left with the small kinetic energy. Therefore, prior to reaching point 2 I reduce my external upward force, $F_{ext}$, to slightly less than $mg$, to give it small acceleration downward. I do this for sufficient time to bring the mass to rest at point 2. Gravity now does a small amount of work downward resulting in a negative change in kinetic energy. The mass continues to rise in height during this period., Re-applying the work-energy principle for this differential amount of time $dt$ and increase in height $dh$: $$dW =-½ m (adt)^2 + (mg)dh$$ The end result in going from 1 to 2 is an increase in potential energy of $$\int_1^2 (mg)dh = (mg)h$$. In order to return from point 2 to point 1 I reverse the above procedure, starting with an upward force slightly less than the gravitational force for a downward acceleration to attain a downward constant velocity $v$ and end with an upward force slightly more than the gravitational force to bring the mass to rest at point 1. The end result is a change in potential energy of $-mgh$. In this case, negative work is done by gravity equal to the positive work done by the external force going from 1 to 2, for a net work of 0. Before closing, a different scenario can consist of moving the mass from 1 to 2 as above, giving it an increase in PE of $mgh$ and then releasing the mass for free fall. Gravity then converts all the potential energy to kinetic energy prior to impact. Ultimately, where does this final kinetic energy come from? It came from the work that was done by the external agent to give the mass its potential energy at point 2, and thus energy is conserved going from 1 to 2 and back to 1. Hope this helps.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9253684878349304, "perplexity": 165.4458562048929}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540569146.17/warc/CC-MAIN-20191213202639-20191213230639-00044.warc.gz"}

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