Students use creative tools in the Seesaw app or website to complete classroom activities like "Noun Group Snowman"
1. Write your noun group. 2. Underline your two adjectives (describing words) in green. 3. Underline your noun (naming word) in blue. 4. Draw your snowman to match you noun group. 5. Voice record your noun group. 5. Post it to our classroom.


All of our decorations are hand-made by our students and staff, to celebrate the spirit of the "Asher Way". The Asher Way is a simple philosophy that is the core of Asher College, and allows each of us to maximize our potential for personal growth and career success. Through evaluating our current reality and creating a vision of our best future, we use goal-setting and affirmations to improve our lives. Our ornaments represent the courses we study, the career paths we pursue, and the inspiration that helps us reach our goals as employer-ready graduates.
Asher College is an accredited and VA approved Career College, providing financial Aid for those who qualify. Asher College provides training in Information Technology, Medical and Business programs. For more than 15 years, we have been empowering motivated professionals to enhance their careers and improve their earning potential. We understand the needs of the students and offer the most flexible schedule combined with hands on learning and one on one instruction. When you graduate from Asher College, you have the skills that the employers are looking which means to get the job done and done well.
We help the community by providing FREE PC Repair to clients from outside. This is done in our PC Clinic which is operated by our students and staff and provides a valuable experience to our students in terms of customer service and technical skills.


A more recent article on peripartum depression is available.
Am Fam Physician. 2010 Oct 15;82(8):926-933.
Patient information: See related handout on postpartum depression, written by the authors of this article.
Postpartum major depression is a disorder that is often unrecognized and must be distinguished from "baby blues." Antenatal depressive symptoms, a history of major depressive disorder, or previous postpartum major depression significantly increase the risk of postpartum major depression. Screening with the Edinburgh Postnatal Depression Scale may be appropriate. Some women with postpartum major depression may experience suicidal ideation or obsessive thoughts of harming their infants, but they are reluctant to volunteer this information unless asked directly. Psychotherapy or selective serotonin reuptake inhibitors may be used to treat the condition. In patients with moderate to severe postpartum major depression, psychotherapy may be used as an adjunct to medication. No evidence suggests that one antidepressant is superior to others. Antidepressants vary in the amount secreted into breast milk. If left untreated, postpartum major depression can lead to poor mother-infant bonding, delays in infant growth and development, and an increased risk of anxiety or depressive symptoms in the infant later in life.
Screening all women for depression during pregnancy or postpartum should be strongly considered; postpartum screening can be performed at the postpartum visit or two-month well-child visit.
The Edinburgh Postnatal Depression Scale may be used to screen for postpartum major depression with acceptable sensitivity and specificity.
Women with postpartum major depression should be asked about suicidal ideation and, if necessary, referred for emergent psychiatric evaluation and possible hospitalization.
Women with suspected postpartum major depression should be asked about a personal or family history of bipolar disorder; a positive history warrants referral to a psychiatrist for further evaluation.
Interpersonal therapy and cognitive behavior therapy in individual or group settings are effective in treating mild to moderate postpartum major depression.
If a woman has previously responded well to a specific antidepressant medication, she should begin treatment for postpartum major depression with that medication unless there is evidence of potential harm to her infant.
The etiology of postpartum major depression remains unclear. Some women may be sensitive to hormonal changes during reproductive events, specifically menses, pregnancy, and menopause.11 The drop in hormone levels after delivery may play a role.12,13 An association between cortisol levels and depressive symptoms during pregnancy and postpartum has been reported.14 Major depression may also begin during pregnancy and continue into the postpartum period.
© 1987 The Royal College of Psychiatrists. The Edinbugh Postnatal Depression Scale may be photocopied by individual researchers or clinicians for their own use without seeking permission from the publishers. The scale must be copied in full and all copies must acknowledge the following source: Cox JL, Holden JM, Sagovsky R. Detection of postnatal depression. Development of the 10-item Edinburgh Postnatal Depression Scale. Br J Psychiatry. 1987;150:782–786. Written permission must be obatined from the Royal College of Psychiatrists for copying and distribution to others or for republication (in print, online or by any other medium).
Translations of the scale, and guidance as to its use, may be found in Cox JL, Holden J. Perinatal Mental Health: A Guide to the Edinburgh Postnatal Depression Scale. London: Gaskell; 2003. Additional information from reference 16.
Postpartum depression is the most common complication of childbearing.16 The 10-question Edinburgh Postnatal Depression Scale (EPDS) is a valuable and efficient way of identifying patients at risk for "perinatal" depression. The EPDS is easy to administer and has proven to be an effective screening tool.
Mothers who score above 13 are likely to be suffering from a depressive illness of varying severity. The EPDS score should not override clinical judgment. A careful clinical assessment should be carried out to confirm the diagnosis. The scale indicates how the mother has felt during the previous week. In doubtful cases it may be useful to repeat the tool after 2 weeks. The scale will not detect mothers with anxiety neuroses, phobias or personality disorders.
Women with postpartum depression need not feel alone. They may find useful information on the web sites of the National Women's Health Information Center (www.womenshealth.gov/) and from groups such as Postpartum Support International (www.postpartum.net/) and Depression after Delivery (www.depressionafterdelivery.com/).
QUESTIONS 1, 2, and 4 (without an *) Are scored 0, 1, 2 or 3 with top box scored as 0 and the bottom box scored as 3.
QUESTIONS 3, 5-10 (marked with an *) Are reverse scored, with the top box scored as a 3 and the bottom box scored as 0.
1. The mother is asked to check the response that comes closest to how she has been feeling in the previous 7 days.
2. All the items must be completed.
Screening can be performed at the four- to six-week postpartum visit or the two-month well-child visit.20,21 The Edinburgh Postnatal Depression Scale is completed by the patient and can be quickly scored by office staff. It is available in several languages and may be downloaded free from multiple sources, including the University of California, San Francisco, Fresno, Web site (http://www.fresno.ucsf.edu/pediatrics/downloads/edinburghscale.pdf).
note: According to the Diagnostic and Statistical Manual of Mental Disorders, 4th ed., text revision, a diagnosis of major depressive disorder requires five symptoms to be present for at least two weeks, causing difficulty in functioning. Either depressed mood or anhedonia must be present.
*—Symptoms may be confused with normal postpartum course.
Postpartum major depression is differentiated from "baby blues" by the severity and duration of symptoms (Table 2). Baby blues begins during the first two to three days after delivery and resolves within 10 days. Symptoms include brief crying spells, irritability, poor sleep, nervousness, and emotional reactivity. Suicidal ideation is not present. Although baby blues was previously considered benign, increasing evidence suggests that women with these symptoms are at risk of progression to postpartum major depression.24 The diagnosis of postpartum major depression should be strongly considered in women who score above 12 on the Edinburgh Postnatal Depression Scale, experience symptoms that cause moderate to severe social dysfunction, report any suicidal ideation, or experience symptoms for more than 10 days.
Many women with postpartum major depression have no psychiatric history and may be reluctant to volunteer symptoms or to seek help. It is important to discuss symptoms, such as obsessive thoughts and suicidal ideation, with these women. Up to 60 percent of women with postpartum major depression have obsessive thoughts focusing on aggression toward the infant.29 These thoughts are intrusive and similar to those in obsessive-compulsive disorder. They do not represent a desire to hurt the infant but over time can lead to avoidance of the infant in an effort to minimize the thoughts. The shame and guilt associated with these intrusive images or thoughts can reduce the likelihood of divulging this symptom. Physicians should ask about these symptoms as part of the diagnosis of postpartum major depression. Nonjudgmental phrases include: "Many women with postpartum major depression have thoughts or images of hurting their baby. Has this happened to you? What did you do in response to them?" By acknowledging that these symptoms are part of postpartum major depression, physicians can help women understand their experiences and seek treatment.
Because suicide is a leading cause of maternal death, physicians should ask women suspected of having postpartum major depression about suicidal ideation.30 Many physicians are familiar with asking about active suicidal ideation, which involves a plan to end one's life. However, women with mild to moderate postpartum major depression may have passive suicidal ideation, defined as a desire to die but no plan. One useful question is, "Have you wished you would go to sleep and not wake up?" A woman with active or passive suicidal ideation may cite her infant or family as a reason not to harm herself. However, as depression worsens, she may view herself as a bad mother and believe that her child would be better off without her. A woman who has active suicidal ideation or thoughts that her child or family would benefit from her death requires emergent psychiatric evaluation and possible hospitalization.
Because hypothyroidism may also cause depressive symptoms, physicians should measure thyroid-stimulating hormone levels in women with suspected postpartum major depression.16 About 8 percent of women develop postpartum autoimmune thyroiditis, which can mimic many symptoms of postpartum major depression. 33 Blood loss during delivery can lead to anemia and considerable fatigue, but does not cause depressed mood or anhedonia.
Light therapy has not been shown to be effective in patients with postpartum major depression.39 Acupuncture, yoga, and exercise have not been studied sufficiently.40 However, exercise, adequate exposure to morning light, and support from others are encouraged by many physicians as adjuncts to other treatments for women with postpartum major depression. A postpartum depression action plan is available at https://familydoctor.org/familydoctor/en/diseases-conditions/postpartum-depression.html.
Selective serotonin reuptake inhibitors have become the mainstay of treatment for moderate to severe postpartum major depression because of their favorable adverse effect profiles and relative safety in overdose compared with tricyclic antidepressants16 (Figure 2). Table 3 lists the most commonly used antidepressants, adverse effect profiles, and starting and target dosages.16 No evidence suggests that one antidepressant is superior to others in treating postpartum major depression.
note: There is no evidence to suggest that one antidepressant is superior to another in treating women with postpartum major depression who are not breastfeeding. The choice of medication should be driven primarily by the patient's history of response and tolerability.
Before prescribing an agent for postpartum major depression, physicians should consider the patient's experience with antidepressants. If the patient has previously responded well to a specific agent, that medication should be the first choice unless there is evidence of potential harm.16,44 Although breastfed infants are unlikely to experience adverse effects from antidepressant medications, infants should be monitored for symptoms, such as persistent irritability, decreased feeding, or poor weight gain. Maternal depression or problems within the mother-infant dyad can also be associated with these symptoms.45,46 Formula feeding should be considered in women with severe postpartum major depression that requires medication implicated in adverse effects for the infant. LactMed is an online, peer-reviewed resource that provides information on the safety of medications in mothers who breastfeed (http://toxnet.nlm.nih.gov/cgi-bin/sis/htmlgen?LACT).
Because postpartum women may have increased sensitivity to adverse effects of medications, a reasonable strategy is to start at low dosages for the first four days and then titrate up.3,16 The Edinburgh Postnatal Depression Scale can be used to monitor progress over time. After symptoms are in remission, treatment is typically continued for six to nine months of euthymia before tapering the medication. Tapering over two weeks, especially for paroxetine, extended-release venlafaxine (Effexor XR), and extended-release desvenlafaxine (Pristiq), can prevent the influenza-like symptoms of discontinuation syndrome.
CHRISTINE Y. MOUTIER, MD, is an associate professor in the Department of Psychiatry and assistant dean of student affairs at the University of California, San Diego, School of Medicine.
Address correspondence to Kathryn P. Hirst, MD, University of California, 402 Dickinson St., Multi Purpose Facility L-012, San Diego, CA 92103 (e-mail: khirst@ucsd.edu). Reprints are not available from the authors.
2. Cooper PJ, Murray L. Course and recurrence of postnatal depression. Evidence for the specificity of the diagnostic concept. Br J Psychiatry. 1995;166(2):191–195.
3. Wisner KL, Perel JM, Peindl KS, Hanusa BH, Piontek CM, Findling RL. Prevention of postpartum depression: a pilot randomized clinical trial. Am J Psychiatry. 2004;161(7):1290–1292.
4. Wisner KL, Perel JM, Peindl KS, Hanusa BH, Findling RL, Rapport D. Prevention of recurrent postpartum depression: a randomized clinical trial. J Clin Psychiatry. 2001;62(2):82–86.
5. Milgrom J, Gemmill AW, Bilszta JL, et al. Antenatal risk factors for postnatal depression: a large prospective study. J Affect Disord. 2008;108(1–2):147–157.
6. Robertson E, Grace S, Wallington T, Stewart DE. Antenatal risk factors for postpartum depression: a synthesis of recent literature. Gen Hosp Psychiatry. 2004;26(4):289–295.
7. Forty L, Jones L, Macgregor S, et al. Familiality of postpartum depression in unipolar disorder: results of a family study. Am J Psychiatry. 2006;163(9):1549–1553.
8. Kozhimannil KB, Pereira MA, Harlow BL. Association between diabetes and perinatal depression among low-income mothers. JAMA. 2009;301(8):842–847.
9. Choi Y, Bishai D, Minkovitz CS. Multiple births are a risk factor for postpartum maternal depressive symptoms. Pediatrics. 2009;123(4):1147–1154.
10. Johnstone S, Boyce PM, Hickey AR, Morris-Yatees AD, Harris MG. Obstetric risk factors for postnatal depression in urban and rural community samples. Aust N Z J Psychiatry. 2001;35(1):69–74.
11. Bloch M, Schmidt PJ, Danaceau M, Murphy J, Nieman L, Rubinow DR. Effects of gonadal steroids in women with a history of postpartum depression. Am J Psychiatry. 2000;157(6):924–930.
12. Green AD, Barr AM, Galea LA. Role of estradiol withdrawal in 'anhedonic' sucrose consumption: a model of postpartum depression. Physiol Behav. 2009;97(2):259–265.
13. Suda S, Segi-Nishida E, Newton SS, Duman RS. A postpartum model in rat: behavioral and gene expression changes induced by ovarian steroid deprivation. Biol Psychiatry. 2008;64(4):311–319.
14. Yim IS, Glynn LM, Dunkel-Schetter C, Hobel CJ, Chicz-DeMet A, Sandman CA. Risk of postpartum depressive symptoms with elevated corticotropin- releasing hormone in human pregnancy. Arch Gen Psychiatry. 2009;66(2):162–169.
15. American College of Obstetricians and Gynecologists. Committee on Obstetric Practice. Committee opinion no. 453: screening for depression during and after pregnancy. Obstet Gynecol. 2010;115(2 pt 1):394–395.
16. Wisner KL, Parry BL, Piontek CM. Clinical practice. Postpartum depression. N Engl J Med. 2002;347(3):194–199.
17. Cox JL, Holden JM, Sagovsky R. Detection of postnatal depression. Development of the 10-item Edinburgh Postnatal Depression Scale. Br J Psychiatry. 1987;150:782–786.
18. Eberhard-Gran M, Eskild A, Tambs K, Schei B, Opjordsmoen S. The Edinburgh Postnatal Depression Scale: validation in a Norwegian community sample. Nord J Psychiatry. 2001;55(2):113–117.
19. Adouard F, Glangeaud-Freudenthal NM, Golse B. Validation of the Edinburgh Postnatal Depression Scale (EPDS) in a sample of women with high-risk pregnancies in France. Arch Womens Ment Health. 2005;8(2):89–95.
20. Peindl KS, Wisner KL, Hanusa BH. Identifying depression in the first postpartum year: guidelines for office-based screening and referral. J Affect Disord. 2004;80(1):37–44.
21. Sheeder J, Kabir K, Stafford B. Screening for postpartum depression at well-child visits: is once enough during the first 6 months of life? Pediatrics. 2009;123(6):e982–e988.
22. American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders. 4th ed., text revision. Washington, DC: American Psychiatric Association; 2000.
23. O'Hara MW, Swain AM. Rates and risk of postpartum depression: a meta-analysis. Int Rev Psychiatry. 1996;8(1):37–54.
24. Henshaw C, Foreman D, Cox J. Postnatal blues: a risk factor for postnatal depression. J Psychosom Obstet Gynaecol. 2004;25(3–4):267–272.
25. Kammerer M, Marks MN, Pinard C, et al. Symptoms associated with the DSM IV diagnosis of depression in pregnancy and post partum. Arch Womens Ment Health. 2009;12(3):135–141.
26. Bernstein IH, Rush AJ, Yonkers K, et al. Symptom features of postpartum depression: are they distinct? Depress Anxiety. 2008;25(1):20–26.
27. Goyal D, Gay C, Lee K. Fragmented maternal sleep is more strongly correlated with depressive symptoms than infant temperament at three months postpartum. Arch Womens Ment Health. 2009;12(4):229–237.
28. Swain AM, O'Hara MW, Starr KR, Gorman LL. A prospective study of sleep, mood, and cognitive function in postpartum and nonpostpartum women. Obstet Gynecol. 1997;90(3):381–386.
29. Wisner KL, Peindl KS, Gigliotti T, Hanusa BH. Obsessions and compulsions in women with postpartum depression. J Clin Psychiatry. 1999;60(3):176–180.
30. Mander R, Smith GD. Saving mothers' lives (formerly why mothers die): reviewing maternal deaths to make motherhood safer 2003–2005. Midwifery. 2008;24(1):8–12.
31. Sharma V, Khan M, Corpse C, Sharma P. Missed bipolarity and psychiatric comorbidity in women with postpartum depression. Bipolar Disord. 2008;10(6):742–747.
32. Munk-Olsen T, Laursen TM, Mendelson T, Pedersen CB, Mors O, Mortensen PB. Risks and predictors of readmission for a mental disorder during the postpartum period. Arch Gen Psychiatry. 2009;66(2):189–195.
33. Nicholson WK, Robinson KA, Smallridge RC, Ladenson PW, Powe NR. Prevalence of postpartum thyroid dysfunction: a quantitative review. Thyroid. 2006;16(6):573–582.
34. Dennis CL, Hodnett E. Psychosocial and psychological interventions for treating postpartum depression. Cochrane Database Syst Rev. 2007; (4):CD006116.
35. Grote NK, Swartz HA, Geibel SL, Zuckoff A, Houck PR, Frank E. A randomized controlled trial of culturally relevant, brief interpersonal psychotherapy for perinatal depression. Psychiatr Serv. 2009;60(3):313–321.
36. Chabrol H, Teissedre F, Saint-Jean M, Teisseyre N, Rogé B, Mullet E. Prevention and treatment of post-partum depression: a controlled randomized study on women at risk. Psychol Med. 2002;32(6):1039–1047.
37. Appleby L, Warner R, Whitton A, Faragher B. A controlled study of fluoxetine and cognitive-behavioural counselling in the treatment of postnatal depression. BMJ. 1997;314(7085):932–936.
38. Crockett K, Zlotnick C, Davis M, Payne N, Washington R. A depression preventive intervention for rural low-income African-American pregnant women at risk for postpartum depression. Arch Womens Ment Health. 2008;11(5–6):319–325.
39. Corral M, Wardrop AA, Zhang H, Grewal AK, Patton S. Morning light therapy for postpartum depression. Arch Womens Ment Health. 2007;10(5):221–224.
40. Dennis CL, Allen K. Interventions (other than pharmacological, psychosocial or psychological) for treating antenatal depression. Cochrane Database Syst Rev. 2008;(4):CD006795.
41. Weissman AM, Levy BT, Hartz AJ, et al. Pooled analysis of antidepressant levels in lactating mothers, breast milk, and nursing infants. Am J Psychiatry. 2004;161(6):1066–1078.
42. Sit DK, Wisner KL. Identification of postpartum depression. Clin Obstet Gynecol. 2009;52(3):456–468.
43. ACOG Committee on Practice Bulletins–Obstetrics. ACOG Practice Bulletin: clinical management guidelines for obstetrician-gynecologists number 92, April 2008 (replaces practice bulletin number 87, November 2007). Use of psychiatric medications during pregnancy and lactation. Obstet Gynecol. 2008;111(4):1001–1020.
44. Payne JL. Antidepressant use in the postpartum period: practical considerations. Am J Psychiatry. 2007;164(9):1329–1332.
45. O'Brien LM, Heycock EG, Hanna M, Jones PW, Cox JL. Postnatal depression and faltering growth: a community study. Pediatrics. 2004;113(5):1242–1247.
46. McGrath JM, Records K, Rice M. Maternal depression and infant temperament characteristics. Infant Behav Dev. 2008;31(1):71–80.
47. Gregoire AJ, Kumar R, Everitt B, Henderson AF, Studd JW. Transdermal oestrogen for treatment of severe postnatal depression. Lancet. 1996;347(9006):930–933.
48. Ahokas A, Kaukoranta J, Wahlbeck K, Aito M. Estrogen deficiency in severe postpartum depression: successful treatment with sublingual physiologic 17beta-estradiol: a preliminary study. J Clin Psychiatry. 2001;62(5):332–336.
49. Sichel DA, Cohen LS, Robertson LM, Ruttenberg A, Rosenbaum JF. Prophylactic estrogen in recurrent postpartum affective disorder. Biol Psychiatry. 1995;38(12):814–818.
50. Moses-Kolko EL, Berga SL, Kalro B, Sit DK, Wisner KL. Transdermal estradiol for postpartum depression: a promising treatment option. Clin Obstet Gynecol. 2009;52(3):516–529.
51. Armitage R, Flynn H, Hoffmann R, Vazquez D, Lopez J, Marcus S. Early developmental changes in sleep in infants: the impact of maternal depression. Sleep. 2009;32(5):693–696.
52. Brand SR, Brennan PA. Impact of antenatal and postpartum maternal mental illness: how are the children? Clin Obstet Gynecol. 2009;52(3):441–455.


Located at Suntec City Mall #03-372 (Between Towers 3 and 4), it operates from 10am to 10pm daily (Last Admission at 9pm). Ticket Prices are reasonably priced at $25 (Adults), $20 (Children 3-12 yrs).
We went on a weekday evening, thus there wasn't much of a queue. 2 hours crazy queue myth now debunked! *phew* Do take note that as there is no washroom inside, it would be advisable to empty your bladder (conveniently located right beside Alive Museum) before heading in.
P.S Just a heads up, redeem a free Burger Kind meal* by "Liking" Alive Museum's Facebook and purchasing a full-priced ticket.
Next Post Summer Beauty with By Terry, Annick Goutal & GlamGlow!


text:
Q: Matplotlib unable to rotate xticks I am new to creating graphs in python.
I have a problem with my x axis "ticks" being displayed vertically:

here is my code:
Counts = {"Hate":hate, "Non-hate":noHate}
graph = pd.Series(Counts)

plt.rcParams['figure.figsize'] = (6,6)
plt.rcParams['font.size'] = (14)

plt.xticks(rotation=20)

graph.plot.bar(yticks=[hate,noHate],color=['tab:red', 'tab:green'])

As you can see, I have tried using plt.xticks(rotation) but I was unsuccessful.
I used the number 20 just for testing, I would like to display the labels horizontally. Any ideas?
Note: I also tried plt.xticks(rotation='vertical') and also horizontal for the sake of it.

A: graph.plot.bar has a parameter "rot" for this purpose, see:
https://pandas.pydata.org/pandas-docs/version/0.15.2/generated/pandas.DataFrame.plot.html
import pandas as pd
import matplotlib as plt

Counts = {"Hate":hate, "Non-hate":noHate}
graph = pd.Series(Counts)

plt.rcParams['figure.figsize'] = (6,6)
plt.rcParams['font.size'] = (14)

graph.plot.bar(yticks=[hate,noHate],color=['tab:red', 'tab:green'], rot=20)

redpajama_set_name: RedPajamaStackExchange


text:
% FILLEDGEGAPS  Fills small gaps in a binary edge map image
%
% Usage: bw2 = filledgegaps(bw, gapsize)
%
% Arguments:    bw - Binary edge image
%          gapsize - The edge gap size that you wish to be able to fill.
%                    Use the smallest value you can. (Odd values work best).
%
% Returns:     bw2 - The binary edge image with gaps filled.
%
%
% Strategy: A binary circular blob of radius = gapsize/2 is placed at the end of
% every edge segment.  If the ends of two edge segments are in close proximity
% the circular blobs will overlap.  The image is then thinned.  Where circular
% blobs at end points overlap the thinning process will leave behind a line of
% pixels linking the two end points.  Where an end point is isolated the
% thinning process will erode the circular blob away so that the original edge
% segment is restored.
%
% Use the smallest gapsize value you can.  With large values all sorts of
% unwelcome linking can occur.
%
% The circular blobs are generated using the function CIRCULARSTRUCT which,
% unlike MATLAB's STREL, will accept real valued radius values.  Note that I
% suggest that you use an odd value for 'gapsize'.  This results in a radius
% value of the form x.5 being passed to CIRCULARSTRUCT which results in discrete
% approximation to a circle that seems to respond to thinning in a 'good'
% way. With integer radius values CIRCULARSTRUCT can produce circles that
% result in minor artifacts being generated by the thinning process.
%
% See also: FINDENDSJUNCTIONS, FINDISOLATEDPIXELS, CIRCULARSTRUCT, EDGELINK

% Copyright (c) 2013 Peter Kovesi
% Centre for Exploration Targeting
% The University of Western Australia
% peter.kovesi at uwa edu au
%
% Permission is hereby granted, free of charge, to any person obtaining a copy
% of this software and associated documentation files (the "Software"), to deal
% in the Software without restriction, subject to the following conditions:
%
% The above copyright notice and this permission notice shall be included in
% all copies or substantial portions of the Software.
%
% The Software is provided "as is", without warranty of any kind.

% PK May 2013

function bw = filledgegaps(bw, gapsize)

    [rows, cols] = size(bw);

    % Generate a binary circle with radius gapsize/2 (but not less than 1)
    blob = circularstruct(max(gapsize/2, 1));

    rad = (size(blob,1)-1)/2;  % Radius of resulting blob matrix. Note
                               % circularstruct returns an odd sized matrix

    % Get coordinates of end points and of isolated pixels
    [~, ~, re, ce] = findendsjunctions(bw);
    [ri, ci] = findisolatedpixels(bw);

    re = [re;ri];
    ce = [ce;ci];

    % Place a circular blob at every endpoint and isolated pixel
    for n = 1:length(re)

        if (re(n) > rad) && (re(n) < rows-rad) && ...
                (ce(n) > rad) && (ce(n) < cols-rad)

            bw(re(n)-rad:re(n)+rad, ce(n)-rad:ce(n)+rad) = ...
                bw(re(n)-rad:re(n)+rad, ce(n)-rad:ce(n)+rad) | blob;
        end
    end

    bw = bwmorph(bw, 'thin', inf);  % Finally thin

    % At this point, while we may have joined endpoints that were close together
    % we typically have also generated a number of small loops where there were
    % more than one endpoint close to an edge.  To address this we identfy the
    % loops by finding 4-connected blobs in the inverted image.  Blobs that are
    % less than or equal to the size of the blobs we used to link edges are
    % filled in, the image reinverted and then rethinned.

    L = bwlabel(~bw,4);
    stats = regionprops(L, 'Area');

    % Get blobs with areas <= pi* (gapsize/2)^2
    ar = cat(1,stats.Area);
    ind = find(ar <= pi*(gapsize/2)^2);

    % Fill these blobs in image bw
    for n = ind'
        bw(L==n) = 1;
    end

    bw = bwmorph(bw, 'thin', inf);  % thin again

redpajama_set_name: RedPajamaGithub


text:
Q: How to run 2 instances of JBoss EPP (Enterprise Portal Platform) on the same machine Is it possible to run 2 instances of JBoss EPP (Enterprise Portal Platform) on the same machine on different ports? JBoss Enterprise Portal Platform version is 5.2.2, which is based on JBoss AS 5.2.0
I'm learning portlets and I need to establish communication via WSRP between 2 portlets running on 2 different JBoss EPP instances.
As a last resort I may end up installing the second JBoss EPP instance on a virtual machine.
Any hints would be highly appreciated.
Update:
There are 2 options:

*

*modify bindings-jboss-beans.xml and set
<parameter>${jboss.service.binding.set:ports-01}</parameter>

*modify jboss startup script (run.conf, run.bat.conf): add
-Djboss.service.binding.set=ports-01 to $JAVA_OPTS

A: Yes you can run 2 instances of JBoss EPP on the same machine, by using differents ports (for example with "run -Djboss.service.binding.set=ports-01" on the second instance).
Did you encounter issues when doing it ?


Most people believe that fish are a rich source of DHA, when in fact fish get DHA from the algae in their foodchain. life'sDHA goes straight to the source, producing DHA from the same microalgae sources fish get it from. Grown in a controlled environment, life'sDHA is a vegetarian and sustainable source of DHA naturally free of ocean-borne contaminants.
life'sDHA's manufacturing plants specialize in the production and refinement of DHA from algae. The process begins with the initial algae cell culture and results in a highly purified DHA oil. The microalgae are grown in fermentors that range in size from 80,000 to 260,000 liters. When grown, the microalgae is then harvested and processed to extract the clear, amber-colored oil rich in DHA.
Recognizing the importance of maternal DHA intake, a workshop sponsored by the National Institutes of Health and International Society for the Study of Fatty Acids and Lipids (NIH/ISSFAL) recommended an intake of 300mg/day of DHA for pregnant and nursing women.


text:
Mark Martin: Latest NASCAR Hall of Fame class features close connections
May 28, 2016 Reid Spencer, NASCAR Wire Service Main Page, NASCAR News Top Story, Top Stories 0
NASCAR Hall of Fame inductee Mark Martin speaks with the media prior during qualifying for the NASCAR XFINITY Series Hisense 300 at Charlotte Motor Speedway on May 28, 2016 in Charlotte, North Carolina. (Photo by Blaine Ohigashi/Getty Images)
CONCORD,N.C. – For Mark Martin, election to the NASCAR Hall of Fame was an unexpected honor, and it's just coincidence that the 40-time NASCAR Sprint Cup Series race-winner has such strong simpatico with his fellow 2017 class members.
"There are so many people in the NASCAR Hall of Fame that are my heroes, that it's pretty incredible to think that I'm going to join them in that Hall," Martin said on Saturday at Charlotte Motor Speedway, where he'll drive the pace car before Sunday's Coca-Cola 600 (6 p.m. ET on FOX).
"It's something that I didn't expect to happen in my lifetime, so it's happened sooner than I would have ever dreamed, if it would happen at all."
Martin is part of a Hall of Fame class that includes Benny Parsons, against whom Martin raced, and Rick Hendrick, who hired Martin to drive his No. 5 Chevrolet. In fact, Martin finished second in the standings – for the fifth time in his career – driving for Hendrick in 2009.
With newly elected Hall of Fame member Richard Childress, Martin has considerable empathy, because Childress, like Martin, had a rough time when trying to break into the sport as a driver. Where Martin ultimately fashioned a Hall of Fame career behind the wheel, Childress made his mark as a car owner.
"This is a real special class," Martin said. "Benny Parsons had a hand in my career, as a person, as a competitor, as a friend – and as a friend to the sport – as well as Mr. Hendrick, the incredible person that he is, the character that he has, the integrity. He had a hand in my getting into the Hall of Fame.
"And, of course, Richard Childress was a driver that was struggling and started out with a $20 car. When I first came to NASCAR, he was driving what I would consider not a really nice car, to put it lightly. And then to see him achieve what he's achieved and be in there as well… it's a real special class.
"Every class is really special but this one has quite a bit of personal connection to it."
SHORT STROKES
Martin Truex Jr. looks like a good bet to win Sunday's Coca-Cola 600, if he can overcome the rotten luck that has plagued him recently. Truex had the fastest lap in Saturday's final practice (186.387 mph) and also posted the best consecutive 10-lap average speed (182.923 mph)…
Dale Earnhardt Jr., suffered a leak in the power steering box of his No. 88 Chevrolet in Saturday's first practice session, the result of a bad seal. The car's handling was an issue, too, but the balance improved during Happy Hour, with Earnhardt 21st fastest (single lap) but sixth best in 10-lap average…
Lest fans forget about Tony Stewart's farewell tour in his final year as a NASCAR Sprint Cup Series driver, there's a banner on the backstretch at Charlotte Motor Speedway that's roughly twice as long as the 200-foot big screen that overlooks the infield. The banner reads "Thank You Tony — Smoke 14" in gigantic lettering.
Written by: Reid Spencer, NASCAR Wire Service, May 28, 2016
XFINITY Series Hisense 4K TV 300
Fresh tires propel Hamlin to NASCAR XFINITY win in Charlotte

redpajama_set_name: RedPajamaCommonCrawl


Cartesian parametrization: (involute of the circle with centre O and radius a, with a contact at (a, 0)).
The definition of this curve is given by its name (see involute)!
The involute of a circle is the curve for which all the normals are tangent to a fixed circle.
More practically, it is the curve traced by a hand unwinding a wire reel held in the other hand.
Conversely, if an involute of a circle turns around the centre of the generating circle, every tangent to the circle remains always normal to the involute.
For large values of t, the Archimedean spiral with equation is the asymptotic curve to the involute of a circle. It also is its pedal with respect to O.
This last property has the following consequence: if an involute of a circle is turned uniformly around its centre, then the tangents to the involute having a given direction move in a uniform translation motion: it is the reason why the profile of cams is an involute of a circle.
The involute of a circle also has a kinematic definition, due to the formulae : it is the curve for which, when it is travelled along in a uniformly varying movement, the rotation speed of the tangent is constant (hence its name anti-clothoid, because the clothoid, is the curve for which, when it is travelled along in a uniform movement, the rotation speed of the tangent is linear).
A person P is walking on a stage turning at constant speed around a point O; the movement of P on the stage is linear and uniform with speed v and does not pass by O. If v = k w /a where a is the distance between the line along which P travels and O, the trajectory of P is given by . Therefore, it is an involute of a circle when k = 1.
Conversely, the trace of a marble rolling freely on the stage (thus with a linear uniform motion in the fixed frame) is such a curve.
Case k = 1: involute of a circle. Case k = -1.
Like any family of involutes of the same curve, the involutes of a circle are parallel to one another (by the way, the involute of a circle is the only curve isometric to its parallel curves).
More precisely, the image by a rotation by an angle of the involute gives a parallel curve at distance .
In the special case , we get the same curve; the involute of a circle is therefore an auto-parallel curve; its coils are at distance from one another.
This implies that if we translate an involute of a circle by perpendicularly to its symmetry axis, we draw the line joining the two cusps, and we spin both the involutes at the same speed around their centres, they will remain in contact, perpendicularly to that line.
This was applied to design gears (composed here of three portions of involutes of circles each) called paradoxical gears, because they turn in the same way.
Make sure to follow the contact points on the black lines.
Let us now apply to the translated involute a symmetry about the common normal; both the involutes remain in contact, but turn in opposite ways.
Remark: the curves slip on one another at the contact point (it is not on the line joining the two centres of rotation) but this is negligible enough so that it does not cause overheating problems in practice. Therefore, these are not mating profiles.
The same gears as above are placed this time on both sides of the line and thus turn in the opposite way.
Another special case of parallel curve: the symmetrical involute about the centre of the circle is parallel to the initial curve, at distance (it is, in a way, the central curve of the initial involute). Therefore, a circle with radius centred on an involute of a circle envelopes the symmetrical involute.
By duality from the previous property, we get the following one: consider an involute of a circle with radius a and centre O; if this involute is translated (in red below) in such a way that its centre describes a circle with centre O and radius , then the envelope of the family of curves obtained is the symmetrical involute of a circle with respect to O and the initial involute. And there are an infinite number of tangency points (2 by coil).
Advice: follow with your eyes a tangency point. This property, discovered by Léon Creux (cf. patent in 1905 and article in 1911) is used in the fabrication of gas compressors called spiral compressors, or scroll compressors.
The family of moving spirals is defined by and the characteristic points are given by .
Animation showing the use of involutes of circles for classic gears. The involutes stay in contact on a tangent common to both the circles.
The section of the developable helicoid by a plane perpendicular to the axis is an involute of a circle.
The projection of an helix of the paraboloid of revolution is an involute of a circle.
The involute of a circle is a sectrix and a quadratrix.
Its inverse with respect to its centre is the tractrix spiral, and its negative pedal is the inverse caustic of a circle.
Its involutes are the curves for which the radius of curvature is equal to the distance to O, plus a constant, one of them being the Norwich spiral.
Gears in involutes of circles made by Vincent Viton.


1) Give. We know the old adage that "it is better to give than to receive." In a test of this, Dunn, Aknin and Norton (2008) assessed the well-being of people who were given money. Those who were instructed to spend the money on others were happier than those who spent it on themselves. So giving money to a charity or treating a friend at your local coffeehouse may increase your happiness more than spending money on yourself.
2) Avoid Debt. Debt can be corrosive to your happiness. Avoid it.
3) Buy Experiences, not Things. Suppose that you have $3,000 cash in your pocket - then you are probably a drug dealer. But seriously, what should you spend that money on? People who spent money on things were just as happy as those who spent money on experiences – but only at first. Within a week or two, the happiness boost from spending money on things faded. The happiness boost that accompanied spending money on experiences lasted longer.
4) Share with Others. Research strongly suggests that our personal relationships are critical to happiness. In fact, it is difficult to find someone who is happy who does not have meaningful interpersonal relationships. Therefore, share your holiday experiences with others.
5) Take Pictures. Taking photographs can increase and prolong our happiness because photos allow us to relive and savor our good times.
6) Go out in Nature. The holidays can be stressful. For example, they are often associated with disruptions to our normal schedules, and increased demands on our finances and time. We recover faster from stressful events, both physically and mentally, when we are exposed to nature. Take a walk outside in a park or forest. Even viewing beautiful nature scenes on your computer may help.
7) Save the Best for Last. Research shows that we remember events more positively when the most positive experiences occur at the end. For example, when the highlight of a holiday or vacation occurs at the very end, we remember the overall experience better than when the highlight occurs at the beginning or in the middle.
8) Emphasize Frequency, not Intensity. Though we often think that it is the intensity of the good events that matters, research shows that it is really the frequency that counts. Aim for more frequent small joys that you can savor (for example, sitting in front of a fire or having a delicious hot chocolate) rather than trying to achieve a few big events.
9) Exercise. Research shows that we underestimate the happiness boost we get from exercise. A brisk walk or a bit of cross-country skiing may increase your happiness more than you anticipate.
10) Watch Your Favorite Holiday Movie. When we watch a movie we have seen many times, the cognitive demands are quite low because we don't have to pay much attention to it. This contributes to our well-being. My two holiday favorites are "It's a Wonderful Life" and "Love Actually."
Research shows that we remember events more positively when the most positive experiences occur at the end. Obat Pelangsing For example, when the highlight of a holiday or vacation occurs at the very end, we remember the overall experience better than when the highlight occurs at the beginning or in the middle.
Children do (and don't) make you happy, depending on where you live.
Research shows that how we choose to spend income can increase our happiness.
Does a Happy Meal Really Make Us Happy?


Innomed Stulberg Hip Positioner 4150-00, full set, and Anterior Trunk Support 4110. Includes three clark sockets, and storage tray.
This set is clean, pads are in great condition, and it is ready to be used. I've included the clark sockets so that the set is complete and ready to be put on a surgical table. Very easy to set up.
Missing a single thumb screw on part of the Anterior Trunk Support, visible in photos. While it would be nice to add one if I can find it (I am searching for one with the correct length and thread size) it is still able to be fastened down in this condition, as there is still one screw where there should be two. I should be able to have the screw before the set is sold, without much trouble, although it may not match the others visually. I can guarantee that it is still a fully functional set.
This is a really great lateral setup.


text:
<?php

use Illuminate\Support\Facades\Schema;
use Illuminate\Database\Schema\Blueprint;
use Illuminate\Database\Migrations\Migration;

class CreateFeedbackTable extends Migration
{
    /**
     * Run the migrations.
     *
     * @return void
     */
    public function up()
    {
        Schema::create('feedback', function (Blueprint $table) {
            $table->increments('id');
            $table->text('content');
            $table->unsignedInteger('uid');
            $table->unsignedTinyInteger('isSolve')->default(0);
            $table->string('ctime', 50);
            $table->string('stime', 50)->default(0);
        });
    }

    /**
     * Reverse the migrations.
     *
     * @return void
     */
    public function down()
    {
        Schema::dropIfExists('feedback');
    }
}


text:
package org.tools4j.mmap.region.impl;

import java.io.Closeable;
import java.io.File;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.io.RandomAccessFile;
import java.lang.reflect.Method;
import java.nio.channels.FileChannel;
import java.util.Objects;

import sun.nio.ch.FileChannelImpl;

public final class MappedFile implements Closeable {
    public static long REGION_SIZE_GRANULARITY = regionSizeGranularity();

    public enum Mode {
        READ_ONLY(OS.ifWindows("rw", "r"),
                  OS.ifWindows(FileChannel.MapMode.READ_WRITE, FileChannel.MapMode.READ_ONLY)),
        READ_WRITE("rw", FileChannel.MapMode.READ_WRITE),
        /** Delete io contents on open*/
        READ_WRITE_CLEAR("rw", FileChannel.MapMode.READ_WRITE);

        private final String rasMode;
        private final FileChannel.MapMode mapMode;
        Mode(final String rasMode, final FileChannel.MapMode mapMode) {
            this.rasMode = Objects.requireNonNull(rasMode);
            this.mapMode = Objects.requireNonNull(mapMode);
        }

        public String getRandomAccessMode() {
            return rasMode;
        }
        public FileChannel.MapMode getMapMode() {
            return mapMode;
        }
    }

    private final RandomAccessFile file;
    private final FileChannel fileChannel;
    private final Mode mode;
    private final long regionSize;

    public MappedFile(final String fileName, final Mode mode, final long regionSize) throws IOException {
        this(new File(fileName), mode, regionSize);
    }

    public MappedFile(final String fileName, final Mode mode, final long regionSize, final FileInitialiser fileInitialiser) throws IOException {
        this(new File(fileName), mode, regionSize, fileInitialiser);
    }

    public MappedFile(final File file, final Mode mode, final long regionSize) throws IOException {
        this(file, mode, regionSize, (c,m) -> {});
    }

    public MappedFile(final File file, final Mode mode, final long regionSize, final FileInitialiser fileInitialiser) throws IOException {
        if (regionSize <= 0 || (regionSize % REGION_SIZE_GRANULARITY) != 0) {
            throw new IllegalArgumentException("Region size must be positive and a multiple of " + REGION_SIZE_GRANULARITY + " but was " + regionSize);
        }
        if (mode == Mode.READ_WRITE_CLEAR && file.exists()) {
            file.delete();
        }
        if (!file.exists()) {
            if (mode == Mode.READ_ONLY) {
                throw new FileNotFoundException(file.getAbsolutePath());
            }
            file.createNewFile();
        }
        final RandomAccessFile raf = new RandomAccessFile(file, mode.getRandomAccessMode());
        this.file = Objects.requireNonNull(raf);
        this.mode = Objects.requireNonNull(mode);
        this.regionSize = regionSize;
        this.fileChannel = raf.getChannel();
        fileInitialiser.init(file.getName(), fileChannel);
    }

    public Mode getMode() {
        return mode;
    }

    public long getRegionSize() {
        return regionSize;
    }

    public long getFileLength() {
        try {
            return file.length();
        } catch (final IOException e) {
            throw new RuntimeException(e);
        }
    }

    public void setFileLength(final long length) {
        try {
            file.setLength(length);
        } catch (final IOException e) {
            throw new RuntimeException("could not set io length to " + length, e);
        }
    }

    public FileChannel getFileChannel() {
        return fileChannel;
    }

    public void close() {
        try {
            file.close();
        } catch (IOException e) {
            throw new RuntimeException(e);
        }
    }

    private static long regionSizeGranularity() {
        try {
            final Method method = FileChannelImpl.class.getDeclaredMethod("initIDs");
            method.setAccessible(true);
            final long result = (long)method.invoke(null);
            method.setAccessible(false);
            return result;
        } catch (final Exception e) {
            throw new RuntimeException(e);
        }
    }
}


When I Was Young is controlled by using the keyboard. Use the arrow keys or WASD keys to move. The P key pauses the game and the R key restarts the game.
Getting old is not fun. Your eyesight diminishes, you get aches and pains, and you cannot move as fast as you could when you were younger. Don't you wish that you could turn back the clock and relive your younger days? You can in When I Was Young. When I Was Young is a Flash platform game featuring colorful cartoon graphics, simple controls, and twenty achievements.
The objective of When I Was Young is to find three artifacts that will help you to regain your youth. These artifacts are a pile of mysterious sand, an hourglass, and a grandfather clock. When you have found all of these artifacts, the game will end. This platform game is played straight through from start to finish. There are no levels, only checkpoints that you will respawn at if you die. You will have to start fresh (which means old and decrepit in this adventure game) at the beginning each time that you play since there is no save system.
Although you have unlimited time and lives in When I Was Young, you should strive to complete the game as quickly as possible and with as few deaths as possible. Completing the game the first time will earn you a speed bonus. You can use this speed bonus to attempt to improve your time on your next playthrough. Unfortunately, nothing is saved in this platform game, so the bonus will not apply if you end your session by exiting the browser, navigating to a different page, or refreshing. This is a shame since When I Was Young would have immense replay value if achievements, time records, and speed bonuses were retained between sessions.
When I Was Young is a fun platform game that can be completed in less then five minutes, so you won't have to worry about time ticking away from you. Turn back the hands of time and relive youth in this amusing adventure game!


While being trapped in a wheelchair, temporarily or permanently, is not a stroll in the park, items don't need to end for the person in question. It's still possible to live a relatively normal life, and we must admit that a wheelchair provides you a part of your independence back. It could have been worse when there weren't wheelchairs in any way. But lucky for some folks, they exist, and they have the capacity of creating your life more bearable. Moreover, as technology progressed, wheelchairs are becoming better and better in providing freedom and support, so let's take a minute and be thankful for this.
It's money out the window — paying rent for a seat in St. Cloud, MN may not be much, but when the numbers accumulate, you are going to realize that purchasing one is a good deal better. Wheelchair rental companies will charge you an quantity of money at the commencement of your contract, and you will have to pay a commission for each and every day or a week of use, based on the company. Delivery and pickup will be charged extra, so remember about that. The complete amount will depend on how far you use the seat. As a result, at the end of your contract, then you are likely to end up with a huge bill which will make you regret that you didn't purchase a wheelchair in the first location. And if you purchase a wheelchair and you also just take great care of this, you can sell it back at a affordable price. That's if you don't need it anymore. If you are permanently disabled, wheelchair rental shouldn't even be an alternative.
You may not get exactly what you need — there is more than one kind of wheelchairs, and you may require a specific model. Let us say that you need one with foot rests, however, the ones which are offered for lease won't provide exactly what you require. Don't settle and proceed with what you'll find. It is advisable for those who listen to your doctor and purchase exactly what you want. Settling can cause additional inconvenience, and the aim is to get out of that chair whenever possible. If eliminating the chair isn't a potential, then you will need to devote your days in a seat that offers maximum comfort and stability. Otherwise, you would not buy a wheelchair at the first location.
You won't get a new wheelchair — wheelchair rental businesses supply chairs for everybody, so if anybody breaks you, it will be 'fixed' and put back in stock. However, some harms can't be undone, and you might wake up having a damaged chair. It may not be serious, however a bent wheel can be extremely inconvenient. You do not want for this, do you? A new seat comes with its integrity intact, and that's what you ought to need for. In the end, using something that someone else broke shouldn't be an alternative for you.
These are just a few reasons why I believe that wheelchair leasing is poor to buy, but definitely if you think about it, you can come back with more.
anybody who's searching for a wheelchair in St. Cloud, MN is concerned about the cost. The wheelchair cost is something that could determine whether you will buy a specific item or not. One thing is for certain, however. Prices can change from 80$ (the simple ones) to 12.000$ (the ones who could raise you up.) Everyone will decide based on funding, but quality is just as important as everybody else.
Most individuals, not just the ones at St. Cloud, MN, do not like to spend considerable quantities of money to get a wheelchair. Yes, there are those who is able to purchase an extremely costly wheelchair, but this is past the point. Nevertheless, it is possible to discover some inexpensive products that are of top quality. And by cheap I suggest that a few hundreds of dollars. On the other hand, a wheelchair above 1000$ could have incredible attributes like elevating footrests and whatnot. At the cost, even electrical wheelchairs are an option. The major idea is not to think just of wheelchair price, but also your comfort and mobility too. Bear in mind that a wheelchair is meant to give back your liberty. It's not the time to go in market mode.
Shed weight is much more significant than you believe. It will impact your freedom considerably. The purpose of a wheelchair would be to move about with ease. In case you opt for a thick wheelchair, that isn't about to occur, is it? If you've got work, or you like to maneuver around the area a lot, you will need a wheelchair that's lightweight, especially in the event that you've got no one to push one.
The wheelchair weight is ordered by the materials used for the structure. Lucky for you, you can find lightweight materials out there that may be as durable as possible. Take aluminum, for example. It's a hard metal, but very lightweight. Hybrids are also an option, but they are a bit pricey. Alloys, on the other hand, are among the cheapest materials. In the end, the choice is yours, and you might base your choice based upon your budget. However, remember that your meal weight is something that you should keep in mind before purchasing a specific item. After all, you are the person who will push it about all day long.
When you buy a wheelchair, it's also wise to consider the accessories that you will need for this. What it is sure is that you want a few things that will improve your mobility, and let's not forget about comfort. Bearing that in mind, here are the most frequent accessories that you will need.
Foam cushions — they are the most commonly purchased because they are the cheapest. While regular foam is comparatively comfortable, my advice is to go with memory foam if you are looking for an affordable solution. High-density foams will keep their shape for more, they will choose the shape of your body, and so much as comfort goes, you will enjoy plenty. However, the disadvantage with foam wheelchair cushions is that they wear fairly quickly, and you will have to purchase a new one pretty soon.
Gel cushions — gel cushions still have a foam foundation, but there is gel bladder someplace in the middle that will avoid pressure sores greater than polyurethane. A great deal of people purchase this and declare themselves satisfied. They can help you with your posture, and you will not experience pain in your lower back and tailbone. The disadvantage with this kind is that they tend to bottom-out. The answer to this would be to knead it at least one time each day.
Air flotation pillow — In case you do not understand what's, allow me to inform you. An air flotation cushion is composed of a certain number of balloons that match to keep you in a comfortable and wholesome posture. You've got the prospect of adjusting the pressure points by attaching a few balloons to prevent them from inflating. The only drawback of the sort of cushions is upkeep. In the end, we're discussing balloons. They could escape air, and you will need to fix that when and if it happens.
Urethane honeycomb — that is the best you can get, but urethane honeycomb futon cushions are quite pricey. The pillow is constructed of many individual cells which can offer exceptional support. The air circulates inside the cushion easily, which heightens the chance of heat retention. As far as glider cushions move, urethane honeycomb is the very best money can purchase.
Access factors are crucial for someone who's bound to a wheelchair. You need one to get outside and within the house, you need one in the office, and anywhere you go. Lucky for handicapped people, it is the law to install a wheelchair ramp into each public area. However, the one that you need at home needs to be installed by you.
Certainly you may discover drops at St. Cloud, MN, and truth be told, they are not so costly. But you will need a person to put in it to you since I don't think you can manage it yourself. 1 aspect which you will need to be cautious about is your wheelchair ramp incline. You would like it to be comfortable for you to use. If the angle is too large, that might be challenging for you to go up, and an extremely low angle requires quite the space for the ramp, which again, it is not convenient. Placing a wheelchair ramp incline is easy, and you should purchase a ramp so.
The capability to go from point A to point B is yet another important aspect. Just because you're in a wheelchair doesn't mean you could move only around your home or neighborhood. Lucky for you, there are a few things that you can try. For one, you may use a wheelchair van. It has an elevating system which will make entering and exiting the car comparatively simple. Such a car might be a little costly, but it is going to assist you with your mobility on long distances. It's possible to go out and socialize more. It'll keep you away from melancholy.
in addition, if you like to travel regardless of your disability, you can use a wheelchair RV. It's pretty much exactly the exact same since the wheelchair van, but instead of a car, you receive an RV. You may travel anywhere you want, and you have the assistance you will need to get in and outside of the car or truck. A wheelchair RV is undoubtedly the best method of travel for a handicapped individual.
Surviving isn't alive, so remember to relish everything in your own life. There are a great deal of means of entertaining that you can try. It is possible to take up a hobby or play a sport. Were you aware that you can play with dinner football? It resembles regular football, except that you will be transferring the ball onto your wheels. You ought to think about doing it.
In case you don't like wheelchair football, then you can always play a few wheelchair basketball. Certainly there are wheelchair basketball clubs in St. Cloud, MN, and also individuals who attend these sports events normally have loads of fun. Who says that person cannot be you. The point is that you may have fun regardless of what state you are in. All you need to do is seek out means of entertainment for handicapped individuals in St. Cloud, MN, and surely your hunt will develop something. Enjoy your life! It's well worth living it.
Once you get used to it, a wheelchair will get a part of you. You are going to discover how to live your own life in it, and you'll be more than nice. Nonetheless, make sure that you make an informed choice when you purchase a wheelchair. Also, do not forget about the accessories, and buy the best wheelchair you can afford. After all, this is the freedom we're discussing, and you deserve the very best.


text:
\section{Alexander--Bishop}

Here we sketch the proof given by Stephanie Alexander and Richard Bishop \cite{alexander-bishop}.
This proof was designed to work for more general ambient spaces.
As a result, it is more elementary.

In the proof, we construct a total-curvature-decreasing deformation of a given knot into a doubly covered
bigon.
The statement follows since the latter has total curvature $4\cdot\pi$.

\parit{Proof of the main theorem.}
Let $\alpha=p_1\dots p_n$ be a nontrivial knot;
that is, one can not get a triangle from $\alpha$ by applying a sequence of triangular isotopies defined in the previous section.

If $n=3$ the polygonal curve $\alpha$ is a triangle.
Therefore, by definition, $\alpha$ is a trivial knot --- nothing to show.

Consider the smallest $n$ for which the statement fails;
that is, there is a nontrivial knot $\alpha\z=p_1\dots p_n$ such that
\[\tc\alpha<4\cdot\pi.
\eqlbl{eq:<4pi}\]
We use the indexes modulo $n$; that is, $p_0=p_n$, $p_1=p_{n+1}$ and so on.
Without loss of generality, we may assume that $\alpha$ is in \emph{general position};
this time it means that no four vertexes of $\alpha$ lie on one plane.

Set $\alpha_0=\alpha$.
If the solid triangle $\solidtriangle p_{0}p_1p_{2}$ intersects $\alpha_0$ only in the two adjacent edges,
then applying the corresponding triangular isotopy, we get a knot $\alpha'_0$ with $n-1$ edges that is inscribed in $\alpha_0$. Therefore
\[\tc\alpha_0\ge \tc\alpha_0'.\]
On the other hand, by the induction hypothesis
\[\tc{\alpha_0'}\ge 4\cdot\pi,\]
which contradicts \ref{eq:<4pi}.

Choose the first point $w'_1$ on the edge $[p_1,p_2]$ so that the line segment $[p_0,w'_1]$
intersects $\alpha_0$.
Denote a point of intersection by $y_1$.

\begin{wrapfigure}{r}{25 mm}
\vskip-0mm
\centering
\includegraphics{mppics/pic-17}
\vskip0mm
\end{wrapfigure}

Choose a point $w_1$ on $[p_1,p_2]$ a bit before $w'_1$.
Denote by $x_1$ the point on $[p_0,w_1]$ that minimizes the distance to $y_1$.
This way we get a closed polygonal curve
$\alpha_1\z=w_1p_2\dots p_n$ with two marked points $x_1$ and $y_1$.
Denote by $m_1$ the number of edges in the arc $x_1w_1\dots y_1$ of $\alpha_1$.

Note that $\alpha_1$ is isotopic to $\alpha_0$;
in particular, $\alpha_1$ is a nontrivial knot.

Now let us repeat the procedure for the adjacent edges $[w_1,p_2]$ and $[p_2,p_3]$ of $\alpha_1$.
If the solid triangle $\solidtriangle w_1p_2p_3$ intersects $\alpha_1$ only at these two adjacent edges, then we get a contradiction the same way as before.
Otherwise, we get a new knot $\alpha_2=w_1w_2p_3\dots p_n$ with two marked points $x_2$ and $y_2$.
Denote by $m_2$ the number of edges in the polygonal curve $x_2w_2\dots y_2$.

Note that the points $x_1,x_2,y_1,y_2$ cannot appear on $\alpha_2$ in the same cyclic order;
otherwise the polygonal curve $x_1x_2y_1y_2$ can be made to be arbitrary close to a doubly covered bigon which again contradicts~\ref{eq:<4pi}.

Therefore we can assume that the arc $x_2w_2\dots y_2$ lies inside the arc $x_1w_1\dots y_1$ in $\alpha_2$
and therefore $m_1>m_2$.

Continuing this procedure we get a sequence of polygonal curves $\alpha_i\z=w_1\dots w_i p_{i+1}\dots p_n$ with marked points $x_i$ and $y_i$ such that the number of edges $m_i$ from $x_i$ to $y_i$ decreases as $i$ increases.
Clearly $m_i>1$ for any $i$ and $m_1<n$.
Therefore it requires less than $n$ steps to arrive at a contradiction.
\qeds

\section{F\'ary}\label{sec:fary}

In this section, we sketch the solution of Istv\'an F\'ary \cite{fary} which was published before Milnor's proof.

We start with Crofton-type formulas for total curvature.
Given a curve $\alpha$ in $\mathbb{R}^3$ and a unit vector $u$, denote by $\alpha_{u^\perp}$
and $\alpha_u$ the projections of $\alpha$ to the plane perpendicular to $u$ and the line parallel to $u$ respectively.
Given a function $f\:\mathbb{S}^2\to\mathbb{R}$, let us denote its average value by $\overline{f(u)}$.

\begin{thm}{Crofton-type formula}\label{prop:tc-crofton}
Let $\alpha$ be a polygonal curve in $\mathbb{R}^3$.
Then
\begin{align*}
\tc\alpha
&=\overline{\tc{\alpha_{u^\perp}}}
=\overline{\tc{\alpha_u}}.
\end{align*}
\end{thm}

\parit{Proof.}
Observe that it is sufficient to check the identities for polygonal curves made of two edges
and in this case, it boils down to very straightforward calculations.
\qeds

The original version of Milnor's proof
used the identity $\tc\alpha=\overline{\tc{\alpha_u}}$;
in Fenner's version of his proof it was hidden under the carpet.

F\'ary's proof is based on the identity $\tc\alpha=\overline{\tc{\alpha_{u^\perp}}}$ and the following inequality for total curvature.
Suppose $\alpha=p_1\dots p_n$ is a simple closed polygonal curve in $\mathbb{R}^3$ and $o\notin\alpha$.
Let us define the \emph{angular length} of $\alpha$ with respect to $o$ as the sum
\[\Psi_o(\alpha)=\measuredangle p_{1} o p_{2}+\dots+\measuredangle p_{n-1} o p_{n}+\measuredangle p_{n} o p_{1}.\]

\begin{thm}{Proposition}\label{prop:angular-length}
For any closed simple polygonal curve and any $o\notin\alpha$, we have
\[\Psi_o(\alpha)\le \tc{\alpha}.\]
\end{thm}

\parit{Proof.}
Let $\alpha=p_1\dots p_n$; for each $i$, set
\begin{align*}
\phi_i&=\pi-\measuredangle p_{i-1}p_ip_{i+1},
&
\psi_i&=\measuredangle p_{i-1} o p_{i},
&
\theta_i&=\measuredangle o p_i p_{i+1}.
\end{align*}
Here we assume that indexes are taken modulo $n$; in particular, $p_{n}=p_0$.

\begin{wrapfigure}{r}{45 mm}
\vskip-0mm
\centering
\includegraphics{mppics/pic-15}
\vskip0mm
\end{wrapfigure}

Note that $\phi_i$ is the external angle at $p_i$;
therefore
\[\tc\alpha= \phi_1+\dots+\phi_n\]

The directions of $p_i-p_{i-1}$, $o-p_i$, and $p_{i+1}-p_i$ make angles
$\psi_i+\theta_{i-1}$, $\theta_i$, and $\phi_i$ to each other.
Applying the triangle inequality for these angles, we get
\[\phi_i\ge \psi_i+\theta_{i-1}-\theta_i.\]
Summing up, we get
\[\phi_1+\dots+\phi_n\ge \psi_1+\dots+\psi_n,\]
and the result follows.
\qeds

\begin{wrapfigure}{r}{44 mm}
\vskip-0mm
\centering
\includegraphics{mppics/pic-13}
\vskip0mm
\end{wrapfigure}

\parit{Proof of the main theorem.}
Consider a projection of the knot to a plane in \emph{general position}
(this time it means that the self-intersections of the projection are at most double and the projection of each edge is not degenerate).
The obtained closed polygonal curve $\alpha_{u^\perp}=p_1p_2\dots p_n$ divides the plane into domains, one of which is unbounded, denote it by $U$, and the others are bounded.

First, note that all domains can be colored in a chessboard order;
that is, they can be colored in black and white in such a way that domains with common borderline get different colors.
If the unbounded domain is colored in white and every other domain is colored in black, then one can untie the knot by flipping these domains one by one.%
\footnote{It is instructive to give a formal proof of the last statement; that is, \textit{show that if there is only one white region, then $\alpha$ is trivial}.}

\begin{wrapfigure}{r}{44 mm}
\vskip-4mm
\centering
\includegraphics{mppics/pic-14}
\vskip0mm
\end{wrapfigure}

Therefore, among the bounded domains there is a white domain, denote it by $D$.
The domain $D$ cannot adjoin
$U$, since they have the same color.
Fix a point $o$ in this domain.

Since any ray from $o$ crosses $\alpha_{u^\perp}$ twice, we get $\Psi_o(\alpha_{u^\perp})\ge 4\cdot\pi$;
that is, the angular length of $\alpha_{u^\perp}$ with respect to $o$ is at least $4\cdot\pi$.
By \ref{prop:angular-length}, we have
\[\tc{\alpha_{u^\perp}}\ge4\cdot\pi.\]
This is true for any $u$ in general position.
The remaining directions contribute nothing to the average value.
It remains to apply the Crofton-type formula $\tc{\alpha}=\overline{\tc{\alpha_{u^\perp}}}$.
\qeds

\section*{Introduction}

The following problem was posted by Karol Borsuk \cite{borsuk}.

\smallskip

\textit{Show that the total curvature of any nontrivial knot is at least $4\cdot\pi$.}

\smallskip

It is known for many proofs based on different ideas.
We sketch several solutions, one solution per section;
each can be read independently.

This problem also has a number of refinements and generalizations;
in particular, a strict inequality holds --- this is the famous \emph{F\'ary--Milnor theorem}.
However, for the sake of simplicity, we stick to the original formulation.

In order to continue, we need to agree on a definition of knot and explain what a \emph{nontrivial knot} is.
Despite the intuitive idea of a knot as a \emph{cyclic rope},
the simplest formal definition uses polygonal curves.
It turns out that this definition is also best suited for our purposes.

A \emph{knot} (more precisely \emph{tame knot}) is a simple closed polygonal curve in the Euclidean space~$\mathbb{R}^3$ (\emph{simple} means \emph{no self-intersections}).

The solid triangle with vertices $a$, $b$, and $c$ will be denoted by $\solidtriangle abc$.
It is defined as the convex hull of the points $a$, $b$, and $c$;
the points $a$, $b$ and $c$ are assumed to be distinct, but they might lie on one line.

We define a \emph{triangular isotopy} of a knot to be the generation of a new knot from the original one by means of the
following two operations:

Assume $[p,q]$ is an edge of the knot and $x$
is a point such that the solid triangle $\solidtriangle pqx$  has no common points with the knot except for the edge $[p,q]$.
Then we can replace the edge $[p,q]$ with the two adjacent edges $[p,x]$ and $[x,q]$.

We can also perform the inverse operation.
That is, if for two adjacent edges $[p,x]$ and $[x,q]$ of a knot the triangle
$\solidtriangle pqx$ has no common points with the knot except for the points on the edges $[p,x]$ and $[x,q]$,
then we can replace $[p,x]$ and $[x,q]$ by one edge $[p,q]$.

Polygons that arise from one another by a finite sequence of
triangular isotopies are called \emph{isotopic}.
A knot that is not isotopic to a triangle (that is, a simple polygonal curve with three vertices) is called \emph{nontrivial}.

\begin{wrapfigure}{r}{22 mm}
\vskip-3mm
\centering
\includegraphics{mppics/pic-18}
\vskip0mm
\end{wrapfigure}

The trefoil knot shown on the diagram to the right gives a simple example of a nontrivial knot.
A proof that it is nontrivial can be found in any textbook on knot theory.
The most elementary and visual proof is based on the so-called \emph{tricolorability} of knot diagrams \cite[Section 1.5]{adams}.

The total curvature of a smooth curve is usually defined as the integral of its curvature.
For polygons, it is defined as the sum of its external angles.
It is well known that the total curvature of a curve cannot be smaller than the total curvature of an inscribed polygonal curve (see, for example, \cite{petrunin-zamora}).
In fact, the total curvature of a curve can be defined as the least upper bound on the total curvature of inscribed polygonal curves \cite{aleksandrov-reshetnyak, sullivan-curves}.
This definition agrees with the definition given for smooth curves, and it makes sense for any simple curve.
The total curvature of a curve $\alpha$ will be denoted by $\Phi(\alpha)$.

It follows that the problem allows the following reformulation which we are going to prove.

\parbf{Main theorem.}
\textit{$\Phi(\alpha)\ge 4\cdot\pi$ for any nontrivial knot $\alpha$.}

\section{Milnor--Fenner}

One of the first solutions to the problem was found by John Milnor \cite{milnor}.
In this section, we present an amusing interpretation of his proof found by Stephen Fenner \cite{ferner}.
Just like the original version, it is based on the following sufficient condition for the triviality of a knot.

\begin{thm}{Proposition}\label{prop:one-max-one-min}
Assume that a height function $(x,y,z)\to z$
has only one local maximum on a closed simple polygonal curve $\alpha$ and all the vertices of the polygonal curve are at different heights.
Then $\alpha$ is a trivial knot.
\end{thm}

The proof is a straightforward construction of a triangular isotopy.

\parit{Proof.}
Let $\alpha=p_1\dots p_n$ be the closed simple polygonal curve
such that the height function $(x,y,z)\to z$ has one local maximum and all vertices have different heights.
Note that the height function also has a unique local minimum,
and $\alpha$ can be divided into two arcs from the min-vertex to the max-vertex with a monotonic height function.

Consider the three vertexes with the largest height;
they have to include the max-vertex and two more.
Note that these three vertices are consequent in the polygonal curve;
without loss of generality, we can assume that they are $p_{n-1}$, $p_n$, and $p_1$.

\begin{wrapfigure}{r}{30 mm}
\vskip-0mm
\centering
\includegraphics{mppics/pic-19}
\vskip0mm
\end{wrapfigure}

Note that the solid triangle $\solidtriangle p_{n-1}p_np_1$ does not intersect any edge $\alpha$ except the two adjacent edges $[p_{n-1},p_n]\cup[p_n,p_1]$.
Indeed, if $\solidtriangle p_{n-1}p_np_1$ intersects $[p_1,p_2]$,
then,
since $p_2$ lies below $\solidtriangle p_{n-1}p_np_1$,
the edge $[p_1,p_2]$ must intersect $[p_{n-1},p_n]$;
the latter is impossible since $\alpha$ is simple.

The same way, one can show that $\solidtriangle p_{n-1}p_np_1$ can not intersect $[p_{n-2},p_{n-1}]$.
The remaining edges lie below $\solidtriangle p_{n-1}p_np_1$, hence they cannot intersect this triangle.

Applying a triangular isotopy, to $\solidtriangle p_{n-1}p_np_1$ we get a closed simple polygonal curve $\alpha'\z=p_1\dots p_{n-1}$ which is isotopic to~$\alpha$.

Since all the vertices $p_i$ have different heights,
the assumption of the proposition holds for $\alpha'$.

Repeating this procedure $n-3$ times we get a triangle.
Hence $\alpha$ is a trivial knot.
\qeds

\parit{Proof of the main theorem.}
Let $\alpha=p_1\dots p_n$ be a nontrivial polygonal knot.
Denote by $v_i$ the unit vector in the direction of $p_{i+1}-p_i$;
we assume that $p_n=p_0$.
Consider the set $U_i$ formed by all unit vectors $u$ such that $\measuredangle(u,v_i)\ge \tfrac \pi 2$ and $\measuredangle(u,v_{i-1})\le \tfrac \pi 2$.
Note that $u\in U_i$ if and only if the function $x\mapsto \langle u,x\rangle$ has a local maximum at $p_i$ on $\alpha$.

Let us choose $(x,y,z)$-coordinates in the space so that $z$-axis points in the direction of $u$.
Then according to \ref{prop:one-max-one-min}, the function $p\mapsto \langle u,p\rangle$ has at least two local maxima on $\alpha$.
It follows that the sets $U_1,\dots,U_n$ cover each point on the unit sphere $\mathbb{S}^2$ twice.

Recall that $\area \mathbb{S}^2=4\cdot\pi$.
Observe that $\phi_i=\measuredangle(v_{i-1},v_i)$ is the external angle of $\alpha$ at $p_i$.
Note that $U_i$ is a slice of the sphere between two meridians meeting at angle $\phi_i$, therefore $U_i$ occupies a $\tfrac{\phi_i}{2\cdot\pi}$ portion of the whole sphere; so, $\area U_i=2\cdot \phi_i$.
Since the sets $U_1, \dots, U_n$ cover $\mathbb{S}^2$ twice, we get
\[\tc\alpha=\phi_1+\dots+\phi_n=\tfrac12\cdot (\area U_1+\dots+\area U_n)\ge \tfrac22\cdot \area\mathbb{S}^2=4\cdot\pi.\]
\qedsf

\section{Ekholm--White--Wienholtz}

In this section, we discuss a solution of the problem based on a theorem of Tobias Ekholm, Brian White, and Daniel Wienholtz \cite{EWW_embed}.
This theorem was a breakthrough in minimal surface theory at the time.
Yet, it was based on an elementary idea that we are going to explain now.

We start with a polygonal curve $\alpha$ with total curvature less than $4\cdot\pi$;
show that an area-minimizing disc spanned by $\alpha$ has no self-intersections, and therefore $\alpha$ has to be a trivial knot.
So in a way the equation for area-minimizing surfaces solves our problem, we only need to understand~it.

The main hero in this proof is the so-called \emph{extended monotonicity theorem}.
We will also apply the \emph{Douglas--Rado theorem} on the existence of area-minimizing discs and reuse the inequality between total curvature and angular length from F\'ary's proof; see \ref{prop:angular-length}.

The image of a map from a domain of $\mathbb{R}^2$ to $\mathbb{R}^3$ will be called a \emph{surface};
it might have self-intersections and singularities, but we assume it is reasonable, say locally Lipschitz; so we can talk about its area.
A point on the surface might refer to a point in $\mathbb{R}^3$, or the corresponding point in the domain of parameters in $\mathbb{R}^2$;
it should be easy to guess from the context.

We denote by $\mathbb{D}$ the closed disc in the plane.
A surface defined by a map $f\:\mathbb{D}\to\mathbb{R}^3$ will be called a \emph{disc}.
The restriction $f|_{\partial \mathbb{D}}$ will be called the \emph{boundary line} of the disc.

A disc $\Sigma$ is called area-minimizing if it has the smallest area among the surfaces with the given boundary line.
The following statement about area-minimizing discs is easy to believe, but not easy to prove; see \cite{white-lectures}.

\begin{thm}{Douglas--Rado theorem}\label{thm:min-exists}
Given a simple closed polygonal curve $\alpha$ in $\mathbb{R}^3$, there is an area-minimizing disc $\Sigma$ with boundary line $\alpha$; it is a smooth surface, possibly with self-intersections and isolated singularities.

Moreover, if $\Sigma$ has no self-intersections, then it is an embedded smooth surface with no singularities (in this case $\alpha$ is a trivial knot).
\end{thm}

\begin{wrapfigure}{r}{35 mm}
\vskip-5mm
\centering
\includegraphics{mppics/pic-51}
\vskip0mm
\end{wrapfigure}

Choose a disc $\Sigma$ in $\mathbb{R}^3$ with boundary line $\alpha$.
Given a point $o\notin \alpha$, let us consider \emph{collared} $\Sigma$ with respect to $o$;
it is a new surface that will be denoted by $\hat\Sigma_o$;
it includes $\Sigma$ and the \emph{collar} formed by all rays that start at points of $\alpha$ and go in the direction opposite to $o$.
Note that $\hat\Sigma_o$ admits a natural parametrization by the whole plane.

\begin{thm}{Extended monotonicity theorem}\label{thm:monotonicity}
Let $\Sigma$ be an area-minimizing disc with boundary line~$\alpha$.
Given a point $o\notin \alpha$, consider the function
\[W_o(r)=\area (\hat\Sigma_o\cap \bar B_r(o)),\]
where $\bar B_r(o)$ denotes the ball of radius $r$ centered at $o$.
Then the function $r\mapsto \frac{W_o(r)}{r^2}$
is nondecreasing. Moreover,

\begin{subthm}{thm:monotonicity:a}
$\lim_{r\to\infty}\frac{W_o(r)}{r^2}=\tfrac12\cdot \Psi_o(\alpha)$, where $\Psi_o(\alpha)$ denotes the angular length of $\alpha$ with respect to $o$; see Section~\ref{sec:fary}.
\end{subthm}

\begin{subthm}{thm:monotonicity:b}
If $o\in \Sigma$, then $\lim_{r\to0}\frac{W_o(r)}{r^2}\ge \pi$.
\end{subthm}

\end{thm}

The classical monotonicity theorem states that the same holds for balls that do not touch the boundary of the surface.
The stated version is due to Brian White \cite{white}.
The same statement holds for minimal surfaces \cite{EWW_embed}; its proof requires deeper diving into differential geometry.
At the same time, the original formulation admits a generalization to a wider class of ambient spaces \cite{St_structure}.

\parit{Proof.}
Denote by $\lambda_o(r)$ the curve of intersection of the sphere $\partial B_r(o)$ with $\hat\Sigma_o$;
set $\ell(r)\z=\length[\lambda_o(r)]$.
Observe that
\[W_o'(r)\ge \ell(r)\]
for almost all $r$.
(Formally speaking, this inequality follows from the so-called \emph{coarea formula}.)

Set $\Delta_r=\hat\Sigma_o\cap \bar B_r(o)$;
it is a surface bounded by $\lambda_o(r)$.
Let $\Delta_r'$ be the cone over $\lambda_o(r)$ with the center at $o$.
Note that $\Delta_r'$ differs from $\Delta_r$ only inside $\Sigma$.
Since $\Sigma$ is area-minimizing, we get that
\[\area \Delta_r'\ge \area \Delta_r\eqlbl{area=<area}\]
for any $r>0$.
Observe that
\begin{align*}
\area \Delta_r&=W_o(r),
&
\area \Delta_r'&=\tfrac12\cdot r\cdot \ell(r).
\end{align*}
Applying \ref{area=<area}, we get
\[r\cdot \ell(r)\ge 2\cdot W_o(r).\]
Therefore, we get
\[r\cdot W_o'(r)\ge 2\cdot W_o(r)\]
for almost all $r$.
If $W_o$ is smooth, then this inequality implies the main statement.
In general, the argument shows that the absolutely continuous part of $W_o(r)/r^2$ is nondecreasing which suffices.

\parit{\ref{SHORT.thm:monotonicity:a}.}
Observe that up to a fixed error we have that $W_o(r)$ is the area of the ball of radius $r$ in the cone over $\alpha$ with the tip at $o$.
It follows that $W_o(r)/r^2$ approaches the area of the unit ball in this cone as $r\to\infty$ --- hence the result.

\parit{\ref{SHORT.thm:monotonicity:b}.}
The statement is evident for smooth points of $\Sigma$.
Since smooth points are dense in $\Sigma$, and $o\mapsto W_o(r)$ is a continuous function,
the main part of the theorem implies that $W_o(r)\ge\pi\cdot r^2$ for \emph{any} point $o\in\Sigma$ --- hence the result.
\qeds

\parit{Proof of the main theorem.}
Suppose that $\tc\alpha<4\cdot\pi$.
Consider an area-minimizing surface $\Sigma$ with the boundary line $\alpha$; it exists by \ref{thm:min-exists}.
If $\Sigma$ has no self-intersections, then $\alpha$ is a trivial knot.

Suppose $\Sigma$ has a self-intersection at a point $o$.
In this case, the intersection $B_r(o)\cap \Sigma$ is covered by two or more small area-minimizing subdiscs of $\Sigma$.
By \ref{thm:monotonicity:b}, we get
\[\lim_{r\to0}\frac{W_o(r)}{r^2}\ge 2\cdot\pi.\]

Applying the main statement in the monotonicity theorem, we get $\frac{W_o(r)}{r^2}\ge 2\cdot\pi$ for any $r>0$.
By \ref{thm:monotonicity:b} and Proposition~\ref{prop:angular-length} in F\'ary's proof, we get
\[\tc{\alpha}\ge \Psi_o(\alpha)\ge 2\cdot \frac{W_o(r)}{r^2}\ge 4\cdot\pi\]
--- a contradiction.
\qeds

\section{Pannwitz--Hopf--Schmitz--Denne}\label{sec:quadrisecant}

F\'ary's paper \cite{fary} is ended by the following note:
\textit{I just received a letter from Mr Borsuk, which says that Theorem 3 has been proved independently by Mr H. Hopf.
It uses the theorem of Miss Pannwitz, which ensures that for any knot there is a line that crosses it at least in four
points.}
This proof is the subject of this section.

\begin{wrapfigure}{r}{22 mm}
\vskip-4mm
\centering
\includegraphics{mppics/pic-41}
\vskip0mm
\end{wrapfigure}

The main step in the proof is the existence of the so-called \emph{alternated quadrisecant} of a knot $\alpha$; that is, there is a line $\ell$ that shares with $\alpha$ four points $a$, $b$, $c$ and $d$
that appear on $\ell$ in the same order and have cyclic order $a$, $c$, $b$, $d$ on $\alpha$.

The question about existence of quadrisecant was asked by Otto Toep\-litz and answered by Erika Pannwitz for tame knots in \emph{general position} \cite{pannwitz}.
Elizabeth Denne had generalized this result to all knots \cite{denne, denne-survey}.
But for us tame knots in general position will be sufficient;
namely, we need the following:

\begin{thm}{Proposition}\label{prop:quadrisecant}
Any nontrivial knot $\alpha$ in general position admits an \emph{alternated quadrisecant}.
\end{thm}

The precise meaning of general position will be clear from the proof;
what is important is that any knot is arbitrary close to a knot in general position.
Erika Pannwitz only proved the existence of quadrisecant, but the existence of alternated quadrisecant can be extracted from her proof.

\parit{Proof of the main theorem modulo \ref{prop:quadrisecant}.}
Let $\alpha=p_1\dots p_n$ be a knot in general position.
Suppose $a$, $b$, $c$ and $d$ be the points as in the definition of an alternated quadrisecant.
Then $acbd$ is an inscribed quadrangle with all external angles equal to $\pi$.
Therefore $\tc{abcd}=4\cdot\pi$.

Since the total curvature of an inscribed polygonal curve cannot be larger than the total curvature of the original curve, we get that $\tc{\alpha}\z\ge 4\cdot\pi$.

Finally, for any knot $\beta=q_1\dots q_n$ there is an arbitrarily close knot $\alpha=p_1\dots p_n$ in general position;
in particular, for any $\eps>0$ we can assume that $\tc{\beta}>\tc\alpha-\eps$.
It follows that $\tc{\beta}>4\cdot\pi-\eps$ for any positive $\eps$ --- hence the result.
\qeds

In the proof of the proposition, we will use the following characterization of the trivial knot.

\begin{thm}{Lemma}
A knot $\alpha$ is trivial if there exists a piecewise linear map $F$ from the disc $\DD$ to $\RR^3$ such that the  restriction of $F$ to the boundary $\partial\DD=\mathbb{S}^1$ is a degree-one map to $\alpha$ and $F$ does not map interior points of $\DD$ to $\alpha$.
\end{thm}

This statement can be deduced from the so-called \textit{Dehn's lemma} --- a heavy weapon of 3-dimensional topology proved by Christos Papakyriakopoulos.
For those who are familiar with Dehn's lemma, it would be an exercise; otherwise, we suggest taking it for granted.
In the following proof, we follow closely the presentation of Carsten Schmitz \cite{schmitz} and the original argument of Erika Pannwitz.

\parit{Proof of the proposition.}
We may assume that $\alpha$ comes with a 1-periodic piecewise linear parametrization by $\RR$;
so the space of oriented chords of $\alpha$ can be identified with the open cylinder $\mathbb{S}^1\times (0,1)$, where $\mathbb{S}^1=\RR/\ZZ$.
Namely, we assume that a pair $(x,y)\in \mathbb{S}^1\times (0,1)$ corresponds to the oriented chord with the ends at $\alpha(x)$ and $\alpha(x+y)$.

\begin{wrapfigure}{r}{42 mm}
\vskip-0mm
\centering
\includegraphics{mppics/pic-43}
\vskip0mm
\end{wrapfigure}

Choose a pair $(x,y)\in \mathbb{S}^1\times (0,1)$.
Let us denote by $r(x,y)$ the ray that starts at $\alpha(x)$ and goes in the direction opposite to $\alpha(x+y)$.
We write $(x,y)\in C_3$ if $r(x,y)$ crosses $\alpha$ at another point.

The points $\alpha(x)$ and $\alpha(x+y)$ divide $\alpha$ into two open arcs $\alpha|_{(x,x+y)}$ and $\alpha|_{(x+y,x+1)}$.
If $(x,y)\in C_3$ and $r(x,y)$ crossed the first arc, then we write $(x,y)\in C_3^+$;
if it crosses the second arc, then $(x,y)\in C_3^-$.
Note that $C_3^+\cup C_3^-=C_3$.

Observe that if $C_3^+$ and $C_3^-$ intersect, then the proposition follows.

The set $C_3^\pm$ might be not closed;
denote by $\bar C_3^\pm$ its closure.
Suppose
\[(x,y)\in (\bar C_3^+\cup\bar C_3^-)\backslash( C_3^+\z\cup C_3^-).\]
Observe that $\alpha(x)$ is a vertex, say $p_i$, of $\alpha$.
Further, the line containing $p_i$ and $\alpha(x+y)$ lies in the plane spanned by $p_{i-1}$, $p_i$, and $p_{i+1}$; and it intersects yet another point of $\alpha$.
The latter is not possible if $\alpha$ in general position.
That is, since $\alpha$ is in general position, $C_3^+\cap C_3^-=\emptyset$ implies that $\bar C_3^+\cap \bar C_3^-=\emptyset$.

Note that there is $\eps>0$ such that for any $x\in\mathbb{S}^1$ we have the following:
if $y<\eps$, then $(x,y)\notin \bar C_3^+$,
and if $y>1-\eps$, then $(x,y)\notin \bar C_3^-$.

\begin{wrapfigure}{r}{22 mm}
\vskip-0mm
\centering
\includegraphics{mppics/pic-46}
\vskip0mm
\end{wrapfigure}

Arguing by contradiction, assume $\bar C_3^+\cap \bar C_3^-=\emptyset$.
Note that in this case there is a large $n$ such that any $\tfrac1n\times\tfrac1n$-square in $\mathbb{S}^1\times (0,1)$ does not intersect both $\bar C_3^+$ and $\bar C_3^-$.
Let us cut $\mathbb{S}^1\times (0,1)$ into $n^2$ such squares; each square defines a closed subset of $\mathbb{S}^1\times (0,1)$.
Color the union of squares that intersect $\bar C_3^+$ or lie in the lowest row of squares for $y\le \tfrac1n$.
Note that we did not color squares in the upper row and the boundary of the colored set contains a simple curve $t\mapsto(x(t),y(t))$ that cuts  the cylinder $\mathbb{S}^1\times (0,1)$ into two cylinders.

Note that $(x(t),y(t))\notin \bar C_3$ for any $t\in \mathbb{S}^1$ and the curve $t\mapsto(x(t),y(t))$ runs along coordinate lines.
Consider the one-parameter family of line segments in $r(x(t),y(t))$ that start at $\alpha(x(t))$ and end on the surface of a large cube that contains $\alpha$ in its interior.
This way we obtain a piecewise linear annulus that connects the curve $t\mapsto \alpha(x(t))$ to a curve on the surface of the cube.
The latter curve can be contracted by a piecewise linear disc in the surface of the cube.
It might have self-intersection, but it cannot contains points of $\alpha$.

Observe that $t\mapsto \alpha(x(t))$ defines a degree-one map $\mathbb{S}^1\to\alpha$.
Applying the lemma, we get the result.
\qeds

\section{Cantarella--Kuperberg--Kusner--Sullivan}\label{sec:2nd-hull}

The following proof is due to Jason Cantarella, Greg Kuperberg, Robert Kusner, and John Sullivan \cite{CKKS};
this is the only proof in our collection that uses knot theory a bit beyond the basic definitions.

Choose a polygonal curve $\alpha=p_1\dots p_n$.
Suppose a plane $\Pi$ is in general position; that is, it does not contain vertices of $\alpha$.
Let us denote by $\cross_\alpha\Pi$ the number of intersections of $\Pi$ and $\alpha$.
Extend the function $\Pi\mapsto \cross_\alpha(\Pi)$ to the minimal upper semicontinuous function defined for all planes;
in other words, $\cross_\alpha(\Pi)$ is the maximal integer $k$ such that there is a plane $\Pi'$ arbitrary close to $\Pi$ that intersects $\alpha$ at $k$ points.
The number $\cross_\alpha(\Pi)$ will be called the \emph{crossing number} of $\Pi$.
Note that for any plane $\Pi$, the crossing number is even, and it cannot exceed $n$.

{

\begin{wrapfigure}{r}{60 mm}
\vskip-3mm
\centering
\includegraphics{asy/trefoil}
\caption*{First and second hull of a trefoil.}
\vskip0mm
\end{wrapfigure}

It is easy to see that the \emph{convex hull} $h_1(\alpha)$ of $\alpha$ can be defined the following way:
\textit{$x\in h_1(\alpha)$ if $\cross_\alpha(\Pi)\ge2$ for any plane $\Pi\ni x$}.
This observation suggests the following definition of the \emph{second hull}:
\textit{$x\in h_2(\alpha)$ if $\cross_\alpha(\Pi)\ge4$ for any plane $\Pi\ni x$}.

\begin{thm}{Theorem}\label{thm:2nd-hull}
The second hull оf any nontrivial knot $\alpha$ has a nonempty interior.
\end{thm}

To show that this theorem implies our main theorem, we will apply the spherical Crofton formula:
\textit{for any spherical polygonal curve $\gamma$ we have
\[\length \gamma=\pi\cdot \overline n,\leqno({*})\]
where $\overline n$ denotes the average number of intersections of $\gamma$ with equators.}
To prove this formula, check it for an arc on an equator and sum it up for all edges of $\gamma$.

}

\parit{Proof of the main theorem modulo \ref{thm:2nd-hull}.}
Choose a point $o\in h_2(\alpha)$; we can assume that $o\notin\alpha$.
Consider the radial projection $\alpha^*$ of $\alpha$ to the sphere centered at $o$;
observe that
\[\length\alpha^*=\Psi_o(\alpha),\]
where $\Psi_o(\alpha)$ denotes the angular length of $\alpha$ with respect to $o$; see Section~\ref{sec:fary}.

By \ref{prop:angular-length}, it is sufficient to show that
\[\length\alpha^*\ge 4\cdot\pi.\leqno({*}{*})\]
Since $o$ is in the second hull, $\alpha^*$ crosses every equator in general position at least 4 times.
It follows that the average number of crossings is at least~4.
Applying $({*})$, we get $({*}{*})$.
\qeds

Suppose that a plane $\Pi$ divides a knot $\alpha$ into two arcs, one on each side; in particular, $\Pi$ intersects $\alpha$ at two points, say $p$ and $q$.
Then we can create two knots $\alpha_1$ and $\alpha_2$ by joining the ends of the two arcs by the line segment $[p,q]$.
In this case, we say that $\alpha$ is a \emph{connected sum} of $\alpha_1$ and $\alpha_2$.

\begin{thm}{Claim}\label{clm:connected-sum}
Suppose that a knot $\alpha$ is a connected sum of knots $\alpha_1$ and $\alpha_2$.
If at least one of the knots $\alpha_1$ or $\alpha_2$ is nontrivial, then so is $\alpha$.
\end{thm}

This claim has an amusing proof via the so-called \emph{infinite swindle} \cite{mazur}.

Suppose $\beta$ is a polygonal curve \emph{inscribed} in $\alpha$;
that is, the vertices of $\beta$ lie on $\alpha$ and they appear in the same cyclic order on $\alpha$ and $\beta$.
If a plane in general position intersects an edge of $\beta$, then it intersects
the corresponding arc of $\alpha$.
Therefore we get the following:

\begin{thm}{Observation}
If a polygonal line $\beta$ is inscribed in $\alpha$.
Then $h_2(\beta)\subset h_2(\alpha)$.
\end{thm}

\parit{Proof of \ref{thm:2nd-hull}.}
Assume the contrary; let $\alpha$ be a nontrivial polygonal knot with the least number of vertices, say $n$, such that $h_2(\alpha)$ has an empty interior.
It is easy to see that $n\ge 6$;
in other words, any simple space 5-gon is a trivial knot.

Suppose $\Pi$ is a plane in general position that divides $\alpha$ in two arcs, so it defines a decomposition of $\alpha$ into a connected sum of two knots $\alpha_1$ and $\alpha_2$.
By the observation, $h_2(\alpha_1)$ and $h_2(\alpha_2)$ have empty interior.
It follows that one of these knots, say $\alpha_1$, is trivial and therefore the other, respectively $\alpha_2$, is isotopic to $\alpha$.
Indeed, if $n_1$ and $n_2$ denote the number of vertices in $\alpha_1$ and $\alpha_2$, then $n_1+n_2=n+4$.
If both knots are nontrivial, then $n_1\ge 6$ and $n_2\ge 6$.
Therefore $n_1<n$ and $n_2<n$ which contradicts minimality of $n$.

The open half-space $H$ bounded by $\Pi$ and containing $\alpha_2\setminus\Pi$ will be called \emph{essential}. Intuitively, an essential half-space cuts off
a trivial knot of $\alpha$. (A half-space containing all of $\alpha$ will be considered essential as well.)
A rather straightforward application of \ref{clm:connected-sum} implies that if $H'$ is another essential half-space for $\alpha$, then it is also essential for $\alpha_2$.
It follows that the intersection of all essential half-spaces, say $W$, has nonempty interior --- roughly speaking, it has to contain the region where knotting of $\alpha$ takes place.

Finally observe that if $\cross_\alpha\Pi=2$ for a plane $\Pi$ in general position, then $\Pi$ bounds an essential half-space.
It follows that if a plane in general position intersects $W$, then it has  crossing number at least 4,
so $h_2(\alpha)\supset W$ --- hence the result.
\qeds

redpajama_set_name: RedPajamaArXiv


Use your Mouse and physics skills to help our cute boy reach his lollipop. Avoid fire and deadly sharp objects on his way of craziness.
Met friendly zombies, vampires, and ghosts. Solve puzzles while been entertained by theirs quirky animations and comments.
Lead the blue dot thru the 3D maze to the red dot as fast as you can.
Escape from the shrinking maze. Avoid walls. Pick up coins to upgrade. Pick up bombs to blow up the walls. Freeze and back the walls.
An addictive and extremely frustrating unique mix of a maze game and a puzzle game.


The pet food recalls continue to roll in. Signed up for Itchmo alerts, the news comes right in my email. Following links and reading more articles in various newspapers around the country, I'm learning that we have good reason to question the human food supply. Much of the food that goes on our own tables comes from out of the country. If shipments from China tainted our pets' food, and at least some of that was marked "human grade," I can't help but wonder about the food that comes in meant specifically for the family meals.
From what I'm reading from Itchmo, both Natural Balance and Blue Buffalo recalled products today that was made by American Nutrition. I just viewed American Nutrition website and they are claiming all their foods are safe. Uh-oh.
We keep waiting for the other shoe to drop and drop it does. Only now more shoes have dropped than an centipede would need.