text:
HOME | ABOUT US | EDITORIAL BOARD | AHEAD OF PRINT | CURRENT ISSUE | ARCHIVES | INSTRUCTIONS | SUBSCRIBE | ADVERTISE | CONTACT
Login | Users online:
Prevalence of most commonly reported tobacco-associated lesions in central Gujarat: A hospital-based cross-sectional study
Manjiri Joshi, Mansi Tailor
Department of Oral Medicine and Radiology, Manubhai Patel Dental College and Hospital, Vadodara, Gujarat, India
Date of Decision 29-May-2016
Date of Acceptance 08-Aug-2016
Background: Oral cancer is a major health problem in tobacco users worldwide and is one of the ten most common cancers. India alone accounts for 1/3 rd of the world's oral cancer and has a high rate of potentially malignant disorders (PMDs). The most common predisposing factors are smoking, smokeless tobacco, betel nut in quid form (pan), alcohol, spicy food, and sharp broken tooth. There are various tobacco-associated lesions (TALs) which can be diagnosed at very early stage. This study was conducted to rule out association of smoking and smokeless tobacco with occurrence of TALs and its dose-response relationship.
Materials and Methods: Total of 60,018 patients attending the Department of Oral Medicine and Radiology from January 2013 to December 2014 with different oral and dental symptoms were screened. Of these, 4795 patients satisfying inclusion and exclusion criteria formed the cohort of the present study. All the patients were informed regarding the study and an informed verbal consent was obtained, following which they were interviewed for tobacco-related habits and examined by the trained dental professionals for the presence of any lesion. Along with patients' demographic details, information regarding the type of habit, duration, and frequency was recorded.
Statistical Analysis: Statistical analysis was performed using STATA 13.1 software (STATA 13.1 software by Stata Corp Ltd India Continent) by applying Pearson's Chi-square test with Fisher's exact test, Independent t-test, and ANOVA test.
Results: The overall study population showed maximum cases having habit of smokeless tobacco (37.9%) and smoking tobacco (36.5%). The overall prevalence of TALs was found to be 7.98%. Our study found strong relation of duration and frequency of habit with respect to occurrence of the lesions.
Conclusion: TALs are often subtle and asymptomatic. Therefore, it is important for the clinician to maintain a high index of suspicion, especially if risk factors such as tobacco habits are present.
Keywords: Duration, frequency, smokeless tobacco and smoking tobacco, tobacco-associated lesions
Joshi M, Tailor M. Prevalence of most commonly reported tobacco-associated lesions in central Gujarat: A hospital-based cross-sectional study. Indian J Dent Res 2016;27:405-9
Joshi M, Tailor M. Prevalence of most commonly reported tobacco-associated lesions in central Gujarat: A hospital-based cross-sectional study. Indian J Dent Res [serial online] 2016 [cited 2022 Jan 21];27:405-9. Available from: https://www.ijdr.in/text.asp?2016/27/4/405/191890
Oral cancer is the 6th most common cancer in the world which accounts for 350,000 new cases and 128,000 deaths annually. Two-third cases of oral cancer are contributed by the developing countries. Oral cancer is almost always preceded by some potentially malignant disorders (PMDs). These PMDs can be detected for up to 15 years before their change to an invasive carcinoma. It usually affects between the ages of 15 and 40 years. India alone accounts for 1/3rd of the world's oral cancer and has a high rate of PMD.[1],[2] It has been estimated that around 43% of cancer deaths are due to predisposing factors such as smoking, smokeless tobacco, betel nut in quid form (pan), alcohol, spicy food, and sharp broken tooth. The most common oral PMDs are oral leukoplakia, erythroplakia, nicotina palati, and oral submucous fibrosis. Tobacco use is common in India and is estimated that 150 million men and 44 million women here use tobacco in various forms.[3] Tobacco has been established as a risk factor for the development of PMDs of oral mucosa. Their overall prevalence among patients attending hospital in certain places of India ranges between 2.5% and 8.4%. It has shown a rate of progression to cancer up to 17% within a mean period of 7 years after diagnosis. The prevalence of oral leukoplakia in India varies from 0.2% to 5.2%. According to various studies, the prevalence of oral submucous fibrosis (OSMF) in India varies between 0.03% and 3.2%.[4] The most important consideration is the relation between the use of tobacco and related products and the development of lesions. The present study is conducted to evaluate the prevalence of tobacco-associated lesions (TALs) among tobacco users and their association with frequency and duration of the said habits.
Total of 60,018 patients attending the Department of Oral Medicine and Radiology from January 2013 to December 2014 with different oral and dental symptoms were screened. Of these, 4795 cases satisfying inclusion and exclusion criteria formed the cohort of the present study. Inclusion criteria were all individuals who had the habit of using smokeless and smoking form of tobacco and had TAL.
Exclusion criteria were patients who were unwilling to give the complete habit details, with no habits, with different oral and dental complaints other than TAL.
All the patients were informed regarding the study and an informed verbal consent was obtained, following which they were interviewed for tobacco-related habits and examined by the trained dental professionals for the presence of any TAL. Along with patient's demographic details, information regarding the type of habit, duration, and frequency was recorded. All the patients were clinically examined using diagnostic instruments with an artificial illumination under aseptic precautions. An ethics clearance was obtained from the institutional ethics and research committee.
Statistical analysis was performed using STATA 13.1 software by applying Pearson's Chi-square test with Fisher's exact test, Independent t-test, and ANOVA test.
Demographic profile of the study population
Age- and gender-wise distribution of total study population showed that 4176 cases were males and 619 were females. A maximum number of study cases fall in the age group of 31-50 years [Graph 1].
Prevalence of tobacco habits in the study population
Overall study population showed maximum cases having habit of smokeless tobacco 1819 (37.9%), smoking tobacco 1748 (36.5%), followed by padiki chewing 1087 (22.7%) and rest of the other habits [Graph 2]. Smokeless tobacco includes tobacco quid, snuff, pan, and pan masala. Padiki is a mixture of smokeless tobacco and betel nut.
Prevalence of tobacco-associated lesions in males and females in the study population
The overall prevalence of TALs was found to be 7.98% whereas prevalence levels of tobacco pouch keratoses, smokers palate, OSMF, leukoplakia, leukoedema, and oral lichen planus were found to be 2.40%, 2.7%, 1.79%, 0.03%, 0.81%, and 0.16%, respectively. Most commonly found TALs in total study population were smokers palate 1648 (34.4%), followed by tobacco pouch keratoses 1444 (30.1%) and OSMF 1078 (22.5%) whereas it was OSMF 248 (40.1%) and tobacco pouch keratoses 242 (39.1%) in females and smokers palate 1608 (38.5%), tobacco pouch keratoses 1202 (28.8%), and OSMF 830 (19.9%) in males [Table 1].
Table 1: Prevalence of tobacco-associated lesions in the study population
Association of tobacco habits and occurrence of tobacco-associated lesions in study population
OSMF 458 (42.1%) and tobacco pouch keratoses 445 (40.9%) were the most common lesions found in cases with habit of padiki chewing. In the cases with habit of smokeless tobacco, lesions most commonly found were tobacco pouch keratoses 965 (53.1%), OSMF 574 (31.6%), and leukoedema 227 (12.5%). Smokers palate 1581 (90.4%) and leukoedema 114 (6.4%) were the lesions found in cases having habit of smoking tobacco. Smoking tobacco includes bidi, cigarette smoking, and hookah. There were cases in the present study having combination of one or the other above-mentioned habits. Among them, most common lesions we found were smokers palate 45 (36%), oral submucous fibrosis 29 (23.2%), tobacco pouch keratoses 25 (20%), and leukoedema 18 (14.4%). There were also few cases in the study population who were giving vague habit history. In this group, oral lichen planus was the most common lesion 8 (50%), followed by smokers palate 3 (18.8%), OSMF 2 (12.5%), and leukoedema 2 (12.5%). All the results were statistically significant as compared to association of habit and occurrence of lesion [Table 2].
Table 2: Association of tobacco habits and occurrence of tobacco-associated lesions in the study population
Dose-response relationship of habit and occurrence of lesion in the study population
In total study population, it was found that mean duration of habit in patients with different lesions was approximately above 10 years of its duration and mean of 6.37 for frequency per day suggesting that they are statistically significant. Pearson's Chi-square test showed statistically significant result (P < 0.00), suggesting that there is a strong relation between duration and frequency of habit as compared to the occurrence of TALs. Furthermore, the descriptive statistics and one-way ANOVA were performed [Table 3] and [Table 4].
Table 3: Dose-response relationship of habit and occurrence of lesion in the study population
Table 4: One-way ANOVA
Oral cancer is a major health problem in tobacco users worldwide. It is one of the ten most common cancers in the world. Despite advances in surgery, radiation, and chemotherapy, the 5-year survival rate for oral cancer has not improved significantly over the past several decades and it remains at about 50-55%.[5],[6] Tobacco and other related habits such as alcohol, betel nut, and poor oral hygiene may contribute to precede oral precancer. Smokeless tobacco has been in use for as long as other forms of tobacco consumption and its use has dramatically increased. In India and Southeast Asia, the chronic use of betel quid (paan) in the mouth has been strongly associated with an increased risk for oral cancer.[7],[8] The quid typically consists of a betel leaf that is wrapped around a mixture of areca nut and slaked lime, usually with tobacco and sometimes with sweeteners and condiments. The slaked lime results in the release of an alkaloid from the areca nut, which produces a feeling of euphoria and well-being in the user. Betel quid chewing often results in a progressive, scarring PMD of the mouth known as OSMF. In India, one study showed a malignant transformation rate of 7.6% for OSMF.[9]
Habitual betel quid chewing is commonly practiced by men and women in Bangladesh, India, Pakistan, and Sri Lanka, while tobacco smoking is much more common among men in these countries compared to women, except for certain small geographic areas.[10]
In the present study, age- and gender-wise distribution of total study population showed that 87.09% (4176) were males and 12.90% (619) were females. This was in accordance with Gupta et al.,[10] Mahawar et al.,[11] Sujatha et al.,[12] and Keluskar and Kale.[13] The reason behind male predominance with precancerous lesions attributes to its association with tobacco and related habits as compared to females. Maximum number of study cases fall in the age group of 31-50 years, which was in accordance with the study conducted by Sujatha et al.[12]
The present study showed that maximum cases had habit of smokeless tobacco (37.9%) or padiki chewing (22.7%) which was in accordance with a study conducted in rural Maharashtra by Kasat et al.[14] The overall prevalence of TALs was found to be 7.98% which was in accordance with the study by Kawatra et al.[4] In the present study, the overall prevalence of various TALs found was as tobacco pouch keratoses (2.40%), smokers palate (2.71%), OSMF (1.79%), leukoplakia (0.03%), leukoedema (0.81%), and oral lichen planus (0.16%) which was slightly higher than in the study by Saraswathi et al.,[15] except leukoedema and oral lichen planus.
In the present study, maximum cases had habit of smokeless tobacco (37.9%) followed by smoking tobacco (36.5%) and padiki chewing (22.7%). The findings of our study were in accordance with Saraswathi et al.[15] and Hashibe et al.[16] We also found strong association between padiki chewing, smokeless tobacco chewing and OSMF, smoking tobacco and occurrence of smokers palate and leukoedema [Table 2]. In the present study, female population was mostly affected with OSMF (40.1%) and tobacco pouch keratoses (39.1%) as compared to males who were more commonly affected by smokers palate (38.5%), tobacco pouch keratoses (28.8%), and OSMF (19.9%). The findings were slightly different from the earlier studies and may be attributed to the fact that in Gujarat, females equally consume chewing tobacco in various forms as compared to males.
Epidemiological studies show that the risk of developing oral cancer is 5-9 times greater for smokers than for nonsmokers, and this risk may increase to as much as 17 times greater for extremely heavy smokers of 80 or more cigarettes per day.[17],[18],[19] The present study also found strong association between duration and frequency of habit and occurrence of the various precancerous lesions. The mean of 12.77 years for duration of habit and mean of 6.37 for frequency per day were found in the present study. The findings of the present study were statistically significant when compared within the group (P < 0.05). Therefore, it suggests strong association of occurrence of precancerous lesions and tobacco and other related habits with respect to their duration and frequency. These observations were in accordance with the studies by Dayal et al.[20] and Gupta et al.[21]
Early precancerous lesion and oral cancers are often subtle and asymptomatic. Therefore, it is important for the clinician to maintain a high index of suspicion, especially if risk factors such as tobacco use or alcohol abuse are present. There is a strong association of these habits with respect to duration and frequency and occurrence of the precancerous lesions. We as diagnostician could be the first one to diagnose these lesions and conditions and can prevent them from malignant transformation.
Narasannavar A, Wantamutte AS. Prevalence of oral precancerous lesions and conditions among tobacco consumers in rural population around Belgaum. A community based cross sectional study. IOSR J Dent Med Sci 2014;13:31-4.
Uplap P, Mishra G, Majumdar P, Gupta S, Rane P, Sadalge P, et al. Oral cancer screening at workplace in India-one-year follow-up. Indian J Community Med 2011;36:133-8.
Rani M, Bonu S, Jha P, Nguyen SN, Jamjoum L. Tobacco use in India: Prevalence and predictors of smoking and chewing in a national cross sectional household survey. Tob Control 2003;12:e4.
Kawatra A, Lathi A, Kamble SV, Sharma P, Parhar G. Oral premalignant lesions associated with Areca nut and tobacco chewing among the tobacco industry workers in area of rural Maharashtra. Natl J Community Med 2012;3:333-8.
Silverman S Jr. Demographics and occurrence of oral and pharyngeal cancers. The outcomes, the trends, the challenge. J Am Dent Assoc 2001; Suppl 132:7S-11S.
Ries LA, Hankey BF, Miller BA, Kosary CL, Clegg L, Edwards BK. Cancer Statistics Review 1973-1988. NIH Publication No. 91-2789. Bethesda: National Cancer Institute; 1991.
Pindborg JJ, Murti PR, Bhonsle RB, Gupta PC, Daftary DK, Mehta FS. Oral submucous fibrosis as a precancerous condition. Scand J Dent Res 1984;92:224-9.
Murti PR, Bhonsle RB, Gupta PC, Daftary DK, Pindborg JJ, Mehta FS. Etiology of oral submucous fibrosis with special reference to the role of Areca nut chewing. J Oral Pathol Med 1995;24:145-52.
Murti PR, Bhonsle RB, Pindborg JJ, Daftary DK, Gupta PC, Mehta FS. Malignant transformation rate in oral submucous fibrosis over a 17-year period. Community Dent Oral Epidemiol 1985;13:340-1.
Gupta PC, Ray CS. Smokeless tobacco and health in India and South Asia. Respirology 2003;8:419-31.
Mahawar P, Anand S, Sinha U, Bansal M, Dixit S. Screening for pre malignant conditions in the oral cavity of chronic tobacco chewers. Natl J Community Med 2011;2:82-5.
Sujatha D, Hebbar PB, Pai A. Prevalence and correlation of oral lesions among tobacco smokers, tobacco chewers, Areca nut and alcohol users. Asian Pac J Cancer Prev 2012;13:1633-7.
Keluskar V, Kale A. An epidemiological study for evaluation of oral pre cancerous lesions, conditions and oral cancer among Belgaum population with tobacco habits. Biosci Biotechnol Res Commun 2010;3:50-4.
Kasat V, Joshi M, Somasundaram KV, Viragi P, Dhore P, Sahuji S. Tobacco use, its influences, triggers, and associated oral lesions among the patients attending a dental institution in rural Maharashtra, India. J Int Soc Prev Community Dent 2012;2:25-30.
Saraswathi TR, Ranganathan K, Shanmugam S, Sowmya R, Narasimhan PD, Gunaseelan R. Prevalence of oral lesions in relation to habits: Cross-sectional study in South India. Indian J Dent Res 2006;17:121-5.
Hashibe M, Sankaranarayanan R, Thomas G, Kuruvilla B, Mathew B, Somanathan T, et al. Alcohol drinking, body mass index and the risk of oral leukoplakia in an Indian population. Int J Cancer 2000;88:129-34.
Neville BW, Damm DD, Allen CM, Bouquot JE. Oral and Maxillofacial Pathology. 2 nd ed. Philadelphia, PA: Saunders; 2002. p. 337-69.
Mashberg A, Boffetta P, Winkelman R, Garfinkel L. Tobacco smoking, alcohol drinking, and cancer of the oral cavity and oropharynx among U.S. veterans. Cancer 1993;72:1369-75.
Lewin F, Norell SE, Johansson H, Gustavsson P, Wennerberg J, Biörklund A, et al. Smoking tobacco, oral snuff, and alcohol in the etiology of squamous cell carcinoma of the head and neck: A population-based case-referent study in Sweden. Cancer 1998;82:1367-75.
Dayal PK, Mani NJ, Bhargava K. Prevalence of oral cancer and precancerous lesions in ′pan′/′supari′ chewers. Indian J Public Health 1978;22:234-45.
Gupta PC. A study of dose-response relationship between tobacco habits and oral leukoplakia. Br J Cancer 1984;50:527-31.
Dr. Manjiri Joshi
Department of Oral Medicine and Radiology, Manubhai Patel Dental College and Hospital, Vadodara, Gujarat
Joshi M
Tailor M
smokeless tobacco and smoking tobacco
tobacco-associated lesions
Contact us | Sitemap | Advertise | What's New | Feedback | Copyright and Disclaimer
© 2007 - Indian Journal of Dental Research | Published by Wolters Kluwer - Medknow
Online since 25th January, 2007


The FORTUNE Global Sustainability Forum program is developed with the guidance of Fortune editors, whose journalistic perspective provides unique insights into the key trends and dynamic people that are driving business. The program is tailored to the immediate needs of its audience, with themes and speakers carefully selected to deliver maximum value.
To suggest a speaker for the FORTUNE Global Sustainability Forum, please send a short biography and a few paragraphs about their expertise to the email above. There are no exact deadlines for speaker submissions.
Participation at the FORTUNE Global Sustainability Forum is by invitation only, and subject to approval. Registration is non-transferrable. Participants are responsible for their own travel and any hotel charges. The registration fee includes the complete program, scheduled meals, and conference materials. Participants are responsible for their own travel and hotel charges.


text:
Käpernick oder Kaepernick ist der Familienname von:

 Colin Kaepernick (* 1987), US-amerikanischer American-Football-Spieler
 Fritz Käpernick (1857–1887), deutscher Schnellläufer
 Harald Käpernick (* 1962), deutscher Fußballspieler und -trainer
Lutz Käpernick (* 1963), deutscher Fußballspieler

redpajama_set_name: RedPajamaWikipedia


journalistically audio. Ms. Mayer also cited unnamed sources who explained they were being advised by a witness that Mr. Trump was fed thoughts in advance of the Fox-sponsored Main discussion in 2015. The network strongly denied that cost, and Ms. Kelly, who served as a discussion moderator, has mentioned she would not feel that her queries were leaked. Mr. Perez said on Wednesday that Ms. Mayer's short article "led me to conclude" that Fox Information was not in shape to sponsor an Formal applicant Discussion board. So far, the Democrats have declared two Main debates for their candidates, which is able to air in June on NBC Information As well as in July on CNN. The decision to exclude Fox Information was very first claimed because of the Washington Put up. Executives at Fox Information say They can be open up to sponsoring A different televised city hall — albeit a single with no imprimatur on the Democratic Social gathering — in the approaching months. The community is free to invite candidates to appear, and candidates are absolutely free to simply accept. Mr. Wallace, Mr. Baier and Ms. MacCallum have Each individual interviewed Mr. Trump throughout his tenure. Fox News did not sponsor haberler a proper discussion during the 2016 Democratic nominating procedure, but it did host a city hall that includes Hillary Clinton and Senator Bernie Sanders of Vermont in March 2016; Mr. Baier served as moderator. Relations amongst political events and news networks will not be often easy. In 2016, Republican officials stripped a debate from NBC Information immediately after alleging that a earlier party, conducted through the network's company sibling CNBC, was "petty and meanspirited in tone," based on the bash's then-chairman, Reince Priebus.


Lead generation is the marketing process of motivating and capturing an interested party into an opportunity or product. In the case of our industry, franchising, the purpose of lead generation is to gain the interest of potential franchise owners to the franchisor's business and model.
Lead generation has seen major changes over the last few years. Specifically, the amount of information that can be found instantly online has seen a rise in 'self-directed buyers'. Today, purchasers can do their own investigation online and can find a mixture of educational resources through search engines, social media, and other online avenues. Through content resources, today's buyer can learn a great deal about a product or service before ever having to even speak to a sales person. So businesses must make sure that they build their digital presence. This, however, is not always a good thing, as the leads are often not focused, lacking any potential to take it any further than an enquiry.
Whether you're a small business or a large national national corporation you should always have a plan for lead generation. In other words, you may have a recognised brand, but if potential owners don't know your recruiting then you will never expand.
There will be hundreds of people out there who will want to be part of your brand, but do not qualify for whatever reason i.e. lack of funding, lack of experience and characteristics. The first thing you must do in this process is to make a list of your ideal franchise owners' attributes.
Different demographics can be found on a number of different platforms. Some people will still be looking for the right opportunity in print format, while others may only be actively searching via the Internet. If you only use one type of recruitment method there is a good chance you may miss out on a quality lead.
If someone is going to invest money into becoming part of your franchise, they are more than likely going to have a interest in your brand or what you offer your clients. You should be sure that your message and the value of your opportunity, no matter what the format, is correct.
It is a different market today, a new economy; the trust is vital to all potential franchise owners. Franchise candidates will choose when and how they want to contact a franchisor and this will be based on the information they have found themselves. This will only happen when they feel they have the sufficient evidence of what the franchisor has to offer. The changes we have seen means that everyone has to change the way they approach lead generation, out with the old ways of selling opportunities and in with the new organic, factual based information that can be verified by a number of avenues.
It's time for franchisors to accept that franchise owner candidates already know the difference between authenticity and untrustworthy sales tactics, and if they don't, they will research enough until they do. They want honest advice about who the franchisor's customers are and why they choose to shop at their business. They wish to learn how the business works and what it takes to be a success before they even talk to a franchise sales representative.


Here Jan Miller shares an activity using two processes in a single 1.5 hr lesson which developed drawing and printing skills together with an introduction to oil painting, with outstanding results.
The focus was Penelope with the Suitors, 1509 by Pintoricchio, displayed in the National Gallery, London. Participants developed an understanding of figures and poses, acknowledging that drawing people is not such a daunting task; irregular and distorted drawings add to the interest. Knowledge of Formal Elements are developed such as line width, shape, tone, composition as well as cropping, overlapping, negative space, size and limited colour.
Penelope at her weaving frame, with her maid in her best dress.
Suitors trying to impress Penelope in their finest clothes, disguise and striking poses.
Symbols included a cat, wool, bird, weaving shuttle, animals, weaving frame, bow & boat etc.
The project began by discussing each of the characters and what clues they could find to describe them. Children analysed how the artist used space in the painting and questioned why he overlapped features and created tension created by cropping. The children thought the colour was quite dull and so wasn't an important consideration – they thought they could eliminate colour from their work and use just one rich colour.
Modern people, including themselves, would feature in their work – inspired by the reflection of gallery visitors in the glass frame, viewing the painting.
Include modern faces and hands only, avoiding bodies or modern costume.
Focus on key figures and eliminate background distraction.
Explore sizes and cropping and drawing from different perspectives.
Use blue background to create mysterious night sky effect to make the figures float around in space or in a dream.
The sessions were run with children aged 8.
The resource could be used with children aged 7 to 11 (older KS 1 children and KS 2 children).
Explore scale, composition, negative space & cropping.
Create a group composition using selected mages.
Using established art to inspire own work.
The session was 1.5 hours with preparation time.
Older children could easily spend more time using smaller brushes or more detail in the mono-printing. Younger children might benefit from two one-hour sessions.
The session below was with the whole class. A helper was beneficial as the children rotated around two activities and because of using white spirit. If managed well, and adequate preparation, there should be no children waiting for a resource or process.
More small portraits could be produced so children could be more selective, considering the quality of line, pressure and amount of ink used.
More washes of colour could be added inside the shapes.
More painted shapes could be added to the background as another layer, as the oil paint, if damp or dry, takes a second layer of any paint well.
The children developed an understanding of process, working spontaneously and with pace. If you follow the sessions you will cover all the areas of exploration listed.
If time is limited, you can easily work on a smaller scale (I have mono-printed on A3-sized floor tiles or used inexpensive plastic table mats). The inked boards can be stacked and used at a later session.
The children could spend time planning in an earlier session by cutting and collaging photocopied elements from the painting to create new compositions to inspire their designs.
Children could work in pairs, each making one of the print methods, then bringing them together.
If you would prefer, use acrylic or poster paint or even oil pastels or crayons for the background.
A historic painting worked well, but any painting could be used as stimulus.
The amount of preparation could be increased. If you are working with a smaller group, the time needed will remain the same.
Collage could be added to the final design from magazines, wallpaper or photographs.
White mono-printed ink drawings could be added by simply inking up thin paper laying it on and drawing through.
Ink up surfaces and tiles. Although children enjoy the energetic rolling of ink, check the amount of ink and that all edges are inked- (easily removed ink by layering newspaper and rubbing to remove excess).
Putting small amounts of oil paint out, small amount of white spirit in labelled glass jar (with hazard label).
Rip out some faces from newspaper, to avoid distraction of other newspaper images. I had some small plastic mirrors available for self-portraits to be reflected in the design.
Where might the sessions might be used?
After school art club or AG&T group.
Have activities on different tables and encourage the children to rotate.
Avoid having a demonstration already completed as they may feel they need to imitate it.
Have wet wipes for hands.
Newspaper on oil painting table.
Oil paint on paper plate/greaseproof paper/foil cake case.
Old jam jars with white spirit so doesn't tip over when brushes in.
Goggles can be made from acetate shapes- hole punched sides with shearing elastic.
Ask parents to donate old tiles.
Small amount of printing ink goes a long way.
A4 dark blue carbon paper is inexpensive and can be masking taped together to give a mono-print effect.
A portable folding drying rack is perfect to dry overnight.
If no old table tops, tiles, mats, plastic sheet from picture frames or off-cuts of Formica boards would be ideal. Don't be afraid to ink the actual table – water based ink easily wipes off, and the children love helping.
The session began by me giving verbal descriptions of the painting we were studying (see above) and with the children making initial pencil drawings in sketchbooks in response to my descriptions, without any visual clues.


text:
\section{Introduction}

Let $(X, d)$ be a metric space. We say $X$ is \textit{doubling} if there
exists an integer $N>0$ such that each ball in $X$ can be covered by $N$
balls of half the radius. Repeated applying this property, it gives that
there exist constants $b,c>0$ and $\alpha >0$ such that for all $r$ and $R$
with $0<r<R<b$, every ball $B(x,R)$ can be covered by $c(\frac{R}{r
)^{\alpha }$ balls of radius $r$. Let $N_{r,R}(X)$ denote the smallest
number of balls with radii $r$ which can cover a ball with radius $R$. The
\textit{Assouad dimension} of $X$, denoted by $\dim _{A}(X)$, is defined as
\begin{equation*}
\dim _{A}(X)=\inf \{\alpha \geq 0\text{ }|\text{ }\exists \text{ }b,c>0\text{
s.t. }N_{r,R}(X)\leq c(\frac{R}{r})^{\alpha }\text{ }\forall \text{
0<r<R<b\},
\end{equation*
which was introduced by Assouad in the late 1970s \cit
{Assouad1,Assouad2,Assouad3}. Now it plays a prominent role in the study of
quasiconformal mappings on $\mathbb{R}^{d}$, and we refer the readers to the
textbook \cite{Heinonen} and the survey paper \cite{Luukkainen} for more
details. It is well known that $\dim _{H}(X)\leq \dim _{P}(X)\leq \dim _{A}X,
$ where $\dim _{H}(\cdot ),\dim _{P}(\cdot )$ are Hausdorff and packing
dimensions respectively.

Suppose that $K$ is a compact subset of $X$ and $s$ is a non-negative real
number. We say $K$ is \textit{Ahlfors-David $s$-regular} if there exists a
Borel measure $\mu $ supported on $K$ and a constant $c\geq 1$ such that,
for all $x\in K$ and $0<r\leq |K|,$
\begin{equation}
c^{-1}r^{s}\leq \mu (B(x,r))\leq cr^{s},\text{ }  \label{AD}
\end{equation
where $B(x,r)$ is the closed ball centered at $x$ with radius $r$ and
|\cdot |$ denotes the diameter of set. Olsen \cite{Olsen} proved that for a
class of fractals with some flexible graph-directed construction, their
Assouad dimensions coincide with their Hausdorff and box dimensions. He also
pointed out that the fractals in \cite{Olsen} are Ahlfors-David regular. It
is well known that self-similar sets and self-conformal sets satisfying the
open set condition (\textbf{OSC}) are always Ahlfors-David regular, see~\cit
{Mattila}. One advantage of such sets is that their dimensions coincide,
namely, for Ahlfors-David\emph{\ }$s$-regular set $K,$ $\dim _{A}K=\dim
_{H}K=\dim _{P}K=s$.

In general, it is difficult to compute the Assouad dimensions of sets which
are not Ahlfors-David regular. Mackay \cite{Mackay} calculated the Assouad
dimensions of two classes of self-affine fractals, namely, Bedford-McMullen
carpets~\cite{Bed} and Lalley-Gatzouras sets~\cite{Mcm},  and he also solved
the problem posed by Olsen \cite{Olsen}. Fraser \cite{Fraser} obtained
Assouad dimensions for certain classes of self-affine sets and
quasi-self-similar sets.

\smallskip

In this paper, we studied the Assouad dimension formula of Moran sets,
Cantor-like sets and homogeneous sets. Moran set was first studied by Moran
in \cite{Moran}, where most cases are not Ahlfors-David regular. First, we
recall the definition of Moran set.

Let $\{n_{k}(\geq 2)\}_{k\geq 1}$ be a sequence of positive integers$.$ For
each $k=0,1,2,\cdots$,let $D_{k}=\{u_{1}u_{2}\cdots u_{k}:1\leq u_{j}\leq
n_{j}\ $, $j\leq k\}$ be the set of words of length $k$, with
D_{0}=\{\emptyset \}$ containing only the empty word $\emptyset$. Let
D=\cup _{k=0}^{\infty }D_{k}$ be the set of all finite words. Suppose that
J\subset \mathbb{R}^{d}$ is a compact set with $\mbox{int}(J)\neq
\varnothing $ (we always write int($\cdot )$ for the interior of set). Let
\{\phi _{k}\}_{k\geq 1}$ be a sequence of positive real vectors where $\phi
_{k}=(c_{k,1},c_{k,2},\cdots ,c_{k,n_{k}})$ and $\Sigma
_{j=1}^{n_{k}}(c_{k,j})^{d}\leq 1$ for each $k\in \mathbb{N}$. We say the
collection $\mathcal{F}=\{J_{\mathbf{u}}:\mathbf{u}\in D\}$ of closed
subsets of $J$ fulfills the \textit{Moran structure} if it satisfies the
following Moran structure conditions (MSC):

(1) For each $\mathbf{u}\in D$, $J_{\mathbf{u}}$ is geometrically similar to
$J$, i.e., there exists a similarity $S_{\mathbf{u}}:\mathbb{R
^{d}\rightarrow \mathbb{R}^{d}$ such that $J_{\mathbf{u}}=S_{\mathbf{u}}(J)
. We write $J_{\emptyset }=J$ for empty word $\emptyset $.

(2) For all $k\in \mathbb{N}$ and $\mathbf{u}\in D_{k-1}$, the elements $J_
\mathbf{u}1}, J_{\mathbf{u}2},\cdots ,J_{\mathbf{u}n_{k}}$ of $\mathcal{F}$
are the subsets of $J_{\mathbf{u}}$ with disjoint interiors, ie., $\mbox{int
(J_{\mathbf{u}i})\cap \mbox{int}(J_{\mathbf{u}i^{\prime }})=\varnothing $
for $i\neq i^{\prime }$. Moreover, for all $1\leq i\leq n_{k} $,
\begin{equation*}
\frac{|J_{\mathbf{u}i}|}{|J_{\mathbf{u}}|}=c_{k,i}.
\end{equation*
We call $E=E(\mathcal{F})=\bigcap\nolimits_{k=1}^{\infty }\bigcup\nolimits_
\mathbf{u}\in D_{k}}J_{\mathbf{u}}$ a \textit{Moran set} determined by
\mathcal{F}$. For all $\mathbf{u}\in D_{k}$, the elements $J_{\mathbf{u}}$
are called \textit{\ $k$th-level basic sets} of $E$. Suppose the set $J$ and
the sequences $\{n_{k}\}$ and $\{\phi _{k}\}$ are given. We denote by
\mathcal{M}=\mathcal{M}(J,\{n_{k}\},\{\phi _{k}\})$ the class of the Moran
sets satisfying the MSC.

For any $k^{\prime }>k,$ let $s_{k,k^{\prime }}$ be the unique real solution
of the equation $\Delta _{k,k^{\prime }}(s)=1$, where
\begin{equation}  \label{eqns}
\Delta _{k,k^{\prime }}(s)=\prod\nolimits_{i=k+1}^{k^{\prime }}\left(
\sum\nolimits_{j=1}^{n_{i}}(c_{i,j})^{s}\right) .
\end{equation}
If the sequence $\{\sup\nolimits_{k}s_{k,k+m}\}_{m=1}^\infty$ converges, we
write
\begin{equation*}
s^{\ast \ast }=\lim_{m\rightarrow \infty }\left(
\sup\nolimits_{k}s_{k,k+m}\right).
\end{equation*}

In Section 2, we prove that the sequence $\{\sup\nolimits_{k}s_{k,k+m
\}_{m=1}^\infty$ is indeed convergent under the assumption $c_{\ast
}=\inf_{i,j}c_{i,j}>0.$ Furthermore, The following theorem indicates that
the limit is the Assouad dimension of Moran sets.

\begin{theorem}
\label{thmMS} Suppose that $\mathcal{M}=\mathcal{M}(J,\{n_{k}\},\{\phi
_{k}\})$ is a Moran class with $c_{\ast }=\inf_{i,j}c_{i,j}>0.$ Then, for
all $E\in \mathcal{M}$,
\begin{equation*}
\dim _{A}E=s^{\ast \ast}.
\end{equation*}
\end{theorem}

As an immediate consequence, we have the following corollary.

\begin{corollary}
Suppose that $\mathcal{M}=\mathcal{M}(J,\{n_{k}\},\{\phi _{k}\})$ is a Moran
class with $c_{\ast }=\inf_{i,j}c_{i,j}>0.$ Let $c_{k,1}=c_{k,2}=\cdots
=c_{k,n_{k}}=c_{k}\ $ for each $k\in\mathbb{N}$. Then, for all $F\in
\mathcal{M}$,
\begin{equation*}
\dim _{A}F=\lim_{m\rightarrow \infty }\left( \sup_{k}\frac{\log (n_{k}\cdots
n_{k+m})}{-\log (c_{k}\cdots c_{k+m})}\right) .
\end{equation*}
\end{corollary}

Let $s_{\ast }$ and $s^{\ast }$ be the upper and lower limits of the
sequence $\{s_{0,m}\}_{m=1}^\infty$, that is,
\begin{equation*}
s_{\ast }=\underline{\lim }_{m\rightarrow \infty }s_{0,m} \text{ and }
s^{\ast }=\overline{\lim }_{m\rightarrow \infty }s_{0,m}.
\end{equation*
It was shown in \cite{Hua,HuaLi,Wen1,Wen2} that, for all $E\in \mathcal{M}$
with $c_{\ast }>0,$
\begin{equation*}
\dim _{H}E=s_{\ast }\text{ and }\dim _{P}E=s^{\ast }.
\end{equation*
In next example, we will construct a Moran set to satisfy $\dim _{H}E<\dim
_{P}E<\dim _{A}E.$

Note that it is also a counter-example to the conclusion in \cite{Li}.
Hereby, Theorem~\ref{thmMS} corrects their main conclusion.

\begin{example}
Let $\{p_{i}\}_{i}$ be an increasing sequence of integers such that
p_{i+1}-p_{i}>i$ for all $i$ and
\begin{equation*}
\lim_{i\rightarrow \infty }\frac{p_{i-1}}{p_{i}-p_{i-1}}=\lim_{i\rightarrow
\infty }\frac{i}{p_{i}-p_{i-1}}=0.
\end{equation*
Let $J=[0,1],n_{k}\equiv 2$ and
\begin{equation*}
c_{k,1}=c_{k,2}=\left\{
\begin{array}{ll}
1/4 & \text{if }k\in \lbrack p_{i}+1,p_{i}+i]\text{ for some\ }i\in \mathbb{
}, \\
1/8 & \text{if }k\in \lbrack p_{i}+i+1,p_{i+1}]\text{ for some\ even }i\in
\mathbb{N}, \\
1/16 & \text{if }k\in \lbrack p_{i}+i+1,p_{i+1}]\text{ for some\ odd }i\in
\mathbb{N}
\end{array
\right.
\end{equation*
Then we have $s_{\ast }=\frac{1}{4},$ $s^{\ast }=\frac{1}{3},$ $s^{\ast \ast
}=\frac{1}{2}.$ Clearly, for all $E\in \mathcal{M}(J,\{n_{k}(\equiv
2)\},\{(c_{k,1},c_{k,2})\})$, the dimensions inequality strictly holds, that
is ,
\begin{equation*}
\dim _{H}E=\frac{1}{4}<\dim _{P}E=\frac{1}{3}<\dim _{A}E=\frac{1}{2}.
\end{equation*}
\end{example}

Suppose $\{a_{n}\}$ is a sequence of positive numbers with
\sum\nolimits_{n}a_{n}<\infty .$ Given sequences $\{c_{k}\}_{k\geq 1\text{ }
} $and $\{n_{k}\}_{k\geq 1}$ such that $c_{k}\in (0,1)$ and $n_{k}\in
\mathbb{N}\cap \lbrack 2,\infty )$ for all $k\in\mathbb{N}.$ We always
assume that $c_{\ast }=\inf_{k}c_{k}>0.$ Let $I$ be the initial set such
that $\mbox{int}(I)\neq \varnothing $. For each $i_{1}\cdots i_{k-1}\in
D_{k-1},$ suppose that $I_{i_{1}\cdots i_{k-1}1}$, $I_{i_{1}\cdots i_{k-1}2}
, $\cdots$, $I_{i_{1}\cdots i_{k-1}n_{k}}\subset I_{i_{1}\cdots i_{k-1}}$
are geometrically similar to $I_{i_{1}\cdots i_{k-1}}$ such that
\begin{equation*}
c_{k}(1-a_{k})\leq \frac{|I_{i_{1}\cdots i_{k-1}j}|}{|I_{i_{1}\cdots
i_{k-1}}|}\leq c_{k}(1+a_{k}), \quad j=1,2,\cdots,n_k,
\end{equation*
where the interiors of $I_{i_{1}\cdots i_{k-1}j}$ are pairwise disjoint. We
call
\begin{equation*}
K=\bigcap \nolimits_{k=1}^{\infty }\bigcup\nolimits_{i_{1}\cdots i_{k}\in
D_{k}}I_{i_{1}\cdots i_{k}}
\end{equation*}
a \textit{Cantor-like set}, and we write $\mathcal{C}(I,\{c_{k}\}_{k}
\{n_{k}\}_{k},\{a_{k}\}_{k})$ for the collection of such sets.

\begin{remark}
Cantor-like sets may not be Moran sets.
\end{remark}

\begin{theorem}
Suppose that $K\in \mathcal{C}(I,\{c_{k}\}_{k},\{n_{k}\}_{k},\{a_{k}\}_{k})$
is a Cantor-like set$.$ Then
\begin{equation*}
\dim _{A}K=\lim_{m\rightarrow \infty }\left( \sup_{k}\frac{\log (n_{k}\cdots
n_{k+m})}{-\log (c_{k}\cdots c_{k+m})}\right) .
\end{equation*}
\end{theorem}

In fact, for Ahlfors-David regular set, using (\ref{AD}), there exist
constants$\ 0<\eta <1\leq \lambda $ and $1<\delta \leq \Delta <\infty $ such
that, for all $x,x^{\prime }\in K$ and $r\leq |K|,$
\begin{equation}
\lambda ^{-1}\leq \frac{\mu (B(x,r))}{\mu (B(x^{\prime },r))}\leq \lambda ,
\label{self1}
\end{equation
\begin{equation}
\delta \leq \frac{\mu (B(x,r))}{\mu (B(x,\eta r))}\leq \Delta .\text{ }
\label{***}
\end{equation
It follows from (\ref{***}) that the measure $\mu $ and set $K$ are
doubling, and $K$ is uniformly perfect \cite{Lv}. We say a compact subset $K$
of $X$ is \textit{homogeneous} if there exists a Borel measure $\mu $
supported on $K$ satisfying $(\ref{self1})$ and $(\ref{***}),$ and we refer
the readers  to~\cite{Lv} for details.

\begin{remark}
All Ahlfors-David regular sets are homogeneous, but homogeneous sets may not
be Ahlfors-David regular.
\end{remark}

Given a point $x\in K,$ we write
\begin{equation}
\alpha _{x}(r)=\frac{\log \mu (B(x,r))}{\log r},  \label{E:equ0}
\end{equation
for $0<r\leq |K|$. Here $\alpha _{x}(r)$ is like the function with respect
to pointwise dimension of measure.

Given $\epsilon>0$, we write
\begin{equation*}
\Omega =\{g(r):(0,\varepsilon )\rightarrow \mathbb{R}^{+}|\text{
0<\inf_{r<\varepsilon }g(r)\leq \sup_{r<\varepsilon }g(r)<\infty \}.
\end{equation*
For each $g\in \Omega ,$ we focus on the behavior of function $g(r)$ as $r$
tends to $0.$ If a mapping $h\in\Omega $ satisfies that, for all
r<\varepsilon ,
\begin{equation}
\left\vert h(r)-g(r)\right\vert \leq C|\log r|^{-1}  \label{E:equ}
\end{equation
for some constant $C,$ we say $h$ and $g$ are \textit{equivalent}, denoted
by $g\sim h$, and we write equivalence class $[g]=\{h:g\sim h\}.$ By the
result of \cite{Lv}, we have $\alpha _{x}(r)\in \Omega .$ Notice that
\alpha _{x}(r)\sim \alpha _{x^{\prime }}(r)$ by (\ref{self1}), we use $h(r)$
to denote any function in the equivalence class $[\alpha _{x}(r)]$ with
x\in K$, and $h(r)$ is called a \textit{scale function} of $K.$

\begin{remark}
For Ahlfors-David $s$-regular set$,$ we can take $h(r)\equiv s.$
\end{remark}

It is easy to check $\dim _{H}K=\liminf_{r\rightarrow 0}h(r)$ and $\dim
_{P}K=\limsup_{r\rightarrow 0}h(r)$, see~\cite{Lv}. Similarly, scale
functions also play an important role in the Assouad dimension formula of
homogeneous sets.

\begin{theorem}
Suppose $K$ is homogeneous with a scale function $h(r)$. Then
\begin{equation*}
\dim _{A}K=\lim_{\rho \rightarrow 0}\left( \sup_{R}\left\vert \frac{h(R)\log
R-h(\rho R)\log (\rho R)}{\log \rho }\right\vert \right) .
\end{equation*}
\end{theorem}

\begin{remark}
Suppose that $h(r)$ is defined on $(0,\varepsilon ).$ Using $(\ref{***}),$
we can obtain that for all $\varepsilon _{1},\varepsilon _{2}\leq
\varepsilon ,$
\begin{equation*}
\lim_{\rho \rightarrow 0}\sup_{R<\varepsilon _{1}}\psi (R,\rho )=\lim_{\rho
\rightarrow 0}\sup_{R<\varepsilon _{2}}\psi (R,\rho ),
\end{equation*
where $\psi (R,\rho )=\left\vert \frac{h(R)\log R-h(\rho R)\log (\rho R)}
\log \rho }\right\vert .$
\end{remark}

For each Cantor-like set $K\in \mathcal{C}(I,\{c_{k}\}_{k},\{n_{k}\}_{k}
\{a_{k}\}_{k}),$ using the approach in \cite{Lv}, it is clear that $K$ is
homogeneous with a scale function
\begin{equation*}
h(r)=\frac{\log n_{1}\cdots n_{k}}{-\log c_{1}\cdots c_{k}}\quad \text{ for
c_{1}\cdots c_{k}|I|<r\leq c_{1}\cdots c_{k-1}|I|.
\end{equation*
Therefore Theorem 2 follows immediately from Theorem 3.

\begin{remark}
Using the result in \cite{Lv}, for every Cantor-like set $K$ as above, we
have
\begin{equation*}
\dim _{H}K=\underline{\lim }_{k\rightarrow \infty }\frac{\log n_{1}\cdots
n_{k}}{-\log c_{1}\cdots c_{k}} , \quad \dim _{P}K=\overline{\lim
_{k\rightarrow \infty }\frac{\log n_{1}\cdots n_{k}}{-\log c_{1}\cdots c_{k}
.
\end{equation*}
\end{remark}

For the rest of the paper, we will prove Theorem 1 and Theorem 3 in Section
2 and Section 3 respectively.

\bigskip

\section{Assouad Dimension of Moran Set}

Suppose that $\mathcal{M}=\mathcal{M}(J,\{n_{k}\},\{\phi _{k}\})$ where
\phi _{k}=(c_{k,1},c_{k,2},\cdots ,c_{k,n_{k}})$, $k=1,2,\cdots.$ Without
loss of generality, we assume that $|J|=1.$

For each word $\mathbf{u}=u_{1}u_{2}\cdots u_{k}\in D_{k}$, we write $
\mathbf{u}|(=k) $ for the length of $\mathbf{u}$. Given $k,k^{\prime }\in
\mathbb{N}$, we write
\begin{equation*}
D_{k,k^{\prime }}= \{\mathbf{v}=v_{k}\cdots v_{k^{\prime }}:1\leq v_{j}\leq
n_{j}\text{ for }k\leq j\leq k^{\prime }\},
\end{equation*
for $k\leq k^{\prime}$, otherwise, $D_{k,k^{\prime }}=\{\emptyset \}$. Note
that $D_{1,k}=D_{k}$. For $\mathbf{v}=v_{k}\cdots v_{k^{\prime }}\in
D_{k,k^{\prime }},$ we write
\begin{equation*}
c_{\mathbf{v}}=c_{k,v_{k}}\cdots c_{k^{\prime },v_{k^{\prime }}},
\end{equation*
with $c_{\mathbf{\emptyset }}=1$. For $\mathbf{u}=u_{1}u_{2}\cdots
u_{k-1}\in D_{k-1}$ and $\mathbf{v}=v_{k}v_{k+1}\cdots v_{k^{\prime }}\in
D_{k,k^{\prime }}$, we write
\begin{equation*}
\mathbf{u}\ast \mathbf{v}=u_{1}u_{2}\cdots u_{k-1}v_{k}v_{k+1}\cdots
v_{k^{\prime }}\in D_{k^{\prime }}.
\end{equation*}
For $\mathbf{v}\in D_{k,k^{\prime }}$, we denote by $\mathbf{v}^{-}$ the
word obtained by deleting the last letter of $\mathbf{v}$. Note that
\mathbf{v}^{-}=\emptyset $ (the empty word) if $k=k^{\prime }-1$. Given
\mathbf{u}\in D$, for $0<\delta <c_{\ast }$, we write
\begin{equation}
\mathcal{A}_{\mathbf{u}}(\delta )=\{\mathbf{u}\ast \mathbf{v}\in D:\text{
c_{\mathbf{v}}\leq \delta <c_{\mathbf{v}^{-}}\}.  \label{2.1}
\end{equation
For $\mathbf{u}=\emptyset ,$ we write $\mathcal{A}(\delta )$ for $\mathcal{A
_{\emptyset }(\delta ).$

Let $\Lambda =\{u_{1}u_{2}\cdots u_{k}\cdots :u_{1}u_{2}\cdots u_{k}\in
D_{k} $ for all $k\}$ be the symbolic system composed of infinite words.
Given a word $\mathbf{i=}i_{1}\cdots i_{n}\in D_{n},$ we call
\begin{equation*}
\lbrack \mathbf{i}]=\{u_{1}\cdots u_{n}\cdots \in \Lambda :u_{1}\cdots
u_{n}=i_{1}\cdots i_{n}\}
\end{equation*}
the \textit{cylinder} with respect to the word $\mathbf{i}.$

\begin{lemma}
\label{lem1} Given $\mathbf{u}\in D,$ we have
\begin{equation}
1=\sum\limits_{\mathbf{u}\ast \mathbf{v}\in \mathcal{A}_{\mathbf{u}}(\delta
)}\frac{(c_{\mathbf{v}})^{s}}{\prod_{p=|\mathbf{u}|+1}^{|\mathbf{u}|+
\mathbf{v}|}\sum\limits_{q=1}^{n_{p}}c_{p,q}^{s}}.  \label{xixi}
\end{equation}
\end{lemma}

\begin{proof}
Fix $\mathbf{u}\in D,$ we have a probability measure $\mu $ supported on $
\mathbf{u}]$ such that
\begin{equation*}
\mu ([\mathbf{u}\ast \mathbf{v}])=\frac{(c_{\mathbf{v}})^{s}}{\prod_{p=
\mathbf{u}|+1}^{|\mathbf{u}|+|\mathbf{v}|}\su
\limits_{q=1}^{n_{p}}c_{p,q}^{s}}\text{ for all }\mathbf{u}\ast \mathbf{v
\in D.
\end{equation*
Since $[\mathbf{u}]=\bigcup\limits_{\mathbf{u}\ast \mathbf{v}\in \mathcal{A
_{\mathbf{u}}(\delta )}[\mathbf{u}\ast \mathbf{v}]$ is a disjoint union, we
obtain
\begin{equation*}
1=\sum\limits_{\mathbf{u}\ast \mathbf{v}\in \mathcal{A}_{\mathbf{u}}(\delta
)}\mu ([\mathbf{u}\ast \mathbf{v}])=\sum\limits_{\mathbf{u}\ast \mathbf{v
\in \mathcal{A}_{\mathbf{u}}(\delta )}\frac{(c_{\mathbf{v}})^{s}}{\prod_{p=
\mathbf{u}|+1}^{|\mathbf{u}|+|\mathbf{v}|}\su
\limits_{q=1}^{n_{p}}c_{p,q}^{s}}.
\end{equation*}
\end{proof}

The following lemma can be obtained directly by using Lemma 9.2 in \cit
{Falconer}.

\begin{lemma}
\label{lem2} \label{lemma1} Suppose $c_{\ast }>0$. Then there exits a
positive integer $l$ such that for all $0<\delta <c_{\ast },$ $\mathbf{u}\in
D$ and $x\in E\cap J_{\mathbf{u}},$ we have
\begin{equation*}
\sharp \{\mathbf{u}\ast \mathbf{v}\in \mathcal{A}_{\mathbf{u}}(\delta )\text{
}|\text{ }B(x,c_{\mathbf{u}}\delta )\cap J_{\mathbf{u}\ast \mathbf{v}}\neq
\varnothing \}\leq l.
\end{equation*
In particular, if $\mathbf{u}$ is the empty word, we have
\begin{equation*}
\sharp \{\mathbf{v}\in \mathcal{A}(\delta )\text{ }|\text{ }B(x,\delta )\cap
J_{\mathbf{v}}\neq \varnothing \}\leq l.
\end{equation*}
\end{lemma}

\medskip

\begin{lemma}
Suppose $c_{\ast }>0$. Let $s_{k,k+m}$ be defined by $(\ref{eqns})$. Then
the sequence $\{\sup_{k}s_{k,k+m}\}_{m=1}^\infty$ is convergent.
\end{lemma}

\begin{proof}
Suppose $E\subset \mathbb{R}^{d}.$ We denote by $\mathcal{L}$ the Lebesgue
measure on $\mathbb{R}^{d}.$ Recall that $c_{\ast }>0$ and $n_{k}\geq 2$.
Since for each $\mathbf{u}\in D_{k-1},$
\begin{equation*}
\text{int}(J_{\mathbf{u}\ast i})\cap \text{int}(J_{\mathbf{u}\ast
j})=\varnothing ,
\end{equation*}
for all $i\neq j\leq n_{k}$, we hav
\begin{equation*}
\sum\nolimits_{i=1}^{n_{k}}\mathcal{L}(\text{int(}J_{\mathbf{u}\ast i}\text
))}\leq \mathcal{L}(\text{int(}J_{\mathbf{u}}\text{)),}
\end{equation*
that is, $\sum\nolimits_{i=1}^{n_{k}}c_{k,i}^{d}\leq 1$. It implie
\begin{equation}
\max_{i}c_{k,i}\leq (1-c_{\ast }^{d})^{1/d}\text{ and }\sup_{k}n_{k}\leq
(c_{\ast })^{-d}.  \label{aa}
\end{equation}

For every $m$, we write
\begin{equation*}
\theta _{m}=\sup_{k}s_{k,k+m}.
\end{equation*
Fix an integer $m\in \mathbb{N}$. Let $s>\theta _{m}$. For each $n\in
\mathbb{Z}\cap \lbrack 0,m-1]$, we have
\begin{equation*}
\Delta _{t,t+pm+n}(s)=\left( \prod\nolimits_{i=0}^{p-1}\Delta
_{t+im,t+(i+1)m}(s)\right) \cdot \Delta _{t+pm,t+pm+n}(s).
\end{equation*
Hence, by (\ref{aa}), $\Delta _{t+pm,t+pm+n}(s)\leq (\sup_{k}n_{k})^{n}\leq
(c_{\ast })^{-nd}$ and
\begin{eqnarray*}
\Delta _{t+im,t+(i+1)m}(s) &\leq &\Delta _{t+im,t+(i+1)m}(\theta _{m})\cdot
(\sup_{k,i}c_{k,i})^{s-\theta _{m}} \\
&\leq &1\cdot (1-c_{\ast }^{d})^{(s-\theta _{m})/d} \\
&=&(1-c_{\ast }^{d})^{(s-\theta _{m})/d}.
\end{eqnarray*
Therefore, for all $t\in \mathbb{N}$, we have
\begin{equation*}
\Delta _{t,t+pm+n}(s)\leq (1-c_{\ast }^{d})^{p(s-\theta _{m})/d}(c_{\ast
})^{-nd}
\end{equation*
which means there exists an integer $p_{0}(s)$ such that for all $p\geq
p_{0}(s),$
\begin{equation*}
\Delta _{t,t+pm+n}(s)\leq 1,
\end{equation*
that is,
\begin{equation*}
s_{t,t+pm+n}\leq s,
\end{equation*
for all $p\geq p_{0}(s)$ and $t\geq 0$. Hence
\begin{equation*}
\overline{\lim }_{p\rightarrow \infty }\theta _{pm+n}=\overline{\lim
_{p\rightarrow \infty }\sup\nolimits_{t}s_{t,t+pm+n}\leq s.
\end{equation*
Since it holds for all $s>\theta _{m}$, we obtain $\overline{\lim
_{p\rightarrow \infty }\theta _{pm+n}\leq \theta _{m}$. Thu
\begin{equation*}
\overline{\lim }_{p\rightarrow \infty }\theta _{pm+n}\leq \inf_{m}\theta
_{m}\leq \underline{\lim }_{m\rightarrow \infty }\theta _{m},
\end{equation*
which implies that $\lim_{m\rightarrow \infty }\theta _{m}$ exists.$\ $
\end{proof}

\medskip

\begin{proof}[Proof of Theorem 1]
$\ $

We first prove that $s^{\ast \ast }$ is an upper bound of $\dim _{A}E$. It
suffices to verify that the inequality $\dim _{A}E\leq s$ holds for all
s>s^{\ast \ast }$.

Since $s>\lim_{m\rightarrow \infty}(\sup_{k}s_{k,k+m})$, there exists a
positive integer $N$ such that, for all $m>N$, we have $s>s_{k,k+m}$.
Therefore, for all $m>N$
\begin{equation}
\prod\limits_{i=k+1}^{k+m}\left( \sum\limits_{j=1}^{n_{i}}c_{i,j}^{s}\right)
\leq \prod\limits_{i=k+1}^{k+m}\left(
\sum\limits_{j=1}^{n_{i}}c_{i,j}^{s_{k,k+m}}\right) =1.  \label{2.2}
\end{equation}

Fix a word $\mathbf{i}\in D$ and $\delta \in (0,c_{\mathbf{i}^{-}}).$ The
fact that $c_{\ast }>0$ implies that the sequence $\{n_{k}\}$ is bounded,
say $\varpi >1,$ that is, $n_{k}\leq \varpi, k=1,2,\cdots.$ Thus, for all
0<m\leq N$,
\begin{equation}
\prod\limits_{i=k+1}^{k+m}\left( \sum\limits_{j=1}^{n_{i}}c_{i,j}^{s}\right)
\leq \varpi ^{N}.  \label{ha}
\end{equation
By (\ref{2.2}) and (\ref{ha}), we have, for all $\mathbf{j}\in D_p$,
\begin{equation}
\prod_{p=|\mathbf{i}|+1}^{|\mathbf{i}|+|\mathbf{j}|}\su
\limits_{q=1}^{n_{p}}c_{p,q}^{s}\leq \varpi ^{N}.  \label{we}
\end{equation}
Combining Lemma~\ref{lem1} with (\ref{we}), we have
\begin{align*}
1& =\sum\limits_{\mathbf{i}\ast \mathbf{j}\in \mathcal{A}_{\mathbf{i
}(\delta )}\frac{(c_{\mathbf{j}})^{s}}{\prod_{p=|\mathbf{i}|+1}^{|\mathbf{i
|+|\mathbf{j}|}\sum\limits_{q=1}^{n_{p}}c_{p,q}^{s}} \\
& \geq \varpi ^{-N}\sum\limits_{\mathbf{i}\ast \mathbf{j}\in \mathcal{A}_
\mathbf{i}}(\delta )}(c_{\mathbf{j}})^{s} \\
& \geq \varpi ^{-N}\sum\limits_{\mathbf{i}\ast \mathbf{j}\in \mathcal{A}_
\mathbf{i}}(\delta )}(c_{\ast }c_{\mathbf{j}^{-}})^{s} \\
& \geq (\varpi ^{-N}c_{\ast }^{s})\cdot \delta ^{s}\cdot \sharp \mathcal{A}_
\mathbf{i}}(\delta ).
\end{align*
It follows that
\begin{equation}
\sharp \mathcal{A}_{\mathbf{i}}(\delta )\leq \frac{\varpi ^{N}}{c_{\ast
}^{s}\delta ^{s}}  \label{2.3}
\end{equation
for all $\mathbf{i}\in D$ and all $0<\delta <c_{\ast }$.

Fix a point $x\in E$ and $r,R$ with $0<r<R$. Since $E$ is doubling, without
loss of generality, we may assume that
\begin{equation*}
0<r<c_{\ast }R<R<c_{\ast }.
\end{equation*
It is clear that
\begin{equation}
B(x,R)\cap E\subset \bigcup\limits_{\mathbf{i}\in \mathcal{A}(R),B(x,R)\cap
J_{\mathbf{i}}\neq \varnothing }J_{\mathbf{i}}\cap E.  \label{2.4}
\end{equation}
For each $\mathbf{i}\in \mathcal{A}(R)$ with $B(x,R)\cap J_{\mathbf{i}}\neq
\varnothing $, we have
\begin{equation*}
J_{\mathbf{i}}\cap E\subseteq \bigcup\limits_{\mathbf{i}\ast \mathbf{j}\in
\mathcal{A}_{\mathbf{i}}(r/R)}J_{\mathbf{i}\ast \mathbf{j}}.
\end{equation*}
Now taking $x_{\mathbf{i},\mathbf{j}}\in J_{\mathbf{i}\ast \mathbf{j}}\cap E
, we have
\begin{equation*}
J_{\mathbf{i}\ast \mathbf{j}}\subseteq B(x_{\mathbf{i},\mathbf{j}},r),
\end{equation*}
due to $c_{\mathbf{i}\ast \mathbf{j}}=c_{\mathbf{i}}c_{\mathbf{j}}\leq
R\cdot r/R=r$ and $|J|=1.$

Thus by (\ref{2.4}), we obtain that
\begin{equation}
B(x,R)\cap E\subset \bigcup\limits_{\substack{ \mathbf{i}\in \mathcal{A}(R)
\\ B(x,R)\cap J_{\mathbf{i}}\neq \varnothing }}\bigcup\limits_{\mathbf{i
\ast \mathbf{j}\in \mathcal{A}_{\mathbf{i}}(r/R)}B(x_{\mathbf{i},\mathbf{j
},r).  \label{2.6}
\end{equation
By (\ref{2.3}) and Lemma \ref{lem2}, we have
\begin{align*}
N_{r,R}(E)& \leq \sum\limits_{\substack{ \mathbf{i}\in \mathcal{A}(R)  \\
B(x,R)\cap J_{\mathbf{i}}\neq \varnothing }}\sharp \mathcal{A}_{\mathbf{i
}(r/R) \\
& \leq \sum\limits_{\substack{ \mathbf{i}\in \mathcal{A}(R)  \\ B(x,R)\cap
J_{\mathbf{i}}\neq \varnothing }}\frac{\varpi ^{N}}{c_{\ast }^{s}}\left(
\frac{R}{r}\right) ^{s} \\
& \leq \frac{\varpi ^{N}}{c_{\ast }^{s}}\left( \frac{R}{r}\right) ^{s}\cdot
\sharp \{\mathbf{i}\in \mathcal{A}(R):B(x,R)\cap J_{\mathbf{i}}\neq
\varnothing \} \\
& \leq \frac{l\varpi ^{N}}{c_{\ast }^{s}}\left( \frac{R}{r}\right) ^{s}.
\end{align*
Hence $s$ is an upper bound, and the arbitrariness implies that
\begin{equation*}
\dim _{A}E\leq s^{\ast \ast }.
\end{equation*}

\medskip

For the rest of the proof, we will verify that $s^{\ast \ast }$ is also a
lower bound.

Since $s^{\ast \ast }$ is the limit of $\{\sup_{k}s_{k,k+m}\}$, there exists
a sequence $\{(m_{k},m_{k}^{\prime })\}_{k=1}^{\infty }$ of integer pairs
with $(m_{k}^{\prime }-m_{k})$ tending to $\infty$ such that
\begin{equation*}
\lim_{k\rightarrow \infty }s_{m_{k},m_{k}^{\prime }}=s^{\ast \ast }.
\end{equation*}

Arbitrarily choose $s<s^{\ast \ast }$. Without loss of generality, we assume
that, for all $k\in \mathbb{N}$.
\begin{equation*}
s_{m_{k},m_{k}^{\prime }}>s.
\end{equation*
Hence, it is clear that
\begin{equation*}
\Delta _{m_{k},m_{k}^{\prime }}(s)>\Delta _{m_{k},m_{k}^{\prime
}}(s_{m_{k},m_{k}^{\prime }})=1.
\end{equation*
Fix an integer $k$, we have
\begin{equation}
\sum\limits_{\mathbf{j}\in D_{m_{k}+1,m_{k}^{\prime }}}c_{\mathbf{j
}^{s}=\Delta _{m_{k},m_{k}^{\prime }}(s)>1.  \label{2.7}
\end{equation
For each $p\in \mathbb{N}\cup \{0\}$, let
\begin{equation}
\mathcal{B}_{p,k}=\{\mathbf{j}\in D_{m_{k}+1,m_{k}^{\prime }}:2^{-p-1}<c_
\mathbf{j}}\leq 2^{-p}\},  \label{test}
\end{equation
and we write
\begin{equation*}
p_{k}=\min \{p:\mathcal{B}_{p,k}\neq \varnothing \}.
\end{equation*
Since
\begin{equation*}
2^{-p-1}\leq c_{\mathbf{j}}\leq (1-c_{\ast }^{d})^{(m_{k}^{\prime
}-m_{k})/d},
\end{equation*
it is obvious that the sequence $\{p_k\}$ tends to infinity, that is,
\begin{equation*}
\lim_{k}p_{k}=\infty ,
\end{equation*
Thus by (\ref{2.7}) and (\ref{test}), we obtain that
\begin{equation}
\sum\limits_{p=0}^{\infty }\sharp \mathcal{B}_{p,k}2^{-ps}>1.  \label{fff}
\end{equation}
Hence, for any $\varepsilon >0$, there exists an integer $q_{k}(\geq p_{k})$
such that
\begin{equation}
2^{-\varepsilon q_{k}}(1-2^{-\varepsilon })\leq \sharp \mathcal{B
_{q_{k},k}(2^{-q_{k}})^{s},  \label{q_k}
\end{equation
otherwise
\begin{equation*}
\sum\limits_{p=0}^{\infty }\sharp \mathcal{B}_{p,k}2^{-ps}<\su
\limits_{p=0}^{\infty }2^{-\varepsilon p}(1-2^{-\varepsilon })=1,
\end{equation*
which contradicts (\ref{fff}). Since $p_{k}$ tends to $\infty $ and
q_{k}\geq p_{k},$ we have
\begin{equation*}
\lim_{k}q_{k}=\infty .
\end{equation*}

Given $\mathbf{i}\in D_{m_{k}},$ we take
\begin{equation*}
R_{k}=|J_{\mathbf{i}}|\text{\ and }r_{k}=\min_{\mathbf{j}\in \mathcal{B
_{q_{k},k}}|J_{\mathbf{i}\ast \mathbf{j}}|\in \lbrack 2^{-q_{k}-1}|J_
\mathbf{i}}|,2^{-q_{k}}|J_{\mathbf{i}}|].
\end{equation*
Since $|J_{\mathbf{i}\ast \mathbf{j}}|\in \lbrack 2^{-q_{k}-1}|J_{\mathbf{i
}|,2^{-q_{k}}|J_{\mathbf{i}}|]$ for all $\mathbf{j}\in \mathcal{B
_{q_{k},k}\ $and int($J_{\mathbf{i}\ast \mathbf{j}}$)$\cap $int($J_{\mathbf{
}\ast \mathbf{j}^{\prime }})=\varnothing $ for all $\mathbf{j}\neq \mathbf{j
^{\prime }\in \mathcal{B}_{q_{k},k}$, by Lemma~\ref{lem2} again, there
exists a positive integer $l^{\prime }$ independent of $k$ such that each
ball with radius $r_{k}(\in \lbrack 2^{-q_{k}-1}|J_{\mathbf{i
}|,2^{-q_{k}}|J_{\mathbf{i}}|])$ intersects at most $l^{\prime }$ elements
in $\{J_{\mathbf{i}\ast \mathbf{j}}\}_{\mathbf{j}\in \mathcal{B}_{q_{k},k}}$.

To prove the lower bound, we need the following inequality
\begin{equation}
\frac{\sharp \mathcal{B}_{q_{k},k}}{l^{\prime }}\leq N_{r_{k},R_{k}}(E).
\label{lll}
\end{equation}
Notice that $J_{\mathbf{i}}\subset B(z,R_{k})$ for all $z\in J_{\mathbf{i}},$
we assume that there exists a smallest number $t$ such that $B(z,R_{k})$ can
be covered by $t$ balls of radius $r_{k},$ e.g.,
\begin{equation*}
B(z,R_{k})\subset B(x_{1},r_{k})\cup \cdots \cup B(x_{t},r_{k}).
\end{equation*}
Notice that $t\leq N_{r_{k},R_{k}}(E)$ and
\begin{equation*}
\bigcup_{\mathbf{j}\in \mathcal{B}_{q_{k},k}}J_{\mathbf{i}\ast \mathbf{j
}\subset J_{\mathbf{i}}\subset B(z,R_{k})\subset B(x_{1},r_{k})\cup \cdots
\cup B(x_{t},r_{k}).
\end{equation*
Then for any $\mathbf{j}\in \mathcal{B}_{q_{k},k},$ there exists at least a
ball $B(x_{i},r_{k})$ $1\leq i\leq t$ such that $J_{\mathbf{i}\ast \mathbf{j
}\cap B(x_{i},r_{k})\neq \varnothing $, that is
\begin{equation*}
\mathcal{B}_{q_{k},k}\subset \bigcup\nolimits_{i=1}^{t}\{\mathbf{j}\in
\mathcal{B}_{q_{k},k}:J_{\mathbf{i}\ast \mathbf{j}}\cap B(x_{i},r_{k})\neq
\varnothing \}.
\end{equation*
Therefore, we have p
\begin{eqnarray*}
\sharp \mathcal{B}_{q_{k},k} &\leq &\sum\nolimits_{i=1}^{t}\sharp \{\mathbf{
}\in \mathcal{B}_{q_{k},k}:J_{\mathbf{i}\ast \mathbf{j}}\cap
B(x_{i},r_{k})\neq \varnothing \} \\
&\leq &t\cdot l^{\prime }\leq N_{r_{k},R_{k}}(E)\cdot l^{\prime },
\end{eqnarray*
which completes the proof of inequality (\ref{lll}).

For any $\zeta >0,$ there exists $C_{\zeta }$ such that for any $k,
\begin{equation}
N_{r_{k},R_{k}}(E)\leq C_{\zeta }(\frac{R_{k}}{r_{k}})^{\dim _{A}E+\zeta }
\label{assouad}
\end{equation
Therefore, using (\ref{q_k}), (\ref{lll})\ and (\ref{assouad}), we have
\begin{eqnarray*}
\frac{2^{q_{k}(s-\varepsilon )}(1-2^{-\varepsilon })}{l^{\prime }}\leq \frac
\sharp \mathcal{B}_{q_{k},k}}{l^{\prime }} &\leq &N_{r_{k},R_{k}}(E) \\
&\leq &C_{\zeta }(\frac{R_{k}}{r_{k}})^{\dim _{A}E+\zeta }\leq C_{\zeta
}(2^{q_{k}+1})^{\dim _{A}E+\zeta }.
\end{eqnarray*
Since $\lim_{k}q_{k}=\infty$, by letting $k\rightarrow \infty ,$ it gives
\begin{equation*}
\dim _{A}E+\zeta \geq s-\varepsilon.
\end{equation*
By taking $\varepsilon \rightarrow 0$ and $\zeta \rightarrow 0,$ we have
\dim _{A}E\geq s$ for all $s<s^{\ast \ast },$ and thus $\dim _{A}E\geq
s^{\ast \ast }.$
\end{proof}

\section{Assouad Dimension of Homogeneous Set}

In this section we will prove the dimension formula for homogeneous sets.

\begin{lemma}
\label{qq} Suppose that $K\subset X$ is doubling. Then
\begin{equation}
\dim _{A}K=\lim_{\rho \rightarrow 0}\left( \sup\limits_{R<\varepsilon }\frac
\log N_{\rho R,R}(K)}{-\log \rho }\right) ,  \label{tttt}
\end{equation
for all $\varepsilon <|K|.$
\end{lemma}

\begin{proof}
First, we prove that
\begin{equation}
\dim _{A}K=\overline{\lim_{\rho \rightarrow 0}}\left( \sup\limits_{R<|K|
\frac{\log N_{\rho R,R}(K)}{\log R-\log (\rho R)}\right) .  \label{def}
\end{equation}
Arbitrarily choose a real $\alpha$ such that $\alpha >\overline
\lim\limits_{\rho \rightarrow 0}}\left( \sup\limits_{R<|K|}\frac{\log
N_{\rho R,R}(K)}{\log R-\log (\rho R)}\right) , $ there exists $\delta \in
(0,1)$ such that, for all $\rho <\delta ,$ we have $\sup\limits_{R<|K|}\frac
\log N_{\rho R,R}(K)}{\log R-\log (\rho R)}<\alpha $, that is ,
\begin{equation*}
N_{r,R}(K)\leq (R/r)^{\alpha },
\end{equation*
for $0<r<\delta R<R<|K|$. On the other hand, there exits a constant
c_{\delta }>0$ such that $N_{r,R}(K)\leq c_{\delta }$, for $\delta R<r<R$.
Hence, for all $0<r<R<|K|$, we have
\begin{equation*}
N_{r,R}(K)\leq c_{\delta }(R/r)^{\alpha },
\end{equation*
which implies that $\alpha \geq \dim _{A}K.$ Since $\alpha$ is arbitrarily
chosen, we have
\begin{equation*}
\overline{\lim\limits_{\rho \rightarrow 0}}\left( \sup\limits_{R<|K|}\frac
\log N_{\rho R,R}(K)}{\log R-\log (\rho R)}\right) \geq \dim _{A}K.
\end{equation*}

Suppose that $\alpha $ is a fixed number such that, for all $R<b$,
\begin{equation*}
N_{\rho R,R}(K)\leq c(R/\rho R)^{\alpha },
\end{equation*
where $b$ is a constant. The
\begin{equation*}
\sup\limits_{R<b}\frac{\log N_{\rho R,R}(K)}{\log R-\log (\rho R)}\leq \frac
\log c}{\log R-\log (\rho R)}+\alpha .
\end{equation*
Taking limit on both sides, we have $\overline{\lim\limits_{\rho \rightarrow
0}}\left( \sup\limits_{R<b}\frac{\log N_{\rho R,R}(K)}{\log R-\log (\rho R)
\right) \leq \alpha .$ Using the doubling property of $K,$ we have
\begin{equation*}
N_{\rho R,R}(K)\leq N_{\rho b/2,b/2}(K)\cdot N_{b/2,|K|}(K),
\end{equation*
for $b\leq R<|K|$, which implie
\begin{equation*}
\overline{\lim\limits_{\rho \rightarrow 0}}\left( \sup\limits_{R<|K|}\frac
\log N_{\rho R,R}(K)}{\log R-\log (\rho R)}\right) =\overline
\lim\limits_{\rho \rightarrow 0}}\left( \sup\limits_{R<b}\frac{\log N_{\rho
R,R}(K)}{\log R-\log (\rho R)}\right) \leq \alpha .
\end{equation*
Since the inequality holds for all $\alpha >\dim _{A}K,$ it follows that
\begin{equation*}
\overline{\lim\limits_{\rho \rightarrow 0}}\left( \sup\limits_{R<|K|}\frac
\log N_{\rho R,R}(K)}{\log R-\log (\rho R)}\right) \leq \dim _{A}K,
\end{equation*
which finishes the proof of (\ref{def}).

We write
\begin{equation*}
t(\rho )=\sup\limits_{R}\frac{\log N_{\rho R,R}(K)}{-\log \rho }.
\end{equation*
To obtain the formula (\ref{tttt}), by (\ref{def}), it is sufficient to show
that the limit of $t(\rho )$ exists as $\rho$ tends to $0$.

Given $\rho>0 $. For any $\rho ^{\prime }<\rho ,$ there exists an integer $m$
such that
\begin{equation*}
\rho ^{m+1}\leq \rho ^{\prime }<\rho ^{m}.
\end{equation*
Since $N_{r_{1},r_{3}}(K)\leq N_{r_{1},r_{2}}(K)N_{r_{2},r_{3}}(K)$ for
r_{1}<r_{2}<r_{3},$ it follows that
\begin{equation*}
N_{(\rho ^{\prime })R,R}(K)\leq N_{\rho ^{m+1}R,R}(K)\leq \left(
\sup_{r}N_{\rho r,r}(K)\right) ^{m+1}.
\end{equation*
Hence, we have that
\begin{equation*}
\left\vert \frac{\log N_{(\rho ^{\prime })R,R}(K)}{\log \rho ^{\prime }
\right\vert \leq \left\vert \frac{\log \left( \sup_{r}N_{\rho r,r}(K)\right)
^{m+1}}{\log (\rho ^{\prime }/\rho ^{m+1})+(m+1)\log \rho }\right\vert ,
\end{equation*
and it implie
\begin{equation*}
\underset{\rho ^{\prime }\rightarrow 0}{\overline{\lim }}t(\rho ^{\prime
})\leq \lim_{m\rightarrow \infty }\left\vert \frac{\log \left(
\sup_{r}N_{\rho r,r}(K)\right) ^{m+1}}{\log (\rho ^{\prime }/\rho
^{m+1})+(m+1)\log \rho }\right\vert =t(\rho )
\end{equation*
due to $1\leq \rho ^{\prime }/\rho ^{m+1}\leq \rho ^{-1}.$ Therefore, we
obtain that
\begin{equation*}
\underset{\rho ^{\prime }\rightarrow 0}{\overline{\lim }}t(\rho ^{\prime
})\leq \inf_{\rho }t(\rho )\leq \underset{\rho ^{\prime }\rightarrow 0}
\underline{\lim }}t(\rho ^{\prime }),
\end{equation*
that is,
\begin{equation*}
\lim_{\rho \rightarrow 0}t(\rho )=\inf_{\rho }t(\rho ).
\end{equation*
On the other hand, since $K$ is doubling, the
\begin{equation*}
\lim_{\rho \rightarrow 0}\left( \sup\limits_{R<\varepsilon _{1}}\frac{\log
N_{\rho R,R}(K)}{-\log \rho }\right) =\lim_{\rho \rightarrow 0}\left(
\sup\limits_{R<\varepsilon _{2}}\frac{\log N_{\rho R,R}(K)}{-\log \rho
\right) .
\end{equation*}
\end{proof}

\begin{proof}[Proof of Theorem 3]
\

Fix a point $x_{0}\in K.$ It is clear that $h\sim \alpha _{x_{0}}.$ By (\re
{E:equ}), we have that, for $r<R,
\begin{eqnarray*}
\left\vert \alpha _{x_{0}}(r)\log r-h\left( r\right) \log r\right\vert &\leq
&C, \\
\left\vert \alpha _{x_{0}}(R)\log R-h\left( R\right) \log R\right\vert &\leq
&C\text{.}
\end{eqnarray*
Henc
\begin{eqnarray}
&&\left\vert \frac{{h\left( R\right) \log R-h\left( r\right) \log r}}{{\log
R-\log r}}-\frac{{\alpha _{x_{0}}(R)\log R-\alpha _{x_{0}}\left( r\right)
\log r}}{{\log R-\log r}}\right\vert  \notag \\
&\leq &\left\vert \frac{{\alpha _{x_{0}}(R)\log R-h\left( R\right) \log R}}{
\log R-\log r}}\right\vert +\left\vert \frac{{\alpha _{x_{0}}(r)\log
r-h\left( r\right) \log r}}{{\log R-\log r}}\right\vert  \label{307} \\
&\leq &\frac{2C}{|\log R/r|}.  \notag
\end{eqnarray}

Suppose $k$ is the smallest number of balls with radius $r$ needed to cover
B(x,R),$ i.e., suppose $B(x,R)$ is covered by $B(y_{1},r),\cdots
,B(y_{k},r). $ In fact, we can choose
\begin{equation}
k=N_{r,R}(K).  \label{wanz1}
\end{equation
Then
\begin{equation*}
\mu (B(x,R))\leq \sum\limits_{i=1}^{k}\mu (B(y_{i},r))
\end{equation*
which implies
\begin{equation*}
\frac{\mu (B(x,R))}{\max_{y\in K}\mu (B(y,r))}\leq k.
\end{equation*
Using (\ref{self1}), we have
\begin{equation}
\lambda ^{-2}\frac{\mu (B(x_{0},R))}{\mu (B(x_{0},r))}\leq k.  \label{wanz2}
\end{equation}

We also assume $p$ is the largest number of disjoint $(r/2)$-balls with
centers in $B(x,R),$ for example, $B(z_{1},r/2),\cdots ,B(z_{p},r/2)$ are
pairwise disjoint. By the routine argument, we have
\begin{equation*}
k\leq p.
\end{equation*
In the same way,
\begin{equation*}
p\min_{y\in K}\mu (B(y,r/2))\leq \sum\limits_{i=1}^{p}\mu (B(z_{i},r))\leq
\mu (B(x,R+r))\leq \mu (B(x,2R)).
\end{equation*
Therefore, using (\ref{self1}), we have
\begin{equation}
k\leq p\leq \frac{\mu (B(x,2R))}{\min_{y}\mu (B(y,r/2))}\leq \lambda ^{2
\frac{\mu (B(x_{0},2R))}{\mu (B(x_{0},r/2))}.  \label{wanz3}
\end{equation}

Using (\ref{***}), the measure $\mu $ is doubling, i.e., there is a constant
$D>0$ such that
\begin{eqnarray*}
\mu (B(x_{0},2R)) &\leq &D\mu (B(x_{0},R)), \\
\mu (B(x_{0},r/2)) &\geq &D^{-1}\mu (B(x_{0},r)).
\end{eqnarray*
Then (\ref{wanz3}) shows that
\begin{equation}
k\leq (\lambda D)^{2}\frac{\mu (B(x_{0},R))}{\mu (B(x_{0},r))}.
\label{wanz4}
\end{equation
Combining (\ref{wanz1}), (\ref{wanz2}) and (\ref{wanz4}), we obtain that
\begin{eqnarray}
&&\frac{\log \lambda ^{-2}}{\log R-\log r}+\frac{\alpha _{x_{0}}(R)\log
R-\alpha _{x_{0}}(r)\log r}{\log R-\log r}  \notag \\
&\leq &\frac{\log N_{r,R}(K)}{\log R-\log r}  \label{306} \\
&\leq &\frac{\log (\lambda D)^{2}}{\log R-\log r}+\frac{\alpha
_{x_{0}}(R)\log R-\alpha _{x_{0}}(r)\log r}{\log R-\log r}.  \notag
\end{eqnarray
By Lemma \ref{qq}, (\ref{307}) and (\ref{306}), we obtain that
\begin{equation*}
\dim _{A}K=\lim_{\rho \rightarrow 0}\left( \sup_{R}\frac{h(R)\log R-h(\rho
R)\log (\rho R)}{-\log \rho }\right) .
\end{equation*}
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
James Marshall Tory (March 7, 1930 – August 19, 2013) was a Canadian corporate lawyer based in Toronto and twin brother to the late John A. Tory.

Early life and education
Tory was born in Toronto, Ontario, to John S. D. Tory, a lawyer, and Jean Tory (née Arnold). He had an older sister, Virginia, and a fraternal twin brother, John Arnold Tory.

Tory and his brother, John, graduated from the University of Toronto Schools in 1946 at the age of 16. They attended Phillips Academy Andover for two years before enrolling at the University of Toronto. Tory was an undergraduate student for two years before he switched streams to attend  the University of Toronto Faculty of Law in 1949. He graduated in 1952, where he was the Gold Medallist in its first graduating class. Tory completed two years of additional training at Osgoode Hall, and with his brother, they joined their father's firm at the age of 24.

Career
Tory and his brother John were members of the firm of Torys, founded by their father John Stewart Donald Tory. He was admitted to the Ontario Bar in 1954, when he joined the family firm Torys. Prior to his passing, he was Chair Emeritus and Senior Counsel of Torys.

He was active on community boards and served as chairman of the board of the Hospital for Sick Children and of the Hospital for Sick Children Foundation.

Personal life
Tory was married to Marilyn Tory ( Yorath), whom he met at the University of Toronto. They had five children Marilyn died in 1999; their son, David, died of brain cancer in 2006.:

 Martha Tory
 James C. Tory (Jim Tory Jr.) is a corporate lawyer
 Suzanne Tory
 David Thomas Tory
 Richard Tory is an executive with Morgan Stanley Canada

References

 James M. Tory

James Marshall
University of Toronto alumni
Osgoode Hall Law School alumni
Lawyers in Ontario
People from Toronto
2013 deaths
1930 births
Canadian twins
Corporate lawyers
University of Toronto Faculty of Law alumni
Phillips Academy alumni
Canadian corporate directors


text:
Go Beavers!!!
Mathematics and Sciences
Humanities, Education, Communication, and Arts
Social Science, Business and Health
Agriculture Power Technology
Electrical Power Technology
High School Career Technical Education
Information Network Technology
Modern Distribution Sales & Management
Wildlife Outfitting & Operations
GoBeaverSports.com
About PCC
Higher Ed Emergency Funds
Online Payment Plan
Nursing Scholarship Application
Bonnie Dudrey Memorial Scholarship Application
Finance Department Downloads
High School Concurrent Enrollment
Find Your Recruiter
Adding and Dropping Classes
Beaver Building Day
2020-2021 Learning Calendar (pdf)
Student Activity Calendar
Pratt Online (Canvas)
Canvas Course Acceptance Steps
EDUKAN
E-Central
Office 365 Info/Help (pdf)
'A Night on Broadway' October 18-20
Pratt Community College Performing Arts department will be presenting 'A Night on Broadway' October 18-20.
The show is the department's annual musical Review show, which features music more than a dozen Broadway classics and modern shows. This year's show will feature a variety of broadway tunes from Waitress to Mary Poppins to Beautiful The Carole King Musical. It will feature solos, duets, and ensembles. The production includes college students and community members as well as youth performers.
Read more about 'A Night on Broadway' October 18-20
Performing arts presents, "Valentine's Dinner and Show" concert
To celebrate the weekend of love, the Pratt Community College musical group Encore! will be holding a special Valentine's Dinner and Show in honor of Valentine's Day.
The dinner and show will be Sunday, Feb. 11 at 6:30 p.m. in the PCC Commons. Enjoy dinner and dessert while being serenaded by Encore! featuring love songs from a variety of genres including solos, duets and ensembles.
Tickets for Dinner and the Show are $15 for adults and children over 5. Reservations can be made with Misty Beck, mistyb@prattcc.edu
Read more about Performing arts presents, "Valentine's Dinner and Show" concert
Performing arts presents, "A Christmas to Remember" Dec. 8-10
Pratt Community College's Performing Arts department will have the annual Christmas show to help kick off the season on Dec. 8-10.
'A Christmas to Remember,' will feature two extraordinary acts encompassing all of your favorite holiday tunes and the true meaning of Christmas.
The show will feature solos, duets, the ensemble and full community choir. There will also be children's numbers.
"It promises to be a fun show with the classic favorites and also some new music," said Misty Beck, show director.
Read more about Performing arts presents, "A Christmas to Remember" Dec. 8-10
Performing arts presents, "A Night on Broadway" Oct. 20-22
On Oct. 21-22 Pratt Community College Performing Arts department will be presenting 'A Night on Broadway'.
The show is the department's annual musical Review show, which features music from dozens of Broadway classics and modern shows. This years show will feature a variety of broadway tunes from Camelot to Addams Family to Les Miserables. It will feature solos, duets, the ensemble and full choir. There will also be children's numbers.
Read more about Performing arts presents, "A Night on Broadway" Oct. 20-22
Students and Friends Showcase concert collecting donations for Texans
The Pratt Community College Performing Arts Department will host their fourth annual Students and Friends Showcase Sunday, Sept. 24 at 2:30 p.m. in Carpenter Auditorium. The show will feature a variety of student performances, solos and small group performances. There will be a variety of music and the show should last approximately an hour and a half. This is a great opportunity for the community to meet the new Encore members and learn more about the department plans for the year. Doors will open at 2 p.m.
Read more about Students and Friends Showcase concert collecting donations for Texans
Performing arts presents the first annual Alumni Concert
Pratt Community College performing arts is bringing back more than 50 alumni to unite with current students for an end of the year concert.
The concert will be held Sunday Apr. 9 at 2:30 p.m. in Carpenter Auditorium. Admission is $5 for community, students and PCC staff and faculty.
Performing arts instructor and show director, Misty Beck, says there was an irresistible demand from alumni and the community to put on a show featuring students and community from years past.
Read more about Performing arts presents the first annual Alumni Concert
PCC Performing Arts presents My Son Pinocchio
The Pratt Community College Performing Arts Department will be performing My Son Pinocchio for their spring show.
The show is a full scale Disney musical based on the movie "Gephetto". The classic tale of toymaker, Geppetto's, little wooden puppet is given new life. This new musical, which retells the classic Disney story from Geppetto's perspective, features the beloved classic songs, "When You Wish upon a Star" and "I've Got No Strings," alongside a host of new songs by Oscar- and Grammy Award winner and master melodist, Stephen Schwartz who also composed the musical "Wicked".
Read more about PCC Performing Arts presents My Son Pinocchio
Encore! Performing Arts Valentines Day concert
To celebrate the weekend of love, come to the Valentine's Day concert hosted Pratt Community College's Encore! on Sunday, Feb. 14 at 2:30 p.m. in Carpenter Auditorium.
The show will feature love songs from a variety of genres featuring solos, duets and ensembles.
There will be a dessert bar at Intermission hosted by Collegiate Farm Bureau for a free will donation.
The concert is a fundraiser for the students' trip to Tennessee in May so it does have a $5 suggested donation per person.
Read more about Encore! Performing Arts Valentines Day concert
Pratt county schools come together for concert
The Pratt Community College Performing Arts Department will host the County Wide Choral Extravaganza on March 13 at 7:30 p.m. in Carpenter Auditorium.
The concert will showcase the musical talents of Pratt county students from PCC and Skyline High School. During the evening, the choirs will sing separately and together, with additional performances by their show choirs.
"It gives all the high school and college students in Pratt County a chance to sing together and create a sense of camaraderie," said J.J. Rupe, PCC music instructor.
Read more about Pratt county schools come together for concert
Valentine's Day concert Sunday
Encore! will be singing love songs from many different genres and music styles.
There will be cookies and punch at intermission.
The concert is a fundraiser for the students' trip to Tennessee in May so it does have a $5 suggested donation.
"We hope everyone will come out and enjoy some great music and treats to finish up the Valentine's weekend," said Misty Beck, theatre instructor.
Read more about Valentine's Day concert Sunday
Virtual & Video Tour
Senior Day Registration
Tech Week Registration
Beaver Building Day Registration
Change of Information Form
College Prep Day
Degree Audit Form Request
Degree Completion and Application for Graduation Form
Final Exam Schedule (pdf)
On-Campus Housing Contract
Student Feedback/Complaints
Taxpayer & Student Transparency Data
College Health
College Start - Concurrent Enrollment
Linda Hunt Memorial Library
Subscribe to the PCC Newsletter
Student Complaint Policy
art performing arts pratt kansas pratt community college rodeo baseball athletics pcc agriculture
Pratt Community College does not discriminate on the basis of race, color, national origin, sex, disability, age or veteran status in its programs and activities. The following person has been designated to handle inquiries regarding the non-discrimination policies: Director of Personnel, Pratt Community College, 348 NE SR 61, Pratt, Kansas 67124, 620.672.5641.
Copyright 2006-20 17 Pratt Community College - 348 NE SR 61 - Pratt KS 67124
620-672-5641 | 1-800-794-3091 (KS Only) | Fax: 620-450-2285
After Hours Service, 620-672-9800 , M-F 5-9pm, Sat/Sun 10am-4pm
Equal Employment Opportunity/Equal Education Opportunity

redpajama_set_name: RedPajamaCommonCrawl


Airtel has finally launched its VoLTE services in most of the states, and if it's still not LIVE in your state than don't worry as it will go Live soon throughout the country. All latest smartphones comes with VoLTE feature enabled, but do you know that you could also use Airtel VoLTE feature in your current smartphone.
Before that let's find out more about what is Airtel VoLTE and what are the benefits of this new feature.
VoLTE is a technology which is being used by Jio, so now Airtel is also on the track of Reliance Jio to provide its customers good quality services at affordable price.
VoLTE denotes Voice over LTE is an advanced service through which customers can experience superior HD quality crystal clear voice with faster call setup time as compared to normal voice calls. While using VoLTE services, users can continue to have un-interrupted high-speed 4G internet experience.
In VoLTE there would be no need to recharge your phone separately with voice and data plans, both voice and data plan would come into one pack. Which would cut down your monthly recharge cost.
Now let's start with the process on how to switch to Airtel VoLTE Feature on your Phone.
Before proceeding further to switch to Airtel VoLTE you should make sure if your device is supporting Airtel VoLTE feature.
You can find the compatible list of smartphone which supports Airtel VoLTE by clicking on below check compatibility button.
So if your smartphone model is mentioned in page than yes you can proceed further to taste Airtel VoLTE on your smartphone.
For Android Device: Settings > Settings > Mobile Network > Turn on VoLTE call. These settings may differ from one handset manufacturer to another. In case you need assistance, please call 121.
If your smartphone is listed under compatible handsets and still you are not getting VoLTE option, than you should make sure if your smartphone device is running on latest software. Just open System Update and update your phone.
The next requirement to switch to Airtel 4G VoLTE is make sure you have 4G SIM, if you have old SIM inserted in your smartphone than you would not be able to switch to 4G Services. You can get 4G SIM free of cost online by clicking here or you can rush into Airtel Store.
If you have passed on all above compatibility tests than here you go to enjoy VoLTE on your smartphone.
Make sure Airtel 4G SIM is inserted into the device and verify that the VoLTE voice calling feature has been enabled on your smartphone.
If you have Dual SIM than make sure the Airtel SIM is inserted in the SIM slot with 4G data capability.
Than set network type to prefer LTE by going to Settings > SIM & Mobile Networks > Select the Airtel SIM > Preferred Network Type > Set it to Prefer LTE.
That's it your device will now show 4G with HD/VoLTE icon on top bar of the screen, indicating that your device is ready for VoLTE calls.
As of now Airtel is also running a promotional launch offer, where on using or switching to Airtel VoLTE Service you can get free 10GB 4G Data with validity of 14 Days. The Free data would be credited in 3 days to customers account after switching to VoLTE service on 4G VoLTE capable handset.


Pay-per-click (PPC) is an effective and clever digital marketing technique for instantly growing the digital outreach & search visibility of your website. While all forms of organic optimization techniques take a considerable amount of time to finally deliver on their promise of increasing traffic on your website, PPC is proven to guarantee similar results in a much shorter time. This becomes increasingly essential for new businesses that want a substantial growth in their digital performance in limited period of time.
Our team of PPC experts at IT by IT Professionals provides best-in-class PPC services in Jaipur, India. Our experience in handling PPC advertising has equipped us with the unique knowledge of resources that we utilize in order to attain the most satisfactory results for our clients.
Targeted Advertising : allowing you to run ads for your potential client base.
Brand Exposure : to create a name for your business.
Search Engine Optimization : for improving your organic search impressions as a direct result of increased traffic through PPC.
Pay Only After Click : to save money that would otherwise be spent on promoting ineffective ads.
All these, and several other factors, make PPC marketing a very effective tool in promoting businesses online. We, as an experienced pay per click company, are qualified to provide the best PPC marketing solutions to a wide range of clients from different sectors.
Explore the dynamic range of PPC services with a reliable PPC company in Jaipur, India: IT by IT Professionals.


Four to five pages of letter size, computerized, double spaced (27 lines per page, 75 characters per line). If possible, use Word or Word Perfect.
In both the form and content, keep in mind that the magazine is aimed at non-Adventist public.
The article must include the name of the author, the position he performs and his address.
Religious news (ecumenism, New Age, gift of tongues, false miracles, dissemination of the Bible, etc.).
Topics of social, economic and political relevance that indicate the need of God, and that show the fulfillment of the prophecies and the imminence of the return of Christ. Along with the problems, offer the biblical solution.
Basic doctrines of the church presented in a positive, Christocentric way, with its application in the individual, family and collective sphere (avoid controversial or dogmatic tone).
Stories of conversion, of answered prayers, of accident protection, or that illustrate the Christian virtues (faith, patience, courtesy, self-control, etc.), ensuring the authenticity of the facts.
Topics for young people: ideals, personality formation, self-image, friendships, youth problems, courtship, etc. (It is convenient to include illustrations and anecdotes).
Articles on health, temperance and disease prevention.
Choose simple and understandable words, avoiding denominational phraseology.
Use the revised Reina-Valera version of 1960, unless it is imperative to use another version for reasons of doctrinal orthodoxy.
Avoid long sentences and very long paragraphs, and vary their length.
Develop ideas in an entertaining and logical way.
Give the article a practical and optimistic approach.
El Centinela® will pay between $100 and $150 per item published. This payment includes electronic rights, since some articles are placed on the website.
The writing of the journal reserves the right to publish or not the article, and to correct or shorten it to meet editorial needs. If possible, before writing the article consult with the editor of the magazine.


text:
The Nicknames of Watch Hands
Brands, Watch 101
Watch Hands and Their Nicknames
Watchmakers are quite possibly some of the most detail-oriented artisans in the world. When you start to dissect the nuances of watches, the intricacies you'll find might astound you. One example is the watch hands. Yes, the hands of a watch, including indicator for the hour, one for the minute, and often one for the second, are one of the most functionally important aspects of a timepiece because they actually indicate the time. However, watch hands are also a stylistic element. There are dozens of unique types of watch hands. Over centuries of the art form, there are certain styles that have become prominent. Each of these has in fact garnered its own moniker. Here, we introduce you to a selection of different types of watch hands and watch hand nicknames.
Types of Watch Hands and Their Nicknames
Watch hands are a stylistic and functional element of a timepiece that might be something you overlook at first glance. However, each of the different types of watch hands are each painstakingly designed with a unique purpose and aesthetic. Below are some of the most notable types of watch hands and watch hand nicknames.
You'll find a number of watch hands are nicknamed for their shape. One example of this is arrow hands. As the name suggests, they feature a pointer in the style of an arrow. Likewise, watchmakers design these hands for at-a-glance timekeeping. While they're highly legible, they're also typically more bulky and bold in size and appearance. You'll most often find arrow hands on technical or sport watches, like the OMEGA Seamaster Planet Ocean.
Other watch hands get their nicknames from their creators. Similarly, Abraham-Louis Breguet, founder of his namesake brand, designed the Breguet hands back in the late 18th century. These elegant and delicate hands feature a ring and a point at the end. The center of the ring is hollow in a way that resembles a crescent moon. For this reason, Breguet hands have a number of other nicknames. Some examples include moon, crescent moon, hollow apple, or pomme, which means apple in French. Of course, you can find this style on a number of Breguet watches, like the Classique.
Dauphine hands are one of the most common and most popular styles. However, their nickname is a bit of a mystery. Many say their name originates from a French monarch's son. Others say that because "Dauphine" is feminine, it was in fact his wife. Their traditional design features an elongated triangle shape that's tapered at the tip and faceted along its length. There are also a number of variations of the Dauphine hands. For example, some are flat while others taper, like on this JLC version. They also come in an array of finishes to enhance their legibility.
Lance hands get their name from the weapon once used by knights and horsemen. The hands feature a similar shape from the instrument for which they draw their name. Their design has a long shaft, wider base, and narrow, pointed head. This A. Lange & Söhne Datograph Flyback has a luminescent version.
Like arrow hands, lollipop hands are nicknamed for their shape. Their design resembles the iconic candy treat, featuring a long stick with a circle at the tip. Depending on the particular watch, the circle may be open or filled. The OMEGA Seamaster is just one example of a model that features this style hands.
The three-pointed star that makes up the Mercedes-Benz logo is one of the most instantly recognizable insignias around the world. In contrast, the Mercedes style hand does not indicate an affiliation between a particular watch model and the famed car manufacturer. However, the Mercedes symbol is where Mercedes hands get their nickname. Rolex is the purveyor of Mercedes hands. They typically only feature the design on the hour hand and additionally, they often treat it with luminescent paint. This helps to distinguish the hour hand in low-light visibility. You'll find this style on the iconic Submariner.
Similar to the arrow and lollipop hands, paddle hands get their shape and nickname from the paddle itself. They feature a straight design with a wider rectangle toward the end and a pointer at the tip. One example of a watch equipped with paddled hands is the Blancpain Fifty Fathoms.
Just as the nickname suggests, stick hands are much like stick hour markers. These long, slim, and straight hands typically feature a blunt edge. You'll most often find this design used on dress watches, like a Glashütte Original or Patek Philippe Calatrava.
Baton hands are similar to stick hands. They are long and straight, but they are thicker and have a more imposing presence. You will find these most often on a watch like the Audemars Piguet Royal Oak.
Alpha hands have a wide base connected by a narrow stem at the center of the dial. Then, they taper out to a point at the tip. One could say their shape resembles the letter "A." Their bold, long, straight design exudes strength, like the implication of the alpha name itself. You can find this style on models like the A. Lange & Söhne Datograph.
One of the most ornate styles of hands is cathedral. Watchmakers have classically used the design in military watches. As the name suggests, cathedral hands feature elaborate geometric details that mimic the stained glass commonly found in church windows. Cathedral hands often consist of a larger hour hand and more slender minute hand. The oversized hour hand typically features a distinctive orb and needle-like tip. Overall, this style adds more visual interest than functionality. The Oris Big Crown Original Pointer Date and Zenith Pilot are two examples of models showcasing this unique style.
Fleur de lys hands are also true to their name. The iconic decorative symbol appears at the pointed end of the hands, toward the tip. The fleur de lys is a relatively common symbol among the French and other European flags and coats of arms. However, it is rarer that you'll find it incorporated into stylistic elements of the watch, like the hands.
Leaf (or Feuille)
Leaf hands or feuille hands, the French word for leaf, are another style whose name indicates the shape. They're wide in the middle and narrow at each end, tapering at the base and the point. The design mirrors the shape of a long, slender leaf. This style is simple, sleek, and understated.
Lozenge hands are somewhat similar to sword-shaped hands. However, they're a bit more symmetrical and elegant. They feature an elongated diamond-like shape that extends for the full length of the watch hands. The surface area provides a perfect opportunity for watchmakers to apply lume, if desired. Models like the Pasha de Cartier showcase this sophisticated design.
Obelisque
Obelisque hands resemble alpha hands to a certain degree. They have a wide base and narrow toward a small point at the tip. However, they're not connected by a short point at the center of the dial. Instead, the wide base extends to the point at which the hour and minute hands meet.
Pencil hands are yet another style whose name informs the shape. Like a classic wooden pencil, these hands are long, thin, and straight with a small point at the tip. Depending on the width of the hands, they may be solid or filled and applied with lume.
It might not be obvious to English speakers. However, plonguer hands draw their nickname from a French word that corresponds to the type of watch showcasing this design. Plonguer means diver in French, and you'll find this style on dive watches, like the OMEGA Seamaster and Ploprof. The hour and minute hands feature a design with a specific combination: a simple, straight hour hand and a more prominent sword-shaped minute hand that's often in a bold color, like orange.
Like a skeleton dial on a watch, skeleton hands feature an open-worked construction. Some skeleton hands will showcase a single see-through element. However, others may feature an intricate design or pattern with several small openings. Audemars Piguet and OMEGA are two examples of brands that employ this style of hands.
Similar to Breguet hands, snowflake hands are specific to one watch brand: Tudor. The brand developed the design for the French Navy. They were looking for something to make their dive watches highly legible, and the snowflake hands were born. The result was a set of incredibly unique, oversized hands. Most notably, the hour hand features a distinctive square pattern at the tip. These robust hands are hefty enough for a good coat of lume for maximum visibility. You can find them on models like the Pelagos.
Spade hands are true to the symbol you find on the playing card suit. The design features a long, tapered hand with a spade shape functioning as a pointer at the tip. Sometimes, you'll only see a spade style hand on the hour hand paired with a simple, slim minute hand. Other times, you'll find the design on both. The Ulysse Nardin Marine Torpilleur is just one example of a model featuring this style.
Sword hands look like plongeur hands to some extent. As the name indicates, they bear resemblance to a sword blade. Unlike plongeur hands, they don't have a stem at the base. Dive and military watches are the most common styles to feature sword hands. However, they're also in a variety of other models. One interesting example is the Cartier Drive.
Last but not least are syringe hands. Just like the medical instrument itself, this style features a similar design. They're comprised of a barrel shape that gradually widens from the center with a needle-like tip. The design of these hands lends itself to precise timekeeping and dressier styles.
A History of Watchmaking in Basel
Waterproof vs. Water Resistance


Here's a new batch of amazing cheesecakes produced by none other than the incomparable Don Folk!!I am not sure how we lucked into these amazing creations but the Workshop students and I are very Grateful! Don tries new recipes on us and we love being guinea Pigs! This bunch includes Banana Cream/walnut, Blueberry, Peaches and Cream, Chocolate Peanut Butter, Coconut Something(wonderful), and one with mandarin oranges(also wonderful). So…….come do a workshop, learn some stuff and eat cheesecake…..who could ask for more?


text:
Robert Joseph "Bob" Cousy (ur. 9 sierpnia 1928 w Nowym Jorku) – amerykański koszykarz, występujący na pozycji rozgrywającego, sześciokrotny mistrz NBA, członek Koszykarskie Galerii Sław im. Jamesa Naismitha.

Mierzący 185 cm wzrostu koszykarz studiował w College of the Holy Cross. Do NBA został wybrany z 4 numerem w drafcie 1950 przez Tri-Cities Blackhawks, został jednak oddany do Chicago Stags. Organizacja wkrótce się rozpadła, a Cousy trafił do Boston Celtics. W organizacji tej spędził większość kariery (1950–1963). Obok Billa Russella był najważniejszym graczem drużyny, która na przełomie lat 50. i 60. zdominowała ligę. W 1957 został wybrany MVP sezonu. Był prawdziwym przywódcą zespołu, doskonale panował nad piłką i świetnie asystował - wielokrotnie był liderem tej klasyfikacji. Tytuły mistrzowskie zdobywał w latach 1957, 1959, 1960, 1961, 1962 i 1963. W sezonie 1969/1970 na krótko wrócił na parkiety NBA w barwach Cincinnati Royals (pracował wówczas jako trener tej drużyny) i rozegrał 7 spotkań.

27 lutego 1959 ustanowił nadal aktualny rekord NBA, notując 19 asyst w trakcie jednej połowy spotkania sezonu zasadniczego. Miało to miejsce podczas konfrontacji z zespołem Minneapolis Lakers.

Trzynaście razy brał udział w meczu gwiazd NBA (MVP w 1954 i 1957). Jako jeden z nielicznych koszykarzy znalazł się we wszystkich historycznych składach najlepszych zawodników NBA: NBA 25th Anniversary Team, NBA 35th Anniversary Team oraz NBA's 50th Anniversary All-Time Team.

Jest jednym z ponad 40 zawodników w historii, którzy zdobyli zarówno mistrzostwo NCAA, jak i NBA w trakcie swojej kariery sportowej.

W 1996 wystąpił w filmie Chluba Boston Celtics.

Osiągnięcia

NCAA
 Mistrz NCAA (1947)
 Wybrany do:
 I składu All-American (1950)
 II składu All-American (1949)
 III składu All-American (1948)
 Akademickiej Galerii Sław Koszykówki (2006)
 Uczelnia Holy Cross zastrzegła należący do niego numer 14

NBA
  6-krotny mistrz NBA (1957, 1959-63)
 Wicemistrz NBA (1958)
 MVP:
 sezonu NBA (1957)
 meczu gwiazd NBA (1954, 1957)
 Uczestnik meczu gwiazd:
 NBA (1951–1963)
 Legend NBA (1964, 1985–1987)
 Wybrany do:
 I składu NBA (1952–1961)
 II składu NBA (1962, 1963)
 grona najlepszych zawodników w historii NBA przy okazji obchodów rocznicowych:
 25-lecia istnienia - NBA 25th Anniversary Team
 35-lecia istnienia - NBA 35th Anniversary Team
 50-lecia istnienia - NBA's 50th Anniversary All-Time Team
 75-lecia istnienia - NBA 75th Anniversary Team (2021)
 Koszykarskiej Galerii Sław im. Jamesa Naismitha (Naismith Memorial Basketball Hall Of Fame - 1971)
 Lider:
 sezonu zasadniczego w asystach (1953-1960)
 play-off w:
 średniej:
 zdobytych punktów (1953–1955)
 asyst (1954–1961)
 skuteczności rzutów:
 z gry (1956)
 wolnych (1955)
 Klub Boston Celtics zastrzegł należący do niego w numer 14

 Trenerskie
 Trener drużyny Wschodu podczas meczu gwiazd Legend NBA (1988, 1990)

Zobacz też
 Bob Cousy Award

Przypisy

Amerykańscy koszykarze
Koszykarze NCAA All-American
Koszykarze Holy Cross Crusaders
Koszykarze Cincinnati Royals
Koszykarze Boston Celtics
Amerykańscy trenerzy koszykarscy
Trenerzy Cincinnati Royals
Członkowie Basketball Hall of Fame
Zawodnicy NBA z zastrzeżonymi numerami
NBA All-Stars
Ludzie urodzeni w Nowym Jorku
Urodzeni w 1928


Welcome to Nerd Nite London, where it's hip to be square. We host a monthly event in East London where we invite nerds to come speak about interesting stuff, while we drink beer and learn things. Evidence based entertainment, in a bar.
We're also on twitter @nerdnitelondon and on Facebook.