Break the ice and build energy in a classroom setting.
Encourage students to think more creatively when problem- solving.
Feel how movement can be an active component of the learning process.
How do you make rain inside a room? This is the opening question for a playful, kinaesthetic, exercise that can help students begin to think more creatively and collaborate as a group. In a round circle of 15 or more, participants work together to create the sound of light rain that then escalates to a powerful storm and back to light rain.
This exercise works for all audiences, undergraduate, graduate, and executive. The exercise is best positioned at the start of a session or a class as a non- traditional opener.
The instructor should begin with the following challenge: "If we wanted to make it rain in this classroom right now, how could this be accomplished?"
Students will offer the more obvious solutions such as: light a fire so the sprinkler system will go off; bring in a bucket of water and throw it up in the air; shake a closed bottle of carbonated water and then open it. Note that many of these solutions are individual based, and they are neither realistic nor innovative. When there are no more ideas, ask the students to form a circle (shoulder to shoulder) in the room and announce the following: "We are going to make rain together as a group. Only do what the person to your right is doing and don't start until the person on your right starts." The "rain" is made during a series of seven rounds. The instructor begins each round, and then the person to the left will do exactly as the instructor does. The rounds flow into one another; there is never a break in the flow of making the rain.
Round 1: rubbing hands together. The instructor begins by rubbing his or her hands together palm to palm. The person on the immediate left should immediately follow. Eventually the entire group will be rubbing their hands together. When hand rubbing reaches the person to the immediate right of the instructor, it's time for round 2.
Round 2: snaps. While everyone is still rubbing their hands together the instructor then begins snapping his or her fingers of both hands. The person on the left should immediately follow. Eventually the entire group will move from rubbing hands to snapping fingers. When the snapping of fingers reaches the person to the immediate right of the instructor, it's time for round 3.
Round 3: slapping hands on thighs. Repeat the format of the previous rounds. Transition to round 4 when thigh slapping reaches the person to the instructor's immediate right.
Round 4: stomping feet while slapping hands on thighs. The storm is at its peak during this round. Repeat the format of previous rounds.
Round 5: return to slapping hands on thighs. Repeat the format of the previous rounds.
Round 6: snaps return, and the storm begins to subside. Repeat the format of the previous rounds.
Round 7: rub hands together so the light rain returns, and then end. Repeat the format of the previous rounds.
Congratulate the students for making rain and creating a storm and initiate a round of applause.
When I asked the question about creating rain, why did your answers not consider "sound" or other ways to create rain?
What does this mean for how you think about creating opportunities? Problem solving?
What was your reaction when I asked you to get into a circle?
How did you feel before, during, and after the exercise?
Every round must flow into the next without stopping.
The instructor sets the tone for each round because he or she is the first to go. For example, it's important to slap hands on thighs rather hard to get the sound needed. The same can be said for stomping.
Students have a tendency to want to all start rubbing their hands together when the first person starts. It's important that each person waits to start (or change) until the person to his or her right starts or changes. The instructor may have to start, stop, and restart in the first round to make the point regarding who does what activity when.
Our frames of reference are the starting point for problem solving and creating, yet these frames are limiting.
There is usually uncertainty associated with asking students to get into a circle, but through collaboration this uncertainty is diminished and something completely unexpected is created.
Entrepreneurship requires action. Simply moving your body can have an immediate impact on emotions, motivation, and confidence to continue.
The following YouTube video illustrates the power of the exercise: http:// www.youtube.com/watch?v5LKDGCgXtETc.
Heidi M. Neck, Patricia G. Greene and Candida G. Brush, 2014. Teaching Entrepreneurship: A Practice-Based Approach (pp.114 – 117). Edition. Edward Elgar Pub.
This exercise has been used by teachers of all levels, though it's not clear who created the exercise. See http://www.teampedia.net/wiki/index. php?title5Make_it_Rain.
Gardner, H. 2011. Frames of Mind: The Theory of Multiple Intelligences. New York: Basic Books.
This exercise is taken from, Heidi M. Neck, Patricia G. Greene and Candida G. Brush, 2014. Teaching Entrepreneurship: A Practice-Based Approach (pp.114 – 117). Edition. Edward Elgar Pub, and is reprinted with the kind permission of the authors.
This exercise is a fantastic way to get people working together to complete a simple challenge. The idea is that they are given a pack of spaghetti, a pen, scissors, a reel of cotton, two chairs and a bar of chocolate for each member of the team. They are asked to build a bridge using the equipment provided that will hold the bars of chocolate. The first team to do this wins.
An example of the tools they will be given is laid out on the facilitators desk where the spaghetti packet is opened and the cotton is cut into a few lengths, providing visual example of what is possible (if potentially misleading!).
The chairs are spread two feet apart and cannot be moved (ideally these will be away from other groups to stop cheating) items are given to the group and they are advised that they are to have a 5 minute team talk to prepare before they start, however the first one to build the bridge wins (and invariably they race ahead and start to build the bridge, but you can reflect upon this approach with the whole group at the end).
Typically the further they race ahead (without planning and discussion) and open the items, the more problems they will face. For example, once the spaghetti is opened it becomes a lot harder but the more creative constructions emerge. The best solution is to leave the packet closed, pierce the ends and use the cotton to secure the packet to the chair and rest the chocolate on the packet of spaghetti. However it is still possible to complete the task once the spaghetti is opened, but they will do well not only to hold the chocolate, but complete the task in 30 minutes.
At the end of the 30 minutes, the groups get a chance to see what the other teams have done and see how creative they have been in creating a bridge.
This could be done with any size group as long as there are sufficient facilitators to split into smaller groups. The optimum numbers in each group would be 4, however multiple groups will be working at the same time. They would have to work at the same time so as not to hear the discussion of other groups.
In advance you need to purchase resources.
Each team will need a pack of spaghetti, scissors, reel of cotton, two chairs and a chocolate bar for each team member. Someone will need to watch the time or you can add time keeping to their responsibilities.
This guide was produced by Michael Marsden.
This example was delivered in a standard lecture theatre for initial interactive workshop, moving (after forty five minutes six tutorial rooms) used for group task in order to deliver to a class of 300.
Within this half day session, second year nurses were introduced to the concepts of enterprise and its relevance to the nursing profession. They had to debate its relevance within the lecture and work in groups to develop solutions to healthcare problems. They then faced a group challenge over a three hour period, which they ultimately needed to present to be assessed by a panel including a lecturer, nursing practitioner and a nursing entrepreneur (externals).
Initial exploration of the concepts of enterprise and entrepreneurship allows for agreement to be formed over "what are the enterprise skills or competencies" being sought by nurses and within healthcare.
Using interactive and discursive techniques to allow for initial thoughts and fears to be shared in small group, it is possible to explore these personal skills and draw out experiences and expertise from the prior knowledge/background of the student group.
The next stage is to collate these responses to create a formal group response (see skill development through reflective practice HOW TO GUIDE ETC SKILL) and ask them, as a group, to debate their relevance to practical nursing. Once this has been explored as the full cohort, put the students into groups and ask them to identify student health problems (issues of personal health that they and their peers ought to consider) and brain storm how to solve or reduce the impact of this issue. Using resources, knowledge and materials they are invited to be ready to pitch to panel for 10 minutes (there is an option to submit group written report) where they will ideally pitch to a panel of externals.
Another option is to provide each team with investment money (monopoly money or similar) and invite them to "fund" the best idea. Once feedback has been given to the group, invite full group reflection and re-examine enterprise skills and behaviours and their confidence in them.
The impact of this approach was clear, with over fifty unsolicited emails providing positive feedback (in addition to the module review). This was measured annually but with the initial 300 students tracked for ten years with their level of Enterprise competency development measure Comments included "the most relevant, best teaching in my entire Nursing degree" and "I now understand how enterprise relates to nursing and intend to make a difference through being an enterprising nurse".
The students in particular enjoyed the interactivity, the focus on social enterprise and the use of fun and drama to get all students involved – and overall the module feedback was the highest for any teaching in the nursing degree.
The initial group of 300 filled in an "efactor" (Gibson 2006) competency questionnaire (both before and after the programme) and then two years five years and ten years afterwards. This clearly showed that all enterprise competencies had increased but in particular resilience, creativity and personal branding had shown the most significant increases over a ten year period.
Wider impact was also demonstrated as five student groups per annum received UNLTD funding and one group in year two won the University business pitch competition, competing against students from all disciplines.
By exploring their own creativity within the context of how social enterprises work, all the nursing students were able to explore their enterprise competences and understand the link between enterprise and nursing. Discussions surrounding the changing Health Environment help to separate out an individual's potential for creativity from perceived increasing corporatisation through privatisation and ensure that the student understand their role in patient care requires enterprising skills and behaviour, working within the NHS framework.
The students all enjoyed the opportunity for creative group work and exploring their profession in a new light.
Further support for the pitching/presentation elements could be have beneficial to some groups, had additional time been available. However the "common currency" of TV Programmes such as "Dragon's Den" allows pitching to be introduced as a fast and effective communication/presentation method to any student group.
This guide was produced by David Gibson OBE ( Head of Entrepreneurship Education, Liverpool John Moores University ). If you would like to contact the author, please use this email address:- d.a.gibson@ljmu.ac.uk.
Option to support future development: to provide the opportunity to identify gaps in learning or development and create a personal action plan for personal development and future learning.
This task provides an opportunity to reflect on the learning gained during particular tasks for activities (ideally should be of "medium" length, such as intense induction programmes, week long activities or longer learning 'events' (modules or years of study). This can be particularly effective in terms of drawing out "change" or learning gain as identified by the learner themselves.
This approach provides an opportunity to reflect upon a wide range of individual development (including emotional development and confidence levels) as well as recognising improvement in the development of skills.
Traditionally physical diaries were issued to encourage students to write regularly and informally, however the wide range of multi-media (through smart phones and tablets) also allows students to select their own format (s) or trial the use of a new media tool for this purpose (ideally agreed in advance with tutor to avoid IT issues in viewing).
A learning diary is therefore a tool of reflection which can take a variety of forms.
NB: Consideration of how to create "value" is key in determining the role/purpose of this approach within an assessment strategy or within a programme. Typically students value activities that the tutor places a value on, and their currency is marks/assessment. However as diary is, by definition, a subjective view, and should reflect what the student has heard, learnt and reviewed, it is the student's own analysis and insights that count, and clear marking parameters and guidance need to be provided to ensure clarity.
Issuing this task should be done at the start of the activity that you wish the learners to reflect upon. Ideally you encourage (or set) answering a range of open-ended questions, delighted to understand their initial position as they approach this learning/task. This may include expanding upon their prior understanding or life experience, as relevant to this work.
Once the activities are being undertaken, reflective models can be issued or sourced by the students to support their thinking. However you may wish to provide a set of reflective questions at regular intervals as prompts to their developing thinking.
This activity can be highly prescriptive, with set timescales at which you expect stages of reflection to be completed (as relating to the task being undertaken) however it is also possible to make this an open task, where the approach and learning is with the student to design and undertake. This allows the learner to explore, source and select their own model for reflection and test its effectiveness as a tool for their development during the process. This additional skills of research, evaluation and comparative analysis but risks diluting the quality of the reflection if the students place the emphasis upon critiquing models rather than the task itself and their personal learning. It is therefore important that you reflect the emphasis you wish to seek within your assessment schedule.
To increase the synthesis, and the ability for personal and confidential reflection, you may wish to create a format in which the students regularly capture thoughts and feelings, but keep this as a personal document (diary, blog or video diary) from which the submission is created. This synthesised version of their learning and reflections build an understanding of their personal development over time and allows for honest and uncensored self-reporting and reflection. Again the structure/control of the format/questions can be loose and open (providing only sources and reference to guide) or highly prescriptive (working within a template or with specific tools/questions) to ensure that the key elements of learning (including emotional elements and confidence) are a required feature of the submission.
Personal reflection is a tremendous skill, but is often difficult for students to develop, particularly during a period of study, with little or no external reference points or practical application. It is therefore recommended that this is an assessed piece, so that the value of reflection is made clear. It is therefore important that you, as the tutor, place importance upon the development of this skill and take class-time to consider what is meant by reflection practice and how to ensure that reflection leads to learning.what is meant by reflective practice and how to ensure that reflection leads to learning. It is also important to consider the formative as well summative assessment within this process, as reflective skills are improved through regular practice, and this form part of your regular teaching. It is important that you 'model' a reflective approach with the students by including reflective questions onto your regular contact with them, and making reflection an explicit aspect of your activity/classroom debrief. Making this explicit within your teaching will reinforce the student's understanding of reflection as an activity to repeated and practiced, as well as help them see how reflective questioning or models can deepen their understanding, and build confidence in their abilities.
Gibb's reflective cycle: from Gibbs, G (1988) "Learning by doing: A guide to teaching and learning methods"
This guide was produced by Peter Brown (University of South Wales). If you would like to contact the author, please use this email address:- peter.brown@southwales.ac.uk.


text:
By: Principle Gallery April 29, 2014 July 6, 2014
Fine Artexhibition, Tourist Season
Tourist Season Spotlight: Marina Dieul and Mark English
We could not be more pleased with the incredible turnout from last Friday's 20th Anniversary celebration, not to mention the outstanding collection of artwork from this invitational. All it takes is one look at the variety and quality of the artwork currently covering the walls, and it's clear why these amazing artists (not to mention the collectors who purchase their work!) have led to twenty successful years at Principle Gallery. Here's to twenty more!
Feel free to check out our website or stop by the gallery in the next few weeks to check out the 20th Anniversary Invitational for yourself!
Switching gears, we thought it might be nice to use the blog as a place to highlight some of the artists who will be featured in May's exhibition, entitled "Tourist Season". Inspired by the influx of visitors to the greater DC area during this time of year, Principle Gallery will also be celebrating visitors, in the form of a stellar group of artists whose work has never been shown at our gallery before! Each blog post leading up to the May 16th opening reception of the show will feature some brief information about a couple of the artists whose work will be included in May's "Tourist Season" show. Today, we're highlighting Marina Dieul and Mark English.
Marina Dieul
Hailing originally from France, Marina has been painting since childhood. Eventually moving to Canada, where she currently resides, Marina has become a well known and highly regarded name in the art world, best known today for her evocative portraits and heart-melting depictions of small animals. Marina has been named a Living Master by the Art Renewal Center, and has accumulated an impressive number of honors and accolades– last year alone, she received awards including First Place in The Artist Magazine Competition, All Media category, Award of Exceptional Merit and Best Portfolio Award at the National Portrait Society of America's International Competition, and Second Place in the People and Figure Competition for International Artist Magazine…just to name a few.
Below is one of Marina's paintings featured in May's "Tourist Season" show, entitled "Minauderie 5":
"Minauderie 5," 8×14.5, oil on panel
Mark English
Born in Texas, self-taught painter Mark English has had an impressive journey as an artist. Originally a painter of signs for small-town businesses, Mark has developed outstanding notoriety in the art world since his graduation from the Art Center College in Los Angeles in 1960. Now both a renowned illustrator and successful painter for galleries worldwide, Mark has been elected to the Illustrators Hall of Fame in New York and can count his work among numerous prestigious private collections around the globe. His unique style in both painting and illustration has garnered him literally hundreds of awards and the status of the most awarded illustrator in the history of the Society of Illustrators in New York. His paintings, typically mixed media portraits and landscapes, feature vibrant color and mesmerizing luminosity.
This landscape, titled "Moreno Valley," is part of the upcoming "Tourist Season" invitational exhibition:
"Moreno Valley," 20×16, mixed media
Continue to check in with the blog and follow our social media pages on Facebook, Twitter, Instagram, and Pinterest for more information and images about the upcoming show!
The Principle Gallery Turns 20
Tourist Season Spotlight: Brett Amory and Amy Lind

redpajama_set_name: RedPajamaCommonCrawl


When it comes to cooking, Rangemaster really are the expert. This smaller 60cm model gives you a massive 96 litres of cooking space inside. Complete with 2 conventional ovens and a ceramic hob, it makes preparing a feast easy. Both the 61 litre and smaller 35 litre ovens are multifunctional, so they can be tailored to your cooking. This means you can braise steak or grill salmon, thanks to the open door grilling function. There's even space to prepare a tray bake of cherry and almond cakes inside the smaller oven. And, the ceramic hob provides lots of space to host an assortment of pans.


text:
About privacy and Location Services in iOS and iPadOS
Learn how privacy settings and Location Services help protect your personal information on your iPhone, iPad, and iPod touch.
Privacy settings in iOS and iPadOS help give you control over which apps have access to information stored on your device. For example, you can allow a social-networking app to use your camera, so you can take and upload pictures to that app. You can also grant access to your contacts, so a messaging app can find any friends that are already using the same app.
In Settings > Privacy, you can see which apps you have allowed to access certain information, as well as grant or revoke any future access. This includes access to:
Media & Apple Music
Motion & Fitness
On your device, you can select a type of data from this list to see which apps have asked for permission to use that data. An app won't appear on the list until it asks permission to use your data. You can add or remove permission from any app that has asked for access to data. An app can use the data type in the setting only if you have given the app permission.
If you sign in to iCloud, apps are granted access by default to iCloud Drive. You can view and manage apps that are allowed to use iCloud under iCloud in Settings as well.
If you allow third-party apps or websites to use your data or your current location, you're subject to their terms, privacy policies, and practices. You should review the terms, privacy policies, and practices of the apps and websites to understand how they use your location and other information. Information that Apple collects will be treated in accordance with Apple's Privacy Policy.
How your device uses Location Services
With your permission, Location Services allows apps and websites (including Maps, Camera, Weather, and other apps) to use information from cellular1, Wi-Fi2, Global Positioning System (GPS)3 networks, and Bluetooth4 to determine your approximate location5.
Apps that can show your location on the screen, including Maps, show your current (approximate) location using a blue marker. In Maps, if your location can't be determined precisely, you'll see a blue circle around the marker. The size of the circle shows how precisely your location can be determined—the smaller the circle, the greater the precision. When Location Services is active, a black or white arrow icon appears in the status bar.
Maps, directions, and location-based apps depend on data services. These data services are subject to change and might not be available in all geographic areas, resulting in maps, directions, or location-based information that might be unavailable, inaccurate, or incomplete. Compare the information provided on the device to your surroundings and defer to posted signs to resolve any discrepancies.
Learn more about Location Services and privacy.
Give apps permission to use your location
The first time an app tries to access your location, it must ask for your permission. You see a prompt explaining which app is asking for permission to use your location as well as the app developer's reason for requesting it.
Some apps ask to use your location only while the app is in use. An app is considered "in use" when you're actively using it in the foreground, or when it is using location in the background, in which case a blue pill will display in your status bar.
If you grant an app While In Use permission, the app may ask you for permission to use your location when it is in the background.
When you allow an app to use your location in the background, your device will remind you from time to time that an app is using your location, and will display those locations on a map. You'll also be asked if you want to continue to allow the app to use your location in the background.
With iOS 13 and iPadOS 13, you can tap Allow Once to let an app access Location Services data only for one session (One Time). If you close and then re-open the app and it tries to access your location again, it will ask for your permission again. Apps won't use your location until they have asked for your permission and you grant permission.
You can change your mind anytime and update your preferences from Settings < Privacy < Location Services.
Turn Location Services on or off
You can turn Location Services on or off at Settings > Privacy > Location Services. You can turn Location Services on either during the Setup Assistant process or later through the Location Services setting. You can individually control which apps and system services have access to Location Services data. When Location Services are off, apps can't use your location in the foreground or background. This will limit the performance of various Apple and third-party apps.
If you would like to reset all of your location settings to the factory default, go to Settings > General > Reset and tap Reset Location & Privacy. When your location and privacy settings are reset, apps will stop using your location until you grant them permission.
Settings > Privacy > Location Services.
Settings > Privacy > Location Services > System Services.
Improve GPS accuracy
GPS accuracy depends on the number of visible GPS satellites. Locating all visible satellites can take several minutes, with accuracy gradually increasing over time. To improve GPS accuracy:
Make sure that you've set the date, time, and time zone correctly on the device in Settings > General > Date & Time. If possible, use Set Automatically.
Keep a clear view in several directions. Walls, vehicle roofs, tall buildings, mountains, and other obstructions can block line of sight to GPS satellites. When this happens, your device uses Wi-Fi or cellular networks to determine your position until the GPS satellites are visible again.
Crowd-sourced Wi-Fi and cellular Location Services
If Location Services is on, your device will periodically send the geo-tagged locations of nearby Wi-Fi hotspots and cell towers to Apple to augment Apple's crowd-sourced database of Wi-Fi hotspot and cell tower locations. If you're traveling (for example, in a car) and Location Services is on, a GPS-enabled iOS device will also periodically send GPS locations, travel speed, and barometric pressure information to Apple to be used for building up Apple's crowd-sourced road-traffic and indoor pressure databases. The crowd-sourced location data gathered by Apple is stored with encryption and doesn't personally identify you.
About Bluetooth, data and Location Services
With iOS 13 and iPadOS 13, an app must ask permission to use Bluetooth functions except to play audio to a Bluetooth device, which doesn't require permission. You can change your device's Bluetooth permissions from Settings > Privacy > Bluetooth.
About your location during emergency calls
For safety purposes, your iPhone's location information may be used when you place an emergency call to aid response efforts regardless of whether you enable Location Services.
Learn more about using Location Services to track your location in Maps and other apps in your product's user guide.
Learn more about Apple and Privacy.
1. iPad (Wi-Fi + Cellular models): Make sure that Cellular Data is on in Settings > Cellular > Cellular Data, even if you're not subscribed to a cellular data plan. This will allow your device to more accurately calibrate itself using network time and location information. In some instances, disabling a SIM PIN may be necessary.
2. iOS and iPadOS devices sold in China mainland and Germany may use the term Wireless LAN (WLAN) instead of Wi-Fi.
3. GPS is available on iPhone and iPad Wi-Fi + Cellular models.
4. Bluetooth interacts with iBeacons to provide a way to create and monitor areas that advertise certain identifying information.
5. iOS and iPadOS devices without a cellular connection use only Wi-Fi for Location Services (if a Wi-Fi network is available). Some third-party apps rely on a Wi-Fi connection for region monitoring. If a device is passcode locked, this feature may be limited or inaccurate.
Published Date: May 13, 2020
Privacy Policy Terms of Use Sales and Refunds Site Map


text:
Q: How many square integers less than $n$? Let $n$ be a positive integer. Let $S(n)$ be the number of positive square integers less than $n$, e.g.
\begin{bmatrix}
n & S(n) \\
\hline
1 & 0 \\
2& 1 \\
3& 1 \\
4& 1 \\
5 & 2 \\
6& 2 \\
7& 2 \\
8& 2 \\
9& 2 \\
10& 3 \\
\end{bmatrix}
Is there a simple formula for $S(n)$ as a function of $n$? I haven't been able to figure it out.

A: The floor function applied to $\sqrt{n-1}$ :
$$S(n) = \lfloor \sqrt {n-1} \rfloor$$
Explation: $k^2 < n \iff k^2 \leq n-1 \iff k \leq \sqrt{n-1}$.

redpajama_set_name: RedPajamaStackExchange


text:
James Francis Browne   was an Anglican Archdeacon in India in the late 19th century.

Browne was educated at the London College of Divinity  and ordained in 1867. After curacies in Middleton Tyas  and  Bath he went as a Chaplain to Madras. He served at Cannanore, Secunderabad, Black Town and Bangalore. He was  Archdeacon of Madras from 1884 to 1891. Returning to England he was the incumbent in Flax Bourton from 1892 to 1894.

References

19th-century Indian Anglican priests
Alumni of the London College of Divinity
Archdeacons of Madras

Year of birth missing
Year of death missing

redpajama_set_name: RedPajamaWikipedia


Most people are familiar with the afternoon and evening liturgies of the Triduum: Mass of the Lord's Supper on Holy Thursday, the Liturgy of the Lord's Passion on Good Friday, and, of course, the Easter Vigil Mass the night before Easter Sunday. But there is also another powerful time of liturgical prayer during the early morning of these three days: Tenebrae.
Rain or Shine, It's Catholic Stanford Time!
We'd been praying for good weather - all week, we'd been blessed with steady rain, but on this day, we hoped for sun! As Mass ended at 9 a.m. on Saturday, March 9th, the rain returned. But that didn't stop about 20 Stanford University students from gathering outside our gate for a morning of grounds work!
Our passions, or our emotions, are truly powerful - they can move us, or drag us down, like nothing else! But can we harness our passion power for good and not evil?
Last Saturday, we had a wonderful group of young women join us for a Come and See Day.
In her 101st year of earthly life and her 64th year of religious profession, our Sister Mary of the Compassion, O.P. has passed to her eternal reward.
By now in the world, Christmas is a distant memory. But in the monastery, we are still celebrating in full swing. Where Christmas tends to be big and flashy, with lots of greetings and gifts between our community and our family, friends, and benefactors, Epiphany is a big feast for us as a community and it is marked in a special way by our novitiate sisters.
Dominican Joy (with Our Latest Video)!
A young woman recently completed her aspirancy with our community. As she was nearing the end of her month-long stay, one of the sisters asked her, "Has anything surprised you during your time here?" The young woman smiled and exuberantly responded, "You're all crazy – in a good way! I didn't expect you to have so much fun!" She's not the first to be surprised by our Dominican joy.


This hand hammered bracelet is an essential element to your accessory collection. Simply put, you'll never want or need to take this bracelet off. As a staple it can be worn on its own or stacked in multiples with other bracelets.
Named after the crescent shape of the moon, which in Alchemy, is a symbol of the metal Silver and known to be the ruler of the brain.
Comes in Ancient Red Bronze and Yellow Bronze. 14K Gold upon request.


text:
Architettura
Casa Pandolfini – edificio storico di Firenze
Palazzo Pandolfini – palazzo rinascimentale di Firenze
Torre Pandolfini – antica torre di Lastra a Signa, città metropolitana di Firenze
Villa Carducci-Pandolfini, nota anche come Villa Carducci di Legnaia – villa di Firenze
Villa Pandolfini – villa nella frazione di Ponte a Signa, comune di Lastra a Signa
Villa Pandolfini di Carraia – villa di Firenze

Persone
Pandolfini – famiglia nobile italiana
Agnolo Pandolfini (1360-1446) – politico fiorentino del XV secolo
Angelica Pandolfini (1871-1959) – soprano italiano
Bruce Pandolfini (1947) – scacchista statunitense
Egisto Pandolfini (1926-2019) – allenatore di calcio ed ex calciatore italiano, di ruolo centrocampista
Francesco Pandolfini (1833-1916) – baritono italiano
Gianfranco Pandolfini (1920-1997) – pallanuotista italiano
Giannozzo Pandolfini (1450 circa–1525) – vescovo cattolico italiano
Niccolò Pandolfini (1440-1518) – cardinale e vescovo italiano
Pier Filippo Pandolfini (...–...) – politico fiorentino del XV secolo
Tullio Pandolfini (1914-1999) – pallanuotista italiano
Turi Pandolfini, propr. Salvatore Pandolfini (1883-1962) – attore italiano

Altro
Via de' Pandolfini – strada del centro storico di Firenze


Edward M. Hallowell, M.D., & John J. Ratey, M.D.
"Driven to Distraction" has been a lifeline to the approximately 18 million Americans who are thought to have ADHD. But most importantly, the authors focus on the positives that can come with this "disorder"—including high energy, intuitiveness, creativity, and enthusiasm.


text:
\section{Introduction}

Physics in extremely high energy regions is particularly of interest to particle physics.
String theory is one of the most successful theoretical frameworks
which overcome the difficulty of ultra-violet divergence in quantum
theory of gravity.
However it has many difficulties in performing practical computations.
Therefore if we construct a field theory which captures some
stringy nature and/or includes stringy corrections, then it would play
a pivotal role in investigating physics in high energy regions
even near the Planck scale.

Some of the stringy corrections appear as $\alpha'$ corrections.
In other words, it often takes the form as higher derivative
corrections i.e. higher order polynomial of momentum.
One of the way to discuss these corrections is deforming the
Heisenberg uncertainty principle to a generalized uncertainty principle (GUP):
\begin{eqnarray}
\Delta \hat{x} \ge \frac{\hbar}{2\Delta \hat{p}} + \frac{\hbar \beta}{2}\Delta \hat{p},
\end{eqnarray}
where $\beta$ is a deforming parameter.
This parameter has the dimension of
$(length)^2$ and its square root gives the minimal length scale of the
theory under consideration.
If it is realized in a certain string theory context, $\beta$ would take a value of order the string scale ($\beta \sim \alpha'$).
This relation comes from various types of studies such as on
high energy or short distance behavior of
strings \cite{Gross:1987kz}, \cite{Konishi:1989wk}, gedanken experiment
of black hole \cite{Maggiore:1993rv}, de Sitter space
\cite{Snyder:1946qz}
and the symmetry of massless particle \cite{Chagas-Filho:2005at}.

There are several canonical commutation algebra which lead to the GUP.
Among these algebra we will focus on the algebra:
\begin{eqnarray}
[\hat{x},\hat{p}]=i\hbar (1+\beta \hat{p}^2).
\label{deformed alg}
\end{eqnarray}
This algebra is investigated in \cite{Kempf:1994su}-\cite{Hossenfelder:2005ed} and an attempt to
construct a field theory with minimal length scale is found in
\cite{Kempf:1996ss}.
In their work they quantize fields using the Bargmann-Fock representation in 1+1 dimensional spacetime.
This has also been used in cosmology, especially in physics at an early universe
(see for example, \cite{Ashoorioon:2004vm}-\cite{Ashoorioon:2005ep} and references therein).

In this paper, we investigate the quantization of fields based on the deformed algebra (\ref{deformed alg}) in the canonical formalism in 1+1 dimensions and in the path integral formalism as well.
In the process of quantization using the latter formalism, we use the
Bjorken-Johnson-Low prescription \cite{Bjorken:1966jh} which states
that T-product of fields must take same value in the two formalism.

This paper is organized as follows.
In section 2, we first introduce a Hilbert space definitely in terms of the maximally localized states which satisfy minimal uncertainty in position space \cite{Kempf:1994su}.
In section 3, we quantize fields in 1+1 dimensions in the
canonical formalism, where the maximally localized states are taken to be one particle states.
In section 4, we construct 1+1 dimensional quantum field theory in the path integral formalism so that it is equivalent to that in the canonical formalism.
We find that higher dimensional extension is straightforward in the path integral formalism.

\section{Generalized Uncertainty Principle}

We start with the following deformed Heisenberg algebra \cite{Kempf:1994su}:
\begin{eqnarray}
[\hat{x},\hat{p}]=i\hbar (1+\beta \hat{p}^2),
\label{GUP}
\end{eqnarray}
which has a representation in momentum space:
\begin{eqnarray}
\hat{p}&=&p , \nonumber \\
\hat{x}&=&i \hbar (1+\beta p^2) \frac{\partial}{\partial p}.
\end{eqnarray}
In general, states which have minimal uncertainty obey the following equation:
\begin{eqnarray}
\bigl(\hat{x}-\langle \hat{x} \rangle + \frac{\langle \left[ \Delta
						       \hat{x}, \Delta \hat{p}
\right]\rangle}{2(\Delta p)^2}\Delta \hat{p}\bigr)|\psi \rangle &=&0,
\end{eqnarray}
where
\begin{eqnarray}
\Delta \hat{A}&\equiv&\hat{A}-\langle \hat{A} \rangle,
 \nonumber \\
\Delta{A}&\equiv&\sqrt{\langle(\Delta\hat{A})^2 \rangle}. \nonumber
\end{eqnarray}
Among these states, we will focus on the states which are subject to the following conditions:
\begin{eqnarray}
\Delta x &=& \hbar \sqrt{\beta}, \\
\Delta p &=& \frac{1}{\sqrt{\beta}}, \\
\langle \hat{p} \rangle &=& 0.
\end{eqnarray}

\begin{figure}[h]
\begin{center}
\setlength{\unitlength}{0.240900pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}%
\begin{picture}(1500,900)(0,0)
\font\gnuplot=cmr10 at 10pt
\gnuplot
\sbox{\plotpoint}{\rule[-0.200pt]{0.400pt}{0.400pt}}%
\put(141.0,123.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(121,123){\makebox(0,0)[r]{ 0}}
\put(1419.0,123.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,246.0){\rule[-0.200pt]{28.818pt}{0.400pt}}
\put(121,246){\makebox(0,0)[r]{ $\sqrt{\beta}$}}
\put(1419.0,246.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,369.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(121,369){\makebox(0,0)[r]{ 2$\sqrt{\beta}$}}
\put(1419.0,369.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,491.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(1419.0,491.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,614.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(121,614){\makebox(0,0)[r]{ 4$\sqrt{\beta}$}}
\put(1419.0,614.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,737.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(121,737){\makebox(0,0)[r]{ 5$\sqrt{\beta}$}}
\put(1419.0,737.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,860.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(121,860){\makebox(0,0)[r]{ 6$\sqrt{\beta}$}}
\put(1419.0,860.0){\rule[-0.200pt]{4.818pt}{0.400pt}}
\put(141.0,123.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(141,82){\makebox(0,0){ 0}}
\put(141.0,840.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(271,123.0){\rule[-0.200pt]{0.400pt}{30.318pt}}
\put(265,246){\makebox(0,0){ $\otimes$}}
\put(300,210){\makebox(0,0){ $\nwarrow$}}
\put(590,190){\makebox(0,0){`Maximal localization states'}}
\put(271,82){\makebox(0,0){ $\frac{1}{\sqrt{\beta}}$}}
\put(401.0,123.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(401,82){\makebox(0,0){ $\frac{2}{\sqrt{\beta}}$}}
\put(401.0,840.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(660.0,123.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(660,82){\makebox(0,0){ $\frac{4}{\sqrt{\beta}}$}}
\put(660.0,840.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(920.0,123.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(920,82){\makebox(0,0){ $\frac{6}{\sqrt{\beta}}$}}
\put(920.0,840.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(1179.0,123.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(1179,82){\makebox(0,0){ $\frac{8}{\sqrt{\beta}}$}}
\put(1179.0,840.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(1439.0,123.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(1439,82){\makebox(0,0){ $\frac{10}{\sqrt{\beta}}$}}
\put(1439.0,840.0){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(141.0,123.0){\rule[-0.200pt]{312.688pt}{0.400pt}}
\put(1439.0,123.0){\rule[-0.200pt]{0.400pt}{177.543pt}}
\put(141.0,860.0){\rule[-0.200pt]{312.688pt}{0.400pt}}
\put(40,491){\makebox(0,0){$\Delta x$}}
\put(790,21){\makebox(0,0){$\Delta p$}}
\put(141.0,123.0){\rule[-0.200pt]{0.400pt}{177.543pt}}
\put(154,737){\usebox{\plotpoint}}
\multiput(154.58,698.52)(0.493,-11.796){23}{\rule{0.119pt}{9.269pt}}
\multiput(153.17,717.76)(13.000,-278.761){2}{\rule{0.400pt}{4.635pt}}
\multiput(167.58,426.45)(0.493,-3.748){23}{\rule{0.119pt}{3.023pt}}
\multiput(166.17,432.73)(13.000,-88.725){2}{\rule{0.400pt}{1.512pt}}
\multiput(180.58,337.96)(0.493,-1.726){23}{\rule{0.119pt}{1.454pt}}
\multiput(179.17,340.98)(13.000,-40.982){2}{\rule{0.400pt}{0.727pt}}
\multiput(193.58,296.74)(0.494,-0.864){25}{\rule{0.119pt}{0.786pt}}
\multiput(192.17,298.37)(14.000,-22.369){2}{\rule{0.400pt}{0.393pt}}
\multiput(207.58,273.80)(0.493,-0.536){23}{\rule{0.119pt}{0.531pt}}
\multiput(206.17,274.90)(13.000,-12.898){2}{\rule{0.400pt}{0.265pt}}
\multiput(220.00,260.93)(0.728,-0.489){15}{\rule{0.678pt}{0.118pt}}
\multiput(220.00,261.17)(11.593,-9.000){2}{\rule{0.339pt}{0.400pt}}
\multiput(233.00,251.94)(1.797,-0.468){5}{\rule{1.400pt}{0.113pt}}
\multiput(233.00,252.17)(10.094,-4.000){2}{\rule{0.700pt}{0.400pt}}
\multiput(246.00,247.95)(2.695,-0.447){3}{\rule{1.833pt}{0.108pt}}
\multiput(246.00,248.17)(9.195,-3.000){2}{\rule{0.917pt}{0.400pt}}
\put(272,245.67){\rule{3.132pt}{0.400pt}}
\multiput(272.00,245.17)(6.500,1.000){2}{\rule{1.566pt}{0.400pt}}
\put(285,246.67){\rule{3.132pt}{0.400pt}}
\multiput(285.00,246.17)(6.500,1.000){2}{\rule{1.566pt}{0.400pt}}
\put(298,248.17){\rule{2.700pt}{0.400pt}}
\multiput(298.00,247.17)(7.396,2.000){2}{\rule{1.350pt}{0.400pt}}
\multiput(311.00,250.61)(2.918,0.447){3}{\rule{1.967pt}{0.108pt}}
\multiput(311.00,249.17)(9.918,3.000){2}{\rule{0.983pt}{0.400pt}}
\multiput(325.00,253.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}}
\multiput(325.00,252.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}}
\multiput(338.00,257.61)(2.695,0.447){3}{\rule{1.833pt}{0.108pt}}
\multiput(338.00,256.17)(9.195,3.000){2}{\rule{0.917pt}{0.400pt}}
\multiput(351.00,260.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}}
\multiput(351.00,259.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}}
\multiput(364.00,264.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}}
\multiput(364.00,263.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}}
\multiput(377.00,268.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(377.00,267.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(390.00,273.60)(1.797,0.468){5}{\rule{1.400pt}{0.113pt}}
\multiput(390.00,272.17)(10.094,4.000){2}{\rule{0.700pt}{0.400pt}}
\multiput(403.00,277.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(403.00,276.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(416.00,282.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(416.00,281.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(429.00,287.59)(1.489,0.477){7}{\rule{1.220pt}{0.115pt}}
\multiput(429.00,286.17)(11.468,5.000){2}{\rule{0.610pt}{0.400pt}}
\multiput(443.00,292.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(443.00,291.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(456.00,297.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(456.00,296.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(469.00,302.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(469.00,301.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(482.00,308.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(482.00,307.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(495.00,313.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(495.00,312.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(508.00,318.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(508.00,317.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(521.00,324.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(521.00,323.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(534.00,329.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(534.00,328.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(547.00,335.59)(1.214,0.482){9}{\rule{1.033pt}{0.116pt}}
\multiput(547.00,334.17)(11.855,6.000){2}{\rule{0.517pt}{0.400pt}}
\multiput(561.00,341.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(561.00,340.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(574.00,346.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(574.00,345.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(587.00,352.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(587.00,351.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(600.00,358.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(600.00,357.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(613.00,363.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(613.00,362.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(626.00,369.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(626.00,368.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(639.00,375.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(639.00,374.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(652.00,381.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(652.00,380.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(665.00,386.59)(1.214,0.482){9}{\rule{1.033pt}{0.116pt}}
\multiput(665.00,385.17)(11.855,6.000){2}{\rule{0.517pt}{0.400pt}}
\multiput(679.00,392.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(679.00,391.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(692.00,398.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(692.00,397.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(705.00,404.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(705.00,403.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(718.00,410.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(718.00,409.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(731.00,416.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(731.00,415.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(744.00,422.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(744.00,421.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(757.00,428.59)(1.378,0.477){7}{\rule{1.140pt}{0.115pt}}
\multiput(757.00,427.17)(10.634,5.000){2}{\rule{0.570pt}{0.400pt}}
\multiput(770.00,433.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(770.00,432.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(783.00,439.59)(1.214,0.482){9}{\rule{1.033pt}{0.116pt}}
\multiput(783.00,438.17)(11.855,6.000){2}{\rule{0.517pt}{0.400pt}}
\multiput(797.00,445.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(797.00,444.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(810.00,451.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(810.00,450.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(823.00,457.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(823.00,456.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(836.00,463.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(836.00,462.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(849.00,469.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(849.00,468.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(862.00,475.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(862.00,474.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(875.00,481.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(875.00,480.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(888.00,487.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(888.00,486.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(901.00,493.59)(1.214,0.482){9}{\rule{1.033pt}{0.116pt}}
\multiput(901.00,492.17)(11.855,6.000){2}{\rule{0.517pt}{0.400pt}}
\multiput(915.00,499.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(915.00,498.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(928.00,505.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(928.00,504.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(941.00,511.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(941.00,510.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(954.00,517.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(954.00,516.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(967.00,523.59)(0.950,0.485){11}{\rule{0.843pt}{0.117pt}}
\multiput(967.00,522.17)(11.251,7.000){2}{\rule{0.421pt}{0.400pt}}
\multiput(980.00,530.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(980.00,529.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(993.00,536.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(993.00,535.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1006.00,542.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1006.00,541.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1019.00,548.59)(1.214,0.482){9}{\rule{1.033pt}{0.116pt}}
\multiput(1019.00,547.17)(11.855,6.000){2}{\rule{0.517pt}{0.400pt}}
\multiput(1033.00,554.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1033.00,553.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1046.00,560.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1046.00,559.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1059.00,566.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1059.00,565.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1072.00,572.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1072.00,571.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1085.00,578.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1085.00,577.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1098.00,584.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1098.00,583.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1111.00,590.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1111.00,589.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1124.00,596.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1124.00,595.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1137.00,602.59)(1.026,0.485){11}{\rule{0.900pt}{0.117pt}}
\multiput(1137.00,601.17)(12.132,7.000){2}{\rule{0.450pt}{0.400pt}}
\multiput(1151.00,609.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1151.00,608.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1164.00,615.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1164.00,614.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1177.00,621.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1177.00,620.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1190.00,627.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1190.00,626.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1203.00,633.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1203.00,632.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1216.00,639.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1216.00,638.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1229.00,645.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1229.00,644.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1242.00,651.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1242.00,650.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1255.00,657.59)(1.026,0.485){11}{\rule{0.900pt}{0.117pt}}
\multiput(1255.00,656.17)(12.132,7.000){2}{\rule{0.450pt}{0.400pt}}
\multiput(1269.00,664.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1269.00,663.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1282.00,670.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1282.00,669.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1295.00,676.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1295.00,675.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1308.00,682.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1308.00,681.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1321.00,688.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1321.00,687.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1334.00,694.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1334.00,693.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1347.00,700.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1347.00,699.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1360.00,706.59)(0.950,0.485){11}{\rule{0.843pt}{0.117pt}}
\multiput(1360.00,705.17)(11.251,7.000){2}{\rule{0.421pt}{0.400pt}}
\multiput(1373.00,713.59)(1.214,0.482){9}{\rule{1.033pt}{0.116pt}}
\multiput(1373.00,712.17)(11.855,6.000){2}{\rule{0.517pt}{0.400pt}}
\multiput(1387.00,719.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1387.00,718.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1400.00,725.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1400.00,724.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1413.00,731.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1413.00,730.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\multiput(1426.00,737.59)(1.123,0.482){9}{\rule{0.967pt}{0.116pt}}
\multiput(1426.00,736.17)(10.994,6.000){2}{\rule{0.483pt}{0.400pt}}
\put(259.0,246.0){\rule[-0.200pt]{3.132pt}{0.400pt}}
\end{picture}
\end{center}
\caption{{Generalized uncertainty relation of $x-p$}}
\label{Fig:GUP}
\end{figure}
As is obvious in Figure \ref{Fig:GUP},
these states realize the minimal value of $\Delta x$
and are called as `maximal localization states' \cite{Kempf:1994su}.
One can easily find the wavefunction of maximal localization states in momentum space as
\begin{eqnarray}
\psi_{\xi}(p)=\frac{N_1}{\sqrt{2\pi \hbar}}\frac{1}{\sqrt{1+\beta p^2}}\exp\left(\frac{\xi}{i\hbar\sqrt{\beta}}
\tan^{-1}(\sqrt{\beta}p)\right), \label{wf}
\end{eqnarray}
where $\xi \equiv \langle \hat{x} \rangle$ and $N_1$ is a normalization
factor.
These maximal localization states are identified as `one particle state' in
the theory based on GUP \cite{Kempf:1994su}.
We will use these states as basic ingredients in constructing fields in the later section.
Note that these states are not eigenstates of the position operator $\hat{x}$.

We here introduce a first quantized Hilbert space as the space equipped with the complete basis $\{|p\rangle\}$ which has the completeness:
\begin{eqnarray}
\vmi{ 1} = \int {\cal D}p \phantom{a} |p\rangle \langle p |.
\end{eqnarray}
We determine the measure ${\cal D}p$ so that the operator
$\hat{x}= i\hbar (1+\beta p^2) \frac{\partial}{\partial p}$ is hermitian
($\langle \chi|\hat{x}\psi\rangle=\langle \hat{x}\chi |\psi\rangle$):
\begin{eqnarray}
{\cal D}p \equiv N_2 \frac{dp}{1+\beta p^2},\label{pmeasure}
\end{eqnarray}
where $N_2$ is a normalization factor independent of $p$.
We set $N_2=1$ by rescaling the basis as $\sqrt{N_2}|p\rangle \to |p\rangle$.
This measure leads to the orthogonal property:
\begin{eqnarray}
\langle p| p' \rangle = (1+\beta p^2)\delta(p-p').
\end{eqnarray}
We write maximal localization states as $|\psi_\xi \rangle$, then we can
rewrite the wavefunction of one particle state (\ref{wf}) in the
bracket notation:
\begin{eqnarray}
\psi_{\xi}(p)=\langle p|\psi_\xi \rangle.
\end{eqnarray}
In this Hilbert space, the inner product of one particle states is
\begin{eqnarray}
\langle \psi_{\xi'} | \psi_\xi \rangle &=&\int^{\infty}_{-\infty}
       \frac{dp}{1+\beta p^2}
       \psi_{\xi'}^{*}(p)\psi_{\xi}(p) \nonumber \\
 &=&\frac{|N_1|^2}{2\pi\hbar}\frac{1}{2\sqrt{\beta}\left(\frac{\xi-\xi'}{2\hbar \sqrt{\beta}}-(\frac{\xi-\xi'}{2\hbar \sqrt{\beta}})^3\right)}\sin\left(\frac{\xi-\xi'}{2\hbar \sqrt{\beta}}\pi\right),
\label{IP}
\end{eqnarray}
which shows they are not completely orthogonal with each other.
This is because these states have $\Delta x_{\mbox{min}} \ne 0$.

Furthermore, we find a completeness of basis $\{|\psi_\xi \rangle\}$ on this Hilbert space as
\begin{eqnarray}
\vmi{ 1} &=& \int \frac{d\xi}{|N_1|^2}(1+\beta
 \hat{p}^2)|\psi_\xi\rangle\langle \psi_\xi| \nonumber \\
 &=& \int \frac{d\xi}{|N_1|^2}|\psi_\xi\rangle\langle \psi_\xi|
(1+\beta \hat{p}^2).
\end{eqnarray}
We set $N_1=1$ by rescaling the basis as
$(N_1)^{-1}|\psi_\xi \rangle \to |\psi_\xi\rangle$.

Here we have two bases $\{|p\rangle\}, \{|\psi_\xi \rangle\}$.
Thus we can perform a `Fourier transformation' which interchanges
the two representations.
We write the $p$-space wavefunction and $\xi$-space wavefunction
for a state $|\phi\rangle$ in this Hilbert space as follows,
\footnote{
We use the same character $\phi$ for $\xi$-space and $p$-space
wavefunctions,
because we can easily recognize on which space a field lives
by noting the argument of the function.
}
\begin{eqnarray}
 \phi(p)&\equiv& \langle p|\phi\rangle, \nonumber \\
 \phi(\xi) &\equiv& \langle \psi_\xi|\phi\rangle.
\end{eqnarray}
The Fourier transformation in this Hilbert space is
\begin{eqnarray}
\label{FT1}
\phi(\xi)&=& \int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}
            \psi_{\xi}^{*}(p)\phi(p), \\
\phi(p)&=&\int^{\infty}_{-\infty} d\xi (1+\beta
 p^2)\psi_{\xi}(p)\phi(\xi).
\label{FT2}
\end{eqnarray}

Here there are statements that are worth while mentioning.
The transformation (\ref{FT2}) is an inverse transformation to the
one (\ref{FT1}) for arbitrary square integrable functions like as in the case of usual Fourier transformation.
However the transformation (\ref{FT1}) is not an inverse transformation to the one (\ref{FT2}) for arbitrary square integrable functions.
This can be seen by inserting (\ref{FT2}) into (\ref{FT1}), then one finds
\begin{eqnarray}
\int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}
            \psi_{\xi}^{*}(p)\phi(p)&=&\int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}
            \psi_{\xi}^{*}(p)\int^{\infty}_{-\infty} d\xi' (1+\beta
 p^2)\psi_{\xi'}(p)\phi(\xi') \nonumber \\
&=&\int^{\infty}_{-\infty}\frac{d\xi'}{\pi}\frac{\hbar}{\xi-\xi'}\sin\left(\frac{(\xi-\xi')\pi}{2\hbar\sqrt{\beta}}\right)\phi(\xi') .
\end{eqnarray}
This is not equal to the original function for arbitrary function $\phi(\xi')$.

Here we must restrict the function $\phi(\xi')$ to linear
combination of $\psi^*_{\xi'}(p)$'s as
$\phi(\xi')=\int \frac{dp}{1+\beta p^2} a(p)\psi_{\xi'}^*(p)$.
For these functions the transformation (\ref{FT1}) is truly the inverse transformation to (\ref{FT2}):
\begin{eqnarray}
\int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}
            \psi_{\xi}^{*}(p)\phi(p)&=&\int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}
            \psi_{\xi}^{*}(p)\int^{\infty}_{-\infty} d\xi' (1+\beta
 p^2)\psi_{\xi'}(p)\phi(\xi') \nonumber \\
&=& \int^{\infty}_{-\infty}\frac{dp'dpd\xi'}{1+\beta p'^2} a(p')
            \psi_{\xi}^{*}(p) \psi_{\xi'}(p)\psi^*_{\xi'}(p') \nonumber \\
&=& \int^{\infty}_{-\infty}\frac{dp'}{1+\beta p'^2} a(p')\psi_{\xi}^*(p') \nonumber \\
&=& \phi(\xi).
\end{eqnarray}
This restriction is nothing but the condition that the state $|\phi\rangle$ is included in our Hilbert space.
Thus the Fourier transformations (\ref{FT1}) and (\ref{FT2}) are well-defined in this Hilbert space.

\begin{figure}[h]
\begin{center}
\resizebox{10cm}{10cm}{\includegraphics{d.eps}}
\end{center}
\caption{{Shape of $\langle\psi_\xi | \psi_{\xi'}\rangle$} in units $\hbar=1$ and $N_1=1$.}
\label{fig:deltax}
\end{figure}

It should be noted that the wavefunction (\ref{wf}) reduces to the plane wave in the limit $\beta \to 0$:
\begin{equation}
\psi_\xi(p) \to \frac{1}{\sqrt{2\pi \hbar}}e^{-ip\xi/\hbar} .
\end{equation}
This is because we have set the normalization constant as $N_1 = 1$
by rescaling $|\psi_\xi \rangle$.
In \cite{Kempf:1994su} the normalization constant is fixed to $N_1=2\sqrt{\hbar\sqrt{\beta}}$ so that the norm of the wavefunction is equal to one.
In this case, the wavefunction vanishes in the limit
because of the extra dependence on $\beta$ in the normalization factor.
In other words, we regard the wavefunction (\ref{wf}) as if it is non-normalizable, because we have taken it as a counterpart of the plane wave in ordinary quantum mechanics.
The plane wave is non-normalizable.
Once we turn on $\beta$, a nonlocality emerges
as one can easily see in Figure 2 which shows the behavior of the
inner product (\ref{IP}).

\section{1+1 dimensional quantum field theory on GUP}

In this section we construct a quantum field theory on GUP in 1+1 dimensional spacetime in the canonical formalism.

First of all, we define the classical field such that it solves the following
Klein-Gordon equation:
\begin{eqnarray}
\label{KG}
0=\left(\left(\hbar\frac{\partial}{\partial t}\right)^2+F(p)+m^2\right)\Phi(p,t).
\end{eqnarray}
In this equation we leave the function of momentum $F(p)$ indefinite,
where $F(p)$ is an arbitrary even function whose explicit form does not have any influence to the following arguments.
This function determines the dispersion relation and depends on what kind of theory we want.

Quantization of a classical field is to provide the set of operator field and Hamiltonian which lead the same Klein-Gordon equation by the Heisenberg equation.
We define a field $\hat{\Phi}(p,t)$:
\begin{eqnarray}
\label{field}
\hat{\Phi}(p,t)\equiv\frac{\hbar}{\sqrt{E(p)}}\hat{\phi}(p)
\exp(\frac{t}{i \hbar} E(p))
+
\frac{\hbar}{\sqrt{E(-p)}}\hat{\phi}^{\dagger}(-p)
\exp(-\frac{t}{i \hbar} E(-p)) ,
\end{eqnarray}
where $E(p) \equiv \sqrt{F(p)+m^2}$.
This field satisfies the Klein-Gordon equation (\ref{KG}).
Its Fourier pair $\hat{\Phi}(\xi,t)$ is written in terms of  our `one particle states':
\begin{eqnarray}
\hat{\Phi}(\xi,t)&\equiv&\int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}\{
                         \psi_{\xi}^{*}(p)\hat{\phi}(p)\frac{\hbar}{\sqrt{2E(p)}}
                         \exp(\frac{1}{i\hbar}tE(p)) \nonumber \\
&&+\psi_{\xi}(p)
                         \hat{\phi}^{\dagger}(p)\frac{\hbar}{\sqrt{2E(p)}}
                         \exp(-\frac{1}{i\hbar}tE(p))\} .
\end{eqnarray}
In the above expression, the operators
$\hat{\phi}(p),\hat{\phi}^{\dagger}(p)$ are annihilation and creation
operators for one particle state with momentum $p$.
They have the following commutation relation:
\begin{eqnarray}
\label{COM}
\left[\hat{\phi}(p),\hat{\phi}^{\dagger}(p')\right]=(1+\beta p^2)\delta(p-p').
\end{eqnarray}
At first sight it is unclear whether the generalized canonical commutation relations is the same form as those of ordinary quantum field theory.
Thus the commutation relation might have an arbitrary power of $(1+\beta p^2)$.
The factor $(1+\beta p^2)$ in front of the delta function is determined
by the form of the measure ${\cal{D}}p$ which is the consequence of the
hermiticity of $\hat{x}$ in the first quantized Hilbert space.

We construct a natural Hamiltonian in terms of the Fock space picture:
\begin{eqnarray}
{\cal H} \equiv \int \frac{dp}{1+\beta p^2}E(p)\hat{\phi}^{\dagger}(p)\hat{\phi}(p).
\end{eqnarray}
This Hamiltonian produces the Heisenberg equation:
\begin{eqnarray}
\frac{1}{i\hbar}\left[\hat{\Phi}(p,t),{\cal
		 H}\right]=\frac{\partial}{\partial t}\hat{\Phi}(p,t),
\end{eqnarray}
which is consistent with the Klein-Gordon equation (\ref{KG}).
We can define conjugate momentum
$\hat{\Pi}(p,t) \equiv \frac{\partial}{\partial t}\hat{\Phi}(p,t)$.
We can write the Hamiltonian
\begin{eqnarray}
 {\cal H}&=& \int^{\infty}_{-\infty}\frac{dp}{1+\beta p^2}
 \frac{1}{2}\left[\hat{\Pi}(-p,t)\hat{\Pi}(p,t)
 +\left(\frac{E(p)}{\hbar}\right)^2\hat{\Phi}(-p,t)\hat{\Phi}(p,t)\right]
\nonumber \\
&&+\mbox{const.}.
\label{HHamil0}
\end{eqnarray}

We compute the following commutation relation by using the commutation relation (\ref{COM}):
\begin{eqnarray}
\left[\hat{\Phi}(p,t),\hat{\Pi}(p',t)\right]&=&i\hbar(1+\beta p^2)
\delta(p+p'), \\
\left[\hat{\Phi}(\xi,t),\hat{\Pi}(\xi' ,t)\right]&=&i\hbar
\langle\psi_\xi|\psi_{\xi'}\rangle.
\end{eqnarray}
As we can see in Figure \ref{fig:deltax}, this commutation relation in $\xi$ space representation does not provide a delta function.
There is a nonlocality in the quantum field theory like as in the first quantized system which is constructed in the previous section.

\section{Higher dimensional algebra}

In this section we extend the previous argument of the quantization of fields to the case in higher dimensional spacetime.
In the process of the extension in a naive way, one would be faced with the  difficulty that we can not identify the maximal localization states.
Therefore we will not use the canonical formalism, instead we use the path integral formalism in which it turns out that we can easily extend to higher dimensional spacetime.

To see the difficulty clearly, we start with higher dimensional GUT algebra.
Jacobi identity determines the full algebra:
\begin{eqnarray}
 \left[\hat{x}^i,\hat{p}_j\right]&=&i\hbar (1+\beta \mbox{\boldmath
  $\hat{p}$}^2)\delta^i_{\phantom{a} j}, \label{GUP2}\\
 \left[\hat{x}^i,\hat{x}^j\right]&=&-2i\hbar \beta(1+\beta \mbox{\boldmath $\hat{p}$}^2)\hat{L}^{ij}.
\label{GUP3}
\end{eqnarray}
Here $\hat{L}^{ij}$ are angular momentum operators
$\hat{L}^{ij}\equiv \frac{1}{1+\beta \mbox{\boldmath $\hat{p}$}^2}(\hat{x}^i\hat{p}^j-\hat{x}^j\hat{p}^i)$
and $i$ runs from 1 to $d$ which is the number of spatial coordinates
and $\mbox{\boldmath$\hat{p}$}^2 \equiv \sum_{i=1}^d (\hat{p}_i)^2$ \cite{Kempf:1994su}.

Let us try to define maximal localization states like
as we did in 1+1 dimensional case.
Such states would satisfy the equation:
\begin{eqnarray}
\left( i\hbar(1+\beta \mbox{\boldmath $p$}^2)\frac{\partial}{\partial p_i}-\langle
 \hat{x}^i \rangle+
 i\hbar\frac{1+\beta (\Delta p_i)^2+\beta \langle p_i \rangle^2}{2\Delta
 p^2}(p_i-\langle p_i\rangle)\right)\psi(p)
=0.
\end{eqnarray}
However these $d$ relations can not coexist because of the noncommutativity of the differential operators acting on the $\psi(p)$.
In general, the wavefunction which obeys the above equations corresponds to $s$-wave which has zero angular momentum.
We can not produce states with non zero angular momentum
from these states only.
Thus we find it difficult to construct a quantum field theory in the same manner as in the 1+1 dimensional case.

A different way of extension of the GUP algebra (\ref{GUP}) is simply to take a direct product of it:
\begin{eqnarray}
\left[\hat{x}^i,\hat{p}_j\right]=i\hbar (1+\beta (p_i)^2)\delta^i_{\phantom{a} j} .
\end{eqnarray}
In this case we can construct a quantum field theory by taking a direct product of fields \cite{Kempf:1996ss}.
However this extension is not a Lorentz invariant deformation.
In particular, the rotational invariance is completely broken.
Thus here we take the algebra (\ref{GUP2})-(\ref{GUP3}) and investigate another way to construct a quantum field theory based on this algebra.

To this end, we reconstruct a quantum field theory on GUP in 1+1 dimensions in a  different manner; the path integral formalism in order to be applicable to higher dimensional spacetime.
In fact, as we shall see later, it is not difficult to extend to higher dimensions in the path integral formalism.
In a way to construct a field theory in the path integral formalism,
we adopt a guiding principle that it produces the same result
as that in the canonical formalism.
The equivalence of the canonical formalism and the path integral one is
guaranteed by the Bjorken-Johnson-Low (BJL) prescription
\cite{Bjorken:1966jh}. At first we introduce deformed factors of the form
$(1+\beta p^2)^{\alpha}$ where $\alpha$ denotes free parameters
in the definitions of Hamiltonian, conjugate momentum fields, Lagrangian and so on.
This is because in generalized canonical commutation relations it is
unclear whether the definition of these phase space structure are the
same as those of ordinary quantum field theory. These free parameters
should be determined so that path integral formalism leads the same
result as that of canonical formalism in 1+1 dimensions.

We define path integral measure and Lagrangian by introducing two parameters $l$ and $m$ as follows,
\begin{eqnarray}
1&=& \int {\cal D}\Phi(p,t)\exp\left(-\frac{1}{2}\int \frac{dtdp}{(1+\beta
 p^2)^{l}}\Phi(p,t)\Phi(-p,t)\right), \label{measure}\\
{\cal L}&=& -\frac{1}{2}\int^{\infty}_{-\infty}dp(1+\beta
    p^2)^{m}\Phi(-p,t)\left[\partial_t^2+\left(\frac{E(p)}{\hbar}\right)^2\right]
\Phi(p,t).\label{lag1}
\end{eqnarray}
Then the action $S\equiv\int dt {\cal L}$ is written as,
\begin{eqnarray}
S&=& -\frac{1}{2}\int^{\infty}_{-\infty}dpdp'dqdq'(1+\beta p^2)^{m}
\delta(p+p')\delta(q+q')\nonumber \\
&&\times  \Phi(p,q)\left[-\left(\frac{q}{\hbar}\right)^2+\left(\frac{E(p)}{\hbar}\right)^2\right]\Phi(p',q').
\label{lag2}
\end{eqnarray}
According to the assumptions (\ref{measure}) and (\ref{lag2}) we can compute the $T^*$-product:
\begin{eqnarray}
\langle T^*\hat{\Phi}(p,q)\hat{\Phi}(p',q')\rangle =\frac{\hbar^3}{i} (1+\beta p^2)^{-m-l}
  \delta(p+p')\delta(q+q')\frac{1}{-q^2+E(p)^2-i\epsilon}.
\label{LT}
\end{eqnarray}
Using the Bjorken-Johnson-Low prescription, from behavior
of
\begin{equation}
\lim_{q \to \infty}\langle T^*\hat{\Phi}(p,q)\hat{\Phi}(p',q')\rangle
\quad {\mbox{and}} \quad
\lim_{q \to \infty}q\langle T^*\hat{\Phi}(p,q)\hat{\Phi}(p',q')\rangle ,
\end{equation}
we find
\begin{eqnarray}
T^*\hat{\Phi}(p,q)\hat{\Phi}(p',q')&=&T\hat{\Phi}(p,q)\hat{\Phi}(p',q'), \\
\left[\hat{\Phi}(p,t) , \hat{\Phi}(p',t)\right]&=& 0.
\end{eqnarray}
In the same manner, from behavior of
\begin{equation}
\lim_{q \to \infty}q^2\langle T^*\hat{\Phi}(p,q)\hat{\Phi}(p',q')\rangle ,
\end{equation}
we also find
\begin{eqnarray}
\left[ \hat{\Phi}(p,t), \dot{\hat{\Phi}}(p',t)\right]=i\hbar
(1+\beta p^2)^{-m-l}\delta(p+p'). \label{Lcomm1}
\end{eqnarray}
If we define conjugate momentum $\Pi(p,t)$ in Lagrangian formalism by using a free parameter $r$ as
\begin{eqnarray}
\Pi(p,t) \equiv (1+\beta p^2)^r\frac{\delta S}{\delta \dot{\Phi}(-p,t)},
\label{Lpi}
\end{eqnarray}
the commutation relation (\ref{Lcomm1}) becomes the following,
\begin{eqnarray}
\left[\hat{\Phi}(p,t),\hat{\Pi}(p',t)\right]=i\hbar(1+\beta p^2)^{-2m-l-r}
\delta(p+p'). \label{Lcomm2}
\end{eqnarray}
The right hand side of the above equation is a symplectic form.
Because the Legendre transformation from Lagrangian to Hamiltonian is
\begin{eqnarray}
\frac{1}{i\hbar}\int dt {\cal H} \equiv (\mbox{symplectic
 form})^{-1}\sum_i \Pi_i d\Phi_i - \frac{1}{i\hbar}\int dt {\cal L},
\end{eqnarray}
we can obtain a Hamiltonian:
\begin{eqnarray}
{\cal H}&=&\int^{\infty}_{-\infty}dp(1+\beta
    p^2)^{m+l}\left[(1-\frac{1}{2}(1+\beta
    p^2)^{-l-2m-2r})\hat{\Pi}(-p,t)\hat{\Pi}(p,t)
    \right.
     \nonumber \\
 &&\left.
 +\frac{1}{2}(1+\beta p^2)^{-l}\left(\frac{E(p)}{\hbar}\right)^2\hat{\Phi}(-p,t)\hat{\Phi}(p,t)\right].
\end{eqnarray}
The condition that the coefficient of $\Pi (-p,t)\Pi(p,t)$ must be homogeneous fixes the free parameter $r$:
\begin{eqnarray}
r=-\frac{1}{2}l-m. \label{rfix}
\end{eqnarray}
Then the Hamiltonian becomes
\begin{eqnarray}
{\cal H}&=&\int^{\infty}_{-\infty}dp(1+\beta
    p^2)^{m+l}\frac{1}{2}\left[\hat{\Pi}(-p,t)\hat{\Pi}(p,t)
    \right.
    \nonumber \\
 &&\left.+(1+\beta p^2)^{-l}\left(\frac{E(p)}{\hbar}\right)^2\hat{\Phi}(-p,t)\hat{\Phi}(p,t)\right].\label{LHamil}
\end{eqnarray}
By comparing this with (\ref{HHamil0}), we obtain the parameters $m$ and
$l$ as the form:
\begin{eqnarray}
l=0, \label{lfix} \\
m=-1. \label{mfix}
\end{eqnarray}

Now it is apparent the two formalism provide consistent structure of
the deformed quantum field theory.
Let us summarize the result.
The Hamiltonian is
\begin{eqnarray}
 {\cal H}&=& \int^{\infty}_{-\infty}\frac{dp}{1+\beta
    p^2}\frac{1}{2}\left[\hat{\Pi}(-p,t)\hat{\Pi}(p,t)
    \right. \nonumber \\
 &&\left.
 +\left(\frac{E(p)}{\hbar}\right)^2\hat{\Phi}(-p,t)\hat{\Phi}(p,t)\right]+
  \mbox{const.}  \label{finalHam}\\
&=& \int^{\infty}_{-\infty}\frac{dp}{1+\beta
    p^2}E(p)
    \left(\hat{\phi}^{\dagger}(p)\hat{\phi}(p)+ \phantom{a} \mbox{const.}\right).
\end{eqnarray}
Annihilation and creation operators for one particle states have commutation relation:
\begin{eqnarray}
 \left[ \hat{\phi}(p) , \hat{\phi}^{\dagger}(p') \right]=(1+\beta p^2)\delta(p-p').
\end{eqnarray}
Canonical commutation relation is
\begin{eqnarray}
\left[\hat{\Phi}(p,t),\hat{\Pi}(p',t)\right]=i\hbar(1+\beta p^2)
\delta(p+p').
\end{eqnarray}
And Heisenberg equation is
\begin{eqnarray}
\frac{1}{i\hbar}\left[ \hat{\Phi}(p,t), {\cal H} \right]
=\frac{\partial}{\partial t}\hat{\Phi}(p,t) =\hat{\Pi}(p,t).
\end{eqnarray}
We find propagators from these quantity as
\begin{eqnarray}
\langle T\hat{\Phi}(p,q)\hat{\Phi}(p',q')\rangle
&\equiv& \frac{1}{2\pi\hbar}
\int dtdt' \exp\left(-\frac{qt+q't'}{i\hbar}\right)\langle T\hat{\Phi}(p,t)\hat{\Phi}(p',t')\rangle \nonumber \\
&=&\frac{\hbar^3}{i} (1+\beta p^2)
  \delta(p+p')\delta(q+q')\frac{1}{-q^2+E(p)^2-i\epsilon}. \nonumber \\
\end{eqnarray}
On the other hand, the Lagrangian is
\begin{eqnarray}
{\cal L} = -\frac{1}{2}\int^{\infty}_{-\infty}\frac{dp}{1+\beta
    p^2}\Phi(-p,t)\left[\partial_t^2+\left(\frac{E(p)}{\hbar}\right)^2\right]
\Phi(p,t). \label{finalLag}
\end{eqnarray}
And the path integral measure which is consistent with the canonical formalism is
\begin{eqnarray}
1= \int {\cal D}\Phi(p,t)\exp\left(-\frac{1}{2}\int dt dp \Phi(p,t)\Phi(-p,t)\right).
\end{eqnarray}

When we add same interaction terms to both Hamiltonian
(\ref{finalHam}) and Lagrangian (\ref{finalLag}) with opposite sign, it
is obvious that Feynman rules in both formalism are same form and then two theories which include the interactions represent the same
physics.

It is easy to see that the quantization procedure developed above
based on the path integral formalism does not have any difficulty
in the generalization to higher dimensions.
Lagrangian and path integral measure in $d+1$ dimensional spacetime can be written
\begin{eqnarray}
{\cal L}&=& -\frac{1}{2}\int^{\infty}_{-\infty}d^dp(1+\beta
    \mbox{\boldmath$p$}^2)^{-1}\phi(-\mbox{\boldmath$p$},t)
    \left[\partial_t^2+\left(\frac{E(\mbox{\boldmath$p$})}{\hbar}\right)^2\right]
\phi(\mbox{\boldmath$p$},t), \\
1&=& \int {\cal D}\Phi(\mbox{\boldmath$p$},t)\exp\left(-\frac{1}{2}\int dtd^dp \Phi(\mbox{\boldmath$p$},t)\Phi(-\mbox{\boldmath$p$},t)\right).
\end{eqnarray}

We would like to make a comment here.
It might be seen that the algebra (\ref{GUP}) is a sort of cut off theory by a change of variables :
\begin{eqnarray}
\rho &\equiv&\frac{1}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}p), \\
\left[ x , \rho \right]&=& i\hbar,
\end{eqnarray}
where
\begin{eqnarray}
-\frac{\pi}{2\sqrt{\beta}} < \rho < \frac{\pi}{2\sqrt{\beta}} .
\end{eqnarray}
However the theory described in this variable can not be regarded as a simple cut off theory of ordinary field theory.
If we set a cut off $\frac{\pi}{2\sqrt{\beta}}$, the parameter $l$
appeared in the integration measure (\ref{measure}) must not take a
value 0 but 1 because of $d\rho=\frac{dp}{1+\beta p^2}$:
\begin{eqnarray}
1&=& \int {\cal D}\Phi(\rho,t)\exp\left(-\frac{1}{2}\int^{\frac{\pi}{2\sqrt{\beta}}}_{-\frac{\pi}{2\sqrt{\beta}}} dt d\rho \Phi(\rho,t)\Phi(-\rho,t)\right) \nonumber \\
 &=&\int {\cal D}\Phi(p,t)\exp\left(-\frac{1}{2}\int^{\infty}_{-\infty}
			       dt \frac{dp}{1+\beta p^2}
			       \Phi(p,t)\Phi(-p,t)\right).
\end{eqnarray}
Thus, our theory is not mere variable transformation of a ordinary quantum field theory.
In other words, the method of quantizing fields considered in this paper
is an another method to the well-known method of quantizing the
particle on finite interval space \cite{Ohnuki:1991uv}.
In that paper, system has periodic boundary condition.
The translation generator $\hat{p}$ is ill-defined on that space, instead the
operator $\exp(i\hat{p})$ is well-defined.
In the same sense, the operator $\hat{\rho}$ is ill-defined and the operator
$\hat{p}\equiv \frac{1}{\sqrt{\beta}}\tan(\sqrt{\beta}\rho)$ is
well-defined in our formalism.
\section{Summary}

In this paper we construct the scaler field theory in arbitrary
dimensions with path integral formalism based on generalized canonical
commutation relations.
We adopt the guideline that canonical formalism and path integral formalism are equivalent by using BJL prescription and a second
quantized Hamiltonian is formed by calculating the `expectation value'
of first quantized Hamiltonian with a `wavefunction' replaced by
the quantized field \cite{Weinberg:1995mt}. We first consider the deformation in 1+1 dimensional canonical formalism, then we construct path integral formalism by BJL
prescription. In the path integral formalism there is not any difficulty in
extension to higher dimension.

\begin{center} \begin{large}
Acknowledgments
\end{large} \end{center}

We would like to thank K. Fujikawa and M. Hayakawa for fruitful discussions.
We also thank T. Tada for careful reading of the manuscript.
One of the author (T. M.) also thanks Y. Kimura, S. Matsuura,
S. Seki and M. Tachibana for valuable discussions and comments.
This work is supported by Special Postdoctoral Researchers Program at RIKEN.

redpajama_set_name: RedPajamaArXiv


text:
'Alive With Birds' Exhibition Brings Together Historic Art to Illustrate the Role of Bird Advocate William Brewster
Black-billed Cuckoo, John James Audubon (1785-1851), 1828. Mass Audubon Collection, Bequest of Anna Bemis Stearns (1995)
"To my great surprise, the Farm was simply alive with birds, the entire morning." William Brewster, April 7, 1893.
The Concord Museum in Concord, Massacusetts, has collaborated with Mass Audubon on the special exhibition, Alive with Birds: William Brewster in Concord, on view March 4, 2022 through September 5, 2022. Alive with Birds is the first and most comprehensive exhibition on William Brewster (1851 -1919), the first president of Mass Audubon and one of the country's earliest advocates for the protection of birds and their habitats.
William Brewster, Daniel Chester French (1850-1931), 1907. Chapin Library, Williams College, Gift of the National Trust for Historic Preservation.
The exhibition also showcases 20 paintings and sculptures from Mass Audubon's Museum of American Bird Art by acclaimed artists, including John James Audubon, Frank Weston Benson, Anthony Elmer Crowell, Charley Harper, David Sibley, Leonard Baskin, and Barry Van Dusen, each of whom depicted birds formerly or currently native to Concord's landscape.
In 2019, Mass Audubon received a gift of 143 acres of the October Farm property, which has been renamed Brewster's Woods Wildlife Sanctuary.
"My colleagues and I have long wanted to curate an exhibit on William Brewster," said Tom Putnam, Concord Museum's Edward W. Kane Executive Director. "This unique collaboration with Mass Audubon allowed us access to an array of artifacts and artwork that will bring Brewster to life and our galleries alive with birds."
Drawn to Concord for its natural beauty and abundant bird life, William Brewster made October Farm, his 300-acre property, into an experimental field laboratory and documented its wildlife for scientific study and public enlightenment. His rich observations and analysis of the local landscape included a concern for environmental changes caused by humans. A prolific writer and amateur photographer, the words and images he left behind form a bridge to the past, just as his lessons provide a key to the future.
Alive with Birds will showcase items drawn from the wide-ranging collection of artwork, photography, and historical objects from Mass Audubon's Museum of American Bird Art, and from the Concord Museum, accompanied by manuscript materials from the Museum of Comparative Zoology at Harvard University and the Chesterwood Archives at Williams College.
"William Brewster stands tall in Mass Audubon's history not only as the first president of our organization, but also through his influence on the advent of the modern conservation movement as evidenced by his writings, his ethic of wildlife protection, and his photography," said Mass Audubon President David J. O'Neill. "We are delighted to share the incredible landscape of Brewster's Woods through this collaboration with the Concord Museum, interpreted through the lens of Brewster's journals and the significant contributions of Robert Gilbert, as well as renowned artists who inspire us to continue to protect the nature of Massachusetts for generations to come."
CHARLEY HARPER (1922-2007) Mass Audubon Collection.
Alive with Birds also explores the contributions of Robert Gilbert, a Black ornithologist who served as Brewster's assistant for over twenty years, as well as Brewster's lifelong friendship with the sculptor Daniel Chester French, for whom Brewster's work served as an inspiration for his art.
"This exhibition weaves together art, science, and history to tell the story of a truly fascinating figure," says Erica Lome, Peggy N. Gerry Curatorial Associate, Concord Museum. "The works of art on display tie everything together, illustrating the diversity of birds species represented by William Brewster in his writings, as well as the artists who, from past to present, have found inspiration in this subject."
River Field Scene, Roger Tory Peterson (1908-1996), about 1949. Mass Audubon Collection, Bequest of John Henry Dick (1995).
David Wood, Curator, Concord Museum, continued, "For thirty years William Brewster labored to make a few hundred acres on a bend on the Concord River a better place for birds. This collaborative exhibition demonstrates that his singular efforts continue to pay rich dividends."
A centerpiece of the show is a meditative exploration of Brewster's Woods in a 10-minute media presentation. William Brewster's own words, taken from his journal entries, guide the visitors in discovering the wonders of this natural landscape and his own photographs, taken in the late 19th century, are compared side-by-side with scenes today showing how little has changed in the intervening years.
As a youth, William Brewster (1851- 1919) explored the fields, meadows, and marshes just beyond his home on Brattle Street in Cambridge, Massachusetts, collecting eggs, nests, and specimens to study. Rather than follow his father's footsteps into banking, Brewster remained devoted to the field of ornithology, of which he was primarily self-taught.
After his marriage to Caroline Kettell in 1878, Brewster built a house in Cambridge which included a library and museum to hold his growing collection of mounted birds. He became the curator of mammals and birds at the Museum of Comparative Zoology at Harvard University in 1885.
In 1890, Brewster bought a tract of woodland on the Concord River with a farmhouse situated on Monument Street dating back to the eighteenth century. The property spanned 300 acres and Brewster renamed it October Farm. Brewster spent the last thirty years of his life exploring October Farm and recording his observations.
Alive with Birds: William Brewster in Concord is a collaboration with Mass Audubon.
Holabird's Pike's Peak or Bust! auction, Jan. 19-22, will honor the Great Colorado Gold Rush of 1859


This virtual no-overhead, high return business has access to multiple databases and proprietary strategies and applications which allow it to find assets for judgment creditors that have exhausted all previous options.
This business collects judgments for a 33% up to a 60% collection fee from bank and investment accounts, employers, third party, and other asset searches.
When the seller first developed this company an initial $100k was invested into the proprietary strategic placement of this company on the web which has allowed it not to spend or have a need to spend for advertising since its inception. This strategy has paid off currently with 20 daily leads and 3 new daily leads by email or phone call. This has generated a $10 million pipeline of fee based judgments, and $325,000 worth of company owned debt for collection which is 100% retained by this money making, collecting company.


text:
About SUSS
Vision, Purpose, Core Values
The SUSS Mace
Patron, Chancellor and Pro-Chancellor
Key Academic Leaders
President Emeritus
Why Choose SUSS
Graduate Employment Survey 2021
The SUSS Difference
S R Nathan School of Human Development
School of Humanities and Behavioural Sciences
School of Science and Technology
College of Interdisciplinary & Experiential Learning
SUSS Core Course Synopses
Core Learning
Centre for Continuing and Professional Education
Centre for Applied Research
Teaching and Learning Centre
Centre for Chinese Studies@SUSS
C-three (Counselling & Life Coaching)
SUSS Node for Inclusive Fintech
Student SUcceSS Centre
Academic Board and Exam & Awards Committee
Online Degree Verification
Campus Location & Facilities
National Service-Learning Clearinghouse
CET Courses
Work-Study Post-Diploma (WSPostDip)
SGUnited Skills Programme
National Silver Academy Courses (NSA)
Alumni Continuing Education (ACE) Scheme
Alumni Continuing Education Plus (ACE+) Scheme
Lifelong Learning Credit (L2C)
Part-time Undergraduate
View all Part-time Programmes
External Scholarships/Sponsorships & Awards
Apply for Admission Now
Full-time Undergraduate
View all Full-time Programmes
Work-Study Degree (WSDeg)
Second Majors
Indicative Grade Profile (IGP)
SUSS Scholarships & Awards
SUSS Student Card
Student Insurance Scheme
After Application
View all Graduate Programmes
Law Programmes
View all Law Programmes
Scholarships/Sponsorships
Apply for Admissions Now
Experience at SUSS
Multiple Pathways at SUSS
SUSS Core
CelebratingSUcceSS
Inclusive and Accessible Learning
Well-Being Support
LEAD Academy
Accomplished Collaboration
UniLEARN
SUCCESS Newsletter
Job Posting Services
SUSS Giving Day 2022
CEX Portal
SIM University To Be Renamed Singapore University Of Social Sciences
Media Resource Type: Media Releases
School/Dept:
17 March 2017 – With effect from today, SIM University (UniSIM) will be renamed the Singapore University of Social Sciences (SUSS), as part of its restructuring into Singapore's sixth Autonomous University (AU).
Commenting on the renaming, Minister for Higher Education and Skills, Ong Ye Kung, said, "The name was chosen to reflect the AU's mission of driving lifelong learning anchored in disciplines with a strong social focus. SUSS will add diversity to Singapore's higher education landscape and complement the existing five AUs. What will be most unique about the University is its tradition of applied education, and outreach to adult learners, all of which will be kept and strengthened.
The University's Distinctive Features
The University will have two distinctive features in its provision of higher education: (i) the social sciences, and (ii) continuing and lifelong education.
(i) Social Sciences
The AU's name reflects its focus on applied education in the domain of the social sciences, as well as in disciplines that have a strong impact on human and community development. The University will continue to offer the range of programmes that UniSIM had offered previously, targeted at both fresh school leavers and adult learners. The focus of its programmes will be on those that prepare individuals for careers in the social sector, such as social work and early childhood education, careers with a strong social orientation, such as human resource management, as well as law (focusing on family and criminal law). At the same time, the University will continue to retain a limited offering in other areas such as business and engineering, especially for adult learners. The name also embodies the AU's social mission, in terms of the educational opportunities it will provide to both students and adult learners.
(ii) Lifelong Learning
SUSS will also focus on lifelong and continuing learning. Since its inception in 2005, UniSIM was focused on providing applied and lifelong education to adult learners, many of whom are working adults, equipping them for career advancement, career transitions, and to serve society. Moving forward, SUSS will strengthen its provision of inclusive educational opportunities to many more students and adult learners, and participate strongly in SkillsFuture – Singapore's national movement for fostering continuing education and skills development.
SUSS will continue to cater to a wide range of learners – students, employees looking to upgrade their skills, and passionate learners keen on gaining new knowledge, while providing flexibility for their learning. We are committed to ensuring that SUSS remains at the forefront of innovative, flexible, and applied tertiary education through close collaboration with industry, employers, and the community in the development and design of our programmes. Leveraging technology, our faculty will be progressive and adaptive in catering to the varied needs, backgrounds, and abilities of learners," added Professor Cheong Hee Kiat, President, Singapore University of Social Sciences.
For media queries on this release, please contact:
Ms Vera Lui
Senior Manager, Communications
Singapore University of Social Sciences
DID: 6248 4305
Email: veralui@unisim.edu.sg
Ms Dorothy Siok
Officer, Media Relations, Communications Division
Email: Dorothy_Siok@moe.gov.sg
TRANSLATIONS FOR THE NAME
"SINGAPORE UNIVERSITY OF SOCIAL SCIENCES"
Language Translated Name
Chinese Complete form
新跃社科大学
跃大
Malay Universiti Sains Kemasyarakatan Singapura
Tamil சிங்கப்பூர் சமூக அறிவியல் பல்கலைக்கழகம்
I want to learn about with in
*Please select at least 1 field
This site uses cookies to offer you a better browsing experience. By continuing to navigate through this site or by clicking "Approve", you are consenting to the use of cookies on your device as described in our privacy policy.
463 Clementi Road
Locate Us on Google Maps
© 2023 Singapore University of Social Sciences. All rights reserved.


2014611 about gyratory crusher principle of operation related products our main products crusher equipment pe series jaw crusher pew series jaw crusher hcs. MORE.
principal operation of Jaw Crusher - allindiatours. principle of operation for a jaw crusher. operation principle of jaw crusher this page is the info jaw crusher home page,you can find the jaw crusher,stone crusher.
Mineral Processing Introduction - Mineral and Metallurgical ... Mineral processing is a major division in the science of Extractive Metallurgy.
operation principle of jaw crusher this page is the info jaw crusher home page,you can find the jaw crusher,stone crusher,rock crusher,cone crusher,vsi crusher .


Paying too much tax is bad. Paying your fair share is good. Black and white statements don't come much truer than that but as we keep finding, people, honest people running good businesses to help their families, are regularly giving more than they need to. Put another way, more than they should. There are many reasons for this but just like a building inspector looking at a house that may be hiding a few issues, we here at Inspire often find that there are structural faults. Structural problems are always expensive – in buildings and especially when building a business.
We often talked about the importance of choosing the right business structure for you before setting out to change the world. If you do this you may well avoid paying a whole lot of tax that you didn't have to pay during the all-important start-up phase. If you realise you've taken a wrong turn during your commercial journey, it may still be best to turn back and get on the right structural path. Sure you may have paid too much during the proceeding period but paying a price to stop the (cash flow) bleeding can still be a lifesaver.
Aaah, strategy! That's the thing. You can have the best tools ever but if you don't know how to use them, you could end up hurting yourself… or your business.
It's important that we get this right so we go live on facebook Wednesdays to do open Q&As and talk through important business solutionsfrom 12.30pm. Don't just think of this as some one-way, free advice. This is your chance to interactively learn and explore ways of improving one of the must-haves in business. Cash flow optimisation. A lot of businesses are giving away half, that's right HALF of their profits in tax because they don't know any better and no one's shown them how to remedy that.
If that's you or you think this may be you, feel free to join our regular Wednesday lunchtime conversations or simply check out the content we post daily.
If you're a small business with questions around structures or other accounting related matters you would like some clarity on, just ask us – we'd love to help.