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When Kepler choose the title "Harmonices Mundi" (The Harmony of the Word) for his book of 1619 reporting the discover of the third law of planetary motion, he aimed at discussing physical phenomena through the lens of an intrinsic harmony at the basis of geometrical forms and physical systems. Also human movements often show an intrinsic harmony, as noted by the neurophysiologist Luria, who introduced the concept of "kinetic melody". Kinetic melodies are dynamic patterns of movements at the basis of a vast repertoire of possible human movements such as walking, speaking, reaching, running, etc. containing an intrinsic harmony and fluidity. Some pathologies can disrupt these patterns, and neurorehabilitation should aim at their recovery, as suggested by Oliver Sacks who talked about the need for patients of a restoration of movements based on their natural rhythm embedded into kinetic melodies.
Walking is probably the most important human motor act that is perfectly embedded in a kinetic melody. In healthy subjects, the symmetry of our body is reflected in symmetric kinematic and kinetic patterns that were performed by lower limbs in counterphase. Also upper limbs are moved in symmetric manner but in counterphase each other and with respect to ipsilateral lower limbs.
For symmetry, it is obvious that one foot starts a stride (id est, its gait cycle) in the moment in which the other foot is in the middle of its own stride, id est at 50% of its gait cycle. It implies that the trunk performed the same movement twice in the sagittal plane during each stride. According to this gait structure, it has been introduced a parameter called Harmonic Ratio that is the ratio between the sum of the amplitudes of even harmonics and that of odds harmonics of a signal (for example trunk acceleration or displacement) evaluated along antero-posterior or cranio-caudal body axis (along medio-lateral axis, it has been computed as the ratio between odds and even harmonics). The Harmonic Ratio has been identified as an important prognostic factor of the risk of fall in elderly. For its name, this parameter has been often related to the harmony of walking, but in this case "harmonic" mainly refers to the fact that it has been computed by the harmonics obtained by means of Fast Fourier Transformation. The harmony that this parameter can assess is the symmetry of two consecutive steps contained into a stride, a feature possible only if each steps cover the 50% of the entire gait cycle.
The gait cycle has been characterized also by another percentage, that is the duration of the stance phase, in which the foot makes contact with the ground, with respect to the swing phase, in which the foot advances in the air. Much of the literature agrees that stance covers the 60-62% of a physiological gait cycle, and swing the remaining 40-38%. This is considerable as an important benchmark of human gait, and an alteration of stance to swing proportion has usually been identified as a sign of pathological gait. Despite it, the reasons for which this proportion is a so reliable parameter of physiological gait remained for years poorly investigated.
In 2013, we noted that the ratio between stance and swing coincides with the golden ratio. This is an irrational number, symbolically indicated with f that is = (1+50.5)/2 (about 1.618034). This number has been related to the problem reported by Euclid in III century b.C. to cut a given straight line so that the proportion between the shorter part to the longer one is the same as the longer part to the whole. The greek letter f was chosen to indicate this number in honor of the sculptor Phidia, who supervised the construction of the Parthenon, the façade of which is a golden rectangle, i.e. a rectangle having lengths of sides in the proportion of f. Henceforth, mathematicians, physicists, biologists, architects, and artists have been interested in the intrinsic symmetric properties of this number, that was also called "divine proportion" during the Renaissance, from the title of a book ("De divina proportione") written by Luca Pacioli and illustrated by Leonardo da Vinci. Could Kepler be not fashinated by a so "harmonic" number? In 1611, he discovered that f is the asymptotic limit of the ratio of consecutive Fibonacci numbers. He stated that geometry has two great treasures: one is the theorem of Pythagoras, and the other the division of a line into extreme and mean ratio, and he combined them in the Kepler's triangle.
f is the simplest example of fractal mathematics, where the term "fractal" refers to structures composed of subunits, in which the larger-scale structure resembles the subunit structure, in accordance to a property called self-similarity.
A wide variety of seemingly disparate physical and biological systems, such as leaf disposition on plant stems and seed arrangement on flower heads, spiral structures of galaxies and mollusks, quantum phase transitions, nucleotide frequencies, cell and shell growth, show similar harmonic characteristics related to f. In human sciences, f has been observed with regard to body proportions and aesthetic preferences.
The fact that stance/swing approximates f, implies an intrinsic fractal frame hidden below the orchestrated repetitive structure of physiological gait that can be represented with the following equation:
As already known since ancient Greek eras, also anthropometry has proportions approximately close to f. In another study we found that the artificial alteration of body segment proportions (obtained with special shoes with high outsoles) affected the gait ratio (stance/swing), whereas no changes occurred during walking with the adjunction of extra masses on body segments. It suggests that the relationship between anthropometric and gait golden proportion may be linked to the pendular mechanism of walking, being the oscillation period of a pendulum dependent only on length and not on mass.
However, the gait ratio is altered in pathological conditions without any alterations in anthropometric proportions. In a clinical study, we found a significant alteration of gait ratio in patients with Parkinson's disease. This alteration resulted quite resistant to dopamine administration. The results reported in that study seemed to suggest that the loss of gait kinetic melody of patients with PD may be related to deficits in the cerebellum – globus pallidus – spinal central pattern generator pathway, that is involved into the perception and production of harmonic features, and less sensitive to dopamine than neural circuits involved in the other motor tasks. The kinetic melody of walking may be related to the production of internal cues that can be altered in a pathology such as Parkinson's Disease, whereas for healthy subjects the equilibrium point could be the same for anthropometry and gait and represent a sort of resonant frequency at which biomechanics and motor control converge. Further studies should investigate the hypothesis that human anthropometry phylogenetically evolved towards proportions favouring the harmony of walking. I am pretty sure that this fascinating hypothesis could have intrigued even Kepler.
Iosa M, Fusco A, Marchetti F, Morone G, Caltagirone C, Paolucci S, Peppe A. The golden ratio of gait harmony: repetitive proportions of repetitive gait phases. Biomed Res Int. 2013;2013:918642
Iosa M, Morone G, Bini F, Fusco A, Paolucci S, Marinozzi F. The connection between anthropometry and gait harmony unveiled through the lens of the golden ratio. Neurosci Lett. 2016;612:138-44.
Iosa M, Morone G, Fusco A, Marchetti F, Caltagirone C, Paolucci S, Peppe A. Loss of fractal gait harmony in Parkinson's Disease. Clin Neurophysiol. 2016;127(2):1540-6.
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In the immediate aftermath of a car crash, there's nothing more important than your health and well-being. This means one thing: You need to receive medical treatment as quickly as possible.
As you recover and begin to ponder the future, you may come to realize that contacting a personal injury attorney would be in your best interest. The primary benefit of hiring an attorney is to have someone on your side to assist with protecting your legal rights.
You may assume that your insurance company will help you recover all the compensation you deserve, but nothing could be further from the truth. You need to take action to ensure that you're taking care of yourself.
How to hire a personal injury attorney
Don't assume that all personal injury attorneys are the same, as each professional has something unique to offer. When making a final decision, here are the types of things to look at:
- Experience and knowledge: The best personal injury attorneys have many years of experience assisting people in your situation. Just the same, they have knowledge of all personal injury laws, thus putting you in position to achieve the desired result.
- Contingency fee: With any personal injury case, you want to find an attorney who takes you on as a client for a contingency fee. This means that they don't get paid unless they win your case. Not only does it ensure the attorney's best performance, but it also takes all the financial risk out of the process for you.
- Time and availability: You don't want to hire a personal injury attorney who doesn't have the time in their schedule to take on your case. It takes many hours to help you receive the compensation you deserve, so you should expect your legal team to be by your side every step of the way.
You never want to find yourself searching for a personal injury attorney, as this means that you or a loved one has suffered some type of injury, perhaps in a motor vehicle accident.
However, you shouldn't shy away from getting professional help if it's something you need. This is the best way to better understand your legal rights and to do your part in holding the negligent driver responsible for their actions.

subset: commoncrawl


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We made a few pretty things this past weekend for a photo session with Mark Van Scyoc. We're looking forward to seeing the end product and be able to use the photos to update our website and promotional material. Here's a glimpse into what we made.
Here's the photographer hard at work! He was awesome to work with!
Working on a simple floral crown
Some happy poppies waiting to be used
One of the bouquets


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Please join us at the PTO general monthly meeting, on Tuesday, January 23rd at 6:30 pm. This month we will have an especially interesting discussion regarding a proposal to change the PS 9 school name. You will find helpful information below in advance of the meeting.
Tuesday, January 23rd, 2017
6:30 to 8:00 pm
In the Book Hive (Library)
Meeting agenda will include:
- Principal's Address
- Discussion about a New School Name
- Upcoming Community Building Opportunities
- Upcoming Fundraising Events
The P.S. 9 PTO is considering changing the name of the school and would like your thoughts.
Don't worry, it will still be P.S. 9. But in the past year, as communities around the country have been considering the historical significance of names and monuments, we at P.S. 9 are also thinking about changing the name of our school from "The Teunis Bergen School" to "The Shirley Chisholm School." The Department of Education regulations for name changes require the PTO to vote on the potential name, the principal to approve it, and then for the school to send it to the Chancellor's office for final approval.
We would like parents to weigh in on this decision, and encourage you to attend the PTO meeting on January 23, 2018, where we will vote on the proposed name change. If you will not be able to attend the meeting, please feel free to share your thoughts with a PTO representative in advance of the meeting.
Who Was Teunis Bergen?
Teunis Bergen was born in 1806 to the influential Bergen family. He was a surveyor and an historian, and served as the town supervisor of New Utrecht from 1836 to 1859, and in the US Congress representing New York's second congressional district from 1861-65. Bergen's papers are collected at the New York Historical Society, and their catalogue can be seen at https://brooklynhistory.org
Who Was Shirley Chisholm?
Shirley Chisholm was born in 1924. Her father was from Guyana and her mother was from Barbados. After graduating with honors from Brooklyn College she worked in early childhood education, obtained a Master's degree from Columbia, and served as a consultant to the New York City Division of Day Care. In 1964, she was elected to the New York State Legislature, and in 1968 she became the first African American woman to serve in the US Congress, representing the 12th Congressional District, which then included Bedford Stuyvesant and Prospect Heights.
Chisholm ran for the democratic nomination for president in 1972, the first African-American (and the second woman) to run for a major party's nomination. She won 10% of the delegates in the 1972 primary season. She continued to serve in Congress until 1983, after which she returned to teaching by joining the faculty of Mount Holyoke before retiring to Florida.
Chisholm was an advocate for racial justice and for ending the Vietnam War. She fought for the poor and for children, and was one of the founders of the Congressional Black Caucus. Moreover, she serves as an inspiration to those who have been left out of the political process and who seek to join it. Just this year, Chisholm's quote, "If they don't give you a seat at the table, bring a folding chair," was embraced by Tamaya Dennard's successful run for the city council of Cincinnati: Dennard created the hashtag #bringafoldingchair for her campaign, and brought a folding chair to her swearing-in.
Why Change the Name Now?
We began looking into the potential name change in the wake of the national debate on monument names and the mayor's commission examining historic monuments and markers. While one factor in the investigation was the fact that the Bergen family owned slaves, Teunis himself was born in 1806, seven years after New York passed the Gradual Emancipation Act, which phased out slavery in the state.
Nevertheless, there is ample preservation of the Bergen family name in New York – Bergen Street and Bergen Beach, to name just two examples. There is no school named after Chisholm (there is a New York State Office building and a day care center), and we thought that the moment was right to consider a change.


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Endocrine System Channel
Related Channels
Humalog Mix75/25 Side Effects
Related MultimediaClick on an icon to view.
Serious Humalog Mix75/25 Side EffectsSome side effects with Humalog Mix75/25 are potentially serious and should be reported immediately to your healthcare provider. Some of the warning signs of these side effects include:
- Symptoms of dangerously low blood sugar levels, such as:
- Extreme hunger
- Blurry vision
- Dizziness
- Shakiness
- Changes in behavior, such as irritability or mood swings
- Loss of coordination
- Cold sweats
- Difficulty speaking
- Sweating
- Confusion
- Seizures
- Unconsciousness
- Signs of an allergic reaction, such as:
- Hives
- Itching
- An unexplained rash
- Swelling of the mouth or throat
- Wheezing or other difficulty breathing.
Make sure that those around you (such as relatives, friends, and coworkers) know how to recognize your particular signs of low blood sugar, as you may be too confused or disoriented to recognize and adequately respond to your symptoms. Different people may have different signs of low blood sugar. Also, make sure you and those around you know exactly how to respond to low blood sugar.
Final Thoughts on Humalog Mix75/25 Side EffectsYou may experience some or none of the Humalog Mix75/25 side effects listed in this article. Unfortunately, there is no way for your healthcare provider to know beforehand if you will have side effects from a medicine that you have never tried. Therefore, make sure to let your healthcare provider know if you develop any side effects while taking the drug or if something "just does not seem right." Even if the side effect is not caused by Humalog Mix75/25, your healthcare provider will be able to diagnose and treat the problem.
Humalog Mix75/25 Medication
Written by/reviewed by: Kristi Monson, PharmD; Arthur Schoenstadt, MD
Last reviewed by: Kristi Monson, PharmD


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Heartland Cooperative Services
331 Industrial Ave. Owen WI. 54460
715-654-5134 or 1-800-521-2021
News & Information-Heartland & Ag
Calf & Heifer - Beef
Country Stores
Location-Contact Info
Customer Account Access
Website Designed
List Your Business for Free
When Must I have a Nutrient Management Plan?
The Department of Ag, Trade, and Consumer Protection, along with the Department of Natural Resources state that farms which apply manure or commercial fertilizer to cropland need to have a Nutrient Management Plan. According to the DATCP website, a Nutrient Management Plan is required when:
A producer voluntarily accepts, or is offered, government cost-share dollars for nutrient management. State law makes enforcement contingent on an offer of cost sharing for this item only.
A producer voluntarily accepts, or is offered, government cost-share dollars for the installation of manure storage.
A producer voluntarily continues participation in the farmland preservation program (FPP).
A producer is regulated under a county manure storage or livestock siting ordinance.
A producer is regulated under a DNR Wisconsin pollution discharge elimination system permit (WPDES). Or if a producer is required to prevent or mitigate imminent harm to waters of the state as an emergency or interim response to a grossly negligent pollution discharge.
For more helpful information about Nutrient Management Planning, take a look at the following links:
http://datcp.wi.gov/Farms/Nutrient_Management/index.aspx for DATCP Nutrient Management Information
http://www.soils.wisc.edu/extension/fertilitybasic.php for Soil Fertility Basics
What is a Nutrient Management Plan?
The USDA Natural Resources Conservation Service defines a Nutrient Management Plan as, "managing the amount, source, placement, form, and timing of the application of nutrients and soil amendments." The main objectives of a Nutrient Management Plan include:
Manure and fertilizer recommendations based on soil test results
Calculating and monitoring soil loss
Calculating and monitoring phosphorus delivery to surface water
Record-keeping systems
How Will I Benefit from a Nutrient Management Plan?
Nutrient management planning will assist the grower in properly supplying the essential nutrients for crop production. Nutrient management planning is also used as a tool to maintain the condition of the soil for future crop productions.
The end goal of an NMP is to minimize the environmental impact of soil loss, nutrient loss to ground and surface water, and meeting the nutrient needs of the crops grown.
Basics of Sampling Soils for Analysis
Why Test Your Soil?
Analyzing soil is a beneficial, easy, and inexpensive diagnostic tool for determining a grower's soil nutrient needs.
Knowing the nutrient levels of the soil helps make decisions on how much fertilizer to apply, therefore saving money on expensive fertilizers.
Knowing the nutrient content of soil also aids in crop production, leading to higher crop yields and soil performance.
Soil sampling is also an essential part of Nutrient Management Planning. Soil test results play a crucial role in the fertilizer recommendations for each field.
Soil tests for Nutrient Management Plans must also be updated every 4 years for accurate nutrient management planning.
What are the Different Soil-Sampling Methods?
Conventional Sampling:
Lower overall sampling cost
Grower gets a general idea of soil fertility
Bulk spreading of fertilizer
Uses existing technology
Grid Sampling:
Visual impact - color maps illustrating fertility variation in a field and on the farm
Consistency - ability to sample the exact same spot each time you sample allowing for a true comparison in fertility from year to year.
Variable Rate Application - precise placement of fertilizer where it is needed.
Data - ability to compare soil fertility to other data such as yield maps, planting maps, soil type, and bio-mas index.
For additional information on soil sampling, take a look at the following links:
http://www.soils.wisc.edu/extension/pubs/A2809.pdf for Nutrient Recommendations
www.soils.wisc.edu/extension/pubs/A2100.pdf for Soil Sampling Guidelines
www.soils.wisc.edu/extension/pubs/A3588.pdf for Management of Wisconsin Soils
uwlab.soils.wisc.edu/soilsummary/ for Soil Test Summary Data
Evaluating Alfalfa Stand for Winter Kill / Consider Shorter Rotations
Although most farm plans are complete and we're gearing up for planting season - keep in mind the health of current alfalfa fields should be considered to make any final changes. If a field has suffered some winter injury, or is just getting old, it may be more profitable to rotate out, take advantage of any Nitrogen credits - rather than limp along a low yielding, unprofitable alfalfa field. Every acre should pull it's weight for overall farm profitability. Call your local Heartland Agronomy office today! UW has some of the best information available and further reading can be found in the links below:
Nutrient Management Plan
Basics of Sampling Soils for Analysis
Evaluating Alfalfa Stand for Winter Kill
Agronomy Staff & Locations
Agronomy News and Information
Future Farm Solutions
Agronomy Products & Services
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text: CEOI 2008 News News for the Central European Olympiad in Informatics 2008 2009-09-15T17:25:26+0200 CEOI 2008 Organization Committee http://www.ceoi2008.de/ CEOI 2008 Website Started website_started 2008-02-22T07:18:00+0100 The CEOI 2008 Website is online, now. CEOI 2008 Schedule online schedule_online 2008-04-29T19:20:00+0200 The CEOI 2008 Schedule overview is online, now. CEOI 2008 Rules online rules_online 2008-06-06T16:00:00+0200 The CEOI 2008 Rules are online, now. CEOI 2008 Online Competition announced online_competition_announced 2008-06-25T22:10:00+0200 Parallel to the CEOI, the CEOI 2008 Online Competition will take place to allow everybody to participate in solving the tasks. Results of the Online Competition online online_competition_results_online 2008-07-17T00:25:00+0200 The results of the CEOI 2008 Online Competition are online.


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Author Jaqueline Leigh joins us at our Parkside Location for a book reading and signing on August 26th from 10:00 - 12:00. There will be a craft for the kids and shopping for the mommas!
This adventurous tale follows a young boy and his stuffed bunny through the exciting world of dreams.
The story is geared towards preschoolers in hopes of promoting healthy and happy sleep.
Jacqueline Leigh holds a BA in Journalism from Valdosta State University. She's
successfully published articles in multiple magazines and online publications. She's
perfected her craft by blending her affection for writing with the thrill of
entertaining children. Jacqueline makes her home in North Carolina with her
husband and two spirited children.


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It's Saturday!!!
Minton Family Farms
We will be at four locations today; come on down and pick up some farm fresh produce! Did we mention that our melons are coming in? Come early and you will have your pick of AMBROSIA, watermelon or cantaloupe!!!
Yuba City
Townsquare Park
8am - noon
California and Booth Street
8am - 1pm
Tamarack Junction Casino
9am - 1pm
Carson City
3rd and Curry St.
8:30am - 1pm


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All members are required to make a CPD Declaration annually, regardless of any previously declared exemptions relating to Continuing Professional Development. A member is eligible to make an annual exemption declaration for one of the following reasons:
NOTE: If you withdraw from the workforce during the year, you still have a full CPD requirement, regardless of how long you worked during the year.
If you are unsure if you qualify for one of these exemptions, click below to view the Exemption Matrix:
Back to CPD Reporting Home Page


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Bsbee is the name of a turquoise stone from Arizona, this stone, along with a red wool string is a sign of good luck and true love for life and nature.
Bsbee promotes a critical message against those behaviors that have lead society and the environment to where they are today: pollution, global warming, wild exploitation of human and natural resources.
Through our clothes we're trying to do our part to contrast this trend.
Wearing sustainable/organic clothing made with non toxic materials helps to actively preserve the planet.
Spring Summer 2020 SHOWING @
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September 13-14-15 2019
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547 West 26th Street
New York NY 10001
Forni Opificio/31
September 19-22 2019
Via Tortona 31
Private Gallery
September 28 –29 –30 - October 1-2 2019
11 Rue Sainte Anastase
75003 – Marais
D&A Los Angeles
October 14-15-16 2019
The New Mart Building
127 East Ninth Street, 3rd floor
Los Angeles, CA 900155
Milano, Viale Col di Lana 8,
tel +39 0284342858, email@example.com
By appointment from September 16th till October 25th 2019
We concentrate on matters and fabrics, which are our primary research focus.
Every fabric is original, exclusively created for our collections, discovered and produced in places where local tradition and craftsmanship have developed the best working skills.
Bsbee pieces are made from premium quality materials and most of them are handmade.
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Imperfections, if any, are part of the value of the handwork itself and prove the uniqueness of this piece.
FALL WiNTER 2018-2019


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1665 E Oak Grove Road, Montrose, CO 81401
- $1,288,888
- 3 Beds
- 2 Full Baths
- 1,766 sq ft
- Year Built: 1953
- Residential
- Listing #757884
Property Description
Beautiful 9.84 acres with great development potential located approximately 1/4 mile east on E Oak Grove at its intersection with Townsend Avenue. Behind Walgreens and the Penn Center Mall. Potential for Motel, residential, or office park with change of zoning. Adjacent to commercially zoned property. The recent widening of E Oak Grove and the expansion of Hillcrest from Niagara to connect with it, has made passing this property a common occurance for many. According to the City, the widening project also included replacement and upsizing of an aged waterline and undergrounding of overhead power lines. This may be the best located parcel with multiple possible uses currently on the local market. Nice sidewalks stretch the length of this property which boast nice views in every direction. Approximately 8.7 shares of UVWUA. The 1766sf 3 bed, 2 bath home is situated at the southeast corner of the property. Currently in Montrose County. Best possible uses would require annexation to City of Montrose.
Listing Details
|3 Beds
|2 Full Baths
|Year Built
|Property Type
|Listing #
|Listing Type
|For Sale
Interior Features
|Square Feet
Exterior Features
|Lot Size
|9.84 acres
Courtesy of Shawn Carroll, NextHome Virtual


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Water sustains your life, but that doesn't mean you need to let it rain on your parade! The Working Person's Store offers you the waterproofing products you need to protect your expensive outdoor gear from water damage. Granger's re-proofing products use the same technology as garment manufacturers, which allows you to restore your important garments, footwear, and equipment to their original quality. Granger's refers to this as 'Original Performance Restored'. Granger's cleaners and waterproofing products are water-based and environmentally friendly. The high concentration means that you have to use less for a long-lasting effect, reducing the cost for you and the manufacturing waste for the environment. Most importantly, Granger's products keep your favorite outdoor products like new: clean and resistant to water. Stop letting water walk all over you. Instead, walk all over water in your newly waterproofed boots thanks to Granger's products from The Working Person's Store.
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ParLeS v3.1—Shareware for chemometric analysis of spectroscopic data
Software capabilities:
- join multiple spectroscopic files from single directory into a single file
- numerous preprocessing and pretreatment algorithms
- performs principal components analysis (PCA)
- performs partial least squares regression (PLSR) with delete-n-cross validation
- performs PLSR1 modelling and prediction
- performs bootstrap aggregation PLSR (bagging-PLSR)
- provides options to save all output for external plotting and analyses
- version 3.1 supports interactive outlier detection
Screenshots and a description of this version are available at http://www.usyd.edu.au/agric/acpa/people/rvrossel/soft01.htm
If you would like to use ParLeS, please send email to Raphael VISCARRA ROSSEL ([email protected]) briefly explaining your application and your current situation (i.e. student, researcher, working at university, commercial organisation etc.).
Raphael VISCARRA ROSSEL ([email protected])
I am Ahmed A. Afifi an egyptian working as a researcher at national Research Centre, in soils and water use Dept., I ahave an idea to use IRS spectroscopy to detect the soil pollution. So I have used some solution to contaminate two types of soils with Cd and Pb. to study the ability of MIR to differentiate between the source of conatminat (under which concentration) and the type of soils. I need to use your program to do the chemometric analysis to buid the model. so I can have your advice and your program I will be appreciated. I do not have a comecial use I only need an acadamic use for your program.
by the way my email adress
[email protected]
thank you in advance ad waitting for your repsonse
Dr. Ahmed A. Afifi


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By Avery Martinez
Staff Writer
The unemployment rate for Archuleta County increased to 3 percent in June, with 7,009 people in the labor force and 208 of those claiming unemployment.
The full version of this story is available in the print edition and e-edition of the Pagosa Springs SUN. Subscribe today by calling (970)264-2100 or click here.
Follow these topics: Archuleta County, Business, News


text:
Odisha Sun Times Bureau
Bhubaneswar, Oct 24:
Siksha 'O' Anusandhan (SOA) University Vice-Chancellor Amit Banerjee today appeared for the second time in a day before the Khandagiri police to answer queries pertaining to the major fire in the Institute of Medical Sciences and SUM Hospital in the Odisha capital on last Monday in which 19 patients were killed due to asphyxiation and eight others have succumbed till date after being relocated to other hospitals.
SUM Hospital is an affiliated institution of SOA University.
Banerjee who had appeared before the Khandagiri police in the morning appeared before it again in the afternoon.
The VC told reporters that he had replied to every question asked of him and had submitted whichever documents were sought by them.
"I have answered whatever questions they asked. Since the matter is under investigation I cannot reveal anything. Further, my revealing it to you will be considered as an unlawful activity since the matter is sub judice," said Banerjee to reporters avoiding to give details.


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On Dasher! On Dancer! On [Name]!, personalize with your own name and photoInside:
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This is the standard notation used on Ordnance Survey Landranger maps.
Each reference consists of two letters identifying a 100,000 metre square block then three digits defining the Easting and finally the three digits defining the Northing with reference to the South West corner of the block.
NN166712 is the grid reference for the summit of Ben Nevis. Where you are given the map number ( For Ben Nevis = 41) it is acceptable to omit the two initial letters e.g. 166712. (Instructions on how to read the references are given on the OS maps).
The UK is covered by 204 Ordnance Survey Landranger 1:50,000 scale maps. Maps numbered 1 to 86 cover Scotland but for the highest mountains (Munros) only 23 maps are required. The name given roughly describes the area covered by the map.
A description of the characteristics of the mountain including any hazards of which you should be aware.
A track from the N leads through a bealach on the NE slopes and this gives easy access to the summit ridge.
Geal Charn viewed from the bulldozed track which runs SE from Dorback Lodge to the watershed at NJ 111 135 allowing an easy ascent SW then W and finally WNW to the cairn.
© Graham Young
Northen Cairngorms viewed from the summit of Geal Charn.
© John Tutt
Geal Charn from the Dorback landrover track 8th Feb 2011
© Peter Sinclair
Summit of Geal Charn 821m with hazy Cairngorm Mountains to the rear. 17/11/2013
© Eileen Stark
A selection of weather forecasts local to #GetMountain.Top_Name#.
Ordnance Survey digital maps are also available to members.
A selection of local accommodation options who advertise with Munromagic.com.
The summary information of one or more ascent routes that include Geal Charn [Cairngorms].
Click on the route title to load the full content for that route.
Pictures submitted by members on the summit of Geal Charn [Cairngorms]
Myself, enjoying a short spell of sunshine at summit cairn of Geal Charn.
James at summit of Geal Charn in a brief period of sunshine. 17/11/2013
Me at the summit
© David Main
Myself and David Peden summit Geal Charn 25/06/11
© John Donnelly
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You can view a members route overlayed on an online map or download the KMZ file for use in Google Earth.
Post a few words about Geal Charn [Cairngorms] or read what others have had to say.
A full written account of a climb submitted by our members.
You can prepare your own write up by first making an entry in your route log and then visiting the logging section above.


text:
Attention: All Players Grade 8-12!
The Fox 40 Prospect Challenge has gained momentum on the West Coast. It has been great to see the competition level at the tryouts. While the unprecedented snow fall has created some challenges and an overall cooling of the events?! It has taken no more than the warm up to get the competitive juices flowing to set the stage that this is the real deal and to trigger the internal drive within the players to show their stuff. The tryout format really allows coaches to coach and work with the players and then put them in drills and competitive situations to assess football skills. It is always impressive to see players improve in a short time and take the coaching and apply it quickly.
While there has been some great defensive back vs. receiver battles I expect to see some top national prospects emerge in the lineman groups.
It has been great to see so many grade 12's come out and compete s this is certainly an opportunity for them to shine and expose themselves if they have fallen somewhat under the radar or if they are simply looking to get better by competing against strong competition. Also great to see many players make the effort to attend more than one tryout (1 fee- multiple tryouts), the echo from many players – "Every time I tryout, I get better!" Bring a teammate so you team gets better!
Be Prepared!
Show up early. Approach it like a game be prepared for the weather, the surface, the coaching, the competition. Get a good night sleep and eat at least an hour before the tryout!
Warm it Up
Use the warm up to get focussed and ready the mind and body for the upcoming coaching and competition.
Make Plays
It is no secret coaches will notice when you "make the play!" Absorb the coaching so you are in the right place at the right time with the right technique.
Fall Down, Get Back Up!
Every player makes mistakes, falls down, slips, drops a ball etc. Show a 'winning attitude' get back up! On to the next play! Do not stop in the drill!
Be a Team Player
Encourage the other player, the competition. Be thankful to your parents for getting you to the tryout and supporting you, and the coaches for challenging you!
A busy schedule ahead this weekend as we head to Vancouver Island for tryouts in Nanaimo and Victoria, while at the same time players will be competing in Burnaby. (See all the details below)
Saturday February 18
Burnaby Lakes
Nanaimo- Merle Logan Field
Victoria – Goudy Field
Coach Dino
Dino Geremia
The Recruiting Academy
About Canadafootballchat.com
CFC is the number one national amateur football website (source: ComScore.com) and most respected source for high school and recruiting news in Canada.
About the 'CFC Insider'
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Become a CFC Insider MEMBER today!SUBSCRIBE NOW


text:
In this section
Insuring your bike may be worthwhile, too. Home insurance policies can often be extended to include your bike when it is at home and more valuable bikes can be insured separately to provide cover when you're out and about. Take a photo of the bike and write down the frame number.
There are three simple ways of cutting the risk of bike theft:
Some locks are stronger than others and price influences their quality, so it pays to invest in a good one. Remember that two locks are better than one. D-locks are sturdy, but consider using a cable lock as well to secure any parts of the bike which are quick release, such as the saddle or front wheel.
Secure the lock round the frame, spokes and bike stand, ensuring there is little room for manoeuvre to make it difficult for thieves to insert their tools.
Where you lock your bike is critical. Leave it on a well-lit and busy street so any potential thieves are conspicuous. Avoid leaving your bike in the same spot every day and lock it to a bike stand or something immovable.
Make sure you take care of your bike when you get home too - according to the British Crime Survey, over two-thirds of bikes were stolen in or near the owner's home. Always keep your bike locked up at home and try to keep it out of view.
It's not just your bike that can go missing – many parts and accessories are also valuable and easy to steal. Follow these tips to keep them safe:
D or U Locks - Shackle locks consist of two parts, the mechanism housing and a U shaped round bar, the ends of which fasten into the mechanism housing.
The bar should be made of hardened steel to resist cutting, as should the housing which contains the lock mechanism.
Shackle locks can be separated into 2 types: Single and Double mechanisms.
Single Mechanism shackle locks anchor one end of the U into the housing and the lock mechanism secures the other end. The key will operate the lock at one end of the housing.Double Mechanism shackle locks have a centrally located mechanism with secures both ends of the bar independantly. The key will operate the lock from a central position (most commonly on the underside) on the housing.
Cable Locks - should be of the multi-wound multi "braided" type. A braided cable with many thin wire strands is far more difficult to cut than a cable with consists of a lesser number of thicker strands. Many cheaper cable locks consist of the basic cable construction whereas better quality cables will feature the finer braided construction.
Chains should prove more secure than cables for longer periods but are considerably heavier. Chains can be virtually any length and because chain collapses into a pile, they can take up less space.
There is a accreditation scheme for locks, in the form of "SOLD SECURE". Sold Secure grant 3 different levels of award according to how secure a lock is. They perform a variety of standard tests on a brand new lock, including a number of attacks with different tools and time the period taken to break into the lock. The award is granted according to how many of these set tests a lock withstands. Bronze is the lowest award and Gold is the highest. A gold award does not mean that the lock is impenetrable, just that it withstood all that Sold Secure threw at it within the allocated time! Look for a Sold Secure logo on the packaging. There is even a loop lock on the British market which is sold secure accredited at a level above many big and bulky cheap locks.
These are the most common locations:


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Jake Hewitt, Teaser 2014 - PhotosThu, 06/05/2014 - 9:09am
Been riding for about 2 years and looking for a sponsor.
Been riding for about 2 years and looking for a sponsor.
Florida Ride put together a street jam in Tampa! So I decided to take a couple pictures! Jumped my first 8 stair! I'll try to get it and post later! Chandler was doing a febel Joe was doing a febel to smith My first edited photos in photoshop!!
A simple turn down out of bank.
Scotia Filming www.youtube.com/tgravesstuff www.facebook.com/scotiafilming instagram: @scotiafilming
kyle grant throwing down one of his first backflips
Chicken noodles yeah i wanna get sponserd woop
nathen parr and james gee at bellingham washinton skatepark
Hi my name is divesh i can nail alot of tricks and im really interested.Im good.Ill scoot my life.i would really like to get sponsor ill help ur company with a lot things and recomandation.im 13
Hi my name is Jack i'm a ametauer scooterer and just want to get a sponsorship because I want to ride for a company and make them proud and just do my best so thanks!
My Deck Grab!!!!! #steezy


text:
28 Apr 2016
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text:
Aldo Customer Service Connection Number
|Connection Number
|118 057
Calls cost £1.50 per minute plus your phone provider's access charge. Minimum charge £1.50. Bill payers permission required. This is a Directory Enquiry Service provided by NewVenture2014. Customer Service Number 0333 009 5159.
Aldo Head Office Address
|Head Office
|2300 Émile-Bélanger
Montreal, Quebec
Aldo Shoes
Aldo is a popular high-street footwear retailer. Shoes are available in a wide variety of styles for both men and women.
For women, you will find high heels, low to mid heels, flats and sandals. Shoes are available in sizes 3-9.5 in certain styles. The shoes all have their own name for each style such as Hanrietta (a high heeled court shoe), Kolito (a kitten heel), Villarosa (a low heeled sandal), and Luisi (a flat loafer). You can purchase Aldo shoes directly from the Aldo website or from your local Aldo store, which you can generally find on the high-street or in shopping centres. If you order directly from the Aldo website you will get free delivery when you spend over £60. Standard delivery takes 2-3 days and costs £5, or you can collect your order from your local Aldo store free of charge. Once you have placed your order, you will be sent a confirmation email with your order number – you will need this along with your postcode in order to track your order's progress.
If you regularly shop at Aldo, you may find it beneficial to join the Aldo A-List where you will be able to enjoy exclusive benefits such as 15% discount off your next order, birthday gifts, free shipping offers and you will receive VIP news before anyone else.
Aldo's return policy allows you to return unworn products within 60 days of purchase. You can return products to any Aldo store, with the exception of another stockist (such as a concession within a department store) as you will need to return such purchases to the place that you bought them from. Alternatively, you can return your order via courier – just submit your order on the Aldo website to receive a returns label and information about how to proceed.
Aldo Boots
Whether you're looking for a casual ankle boot, a glamorous over-the-knee style or something different, Aldo has a great selection of boots available. You will find Chelsea boots in a leather style for a classic take on the boot, or knee-high boots which go perfectly with jeans. Over the knee styles look ideal with knitted dresses and tights for the winter months. As with the shoes, the boots all have their own names such as Scotch (Chelsea boot), Keesha (knee high boot) and Arabia (over the knee boot) and they are available in sizes 3-9.5 in certain styles.
Aldo Accessories
If you're looking for the perfect accessory to set off an outfit, Aldo has a lot of choice. You will find cosy hats, gloves and scarves, necklaces, stylish sunglasses, earrings, bracelets, rings, watches, leggings, tights and socks. You can search by the style of accessory that you are looking for, or you can look what is currently trending. At the moment, the current trending accessory is chokers and pin badges.
Aldo Shoes for Men
As with women, men will find a wide variety of styles to choose from at Aldo. There are trainers and plimsolls, casual shoes, formal shoes, oxfords, brogues, slip-ons and loafers. The shoes all have their own individual names such as Swithbert (a desert boot), Lentina (a leather brogue) and Fildes (a leather loafer). Men's shoes are available in sizes 6-13 in certain styles. There is also a premium collection – Mr B's for Aldo – which focuses on quality construction and high-quality leather, using expert design to craft stylish pieces for the modern man.
Aldo Handbags
Whether you need a cross-body style for on-the-go or a handbag to carry around your work essentials, Aldo's handbag collection is varied yet stylish. You will find shoulder bags, totes, satchels, cross-body bags, clutches, backpacks, laptop bags, and wallets. As with Aldo shoes, each style of bag has its own name. There is Evesham (a metallic clutch bag), Game (a quirky appliqué handbag), Colby (a crossbody bag) and Acawien (a shoulder bag).
Aldo Outlet
There is an Aldo Outlet in London. The address of the store is 231-233 Camden High Street, Camden, NW1 7BU. It sells Aldo footwear at a lower price – this is usually because the products are last season's styles. The store is open from 10am-7pm every day except Sundays when it is open from 11am-6pm and it opens at 9am on Fridays.
Aldo Locations
If you are looking to find your closest Aldo store, you can do so by using the online store locator tool. You can search by city or postcode and you will be presented with a list of the stores closest to you. You will be given the store's full address as well as related information such as opening times. If you enter a postcode, you will also be given the distance in KM that the store is from the postcode that you gave.
About Aldo
Aldo was first founded in 1972 by Aldo Bensadoun in Montreal, Canada. It was first founded as a concession within Le Chateau, a fashion company but soon established a name for itself. Today, the Aldo Group is one of the largest footwear retailers in the world with almost 3,000 stores. The Aldo Group owns the UK, US and Ireland stores, whilst stores in other countries are franchised. The company headquarters can still be found in Montreal.
For more information about Aldo products and services, call the Aldo customer service number.


text:
Summer of Steam and Smoke
"Decide what the theme of your story is." This excellent writing advice sometimes applies to life, too. Summer began with an impromptu trip to the Railroad Museum of Pennsylvania, to talk … Continue reading
August 29, 2015 · Leave a comment
There is a tide in the affairs of men…
which reminds one to wake up and smell the coffee. As Gary wrote to his kids that afternoon: "This is sooo embarrassing! The adventure was a little too intense. "We … Continue reading
August 8, 2015 · 2 Comments


text:
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Over 20 Classes Available Weekly
Our classes produce exceptional results by effectively combining modern martial arts techniques and other functional fitness movements. Our classes are universally scalable – meaning the class sessions are designed to be equally challenging to all athletes regardless of past experience or level of fitness. Sessions focus on the proper execution of all movements to achieve an all round skill level of Brazilian Jiu Jitsu, Kickboxing, Wrestling, Judo and MMA.


text:
It was just a matter of time before iPads went from helping out college marching bands to the NFL sidelines.
Microsoft has created a "Sideline Viewing System" that will allow NFL teams to see instant replays from the sidelines, which could prove to be a major coaching advantage. Players and coaches will also be able to draw on photos, which will be featured on Surface Pro 2 tablets.
NFL players and coaches won't be forced to use the technology, however, and the option is available to all 32 teams.
This isn't the only new technology making its way onto football fields this fall. Click here to see which NFL team will be the first to debut Google Glass on the field.


text:
In addition to what's in Anaconda, this lecture will need the following libraries:
!pip install --upgrade quantecon
!pip install interpolation
In this lecture, we're going to study a simple optimal growth model with one agent.
The model is a version of the standard one sector infinite horizon growth model studied in
The technique we use to solve the model is dynamic programming.
Our treatment of dynamic programming follows on from earlier treatments in our lectures on shortest paths and job search.
We'll discuss some of the technical details of dynamic programming as we go along.
Let's start with some imports.
We use an interpolation function from the interpolation.py package because it comes in handy later when we want to just-in-time compile our code.
This library can be installed with the following command in Jupyter:
!pip install interpolation.
Let's start with some imports:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from interpolation import interp
from numba import njit, prange
from quantecon.optimize.scalar_maximization import brent_max
The Model¶
Consider an agent who owns an amount $ y_t \in \mathbb R_+ := [0, \infty) $ of a consumption good at time $ t $.
This output can either be consumed or invested.
When the good is invested it is transformed one-for-one into capital.
The resulting capital stock, denoted here by $ k_{t+1} $, will then be used for production.
Production is stochastic, in that it also depends on a shock $ \xi_{t+1} $ realized at the end of the current period.
Next period output is$$ y_{t+1} := f(k_{t+1}) \xi_{t+1} $$
where $ f \colon \mathbb R_+ \to \mathbb R_+ $ is called the production function.
The resource constraint is
$$ k_{t+1} + c_t \leq y_t \tag{1} $$
and all variables are required to be nonnegative.
Assumptions and Comments¶
In what follows,
- The sequence $ \{\xi_t\} $ is assumed to be IID.
- The common distribution of each $ \xi_t $ will be denoted $ \phi $.
- The production function $ f $ is assumed to be increasing and continuous.
- Depreciation of capital is not made explicit but can be incorporated into the production function.
While many other treatments of the stochastic growth model use $ k_t $ as the state variable, we will use $ y_t $.
This will allow us to treat a stochastic model while maintaining only one state variable.
We consider alternative states and timing specifications in some of our other lectures.
Taking $ y_0 $ as given, the agent wishes to maximize
$$ \mathbb E \left[ \sum_{t = 0}^{\infty} \beta^t u(c_t) \right] \tag{2} $$
subject to
$$ y_{t+1} = f(y_t - c_t) \xi_{t+1} \quad \text{and} \quad 0 \leq c_t \leq y_t \quad \text{for all } t \tag{3} $$
- $ u $ is a bounded, continuous and strictly increasing utility function and
- $ \beta \in (0, 1) $ is a discount factor.
In (3) we are assuming that the resource constraint (1) holds with equality — which is reasonable because $ u $ is strictly increasing and no output will be wasted at the optimum.
In summary, the agent's aim is to select a path $ c_0, c_1, c_2, \ldots $ for consumption that is
- nonnegative,
- feasible in the sense of (1),
- optimal, in the sense that it maximizes (2) relative to all other feasible consumption sequences, and
- adapted, in the sense that the action $ c_t $ depends only on observable outcomes, not on future outcomes such as $ \xi_{t+1} $.
In the present context
- $ y_t $ is called the state variable — it summarizes the "state of the world" at the start of each period.
- $ c_t $ is called the control variable — a value chosen by the agent each period after observing the state.
The Policy Function Approach¶
One way to think about solving this problem is to look for the best policy function.
A policy function is a map from past and present observables into current action.
We'll be particularly interested in Markov policies, which are maps from the current state $ y_t $ into a current action $ c_t $.
For dynamic programming problems such as this one (in fact for any Markov decision process), the optimal policy is always a Markov policy.
In other words, the current state $ y_t $ provides a sufficient statistic for the history in terms of making an optimal decision today.
This is quite intuitive but if you wish you can find proofs in texts such as [SLP89] (section 4.1).
Hereafter we focus on finding the best Markov policy.
In our context, a Markov policy is a function $ \sigma \colon \mathbb R_+ \to \mathbb R_+ $, with the understanding that states are mapped to actions via$$ c_t = \sigma(y_t) \quad \text{for all } t $$
In what follows, we will call $ \sigma $ a feasible consumption policy if it satisfies
$$ 0 \leq \sigma(y) \leq y \quad \text{for all} \quad y \in \mathbb R_+ \tag{4} $$
In other words, a feasible consumption policy is a Markov policy that respects the resource constraint.
The set of all feasible consumption policies will be denoted by $ \Sigma $.
Each $ \sigma \in \Sigma $ determines a continuous state Markov process $ \{y_t\} $ for output via
$$ y_{t+1} = f(y_t - \sigma(y_t)) \xi_{t+1}, \quad y_0 \text{ given} \tag{5} $$
This is the time path for output when we choose and stick with the policy $ \sigma $.
We insert this process into the objective function to get
$$ \mathbb E \left[ \, \sum_{t = 0}^{\infty} \beta^t u(c_t) \, \right] = \mathbb E \left[ \, \sum_{t = 0}^{\infty} \beta^t u(\sigma(y_t)) \, \right] \tag{6} $$
This is the total expected present value of following policy $ \sigma $ forever, given initial income $ y_0 $.
The aim is to select a policy that makes this number as large as possible.
The next section covers these ideas more formally.
The $ \sigma $ associated with a given policy $ \sigma $ is the mapping defined by
$$ v_{\sigma}(y) = \mathbb E \left[ \sum_{t = 0}^{\infty} \beta^t u(\sigma(y_t)) \right] \tag{7} $$
when $ \{y_t\} $ is given by (5) with $ y_0 = y $.
In other words, it is the lifetime value of following policy $ \sigma $ starting at initial condition $ y $.
The value function is then defined as
$$ v^*(y) := \sup_{\sigma \in \Sigma} \; v_{\sigma}(y) \tag{8} $$
The value function gives the maximal value that can be obtained from state $ y $, after considering all feasible policies.
A policy $ \sigma \in \Sigma $ is called optimal if it attains the supremum in (8) for all $ y \in \mathbb R_+ $.
The Bellman Equation¶
With our assumptions on utility and production function, the value function as defined in (8) also satisfies a Bellman equation.
For this problem, the Bellman equation takes the form
$$ v(y) = \max_{0 \leq c \leq y} \left\{ u(c) + \beta \int v(f(y - c) z) \phi(dz) \right\} \qquad (y \in \mathbb R_+) \tag{9} $$
This is a functional equation in $ v $.
The term $ \int v(f(y - c) z) \phi(dz) $ can be understood as the expected next period value when
- $ v $ is used to measure value
- the state is $ y $
- consumption is set to $ c $
As shown in EDTC, theorem 10.1.11 and a range of other texts
The value function $ v^* $ satisfies the Bellman equation
In other words, (9) holds when $ v=v^* $.
The intuition is that maximal value from a given state can be obtained by optimally trading off
- current reward from a given action, vs
- expected discounted future value of the state resulting from that action
The Bellman equation is important because it gives us more information about the value function.
It also suggests a way of computing the value function, which we discuss below.
Greedy Policies¶
The primary importance of the value function is that we can use it to compute optimal policies.
The details are as follows.
Given a continuous function $ v $ on $ \mathbb R_+ $, we say that $ \sigma \in \Sigma $ is $ v $-greedy if $ \sigma(y) $ is a solution to
$$ \max_{0 \leq c \leq y} \left\{ u(c) + \beta \int v(f(y - c) z) \phi(dz) \right\} \tag{10} $$
for every $ y \in \mathbb R_+ $.
In other words, $ \sigma \in \Sigma $ is $ v $-greedy if it optimally trades off current and future rewards when $ v $ is taken to be the value function.
In our setting, we have the following key result
- A feasible consumption policy is optimal if and only it is $ v^* $-greedy.
The intuition is similar to the intuition for the Bellman equation, which was provided after (9).
See, for example, theorem 10.1.11 of EDTC.
Hence, once we have a good approximation to $ v^* $, we can compute the (approximately) optimal policy by computing the corresponding greedy policy.
The advantage is that we are now solving a much lower dimensional optimization problem.
The Bellman Operator¶
How, then, should we compute the value function?
One way is to use the so-called Bellman operator.
(An operator is a map that sends functions into functions)
The Bellman operator is denoted by $ T $ and defined by
$$ Tv(y) := \max_{0 \leq c \leq y} \left\{ u(c) + \beta \int v(f(y - c) z) \phi(dz) \right\} \qquad (y \in \mathbb R_+) \tag{11} $$
In other words, $ T $ sends the function $ v $ into the new function $ Tv $ defined by (11).
By construction, the set of solutions to the Bellman equation (9) exactly coincides with the set of fixed points of $ T $.
For example, if $ Tv = v $, then, for any $ y \geq 0 $,$$ v(y) = Tv(y) = \max_{0 \leq c \leq y} \left\{ u(c) + \beta \int v^*(f(y - c) z) \phi(dz) \right\} $$
which says precisely that $ v $ is a solution to the Bellman equation.
It follows that $ v^* $ is a fixed point of $ T $.
Review of Theoretical Results¶
One can also show that $ T $ is a contraction mapping on the set of continuous bounded functions on $ \mathbb R_+ $ under the supremum distance$$ \rho(g, h) = \sup_{y \geq 0} |g(y) - h(y)| $$
See EDTC, lemma 10.1.18.
Hence it has exactly one fixed point in this set, which we know is equal to the value function.
It follows that
- The value function $ v^* $ is bounded and continuous.
- Starting from any bounded and continuous $ v $, the sequence $ v, Tv, T^2v, \ldots $ generated by iteratively applying $ T $ converges uniformly to $ v^* $.
This iterative method is called value function iteration.
We also know that a feasible policy is optimal if and only if it is $ v^* $-greedy.
It's not too hard to show that a $ v^* $-greedy policy exists (see EDTC, theorem 10.1.11 if you get stuck).
Hence at least one optimal policy exists.
Our problem now is how to compute it.
Unbounded Utility¶
The results stated above assume that the utility function is bounded.
In practice economists often work with unbounded utility functions — and so will we.
In the unbounded setting, various optimality theories exist.
Unfortunately, they tend to be case-specific, as opposed to valid for a large range of applications.
Nevertheless, their main conclusions are usually in line with those stated for the bounded case just above (as long as we drop the word "bounded").
Consult, for example, section 12.2 of EDTC, [Kam12] or [MdRV10].
Fitted Value Iteration¶
The first step is to compute the value function by value function iteration.
In theory, the algorithm is as follows
- Begin with a function $ v $ — an initial condition.
- Solving (11), obtain the function $ Tv $.
- Unless some stopping condition is satisfied, set $ v = Tv $ and go to step 2.
This generates the sequence $ v, Tv, T^2v, \ldots $.
However, there is a problem we must confront before we implement this procedure: The iterates can neither be calculated exactly nor stored on a computer.
To see the issue, consider (11).
Even if $ v $ is a known function, unless $ Tv $ can be shown to have some special structure, the only way to store it is to record the value $ Tv(y) $ for every $ y \in \mathbb R_+ $.
Clearly, this is impossible.
What we will do instead is use fitted value function iteration.
The procedure is to record the value of the function $ Tv $ at only finitely many "grid" points $ y_1 < y_2 < \cdots < y_I $ and reconstruct it from this information when required.
More precisely, the algorithm will be
- Begin with an array of values $ \{ v_1, \ldots, v_I \} $ representing the values of some initial function $ v $ on the grid points $ \{ y_1, \ldots, y_I \} $.
- Build a function $ \hat v $ on the state space $ \mathbb R_+ $ by interpolation or approximation, based on these data points.
- Obtain and record the value $ T \hat v(y_i) $ on each grid point $ y_i $ by repeatedly solving (11).
- Unless some stopping condition is satisfied, set $ \{ v_1, \ldots, v_I \} = \{ T \hat v(y_1), \ldots, T \hat v(y_I) \} $ and go to step 2.
How should we go about step 2?
This is a problem of function approximation, and there are many ways to approach it.
What's important here is that the function approximation scheme must not only produce a good approximation to $ Tv $, but also combine well with the broader iteration algorithm described above.
One good choice from both respects is continuous piecewise linear interpolation (see this paper for further discussion).
The next figure illustrates piecewise linear interpolation of an arbitrary function on grid points $ 0, 0.2, 0.4, 0.6, 0.8, 1 $
def f(x):
y1 = 2 * np.cos(6 * x) + np.sin(14 * x)
return y1 + 2.5
def Af(x):
return interp(c_grid, f(c_grid), x)
c_grid = np.linspace(0, 1, 6)
f_grid = np.linspace(0, 1, 150)
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(f_grid, f(f_grid), 'b-', label='true function')
ax.plot(f_grid, Af(f_grid), 'g-', label='linear approximation')
ax.vlines(c_grid, c_grid * 0, f(c_grid), linestyle='dashed', alpha=0.5)
ax.set(xlim=(0, 1), ylim=(0, 6))
Another advantage of piecewise linear interpolation is that it preserves useful shape properties such as monotonicity and concavity/convexity.
Optimal Growth Model¶
We will hold the primitives of the optimal growth model in a class.
The distribution $ \phi $ of the shock is assumed to be lognormal, and so a draw from $ \exp(\mu + \sigma \zeta) $ when $ \zeta $ is standard normal
class OptimalGrowthModel:
def __init__(self,
f, # Production function
u, # Utility function
β=0.96, # Discount factor
self.β, self.μ, self.s = β, μ, s
self.f, self.u = f, u
# Set up grid
self.grid = np.linspace(1e-5, grid_max, grid_size)
# Store shocks
self.shocks = np.exp(μ + s * np.random.randn(shock_size))
import numpy as np
from interpolation import interp
from numba import njit, prange
from quantecon.optimize.scalar_maximization import brent_max
def operator_factory(og, parallel_flag=True):
A function factory for building the Bellman operator, as well as
a function that computes greedy policies.
Here og is an instance of OptimalGrowthModel.
f, u, β = og.f, og.u, og.β
grid, shocks = og.grid, og.shocks
def objective(c, v, y):
The right-hand side of the Bellman equation
# First turn v into a function via interpolation
v_func = lambda x: interp(grid, v, x)
return u(c) + β * np.mean(v_func(f(y - c) * shocks))
def T(v):
The Bellman operator
v_new = np.empty_like(v)
for i in prange(len(grid)):
y = grid[i]
# Solve for optimal v at y
v_max = brent_max(objective, 1e-10, y, args=(v, y))[1]
v_new[i] = v_max
return v_new
def get_greedy(v):
Computes the v-greedy policy of a given function v
σ = np.empty_like(v)
for i in range(len(grid)):
y = grid[i]
# Solve for optimal c at y
c_max = brent_max(objective, 1e-10, y, args=(v, y))[0]
σ[i] = c_max
return σ
return T, get_greedy
The function
operator_factory takes a class that represents the growth model
and returns the operator
T and a function
get_greedy that we will use to solve the model.
Notice that the expectation in (11) is computed via Monte Carlo, using the approximation$$ \int v(f(y - c) z) \phi(dz) \approx \frac{1}{n} \sum_{i=1}^n v(f(y - c) \xi_i) $$
where $ \{\xi_i\}_{i=1}^n $ are IID draws from $ \phi $.
Monte Carlo is not always the most efficient way to compute integrals numerically but it does have some theoretical advantages in the present setting.
(For example, it preserves the contraction mapping property of the Bellman operator — see, e.g., [PalS13])
An Example¶
Let's test out our operator when
- $ f(k) = k^{\alpha} $
- $ u(c) = \ln c $
- $ \phi $ is the distribution of $ \exp(\mu + \sigma \zeta) $ when $ \zeta $ is standard normal
As is well-known (see [LS18], section 3.1.2), for this particular problem an exact analytical solution is available, with
$$ v^*(y) = \frac{\ln (1 - \alpha \beta) }{ 1 - \beta} + \frac{(\mu + \alpha \ln (\alpha \beta))}{1 - \alpha} \left[ \frac{1}{1- \beta} - \frac{1}{1 - \alpha \beta} \right] + \frac{1}{1 - \alpha \beta} \ln y \tag{12} $$
The optimal consumption policy is$$ \sigma^*(y) = (1 - \alpha \beta ) y $$
We will define functions to compute the closed-form solutions to check our answers
def σ_star(y, α, β):
True optimal policy
return (1 - α * β) * y
def v_star(y, α, β, μ):
True value function
c1 = np.log(1 - α * β) / (1 - β)
c2 = (μ + α * np.log(α * β)) / (1 - α)
c3 = 1 / (1 - β)
c4 = 1 / (1 - α * β)
return c1 + c2 * (c3 - c4) + c4 * np.log(y)
A First Test¶
To test our code, we want to see if we can replicate the analytical solution numerically, using fitted value function iteration.
First, having run the code for the general model shown above, let's generate an instance of the model and generate its Bellman operator.
We first need to define a jitted version of the production function
α = 0.4 # Production function parameter
def f(k):
Cobb-Douglas production function
return k**α
Now we will create an instance of the model and assign it to the variable
This instance will use the Cobb-Douglas production function and log utility
og = OptimalGrowthModel(f=f, u=np.log)
We will use
og to generate the Bellman operator and a function that computes
greedy policies
T, get_greedy = operator_factory(og)
Now let's do some tests.
As one preliminary test, let's see what happens when we apply our Bellman operator to the exact solution $ v^* $.
In theory, the resulting function should again be $ v^* $.
In practice, we expect some small numerical error
grid = og.grid
β, μ = og.β, og.μ
v_init = v_star(grid, α, β, μ) # Start at the solution
v = T(v_init) # Apply the Bellman operator once
fig, ax = plt.subplots(figsize=(9, 5))
ax.set_ylim(-35, -24)
ax.plot(grid, v, lw=2, alpha=0.6, label='$Tv^*$')
ax.plot(grid, v_init, lw=2, alpha=0.6, label='$v^*$')
The two functions are essentially indistinguishable, so we are off to a good start.
Now let's have a look at iterating with the Bellman operator, starting off from an arbitrary initial condition.
The initial condition we'll start with is $ v(y) = 5 \ln (y) $
v = 5 * np.log(grid) # An initial condition
n = 35
fig, ax = plt.subplots(figsize=(9, 6))
ax.plot(grid, v, color=plt.cm.jet(0),
lw=2, alpha=0.6, label='Initial condition')
for i in range(n):
v = T(v) # Apply the Bellman operator
ax.plot(grid, v, color=plt.cm.jet(i / n), lw=2, alpha=0.6)
ax.plot(grid, v_star(grid, α, β, μ), 'k-', lw=2,
alpha=0.8, label='True value function')
ax.set(ylim=(-40, 10), xlim=(np.min(grid), np.max(grid)))
The figure shows
- the first 36 functions generated by the fitted value function iteration algorithm, with hotter colors given to higher iterates
- the true value function $ v^* $ drawn in black
The sequence of iterates converges towards $ v^* $.
We are clearly getting closer.
We can write a function that iterates until the difference is below a particular tolerance level.
import numpy as np
def solve_model(og,
T, _ = operator_factory(og, parallel_flag=use_parallel)
# Set up loop
v = np.log(og.grid) # Initial condition
i = 0
error = tol + 1
while i < max_iter and error > tol:
v_new = T(v)
error = np.max(np.abs(v - v_new))
i += 1
if verbose and i % print_skip == 0:
print(f"Error at iteration {i} is {error}.")
v = v_new
if i == max_iter:
print("Failed to converge!")
if verbose and i < max_iter:
print(f"\nConverged in {i} iterations.")
return v_new
We can check our result by plotting it against the true value
v_solution = solve_model(og)
fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(grid, v_solution, lw=2, alpha=0.6,
label='Approximate value function')
ax.plot(grid, v_star(grid, α, β, μ), lw=2,
alpha=0.6, label='True value function')
ax.set_ylim(-35, -24)
Error at iteration 25 is 0.4145822817688227. Error at iteration 50 is 0.14940965445785537. Error at iteration 75 is 0.05384674893070951. Error at iteration 100 is 0.01940619152821199. Error at iteration 125 is 0.006993927713537573. Error at iteration 150 is 0.0025205885859058696. Error at iteration 175 is 0.0009084118509896655. Error at iteration 200 is 0.00032738864873138596. Error at iteration 225 is 0.00011798979415900135. Converged in 230 iterations.
The figure shows that we are pretty much on the money.
The Policy Function¶
To compute an approximate optimal policy, we will use the second function
returned from
operator_factory that backs out the optimal policy
from the solution to the Bellman equation.
The next figure compares the result to the exact solution, which, as mentioned above, is $ \sigma(y) = (1 - \alpha \beta) y $
fig, ax = plt.subplots(figsize=(9, 5))
ax.plot(grid, get_greedy(v_solution), lw=2,
alpha=0.6, label='Approximate policy function')
ax.plot(grid, σ_star(grid, α, β),
lw=2, alpha=0.6, label='True policy function')
The figure shows that we've done a good job in this instance of approximating the true policy.
Exercise 1¶
Once an optimal consumption policy $ \sigma $ is given, income follows (5).
The next figure shows a simulation of 100 elements of this sequence for three different discount factors (and hence three different policies)
In each sequence, the initial condition is $ y_0 = 0.1 $.
The discount factors are
discount_factors = (0.8, 0.9, 0.98).
We have also dialed down the shocks a bit with
s = 0.05.
Otherwise, the parameters and primitives are the same as the log-linear model discussed earlier in the lecture.
Notice that more patient agents typically have higher wealth.
Replicate the figure modulo randomness.
def simulate_og(σ_func, og, α, y0=0.1, ts_length=100):
Compute a time series given consumption policy σ.
y = np.empty(ts_length)
ξ = np.random.randn(ts_length-1)
y[0] = y0
for t in range(ts_length-1):
y[t+1] = (y[t] - σ_func(y[t]))**α * np.exp(og.μ + og.s * ξ[t])
return y
fig, ax = plt.subplots(figsize=(9, 6))
for β in (0.8, 0.9, 0.98):
og = OptimalGrowthModel(f, np.log, β=β, s=0.05)
grid = og.grid
v_solution = solve_model(og, verbose=False)
σ_star = get_greedy(v_solution)
# Define an optimal policy function
σ_func = lambda x: interp(grid, σ_star, x)
y = simulate_og(σ_func, og, α)
ax.plot(y, lw=2, alpha=0.6, label=rf'$\beta = {β}$')
ax.legend(loc='lower right')


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text:
Students who take Government 1061, "Modern Political Thought," this semester, won't get just one grade from Kenan Professor of Government Harvey C. Mansfield Jr. '53, they will get two.
In lecture Thursday morning, Mansfield announced he will distribute two sets of grades to his students: an initial grade he thinks they deserve, and then a second grade--the one that will go on their transcript--which will be "based on Harvard's system of inflated grades."
Mansfield is one of Harvard's most ardent opponents of grade inflation. He has voiced concern about grade elevation at Faculty meetings and even with University President Neil L. Rudenstine.
"[But] the administration has done nothing about this glaring flaw in education," Mansfield said in an interview Friday.
Mansfield said that when he compares the grades he assigns in his courses with grades students receive in other classes, his marks are usually a half-letter lower.
Mansfield's reputation as a tough grader has earned him the nickname "C-Minus Mansfield." And he said this semester's grading experiment will let him evaluate students rigorously and accurately, but not unfairly lower students' grade point averages.
"I don't want to continue punishing students for a situation that is not their fault," he said. "I've had to adjust my grades upward over the years, and the strain on my conscience has become too great."
Mansfield has asked the registrar
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text:
A former Houma police officer is suing the Houma police, alleging violation of his civil rights in regard to alleged unlawful employment practices.
Joseph Thompson Jr. of Houma filed a lawsuit Aug. 21 against Terrebonne Parish Consolidated Government, doing business as the a Houma Police Department, in U.S. District Court for the Eastern District of Louisiana, alleging noncompliance with civil rights law in his 2014 discharge.
According to the lawsuit, Thompson, who was employed by the defendant, was terminated along with eight other officers due to budgetary constraints on May 14, 2014. The suit says all the other officers were immediately rehired, but Thompson was denied re-employment in retaliation for having filed a previous unsuccessful complaint of racial discrimination against the defendant with the EEOC. The lawsuit states the defendant's actions were racially motivated because Thompson is black.
The suit states that the police department's actions were intentional and the plaintiff is entitled to back and front pay. He alleges embarrassment, loss of self-esteem, pain, suffering and humiliation.
Thompson seeks compensation of more than $75,000, fees, expenses and court costs. The plaintiff is self-represented in the case.
U.S. District Court for the Eastern District of Louisiana case number 2:15-cv-03691-HGB-SS.