Hip bursitis treatment, rehabilitation, exercises. Bursitis of the hip or trochanteric bursitis causes pain on the outside of the hip. Symptoms, causes, treatment and hip bursitis exercises explained.
Hip pain causes and treatment bootswebmd. Hip pain can be caused by wearandtear over the years, or by a fall or injury.
Hip bursitis treatment, rehabilitation, exercises. Bursitis of the hip or trochanteric bursitis causes pain on the outside of the hip. Symptoms, causes, treatment and hip bursitis exercises explained. Hip pain reasons, signs, prognosis, and treatment. Some common reasons of hip ache encompass bursitis, sciatica, it band syndrome, and arthritis. Read approximately associated signs and symptoms, and study prognosis. Hip ache when walking buzzle. Relying on the reason of the hip ache, diverse treatment alternatives are available. However, if hip ache at the same time as taking walks is chronic, it's miles advisable to seek advice from a health practitioner.
Hip ache causes and treatment bootswebmd. Hip pain can be as a result of wearandtear over time, or with the aid of a fall or injury. Hip joint injections for ache relief spinehealth. To address hip ache, a hip joint injection affords joint ache alleviation via a method that involves injecting medication into the joint positioned wherein the leg and. Groin & hip ache upon status & walking livestrong. · hip osteoarthritis. Hip osteoarthritis is the sporting or degeneration of the cartilage for your hip. Together with hip and groin ache, you can revel in joint. Hip ache whilst taking walks buzzle. Depending at the cause of the hip ache, various treatment options are to be had. However, if hip pain at the same time as on foot is chronic, it's miles advisable to consult a doctor. Osteoarthritis knee ache treatment with injections inside the united kingdom. We provide viscosupplementation, a groundbreaking remedy for osteoarthritis knee pain that gives durable alleviation. Hip damage hip joint injury hip fracture. The hip joint is susceptible to numerous sorts of accidents. Learn information about types of hip damage like hip fracture or broken hip, bursitis, dislocation, labral tear. Hip ache causes and remedy bootswebmd. Hip ache may be because of wearandtear over time, or through a fall or harm. Osteoarthritis hip ache remedy with injections within the uk. We offer viscosupplementation, a groundbreaking treatment for osteoarthritis hip pain that provides durable comfort.
Hip ache & harm physioworks. Name physioworks. Book online. Hip ache treatment options. Your hip physiotherapist may additionally take into account the usage of guide remedy to improve your joint mobility, flexibility or ache. Acupuncture treatment for hip bursitis. Ache or discomfort around the hip vicinity is a commonplace trouble among each genders and every age. Welcome to energetic bodily therapy health facility in clinton laurel. Recommendations from nearby energetic athletics advocated me visit energetic bodily therapy for lingering knee ache. Sacroiliac joint disorder si joint ache remedy. Read about sacroiliac joint (si) pain, or sacroiliac joint dysfunction, caused by osteoarthritis, being pregnant, and strange walking sample. The main symptom of si. Hip joint injections for ache remedy spinehealth. To deal with hip pain, a hip joint injection gives joint pain relief via a procedure that entails injecting remedy into the joint located wherein the leg and.
prognosis and treatment of low back ache annals of. Analysis and treatment of low returned ache a joint scientific practice tenet from the yank college of physicians and the yankee ache society. Hip, pelvic,lower lower back ache on left side with pain down left. I have pain in lower returned on left facet, left hip ache, left pelvic pain and pain down the left leg at times. It comes and is going. Receives actual terrible,(after I put weight on. Hip ache reasons, signs, remedy what tests do. What tests do healthcare experts use to diagnose hip ache? Hip pain & damage. Call physioworks. E-book online. Hip pain remedy options. Your hip physiotherapist may additionally consider using manual therapy to improve your joint mobility, flexibility or pain. Diagnosis and treatment of low returned ache annals of internal. Prognosis and remedy of low returned ache a joint scientific practice guiding principle from the american college of physicians and the yank pain society. Ankle surgery ache treatment, fusion or joint. Find out about shoulder surgical procedure the usage of arthroscopic remedy for ache and joint remedy through orthopedic surgery.
Ankle surgery ache treatment, fusion or joint substitute. Study shoulder surgery using arthroscopic treatment for ache and joint remedy thru orthopedic surgical operation. Acupuncture remedy for hip bursitis. Pain or pain around the hip area is a not unusual problem amongst both genders and all ages. Groin & hip ache upon status & taking walks livestrong. Aug thirteen, 2017 hip osteoarthritis. Hip osteoarthritis is the carrying or degeneration of the cartilage on your hip. Along side hip and groin ache, you can experience joint. returned ache sciatica hip pain natural treatment & returned. Natural and effective ways to cope with low returned pain, hip ache, and sciatica, as well as what is known as piriformis syndrome and sciatic nerve troubles. Hip ache reasons, symptoms, prognosis, and remedy. A few common reasons of hip ache consist of bursitis, sciatica, it band syndrome, and arthritis. Examine approximately related symptoms and signs and symptoms, and study diagnosis. Treatment alternatives for degenerative joint sickness of. 286 fig. 14.5 ere hip v sedjd with subchondral cysts, osteophytes, subchondral sclerosis, and joint area loss in a position t14.1 nonoperatie control v. Hip ache reasons, signs and symptoms, remedy emedicinehealth. What checks do healthcare specialists use to diagnose hip pain? Hip bursitis remedy, rehabilitation, exercises. Bursitis of the hip or trochanteric bursitis reasons pain at the outdoor of the hip. Signs and symptoms, reasons, remedy and hip bursitis exercises explained.
Si joint disorder treatment continual si joint ache. Find out greater approximately continual si joint ache and one-of-a-kind treatment strategies available from the professionals at virginia spine institute! Contact us nowadays. Welcome to active physical therapy health facility in clinton. Suggestions from local lively athletics recommended me visit lively physical remedy for lingering knee ache. Treatment alternatives for degenerative joint disease of the hip. 286 fig. 14.Five ere hip v sedjd with subchondral cysts, osteophytes, subchondral sclerosis, and joint space loss in a position t14.1 nonoperatie control v. Hip injury hip joint damage hip fracture ache relief advice. The hip joint is liable to several forms of injuries. Research information about varieties of hip harm like hip fracture or damaged hip, bursitis, dislocation, labral tear. Sacroiliac joint dysfunction si joint pain remedy. Read approximately sacroiliac joint (si) ache, or sacroiliac joint dysfunction, as a result of osteoarthritis, being pregnant, and odd walking sample. The main symptom of si. Avascular necrosis of femur head causing hip joint ache. Concise evaluate of avascular necrosis (avn) of hip, such as definition, diagnosis and treatment. Osteoarthritis hip pain remedy with injections inside the uk. "i write to offer my honest thank you for the treatment you encouraged on my left hip. I'm thrilled to mention i now haven't any ache and am on foot typically again without. returned ache sciatica hip pain natural treatment & back. Herbal and effective ways to deal with low back pain, hip pain, and sciatica, in addition to what's called piriformis syndrome and sciatic nerve problems.


Everyone wants pearly white teeth – but what if you notice that your teeth have white spots? Those irregular milky opaque white patches that can be both aesthetically troublesome could signify dental problems.
Most people don't pay too much attention to white spots, because unlike the brown spots; they are less noticeable and unattractive. However, like any other tooth discolouration, white spots can be a result of some oral problems that can warrant treatment.
A number of reasons can cause white spots. Too much fluoride, known as fluorosis can cause decay and white discolouration. Similar to Fluorosis, Enamel Hypoplasia is seen as spots varying from milky white to yellowish or brown. Enamel Hypoplasia happens due to nutritional deficiencies, medicinal side effects, and high fever. When plaque accumulates on your teeth and causes demineralization, this could lead to permanent white areas of decalcification. A diet high in acid and sugar as well as sleeping with your mouth open can also lead to white spot formation on your teeth.
There are different treatment options available for white spots, depending on the severity of your condition. Your dentist will usually suggest microabrasion, bleaching, the use of dental veneers, or the use of composite resin to help treat white spots on your teeth.
Don't let white spots interfere with your smile! If you notice some white milky discolouration on your teeth, contact or visit us at Southwood Dental Studio in London. We will gladly assist you with your dental concerns!


Posted: 2014-11-21 06:50:00	by Metro Networks, Inc.
(Portland, OR) -- Multnomah County residents are being warned of a phone scam in which the caller claims to be with the sheriff's office. Deputies say the caller demands payment for an outstanding arrest warrant and tells the intended victim to purchase a Green Dot, MoneyPak, or Vanilla Network reloadable card to make the payment. Authorities say none of those who reported the scam lost any money. The Multnomah County Sheriff's Office says it would never call anyone to ask for a payment over the phone for warrants or traffic fines.


KIA: 2019 Super Bowl TV Commercial – extended Millions of dollars will be spent on celebrity endorsements for Super Bowl LIII said KIA.
KIA unveiled its 2019 Super Bowl TV Commercial, an advert created by the advertising agency David&Goliath. Named "Give It Everything," a :30 teaser ad aired during Sunday's NFC Championship game.
The message is: "Millions of dollars will be spent on celebrity endorsements for Super Bowl LIII. But what if that money could be spent differently?


Your aura is the non-physical part of your being, depicted visually as a halo of colours around your physical body. For some pictures of auras click HERE.
If you are uncomfortable with the idea of the aura being real, think of the aura as a visual metaphor for the mind-body. For example, a blue aura could be a visual metaphor for a very calm and peaceful mind-body state, whereas a red aura could be a visual metaphor for a highly energised and passionate mind-body state.
Aura colours are created by chakras. Chakras are energy centres in the aura. Each chakra generates a specific colour, as per the picture to the left. To find out more about different chakras and their colours, click HERE.
Aura colours found above the head reflect higher guidance from your spirit guides, angels, deceased loved ones, higher self and so on. Colours in this area can also reflect your deeper, more spiritual values, as well as the personality and history of your soul.
Aura colour found around the head area represent your perspectives, attitudes, thinking style and 'what is on your mind', especially your current or habitual thoughts. You could also think of this area as representing your eyes and how they see the world around you, almost as though each aura colour is colouring your view of reality.
Aura colours found in your throat area reflect your general communication style, recent or habitual topics of conversation and specific communication challenges you might be facing. To some extent, colours around the throat chakra give you an insight into how you come across to other people or how other people see you.
Aura colours around the arms represent emotions, relationships and how we relate to others, because the arms are an extension of the heart chakra. These colours can reflect specific relationships, recurring themes or patterns in your relationships and habitual emotional states. Aura colours around the arms sometimes belong to people we are close to, especially when we are getting over involved with them.
Aura colours found around the hands represent your work, career, role and self-identity. Finances, study, hobbies and major projects also feature quite strongly in this area. Colours found around the hands can be influenced by what you do with your time and how you feel about your job, place of employment, employer and work mates.
Aura colours found around the hips provide insight into your deeper emotions, wounds and personality traits, as shaped by your childhood and the past. This position reflects your private emotional world, unexpressed emotions and potential talents that lay unrealised within you. Colours in this area draw your attention to what requires healing, releasing, uncovering, resolving and expressing in order to move forward.
Rather than representing personality traits, aura colours in the leg area reflect temporary circumstances and future possibilities. Colours in this area predict the most likely outcomes based on current circumstances, but remember that predictions can sometimes be difficult to interpret correctly and the future is always changing.


text:
import * as React from 'react';
import { styled } from '@material-ui/core/styles';
import Box from '@material-ui/core/Box';
import List from '@material-ui/core/List';
import ListItem from '@material-ui/core/ListItem';
import ListItemAvatar from '@material-ui/core/ListItemAvatar';
import ListItemIcon from '@material-ui/core/ListItemIcon';
import ListItemText from '@material-ui/core/ListItemText';
import Avatar from '@material-ui/core/Avatar';
import IconButton from '@material-ui/core/IconButton';
import FormGroup from '@material-ui/core/FormGroup';
import FormControlLabel from '@material-ui/core/FormControlLabel';
import Checkbox from '@material-ui/core/Checkbox';
import Grid from '@material-ui/core/Grid';
import Typography from '@material-ui/core/Typography';
import FolderIcon from '@material-ui/icons/Folder';
import DeleteIcon from '@material-ui/icons/Delete';

function generate(element: React.ReactElement) {
  return [0, 1, 2].map((value) =>
    React.cloneElement(element, {
      key: value,
    }),
  );
}

const Demo = styled('div')(({ theme }) => ({
  backgroundColor: theme.palette.background.paper,
}));

export default function InteractiveList() {
  const [dense, setDense] = React.useState(false);
  const [secondary, setSecondary] = React.useState(false);

  return (
    <Box sx={{ flexGrow: 1, maxWidth: 752 }}>
      <FormGroup row>
        <FormControlLabel
          control={
            <Checkbox
              checked={dense}
              onChange={(event) => setDense(event.target.checked)}
            />
          }
          label="Enable dense"
        />
        <FormControlLabel
          control={
            <Checkbox
              checked={secondary}
              onChange={(event) => setSecondary(event.target.checked)}
            />
          }
          label="Enable secondary text"
        />
      </FormGroup>
      <Grid container spacing={2}>
        <Grid item xs={12} md={6}>
          <Typography sx={{ mt: 4, mb: 2 }} variant="h6" component="div">
            Text only
          </Typography>
          <Demo>
            <List dense={dense}>
              {generate(
                <ListItem>
                  <ListItemText
                    primary="Single-line item"
                    secondary={secondary ? 'Secondary text' : null}
                  />
                </ListItem>,
              )}
            </List>
          </Demo>
        </Grid>
        <Grid item xs={12} md={6}>
          <Typography sx={{ mt: 4, mb: 2 }} variant="h6" component="div">
            Icon with text
          </Typography>
          <Demo>
            <List dense={dense}>
              {generate(
                <ListItem>
                  <ListItemIcon>
                    <FolderIcon />
                  </ListItemIcon>
                  <ListItemText
                    primary="Single-line item"
                    secondary={secondary ? 'Secondary text' : null}
                  />
                </ListItem>,
              )}
            </List>
          </Demo>
        </Grid>
      </Grid>
      <Grid container spacing={2}>
        <Grid item xs={12} md={6}>
          <Typography sx={{ mt: 4, mb: 2 }} variant="h6" component="div">
            Avatar with text
          </Typography>
          <Demo>
            <List dense={dense}>
              {generate(
                <ListItem>
                  <ListItemAvatar>
                    <Avatar>
                      <FolderIcon />
                    </Avatar>
                  </ListItemAvatar>
                  <ListItemText
                    primary="Single-line item"
                    secondary={secondary ? 'Secondary text' : null}
                  />
                </ListItem>,
              )}
            </List>
          </Demo>
        </Grid>
        <Grid item xs={12} md={6}>
          <Typography sx={{ mt: 4, mb: 2 }} variant="h6" component="div">
            Avatar with text and icon
          </Typography>
          <Demo>
            <List dense={dense}>
              {generate(
                <ListItem
                  secondaryAction={
                    <IconButton edge="end" aria-label="delete">
                      <DeleteIcon />
                    </IconButton>
                  }
                >
                  <ListItemAvatar>
                    <Avatar>
                      <FolderIcon />
                    </Avatar>
                  </ListItemAvatar>
                  <ListItemText
                    primary="Single-line item"
                    secondary={secondary ? 'Secondary text' : null}
                  />
                </ListItem>,
              )}
            </List>
          </Demo>
        </Grid>
      </Grid>
    </Box>
  );
}

redpajama_set_name: RedPajamaGithub


text:
1968 The Puerta 12 Tragedy
RIVER VS. BOCA '68 – THE TRAGEDY OF LA PUERTA 12
footballpink.net
The Superclasico between River Plate and Boca Juniors is a fixture in the world football calendar that's steeped in history, one that halts all activity in Buenos Aires but also captures the imagination of football fans around the globe. It has witnessed some of the game's greats take part on either side: Alfredo Di Stefano, Diego Maradona, Enzo Francescoli, Juan Roman Riquelme and Gabriel Batistuta to name just a handful and is credited with being one of the most incredible spectacles in sport.
23 June 1968 – The Puerta 12 Tragedy
tdifh.blogspot.com
On 23 June 1968, 74 people died and over 150 were injured trying to exit the stadium after a match between River Plate and Boca Juniors. It remains the largest football-related disaster in Argentina's history.
The tragedy of Gate 12
hastaelgolsiempre.com
Monday marked the fourtieth anniversary of a dark day in the history of football in Argentina and, indeed, across South America. On the 23rd June 1968, as Boca Juniors fans left the Estadio Monumental at the end of a mid-season superclásico, a gate was left less than fully opened under circumstances that remain mysterious to this day, and the result was 71 deaths: the highest death toll of any of Argentina's stadium disasters
INVESTIGATION: 32 YEARS AFTER THE GREATEST TRAGEDY LIVED BY ARGENTINE FOOTBALL
Gate 12: Memories of Horror
clarin.com
It was June 23, 1968. A River-Boca had finished. At Gate 12 of the Monumental there was an avalanche. 71 fans died, most of them minors. Justice never found guilty. Clarín reconstructed that painful afternoon with new testimonies.

redpajama_set_name: RedPajamaCommonCrawl


3.1 What is a Heat Pump?
3.3 Now, what are Furnaces?
With the exception of a couple of alternatives, there are generally two kinds of heating systems from which to choose: Furnaces or heat pumps.
No matter whether you reside in a mild or a frigid clime, deciding on the type of heating system for your home can nonetheless be challenging.
To make it a bit smoother, this article will explore the pros and cons of both and, ideally, resolve the heat pump vs furnace dilemma.
Arguably, the first step in determining the answer to this question is by considering where in the country you live.
For example, heat pumps are best designed for regions in which mild winters are the norm.
They are well suited for coastal areas, particularly in the Gulf States.
On the other hand, furnaces enable homeowners to endure even the most challenging of winter weather.
Again, heat pumps are essentially air conditioners that work in reverse to provide warmth for the home.
Unlike a furnace that creates heat at the source via a flame, a heat pump moves heat from one location to another.
Simply put, during the warmer months, a heat pump will move heat from your home to the outside. In the colder months, the process is reversed and heat is transferred from the outdoors to your home's interior.
Heat pumps are beneficial in that they function efficiently year-round, which means that a single outlay of one's hard-earned funds takes care of all a home's interior environmental comfort needs while assisting in the lowering of both heating and cooling bills.
They can also satisfy one's desire to be "green" by putting out up to three times more heat than the energy consumed.
As also previously mentioned, there is the issue with heat pumps of sometimes needing a backup heating source.
When one experiences extra cold nights, an additional heat source (such as a small furnace) may be needed.
Arguably, the most common kind of home heating is that which is provided by natural gas furnaces, creating heat via the burning of the gas.
The heat produced from forced air furnaces is then pushed out through registers situated throughout the home.
The good news here is that a furnace is less costly than a heat pump, though one will still need to fork out the money for an air conditioner separately so as to enjoy interior cooling when the weather gets hot.
However, a furnace will always work, even on the chilliest of nights with no backup heat source being required.
And, in this day and age, even furnaces have attained high levels of energy efficiency.
Keep in mind that buying the latest in furnace technology can save a homeowner serious money over older models.
At the end of the day, the decision of either a furnace or a heat pump boils down to the climate in which you reside, how expensive it is to use electricity and gas and the kind of heating method that you fancy.
Prior to getting into the heat pump vs furnace debate directly, let's first define what they each are and how they operate.
Essentially, a heat pump is like a reverse air conditioner that, as opposed to cooling, rather heats your home.
In the heat pump's case, it utilizes the same methodology to increase the temperature (rather than decreasing it). In other words, a heat pump is an air conditioner, but an air conditioner is, obviously, not always a heat pump, despite the fact that mechanically they are the same.
Heat pumps actually move in air from the outside, rather than boosting a home's temperature utilizing a flame, like a furnace does.
There's an outdoor compressor, like all typical central A/C systems, that is comprised of copper tubing along with aluminum fins that work together in drawing in the air and compressing it.
Next a chemical agent works to evaporate the compressed and now heated air and transforms it into a gas; it then moves it to a coil inside the home where it is condensed back into liquid form and distributed throughout the home.
Heat pumps are best used in more mild climates where one can be afforded the use of electricity at a lower cost- this can make a heat pump a viable alternative to a fuel-fired furnace.
Where temperatures regularly fall under 40 degrees (Fahrenheit), compressing heat from the exterior air, as heat pumps do, can be problematic.
Keep in mind, though, that there is an optional component that works in conjunction with either the electric or gas resistance coils that can help make the heat pump function better.
Simply put, it's a miniature furnace that boosts the heat pump's efficiency.
A furnace forces air into the home that is heated by a flame, normally (but not always) using a fan to push warmth through ducting and out a litany of vents that are strategically placed throughout the dwelling.
It is wise to consider replacing older model furnaces, regardless of the heat pump vs furnace debate, as they can be grossly inefficient.
Keep in mind that newer models can be vastly more environmentally and economically friendly, boasting an eye-popping efficiency of up to 98%.
This kind of home heating method depends on a fluid medium, normally air but sometimes steam or hot water, to distribute the warmth.
There is a myriad of types of furnaces, such as gas, oil, and electric, but they are distinguished from heat pumps in that they burn a fuel (or use electricity) to directly heat the air inside the home.
The "Annual Fuel Utilization Efficiency" (AFUE) rating is used to determine the efficiency of furnaces and heat pumps.
All heaters, whether gas or oil fueled, are measured using AFUE. It's an objective method of comparison and is measured as a percentage.
The higher the AFUE percentage, the more efficient is the furnace or heat pump.
SEER is an acronym meaning "Seasonal Energy Efficiency Ratio" and is the way in which to measure the efficiency of air conditioners and heat pumps throughout the year.
(The higher the SEER value, the more efficient the appliance.) It's not a perfect way to measure efficiency in the real world because it is determined in the laboratory where the environment is controlled.
Understand that, like the old EPA gas mileage standard, the efficiency claims can be overblown.
The scale is ranked currently between SEER 7 and SEER 21 to 25 (depending on the highest SEER rated air conditioner available on the market at any given time).
The heat pump vs. furnace dilemma can be tricky for those in the market for a good home heating system.
Now, armed with knowledge, you can make an informed decision.
Basically, if you brave frigid and snow-filled winters, perhaps decide upon a furnace.
If you're living in a mild climate, like Southern California or Florida, then a heat pump might be the way to go.
Now, it's in your hands. Good Luck!


The Florida Marlins are owners of baseball's best record after playing a dozen games (11-1).
And for the first time in 11 years, the Marlins swept a series in which they were trailing in all three games heading into their last at bat.
The last team to win three consecutive games after trailing heading into their last at bat were the 1998 Detroit Tigers.
The Marlins won the World Series in 1997. The Marlins won the World Series in 2003. If they continue their six-year trend, they are due to win another World Series in ... 2009.
With only 150 games to go, there's still a chance that the Mets, Braves and Phillies will pass the red-hot Marlins. But, then again, could they be this year's Rays?


text:
Lawyers For Strauss-Kahn Ask Judge To Drop Civil Suit
Filed Under:Dominique Strauss-Kahn, International Monetary Fund, Nafissatou Diallo, Sofitel Hotel
NEW YORK, NY - JULY 01: Former International Monetary Fund leader Dominique Strauss-Kahn (R) and his wife Anne Sinclair leave New York State Supreme Court for a hearing on July 1, 2011 in New York City. Strauss-Kahn, arrested on May 14 on sexual assault charges, is back in court for a bail hearing amid new reports surrounding the accuser's credibility. (Photo by Mario Tama/Getty Images) Former International Monetary Fund leader Dominique Strauss-Kahn (R) and his wife Anne Sinclair. (credit: Mario Tama/Getty Images)
NEW YORK (CBSNewYork) — Former International Monetary Fund chief Dominique Strauss-Kahn has asked a New York judge to dismiss the civil lawsuit filed by the maid who accused him of sexual assault in May.
Lawyers for Strauss-Kahn filed the motion in the Bronx late Monday. They argued Strauss-Kahn is immune from a civil suit under international law.
In August, a Manhattan judge dropped sexual assault charges against Strauss-Kahn amid concerns about his accuser's credibility.
Former Sofitel hotel housekeeper Nafissatou Diallo then filed a civil suit in the Bronx, where she lives.
Strauss-Kahn returned to France last month after the charges were dropped. He recently called his encounter with Diallo a "moral failing."
"What happened does not include violence, no force, no aggression, nor any unlawful act. It is the prosecutor who says this, not me. What happened is not only an inappropriate relationship but, more than that, an error," Strauss-Kahn told a French television station.
Do you think the civil suit against DSK will be dropped? Tell us in the comments section below…


In doing so, we concentrate on the current trends in animal nutrition, and we listen to the concerns and desires of the consumer. For example, the Nonstructural Carbohydrates (NSC) in each formula have been adjusted to optimum levels to maintain good health in all classes of horses.
We are pleased to introduce Lone Star Infinity Pelleted Equine Rations.
Infinity provides four newly developed formulas to choose from that feature Chelated Minerals, Sel-Plex® Organic Selenium, Zinpro® zinc methionine, Probiotics and Omega Fatty Acids, and Infinity rations are formulated with controlled levels of nonstructural carbohydrates (NSC). Plus, TFP Nutrition has over 88 years of experience producing feeds for the Texas and Louisiana area so your customers can be assured our horse feeds will be right for their conditions.
Check out the new Infinity formulas in the Horse Feeds section of the website.


An elegantly subtle woven pattern. Our easy care, 100% premium cotton shirt.
Ultra fine 100-count double-twist yarns create an exceptionally smooth, premium texture.
The slightly raised pattern created by the dobby weave gives it a solid look that's easy to coordinate.
Pocket designed for both style and practicality, sized just right to fit an ID card or passport.


I have listed a new property at 207 Academy RD in WINNIPEG.
Beautiful unit with loads of character throughout just steps away from Wellington Crescent! This stunning condo is perfect for a single person, a couple or revenue property (see Realtor remarks for more information). Owner is willing to sell fully furnished. Unit features bright open spaces with newer windows thought. Newer Eat-in kitchen with dormer window. Newer bathroom. spacious open concept living room with gleaming original hardwood floors and bright dormer window, perfect for entertaining. Newer carpeting. Close to all amenities. With shopping, bus routes and the beautiful jogging paths of Wellington Crescent only steps away! Shared laundry in basement with newer washer and dryer. Don't miss your opportunity to own this one of a kind condo in desirable River Height.


Like most everyone, I know how important pollinators are to the food chain. What I didn't realize, until I read this book, was how artisanal honeys, like fine wines or single source coffees, have distinctive flavors rooted in terroir.
To find a spot on my shelf these days a cookbook needs to be inspirational, have a strong point of view, and help me explore new culinary cultures. The Fresh Honey Cookbook, 84 Receipes from a Beekeeper's Kitchen by Laurey Masterton (Storey Publishing, North Adams, MA, 2013) meets all of my criteria. The book is at once a manifesto on the importance of bees, a resource for hobby beekeeping, an introduction to honey varietals, and a collection of tasty, yet straightforward, seasonal recipes centered on honey. Masterton manages to pack all of this into 206 well-designed pages of clear prose and beautiful photography by (Asheville, NC based) Johnny Autry.
Laurey Masterton writes in her introduction that she hadn't really thought much of bees until she was asked to prepare a meal for schoolchildren who were learning about the importance of honeybees. Her interest was piqued and, when she learned that honeybees are responsible for every third bite we eat, she decided to become a beekeeper.
As a chef, Masterton's beekeeping opened up a whole new world of food connoisseurship. In a beautifully produced chapter early in the book she unpacks the world of honey through sight, smell, flavor, and species. It turns out that the flavor profiles of honey produced by various plant species are as individual as wine is to grape varietals.
Masterson's book follows the calendar. Each month includes a menu and featured honey varietal. Along the way we learn about the rhythms of beekeeping and honey trivia.
For example, the October chapter features eucalyptus honey, which Masterton says is on the dark side and is perfect for the chillier days of autumn. There's also fascinating information in this chapter about beeswax: Bees eat their own honey to make beeswax and it takes 6-8 pounds of honey to make 1 pound of wax.
Other months each feature another honey varietal: January, orange blossom; February, tupelo; March, acacia; April, avocado; May, raspberry; June, tulip poplar; July, sourwood; August, blueberry; September, sage; November, cranberry; and December, chestnut.
But of course, a cookbook is judged on its menus and recipes. And I tried several.
Though Masterton does give a helpful list of sources for her honey varietals, I decided to see what I could find locally. Most stores offer a generic "wildflower" honey, and I already have a number of wildflower honeys from local beekeepers, which are organized by season, but not by specific flower.
After visiting several grocery stores, I was finally able to find three distinctive honeys: cranberry, orange blossom and acacia — all at the Honest Weight Food Co-op in Albany.
I wasn't sure that the honey varietal would make that much difference in these recipes until I worked with cranberry honey, which is the chosen honey for the November chapter. Masterton describes the taste as warm with cooked fruit hints and no aftertaste. It did have a delicious, mild yet distinct fruity taste. It made a fantastic dressing for the cookbook's recipe for California endive salad with pomegranate seeds and shaved parmesan.
Because the November menu was so focused on Thanksgiving-type food (think cranberries), I also substituted the cranberry honey for orange blossom honey in a marinade recipe for Meyer lemon and honey marinated chicken skewers included in the January menu. I don't know whether Masterton's preference for using orange blossom honey as suggested in her recipe would have varied the flavor substantially, but I do know that all the recipes I tried were quick, easy, and delicious!
But interestingly, I saved the leftover salad dressing and used it in one of my favorite kale salad recipes the next day with disappointing results. The kale salad needed a brighter, lighter, less fruity honey than the cranberry honey. From that accidental experiment, I realized that these recipes, though simple to prepare, feature carefully balanced flavors that connect with the distinctive qualities of the honey varietals.
I'm looking forward to adding more honey to my shelf and working through more of the recipes.


text:
#ifndef PHALCON_TRANSLATE_ADAPTER_GETTEXT_H
#define PHALCON_TRANSLATE_ADAPTER_GETTEXT_H

#include "php_phalcon.h"

extern zend_class_entry *phalcon_translate_adapter_gettext_ce;

PHALCON_INIT_CLASS(Phalcon_Translate_Adapter_Gettext);

#endif /* PHALCON_TRANSLATE_ADAPTER_GETTEXT_H */


text:
Felsa
1 Felsa
3 Notable Movements
Felsa was a Swiss watch movement manufacturer founded in Grenchen in 1918 and becoming part of Ebauches SA in 1928.
Felsa was a relative newcomer to Grenchen, established in 1918 close to the huge A. Schild factory. The company joined A. Schild, Adolphe Michel, FHF, and others under the Ebauches SA banner in 1928. Felsa became an innovator in the 1930s with designer Friedrich Meyer, who patented an automatic winding mechanism in 1942. A bi-directional winding mechanism was patented in 1947 by Meyer. This was used in the well-known Bidynator movement which was used by Breitling, among other brands. Because of Felsa's successful bi-directional automatic winding system, they were allowed to specialize in automatic movements by the cartel of Swiss movement makers, Ebauches SA, along with ETA and A. Schild.
Notable Movements
Bydantor
Felsa 690
Retrieved from "https://chronopedia.club/index.php?title=Felsa&oldid=2240"


text:
\section{Introduction}

\noindent
Let $G$ be a connected graph with vertex set $V(G)$. The {\em eccentricity} $e(v)$ of a vertex is the maximum distance between $v$ and a vertex in $G$.
The {\em average eccentricity} ${\rm avec}(G)$ of $G$ is the average of the eccentricities
of the vertices of $G$, i.e., ${\rm avec}(G) = \frac{1}{|V(G)|}\sum_{u\in V(G)} e(u)$. The
average eccentricity was first introduced by Buckley and Harary \cite{BucHar} under the
name eccentric mean. Its systematic study was initiated by Dankelmann, Goddard and Swart
\cite{DanGodSwa2004},  who established an upper bound on the average eccentricity  in terms
of order and minimum degree, and further showed among other results, that the path
maximizes the average eccentricity among all connected graphs of given order.
\begin{proposition}[\cite{DanGodSwa2004}]\label{prop1}
Let $G$ be a connected graph of order $n$. Then
\begin{equation*}
	{\rm avec(G)} \leq {\rm avec}(G) \leq \frac{1}{n} \left\lfloor \frac{3n^2}{4} -\frac{n}{2} \right\rfloor,
\end{equation*}
and this bound is sharp.
\end{proposition}
We first recall some upper bounds on the average eccentricity of a connected graph
involving the minimum degree. Dankelmann, Goddard and Swart \cite{DanGodSwa2004}
showed that if $G$ is a graph of order $n$ and minimum degree $\delta \geq 2$, then
\begin{equation}\label{equ-DGS}
{\rm avec}(G) \leq \frac{9n}{4(\delta +1)} + \frac{15}{4},
\end{equation}
and this bound is sharp apart from a small additive constant. By applying a similar technique, it was recently proved in \cite{DanMukOsaRod} that if $G$ is $K_3$-free, then the bound \eqref{equ-DGS} can be improved to
\begin{equation}\label{eqn-DMOR-K3}
{\rm avec}(G) \leq 3\left\lceil \frac{n}{2\delta} \right\rceil +5,
\end{equation}
and for $C_4$-free graphs to
\begin{equation}\label{eqn-DMOR-C4}
	{\rm avec}(G) \leq \frac{15}{4} \left\lceil \frac{n}{\delta^2 -2\lfloor \frac{\delta}{2} \rfloor +1} \right\rceil +\frac{11}{2}.
\end{equation}
Moreover, they showed that the bound in \eqref{eqn-DMOR-K3} is sharp apart from a small
additive constant and that, for $\delta +1$ a prime power, there exists an infinite number
of $C_4$-free graphs of minimum degree at least $\delta$ with
\begin{equation*}
	{\rm avec}(G) \geq \frac{15}{4} \frac{n}{\delta^2 +3\delta +2} -\frac{5}{2}.
\end{equation*}
Other recent results on the average eccentricity of graphs can be found, for example, in
 (\cite{AliDanMorMukSwaVet2018}, \cite{DanMuk2014}, \cite{DanOsa-manu}, \cite{DanMukOsaRod},
 \cite{DarAliKlaDas2018},
  \cite{DuIli2013}, \cite{DuIli2016}, \cite{Ili2012}, \cite{SmiSzeWan2016}, \cite{TanZho2012}).

In the bounds \eqref{equ-DGS}, \eqref{eqn-DMOR-K3} and \eqref{eqn-DMOR-C4}, the examples that
show that these bounds are sharp or close to being sharp, all have vertex degrees close to the
minimum degree. That suggests that these bounds can be improved for graphs containing a vertex
of large degree. In this paper, we show that this is indeed the case by proving asymptotically
sharp upper bounds on the average eccentricity of a graph of given order, minimum degree and
maximum degree. We also give corresponding bounds for triangle-free graphs and $C_4$-free
graphs. Our bounds improve on the inequalities \eqref{equ-DGS}, \eqref{eqn-DMOR-K3}
and \eqref{eqn-DMOR-C4}. Moreover, we construct graphs to show that our bounds for connected
graphs and triangle-free graphs are sharp apart from a small additive constant, and that for
$C_4$-free graphs our bound is close to being best possible.

\section{Definitions, notation and preliminaries}

\noindent
We use the following notation. Let $G$ be a connected graph of order $n$ with vertex set
$V(G)$, edge set $E(G)$. The \textit{distance} between two vertices $u$ and $v$, denoted
by $d_G(u,v)$ is the length of a shortest $(u,v)$-path in $G$.
The diameter ${\rm diam}(G)$ and radius ${\rm rad}(G)$ are
the largest and smallest, respectively, of all eccentricities of vertices of $G$. The {\em
total eccentricity} $EX(G)$ is the sum of the eccentricities of the vertices of $G$.
If $x$ is a vertex of $G$, $e$ an edge of $G$, and $A$ a subset of $V(G)$, then
$d_G(x, A)$ is the minimum of the distances between $x$ and a vertex in $A$, and
$d_G(e,A)$ is the minimum of the distances between a vertex incident with $e$ and the
vertices in $A$.

The neighbourhood of a vertex $v$, denoted by $N_G(v)$, is the set of vertices adjacent to $v$. The closed neighbourhood $N_G[v]$ is defined as $N_G(v) \cup \{v\}$. For
$k\in \mathbb{N}$, the $kth$ neighbourhood of a given subset $A\subset V$, denoted by
$N_G^k(A)$, is the set of all vertices $x$ of $G$ with $d_G(x,a) \leq k$ for some $a\in A$.
If $k=1$, then we simply write $N_G[A]$. If
no confusion can occur, we drop the argument or subscript $G$. The degree
${\rm deg}(v)$ of a vertex $v$ is defined  as $|N_G(v)|$. We denote the minimum and
maximum degree of $G$ by $\delta(G)$ and
$\Delta(G)$, respectively. For
$k\in \mathbb{N}$, the $k^{th}$ power $G^k$ of $G$ is the graph with the same vertex set of $G$, in which two distinct vertices $u$ and $v$ are adjacent if $d(u,v)\leq k$. If $A\subset V$, then $G^k[A]$ is the subgraph of $G^k$ induced by $A$. If $H$ is a subgraph of $G$, we write $H\leq G$.

The line graph $L(G)$ of $G$ is the graph whose vertices are the edges of $G$ such that two vertices of $L(G)$ are adjacent if they share a vertex as edges of $G$. Let $M\subseteq E(G)$, then $V(M)$ is the set of vertices incident with at least an edge of $M$.  For $k\in \mathbb{N}$, a $k$-packing of $G$ is a subset $A\subseteq V$ with $d_G(u,v) >k$ for all $u,v \in A$. The sequential sum $G_1 +G_2 +\ldots +G_k$ is the graph obtained from the disjoint union of the graphs $G_1$, $G_2, \ldots, G_k$ by joining every vertex of $G_i$ to every vertex of $G_{i+1}$ for $i=1,2,\ldots, k-1$.

A set of edges $M\subset E(G)$ is a \textit{matching} in $G$ if no two edges of $M$ are
incident.  A \textit{maximum matching} is a matching of maximum size. A graph $G$ is said to be \textit{complete} if all vertices of $G$ are pairwise adjacent. A path, cycle or complete graph of order $n$ is denoted by $P_n$, $C_n$ or $K_n$. We refer to $K_3$ also as a \textit{triangle}. For a positive integer $k$, $kK_1$ is an edgeless graph of $k$ isolated vertices. If $F$ is a graph, then $G$ is said to be $F$-free if $G$ does not contain $F$
as a subgraph.

An important tool in the proofs of our main results is the weighted eccentricity of a graph defined in \cite{DanGodSwa2004}.
\begin{definition}[\cite{DanGodSwa2004}]\label{def1}
	Let $G$ be a connected graph and $c:V(G) \rightarrow  \mathbb{R}$ a
	nonnegative weight function
	on the vertices of $G$. Then the eccentricity of $G$ with respect to $c$ is defined by
	\[ EX_c(G)= \sum_{x\in V(G)} c(x) e_G(x). \]
	Let $N=\sum_{x\in V(G)} c(x)$ be the total weight of the vertices in $G$. If $N>0$, then the average eccentricity of $G$ with respect to $c$ is
	\[ {\rm avec}_c(G) = \frac{EX_c(G)}{N}.   \]
\end{definition}
The following result from \cite{DanGodSwa2004} generalises the bound in Proposition \ref{prop1}.
\begin{proposition}[\cite{DanGodSwa2004}]\label{prop2}
	Let $G$ be a connected graph, $c$ a weight function on the vertices
	of $G$, and $N = \sum_{v\in V(G)} c(v)$ the total weight of the vertices
	of $G$. If $c(v) \geq 1$ for all $v\in V(G)$, then
	\[
	{\rm avec}_c(G) \leq {\rm avec}(P_{\lceil N \rceil}).
	\]
\end{proposition}

\section{Main Results}

\noindent
We begin by presenting a result on an upper bound on the average eccentricity in terms of order, minimum degree and maximum degree. The approach used throughout this section is a slight modification of that used in \cite{DanGodSwa2004} and \cite{DanMukOsaRod}.

\subsection{A bound on the average eccentricity of a graph}
\begin{theorem}\label{thm5-1}
	Let $G$ be a connected graph of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. Then,
	\begin{equation*}\label{label1}
	{\rm avec}(G)\leq   \frac{9}{4} \frac{n-\Delta-1}{\delta+1}
	     \Big( 1 + \frac{\Delta-\delta}{3n} \Big) + 7.
	\end{equation*}
	This bound is sharp apart from the value of the additive constant.
\end{theorem}

\noindent
{\bf Proof:}
Let $v_1$ be a vertex of degree $\Delta$. We find a maximal 2-packing $A$ of $G$ using the
following method: Let $A=\{v_1\}$. If there exists a vertex $a_1$ with $d_G(a_1,A)=3$, add
$a_1$ to $A$. Add vertices $a_i$ with $d_G( a_i,A)=3$ to $A$ until every vertex not in $A$ is
within distance two of $A$. Then $A=\{v_1,a_1,a_2,\ldots, a_r\}$ and $|A|=r+1$. Let $T_1$ be
the forest with vertex set $N[A]$ and whose edge set consists of all edges incident with
vertices in $A$.

By our construction of $A$, there exist $r$ edges in $G$, each joining two neighbours of
distinct vertices of $A$, such that the addition of these edges to $T_1$ yields a subtree
$T_2 \leq G$. Now every vertex $u\in V(G)-V(T_2)$ is adjacent to some $u'\in V(T_2)$. Let
$T$ be the spanning tree of $G$ with edge set $E(T) = E(T_2) \cup \{uu'| u\in V(G)-V(T_2)\}$.
Since ${\rm deg}_T(v_1) = {\rm deg}_G(v_1)$, it follows that $T$ has the same maximum
degree as $G$.

Since deleting edges does not decrease the average eccentricity, we have
${\rm avec}(G) \leq {\rm avec}(T)$.  Thus it suffices to prove the result for $T$.
We think of $T$ as a weighted tree in which each vertex has weight $1$. We now obtain a
new weight function by moving the weight
every vertex to a nearest vertex in $A$. More specifically, for every vertex $x$ of $T$
let $x_A$ be a vertex in $A$ closest to u in $T$.
Let $c:V(T)\rightarrow \mathbb{N} \cup \{0\}$ be the weight function defined by
\[c(u)=|\{x\in V(T)|x_A=u\}|\] for each $x\in V(T)$. Then $c(u)=0$ if $u\not \in A$, $c(u) \geq  \delta +1$ if $u\in A-\{v_1\}$, and $c(u) \geq \Delta +1$ if $u = v_1$. Note that $\sum_{u\in V(T)} c(u) =n$. It follows that
\[
n = \sum_{u\in V(T)} c(u) = \sum_{u\in A}c(u)
 \geq \Delta +1 + (|A|-1)(\delta +1)
 =|A|(\delta +1)+\Delta -\delta.
\]
By rearranging, we have
\begin{equation}\label{eqn5.1*}
|A| \leq \frac{n-\Delta +\delta}{\delta +1}.
\end{equation}
For every vertex $x$ of $T$ we have $d(x,x_A) \leq 2$ and thus $|e_T(x) - e_T(x_A)| \leq 2$. Hence
\begin{eqnarray}
{\rm avec}(T) & = & \frac{1}{n} \sum_{x \in V(T)} e_T(x) \nonumber \\
& \leq & \frac{1}{n} \sum_{x\in V(T)} (e_T(x_A) + 2) \nonumber \\
& = & \big( \frac{1}{n} \sum_{u \in A} c(u) e_T(u) \big) + 2 \nonumber \\
& = & {\rm avec}_{c}(T) + 2.    \label{eqn-5b}
\end{eqnarray}
Note that the weight of $c$ is concentrated in $A$. Thus we consider the induced subgraph
$T^3[A]$, which by the construction of $A$ is connected.
Clearly for all $a_i,a_j \in A$, $d(a_i,a_j)\leq 3d_{T^3[A]}(a_i,a_j)$. Since every vertex
of $T$ is within distance two of some vertex in $A$, we have for each $a_i\in A$,
$e_{T}(a_i) \leq 3e_{T^3[A]}(a_i) +2$. Thus,
\begin{equation}\label{eqn5a}
{\rm avec_c}(T) \leq 3{\rm avec_c}(T^3[A]) +2.
\end{equation}
We now modify the weight function $c$ to obtain a new weight function $c'$
which satisfies $c'(a) \geq 1$ for all $a\in A$.
Define $c'(u) = \frac{c(u)}{\delta+1}$ for $u\in A-\{v_1 \}$, and
$c'(v_1) = \frac{c(v_1)-\Delta+\delta}{\delta+1}$.
Since $c(u) \geq {\rm deg}_G(u)+1$ for all $u\in A$, and since ${\rm deg}_G(v_1)=\Delta$
while ${\rm deg}_G(a_i) \geq \delta$ for $a_i \in A - \{v_1\}$, we have
$c'(u) \geq 1$ for all $u\in A$.

Since  $\sum_{u\in A}c(u) = n$, we have for  the total weight $N$ of $c'$,
\[N= \sum_{u\in A}c'(u)=\frac{(\sum_{u\in A}c(u))-\Delta +\delta}{\delta +1}
=  \frac{n-\Delta +\delta }{\delta + 1}.  \]
We now express ${\rm avec_c}(T^3[A])$ in terms of ${\rm avec}_{c'}(T^3[A])$. We have
\begin{eqnarray*}
	{\rm avec}_{c'}(T^3[A]) &=&\frac{EX_{c'}(T^3[A])}{N}\\
	&=& \frac{\frac{1}{\delta +1} [\sum_{u\in A-\{v_1\}}c(u) e_{T^3[A]}(u)+ (c(v_1)-\Delta +\delta)e_{T^3[A]}(v_1) ]}{\frac{1}{\delta +1} [n-\Delta +\delta]}   \\
	&=& \frac{EX_{c}(T^3[A])- e_{T^3[A]}(v_1)(\Delta -\delta)}{n-\Delta+\delta} \\
	&=& \frac{n}{n-\Delta+\delta} {\rm avec}_{c}(T^3[A])
	- \frac{\Delta-\delta}{n-\Delta+\delta}  e_{T^3[A]}(v_1),
\end{eqnarray*}
and thus, by rearranging,
\begin{equation} \label{eq:avec(Tc)-in-terms-of-avec(Tc')}
{\rm avec_c}(T^3[A]) = \frac{n-\Delta+\delta}{n} {\rm avec_{c'}}(T^3[A])
+\frac{\Delta -\delta}{n} e_{T^3[A]}(v_1).
\end{equation}
We bound the two terms of the right hand side of \eqref{eq:avec(Tc)-in-terms-of-avec(Tc')}
separately. Since $T^3[A]$ has order $|A|$, and since $|A| \leq \frac{n-\Delta+\delta}{\delta+1}$
by \eqref{eqn5.1*}, we have
\begin{equation}  \label{eq:bound-on-e(v_1)}
e_{T^3[A]}(v_1) \leq {\rm diam}(T^3[A]) \leq |A|-1 \leq \frac{n-\Delta-1}{\delta+1}.
\end{equation}
To bound ${\rm avec_{c'}}(T^3[A])$ note that $c'(u) \geq 1$ for all $u\in A$.
Hence we have by Proposition \ref{prop2},
\begin{equation}\label{eqn-5a}
{\rm avec_{c'}}(T^3[A])
\leq {\rm avec}(P_{\lceil N\rceil})
\leq \frac{3}{4}\lceil N\rceil-\frac{1}{2}
< \frac{3}{4} \frac{n - \Delta-1}{\delta+1} + 1,
\end{equation}
with the last inequality holding since
$\lceil N \rceil = \lceil \frac{n-\Delta+\delta}{\delta+1} \rceil < \frac{n-\Delta-1}{\delta+1} + 2$.
Substituting  \eqref{eq:bound-on-e(v_1)} and \eqref{eqn-5a} into
\eqref{eq:avec(Tc)-in-terms-of-avec(Tc')} yields, after simplification,
\begin{eqnarray}
{\rm avec_{c}}(T^3[A])  & < &
\frac{n-\Delta+\delta}{n} \Big( \frac{3}{4} \frac{n - \Delta-1}{\delta+1} + 1 \Big)
+\frac{\Delta -\delta}{n} \Big( \frac{n-\Delta-1}{\delta+1} \Big) \nonumber \\
& = & \frac{3}{4} \frac{n-\Delta-1}{\delta+1}
  \big[ 1 + \frac{\Delta-\delta}{3n} \big] + 1 -  \frac{\Delta-\delta}{n} \nonumber \\
& \leq & \frac{3}{4} \frac{n-\Delta-1}{\delta+1}
  \big[ 1 + \frac{\Delta-\delta}{3n} \big] + 1.
\label{eqn5-3a}
\end{eqnarray}
Combining inequalities \eqref{eqn-5b}, \eqref{eqn5a}, and \eqref{eqn5-3a}, we obtain
\begin{eqnarray*}
{\rm avec}(T)
    &\leq & {\rm avec_{c}}(T) +2 \\
	&\leq &  3 {\rm avec_{c}}(T^3[A]) +4 \\
	& < &  \frac{9}{4} \frac{n-\Delta-1}{\delta+1}
	     \Big( 1 + \frac{\Delta-\delta}{3n} \Big) + 7.
\end{eqnarray*}
The bound on ${\rm avec}(G)$ now follows since ${\rm avec}(G) \leq {\rm avec}(T)$.

To see that the bound in Theorem \ref{thm5-1} is sharp apart from an additive constant, let $\delta$, $\Delta$ and $k$ be positive integers with $\Delta \geq \delta +1$ and consider the graph  $G_{\delta,\Delta,k}$ obtained as follows.

Let $G_1,G_2,\ldots , G_{k-1}$ be disjoint copies of the complete graph $K_{\delta +1}$ and let $G_k$ be the complete graph $K_{\Delta}$. For $i=2,3,\ldots,k-1$ let
$u_iv_i$ be an edge of $G_i$. Let $G_{\delta,\Delta,k}$ be the graph with vertex set
$V(G_{\delta,\Delta,k})=V(G_1) \cup V(G_2)\cup \ldots \cup V(G_k)$,
and edge set
$E(G_{\delta,\Delta, k})= E(G_1) \cup E(G_2) \cup \cdots \cup E(G_k)
  - \{u_2v_2, u_3v_3,\ldots,u{k-1}v_{k-1}\} \cup \{v_1u_2, v_2u_3,\ldots, v_{k-1}u_k\}$.
The graph $G_{3,8,6}$ is shown in  Figure \ref{fig:sharpness-example-for-general-graphs}.

It is easy to see that $G_{\delta,\Delta,k}$ has minimum degree $\delta$,
maximum degree $\Delta$, order $n=\Delta+(\delta +1)(k-1)$,
${\rm diam}(G_{\delta,\Delta, k}) = 3(k-1)$ and
${\rm rad}(G_{\delta,\Delta, k}) = \lceil \frac{3}{2}(k-1)\rceil$.
A straightforward calculation, whose details we omit, shows that
\[ EX(G_{\delta,\Delta,k}) =
  3(k-1)(\Delta+\delta+1) - 2
   + 6(\delta+1)[\frac{3}{8}k^2 - \frac{5}{4}k + 1]. \]

Now $k-1=\frac{n-\Delta}{\delta+1}$. Hence
$3(k-1)(\Delta+\delta+1) = 3 \frac{n-\delta}{\delta +1}(\Delta+ \delta+1)
       =  3\frac{(n-\Delta)\Delta}{\delta+1} + O(n)$
and
$6(\delta+1)[\frac{3}{8}k^2 - \frac{5}{4}k + 1] = 6(\delta+1)[\frac{3}{8}(k-1)^2 +O(k)]
     = \frac{9}{4} \frac{(n-\Delta)^2}{\delta+1} +O(n)$,
and thus
\begin{eqnarray*}
{\rm avec}(G_{\delta,\Delta,k})
 &= &\frac{1}{n} EX(G_{\delta,\Delta,k})  \\
 & = &  \frac{1}{n}
    \Big( 3 \frac{(n-\Delta)\Delta}{\delta+1}
    + \frac{9}{4} \frac{(n-\Delta)^2}{\delta+1} +O(n) \Big)   \\
 & = & \frac{9}{4} \frac{n-\Delta}{\delta+1} (1 + \frac{\Delta}{3n}) + O(1).
\end{eqnarray*}
Since the upper bound on ${\rm avec}(G)$ is also
$\frac{9}{4} \frac{n-\Delta}{\delta+1} (1 + \frac{\Delta}{3n}) + O(1)$,
we conclude that the bound is sharp apart from an additive constant.
\qed \\

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	\draw[black, very thick] (d3)--(d4);
	\draw[black, very thick] (d3)--(d5);
	\draw[black, very thick] (d3)--(d6);
	\draw[black, very thick] (d3)--(d7);
	\draw[black, very thick] (d3)--(d8);

	\draw[black, very thick] (d4)--(d5);
	\draw[black, very thick] (d4)--(d6);
	\draw[black, very thick] (d4)--(d7);
	\draw[black, very thick] (d4)--(d8);

	\draw[black, very thick] (d5)--(d6);
	\draw[black, very thick] (d5)--(d7);
	\draw[black, very thick] (d5)--(d8);

	\draw[black, very thick] (d6)--(d7);
	\draw[black, very thick] (d6)--(d8);

	\draw[black, very thick] (d7)--(d8);

	\draw[black, very thick] (a0)--(a1);
	\draw[black, very thick] (a1)--(a2);
	\draw[black, very thick] (a3)--(a4);
	\draw[black, very thick] (a5)--(a6);
	\draw[black, very thick] (a7)--(a8);

	\draw[black, very thick] (a0)--(b1);
	\draw[black, very thick] (a0)--(c1);
	\draw[black, very thick] (b1)--(a1);
	\draw[black, very thick] (c1)--(a1);
	\draw[black, very thick] (b1)--(c1);

	\draw[black, very thick] (a2)--(b2);
	\draw[black, very thick] (a2)--(c2);
	\draw[black, very thick] (b2)--(a3);
	\draw[black, very thick] (c2)--(a3);
	\draw[black, very thick] (b2)--(c2);

	\draw[black, very thick] (a4)--(b3);
	\draw[black, very thick] (a4)--(c3);
	\draw[black, very thick] (b3)--(a5);
	\draw[black, very thick] (c3)--(a5);
	\draw[black, very thick] (b3)--(c3);

	\draw[black, very thick] (a6)--(b4);
	\draw[black, very thick] (a6)--(c4);
	\draw[black, very thick] (b4)--(a7);
	\draw[black, very thick] (c4)--(a7);
	\draw[black, very thick] (b4)--(c4);

	\draw[black, very thick] (a8)--(b5);
	\draw[black, very thick] (a8)--(c5);
	\draw[black, very thick] (b5)--(a9);
	\draw[black, very thick] (c5)--(a9);
	\draw[black, very thick] (b5)--(c5);

	\end{scope}
	\end{tikzpicture}
\end{center}
\caption{$G_{\delta,\Delta, k}$ with $\delta =3$, $\Delta =8$ and $k=6$. }
  \label{fig:sharpness-example-for-general-graphs}
\end{figure}

We note that for fixed $\delta$ and large $n$, the value $avec(G_{\delta,\Delta,k})$ and
the bound in Theorem \ref{thm5-1} differ only by $O(1)$, irrespective of whether $\Delta$ is
constant or grows with $n$. In addition, since $\Delta \geq \delta$ and since the bound
in Theorem \ref{thm5-1} is decreasing in $\Delta$, replacing
$\Delta$ by $\delta$ in our bound in Theorem \ref{thm5-1} yields a bound on the
average eccentricity in terms of order and minimum degree that differs from
\eqref{equ-DGS} only by
a small additive constant. Hence Theorem \ref{thm5-1} is, in some sense, a generalisation
of \eqref{equ-DGS}.

\subsection{A bound on the average eccentricity of a triangle-free graph}

We now show that the bound in Theorem \ref{thm5-1} can be improved for triangle-free graphs.

\begin{theorem}\label{thm5-2}
	Let $G$ be a connected triangle-free graph with $n$ vertices, minimum degree $\delta$ and maximum degree $\Delta$. Then,
	\begin{equation*}\label{eqn5Dd}
	{\rm avec} (G) \leq \frac{3}{2} \frac{n-\Delta}{\delta}
    \Big[ 1 + \frac{\Delta-\delta}{3n} \Big]
    + \frac{19}{2}.
	\end{equation*}
	This bound is sharp apart from the value of the additive constant.
\end{theorem}

\noindent
\textbf{Proof.}
Let $v_1$ be a vertex of degree $\Delta$ in $G$ and let $e_1$ be an edge incident with $v_1$.
We obtain a maximal matching $M$ of $G$ as follows.
Let $M=\{ e_1\}$. Let $V(M)$ be the set of vertices incident with an edge of $M$. Recall that
$d_G(e_1,V(M))$ is the minimum of the distances between a vertex incident with edge $e_1$ and
a vertex in $V(M)$.
If there exists an edge $e_2$ in $G$ with $d_G(e_2,V(M))=3$, add $e_2$ to $M$. Add edges $e_i$
with $d_G(e_i,V(M))=3$ to $M$ until each of the edges not in $M$ is within distance two of
$M$.

Let $T_1\leq G$ be a subforest of $G$ with vertex set $N[V(M)]$ and edge set consisting of all
edges incident with a vertex in $V(M)$. By our construction of $M$, $G$ contains $|M|-1$
edges, each joining two distinct components of $T_1$, whose addition to $T_1$ yields a
subtree $T_2 \leq G$ so that $T_2$ contains $T_1$ and $|V(T_1)| = |V(T_2)|$.

Now every vertex $x$ not in $T_2$ is within distance at most three from some vertex $x'$ in
$T_2$. Let $T$ be a spanning tree of $G$ which contains $T_2$ and which is distance preserving
from $V(M)$, i.e. $d_T(x,V(M))=d_G(x,V(M))$, for every vertex $x\in V(G)$. Since
${\rm deg}_T(v_1) = {\rm deg}_G(v_1)$, tree $T$ has the same maximum degree as $G$.
Furthermore, since
${\rm avec}(G)\leq {\rm avec}(T)$, it suffices to prove the bound for $T$.

For every vertex $u \in V(T)$, let $u_M$ be a vertex in $V(M)$ closest to $u$ in $T$. We
can view $T$ as a weighted tree in which each vertex has weight $1$. We now move the weight
of $u$ to $u_M$. That is, we define a weight function
$c:V(T) \rightarrow \mathbb{N} \cup \{0\}$ by
\[ c(u) = |\{x \in V(T)|x_M=u\}|  \]
for $u \in V(T)$. Note that $c(u) =0$ if $u \not \in V(M)$ and $\sum_{u\in V(M)} c(u) = n$,
where $n=|V(G)|$. Since $G$ is triangle-free, no two incident vertices of an edge in $M$ have
a common neighbour. Hence, ${\rm deg}_T(u) \geq \delta$ implying that $c(u)\geq \delta$ for
$u\in V(M)-\{v_1\}$  and ${\rm deg}_T(v_1) = \Delta$.

Now $d_T(x,x_M)\leq 3$ for every vertex $x$ of $T$. The same argument as in the proof
of Theorem \ref{thm5-1} (see \eqref{eqn-5b}) shows that
\begin{equation}\label{eqn5-2a}
{\rm avec}(T) \leq {\rm avec_c}(T) +3.
\end{equation}
Let $\bar{c}$ be the weight function on the vertex set $E(T)$ of the line graph $L= L(T)$
defined by:
\[
\bar{c}(uv) =
\begin{cases}
c(u) +c(v) & \text{if } uv \in M, \\
0
& \text{if } uv \not \in M.
\end{cases}
\]
Since $e_1$ is an edge incident with $v_1$ in $T$, we have $\bar{c}(e_1)\geq \Delta +\delta$
and $\bar{c}(e) \geq 2\delta$ for $e\in M-\{e_1\}$.
Note that $\sum_{e\in M}\bar{c}(e) = \sum_{v\in V(T)}{c}(v)=n.$ It follows that
\begin{equation*}
\begin{aligned}
n &\geq \Delta +\delta + \sum_{x\in M-\{e_1\}} 2\delta = \Delta +\delta +2\delta (|M|-1),
\end{aligned}
\end{equation*}
and rearranging yields
\begin{equation}\label{eqn5-m}
|M| \leq \frac{n-\Delta +\delta}{2\delta}.
\end{equation}
It is easy to show that if $u, v\in V(T)$ and $e_u, e_v$ are edges of $T$ incident with $u$ and $v$, respectively, then
\[|d_L(e_u,e_v) - d_T(u,v)|\leq 1. \]
Hence, if $u$ is an eccentric vertex of $v$, i.e., a vertex with $d_T(v,u)=e_T(v)$, then
\[ e_T(v) = d_T(u,v)\leq d_L(e_u,e_v) +1 \leq e_L(e_v) +1,  \]
where $e_u$ and $e_v$ are edges of $T$ incident with $u$ and $v$, respectively.
Summation over all vertices of $T$ yields that
\begin{equation*}
\begin{aligned}
\sum_{v\in V(T)}{e_T}(v) c(v) &= \sum_{uv\in M}\big[e_T(u)c(u) + e_T(v)c(v)\big]\\
&\leq \sum_{uv \in M}\Big[\bar{c}(uv)\big(e_L(uv) +1\big)\Big]\\
&= \sum_{e \in M}e_L(e)\bar{c}(e) + \sum_{e \in M}\bar{c}(e).
\end{aligned}
\end{equation*}
Therefore,
\[ EX_{c}(T) \leq EX_{\bar{c}}(L)+ \sum_{e \in M}\bar{c}(e),\]
and since $\sum_{v\in V(T)} c(v) = \sum_{e \in M} \overline{c}(e) = n$,
division by $n$ yields
\begin{equation}\label{eqn5-2b}
{\rm avec_c}(T) \leq {\rm avec}_{\bar{c}}(L) +1.
\end{equation}
If $e_1$ and $e_2$ are two matching edges with $d_T(e_1, e_2)=3$, then $d_L(e_1, e_2) \leq 4$.
Now the weights lie solely on $M$. Thus we construct the induced subgraph $L^4[M]$ which is
connected and has exactly $|M|$ vertices.
Clearly, for every two edges $e_1,e_2 \in M$, $$d_L(e_1,e_2) \leq 4d_{L^4[M]}(e_1,e_2).$$ Furthermore, for every edge $e' \in E(T)$ there exists an edge $e'' \in M$ with $d_L(e',e'') \leq 3$, so that $e_L(e) \leq 4e_{L^4[M]}(e) +3$ for every $e\in M$, and so
\begin{equation}\label{eqn5-2c}
{\rm avec}_{\bar{c}}(L) \leq 4{\rm avec}_{\bar{c}}(L^4[M]) +3.
\end{equation}
We now modify the weight function $\bar{c}$ to obtain a weight function
$\overline{c}'$ on $M$ with $\bar{c}'(e) \geq 1$ for all $e\in M$.
Define $\bar{c}'(e) =\frac{\bar{c}(e)}{2\delta}$ for $e\in M-\{e_1\}$, and $\bar{c}'(e_1)=\frac{\bar{c}(e_1)-\Delta +\delta}{2\delta}$.
Since $\bar{c}(e_1)\geq \Delta +\delta$ and $\bar{c}(e) \geq 2\delta$ for
$e\in M-\{e_1\}$, weight function $\bar{c}'$ assigns a weight of at least $1$
to every edge of $M$.
Let $N' = \sum_{e\in V(M)} \bar{c}'(e)$. Then $N'=\frac{n-\Delta +\delta}{2\delta}$.
As in the proof of Theorem \ref{thm5-1} (see \eqref{eq:avec(Tc)-in-terms-of-avec(Tc')}),
we show that
\begin{equation} \label{L1}
avec_{\bar{c}}(L^4[M])  =
\frac{n-\Delta+\delta}{n}
avec_{\bar{c}'}(L^4[M])
+ \frac{\Delta-\delta}{n} e_{L^4[M]}(e_1).
\end{equation}

Since $L^4[M]$ has order $|M|$, and since $|M| \leq \frac{n-\Delta+\delta}{2\delta}$
by \eqref{eqn5-m}, we have
\begin{equation}  \label{L2}
e_{T^4[M]}(v_1) \leq {\rm diam}(L^4[M]) \leq |M|-1 \leq \frac{n-\Delta}{2\delta} -\frac{1}{2}.
\end{equation}
Since $ \lceil N' \rceil =   \lceil \frac{n-\Delta+\delta}{2\delta} \rceil < \frac{n-\Delta}{2\delta} + \frac{3}{2}$, we have by Proposition \ref{prop2},
\begin{equation}\label{L3}
{\rm avec_{\bar{c}'}}(L^4[M])
\leq {\rm avec}(P_{\lceil N'\rceil})
\leq \frac{3}{4}\lceil N'\rceil-\frac{1}{2}
< \frac{3}{4} \frac{n - \Delta}{2\delta} + \frac{5}{8}.
\end{equation}

Substituting  \eqref{L2} and \eqref{L3} into
\eqref{L1} yields, after simplification,
\begin{eqnarray}
{\rm avec_{\bar{c}}}(L^4[M]) &\leq &
\frac{n-\Delta+\delta}{n} \Big( \frac{3}{4} \frac{n - \Delta}{2\delta} + \frac{5}{8} \Big)
+\frac{\Delta -\delta}{n} \Big( \frac{n-\Delta}{2\delta} -\frac{1}{2} \Big) \nonumber \\
& = & \frac{3(n-\Delta)}{8\delta}
+ \frac{(n-\Delta)(\Delta-\delta)}{8n\delta} - \frac{9(\Delta-\delta)}{8n} +\frac{5}{8}.
\label{eqn5-3}
\end{eqnarray}
Hence combining these inequalities \eqref{eqn5-2a}, \eqref{eqn5-2b}, \eqref{eqn5-2c} and \eqref{eqn5-3} yields
\begin{eqnarray*}
{\rm avec}(T) &\leq & {\rm avec_c}(T) +3 \\
& \leq & {\rm avec}_{\bar{c}}(L) +4 \\
&\leq 4 & {\rm avec}_{\bar{c}}(L^4[M]) +7\\
&\leq & 4\big( \frac{3(n-\Delta)}{8\delta}
+ \frac{(n-\Delta)(\Delta-\delta)}{8n\delta} - \frac{9(\Delta-\delta)}{8n} +\frac{5}{8} \Big) +7\\
&=&  \frac{n-\Delta}{2\delta}
    \big( 3 + \frac{\Delta-\delta}{n} \big)
    - \frac{9(\Delta-\delta)}{2n}
    + \frac{19}{2} \\
& \leq &  \frac{3}{2} \frac{n-\Delta}{\delta}
    \big( 1 + \frac{\Delta-\delta}{3n} \big)
    + \frac{19}{2}.
\end{eqnarray*}
The bound now follows since ${\rm avec} (G) \leq {\rm avec} (T)$.

To see that the bound is sharp apart from an additive constant, let $n, \delta, \Delta$ and $k$ be positive integers with $\Delta \geq \frac{3}{2}\delta$ and $k$ a multiple of 4. Consider the graph $G'= G_{\delta, \Delta,k}'$ defined by
\[G'=G_1+G_2+ \ldots +G_k,   \] where $G_2= (\Delta -\lceil \frac{\delta}{2}\rceil) K_1$, $G_{k-1} = \delta K_1$, and for $1\leq i\leq k$, $i \neq 2,k-1$,
\[
G_i =
\begin{cases}
\lceil \frac{\delta}{2}\rceil K_1 & \text{if}\,\, i=0\,\, \text{or}\,\, 3\,\mod 4,  \\
\lfloor \frac{\delta}{2}\rfloor K_1 & \text{if} \,\, i=1\,\, \text{or}\,\, 2 \,\mod 4.
\end{cases}
\]
The graph $G'_{10,4,12}$ is shown in Figure \ref{fig:sharpness-example-triangle-free}.

By inspection, $G_{\delta, \Delta,k}'$ has minimum degree $\delta$, maximum degree $\Delta$,
and it is triangle-free. Moreover, $G_{\delta, \Delta,k}'$ has exactly
$\Delta + \delta (\frac{k}{2}) -\lceil \frac{\delta}{2}\rceil$ vertices. A simple
calculation, whose details we omit, yields that
\begin{equation*}
EX(G'_{\delta,\Delta,k})
= \delta\Big( \frac{3k^2}{8}-\frac{k}{4}\Big)
     +(\Delta -\Big\lceil \frac{\delta}{2}\Big\rceil)(k-2).
\end{equation*}
Let $n= n(G'_{\delta,\Delta,k})
=\Delta + \delta (\frac{k}{2}) -\lceil \frac{\delta}{2}\rceil$. Then
$k=\frac{2}{\delta}(n-\Delta+\lceil \frac{\delta}{2} \rceil)
      = \frac{2}{\delta}(n-\Delta) + O(1)$.
Hence
$\delta\Big( \frac{3k^2}{8}-\frac{k}{4}\Big)
 =  \frac{3}{8}\delta k^2 +O(n)
 = \frac{3}{2\delta} (n-\Delta)^2 + O(n)$
and
$ (\Delta -\Big\lceil \frac{\delta}{2}\Big\rceil)(k-2)
  = \Delta k +O(n)
  = \frac{2}{\delta}(n-\Delta)\Delta + O(n)$.
Substituting this into the above expression for  $EX(G'_{\delta,\Delta,k})$
and dividing by $n$ we get
\begin{eqnarray*}
{\rm avec}(G_{\delta,\Delta,k})
   &=&  \frac{1}{n}
    \Big( \frac{3}{2\delta} (n-\Delta)^2   +  \frac{2}{\delta}(n-\Delta)\Delta + O(n) \Big) \\
    &=& \frac{3}{2} \frac{n-\Delta}{\delta} \big(1 + \frac{\Delta}{3n}\big) + O(1).
\end{eqnarray*}
Hence the bund in Theorem \ref{thm5-2} and ${\rm avec}(G_{\delta,\Delta,k})$ is
bounded by a constant.
\qed

	\begin{figure}
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		\qbezier(1,1.5)(1.5,1)(1.5,1)

		\qbezier(1.5,1)(2,1.5)(2,1.5)
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		\end{picture}
\caption{$G'_{\delta,\Delta,k}$ with $\Delta=10$, $\delta=4$ and $k=12$.}
\label{fig:sharpness-example-triangle-free}
\end{figure}

\subsection{A bound on the average eccentricity of a $C_4$-free graph}

\noindent
In this section we show that the bound in Theorem 1 can be improved for graphs not
containing $C_4$ as a (not necessarily induced) subgraph.

\begin{theorem}\label{theo:C4-free}
Let $G$ be a $C_4$-free graph of order $n$, minimum degree $\delta$ and maximum degree
$\Delta$. Then
\[{\rm avec}(G)\leq
\frac{15}{4} \frac{n-\varepsilon_{\Delta} + \varepsilon_{\delta}}{\varepsilon_{\delta}}
\Big[ 1 + \frac{\varepsilon_{\Delta} - \varepsilon_{\delta}}{3n} \Big]  + \frac{37}{4},
	\]
where
$\varepsilon_{\Delta}:= \Delta \delta -2\Big\lfloor\frac{\Delta}{2}\Big\rfloor +1$
and
$\varepsilon_{\delta} := \delta^2 -2\lfloor \frac{\delta}{2} \rfloor +1$.
\end{theorem}

\noindent
{\bf Proof.} Let $v_1$ be a vertex in $G$ with ${\rm deg}_G(v_1)=\Delta$. We find a $4$-packing $A$ of $G$ using the following method. Let $A=\{v_1\}$. If there exists a vertex $a_1$ with $d_G(a_1, A)=5$, add $a_1$ to $A$. Add vertices $a_i$ satisfying $d_G(a_i,A)=5$ to $A$ until each of the vertices not in $A$ is within distance $4$ of $A$. For each $a\in A$, let $T_{a}$ be a subtree of $G$ with vertex set $N_G^2[a]$ which is distance preserving from $a$. Let $T_1= \bigcup_{a\in A} T_{a}$. Then $T_1$ is a subforest of $G$. By the way $A$ is constructed, there exists $|A|-1$ edges in $G$, each joining two components of $T_1$, whose addition to $T_1$ yields a subtree $T_2$ of $G$.

Let $T$ be a spanning tree of $G$ obtained from $T_2$ and which satisfies $d_T(x,A) =d_G(x,A)$ for each $x\in V(G)$.
Since ${\rm deg}_T(v_1) = {\rm deg}_G(v_1)$, tree $T$ has the same maximum degree as $G$.
For every vertex $x$ of $T$ let $x_A$ be a vertex in $A$ closest to $x$.
We define a weight function $c: V(T)\rightarrow \mathbb{N}\cup \{0\}$ for each $u\in V(T)$ as
\[c(u) =|\{x\in V(T)|x_A=u \} |.\]
Since $G$ is $C_4$-free, no two neighbours of $v_1$ have a common neighbour apart from $v_1$. It follows that for $\Delta$ even, we have
$|N_G^2[v_1]| \geq 1 + \Delta + \Delta(\delta -2)=\Delta \delta -\Delta +1$.
For $\Delta$ odd, the handshake lemma yields that at least one of the neighbours
of $v_1$ is not adjacent to any other neighbour of $v_1$, and so
$|N_G^2[v_1]| \geq 1 + \Delta + (\Delta-1)(\delta -2) +(\delta -1) =\Delta \delta -\Delta +2$.
Combining these two bounds on $|N_G^2[v_1]|$ yields that
$|N_G^2[v_1]| \geq \Delta \delta -2\Big\lfloor\frac{\Delta}{2}\Big\rfloor +1
     = \varepsilon_{\Delta}$.
The same reasoning as for $v_1$ shows that for vertices $a_i \in A-\{v_1\}$,
$|N_G^2[a_i]| \geq \delta^2 -2\lfloor \frac{\delta}{2} \rfloor +1 =\varepsilon_{\delta}$.
This implies that
\begin{equation}  \label{eq:lower-bound-on-c}
c(v_1) \geq \varepsilon_{\Delta}, \quad
\textrm{and $c(a_i) \geq \varepsilon_{\delta}$ for $a_i\in A-\{v_1\}$},
\end{equation}
while $c(u)=0$ whenever $u\not \in A$.
Now for every vertex $x \in V(T)$, $d(x,x_A) \leq 4$. Thus we have
$|e_T(x) - e_T(x_A)| \leq 4$, and an argument similar to \eqref{eqn-5b} shows that
\begin{equation}\label{eqn5.1.3c}
{\rm avec}(T)\leq {\rm avec_c}(T) +4.
\end{equation}
By the way $A$ was constructed, $T^5[A]$ is connected and so for $a_i,a_j\in A$, $d(a_i,a_j)\leq 5d_{T^5[A]}(a_i,a_j)$. Since every vertex of $T$ is within distance 4 of $A$, we have
\begin{equation}\label{eqn5.1.3d}
{\rm avec_c}(T) \leq 5{\rm avec_c}(T^5[A]) +4.
\end{equation}
We modify $c$ to obtain a weight function $c''$ for which every vertex of $A$ has weight
at least $1$.
Define $c''(a_i) = \frac{c(a_i)}{\varepsilon_{\delta}}$
for all $a_i \in A-\{v_1\}$ and
$c''(v_1)=
\frac{c(v_1)-\varepsilon_{\Delta} + \varepsilon_{\delta}}{\varepsilon_{\delta}}$.
By \eqref{eq:lower-bound-on-c}
we have  $c''(a) \geq 1$ for all $a\in A$.

Let $N'' = \sum_{u\in A}c''(u)$. Then
\[ N''= \frac{n-\varepsilon_{\Delta} + \varepsilon_{\delta}}{\varepsilon_{\delta}}.\]
As in the proof of Theorem 1 (see \eqref{eq:avec(Tc)-in-terms-of-avec(Tc')}),
we express ${\rm avec_{c}}(T^5[A])$ in terms of ${\rm avec_{c''}}(T^5[A])$. Clearly,
\begin{eqnarray*}
{\rm avec_{c''}}(T^5[A])
&=&  \frac{EX_{c''}(T^5[A])}{N''} \\
&=& \frac{ \frac{1}{\varepsilon_{\delta}}  EX_{c}(T^5[A])
	  - \frac{\varepsilon_{\Delta} - \varepsilon_{\delta}}{\varepsilon_{\delta}}
	   e_{T^5[A]}(v_1)}{(n-\varepsilon_{\Delta}+ \varepsilon_{\delta}) / \varepsilon_{\delta}}  \\
&=& \frac{n}{n-\varepsilon_{\Delta} +\varepsilon_{\delta}} {\rm avec_c}(T^5[A])
  - \frac{\varepsilon_{\Delta} - \varepsilon_{\delta}}{n-\varepsilon_{\Delta} +\varepsilon_{\delta}} e_{T^5[A]}(v_1).
\end{eqnarray*}
By rearranging, we obtain
\begin{equation}\label{t5}
{\rm avec_c}T^5[A]
 = \frac{n-\varepsilon_{\Delta} + \varepsilon_{\delta}}{n} {\rm avec_{c''}}(T^5[A])
   + \frac{\varepsilon_{\Delta} - \varepsilon_{\delta}}{n} e_{T^5[A]}(v_1).
\end{equation}
We bound the terms on the right hand side of \eqref{t5} separately.
Since $T^5[A]$ has order $|A|$, we have $e_{T^5[A]}(v_1) \leq |A|-1$.
Also $|A| = \sum_{a\in A} 1 \leq  \sum_{a\in A} c''(a) = N''$, and so
 \begin{equation}\label{t6}
 e_{T^5[A]}(v_1) \leq |A|-1  \leq N'' - 1
   = \frac{n-\varepsilon_{\Delta}+\varepsilon_{\delta}}{\varepsilon_{\delta}} - 1.
 \end{equation}
Since $\lceil N'' \rceil < N''+1
= \frac{n - \varepsilon_{\Delta} +\varepsilon_{\delta}}{\varepsilon_{\delta}}
  + 1$
it follows by Proposition \ref{prop2} that
\begin{equation}\label{eqn5.1.3g}
{\rm avec_{c''}}(T^5[A])
\leq {\rm avec}(P_{\lceil N''\rceil})
\leq \frac{3\lceil N''\rceil}{4}-\frac{1}{2}
<  \frac{3}{4} \frac{n-\varepsilon_{\Delta} + \varepsilon_{\delta}}{\varepsilon_{\delta}}
  +\frac{1}{4}.
\end{equation}
Thus substituting \eqref{eqn5.1.3g} and \eqref{t6} in \eqref{t5} yields, after simplification,
\begin{eqnarray*}
{\rm avec_c}T^5[A]
 & < &\frac{n-\varepsilon_{\Delta} + \varepsilon_{\delta}}{n}
  \Big[ \frac{3}{4} \frac{n-\varepsilon_{\Delta} + \varepsilon_{\delta}}{\varepsilon_{\delta}}
  +\frac{1}{4} \Big]
   + \frac{\varepsilon_{\Delta} - \varepsilon_{\delta}}{n}
   \big( \frac{n-\varepsilon_{\Delta}+\varepsilon_{\delta}}{\varepsilon_{\delta}}
      - 1 \Big) \nonumber \\
& = & \frac{3}{4} \frac{n - \varepsilon_{\Delta} + \varepsilon_{\delta}}{ \varepsilon_{\delta}}
      \Big[ 1 + \frac{ \varepsilon_{\Delta} -\varepsilon_{\delta}}{3n} \Big] + \frac{1}{4}
        - \frac{5}{4} \frac{\varepsilon_{\Delta}-\varepsilon_{\delta}}{n}  \nonumber  \\
& \leq & \frac{3}{4} \frac{n - \varepsilon_{\Delta} + \varepsilon_{\delta}}{ \varepsilon_{\delta}}
      \Big[ 1 + \frac{ \varepsilon_{\Delta}-\varepsilon_{\delta}}{3n} \Big] + \frac{1}{4}.    \label{t7}
\end{eqnarray*}
Combining inequalities \eqref{eqn5.1.3c}, \eqref{eqn5.1.3d} and \eqref{t7}, we obtain with further simplification,
\begin{eqnarray*}
{\rm avec}(T) & \leq &{\rm avec_c}(T) +4 \\
&\leq &  5 {\rm avec_{c}}(T^5[A]) +8 \\
   & \leq & \frac{15}{4}  \frac{n- \varepsilon_{\Delta} + \varepsilon_{\delta}}{\varepsilon_{\delta}}
         \big[1 + \frac{\varepsilon_{\Delta} - \varepsilon_{\delta}}{3n} \big]
          + \frac{37}{4},
\end{eqnarray*}
as desired.
\qed   \\

The following theorem shows that the bound in Theorem \ref{theo:C4-free} is not far
from being sharp if $\delta +1$ is a prime power.  In our construction below, the
maximum degree and order can be chosen almost arbitrarily, they only have to satisfy
certain conditions regarding their values modulo $\delta+1$ or $\delta+2$.
Our construction is a modification of a graph first constructed  by
Erd\"{o}s et al.\ \cite{ErdPacPolTuz1989}.

\begin{theorem}
Let $\delta \geq 3$ be an integer such that $\delta +1$ is a prime power.
Then for $n, \Delta \in \mathbb{N}$ with
$2\delta-3 \leq \Delta <n$ and $n \equiv 0 \pmod{(\delta+1)(\delta+2)}$
and $\Delta \equiv \delta +1 \pmod{\delta+2}$ there exists a $C_4$-free graph
$G$ with minimum degree $\delta$, maximum degree $\Delta$, order
$n$ whose average eccentricity satisfies
\[{\rm avec}(G)\geq
 \frac{3}{4} \frac{n - \varepsilon_{\Delta} - \Delta}{\varepsilon_{\delta}'}
    \big( 1 + \frac{\Delta (\delta+1)}{3n} \big) + O(1), \]
where $\varepsilon_{\Delta}$ is as in Theorem \ref{theo:C4-free},
and $\varepsilon_{\delta}' =(\delta+1)(\delta+2)$.
\end{theorem}

\begin{figure}
  \begin{center}
\begin{tikzpicture}
  [scale=0.6,inner sep=1mm,
   vertex/.style={circle,thick,draw},
   thickedge/.style={line width=2pt}]

    \node[vertex] (a1) at (0.8,1.3) [fill=black] {};
    \node[vertex] (a2) at (3.5,1.3) [fill=black] {};
    \node[vertex] (a3) at (5.1,1.3) [fill=black] {};
    \node[vertex] (a4) at (7.7,1.3) [fill=black] {};
    \node[vertex] (a5) at (9.3,1.3) [fill=black] {};
    \node[vertex] (a6) at (11.9,1.3) [fill=black] {};
    \node[vertex] (a7) at (13.5,1.3) [fill=black] {};
    \node[vertex] (a8) at (16.1,1.3) [fill=black] {};

\draw[black] (0,0) circle [radius=1.8];
\draw[black] (0,2.6) circle [radius=1.8];
\draw[black] (2.2,1.3) circle [radius=1.8];
\draw[black] (6.4,1.3) circle [radius=1.8];
\draw[black] (10.6,1.3) circle [radius=1.8];
\draw[black] (14.8,1.3) circle [radius=1.8];

\draw[thick] (a2)--(a3)  (a4)--(a5)   (a6)--(a7);

\node[right] at (0.3,2) {$u_1^*$};
\node[right] at (3,2) {$v_1^*$};
\node[right] at (4.6,2) {$u_2^*$};
\node[right] at (7.2,2) {$v_2^*$};
\node[right] at (8.8,2) {$u_3^*$};
\node[right] at (11.4,2) {$v_3^*$};
\node[right] at (13,2) {$u_4^*$};
\node[right] at (15.6,2) {$v_4^*$};

\node at (2.4,1.3) {$H_1^*$};
\node at (6.6,1.3) {$H_2^*$};
\node at (10.8,1.3) {$H_3^*$};
\node at (15,1.3) {$H_4^*$};
\node at (0.2,0.0) {$H_1$};
\node at (0.2,2.6) {$H_2$};
\end{tikzpicture}
\end{center}
\caption{$H_{\delta,\Delta,k,m}$ for $k=4$ and $m=2$.}
\label{fig:C4-free-sharpness-example}
\end{figure}

\noindent
{\bf Proof:} Our construction is a modification of a construction given
in \cite{ErdPacPolTuz1989}.
Let $q=\delta+1$, so $q$ is a prime power. Denote be $GF(q)$ the field of
order $q$ and by $GF(q)^3$ the
$3$-dimensional vector space over $GF(q)$ of all triples of elements of $GF(q)$.
Let $H$ be the graph whose vertices are the $1$-dimensional subspaces of $GF(q)^3$.
Two vertices are adjacent if, as subspaces of $GF(q)^3$, they are orthogonal.
It is easy to verify that $H$ is $C_4$-free, has $q^2+q+1$ vertices and that
${\rm diam}(H)=2$. Each vertex of $H$
has degree either $q+1$ (if the corresponding subspace is not self-orthogonal) or
$q$ (if the corresponding subspace is self-orthogonal).

Let $H^*$ be the graph obtained from $H$ by choosing a vertex $z$ that corresponds to
a self-orthogonal subspace, two neighbours $u$ and $v$ of $z$, and deleting vertex
$z$ as well as all edges joining a neighbour of $u$ to a neighbour of $v$. It is
easy to show that $H^*$ has order $q^2+q$, that $\delta(H^*) \geq q-1$, and that
$d_{H^*}(u,v)= {\rm diam}(H^*)=4$.

For $i=1,2,\ldots,k$ let $H^*_i$ be a copy of $H^*$. We denote the vertices of $H^*_i$
corresponding to the vertices $u$ and $v$ of $H^*$ by $u_i^*$ and $v_i^*$, respectively.
For $j=1,2\ldots,\ell$ let $H_i$ be a copy of $H$, and let $w_i$ be a vertex of
$H_i$ of degree $q+1$.  Let $H_{\delta,\Delta,k,\ell}$ be the graph obtained
from the disjoint union
$H^*_1 \cup H^*_2 \cup \ldots,\cup H^*_k \cup H_1\cup H_2\cup \ldots \cup H_{\ell}$
by adding the edges $v_1^*u_2^*, v_2^*u_3^*,\ldots,v_{k-1}^*u_k^*$ and by identifying the
vertices $w_1, w_2,\ldots,w_{\ell}$ and $u_1^*$ to a new vertex $y$.
Then $H_{\delta,\Delta,k,\ell}$ has order $n=(m+k)(q^2+q)$, minimum degree
$\delta=q-1$, and maximum degree $\Delta=(m+1)(q+1)-1$, which is attained by vertex $y$.

We now bound the average eccentricity of this graph from below.
For $i=0,1,\ldots,\frac{k}{2}-1$, each of the $2(q^2+q)$ vertices in
$V(H^*_{k/2-i}) \cup V(H^*_{k/2+1+i})$ has eccentricity at least $5(i+k/2)$.
The remaining $m(q^2+q)$ vertices, which are in $(V(H_1) \cup \ldots V(H_m))-\{u_1^*\}$,
have eccentricity at least $5k$. Hence,
\begin{eqnarray*}
EX(H_{\delta,\Delta,k,m})
  &=& \big( \sum_{i=0}^{k/2-1} 2(q^2+q) 5 (\frac{k}{2}+i) \big) + m(q2+q) 5k  \\
  &=& 5k(q^2+q) \big( \frac{3}{4}k + m - \frac{1}{2} \big).
\end{eqnarray*}
We now express the two factors of the above term for $EX(H_{\delta,\Delta,k,m})$
separately in terms of $n$, $\Delta$ and $\delta$.
Let $\varepsilon_{\Delta}$ and $\varepsilon_{\delta}$ be as in Theorem \ref{theo:C4-free},
and define $\varepsilon_{\delta}' =(\delta+1)(\delta+2)$.

Since $q=\delta+1$ and $q$ is constant, we have
$n-\varepsilon_{\Delta} + \varepsilon_{\delta} -2\Delta
 = n - \Delta \delta +O(1)
 = (m+k) (q^2+q) - (m+1) - [(m+1)(q+1)-1](q-2) - 2(m+1)(q+1) - 2 +O(1)
 =k(q^2+q) +O(1)$. Hence the first factor is
 $k(q^2+q) = n-\varepsilon_{\Delta}+\varepsilon_{\delta} - 2\Delta +O(1)$.
 Now consider the second factor.
$\frac{3}{4}k + m -\frac{1}{2} = \frac{3}{4}(k+m) + \frac{1}{4}m +O(1)
   = \frac{3}{4} \frac{n}{q^2+q} + \frac{1}{4} \frac{\Delta}{q+1} +O(1)
   = \frac{3}{4} \big( \frac{n}{\delta+1)(\delta+2)} + \frac{\Delta(\delta+1)}{3(\delta+1)(\delta+2)} \big) +O(1)$. Combining these terms we obtain
\[ EX(H_{\delta, \Delta,k,m})
  = \frac{3}{4} \frac{n - \varepsilon_{\Delta} - \Delta}{\varepsilon_{\delta}'}
    \big( n + \frac{\Delta (\delta+1)}{3} \big) + O(n). \]
Dividing by $n$ now yields the statement of the theorem.
\qed

redpajama_set_name: RedPajamaArXiv


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text:
The Culling is a first-person action
battle royale game developed and published by American studio Xaviant. Following an early access beta phase, the game was released for Linux, Microsoft Windows, and Xbox One on October 5, 2017.

The Culling initially was popular, but in the wake of two other battle royale games, PlayerUnknown's Battlegrounds and Fortnite Battle Royale, the player base for The Culling waned, and Xaviant halted future development in December 2017. The company later announced a sequel, The Culling 2, which was released in July 2018, amid the popularity of Fortnite Battle Royale. It was panned by players and lacked sufficient players for full matches within 19 hours, leading Xaviant to pull the sequel from marketplaces, and instead restart work on adapting The Culling as a free-to-play title. The game servers were finally shut down on May 15, 2019, with the game no longer being purchasable on any platforms.

Gameplay
Sixteen players are placed in an arena (either a small island or a prison) full of various buildings containing loot. The maps contain caves, bridges, poison gas, explosive barrels, and more. Upon completing various tasks such as living for a few minutes, killing other players, traveling certain distances or interacting with the environment, players receive "F.U.N.C", which is the name of the in-game currency used to craft or purchase equipment and upgrades each match. When particular perks and air drops are put together they provide unique strengths and weaknesses. Players call these combinations "builds". Selecting the correct perks can provide players with an edge based on their play style.

Development
The Culling was developed by Xaviant. Having finished Lichdom: Battlemage, they saw the popularity of mods that created a battle royale game atop existing games. The Culling was envisioned to be the first standalone battle royale game on Steam, with the potential to tap into the esports market. The game, as developed, required players that had been killed off to remain in the match, which necessitated the need for Xaviant to provide a robust spectator mode for these players to watch the remaining match play out. Late in the development process, they also recognized this would help the popularity of the game in video game live streaming.
Xaviant first announced The Culling in February 2016. After a brief closed alpha period, the game was released into early access on Windows via Steam on March 4, 2016, where it quickly rose to one of the top ten games played on Steam according to Steam Spy.

In March 2017, PlayerUnknown's Battlegrounds was released on Steam in early access. Battlegrounds was another battle royale game which gained popularity quickly over the next year. This caused The Cullings player counts to drop significantly after May 2017. In December 2017, Xaviant announced that development for the game was halting, so that they could work on "a new title". Xaviant left The Culling servers open for players to continue to use.

Sequel and reboot
In mid-June 2018, Xaviant announced it was preparing to release a sequel, The Culling 2, which they had been working on since stopping development of the first game. At this point in time, Fortnite Battle Royale had been released and surpassed the popularity of PlayerUnknown's Battlegrounds, and Xaviant's decision to release The Culling 2 in the oversaturated market was questioned. Xaviant's director of operations Josh Van Veld believed that the timing was right, as it had been about two years since The Cullings initial release, making it a fresh title, and that The Culling 2 would be released without an early access period. The announcement came around the same time that leaked images from The Culling 2 had been discovered, with fans of the first game criticizing the look and feel for being too close to PlayerUnknown's Battlegrounds. Van Veld said the choice to make the game more similar to Battlegrounds was to reflect the trend that the battle royale genre had seen over the last two years. Within The Culling 2, match sizes were increased to 50 players.

The Culling 2 was released on July 10, 2018 for Windows, PlayStation 4, and Xbox One. The game was criticized by players for appearing to be an unfinished title, its release timed to hit the middle of the popularity of battle royale games. Additionally, the release occurred on the same day as the start of a new season within Fortnite: Battle Royale. Steam peak player counts for the game on its first day were around 250 players and within 40 hours the game dropped to a single player.

On July 18, 2018, Xaviant decided to pull The Culling 2 from storefronts, close down its servers and refund all purchases. Additionally, Xaviant stated they would instead retake up development and support of the original game towards making it a free-to-play title. The rebooted game, The Culling: Origins, returned The Culling to a state similar to when it was first released on early access in March 2016, including bringing back features that had been removed during its release development, as well as new graphics, features and optimizations featured in the latest build. Anyone who owns the game is granted access to The Culling: Day One, an exact replica of the game in its original state, when it was released in March 2016, with a few bug fixes. The Culling: Day One allows players to replay the game in its earliest form and is to be used as a reference point for the free-to-play release to ensure the game is as fun as its first release so it does not lose what makes it unique.

Xavient announced in March 2019 that revenue from the rebooted The Culling was not sufficient to cover their ongoing expenses, and were thus planning on closing down the servers in May 2019; offline portions of the game would otherwise remain playable.

On March 25, 2019, the title was made unavailable for purchase on Steam and de-listed from the Xbox Store.

On May 12, 2020, Xaviant announced that The Culling would return to Xbox One, re-branded as The Culling: Origins, with a "pay-per-match" model. Originally, Xaviant Games announced that players could only play one match per day without a token. However, after considerable backlash, it was announced that this would be changed to ten games per day. By Nov 13th, 2020 the servers had been shut down and Xaviant had few to no employees supporting the game.

References

External links

2017 video games
Battle royale games
Early access video games
First-person adventure games
Linux games
Video games developed in the United States
Windows games
Xbox One games
Xbox Play Anywhere games
Multiplayer online games
Unreal Engine games

redpajama_set_name: RedPajamaWikipedia


It seems like every few months, news outlets are covering another macro-level destination site that's set to transform real estate search. While I welcome disruption in our industry, I can't help but think that these folks are fundamentally misinterpreting the kind of innovation real estate professionals need, as well as the direction the business is headed.
Perhaps it's natural that real estate professionals are looking for someone to challenge Zillow. After all, Zillow wants to own real estate search on the Web, and it's well on its way to that goal: The site now accounts for over 17 percent of all real estate-related traffic, and powers the search results of several other leading sites, including Yahoo! Homes, HotPads and, most recently, MSN Real Estate.
At the same time, Zillow has also been criticized for a number of its practices, including consistently displaying inaccurate property data and pushing lead-buying platforms that both agents and consumers find disingenuous.
Nevertheless, replacing Zillow with a different real estate mega-portal is akin to overthrowing one dictator and installing another in his place. Whether it's a startup, or Trulia, or any other company, giant destination sites will always be middlemen.
Search innovations, gamified user experiences, new agent marketing schemes — none of these change the fact that destination sites are forcing themselves between real estate consumers and agents and holding both sides hostage with their massive market share.
Of course, destination sites paint predatory agents as the problem, suggesting that it's their job to "insulate" their users until they're ready for a transaction. I, for one, am tired of hearing that homebuyers and sellers need any kind of protection from real estate professionals.
Year after year, consumers have reaffirmed the value of working closely with agents and brokers. In fact, the share of homebuyers who purchased their home through an agent or broker has actually increased by almost 20 percent since 2001.
Perhaps more important, the vast majority of consumers are satisfied with their agents. In 2013, two-thirds of buyers interviewed only one agent, and 88 percent said they'd use their agent again or recommend them to others.
Clearly, consumers don't want to be insulated from agents. Yet despite this, destination sites insist on hijacking the process and keeping them at arm's length. In turn, agents have no choice but to buy into advertising and lead generation programs that favor the portal, not the professional.
Perhaps that's why consumers aren't especially loyal to any specific destination site. Sure, big real estate portals make finding a home easier. But they also make finding the right agent more difficult.
Instead of trying to get in the middle of every real estate transaction, real estate technology providers should be acting as partners, empowering agents so that they can connect with consumers on their own terms. The future of real estate technology isn't about building a dictatorship on one real estate search portal and disenfranchising everyone else.
Popular opinion aside, I'd guess even the major destination sites wouldn't want that. Our industry is too diverse, too locally driven for that approach.
Instead, it should be about democratizing the tools and strategies that destination sites have piloted, about creating powerful, flexible platforms that agents and brokers can adapt and apply to suit the needs and wants of their clients.
Agents and brokers are the true movers and shakers of the real estate industry. They're involved in their communities, they know their markets, and more than ever, they're the people consumers depend on to find the right home.
Real estate has always been and will always be a local business.
It's time technology providers see it that way, too.
Matt Barba is the co-founder and CEO of Placester, a Boston-based real estate technology company that helps real estate professionals, developers, and their partners create real estate websites.