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What do you do when your house is on fire? Play fiddle, perhaps sing that Billy Joel song? When the planet began simmering with trapped heat, young people fighting off eco-anxiety organised school strikes, they came out on the streets in protests and dharnas demanding a better future. As humanity writhed and contorted with pain unleashed by a virus that emerged from the darkest desires of an exploitative economic order or perhaps from the blind spots of scientific arrogance, we noticed headless chickens, and we witnessed fire-fighting angels. As the Fahrenheit of suffering climbed skyward we encountered untold despair but we also met hope in the shape of a stranger with an oxygen cylinder.
Where from did hope and resilience arise? How do we continue with a positive frame of mind in the face of epochal upheavals that scar and disrupt millions of lives?
The ninth World Happiness Report published earlier this year, which ranked countries by happiness in the backdrop of Covid-19, found a great amount of resilience globally with considerable national variation. It quantified happiness using 'life evaluation', a stable measure wherein respondents were asked to evaluate their current life on a scale of zero to 10, imagined as the steps of a ladder.
To understand how the pandemic affected lives, the report went deeper to study positive and negative emotions which, clubbed with 'life evaluation', arrives at a measure of subjective 'well-being'. Elsewhere, psychologist Martin Seligman, who has studied well-being, counts positive emotion, engagement, positive relationships, meaning, and accomplishment as its measurable elements. Well-being interests us most because of its policy implications.
The happiness report found trust and benevolence among the important drivers of well-being and that they also go a long way in fighting the pandemic. Think of the angels and the fire-fighters. Think of the assurance to know that you are not alone. The study asserts, 'The extent to which people trust their governments, and have trust in the benevolence of others, supports not only their ability to maintain their happiness before and during the pandemic but also reduces the Covid-19 death toll....'
However in a world of bare-knuckle competition, where existence pivots on material acquisition, trust and its reflection in happiness remains a neglected virtue. In the sphere of government and politics, authoritarian rulers spend sleepless nights mistrusting their subjects, nation-states are divided into clubs hardly trusting each other as they speed along the autobahns of economic growth while corporations chase profits. All of this has an impact on happiness which the pandemic aggravated.
So from the level of the individual right up to the state and larger groupings, the problem of trust is coded into an economic order pegged on exploitation of limited resources and the illusion of endless growth. In The Conquest of Happiness, Bertrand Russell writes, "...to be without some of the things you want is an indispensable part of happiness." This applies as much to people as it does to collectives.
Today, Russell's comment about happiness or Gandhi's oft-quoted words about need, sufficiency and greed are increasingly echoed in the works of the proponents of degrowth who warn us about the limits to which the planet can bear our hyper-competitive, material intensive and climate-disrupting lifestyles. Noted theorist of de-growth, Giorgos Kallis writes, "Limits are not a constraint for freedom, limits are a condition for freedom," meaning that the setting of limits to growth and respecting planetary boundaries, does not curtail freedom but is necessary for freedom.
Readers of dystopian fiction know well that the trade-off between freedom and happiness is complicated but we all realise that GDP and growth figures are not the panacea for suffering. In fact, as advocates of de-growth would argue, GDP and growth constitute a wrong scale and it is possible to augment quality of life and happiness while limiting economic expansion.
The 2021 report on happiness has devised a metric called Well-Being-Adjusted Life-Years (or WELLBYs) for individuals and nations, which could provide one entry point to challenge the dominance of economic growth-based approaches to policy. WELLBYs marry well-being (from any reason) to the length of life and among other things assign a lower than usual value to money compared to the number of years one lives.
In the pandemic's context, trust, connectedness, volunteering, community and equality emerged among other conditions that foster well-being. From climate activist to dreamy-eyed solar punk, many believers of a hopeful future extol a similar set of values. It is time now to explore this common ground, to seek the spot where the trail of happiness meets the stream of a clean green tomorrow.
Rajat Chaudhuri
Writer and climate activist


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Fork Horn
Thread Starter
Join Date: Jul 2009
Location: Oh
Posts: 193
I see the new Parkers are starting to let out information. The Tornado is about 2.5 inches wider and the Cyclone has been shortened and shoots about 10 fps slower due to the changes. A new camo has been added also I see from reading the article on another site.
You been gleaning the sites for "experience" again, eh whatsyourname? The Parker fans already know that. But since you're not one of them, why bother to post, dufus?
Last edited by Cossack; 12-31-2009 at 06:16 AM.

subset: commoncrawl


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Only two (2) tiny portable 1-pound disposable propane cylinders are allowed to be stored within your home or attached garage. All other propane cylinders must be stored in an unattached garage or shed outside your home.
Is it a good idea to keep propane in the house?
Propane tanks should not be kept in a shed, garage, basement, or attic, or carried indoors for any reason; they should only be kept in a dry, open, well-ventilated place outside. It's only safe to keep your propane tank inside if it's been entirely depleted. Keeping a tank indoors might cause the temperature inside to rise, posing a fire risk. It's never a good idea to leave your propane tank in your car.
Keep dust and debris off your barbecue during the off-season by covering it with a grill cover or a tarp fastened with rope. To protect a freestanding grill from rusting, place it in a dry, cool location of the garage or shed. If you're looking for a storage shed for a grill, go no farther than LP Outdoor Building Solutions' extraordinarily durable solutions, which may elevate your storage building or other structure above the rest.
Grilling causes a yearly average of 8,900 home fires, according to the National Fire Protection Association, with the majority of incidences occurring from May to August. Don't let a fire stop you from grilling this summer! Additional grilling safety advice can be found here.
Get Outdoor Grill Storage From LP
The shed components from LP Outdoor Building Solutions can be combined in practically unlimited ways to compliment any property. Choose from a variety of sheds, barns, and other structures. We invite you to speak with a local shed dealer near you to get the building that's ideal for you.
How long can a 20-pound propane tank be stored?
We get asked all the time how long propane can be stored before it spoils.
The basic reason is that propane has a long lifespan. It is, in fact, one of the most storage-friendly, reliable, and durable energy sources currently available. Propane has an unusually long shelf life because it has no expiration date and does not degrade or lose its potency over time. To clarify, "shelf life" is defined as "the amount of time a commodity can be stored before it becomes unfit for use, consumption, or sale" (Wikipedia). Unlike diesel, which has a one-year shelf life and gasoline, which has a three- to six-month shelf life, propane can be stored indefinitely, making it a highly effective and convenient fuel for both residential and commercial applications, as well as an excellent partner in emergency preparedness plans.
Furthermore, the cylinders in which propane is kept have a very long shelf life. This is why propane cylinders must be recertified by a skilled propane technician on a regular basis. When regular cylinder maintenance is followed, it is possible to get 30 or 40 years out of a stored propane tank.
Home heating, portable cylinders for BBQ and gardening/landscaping, space heating, forklift, autogas, and emergency preparedness are just a few instances of how propane outperforms other fuels in terms of convenience and longevity.
Oil-fired heating systems are colossal, filthy behemoths that take up valuable space and necessitate regular maintenance. If heating oil is stored for an extended period of time, it might become dangerous. What is "too much time"? Think 18 months or fewer, depending on how properly it was stored in the first place.
BBQ cylinders are widely used. They're lightweight, portable, refillable, and easy to store. They can last for years if stored properly, with no need for replacements or additives.
Propane in portable, tiny cylinders up to and including 1,000 gallon cylinders can be used for emergency preparedness.
Space heaters and generators powered by diesel or gasoline emit a foul odor, can cause messy spills and stains, and must be replaced on a regular basis. However, not propane! Its long shelf life makes it an ideal fuel for portable use or powerful backup generator systems that can power a whole house.
It's no surprise that propane is popular among homeowners and businesses because of its convenience, versatility, and peace of mind.
Visit our Residential and Commercial sections to learn more about all the wonderful things propane can do for you.
At Paraco, we've got you covered with daily fuel.
Is it possible to keep a 20-pound propane tank indoors?
If you're winding up your barbeque season with your propane tank and storing your grill outside, you can keep the gas connected as long as you turn the tank off. Turn off your propane tank and remove it from the grill if you want to store it indoors.
Because propane gas is heavier than air, it's critical that you never store your propane tank indoors. This implies that if your propane tank leaks, there is a considerable risk of serious injury.
When you're putting your tank outside, you'll want to make sure it's flat and protected from the elements as much as possible. You can store your propane tank in a milk box to guarantee that it stays upright while being stored, which will boost the stability of the space where you're storing it.
Check your propane tank before storing it away
Regardless of how old your propane tank is, you must inspect it before selecting to store it. Look for any evidence of wear or leaks in the valves, the tank itself, and the connector hose.
After the tank has been detached, you can check for leaks by putting a bottle of non-abrasive soap on it. The non-abrasive soap combination should be sprayed on the tank, around the valves, and around the hose.
If you notice any bubbles, it means there's a leak in the tank accessories or the tank itself. You can spray the tank again to make sure there are no leaks after double-checking that the cylinder valve is completely shut off and that you've disconnected all of the connections.
If bubbles continue to occur on the connections or the tank, call a propane supplier to find out where you can safely dispose of the tank.
How do you keep propane in your home?
- Propane cylinders should be stored outside. Never store them inside or in a confined space like a garage, shed, or basement.
- Keep them somewhere cool. Heat should not be applied to the cylinders above 120 degrees. This temperature could result in a fire or a leak.
- Propane cylinders should be kept away from open fires. This includes smoking and the use of spark-producing devices. Any flame has the potential to induce combustion.
Is it permissible to keep a propane tank in the garage?
Do you use cylinder-style propane tanks for your grill? If that's the case, you're well aware of the many benefits of grilling with propane, including its ease, convenience, and excellent outcomes.
These propane tanks can be used for a variety of purposes around your home in the Seacoast region. Outdoor fireplaces and firepits, pool and spa heaters, patio and deck warmers, insect traps, and portable generators can all be powered by them.
However, there are a few things you should know about storing them safely. We've compiled a simple, user-friendly dos and don'ts list.
Dos and Don'ts of propane tank storage
DO NOT expose propane tanks to temperatures over 120 degrees Fahrenheit. It could result in a leak or, even worse, combustion.
DO NOT keep propane tanks in a garage, basement, carport, shed, sunporch, or carport.
DO NOT put propane tanks on their sides, since this might cause liquid and vapor leakage.
DO NOT put propane tanks on wet ground or any other wet surface, since this can cause the tank to rust and pit, rendering it ineffective if not lethal.
When taking propane tanks to be refilled or exchanged, keep them upright and secure.
DO NOT STORE PROPANE TANKS IN CLOSED AUTOMOBILES. That implies you can't conduct any other errands while getting your propane tank exchanged or refilled.
Make sure you have your gas tank professionally refilled by a licensed propane distributor.
REFILLING OUTDATED TANKS WITH EXPIRED CERTIFICATION IS NOT A GOOD IDEA. It's against the law to do so. Propane cylinders must be recertified 12 years after their manufacture date, and then every five years thereafter.
Is it permissible to leave a propane tank outside during the summer?
Propane tanks can be safely stored outside, but it's important to find a location that's not too close to your home. When it comes to storing your propane tanks in the winter, it's crucial to remember that freezing temperatures aren't a concern for propanein fact, you don't even need to cover your tank if you're storing it outside.
Place the tank on a solid surface, such as a piece of wood or a slab of concrete, and store it open or closed.
It's also simple to store propane tanks during the summer. During the summer, your propane tank can be kept outside on a flat, firm surface. Keep the tank in a shady area so it doesn't sit in direct sunlight for lengthy periods of timethis will keep the tank at a safe temperature, which should not exceed 120 degrees Fahrenheit (49 degrees Celsius).
Is it true that propane tanks can explode?
Although propane is flammable and can cause an explosion, a propane-LPG tank explosion is extremely unusual. Propane tanks (gas cylinders) can explode, however this does not happen frequently. It is extremely difficult for a propane tank to explode.
Is it possible to leave the propane tank connected to the grill?
You'd be shocked how many calls we get from customers who forget to turn off their propane tank valve while they're not using their barbecue.
Similarly, many customers who own natural gas grills leave the supply line shut-off valve open when the grill is not in use.
Natural gas lines and propane tanks are both available "For gas grills, there are a variety of fuel sources. Many consumers abandon their fuel supply "do so for the sake of convenience. Before and after grilling, there's one less thing to switch on or off.
Regardless of the fuel source, it's critical to switch off the gas supply to the grill when it's not in use for safety reasons.
If someone or something turned on one of the grill's control knobs, it would fill with gas, perhaps resulting in an extremely deadly situation. In the best-case scenario, the buyer would have squandered his or her petrol.
Rodents and wildlife have a role in another scenario. This one may appear amusing or unlikely until it occurs to you. It's not uncommon for wildlife to gnaw on gas hoses, and if an animal does so on a hose that's linked to a fuel source that hasn't been shut off, the consequence may be disastrous "When a gas valve is turned off, a gas leak can occur, which is a dangerous condition.
Using a grill as a fuel source "The convenience of "on" is overwhelmed by the risk of danger, and how long does it take to flip a valve in the first place?
Would you open your car's gas tank hatch to save a few seconds at the petrol station? Would you leave your front door wide open while you're gone because turning the door knob takes too long when you return? Don't be a fool…turn off the gas to your Weber Gas Grill!
Is it possible for propane tanks to explode in the sun?
Yes, they are capable. Temperatures can quickly soar on a hot summer day. The pressure within the propane tank will rise as the tank heats up. Despite the fact that portable propane tanks include safety relief valves to release pressure, the best place to store a tank is out of direct sunlight. Outdoors, in the shade, and in cool temps is the safest place to store a propane tank.
Is it possible to stack propane tanks?
Propane cylinders should always be placed in such a way that the relief valve is in direct contact with the container's vapor space. The relief valve on cylinders that are stored vertically upright will communicate with the vapor space. When it comes to the cylinder relief valve, it's critical to keep it aimed away from anything that escaping propane can come into touch with, perhaps creating a fire or disaster. If cylinders are placed incorrectly, such as on their side with the relief valve in touch with propane liquid, and the safety relief valve opens for whatever reason, liquid instead of vapor will escape, providing a considerably larger risk. In a nutshell, make sure the relief valve is connected to the tank's vapor space. In all cases, consumer propane cylinders should be stored properly upright.


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Time going very fast growing social thousand simply arenвt available in the world with convenience. Badoo also allows you to make any of that and instead, we allow guest chatting. The site it isdesigned solely to find a soul mate of your knowledge, maintaining your online pals can like, remark and share them in real life or just does not offer an innovative and exciting friendships. Once you try to impersonate you.
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Aloe vera plants are succulents, so use a well-draining potting mix, such as those made for cacti and succulents. Do not use gardening soil. A good mix should contain perlite, lava rock, chunks of bark, or all three.
Furthermore, how do you make cactus soil for aloe vera?
Mix 2 parts washed construction sand or pumice, 1 part manure or compost and 1 part shredded leaves, bark or other organic matter. To ensure a continuous supply of nutrients, fertilize regularly or plan to re-pot your aloe vera every year to two years.
In this manner, how do you prepare the soil for an aloe vera plant?
You have to make a few changes to use this soil for the aloe vera plant.
- Mix 3 part cactus mix with 1 part cocopeat and 1 part dry compost.
- Add 1 tea-spoon low nitrogen NPK 5-5-5 or similar to this soil mix.
- Spray little water to this soil mix. …
- Leave the soil mix to completely dry for 3 days.
Are coffee grounds good for aloe plants?
Are coffee grounds good for my Aloe vera plants? No, Aloe vera do not like coffee grounds. Aloe veras tolerate soils that are slightly acidic to slightly alkaline, but seem to do better in neutral to slightly alkaline soils.
Do aloe plants need direct sunlight?
Aloe vera is a succulent plant species of the genus Aloe. … Before you buy an aloe, note that you'll need a location that offers bright, indirect sunlight (or, artificial sunlight). However, the plant doesn't appreciate sustained direct sunlight, as this tends to dry out the plant too much and turn its leaves yellow.
Can I use normal soil for aloe vera?
Like cacti, succulents do best in dry conditions. When growing aloe vera plants, plant them in a cactus potting soil mix or a regular potting soil that has been amended with additional perlite or building sand. Also, make sure that the pot has plenty of drainage holes. Aloe vera plants cannot tolerate standing water.
Do aloe vera plants need big pots?
Aloe plants are a helpful succulent to keep around the house, not only for their beauty, but also for their well-known healing properties. … Aloe plants can range in size from very small ones that can fit comfortably in a 3-inch pot to large plants that need a 6-inch diameter pot or larger.
What does an overwatered aloe plant look like?
Overwatering Aloe Vera
When an aloe plant is being overwatered, the leaves develop what are called water-soaked spots that look soggy and soft. It is almost as though the entire leaf becomes saturated with water, then it turns to mush.
Are eggshells good for aloe vera plants?
So, are eggshells good for Aloe Vera plants? Yes, as long as you wash away the yolk/white then they are perfectly fine to give to your Aloe Vera plants. They will even give your succulent an extra boost of potassium.
Should I cut the brown tips off my aloe plant?
Trim off any leaf tips or whole leaves that have turned pinkish-brown. These parts are dying, so removing them helps the aloe plant stay healthy and green. Use a knife for small and medium-sized plants, or sheers for large, thick leaves.
When should you repot an aloe plant?
These are smaller offshoots of the mother plant that are still attached to the main root system but can live on their own as full plants. If your main aloe plant is starting to look leggy and droopy and is surrounded by smaller pups, it's definitely time to transplant.
How much sun do aloe plants need?
Light. Light is where many succulent gardeners fall short of the needs of their plants. It is critical that you place your aloe in a window where it will receive a minimum of six hours of sunlight per day. Without extended, direct light, your succulent will begin to stretch and lose its attractive, compact form.
Why does my aloe plant not stand up?
Too much water can also be an issue and lead to an aloe plant flopping over. A simple watering strategy for aloe is to wait for the soil to dry out entirely and then wet it completely. Tip out any excess water. … A shallow container won't allow the plant to develop enough strong roots to remain upright.
How much do aloe plants cost?
Prices typically range from just $4.50 to $10 for one of their baby Aloe plants under 3" tall. Please note you'll need appropriate pot and soil mix ready to plant on delivery.


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JEAN-GABRIEL DOMERGUE (1889-1962) ALAIN SAINT-OGAN (1895-1974) Original drawings signed
Лот 930
1p.18 x 6 cm
Nude, original drawing signed "Jean Gabriel Domergue" lower right, brown ink and pen.
On the back, a comic strip by Alain Saint-Ogan. Two original drawings, blue ink, pen and lavis, with autograph captions, representing a horse, lying down and then running. Signed and dated lower right "Alain Saint-Ogan March, 1949 : "Marcel, the horse of Zig et Puce is lazy, but he likes good music and... to go and listen to Pasdoc he does what everyone else does, he runs down rue des Petits-Champs...".
Joseph Charles Louis Jean Gabriel Domergue, a French painter and engraver often described as the "inventor of the pin-up" because of his portraits of wide-eyed, slender-necked Parisian women in the 1920s, was a sought-after portraitist in aristocratic circles and also worked as a designer for famous couturiers including Paul Poiret and Henry Marque. In 1955, Domergue was appointed curator of the Jacquemart-André Museum in Paris, and was later made a Chevalier de la Légion d'honneur and a member of the Académie des beaux-arts.
Alain Saint-Ogan, French comic strip author and illustrator. A major figure in European comics, he began writing comics in 1925, creating Zig et Puce (two young heroes and their companion Alfred the penguin, constantly travelling around the world, trying in vain to get to America to make their fortune) for the family weekly Dimanche-Illustré. This was the first important European series with a text written only in speech bubbles, and from 1927 onwards it was to enjoy unprecedented success. None of the many other characters he created, such as L'ours Prosper or Monsieur Poche, have achieved comparable fame.
The singer André Pasdoc, whose real name was Dimitri de Salkoff, ran the Parisian cabaret L'Echanson from 1948 to 1960, at 49 rue des Petits-Champs.


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Jaimie De Lutiis in the Junior World Final
September 8, 2023
Exciting news from one of TIGS star swimmers. Jaimie De Lutiis has received her first World Junior Aquatic Championship medal, receiving a silver with her 4 x 200 relay team and has qualified for the final of the 200m Freestyle.
Congratulations to Jaimie and our best wishes in the final.


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If your small business does not yet have a website, you may find this post helpful in gaining understanding of where and how you need to start and what to expect. With today's heavily digitized society, it is no longer enough to just have a yellow page or newspaper listing. To gain new customers, your business will need some dedicated presence online. Otherwise, most online users won't know if you have a business.
What we going to suggest may sound like a lot of information but it is necessary to prepare yourself an effective plan for your website. Planning is everything. Without it success is left to a chance. Who runs a business based on random events? Any website owner who wants to succeed online will need a plan that promises success. A successful plan will give you a clearer direction of accomplishing your online business objectives.
We believe developing an effective plan for a new website need to take minimally into account the following:
Obviously, you may not have the knowledge and experience in gathering such information so you may consider hiring a web agency. Although this post focuses on pointing out the tasks involved or should be involved in planning a website, we will try our best to indicate when help from a third party will be helpful. If you can do all the tasks in the planning stage of the website development, it's even better because you don't have to pay someone do that for you.
You want a website for your business for a reason or a number of reasons. Right? No one knows your business and its web requirements more than you do. So you develop a clear objective for your website. Think of what you want your website to do. Do you want it simply present information (such as the services or products you offer, business hours, contact information, etc.)? Do you want to sell on your website? Do you want your website to let your patients make and mange appointments online?
Be clear about what your website needs to accomplish. Make sure it is achievable. A goal that just simply states I want a new website because I want my business to make more money is not probably sufficient. First of all, most websites are created for some financial gain. If you primary driver for your new website is to make more money, you may be disappointed if your website fails to make you money as you anticipated.
The fact is a new website does not start making you money immediately. A new website does not mean more immediate business or money. There are no short-cuts to making money from an online presence. Think of your business. If you built it from scratch, think of the hard work you put into it. The return on your investment was not immediate. It took time and effective management skills to build the customer base. Similarly, a new website takes time to build online recognition of your business before you see any real return on your investment.
To keep disappointments to minimum, we suggest set the right objective from the start. For a new website, define your primary objective in terms of what you want your website to do (and to what extent) for your customers, not what the website do for you, especially financially – as we pointed out in the previous paragraph.
The success of your business is dependent on your customers. If your customers come to you because they rely on the quality of your product or services, you can expect to stay in business for long time to come. Similarly, your online success is driven by your online customers or audience. Your online customers will come back to your website if your site convinces them you have what they need now (or in the future).
A website should be targeted to the right audience to grow online, as much as it is true off-line. Are your customers residential or commercial? Are they doctors? Are they car owners? Are they senior citizens? Are they patients? Are they children? Are they teens? Are they listeners (radio listeners) or viewers (TV viewers)?
By knowing who you want to serve online, you have a better idea of what content to serve and in what format. For example, website that is targeted to doctors should assume the doctors know more than patients do. So there is no need to explain medical terms that the average person does not know. On the other hand, a website targeting patients should keep medical jargon to minimum.
It is often said the content is king or content is everything. Content is what your potential customers see, hear, view, do, or even feel when they come to your proposed website. For your website to be successful, your website content has to be relevant to your customer needs or desires.
It is worth mentioning studies indicate on average people spend no more than 10 seconds to browsing a website they visit the first time. During this first brief visit, they can quickly decide they want to stay or leave the website. Most site users leave within 10 seconds because they did not immediately find what they were looking for. Suppose your website is called EzPainting.com and a customer lands on this website. He wants to have his house painted but he quickly discovers the site is targeting only commercial building clients, not residential customers. So the customer immediately moves to the next website.
So how does the customer know that the website is not for residential clients? There probably is more than one clue on the home page of the website. First, the site slogan may indicate the targeted customers. For instance, if the slogan reads "for all of your residential painting needs" or "for all of your commercial painting needs" is a very clear way to indicate the targeted customer. Also, there are other clues, such as images. The images may show the employees painting large buildings and commercial structures. Finally, a quick glance at the text on the home page can indicate the targeted audience of the website of this business.
So how do you make your website relevant to your customers or their needs? Regardless of your business, assume you want to serve three special audiences:
Your website's content should address the needs of minimally of the first two audiences because they offer you the best chance of brining you a positive return on your online investments. It is true that initially what you add to the website will be new to everyone. However, you want to do your best to write content that best addresses the needs of your potential customers. One of the ways, you can do this is by reviewing your offline selling practices. How do you sell to your new customers? Think of all the steps you take to ensure why you are better than your competitors. You take a similar approach but with written words, videos, animations, and graphics to attract, retain, and hopefully sell to your potential online customers.
If you have trouble coming up with what content you need to have for your website, try visiting your competitors' websites for ideas. As you observe these websites make a note of the things you like and dislike. Obviously, you want to keep your website content to be unique so we here are not suggesting that your website has to be similar to what your competitors have.
As we briefly discussed in the previous section, keep an eye on your competitor websites. Why should you have to visit your competitor websites? The short answer is that you should know what they have done on their website so you don't do something similar, particularly what has not worked for them. The bottom line is you want to avoid using techniques, technologies, and suppliers for your website that don't work for you competitors. A few years ago there was a well known European company that used a web content management system that made it harder for the website to appear in top search results. There were other companies that experienced the same problem and had to invest additional recourses to rectify the problem.
In your planning stages, if you avoid considering solutions that are known to be problems for others, you can help yourself stay ahead of competition and reduce risks of failures. Talk with your web agency about clarification on technical jargon or concepts to get a better understanding of what's out there that works and does not work.
You should have some understanding of what hardware and software you will need to successfully run your website. Ideally, your web development agency will discuss with you what technical solutions you will need based on what you want to accomplish online.
When it comes to choosing a particular web solution, consider:
It should not be a surprise that hardware, software, and internet access costs have reduced dramatically and continue to decrease with time. But it does not mean running a website is free or almost free. At the very minimum, you will incur all or most of the costs shown in table 1. In constructing this table, we are basing the costs on personal experience for a small business website. Depending on the complexity involved, your actual costs of running a website could be substantially lower or higher than what is represented in the table.
Table 1 typical costs of running a website | ||
Costs | Low, medium or high? | Ongoing? |
Acquiring a domain name | Low | Yes – typically annually |
Web server (also web hosting) | Low – medium | Yes – monthly, semiannually, or yearly |
Initial web site design | Medium to high | No |
Marketing the website | Low to medium | Yes- daily, weekly, monthly, and so on depending on the advertising media. |
Website maintenance and monitoring | Medium to high | Yes |
Low = $15 Medium = $100 High = greater than $100 |
Don't forget the launch date for your website. If you want to launch your website next week, it's probably too late to do any real planning! You need ample time not only to do your serious planning but also a consideration for the development, testing, and launch of the website. Based on our experience, we expect the time allocation provided in table 2 for each task leading up to the launch of the website.
Table 2 time guide to website launch date | ||
Task | Suggested time required | Who performs this task? |
Website planning: purpose/concept, functions/features, budgeting, etc. | Low to medium | You |
Developing content | High | You |
Website design, development, and testing | Medium to high | Your web agency |
Website review | Low to medium | You |
Implementing the changes you propose. | Low to medium | Your web agency |
Launching | Low to medium | Your web agency |
Low = a few weeks Medium = month High = over a month You = small business owner |
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After 'Kristallnacht' letter, venture capitalist issues apology, but no retraction
Tom Perkins, co-founder of the Silicon Valley firm Kleiner Perkins Caufield & Byers, earned scorn for comparing recent protests in and around San Francisco to Nazi Germany.
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text:
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On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically
Journal of Inequalities and Applications volume 2023, Article number: 4 (2023)
In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the \(\Delta ^{2}\), which will be useful to obtain the convexity results. We examine the correlation between the positivity of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\) and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of \((2,3)\), \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\). The decrease of these sets allows us to obtain the relationship between the negative lower bound of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\) and convexity of the function on a finite time set \(\mathrm{N}_{w_{0}}^{\mathrm{P}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots , \mathrm{P}\}\) for some \(\mathrm{P}\in \mathrm{N}_{w_{0}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots \}\). The numerical part of the paper is dedicated to examinin the validity of the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\) for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem.
1 Introduction
The recent development of pure and applied mathematics is characterized by increasing attempts to use mathematical modeling tools of fractional order in different engineering, medicinal, and biological fields. It is known that this mathematical modeling of fractional order attempts to improve our understanding of more and more complicated phenomena, and in general, it is based on ordinary and partial differential equations. In the past several decades, some algorithms have been proposed for solving such fractional problems, which can be classified into different categories (see [1–4]).
Discrete fractional problems and discrete fractional operators are worth studying in the setting of fractional calculus and have attracted much attention from scholars. Furthermore, the concept of discrete fractional calculus was rooted in the few decades; it gets attention of many researchers recently because of their inquisitive thinking (see [5–7] and references therein). The main reason is that these problems and operators have a wide range of practical applications, such as mathematical analysis [8, 9], stability analysis [10–12], probability and statistics [13–15], geometry [16, 17], ecology [18, 19], and topology [20–22].
Positivity, monotonicity and convexity analysis plays a crucial role in discrete fractional calculus theory. Although a number of papers have been contributed to the analysis of discrete fractional operators with singular and nonsingular kernel type, the question of positivity, monotonicity, and convexity of discrete fractional operators of Riemann–Liouville type on a time set still remains open. Furthermore, the positivity, monotonicity, and convexity analysis is important in understanding the nature of the discrete fractional problems from the perspective of continuous fractional problems. Many authors have developed many interesting results on optimality and duality in the setting of Riemann–Liouville and Liouville–Caputo fractional differences; see, for instance, [5, 23–26] and the references therein.
Let \(\mathrm{N}_{w_{0}}:=\{w_{0},w_{0}+1,w_{0}+2,\ldots \}\) and \({{\mathscr{G}_{w_{0}}(\mathrm{f}):=\{ \mathrm{f}:\mathrm{N}_{w_{0}}\rightarrow \mathcal{R}\text{ for } w_{0}\in \mathcal{R} \}}}\). There is a clean and clear correlation between the convexity of a function \(\mathrm{f}\in \mathscr{G}_{w_{0}}(\mathrm{f})\) and the nonnegativity of the \(\Delta ^{2}\) difference of f, given in the following relationship:
Particularly, convexity has been studied in fractional calculus quite extensively. However, with discrete fractional calculus theory and discrete operators, this area has not received a lot of attention so far. Moreover, in terms of numerical simulations, some works can be found for monotonicity analysis of the discrete fractional calculus operators (see [27–29] and references therein) or convexity analysis of these operators (see [30–32] and references therein).
The main objective of this paper is to provide a relationship between the positive lower bound of the fractional difference operators of delta Riemann–Lioville type \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\) and the convexity of a function f on an infinite time set \(\mathrm{N}_{w_{0}}\) and a relationship between the negative lower bound of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\) and convexity of a function f on a finite time set \(\mathrm{N}_{w_{0}}^{\mathrm{P}}\). Besides, we present some numerical simulations for the negative lower bound case to demonstrate the solution spaces of the defined sets on the negative lower boundedness of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\).
The paper is organized as follows. Section 2 contains main properties and notations of discrete fractional operators of delta Riemann–Lioville type. In Sect. 3, we state and prove our analytical results by defining the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\), studying the main lemmas and theorems on the sets, and examining the delta convexity results of the proposed difference operators. Section 4 deals with numerical results including eight illustrative figures, in which the time steps will be applied on the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\) as applications. Also, the domains of the solutions are determined for both sets. Finally, in Sect. 5, we provide a conclusion along with future work possibilities in this field.
2 Basic tools and results
In this section, we briefly consider the discrete fractional sums and differences in the setting of Riemann–Liouville. We refer the reader to [5, 12, 33, 34] for the relevant details.
For \(\mathrm{f}\in \mathscr{G}_{w_{0}+\alpha}(\mathrm{f})\) with \(\alpha >0\), the Δ sum operator of order α can be defined as follows:
where \(\mathrm{t}^{\underline{\alpha}}\) is defined by
such that the right-hand side of this identity is well defined. Besides, we use \(\mathrm{t}^{\underline{\alpha}}=0\) when the numerators in each identity is well-defined but the denominator is not. Further, we have
Definition 2.1
Let \(\mathrm{f}\in \mathscr{G}_{w_{0}}(\mathrm{f})\). Then \((\Delta \mathrm{f} )(\mathrm{t}):=\mathrm{f}( \mathrm{t}+1)-\mathrm{f}(\mathrm{t})\) for \(\mathrm{t}\in \mathrm{N}_{w_{0}}\) is the Δ difference operator. In addition, the Δ fractional difference of order α (\(\aleph -1< \alpha <\aleph \)) of the Riemann–Liouville type is defined by
The following theorem is an alternative representation of the Δ fractional difference (2.4), which is also a generalization of the result established for \(0<\alpha <1\) in [25].
Theorem 2.1
For \(\mathrm{f}\in \mathscr{G}_{w_{0}+\alpha}(\mathrm{f})\) with \(\aleph -1<\alpha <\aleph \), the Δ fractional difference of order α of the Riemann–Liouville type can be given by
The result was proved by Mohammed et al. [25, Theorem 1] for \(\aleph =1\) (that is, for \(0<\alpha <1\)), and their result is
For \(\aleph =2\) (that is, for \(1<\alpha <2\)), by Definition (2.4) we find that for each \(\mathrm{t}\in \mathrm{N}_{w_{0}+2-\alpha}\),
where we have first used (see [25, Lemma 1])
and then used
The same procedure can be repeated \(\aleph -1\) times to obtain the required result stated by Theorem 2.1. □
3 Negative lower bound results
Let us state and prove our main lemma concerning the \(\Delta ^{2}\) fractional difference.
Lemma 3.1
Let \(\mathrm{f}\in \mathscr{G}_{w_{0}}(\mathrm{f})\), \(\alpha \in (2,3)\), and \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha}\mathrm{f} )( \mathrm{t})\geqq 0\) for \(\mathrm{t}\in \mathrm{N}_{w_{0}+3-\alpha}\). Then, for \(\mathrm{t}:=w_{0}-\alpha +3+\mathrm{k}\) with \(\mathrm{k}\in \mathrm{N}_{0}\),
By Theorem 2.1 and (2.3) we have
where we have used \((-\alpha -1)^{\underline{-\alpha}}=0\). By the same technique as before, we can deduce
where we have used \((-\alpha -1)^{\underline{1-\alpha}}=0\). Since \((\mathrm{t}-\mathrm{s}-1)^{\underline{1-\alpha}}=0\) at \(\mathrm{s}=\mathrm{t}+\alpha , \mathrm{t}+\alpha -1\), (3.4) becomes
By the assumption \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha}\mathrm{f} )( \mathrm{t})\geqq 0\) it follows that
For \(\mathrm{t}:=w_{0}-\alpha +3+\mathrm{k}\) with \(\mathrm{k}\in \mathrm{N}_{0}\), it becomes
which is the required (3.1). Now it is clear that for \(2<\alpha <3\),
for \(\imath =0, 1,\ldots ,\mathrm{k}\) and \(\mathrm{k}\in \mathrm{N}_{0}\). Thus our proof is complete. □
Based on this lemma, we now can present the Δ convexity result.
Theorem 3.1
If \(\alpha \in (2,3)\) and \(\mathrm{f}\in \mathscr{G}_{w_{0}}(\mathrm{f})\) satisfies \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha}\mathrm{f} )( \mathrm{t})\geqq 0\) for all \(\mathrm{t}\in \mathrm{N}_{w_{0}+3-\alpha}\), \(\mathrm{f}(w_{0})\leqq 0\), \((\Delta \mathrm{f})(w_{0})\geqq 0\), and \((\Delta ^{2} \mathrm{f} )(w_{0})\geqq 0\), then \((\Delta ^{2} \mathrm{f} )(\mathrm{t})\geqq 0\) for \(\mathrm{t}\in \mathrm{N}_{w_{0}}\).
We will prove by using strong induction. From the assumption we know that \((\Delta ^{2} \mathrm{f} )(w_{0})\geqq 0\). We assume that \((\Delta ^{2} \mathrm{f} )(w_{0}+\imath )\geqq 0\) for \(\imath =0,1,\ldots ,\mathrm{k}\). Then Lemma 3.1 gives that \((\Delta ^{2} \mathrm{f} )(w_{0}+\mathrm{k}+1)\geqq 0\). Thus the proof is done. □
This theorem has demonstrated a correlation between the nonnegativity of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha}\mathrm{f} )( \mathrm{t})\) and the convexity of f. Now we wish to investigate what happens by replacing the zero lower bound of Theorem 3.1 with a negative lower bound −ϵ for some \(\epsilon >0\).
Firstly, we need a lemma concerning the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\) defined by
respectively, for some \(\epsilon >0\) and \(\mathrm{k}\in \mathrm{N}_{0}\).
Lemma 3.2
The collections \(\{\mathscr{H}_{\mathrm{k},\epsilon} \}_{\mathrm{k}=0}^{ \infty}\) and \(\{\mathscr{M}_{\mathrm{k},\epsilon} \}_{\mathrm{k}=0}^{ \infty}\) are decreasing for all \(\epsilon >0\). Moreover,
Suppose that \(\alpha \in \mathscr{H}_{\mathrm{k}+1,\epsilon}\). Then we have
Also, we know that
since \(\alpha \in (2,3)\),
and by Lemma 3.1,
Thus \(\mathscr{H}_{\mathrm{k}+1,\epsilon} \subseteq \mathscr{H}_{ \mathrm{k},\epsilon}\), and hence the collection \(\{\mathscr{H}_{\mathrm{k},\epsilon} \}_{\mathrm{k}=0}^{ \infty}\) is decreasing.
Similarly to the above proof, we can prove that \(\{\mathscr{H}_{\mathrm{k},\epsilon} \}_{\mathrm{k}=0}^{ \infty}\) is decreasing as well.
To end the lemma, following Theorem 3.4-1 in [35], we have
Hence, for \(\epsilon >0\), there exist \(\mathrm{k}_{1}:=\mathrm{k}_{1}(\epsilon )\geqq 0\) and \(\mathrm{k}_{2}:=\mathrm{k}_{2}(\epsilon )\geqq 0\) such that \(\mathscr{H}_{\mathrm{k},\epsilon}=\emptyset \) for all \(\mathrm{k}\geqq \mathrm{k}_{1}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}=\emptyset \) for all \(\mathrm{k}\geqq \mathrm{k}_{2}\). Consequently,
as required. Thus the proof is done. □
Based on the above lemma, we now state and prove our convexity results.
Theorem 3.2
Suppose \(\alpha \in (2,3)\) and \(\mathrm{f}\in \mathscr{G}_{w_{0}}(\mathrm{f})\) satisfies
and some \(\mathrm{P}\in \mathrm{N}_{w_{0}+3-\alpha}\) such that \(\epsilon >0\) and \(\mathrm{f}(w_{0})\leqq 0\). If
\((\Delta \mathrm{f})(w_{0})\geqq 0\),
\((\Delta ^{2} \mathrm{f} )(w_{0})\geqq 0\), and
\(\alpha \in \mathscr{H}_{\mathrm{P}-w_{0}+\alpha -3,\epsilon}\),
then \((\Delta ^{2} \mathrm{f} )(\mathrm{t})\geqq 0\) for all \(\mathrm{t}\in \mathrm{N}_{w_{0}}^{\mathrm{P}-3+\alpha}\).
By the assumption that \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha}\mathrm{f} )( \mathrm{t})\geqq \epsilon \mathrm{f}(w_{0})\) in (3.5), changing the variable \(\mathrm{t}:=w_{0}-\alpha +3+\mathrm{k}\) for \(\mathrm{k}\in \mathrm{N}_{0}\), we have
From the condition (i) and (3.3) it follows that
Since \(\alpha \in \mathscr{H}_{\mathrm{P}-w_{0}+\alpha -3,\epsilon}\) by condition (iii) and
by Lemma 3.2, it follows that
for \(\mathrm{k}\in \mathrm{N}_{0}^{\mathrm{P}-w_{0}+\alpha -4}\). Consequently, by condition (ii), the fact that \([-\mathrm{f}(w_{0}) ]\geq 0\), (3.2), and (3.13) in (3.12), we obtain
and thus \((\Delta ^{2} \mathrm{f} )(\mathrm{t})\geqq 0\) for all \(\mathrm{t}\in \mathrm{N}_{w_{0}}^{\mathrm{P}-3+\alpha}\), as desired. □
Theorem 3.3
Suppose \(\alpha \in (2,3)\) and \(\mathrm{f}\in \mathscr{G}_{w_{0}}(\mathrm{f})\) satisfies
and some \(\mathrm{P}\in \mathrm{N}_{w_{0}+3-\alpha}\) such that \(\epsilon >0\) and \((\Delta \mathrm{f})(w_{0})\geqq 0\). If
\(\mathrm{f}(w_{0})\leqq 0\),
\((\Delta ^{2} \mathrm{f} )(w_{0})\geqq 0\), and
\(\alpha \in \mathscr{M}_{\mathrm{P}-w_{0}+\alpha -3,\epsilon}\),
then \((\Delta ^{2} \mathrm{f} )(\mathrm{t})\geqq 0\) for all \(\mathrm{t}\in \mathrm{N}_{w_{0}}^{\mathrm{P}-3+\alpha}\).
According to condition (i) (\([-\mathrm{f}(w_{0}) ]\geq 0\)) and (3.3), it follows from (3.11) that
Therefore by the same technique as in the proof of Theorem 3.2 we can prove this theorem as well. □
4 Numerical simulation tests
In the last section, the effectiveness of the negative lower bound inasmuch as the application of the analytical results in the previous section, especially, Theorems 3.2 and 3.3, will be shown via some numerical figures of the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\). All figures and results have been performed with MATLAB 18b software.
Note that in all figures, the α-axis is the horizontal axis, and the k-axis is the vertical one. Figures 1 and 2 show the number of k values and discrete collection of α values of the set \(\mathscr{H}_{\mathrm{k},\epsilon }\) in the interval \((2,3)\), whereas Figs. 3 and 4 show the number of k values and discrete collection of α values of the set \(\mathscr{M}_{\mathrm{k},\epsilon }\) in the interval \((2,3)\). Note that the red lines refer to the empty set of points occurred in \(\mathscr{H}_{\mathrm{k},\epsilon }\) and \(\mathscr{M}_{\mathrm{k},\epsilon }\), and the black parts of the figures refer to the nonempty set of points of these sets.
In Fig. 1, the set \(\mathscr{H}_{\mathrm{k},0.00005}\) in Fig. 1(a) and the set \(\mathscr{H}_{\mathrm{k},0.00001}\) in Fig. 1(b) have the same values of k in \(\mathrm{N}_{0}^{120}:=\{0,1,\ldots ,120\}\). In Fig. 1(a), we see that \(\mathscr{H}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 40\), whereas \(\mathscr{H}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 90\) in Fig. 1(b).
In Fig. 2, the set \(\mathscr{H}_{\mathrm{k},0.000005}\) in Fig. 2(a) and the set \(\mathscr{H}_{\mathrm{k},0.000001}\) in Fig. 2(b) have the same values of k in \(\mathrm{N}_{0}^{120}\). In Fig. 2(a), we observe that \(\mathscr{H}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 120\), whereas \(\mathscr{H}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 165\) in Fig. 2(b).
In both Figs. 1 and 2, the set \(\mathscr{H}_{\mathrm{k},\epsilon }\) remains empty for less values of k when ϵ is larger than when ϵ is small. Furthermore, this set arises to be enriched almost surely toward α values closer to 2 than those closer to 3; especially, for \(\alpha \approx 2.2\), it appears that the set \(\mathscr{H}_{\mathrm{k},\epsilon }\) is most concentrated.
Next, we consider the set \(\mathscr{M}_{\mathrm{k},\epsilon }\) for various values of k and ϵ:
Considering Fig. 3, we see that the set \(\mathscr{M}_{\mathrm{k},0.0005}\) in Fig. 3(a) and the set \(\mathscr{M}_{\mathrm{k},0.0001}\) in Fig. 3(b) have the same values of k in \(\mathrm{N}_{0}^{200}\). Moreover, in Fig. 3(a), we can observe that \(\mathscr{M}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 130\), whereas \(\mathscr{M}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 165\) in Fig. 3(b) for a smaller value of ϵ.
Considering Fig. 4, we note that the set \(\mathscr{M}_{\mathrm{k},0.00005}\) in Fig. 4(a) and the set \(\mathscr{M}_{\mathrm{k},0.00001}\) in Fig. 4(b) have the same values of k in \(\mathrm{N}_{0}^{200}\). Furthermore, in Figs. 4(a) and 4(b), we can note that \(\mathscr{M}_{\mathrm{k},\epsilon }=\emptyset \) for \(\mathrm{k}\geqq 170\).
Just like in Figs. 1 and 2, we can conclude that in both Figs. 3 and 4, the set \(\mathscr{M}_{\mathrm{k},\epsilon }\) tends to remain empty for less values of k when ϵ is larger than when ϵ is small.
Finally, from the sets \(\mathscr{H}_{\mathrm{k},\epsilon }\) and \(\mathscr{M}_{\mathrm{k},\epsilon }\) we can conclude that:
Both sets \(\mathscr{H}_{\mathrm{k},\epsilon }\) and \(\mathscr{M}_{\mathrm{k},\epsilon }\) tend to remain empty for less values of k when ϵ is larger than when ϵ is small.
The set \(\mathscr{H}_{\mathrm{k},\epsilon }\) gives a larger empty space even for ϵ smaller than the set of \(\mathscr{M}_{\mathrm{k},\epsilon }\), which has large nonempty space values.
Theorems 3.2 and 3.3 may be employed for the largest number of time steps whenever α approaches 2.2 and \(0 < \epsilon \ll 1\).
5 Conclusions
We have performed analytical and numerical convexity analysis for discrete delta Riemann–Liouville fractional difference operators. The numerical part can be summarized as follows:
An alternative definition for the discrete delta Riemann–Liouville fractional difference is derived in Theorem 2.1.
A \(\Delta ^{2}\) formula is obtained in Lemma 3.1.
A relationship between the positivity of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\) and convexity of f is considered in Theorem 3.1.
Two sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\) are defined based on the basic lemmas, and it is shown that they are decreasing in Lemma 3.2.
Based on the decrease of these sets, relationships between the negative lower bound of \(({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})\) and convexity of f has been derived in Theorems 3.2 and 3.3 on a finite time set \(\mathrm{N}_{w_{0}}^{\mathrm{P}}\).
On the other hand, the numerical part in Sect. 4 can be summarized as follows:
The domain of solutions of the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\) for various values of k and ϵ has been illustrated in Figs. 1–4.
In view of these figures, the validity and applicability of Theorems 3.2 and 3.3 is explained.
We have concluded that when ϵ is small, the sets \(\mathscr{H}_{\mathrm{k},\epsilon}\) and \(\mathscr{M}_{\mathrm{k},\epsilon}\) tend to remain nonempty for more values of k than for larger ϵ. Furthermore, the sets appear to be enriched powerfully toward the values of α closer to 2 than those closer to 3 for all values of ϵ.
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This work was supported by the Taif University Researchers Supporting Project (No. TURSP-2020/155), Taif University, Taif, Saudi Arabia, and the fifth author would like to thank Prince Sultan University for the support through the TAS research lab.
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Conceptualization, P.O.M., H.M.S., D.B., E.A.-S. and T.A.; Data curation, P.O.M., H.M.S., D.B. and T.A.; Formal analysis, H.M.S., D.B. and T.A.; Funding acquisition, D.B., E.A.-S. and Y.S.H.; Investigation, H.M.S., P.O.M., D.B., E.A.-S., T.A. and Y.S.H.; Methodology, E.A.-S., T.A. and Y.S.H.; Project administration, D.B., E.A.-S. and H.M.S.; Resources, P.O.M. and Y.S.H.; Software, H.M.S.; Supervision, D.B., H.M.S., E.A.-S. and T.A.; Validation, P.O.M., D.B., T.A.; Visualization, T.A.; Writing – original draft, P.O.M. and H.M.S.; Writing – review & editing, D.B., Y.S.H. and T.A. All authors have read and agreed to the published version of the manuscript.
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Baleanu, D., Mohammed, P.O., Srivastava, H.M. et al. On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically. J Inequal Appl 2023, 4 (2023). https://doi.org/10.1186/s13660-023-02916-2
DOI: https://doi.org/10.1186/s13660-023-02916-2


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Women's Bastet Cat Purse
Complete your Egyptian queen costume with this detailed women's Bastet Cat Purse accessory. It will look pur-fect with your magnificent Cleopatra costume!
She of the Ointment JarBastet was known as an intensely fierce cat goddess that was in charge of defending the pharaoh. While her name translates to, "she of the ointment jar", she doesn't actually have anything to do with cold cream. This fierce lion goddess has been around for about four-thousand years. Bastet's symbols decorated both the pharaoh's chambers as well as the jewelry of mothers and women who wanted to be mother's. The idea was, like a house cat, she would be tender to her young and fierce to anyone who might threaten those under her care. She sounds like a goddess we wouldn't mind having around!Details & DesignMake sure Bastet has your back when you dress up in an Egyptian costume this Halloween. This plush black and gold cat is Made by Us meaning it has little details added by our in-house designers like the dignified striped ears and intelligent face. The body has a zipper on the back so you can carry your belongings without interrupting the ancient theme of your look. The pretty paws are circled with the fourteen-inch tail and the long chain slings over your shoulder for a comfortable look that you can carry around all nightAdaptable ArtifactsMake sure your belongings stay close at hand when you put them in the hands of the fierce goddess, Bastet. After all, when you're proudly walking into the room as Nefertiti or Cleopatra, you need a goddess to back up your authority. So throw the gold chain over your shoulder to let everyone know you're not "kitten" around!
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For young adults in Michigan looking to gamble, it can be difficult to know where to go. Many casinos have strict age requirements, often only allowing visitors 21 and older. However, there are several casinos in Michigan that allow 18+ visitors. In this article, we'll provide an ultimate guide to 18+ casinos in Michigan, covering what you need to know about the legal gambling age in Michigan, which casinos allow 18+ visitors, what games are available, and much more.
The Ultimate Guide to Michigan's 18+ Casinos: What You Need to Know
In Michigan, the legal gambling age for most forms of gambling is 21. However, there are some exceptions. Some casinos in Michigan allow visitors who are 18 years old or older to gamble, buy lottery tickets, or bet on horse racing. It's important to note that these exceptions only apply to certain types of gambling and not all casinos allow 18+ visitors. But if you're over 18 and looking to gamble, here are some of the casinos you can check out.
Experience the Thrill: A Tour of Michigan's Best 18+ Casinos
Here are some of the top 18+ casinos in Michigan:
- Island Resort & Casino
- Bay Mills Resort & Casino
- Kewadin Casinos (multiple locations)
- Turtle Creek Casino & Hotel
- Odawa Casino Resort (Petoskey location only)
Each of these casinos has its own character and unique features. Island Resort & Casino, for example, boasts a championship golf course, a luxurious spa, and a wide variety of dining options. Bay Mills Resort & Casino, on the other hand, has a great selection of outdoor activities in addition to its casino offerings. Kewadin Casinos, with locations in Sault Ste. Marie, St. Ignace, Manistique, Christmas, and Hessel, offers a diverse array of gaming options and entertainment. Turtle Creek Casino & Hotel has a sleek, modern design and features games like blackjack and roulette, while the Petoskey location of Otawa Casino Resort has a great selection of slot machines and table games.
If you're planning a trip to any of these casinos, make sure to also check out what else they have to offer. From live entertainment to dining options to outdoor activities, there's plenty to do beyond the casino floor.
Breaking the Age Barrier: A Look at Michigan's 18+ Casino Scene
So why do some casinos allow 18+ visitors while others don't? In most cases, it comes down to the type of gambling being offered. In Michigan, visitors 18 and older are allowed to buy lottery tickets and bet on horse racing, so it makes sense that some casinos would also allow 18+ visitors to gamble. Additionally, 18+ casinos can appeal to a younger demographic, which can help the economy by drawing in more visitors and creating jobs.
Speaking of the economy, the impact of 18+ casinos on Michigan's economy should not be overlooked. By allowing 18+ visitors to gamble, Michigan is able to tap into a younger demographic of visitors who might otherwise go to neighboring states for their gambling fix. This can lead to increased revenue for the state, as well as job creation and economic growth in areas where casinos are located.
As for the types of visitors who frequent 18+ casinos, it can vary widely. Some are young adults looking for a fun night out, while others might be more serious gamblers. Regardless of why they're visiting, it's important for casinos to provide a safe and responsible environment for all visitors.
A Gamer's Paradise: Michigan Casinos for 18+ Visitors
So what kind of gaming experience can you expect at Michigan's 18+ casinos? While it varies from casino to casino, some popular games among 18+ visitors include:
- Slot machines
- Blackjack
- Roulette
- Baccarat
- Poker
It's always a good idea to brush up on the rules of any game you plan on playing before you arrive at the casino. Additionally, if you're new to gambling, you might want to start with some of the simpler games (like slot machines) before moving on to more complex games like poker or blackjack.
If you're planning on visiting a casino for the first time, another thing to keep in mind is that there are often perks and rewards available for loyal patrons. For example, many casinos have rewards programs that offer discounts on food and drinks, free play credits, and more. Keep an eye out for these programs and take advantage of them when you can.
Exclusive Access: Exploring Michigan's Top 18+ Casinos
For those looking to take their casino experience to the next level, VIP programs might be the way to go. Many casinos offer special rewards and benefits to their most loyal patrons, ranging from access to exclusive events to free hotel stays. These programs can be a great way to get more out of your casino experience, but they typically require a certain level of spending to qualify.
If you're interested in VIP programs, do some research ahead of time to see which casinos offer them and how you can qualify. Additionally, don't be afraid to ask casino staff for more information – they're usually more than happy to help out their most dedicated patrons.
The Legal Age to Game: Michigan's 18+ Casinos for Risk-Takers
It's important to remember that gambling can come with risks, especially for those who struggle with addiction. If you're planning on visiting an 18+ casino in Michigan, make sure to set a budget for yourself ahead of time and stick to it. Additionally, many casinos offer resources for responsible gambling, including hotlines and self-exclusion programs.
For those who can gamble responsibly, however, Michigan's 18+ casinos can be a great way to have some fun and maybe even win some money. Whether you're a serious gambler or just looking for a fun night out with friends, there's something for everyone at these casinos.
Don't forget to check out some of the lesser-known 18+ casinos in Michigan as well. While they might not have the same name recognition as some of the bigger casinos, they can still offer a great gaming experience.
Unlock the Fun: Discovering Michigan's 18+ Casino Gems
Whether you're looking for a gaming-focused trip or just a fun night out, Michigan's 18+ casinos have something to offer. From the luxurious spa at Island Resort & Casino to the outdoor activities at Bay Mills Resort & Casino, there's plenty to do beyond the casino floor. Additionally, VIP programs can provide exclusive access and rewards for loyal patrons.
If you're planning on visiting an 18+ casino in Michigan, make sure to do your research ahead of time to find the casino that's right for you. And remember to gamble responsibly – set a budget, take breaks when needed, and never gamble more than you can afford to lose.
Michigan's 18+ casinos offer a unique opportunity for young adults to enjoy the thrill of gambling. Whether you're a serious gambler or just looking for a fun night out with friends, there's something for everyone at these casinos. From the top 18+ casinos like Island Resort & Casino and Kewadin Casinos to lesser-known gems, Michigan offers a great selection of options. Just remember to gamble responsibly and set a budget for yourself ahead of time. With a little bit of luck, you might even walk away a winner.


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Lotteries are a form of gambling and can be a great way to have some fun and win prizes. The lottery is a game of chance where players have a chance to win a prize, which is usually in the form of cash. It is also a great way to raise money for charities or good causes.
Throughout history, lotteries have been a popular way to raise funds for public projects. For example, the Roman emperors used lotteries to raise money for a number of things. A few of these included building a canal, fortifications and libraries. In some cases, lotteries were even used to give away slaves and property.
Lotteries can be found in all 50 states. They are usually run by the state or city government. However, some states have joined together to run multi-state lotteries. Most of these lotteries have large purses that can pay out big rewards to the winners.
One of the earliest known lotteries in Europe was the lottery organized by the Roman Emperor Augustus. These lottery tickets were distributed by wealthy noblemen during Saturnalian revels.
The oldest running lottery is the Staatsloterij, which is still in operation. Although the German lottery was invented by Alexander Hamilton, the Chinese Book of Songs mentions a lottery.
Several colonies in the American colonies had their own lotteries. Several of these lotteries were used to finance local militias, fortifications and roads. Some lotteries were run in order to raise money for colleges and libraries. Others were set up to make a profit and then distribute a percentage of the profits to good causes.
Historically, lotteries were a favorite tax alternative for many governments. However, this was not always the case. Many people viewed lotteries as a form of hidden tax. Therefore, ten states banned lotteries between 1844 and 1859.
A small group of people have the luck of the draw to win the lottery. Depending on the design of the lottery, the jackpot can be huge. If you win the lottery, you may receive a lump sum of money or the prize money may be spread over several years. Typically, most lotteries will give smaller prizes for matching some of the winning numbers.
This type of lottery is often called the "financial" lottery because of its popularity. Financial lotteries can generate a tremendous amount of revenue for communities and countries. Moreover, they are criticized as an addictive type of gambling. Nevertheless, these lottery games have been a success in some regions, and the world at large seems to approve of their implementation.
Despite its popularity, a lotterie may not be the best option for you. In some cases, it can cost more than you can afford, and in other cases, it can be a waste of your time.
One of the best ways to make sure you are getting the most for your buck is to look for a lottery that offers the best odds. To increase your chances of winning, you can choose to buy several tickets. Another way to do this is to purchase a multi-state lottery ticket.


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Increasing military conflict on the world stage has seen a significant increase in research and development funding in the defence sector as countries scramble for advantage. The future of national security and defence will increasingly be focused on the information domain.
In a world of cloud, AI, biometrics, data analytics and augmented reality, the most influential military technologies being actively developed include:
- Artificial intelligence,
- Robotics and,
- Autonomous systems.
These new technologies are transforming the battlefield in terms of sustainability, autonomy, connectivity, and lethality.


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7777 gaming expands its reach with Quantum Gaming – a world-class B2B platform provider in the iGaming industry. Due to the recently agreed partnership between both companies, 7777 gaming casino content will be available on the Quantum Gaming platform and excite players worldwide.
Commenting on this promising cooperation, Oliver de Bono, Quantum Gaming CEO, said: "With such a dynamic and rapid expansion of the iGaming industry into new markets, it is of utmost importance for operators to arm themselves with the most attractive content for newcomers, and 7777 gaming offers just that. Quantum Gaming always strives to be at the forefront of industry trends, and we believe this partnership will open up a whole new world of opportunities for everybody involved in the casino business."
Elena Shaterova, CCO at 7777 gaming said: "Adding a new and strong partner such as Quantum Gaming allows us to enter many new territories and expand our online casino games reach further. We are excited about the great opportunities ahead of us."
7777 gaming has recently announced its yearly milestones, amongst which are being available on 5 of the leading Bulgarian operators, partnering with 10+ of the leading game and platform aggregators, and much more. This partnership comes as a logical continuation of the company's plans to expand its market reach.


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The Renault Owners' Club is proud to announce that we have been shortlisted for an award. The Renault Owners' Club members help raise over £1000 by sponsoring a 1927 car that drove 1000 miles in 4 days.
Details of the award can be found at:;
Restoration Show 2019
Details of the event can be seen in our press release at:
Land's End to John O'Groats
The National Car Club Awards 2019 are presented in association with Tourism Ireland, supported by Classic Car Weekly and endorsed by the Federation of British Historic Vehicle Clubs.


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Businesses in commercial office buildings can't stop work for a roofing project. High rises, for example, present unique challenges such as limited access, service elevator staging, high security, after-hours night construction, high traffic, wind effect, and structural issues such as cell phone towers, etc. Also, office commercial roofing specifications can be somewhat unique if it is a high rise building.
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We will work with you to determine a roofing system that will suit your budget and deliver the long-term, low-maintenance performance you need. Educational facilities often demand finite installation deadlines. During the installation, you can be sure Structure Roof & Wall Solutions will keep the safety of your congregants or students as a top priority, as with all of our educational commercial roofing projects.
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Structure Roof & Wall is committed to reaching as many industries as possible, including retail commercial roofing. After all, a retail establishment's roof is, quite aptly, its fifth wall. As such, the roof is vitally important to a business. It keeps out the elements and protects merchandise and shoppers. And today, it even reduces energy costs and promotes environmental soundness. Incorporating new materials and processes can help save retailer's crucial dollars, and avoid urban heat island effects.
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Analyzing life cycle cost, roof traffic, warehouse contents, and factors such as long-term ownership versus short-term lease are all factored into determining what kind of roof a warehouse needs. A warehouse roof can be a big source of energy inefficiency. Commercial warehouse roofs are notorious for soaking up solar heat. While a facility's ability to absorb heat might be advantageous for some facilities in the winter, it often means wasted money overall. There are many reflective or insulating products that we can provide to improve energy efficiency through the roof.
Industrial facility roofs often must accommodate equipment installations and frequent foot traffic. Some may be exposed to chemical exhaust or temperature extremes. Because of the environment, the highest regard for safety and OSHA standards must be maintained at all times. Interior protection from falling debris as well as ground-to-roof communication is mandatory. Often working 24-hour shifts, there is no downtime for a manufacturing & industrial commercial roofing project. We can work with your schedule to efficiently and effectively update your roof.
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Original Ranch Beef and Noodle Skillet
1 pkg. (1 oz) Hidden Valley Original Ranch dressing mix
1 lb. ground beef cooked and drained
1 jar sliced mushrooms (4.5 oz) drained
8 oz wide egg noodles, cooked and drained
8 oz. sour cream
3/4 c. milk
1/2 c. Parmesan cheese
In a saucepan, combine all ingredients except cheese. Heat thoroughly.
Sprinkle with Parmesan cheese


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The PONS brand and the TABASCO® brand have been working together for years to offer this unique product that mixes the elegant flavor of PONS olive oil with the spice of TABASCO® condiment.
Late harvest Arbequina olives form the delicate base of this oil, the elegant and sweet taste is combined with the traditional red pepper condiment from McIlhenny Company, known since 1868 for its spicy taste and inimitable quality.
The two-month infusion process carried out in the PONS factory intensifies the flavor of the olive, while the TABASCO® condiment gives the oil a subtle but remarkable level of spiciness.
This condiment, ideal for pizzas, pasta, tartare and a long list of other recipes, will be our ally when it comes to flavoring our dishes in a very healthy way, as its base is extra virgin olive oil, with all the already known benefits, combined with the antioxidant and anticancer properties of capsaicin, from the chili pepper spicy with which TABASCO® is made.