reductions – How to convert Bipartite Perfect Matching to SAT?

Since 3-SAT is NP-complete you can take any algorithm that decides bipartite perfect matching and use the construction from Cook-Levin’s theorem. A direct construction is as follows.

Let $G=(U+V, E)$ be the bipartire graph with $|U| le |V|$ and no isolated vertices.
Create a boolean variable $x_e$ for each edge $e in E$. Intuitively, $x_e$ is set to true iff $e$ is selected in the matching.

Then, for all (unordered) pairs of distinct edges ${e, f}$ such that $e$ and $f$ share an endpoint, add the clause $(overline{x}_e vee overline{x}_f)$. These clauses ensure that the selected edges form a matching.

Finally, for each vertex $u in U$, create a new clause $bigveelimits_{ e=(u,v) in E} x_e$. Intuitively, this clause is satisfied iff $u$ is matched.

The SAT formula $varphi$ obtained as the conjunction of all the above clauses is satisfiable iff $G$ admits a perfect matching. To obtain a 3-SAT formula you can apply the standard transformation from SAT to 3-SAT.
Essentially, you can suitably duplicate and pad clauses with less than $3$ literals, while a generic clause $(ell_1 vee ell_2 vee dots, ell_k)$ with $k > 3$ literals can be replaced by the two clauses $(ell_1 vee ell_2 vee y) wedge (overline{y} vee ell_3 vee ell_4 vee dots vee ell_k)$, where $y$ is a new variable. Repeat the above transformation until all clauses have exactly 3 literals.

How to convert Bipartite Perfect Matching to 3SAT?

3SAT is $NP$-complete while Bipartite Perfect Matching is in NC under derandomization assumptions. How to convert Bipartite Perfect Matching to 3SAT?

Benefits of Pet Beds & How to Pick the Perfect One!

Benefits of Pet Beds & How to Pick the Perfect One!
There’s nothing we wouldn’t do for our pets. They are a huge part of our lives, they bring us so much joy and they are, most definitely, honorary members of our families.
We’ve already covered about a few different pet-related topics for our apartment residents, like the benefits of making homemade dog treats, how to calm hyperactive dogs, apartment pet etiquette and even ways to be an eco-friendly pet owner. Our apartments are pet-friendly, and we provide plenty of amenities to make you and your pets live the best life possible here at AMLI Residential. Bottom line, we love our residents, and we love our pets, and so we love to make sure both are as comfortable as possible!
Pet bed is just one facet of pet ownership, and it’s one that provides a lot of benefit to both the pet and the owner. We’ll delve into what makes a good pet bed, what the benefits of pet beds are and, finally, how you can make some of your own DIY pet beds!
What are the benefits of pet beds?
If your pets are the kind that sleep anywhere and everywhere (looking at you, cats!), then you may be wondering if it’s even necessary to provide them with a designated pet bed. As it turns out, however, having a designated sleeping area is as important for pets as it is for us pet owners!
Support
Think about it. We could fall asleep on the couch, on a sofa, a recliner or even on the floor if we are tired enough, but our bodies benefit from a comfortable and supportive bed. Both dogs and cats need beds that support their bones and muscles enough to allow adequate sleep and rest without stiffness or achiness. As your pets get older, it’s even more important for them to have proper support.
Warmth
Having a warm bed that’s situated away from drafts can help your pets avoid catching colds. Floors can get cold, sofas next to air vents can get chilly and even your own bed can be too hot or cold for your pets’ liking. If your cat or dog has its own pet luxury bed situated in a warm, comfortable spot, then your pets will be able to rest comfortably and better regulate their internal temperatures.
Security
It’s important for your pet to have a space that they can call their own, especially if you have multiple pets. Your pet should feel safe and secure in their bed, and it should be a place where they can truly relax.
Health
We love snuggling with our pets in bed, we also need to have a proper night’s sleep without a paw in our faces or a tail across our nose. Having clean bedsheets and sleep areas are good for our own health and wellness, especially if you’re prone to allergies.
This doesn’t necessarily mean that sleeping next to your pet is bad, per se. It just means that if your pet is negatively affecting your sleep and health, then a pet bed is a great alternative.
What type of bed does your pet need?
Veterinarians are the best resource you have for finding out what’s best for your pet. Since every animal is so different and has such different needs, your vet will be able to give you the most accurate information that’s tailored specifically to your pet.
In general, though, there are a few different things you can be aware of that will help inform your decision on what type of bed your pet needs.
Size
Your pet needs a bed that’s big enough for it to shift, turn and settle in comfortably. No heads and/or limbs should have to hang off the edge of the bed.
Washable
You want your dog’s bed to be washable, trust us, especially if you have young puppies or senior dogs. Pets also bring food, fleas, urine and other regular pet-related smells and sights into their beds with them, so being able to clean the beds regularly will make a huge difference. Plus, you wouldn’t want to sleep in filth for months on end, right? Neither do your pets.
Stuffing/support
Keeping your pets warm and well-supported is crucial for their health, especially as they age. Orthopedic foam is usually a good option for pets of any age, as it’s sturdy and reliable over long periods of time.
Eco-friendly
There are a lot of factors when it comes to choosing an eco-friendly pet bed for your pet. Sustainable pet beds could mean chemical-free fabrics or recycled material, or it could mean upcycling something you already had so it gets more use. Whatever your priorities are, there’s definitely something out there already that’s exactly what you’re looking for.
Here are a few examples of eco-friendly dog beds!
Making your own pet bed
If your pet is healthy and young (i.e. they don’t need special orthopedic support), then making your own pet soft bed can be a great way to upcycle old materials and give your furry friend their own space.
As long as the bed is washable and supportive enough for your pet, then it’s a great way to avoid buying new products and keep old products in use for much longer!
How To Choose a Dog Bed
Choosing the right dog bed is an important decision. After all, a dog can easily spend 14 hours a day or more sleeping. And as a cherished member of the family, your pet deserves no less than the very best sleeping accommodations. Even a dog who shares your bed at night needs a special place to claim as his own: a quiet sanctuary for peaceful, uninterrupted naps.
Before you make this important purchase, consider your dog’s age, size, health, breed, activity level and sleeping habits. Think about how the dog bed will look in your home. Will the color blend with your décor? Does the size fit your dog and your room? Is the bed easy to launder?
With so many choices available in the marketplace today, you can rest easy knowing that we’ve curated an extensive sleep selection that reflects our eye for smart design, uncompromised quality, dog-approved comfort and owner-approved ease.
For lasting value and satisfaction, we recommend choosing the highest quality dog bed your budget allows, rather than a bargain bed that will soon need replacing.
Choosing the Shape
This is an important decision, and one your dog can help you with. Observe him as he sleeps. Does he like to curl up, burrow in, sprawl out or lean back? Most dogs fall into one of these four “sleep personality” categories.
Choosing the Support
While young and healthy dogs will slumber soundly on a bed plumped with resilient, high-quality polyfiberfill, polyfill or foam, many dogs require the extra support that only an orthopedic dog bed can provide.
Dogs who are aging or arthritic, suffer from hip dysplasia or are recovering from injury will benefit from sleeping on an orthopedic dog bed. High-density orthopedic foam soothes aches and pains, reduces stress on joints, hips and pressure points, and provides superior support. For exceptional comfort and therapeutic relief, look for beds made with authentic, NASA-certified orthopedic memory foam.
Choosing the Design
Because a dog bed is a visible part of your home décor, it should reflect your own sense of style as well as your dog’s. Our collection includes a beautiful selection of luxe, upholstery-grade fabrics—plush microvelvet, sumptuous chenille, cuddly Sherpa fleece, shaggy faux fur, rugged canvas and lots more—in the newest decorator colors and designer patterns to enhance any room.
Determining the Correct Size
Before ordering a dog bed, make sure to measure your dog from nose to base of tail while he’s in his sleeping position, then add 8″ to 12″ to determine the bed length or diameter.
If in doubt, always round up in size to ensure maximum comfort. Then doublecheck to make sure the bed you’ve selected will fit into the spot you’ve chosen in your home.
Easy Clean Features
Because every dog bed eventually gets dirty, wet or smelly, we’ve made sure that every bed we sell is machine washable or has a removable, machine-washable cover.
Special Beds for Special Dogs
Is your dog a senior with mobility issues, or one who suffers from arthritis or hip dysplasia? Simple, low-profile dog pet bedssuch as pillows, nappers, pads and mats are easily accessible to an aging or aching pet.
If your dog is elderly, incontinent or prone to accidents, look for an easy-to-launder bed with a waterproof liner.
For the dog-on-the-go, portable travel beds provide instant, familiar comfort and convenience when you’re on the road.
Need a bed you can use outdoors? Our weatherproof dog beds, cots and futons let your dog camp out in comfort on your deck or porch, join you by the pool or relax under a shady tree—and they’re so attractive, you can move them into your home as the seasons change.

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Titanium is the perfect metal to make replacement human body parts

Titanium is the perfect metal to make replacement human body parts

Titanium material is expensive and can be problematic when it comes to traditional processing technologies. For example, its high melting point (1,670℃, much higher than steel alloys) is a challenge.
The relatively low-cost precision of 3D printing is therefore a game-changer for titanium. 3D printing is where an object is built layer by layer and designers can create amazing shapes.
This allows the production of complex shapes such as replacement parts of a jaw bone, heel, hip, dental implants, or cranioplasty plates in surgery. It can also be used to make golf clubs and aircraft components.
The CSIRO is working with industry to develop new technologies in 3D printing using titanium. (It even made a dragon out of titanium.)
Advances in 3D printing are opening up new avenues to further improve the function of customised bodypart implants made of titanium.
Such implants can be designed to be porous, making them lighter but allowing blood, nutrients and nerves to pass through and can even promote bone in-growth.
Safe in the body
Titanium is considered the most biocompatible metal – not harmful or toxic to living tissue – due to its resistance to corrosion from bodily fluids. This ability to withstand the harsh bodily environment is a result of the protective oxide film that forms naturally in the presence of oxygen.
Its ability to physically bond with bone also gives titanium an advantage over other materials that require the use of an adhesive to remain attached. Titanium implants last longer, and much larger forces are required to break the bonds that join them to the body compared with their alternatives.
Titanium alloys commonly used in load-bearing implants are significantly less stiff – and closer in performance to human bone – than stainless steel or cobalt-based alloys.
Aerospace applications
Titanium weighs about half as much as steel but is 30% stronger, which makes it ideally suited to the aerospace industry where every gram matters.
In the late 1940s the US government helped to get production of titanium going as it could see its potential for “aircraft, missiles, spacecraft, and other military purposes”.
Titanium has increasingly become the buy-to-fly material for aircraft designers striving to develop faster, lighter and more efficient aircraft.
About 39% of the US Air Force’s F22 Raptor, one of the most advanced fighter aircraft in the world, is made of titanium.
Civil aviation moved in the same direction with Boeing’s new 787 Dreamliner made of 15% titanium, significantly more than previous models.
Two key areas where titanium is used in airliners is in their landing gear and jet engines. Landing gear needs to withstand the massive amounts of force exerted on it every time a plane hits a runway.
Titanium’s toughness means it can absorb the huge amounts of energy expelled when a plane lands without ever weakening.
Titanium’s heat resistance means it can be used inside modern jet engines, where temperatures can reach 800℃. Steel begins to soften at around 400℃ but titanium can withstand the intense heat of a jet engine without losing its strength.
Where to find titanium
In its natural state, titanium is always found bonded with other elements, usually within igneous rocks and sediments derived from them.
The most commonly mined materials containing titanium are ilmenite (an iron-titanium oxide, FeTiO3) and rutile (a titanium oxide, TiO2).
Ilmenite is most abundant in China, whereas Australia has the highest global proportion of rutile, about 40% according to Geoscience Australia. It’s found mostly on the east, west and southern coastlines of Australia.
Both materials are generally extracted from sands, after which the titanium is separated from the other minerals.
Australia is one of the world’s leading producers of titanium, producing more than 1.5 million tonnes in 2014. South Africa and China are the two next leading producers of titanium, producing 1.16 and 1 million tonnes, respectively.
Being among the top ten most abundant elements in Earth’s crust, titanium resources aren’t currently under threat – good news for the many scientists and innovators constantly looking for new ways to improve life with titanium.
Titanium Forgings Shapes
Titanium forgings refer to products manufactured by the process of shaping metal utilizing compressive forces. The compressive forces used are generally delivered via pressing, pounding, or squeezing under great pressure. Although there are many different kinds of forging processes available, they can be grouped into three main classes:
Forging produces pieces that are stronger than an equivalent cast or machined part. As the metal is shaped during the forging process, the internal grain deforms to follow the general shape of the part. This results in a grain that is continuous throughout the part, resulting in its high strength characteristics. Forgings are broadly classified as either cold, warm or hot forgings, according to the temperature at which the processing is performed.
Iron and steel are nearly always hot forged, which prevents the work hardening that would result from cold forging. Work hardening increases the difficulty of performing secondary machining operations on the metal pieces. When work hardening is desired, other methods of hardening, most notably heat treating, may be applied to the piece. Alloys such as aluminum and titanium that are amenable to precipitation hardening can be hot forged, followed by hardening. Because of their high strength, forgings are almost always used where reliability and human safety are critical such as in the aerospace, automotive, ship building, oil drilling, engine and petrochemical industries.
For more information or to receive a prompt aluminum price quote, please contact us at 800 398-4345 or submit the Request Information form on the right side of this page.
Titanium rod and bar are made from a corrosion-resistant material that has one of the highest strength-to-weight ratios of all metals. Due to the wear resistance, corrosion resistance, high-temperature resistance, and non-magnetic properties of titanium rods, it is used in the main parts of equipment, shaft body, solid parts, mixing shaft, etc.
Titanium Rods’ Characteristics
In addition, titanium rods have the characteristics of high strength, good toughness, low modulus of elasticity, compatibility with the human body, and are widely used in the medical industry.
The forging material of the titanium rod is mainly pure titanium and titanium alloy of various compositions, and the original state of the material is titanium rod, titanium ingot, metal powder, and liquid metal.
The ratio of the cross-sectional area of the metal before deformation to the cross-sectional area after deformation is called the forging ratio. Proper selection of forging ratio, reasonable heating temperature and holding time, reasonable initial forging temperature, and final forging temperature, reasonable deformation, and deformation speed is closely related to improving product quality and reducing cost.
Generally, small and medium-sized forgings use round or square bars as blanks. If the grain structure and mechanical properties of the bar are uniform and good, the shape and size are accurate, and the surface quality is good, it is convenient for mass production. As long as the heating temperature and deformation conditions are properly controlled, high-quality forgings can be forged without requiring large forging deformation.
On the aircraft, titanium alloy is mainly used to manufacture the main force members such as girders, landing gears, hubs, and joints. Titanium alloy is mainly used to manufacture adapter rings, scraper fans, compressor discs, and blades on the engine.
In the metalworking world, every metal part possesses its own unique set of benefits. No matter what industry you’re in, it’s important to know the benefits of different metal components so that you can choose the part that will be the most valuable for your specific project. One metal part that is used in various industries for it’s advantages is titanium tubing! What are the benefits of titanium tubing? Keep reading to find out!
4 Benefits of Titanium Pipe and Tube
1. Lower Density
The density of Titanium pipe and tube is significantly lower than steel, copper, or nickel products. Despite their low density, they are very strong and rigid when compared to other alloy components.
2. Resistant to Corrosion
Another benefit of utilizing titanium tubing is that titanium alloys are resistant to corrosion. This makes these tubes an appealing option if you need a part to work efficiently in a highly corrosive environment.
 3. Resistant to Chemicals
 Corrosion is not the only thing that titanium tubes are resistant to. They are also resistant to chemicals. These titanium parts can withstand different chemical compounds while still preforming effectively.
 4. Great Heat Transfer Properties
Titanium tubes have great heat transfer properties thanks to its thermal conductivity and resistance. The same can’t be said for copper and carbon steel tubes. Their resistance to heat also allows them to work successfully at temperatures up to 600 degrees or higher.
These are just a few of the many benefits that come from utilizing titanium tubes. Because of these benefits, you can find titanium tubing in countless important industries such as power generation, sporting goods, marine, nuclear, and paper industries.
If you’re interested in using titanium tubes for your next project, Ferralloy, Inc. can help! We also have the infrastructure and facilities to supply raw materials in numerous grades and forms! Visit our metalworking foundry online today!

Titanium is sliver grey, colored transition metal found in abundance among all minerals. Titanium has high melting point and offers very good corrosion resistant property, heat properties and strength to weight ratio. Titanium is extracted from ores of rutile and ilmenite. Aerospace & aviation industry is the major end user of the titanium product.
Titanium is used in production of super light high speed aircrafts, satellites and spacecrafts, and ships. Apart from aerospace & aviation, some other major end user industries of titanium products include paper, plastic, and paints & coatings.
Titanium products are also popular in various healthcare applications such as pacemakers, and defibrillators due to chemical properties of titanium such as inertness to UV rays and self-cleaning properties.
Based on the different product type, the global titanium products market can be broadly categorized as titanium concentrate, titanium tetrachloride, titanium sponge, ferrotitanium, titanium pigment and other. Based on the various applications of the titanium product, the market can be segmented in seven broad categories namely, aerospace & marine, industrial, medical, energy, pigments, additives & coatings, and others.
Rising demand of titanium products in aircraft carriers, defense equipments and various other chemical processing industries such as oil and gas is driving the global titanium product market. Moreover, the recent development in cost effective manufacturing technology coupled with superior weight-to strength ratio compare to some of the other traditional product such as steel is expected to boost the market in upcoming years.
 Unavailability of raw material, fluctuating price of input materials, and high cost of titanium product are some of the major challenge for the titanium product market.
North America is the largest market of titanium product followed by the Europe and Asia pacific. Asia Pacific is the fasted growing market. The major end user industry such as healthcare, power, automotive and aerospace industries of the titanium product is growing which in turns helping the titanium product market in this region.
Some of the major companies operating in global titanium products market include, Huntsman International, DuPont, Ineos, Iluka Resources Ltd., Sumitomo Corporation VSMPO-AVISMA Corporation., Toho Titanium Co., Ltd., RTI International Metals, Allegheny Technologies Incorporated, Titanium Metal Corporation., Tronox Limited (U.S)., Indian Rare Earths Limited (India)., and Sierra Rutile Limited (U.K)

Titanium is a well-known material to be characterised as flammable under certain morphologies. Titanium and its alloys have a great affinity for oxygen and will form a native 2-7 nm TiO2 layer instantly if a clean metallic surface is exposed to air at room temperature. This film prevents further oxidation from taking place and protects the underlying metal powder. When heat is applied, either through a thermal source or a spark, the powder can generally or locally heat to the point of thermal runaway or burning. The consensus mechanism of self-sustaining thermal runaway of titanium powder occurs by means of ion diffusion through this native TiO2 film on the titanium powder (1, 2). As the micron size of the powder decreases, the specific surface area (in units of m2/g) increases at a rate of 6/d where d=particle diameter. In context, to fill a typical AM machine with 45 kg of titanium powder, with an average particle size of 20 μm, this powder will have enough surface area to cover over 3000 m2. Generally, titanium powders with a particle size < 45 μm are considered a flammability hazard.
When describing a reaction of any metal powder, there are three categories into which each reaction may fall: 1) stagnant, 2) freely aspirated and 3) conveyed. Stagnant powder reactions generally are a result of powder that collects on a horizontal surface and ignition is typically from a heat source, as a more significant source is necessary to ignite a stagnant bed of powder. When powder is dispersed in the air, the fine powders may stay aloft creating a cloud. Aspiration of powder, and specifically titanium powder, does not automatically mean the cloud will ignite spontaneously. However, if the temperature threshold or spark energy necessary for ignition is met, rapid oxidation of powder can occur as it mixes with oxygen from the air. This is a result of no thermal heat sink of other powders or materials in near proximity to the powder cloud allowing it to reach a much higher temperature and propagate to other powders, which may result in a large pressure increase and possible explosion.
Ignition can come from a variety of sources, which will be discussed throughout this article. Thermal exposure to temperatures of 300-700°C can cause ignition of titanium powder despite the native oxide layer (i.e., minimum ignition temperature or MIT). Spark ignition can come from a variety of sources including static electricity build-up, electric components and friction/impact of metal components. Titanium powder can have minimum (spark) ignition energies (MIE) of 3-30 mJ.
Powder production
After atomisation, Titanium powder is traditionally collected in a cyclone system. These powders are typically non-passivated. The transfer of these non-passivated powders from the atomisation cyclone to ancillary process containers is considered to present a high risk of thermal runaway, which may require breaking of the inert gas seal and exposure to oxygen with high potential for powder aspiration. To overcome this problem, non-passivated powder requires exposure to air (or a reactive gas) to passivate at room temperature, a very time consuming and potentially dangerous process. As an example, passivation of 215 kg of aluminium powder was conducted in a powder collection canister after atomisation, requiring a 20 hour cool down (below MIT), followed by a 1.5 hour passivation period (3). While canisters can be isolated and moved for passivation, this process concentrates a large quantity of nascent surface powders (i.e. highly reactive) in a confined vessel, which is not ideal.
A novel passivation approach
As a solution to this problem, Praxair Surface Technologies, Inc. uses a novel in-situ passivation process that prevents further oxidation of the powder during exposure to air, thus minimising any exothermic reaction, thereby greatly diminishing the possibility of thermal runaway or burning of powder. Using Praxair’s in-situ process, titanium powders are passivated prior to reaching the cyclone collection and are deemed safe to handle after dropping below the aforementioned MIT (300-700°C in air). This not only increases the productivity of titanium powder production, but also greatly diminishes the hazards of the powder.
The ability to add a specific passivation layer to the titanium powder without greatly affecting the powder making process requires the formation of an oxide shell in-situ after the powders initially solidify and descend downwards within the atomisation chamber. The most important aspect of in-situ passivation is the generation of a layer similar (in thickness and chemistry) to the native oxide film that will form on the surface of titanium at room temperature (i.e., a 2-7 nm thick oxide) (4-7). Oxide thickness becomes extremely important because of the extremely large surface area described above. Ideally, the total oxygen content should stay below 1300 ppmw (0.13 wt.%) for a 20 μm particle, which requires a target titanium oxide thickness of ~2-3 nm if the bulk material contains less than 1000 ppmw of O2. If the target oxide shell thickness of 1-3 nm can be produced then no additional oxidation should take place when exposed to air at room temperature for extended periods of time.
The post-processing of titanium powder undoubtedly will utilise electrical equipment from sieves, blenders, feeders, etc. This challenge also presents itself to users of AM equipment. When considering electrical installations involving any flammable substance, it is highly recommended to reference National Electrical Code, NFPA 70, particularly articles 500 to 504. These sections describe the recommended best installation practices for electrical equipment in the presence of a hazardous material. Class II is relevant to combustible dusts (e.g., metal powders) and, within Class II locations, there are Divisions I and II. To determine which division a process/material may fall into, the reader is directed to read these descriptions carefully.

performance – Does a good but less than perfect Pagespeed Insights score hurt the chances of a page ranking in Google?

If the 86/100 is the mobile score, the desktop score is likely ~95-100.

I have many sites that score like this, and have no problems ranking content on either desktop or mobile.

This will not hinder pages on your site from ranking.

However, sinces Google’s recent Page Experience Update, it does mean that you may lose a tie breaker where the decision is between your site and another for a certain position in search results on mobile


Page Experience and Search

Q: What is the page experience update and how important is it compared to other ranking signals?

A: The page experience update introduces a new signal that our search algorithms will use alongside hundreds of other signals to determine the best content to show in response to a query. Our systems will continue to prioritize pages with the best information overall, even if some aspects of page experience are subpar. A good page experience doesn’t override having great, relevant content.

This is similar to changes we’ve had in the past, such as our mobile-friendly update or our speed update. As with those signals, page experience will be more important in “tie-breaker” types of situations. If there are multiple pages of similar quality and content, those with better page experience might perform better than those without.

In short, publishers shouldn’t worry that when we begin using page experience, that they may suffer some immediate significant drop, if they’re still working on making improvements. But publishers should be focused on making those improvements a relative priority over time. This is because as more and more sites continue to improve their page experience, it will be the norm that publishers will want to match.

Full Q&A


Lab vs. Field Data (Additional Thought)

  • PageSpeed Insights gives us “Lab Data”. To gauge true speed, Google looks at actual timings for users aka “Field Data”. Depending on how much traffic you get, Google will report to you these timings in Google Search Console
  • PageSpeed Insights/Lighthouse also use a “Fast 3G” connection and an emulated Moto G4 (last I checked). This might not be realistic for many of your users depending on the Demographics/Geography of your audience.

nt.number theory – Is my proof of even almost perfect numbers valid?

I think that I’ve discovered a proof that if an even almost perfect number is not of the form $2^k$ it has to have at least 3 distinct primes.

Let $σ(2^k.y^m) $ be an even almost perfect number where $y^m$ is odd.

$σ(2^k.y^m) = (2^{k+1}-1)(y^m+y^{m-1} … + y + 1) = 2^{k+1} . y^m – 1 $

$2^{k+1} . y^m + 2^{k+1}(y^{m-1}+y^{m-2} … + y + 1) – 1(y^m+y^{m-1} … + y) – 1 = 2^{k+1}.y^m-1$.

$2^{k+1}(y^{m-1}+y^{m-2} … + y + 1) – 1(y^m+y^{m-1} … + y) = 0$

$2^{k+1}(y^{m-1}+y^{m-2} … + y + 1) = y(y^{m-1} + y^{m-2} … + 1)$

$2^{k+1} = y$ which is a contradiction because y is an odd prime.

Is this proof valid?

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Creating “realistic” pixel perfect physcics rendering – C++ – Box2D and SDL2

Whats the best way to be rendering the locations of objects in box2D in such a way that there no stuttering on movement beyond 8-directions. I’m looking to have object strictly snap to the pixels but still keep objects moving smoothly without jittering.

every game loop iteration calls an Update() method on all GameObjects which rounds the objects position in the Physics world to a corresponding pixel coordinate and scaled up.

destRect.x = std::floor(getX() * scale);
destRect.y = std::floor(getY() * scale);

drawn to the screen afterward in GameObject::Draw()

SDL_RenderCopy(targetRenderer, objectTexture, &srcRect, &destRect);

this works fine and positions the object perfectly on the screen–too perfectly.

Moving straight in any direction is smooth and doesn’t stutter. I quickly wrote the following code to fix jittering on diagonals (45 degrees) by ensuring that each row/column has only one pixel drawn on it. This makes 8-way movement smooth.

if (IsMoving())
    {
        // Which direction is the object traveling
        float angleAbs = std::abs(GetAngleDegrees());
        if (angleAbs <= 135 && angleAbs >= 45)
        {
            CurrentCol = CurrentColY = NULL;
    
            // set y to floored object->x in scaled pixels
            destRect.y = std::floor(getY() * scale);
    
            // has this row been rendered to yet?
            if (CurrentRow == std::floor(getY()))
            {
                destRect.x = CurrentRowX;
            }
            else
            {
                CurrentRow = std::floor(getY());
                destRect.x = CurrentRowX = std::floor(getX() * scale);
            }
        }
        else
        {
            CurrentRow = CurrentRowX = NULL;
    
            // set y to floored object->x in scaled pixels
            destRect.x = std::floor(getX() * scale);
    
            // has this column been rendered to yet?
            if (CurrentCol == std::floor(getX()))
            {
                destRect.y = CurrentColY;
            }
            else
            {
                CurrentCol = std::floor(getX());
                destRect.y = CurrentColY = std::floor(getY() * scale);
            }
        }
    }
    else 
    {    
        // update any changes made to the coordinates
        destRect.x = std::floor(getX() * scale);
        destRect.y = std::floor(getY() * scale);
    }

This code tries to smooths out other angles that objects might move in (jumping forward, for, example, forms a parabola like shape) but doesn’t look smooth when speed up to normal frame rates. The motion doesn’t render seamlessly.

I think I’m rendering the locations of the objects almost too precisely, making the objects movement (ironically) pixelated and mechanical, but I’m lost on how to make pixel perfect motion on varied angles seem–or at lease give the illusion of–smooth “realistic” motion.

I’m looking for advice on how to approach this problem and find a working solution rather than make it work with my engine directly–all the same, the code for my game engine can be found on GitHub here if it helps.

Which are some perfect medical directory submission site?

Hi, For SEO purpose I need some Free local directory for USA-California " Medical niche" , I would like to know If someone have a source or experience share please.