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Why Non-Linear Differential Equations are Hard to Solve

In the study of ordinary differential equations, it's well known that most nonlinear equations are pretty much impossible to solve (at least analytically). On the contrary, linear equations are always solvable, and have nice closed form solutions. This is a consequence of the structure of a general solution to the linear equation: it's a sum of two solutions that are easier to find. Solutions to non-linear equations do not always abide by this property. To understand this more deeply, we will briefly study and attempt to solve a small variation of a linear equation and see where the linear method fails.  For first order equations, equations involving just the first derivative, a linear equation looks like y' + a(t)y = q(t). Note that there are no weird functions of y in the expression like y^3/2 or sqrt(y), there is just multiplication by functions of t and y and addition of terms, both linear operations. To solve this equation, we would start by looking for two different...

Ancient Footprints in New Mexico may place Human Migration into the Americas back to the Ice Ages

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Footprints of ancient humans were uncovered in a trench in White Sands National Park in New Mexico. Carbon dating using seeds of ancient plants surrounding the prints date them to about 23,000 years ago. That implies humans occupied the Americas about 8000 years earlier then what most archeologists believed. While these results are controversial, they are the first significant piece of evidence for building a case that could completely change our understanding of human history. In 2009, David Bustos, the resource program manager of the park, discovered the footprints for the first time. Since then, Bustos along with several teams of scientists and archeologists have uncovered thousands of prints across the park. The prints are thought to have been formed by ancient people walking along lakes, their feet imprinting the ground. Sediment would fill these holes and as the ground hardened, the prints would remain in tack, lost under layers of white sand. However, erosion has uncovered the...

Genetically Engineered Mosquitoes Combat Spread Of Deadly Diseases

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Mosquitoes are those annoying pests that cause mild discomfort on a warm summer’s day, but they are also responsible for the transmission of deadly diseases such as dengue fever and the zika virus. Aedes   aegypti , an invasive mosquito species that calls the Florida Keys its home, causes most of these infections each year. With the help of genetic engineering, U.K biotechnology company Oxitech have developed a genetically altered variant of Aedis aegypti that can destroy the species from the inside out. The deadly Aedis Aegypti  or yellow fever mosquito, notorious for spreading deadly viruses such as dengue fever.  Mosquito transmitted diseases account for tens of millions of cases of illness every year. Rises in global temperature and urbanization have increased their population sizes. In addition, mosquitoes are becoming more resistant to the insecticides that have kept them at bay for years. A new approach was necessary. Well, if we can’t reduce their current popul...

Video Game helps Scientists Researching Alzheimer's Disease

Alzheimer’s is an incurable disease that affects those in their elderly years and results in loss of many cognitive skills including memory and problem-solving skills. People with Alzheimer’s tend to show a loss of navigational skills before their rapid loss of memory. Keeping track of this may allow people to identify and begin treatment earlier. But to understand the difference in navigational abilities, it's necessary to develop a baseline of data: what are the expected navigational skills of a 75-year-old in America? To gather this data, researchers proposed a video game that would test navigational skills and collect demographic data. They partnered with a subsidiary of T-Mobile to develop a video game called Sea Hero Quest, which has players navigate varying complexity maps to complete objectives. The researchers expected to collect data on 100,000 people from Western Europe, but due to an ad-campaign and a partnership with one of the largest youtubers in the world named Pe...

This Simple Circuit Taught Itself To Recognize Flowers

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AI is a revolutionary technology that has applications in many diverse fields. From autonomous cars to optimal stock trading and beyond, AI has a profound impact on the world around us. One facet of AI is machine learning, where a computer system can learn to perform a task without explicit instruction via algorithms and statistical analysis of data sets. Due to these requirements, machine learning requires a lot of computational power. However, scientists have recently developed a simple electrical circuit, based loosely on how human brains function, that can teach itself to perform various AI tasks, such as recognition, without a computer aid. To understand how this circuit works like an AI, we need to understand a basic neural network structure. Neural networks can be modeled as a collection of nodes connected by weighted edges that numerically represent the correlation between nodes. Several networks are layered on top of one another, where the first network takes input data, an...

Research suggests a healthy gut microbiome can help you retain muscle mass

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  During hibernation, mammals will go without food or water and rest in a catatonic state for months on end. However, at the end of it all, these animals will continue about their business as if nothing had happened. Part of understanding this mystery is to find out how mammals prevent the degradation of their muscles during long periods of inactivity. New research on thirteen-line squirrels suggests that microbes in the gut are responsible. A thirteen-lined squirrel presenting its signature stripes. They hold the key to understanding the mechanisms that make hibernation possible for mammals.  Research was conducted to investigate how thirteen-lined squirrels can maintain their muscle mass over long periods of hibernation in the wintertime. They hypothesized that squirrel's maintain the energy by  breaking down muscle proteins throughout their body. This process produces ammonium, a nitrogen containing substance, which their body converts to urea. Urea is transported t...

Insights into Complex Group Behavior in Animals

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Collective animal behavior is a curious phenomenon in the animal kingdom. Animals of varying complexity and intelligence have shown the capacity to move in highly coordinated groups. This includes flocks of birds and foraging parties in ant colonies. Scientists haven’t been able to figure out exactly how these complex groups function. However, recent experiments have generated insight into some key mechanisms underlying this behavior. Researchers wanted to better understand the flocking behavior of jackdaws, birds that closely resemble crows in physicality and intelligence. They began tracking the birds flight patterns using high-speed cameras to create three-dimensional maps that show the flight path of every bird in the flock. In the winter months after foraging for food, the birds will travel together back to their nests; scientist call this a transiting flock. To test how their group behavior changed given different environmental conditions, researchers placed a robotic fox in a...