You can probably remember a time whenever you used math to unravel an on a regular basis problem, equivalent to calculating suggestions at a restaurant or determining the square footage of a room. But what role does mathematics play in solving complex problems equivalent to curing an illness?

In my job as Applied mathematicianI take advantage of mathematical tools to review and solve complex problems in biology. I actually have worked on problems involving genes and neural networks equivalent to: Interactions between cells And Decision making. To do that, I create descriptions of an actual situation in mathematical language. The strategy of converting a situation right into a mathematical representation known as modeling.

Translating real situations into mathematical terms

If you've ever done a math problem in regards to the speed of trains or the fee of food, that's an example of mathematical modeling. But for tougher questions, it could actually be complicated to jot down the true scenario as a mathematical problem. This process requires rather a lot Creativity and understanding of the issue in query and is usually the results of collaboration between applied mathematicians and scientists from other disciplines.

As an example, we could represent a Sudoku game as a mathematical model. In Sudoku, the player fills in empty squares in a puzzle with numbers between 1 and 9, with certain rules in place, equivalent to that numbers can’t be repeated in any row or column.

The puzzle starts with just a few pre-filled boxes and the goal is to work out which numbers fit within the remaining boxes.

Imagine that a variable, say x, represents the number that appears in one in every of these empty fields. We can guarantee that x is between 1 and 9 by saying that x solves the equation (x-1)(x-2) … (x-9)=0. This equation is true provided that one in every of the aspects on the left side is zero. Each of the aspects on the left is zero provided that x is a number between 1 and 9; for instance (x-1)=0 when x=1. This equation encodes a fact about our Sudoku game, and we will do this Code the sport's other features in an identical manner. The resulting Sudoku model consists of a series of equations with 81 variables, one for every box within the puzzle.

Another situation we could model is the concentration of a drug, equivalent to aspirin, in an individual's bloodstream. In this case, we can be interested by how the concentration changes after we take aspirin and the body metabolizes it. Just like Sudoku, one can create a series of equations that describe how the concentration of aspirin evolves over time and the way additional intake affects the dynamics of this drug. However, unlike Sudoku, the variables that represent concentrations usually are not static but change over time.

But the act of modeling is just not at all times that easy. How would we model diseases like cancer? Is it enough to model the dimensions and shape of a tumor, or do we want to model each blood vessel within the tumor? Every single cell? Every single chemical in every cell? Loads is unknown about cancer. So how can we model such unknown features? Is that even possible?

Applied mathematicians must discover a balance between models which might be realistic enough to be useful and straightforward enough to implement. These models can take several years to create, but in collaboration with experimental scientists, trying to seek out a model often provides recent insights into the real-world problem.

Mathematical models help to seek out real solutions

After a mathematical problem is written to represent a situation, the second step within the modeling process is to unravel the issue.

For Sudoku we’ve got to unravel a group of equations with 81 variables. For the aspirin example, we want to unravel an equation that describes the speed of change in concentrations. This is where all of the mathematics that was invented and continues to be being invented comes into play. Areas of pure mathematics equivalent to algebra, evaluation, Combinatorics and lots of others might be used – in some cases combined – to unravel the complex mathematical problems that arise from the applying of mathematics in the true world.

The third step of the modeling process is to translate the mathematical solution into the answer of the applied problem. In the case of Sudoku, solving the equations tells us which number must fit into which box to unravel the puzzle. In the case of aspirin, the answer can be a series of curves that tell us the concentration of aspirin within the digestive system and bloodstream. This is how applied mathematics works.

When making a model is just not enough

Or is it? While this three-step process is the perfect strategy of applied mathematics, the fact is more complicated. Once I reach the second step where I need the answer to the mathematics problem, fairly often, if not more often than not, it seems that nobody knows tips on how to solve the mathematics problem within the model. In some cases, mathematics is used to research the issue doesn't exist in any respect.

For example, it’s difficult to investigate cancer models since the interactions between genes, proteins, and chemicals usually are not as clear because the relationships between boxes in a Sudoku game. The predominant difficulty is that these interactions are “nonlinear,” meaning that the effect of two inputs is just not simply the sum of the person effects. To address this problem, I actually have been working on novel methods for studying nonlinear systems, equivalent to: Boolean network theory and polynomial algebra. Using this and traditional approaches, my colleagues and I actually have examined questions in areas equivalent to
Decision making,
Gene networks,
Cell differentiation And
Limb regeneration.

When approaching unsolved problems in applied mathematics, the excellence between applied and pure mathematics often disappears. Areas that were once considered too abstract were exactly what is required for contemporary problems. This highlights the importance of mathematics for all of us; Current areas of pure mathematics can grow to be the applied mathematics of tomorrow and be the tools needed for complex, real-world problems.

So intense Heat breaks records Around the world, a little-reported fact offers hope for urban cooling: Even in periods of maximum heat, some city blocks never experience heatwave temperatures.

How is that this possible?

Civilizations have recognized the flexibility of cities to heat and funky themselves for hundreds of years. Urban architects in ancient Rome called for Narrowing of streets to mitigate late afternoon temperatures. Narrow streets have been found to chill the air by limiting the realm exposed to direct sunlight.

The whitewashed architecture of the Greek islands is one other long-standing strategy. Bright partitions and roofs might help cool cities by reflecting incoming sunlight.

In hot and humid regions within the southern United States Thomas Jefferson proposed a unique approach for cooling: All latest settlements must have a checkerboard pattern of heavily vegetated city blocks interspersed with dense buildings. This could promote cooling through convective air movement between cool and warm zones.

As I explain in my latest book, “Radical Adaptation: Transforming Cities for a Climate Changed World”, Modern cities unintentionally increase their very own temperatures and create the so-called “urban heat island effect”.

How cities heat up

Cities increase their temperature in Four essential paths:

In combination, these 4 aspects of the urban heat island effect can increase temperatures in cities by 10 to twenty degrees Fahrenheit (5.6 to 11 degrees Celsius) on a hot summer afternoon – a big man-made change within the weather that may pose a serious health risk to people without air con.

In conjunction with the design of the built environment, a city's natural topography can amplify temperature differences between neighborhoods. San Francisco's hills and fog patterns, for instance, consistently divide neighborhoods into different climate zones. And the extensive Use of garden irrigation systems In hot and dry climates, temperatures in cities may be lower than in the encompassing desert. Hence the so-called “cool city islands”.

Simple steps to chill down cities

By understanding the extent to which cities can heat up, we will find effective tools to chill them down when human-caused global warming causes baseline temperatures to rise.

First, it is crucial that cities drastically reduce their greenhouse gas emissions to stop climate change from accelerating. Urban areas world wide, with their industries, vehicles and buildings, greater than 70% of greenhouse gas emissions from energy consumption and Their population is growing rapidlyEven a globally coordinated reduction in greenhouse gas emissions requires many many years to measurably decelerate the warming trendsdue to this fact, cities will still need to adapt.

Cities may slow urban heat island warming by taking sometimes easy measures. Research shows that Health advantages of reducing urban heat islands may very well be significant.

On Georgia Tech Urban Climate Labmy colleagues and I are working with cities to estimate the cooling potential of urban heat management—strategies designed to reverse the urban heat island effect. We do that by measuring the direct health advantages of actions comparable to expanding tree cover and other green infrastructure and using cool materials for streets and roofs.

Our work shows that planting trees on just half the realm available for tree cover growth—comparable to along roads, in parking lots, and in residential gardens—can reduce summer afternoon temperatures by 5-10 F (2.8-5.6 C) and reduce heat-related deaths by 40%-50% in some districts.

Recognizing these significant advantages, New York City set a goal of 1 million trees in its five districts.

Cool roofing material and light-colored finishes may help lower the temperature. Wearing a black shirt within the sun on a hot day will heat you up greater than wearing a white shirt. Likewise, light-colored constructing materials, roof coatings and shingles reflect more incoming solar heat than darkand absorb less heat. This is particularly effective in the warmth of the day when the sun's rays are at their strongest.

To benefit from this cooling effect, Los Angeles was the primary major city to require cool roofs for all latest buildings.

What cities can do now

Aggressive strategies to extend green tree cover in cities, a rapid transition to cooling roofing materials and even Replacing some parking lanes on the road and other unused sealed areas with biofilter troughs crammed with vegetation can significantly reduce temperatures in cities. In this manner, the resilience of a city to rising temperatures may be increased.

We have conducted heat risk assessments in quite a few U.S. cities, including Atlanta; Dallas; Louisville, Kentucky; And San Franciscoshow that a mixture of urban heat management strategies could reduce neighborhood temperatures by greater than 10 °F (5.6 °C) on hot days and reduce heat-related premature deaths by 20 to 60%.

A cooler city is a safer city. And it’s clearly as much as the communities to realize this.

The charges – and in some cases the convictions – against Hundreds of individuals Charges of participating within the riot on the U.S. Capitol on January 6, 2021, are resulting from be reconsidered and possibly dropped resulting from a U.S. Supreme Court ruling on June 28, 2024. Among those charged under a broad interpretation of the obstruction of justice statute that has now been narrowed by the Supreme Court is former President Donald Trump.

In its decision of Fischer v. United StatesThe Supreme Court ruled that a federal law prohibiting obstruction of an official proceeding may not apply to 3 defendants accused of participating within the revolt on the U.S. Capitol. Although former President Donald Trump will not be a defendant within the case, Special investigator Jack Smith has charged him individually with the violation of the identical law.

As a law professor I’ll join Mr. Smith, who teaches and writes within the areas of constitutional law and federal courts, to elucidate what the court's decision means for the January 6 defendants – and for Smith's case against Trump.

Charges against Capitol rioters

According to their indictment Joseph Fischer, Edward Long And Garret Miller were present on the Capitol on January 6, 2021. Prosecutors say all three men entered the Capitol through the riot and attacked cops. One of the boys, Lang, brandished a baseball bat and a stolen police shield, and one other, Miller, later called on social media for the assassination of U.S. Rep. Alexandria Ocasio-Cortez.

The Federal Prosecutor’s Office charged the three men with various crimes, including Attack on a federal official, inappropriate behavior on Capitol grounds And Obstruction of a congressional procedureThis last allegation is the topic of the appeal before the Supreme Court.

Before trial, the defendants argued that the statute prosecutors had relied on to charge them with obstruction of justice applied only to tampering with evidence, to not violently disrupting a session of Congress. The district court agreed and dismissed the costs, however the US Court of Appeals for the DC Circuit overturned ruling and sent the case back for trial.

The Supreme Court then agreed to listen to the case, but adjourned the hearing while it considered the dispute over the scope of the obstruction of justice law.

Definition of a collective term

In a 6-3 opinion by Chief Justice John Robertsthe Supreme Court ruled in favor of the defendants, holding that the law only prohibited tampering with evidence. It then remanded the case to the Court of Appeals to choose whether the defendants had violated the law in that narrow interpretation by attempting to stop Congress from obtaining and certifying the states' true electoral votes.

The court began with the text of the obstruction of justice statute. The law punishes anyone who “alters, destroys, mutilates, or conceals a record, document, or other item” or “otherwise obstructs, influences, or prevents any official proceeding.” The government argued that the defendants “otherwise obstructed” the strategy of certifying the outcomes of the 2020 election in Congress.

The court rejected that argument, nevertheless, noting that the phrase “otherwise impedes” refers only to impediments that — resembling altering, destroying, mutilating, or concealing a record, document, or object — impair the provision or integrity of evidence to be used in an official proceeding. The statutory catch-all term for “otherwise impedes” an official proceeding should be read along with the list of acts that precede it, the court explained. Otherwise, the list can be redundant.

The court also pointed to the historical background of the law. Congress, the court explained, had passed this special disability law in 2002 within the wake of the Enron accounting fraud scandalThe aim was to shut a loophole within the country's existing obstruction of justice laws, which on the time prohibited commissioning third parties to destroy incriminating evidence but stopping the destruction of the evidence oneself.

The court explained that the federal government's interpretation of the law went far beyond that purpose since it prohibited types of obstruction of evidence that Congress never intended to criminalize.

What this implies for the defendants of January 6 – and for Trump

The Supreme Court's decision doesn’t end the proceedings against the defendants within the Fischer criminal case; they’ll probably be tried for assault and disturbing the peace.

But it could end in the obstruction of justice charge being dismissed or the obstruction of justice convictions for other defendants from January 6 being overturned. According to an NPR databaseThe Federal Prosecutor's Office brought charges against no less than 250 other defendants for obstructing an official proceeding; 128 were convicted.

The ruling could also undermine special counsel Jack Smith’s case against former President Donald Trump, which Smith charged with obstruction under the identical lawIf that's the case survives a separate pending appeal before the Supreme Courtthe previous president will likely seek to have these charges dismissed.

Trump may not succeed, nevertheless, because the obstruction of justice charge is predicated partially on the allegation that he organized electoral lists to substantiate false election results to CongressThis may compromise the integrity of the evidence utilized in the certification process.

And obstruction of justice will not be the one charge the previous president faces. But the ruling could limit the case and make it tougher for the special counsel to present evidence of the Jan. 6 violence to the jury. Under this recent ruling, that violence alone couldn’t qualify as obstruction of justice.

The Fischer case also shows how judges can sometimes, especially in high-stakes cases, use methods of legal reasoning that they’re quick to criticize in other contexts. In the opinion, members of the Supreme Court's conservative majority pointed to the history of the obstruction of justice law – evidence that conservative jurists like the late Justice Antonin Scalia often described as unreliable.

The Supreme Court's decision in Fischer could have profound implications for the special counsel's historic prosecution of former President Trump.

But even when that will not be the case, the report still sheds essential light on the court's internal workings and the federal government's power to guard the integrity of its proceedings.