The significant tornado parameter (STP) is one of many models recently created by the National Storm Prediction Center (SPC) to predict the possibility of severe weather. This parameter looks at the winds, potential energy, and rotation in the atmosphere and performs calculations based on its predicted state. As more research on tornadoes has become available, this relatively new model has become more refined. The purpose of this project was to learn how physics and math work in meteorology, specifically in severe weather systems. Derivation and learning of each component of the STP was found by researching several documents and scientific papers released by the SPC and putting each component together into one equation. The results found from this project was that each function was a basic multivariable calculus calculation using concepts from thermodynamics. The data is gathered from modeled Skew-T graphs, which show various points of temperature, humidity, as well as wind direction and speed with height, as well as Hodographs which show a different view of the winds in a circular motion relative to a starting position.

Nonlocal methods have been greatly expanded upon since the beginning of the 21st century. For example, their applications in peridynamics have been invaluable in modeling fracture mechanics. However, integro-differential equations can be expensive to solve numerically. Additionally, how best to calibrate the kernel of a peridynamics operator against experimental data remains an open question. We apply a machine learning approach to address these topics. It has been shown that through providing information about a given function and information from the corresponding partial differential equation or fractional partial differential equation, a neural network can return the solution to some degree of accuracy. Additionally, a network can also invert for model parameters. We apply these methods to nonlocal equations by using the recently introduced approach "nPINNs" (nonlocal Physics Informed Neural Networks). We consider both the forward problem -- solving for the solution to the nonlocal equation, and the inverse problem -- solving for certain parameters in the kernel. We further test in the forward problem how the neural network parameters affect the solution accuracy.

It is well-known that geometry is important (for example, in engineering). It is also well-known the geometry has a long history. Here are some of the famous people in that history: Euclid is regarded as the founder of geometry. Descartes provided a relation between geometry and algebra. Hilbert (1899) simplified the foundations of geometry. (Note that Hilbert wrote on logic. See following Wikipedia articles: "Hilbert's axioms" and "David Hilbert.) Tarski provided a first-order (or elementary) theory of geometry. In Tarski's geometry the only primitive relations (predicates) are "betweenness" and "congruence" among points. In other words, Tarski simplified the foundations of geometry. Tarski also proved that his elementary theory of his geometry is decidable. But the decision procedure is computationally difficult. (See Tarski & Givant (1999) and Wikipedia "Tarski's geometry”.) Szmielew was a follower of Tarski. She has a number of geometries and/or algebras. Her elementary geometries are even simpler than Tarski's. (She also proved that the first theory of Abelian groups is decidable. See the Wikipedia article "Wanda Szmielew". Also see Szmielew (1983). She died before she finished this book. Her colleague M. Moszynska finished it.) We now consider a few of Szmielew's geometries/algebras. In particular, we consider midpoint algebras and parallelity planes. These notions appear in Szmielew (1983). We shall see that the notion of "parallel" is definable in midpoint algebras. (We believe that Szmielew wanted to explicitly state this result in her book. In other words, in this paper we are solving a historical mystery.) Here is the definition of a parallelity plane: Let S be a set and let a || be a quaternary relation on S. (We write ab || cd rather that ||(abcd).) Then the structure (S, ||) is said to be a "parallelity plane" whenever the following axioms are satisfied in this structure: 1. ab || ba; 2. ab || cc; 3. if a is different than b and ab || pq and ab || rs , then pq || rs; 4. if ab || ac then ba || bc; 5. there is a triple abc of points, such that not (ab || ac); 6. for every triple abp, there exists q different than p, such ab || pq; 7. if not (ab || cd) then there exists a p such (pa || pb and pc || pd). Recall that a "groupoid" is a set G together a binary operation + on G. Here is the definition of a midpoint algebra: A groupoid (G, +) will be called a "midpoint algebra" if it satisfies the following axioms: 1. a + a = a (idempotency); 2. a + b = b +a (commutativity); 3. (a+b) + (c+d) = (a+c) + (b+d) (bi-commutativity); 4. for all a and b, there exists x such that x + a = b (solution). The point a + b is referred to as a "midpoint of the pair (a,b)" and + is the midpoint operation. Here we provide a definition of parallel within a midpoint algebra. Recall that in Euclidean geometry a quadrilateral is a parallelogram iff the diagonals bisect each other. (See Wikipedia "Parallelogram".) This result provides motivation for this definition of parallel. In particular, we use this result to define a relation || on the midpoint algebra (G,+). Consider four distinct points a, b, c, d in a midpoint algebra (G, +). Then we define ab || cd iff a + d = b + c. Proposition: Let (G,+) be a midpoint algebra and let the relation parallel || on G be defined as above. Then the structure (G,||) is parallelity plane. Here is an additional question: Are midpoint algebras decidable? I conjecture that the answer is "yes".

We modify the proof of Natterer for the relaxed Kaczmarz algorithm with nonuniform relaxation parameters. Our method then extends results for alternative Kaczmarz methods.

Measles was declared eliminated in the United States in 2000, however, the number of measles cases has reached a new height in 2019. As the number of measles cases in the United States continues to rise due to unvaccinated individuals in the population and the influence of antivaxxers it becomes important to track their influence. The ability to track diseases using social media sites is an emerging method in the public health field. In this study, we collected 105,588 tweets from May to July 2019 pertaining to vaccinations and measles and analyzed the social sentiment emanating from these tweets. We found that when there were more measles cases, we have a more negative sentiment. Network analysis was then used to analyze the connection between users in the month of May by measuring their interactions in retweets and mentions. Of the 31,725 users, a majority were found to have a low degree of interaction while four were found to have a high degree of interaction. Three of the four users identified were tweeting with an antivaccination sentiment thus spreading their views to a large number of other users.

Models using nonlocal operators can potentially be used with a wider range of functions and applications than their classical differential equation counterparts. However, the well-posedness of the associated systems must be verified for each specific formulation of the nonlocal operators. Here, well-posedness of systems involving unweighted nonlocal operators is leveraged to propose well-posedness of associated systems with weighted nonlocal operators.

Most plant-vector-virus diseases found in nature are caused by the interaction of multiple viruses in organisms, called co-infection. Three main types of co-infection have been identified: helper-dependence, cross protection, and synergism. The vectors that are able to carry the diseases between hosts also tend to prefer certain categories of hosts over others. In this study, we developed a general compartmental disease model that incorporates all three types of co-infections, vector preference, and the transmission of disease in vectors in addition to in hosts. The model was utilized to evaluate the effectiveness of three common disease control strategies that are applied in agriculture: increasing planting of healthy hosts, rouging (removal of diseased plant hosts), and pesticides. When vectors preferred healthy hosts, for all types of co-infections none of these control measures eliminated the disease. For other types of preference, increasing the planting of healthy hosts rarely significantly lowered disease incidence, whereas planting more protected hosts when cross protection was present did lower disease occurrence. Also, when vectors did not prefer healthy hosts, pesticide was the most effective control. Rouging completely eliminated disease in the presence of cross protection but was largely ineffective against helper-dependence or synergism.

Flocking can be described as the collective behavior of a large number of interacting agents which eventually leads towards specific collective motion on a large scale. These types of interactions are ubiquitous in nature, finding relevance in large-scale phenomena such as the flocking of birds and in small-scale phenomena such as biological organism’s formation behavior. In the last decade, there has been significant progress in the field of flocking models. The Cucker-Smale (CS) model is the most widely accepted discrete model for describing microscopic flocking properties and paved the way for the Shvydkoy-Tadmor (ST) continuum model, which provides a description for macroscopic flocking dynamics. Two key properties present in all flocking models are the alignment of velocities and the cohesion of agents in the system.

In this study there was development of 2D CS and ST simulations. Alignment and Cohesion was found in both models under multiple trials consisting of various parameter regimes. The relative disorder of flocks grant the ability to quantify the average cohesion, as found in recent developments of the ST model’s long term entropy behavior. In further simulations of the ST model, an additional force in the analog for wind has been implemented and resulted in mostly anticipated behavior in the disorder of undisturbed flocks.