https://brooksemerick.com/gallery daily 0.75 2021-12-30 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1442542306946-36CDIZB2HXEWTZ33ILEQ/PhD_Phase_Portrait_Half.jpg Research Gallery - Wnt Pathway Dynamics with APC and Axin Regulation The Wnt Pathway is a cascade of chemical reactions within a cell that ultimately determines the destruction or accumulation of a proliferation promoting protein called beta-catenin. This figure shows a single trajectory in the phase space of a system of nonlinear differential equations that describes the interactions of three key proteins involved the Wnt pathway: beta-catenin, Axin, and APC. When the concentration of Axin is relatively small, the three concentrations become periodic in time. This indicates that a cell switches, periodically, between a proliferative and quiescent state. The paper is here. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1484763431952-CCFMHL5OCIAYXCUKLDMV/image-asset.jpeg Research Gallery - Normal Colonic Crypt Fission The colonic crypt is a small fold or invagination in the epithelial layer of the colon, which is responsible for consistent cell renewal. The normal crypt cycle is a mechanism by which the crypt itself divides to form two identical crypts. This process is characterized by an initial growth of the crypt followed by a budding at the base. The crypt bifurcation elongates and extends upwards until it reaches the lumen, finishing the crypt fission phase. The crypt cycle is a continuous process responsible for epithelial maintenance. To model this, we consider the crypt as a stack of cell rings such that each ring divides into two identical rings in a cascade-type fashion that begins with one cell at the base. The evolution of each cell-ring-division is modeled using the Level Set Method. The video above is the result of our working model. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1594836475063-U45OHH3MXDAB3Z8AJU2S/Surface_Stability.png Research Gallery - Host-Parasitoid Semi-Discrete Model with Parasitoid Migration Using the semi-discrete framework, we investigate the migration of parasitoids between two locations. Typically, hosts are immobile and parasitoids can fly. We consider the idea of redistribution of hosts between yearly updates. This means that each patch has a similar proportion of hosts. This figure shows the stability region for different values of alpha (the proportion of hosts at the first patch) versus the log of the local migration rates. We see that much of the space is stable, indicating that coexistence at both patches occurs. However, there is a sliver of unstable possibilities. The paper is here. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1442542311497-9SEF3197AQYYFEYQF7Y2/HP_Stability_Region_Host_Mortality.jpg Research Gallery - Host-Parasitoid Semi-Discrete Model with Host-Feeding Using the semi-discrete framework, we address the tendency of parasitoids to feed on host larvae. This figure compares the stability region of the non-zero equilibrium of the semi-discrete model with and without host-feeding when coupled with a density dependent host-mortality. The non-zero equilibrium corresponds to the system that yields coexistence between the host and the parasitoid, which is observed in nature. Host-feeding burdens the parasitoid population since the no-parasitoid equilibrium stability region is larger. Hence, host-feeding causes an inefficiency in the parasitoid's yearly reproductive habits, which yields a higher population of hosts per generation. Find the paper here. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1449102941351-6S5VLQYXNEIGM88T88TW/IFH_Bifurcation.png Research Gallery - Host-Parasitoid Semi-Discrete Model with Infected Host-Feeding Using the semi-discrete framework, we incorporate an infected host-feeding scenario, which assumes that the parasitoids feed on already infected hosts. Essentially, this means a parasitoid does not know a host has already been oviposited, resulting in a loss of potential adult parasitoids via host-feeding. This, interestingly, results in an oscillatory behavior for a critical value of R (viable eggs per adult host), which produces chaotic behavior for even larger values of R. This figure shows a period doubling bifurcation in the parameter R for the parasitoid population. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1475282845931-BNNRXN72QVOMRGKNG83H/Traj_No_Overlap_P_Var_vp2.png Research Gallery - Variability of Risk in Host-Parsitoid Models Using the traditional discrete structure of host-parasitoid models, we assume that the parasitoid has a different attack rate for each host, which means each host has variable risk. The risk, x, is assumed to be independent of local host density and distributed according to the continuous probability distribution, p(x). We implement the variability in existing host-parasitoid models to search for the conditions of coexistence of the host larvae population and two parasitoid population, which have different time frames of attack and different rates of attack. The figure shows a stable trajectory in which all three populations coexist. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1592884218187-5QVW54BTZ429D8KPJP50/Predicting_Opt_Sols.png Research Gallery - Predicting the Number of Optimal Solutions in MCSCP A minimium cardinality set covering problem is a well-known integer programming problem. Given a random matrix of ones and zeros with fixed dimensions and density, it is possible to determine the the smallest set of columns from that matrix that covers all rows. However, once this minmal set of columns is found, is there any optimal solutions of the same cardinality? The answer is sure, in some cases. In fact, there may be a large number of alternative solutions of the same cardinality. Is there a way to predict this number? We seek an answer to this question using statistics and machine learning. The plot above shows a boxplot of the number of optimal solutions for each minimal cardinality for 10x20 matrices with 20% density. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1475285669481-IW8XNBQ6Z0S6H5EL2P6Y/Dictyostelium.png Research Gallery - Compartment-based Model for Peptide Degradation This project considered a system of linear ODEs to model the degradation of a peptide substrate reporter for protein kinase B (VI-B) in five different cell cultures from data provided by the Chemistry department. After solving these equations one by one, we find the best fit parameters that match the data using least squares tools in Matlab with an iterative approach. Drawing from the histogram plots of parameter distributions, we conclude that the most popular values which yield the smallest residuals are best fit parameters. The figure shows the best fit of our model to the given data. The paper can be found here. https://brooksemerick.com/trinity-teaching-gallery daily 0.75 2021-01-17 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452638707766-OCJLRCV9VRGTY5C2NH8A/MATH305_Pic.jpg Trinity Teaching Gallery - MATH 305 -- Probability This course is an upper level math class focusing on discrete and continuous probability, combinatorial analysis, random variables, random vectors, density and distribution functions, moment generating functions, and particular probability distributions including the binomial, hyper-geometric, gamma, and normal. The weak and strong law of large numbers is discussed and the proof of the Central Limit Theorem is covered. The figure to the left illustrates the Central Limit Theorem. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1496262747116-45UYF31M6A8EIXA8OJ7L/MATH254_Fig.png Trinity Teaching Gallery - MATH 254 -- Mathematical Modeling II This course is designed as a companion course to MATH 252, with an alternate set of topics that emphasize mathematical applications from the social sciences, especially economics. In this course, we analyze and simulate discrete, continuous, and stochastic models that relate to economics and finance. Basic fixed/equilibrium point analyses are covered for single equations and systems. An emphasis on numerical simulation is also covered using MATLAB. The figure to the left shows a bivariate histogram for two best fit parameters of a continuous Allee-Effect model. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1491242275498-AOIGXNB95P7SQSDDEDRY/image-asset.jpeg Trinity Teaching Gallery - MATH 234 -- Differential Equations An introduction to the theory of ordinary differential equations and their applications. Topics include analytical and qualitative methods for analyzing first-order differential equations, second-order differential equations, and systems of differential equations. Analytical methods for finding solutions to differential equations will range from separation of variables and variation of parameters to Laplace transforms. Special emphasis will be placed on equilibria, stability analysis, and phase portraits of both linear and nonlinear equations and systems. The movie to the left shows a bifurcation of a one-parameter family of first-order linear systems. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1476805516063-SWAY4CXIHNYQYC7HSZSW/MATH231_Gradient.png Trinity Teaching Gallery - MATH 231 -- Calculus III A typical multivariate calculus course covering vector-valued functions; partial derivatives; double and triple integrals; line and surface integrals of scalar functions and vector fields; polar, cylindrical, and spherical coordinate systems; and Green's Theorem, Stokes' Theorem, and the Divergence Theorem. The figure to the left shows the contour plot of a smooth surface with the gradient vector field overlaid. This shows how the gradient is perpendicular to level curves of the surface. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452653708816-I1ZBTN28YRY3RE9FY6LH/MATH228_Pic.jpg Trinity Teaching Gallery - MATH 228 -- Linear Algebra MATH 228 is a proof-based course in linear algebra, covering systems of linear equations, matrices, determinants, finite dimensional vector spaces, and linear transformations. The least-squares problem and eigenvalues/eigenvectors are typically covered towards the end of the class. Although there is a computational component to the class, a stronger focus is put on developing well written proofs, which may include the properties of matrix algebra, set containment, linear independence, and properties of vector spaces and subspaces. The figure to the left shows the intersection of three planes in three dimensional space, i.e. a 3x3 linear system with a unique solution. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452708866640-1O30V2VOHKUCUIESD5H1/MATH210_Mariam.jpg Trinity Teaching Gallery - MATH 210 -- Scientific Computing in MATLAB This course is a computational workshop designed to introduce the student to Matlab, a powerful scientific computing software package. The workshop will focus on visual learning based on graphical displays of scientific data and simulation results from a variety of mathematical subject areas, such as calculus, differential equations, statistics, linear algebra, and numerical analysis. No prior computer language skills are required as basic programming tools such as loops, conditional operators, and debugging techniques will be developed as needed. The workshop prepares the student for future courses in applied mathematics as well as courses in other disciplines where scientific computing is essential. The figure to the left is a fractal map called the Julia set generated by Matlab with additional lighting/smoothing features enabled. Figure credit goes to Mariam Avagyan. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452640175220-26X4AWIM38SA4KASPPMR/C14_Prob_Dist_2.jpg Trinity Teaching Gallery - MATH 207 -- Statistical Data Analysis This class is an introductory course to modern statistical techniques that moves more quickly than its MATH 107 counterpart. Techniques of data analysis are explored including graphical methods, random variables, discrete and continuous probability models, regression analysis, sampling distributions for sample proportions and means, and inference for one and two population proportions and means. The figure to the left shows the binomial probability distribution. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452640602632-4CHZ9D7XXHSB4DKMBQ4T/C7_Discont_Integrand.jpg Trinity Teaching Gallery - MATH 132 -- Calculus II A typical calculus II course for students in the natural sciences that focuses primarily on the Riemann integral and its applications, techniques of integration, first-order ordinary differential equations, and infinite sequences and series. The figure to the left shows the infinite area under a discontinuous function. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452640471443-L4I2SRACHOOLJDGZD8JT/Sec_4_NM.jpg Trinity Teaching Gallery - MATH 131 -- Calculus I This course is a typical first calculus class designed for mathematics, natural science, and computer science majors. The course introduces important functions and graphs, the concept of limits and continuity, derivatives and their applications, and anti-derivatives with a brief introduction to the fundamental theorem of calculus. The figure to the left demonstrates Newton's Method for approximating roots of functions. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452639812208-V8KZ0L9AQ5STT80Y04TW/Coin.jpg Trinity Teaching Gallery - MATH 107 -- Elements of Statistics This course is a light introduction to statistical methods primarily for students in the social and natural sciences. The course typically covers the general measures of central tendency and dispersion, graphical displays of categorical and quantitative data, basic probability, random variables, sampling distributions, and inference for one and two population means. The figure to the left demonstrates the law of large numbers by keeping track of the proportion of heads that are flipped over a trial of 100 coin flips. https://static1.squarespace.com/static/55189425e4b017e0133cc3c2/5679e4c125981d64c1c05e24/57029fb080b19ac4c0cedb96/1459434851938/ Trinity Teaching Gallery - MATH 234 -- Differential Equations An introduction to the theory of ordinary differential equations and their applications. Topics will include analytical and qualitative methods for analyzing first-order differential equations, second-order differential equations, and systems of differential equations. Analytical methods for finding solutions to differential equations will range from separation of variables and variation of parameters to Laplace transforms. Special emphasis will be placed on equilibria, stability analysis, and phase portraits of both linear and nonlinear equations and systems. The movie to the left shows a bifurcation of a one-parameter family of first-order linear systems. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1452635118131-9KE4R2DVO5KXFLRBP4GR/MATH210_Pic.jpg Trinity Teaching Gallery - MATH 210 -- Scientific Computing in Matlab This course is a computational workshop designed to introduce the student to Matlab, a powerful scientific computing software package. The workshop will focus on visual learning based on graphical displays of scientific data and simulation results from a variety of mathematical subject areas, such as calculus, differential equations, statistics, linear algebra, and numerical analysis. No prior computer language skills are required as basic programming tools such as loops, conditional operators, and debugging techniques will be developed as needed. The workshop prepares the student for future courses in applied mathematics as well as courses in other disciplines where scientific computing is essential. Here is a gallery of some of the students' final projects. The figure to the left is a surface plot generated by Matlab with additional lighting/smoothing features enabled. https://brooksemerick.com/kutztown-teaching-gallery daily 0.75 2024-05-15 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1611243048462-QA8XVEOXEI0NCKGRC5W4/Kutztown_PDEs.png Kutztown Teaching Gallery - MAT 473 -- Partial Differential Equations This course is an introduction to the theory of and solutions to partial differential equations. The three big PDEs are the motivation: the heat equation, the wave equation, and Laplace’s equation. A solid understanding of linear systems of IVPs is assumed, and a strong focus on BVPs and Cauchy-Euler problems is applied initially. The theory and application of Fourier Series is motivated by finding the solutions to the three PDEs with Dirichlet and/or Neumann boundary conditions. Solutions of nonhomogeneous PDES on finite domains are considered as well as integral transforms for semi-infinite and infinite domain problems. Other topics include the method of characteristics, existence theorems, and Green’s Functions. The figure depicts the solution to the heat equation with no flux boundary conditions. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1611243048462-QA8XVEOXEI0NCKGRC5W4/Kutztown_PDEs.png Kutztown Teaching Gallery - MAT 473 -- Partial Differential Equations This course is an introduction to the theory of and solutions to partial differential equations. The three big PDEs are the motivation: the heat equation, the wave equation, and Laplace’s equation. A solid understanding of linear systems of IVPs is assumed, and a strong focus on BVPs and Cauchy-Euler problems is applied initially. The theory and application of Fourier Series is motivated by finding the solutions to the three PDEs with Dirichlet and/or Neumann boundary conditions. Solutions of nonhomogeneous PDES on finite domains are considered as well as integral transforms for semi-infinite and infinite domain problems. Other topics include the method of characteristics, existence theorems, and Green’s Functions. The figure depicts the solution to the heat equation with no flux boundary conditions. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1640185003693-IR4Q8AFS60LE37AOO9BN/MAT361_Website_Figure.jpg Kutztown Teaching Gallery - MAT 361 -- Operations Research Operations Research is a classic study of mathematical optimization and its applications to a wide array of real-world problems. Operations Research uses quantitative methods to determine the best decision for an operating system. A mathematical approach to studying methods as applied to the decision process in industry is taken. The methods studied are selected from among: linear programming, game theory, graph theory and network analysis. Our focus in this course will be on linear programming and sensitivity analysis. Students will be required to use appropriate computer software, including Excel. The figure depicts the optimal solution to a two-dimensional linear programming problem. The planar surface shows the objective function reaching it’s maximum value at a corner point of the feasible space. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1611243148184-H89N68B83W3N2MJAPZI9/Kutztown_ODEs.png Kutztown Teaching Gallery - MAT 340 -- Differential Equations An introduction to the theory of ordinary differential equations and their applications. Topics include analytical and qualitative methods for analyzing first-order differential equations, second-order differential equations, and systems of differential equations with an emphasis on modeling predator-prey and competing species systems. Analytical methods for finding solutions to differential equations will range from separation of variables and variation of parameters to Laplace transforms. Special emphasis will be placed on equilibria, stability analysis, and phase portraits of both linear and nonlinear equations and systems. The figure shows the phase-portrait with several trajectories of a nonlinear system that undergoes a subcritical Hopf bifurcation. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1611243187060-B2JA8NXIBWQRJEUG9EC9/Taylor_Polys_2.png Kutztown Teaching Gallery - MAT 182 -- Calculus II This course is one of a series intended for students who major in mathematics, the sciences, or engineering. The topics include the definition, properties, and applications of definite integrals; properties, derivatives, and integrals of exponential, logarithmic, trigonometric, inverse trigonometric, and hyperbolic functions with applications; techniques of integration; indeterminate forms and improper integrals; sequences, series, and convergence tests; differentiation and integration of power series; and polar integrals. This figure shows the function f(x) = cos(x) along with several approximating Taylor polynomials centered at a = pi/3. We can see that as the degree of the polynomial gets larger, the Taylor polynomials converge to the function. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1611243444703-HMGF4RUFKCNXCQFO82J7/MAT181_Figure.png Kutztown Teaching Gallery - MAT 181 -- Calculus This course is one of a series intended for students who major in mathematics, the sciences, or engineering. The topics include the definition and calculation of limits, continuity and differentiability, differentials, derivatives of algebraic and transcendental functions, the application of derivatives to graphing, antiderivatives, and the introduction of the definite integral, applications of definite integrals; and some techniques of integration. The figure above shows a midpoint Riemann sum approximation to the area under the curve y = x(x-1)(x+3). https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1516201470386-2DW0G7G3SE202J3U9ZNP/MAT140_Pic.png Kutztown Teaching Gallery - MAT 140 -- Applied Statistical Methods This course is an introduction to quantitative methods as applied to statistical reporting and data analysis. It incorporates some or all of the following: Techniques for obtaining, analyzing and presenting data in numerical form; measures of central tendency and dispersion; the normal distribution curve; standard scores; applicability of probability and sampling theory to statistical research; interpretation of confidence intervals; hypothesis testing; correlation; linear regression. The figure to the left gives a simple population distribution and the corresponding sampling distribution of sample means from samples of size 15. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1611245678600-01AHXR26APEOA1XWGNHE/MAT115_Figure_2.png Kutztown Teaching Gallery - MAT 115 -- Precalculus This course is designed to give students a thorough review of the mathematics background needed for calculus courses. Topics generally include properties of the real numbers, problem-solving using equations and inequalities, algebraic functions, graphing, and systems of equations. A treatment of trigonometry is also provided as well as applications of exponential functions. The figure shows trajectories of different exponential functions. Each function produces the amount of money at time t in an account if yearly interest is compounded n times throughout the year. The black curve shows the limit as n -> infinity, which indicates that interest is compounded continuously. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1548132369442-QYQHPHPXXGHCIYC3V1ZV/Trans_Cosine.png Kutztown Teaching Gallery - MAT 106 -- Trigonometry This course is intended for students with an elementary knowledge of algebra who need more work in trigonometric topics before taking more advanced mathematics courses. Topics include properties of and operations of functions, inverse functions, exponential and logarithmic functions, angle measurement, trigonometric functions and their inverses, graphing functions, and problem solving with equations. The figure shows the graph of the cosine function for increasing amplitude and increasing frequency given by f(x) = kcos(kx/2pi), for k ranging from -15 to 15. https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1534991926423-Z4PCTDLY2S1UFV5ZCZGM/image-asset.jpeg Kutztown Teaching Gallery - MAT 182 -- Calculus II This animation shows the convergence of the Taylor series centered at a = 3 of the function f(x) = x^2 exp(-2x) cos(x). https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1571098125300-AWG82X22QLAI7A6VIT3W/Kutztown_ODEs.png Kutztown Teaching Gallery https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1571097848707-OSX6BWZ6XYEBBTTSDW7O/Kutztown_ODEs.png Kutztown Teaching Gallery https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1516201280585-7W5GFR9OAQEY1RE64G0S/MAT140_Pic.png Kutztown Teaching Gallery https://brooksemerick.com/brooks-emerick daily 1.0 2025-08-21 https://brooksemerick.com/contact daily 0.75 2025-08-21 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/9581269b-39ae-49d9-b34b-dde3306d315c/Stamp+Background+Removed+3.png Contact - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://brooksemerick.com/teaching daily 0.75 2026-01-28 https://brooksemerick.com/research daily 0.75 2020-06-23 https://brooksemerick.com/math-362 daily 0.75 2026-03-17 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/2cc608e7-06d3-4ace-b508-b1cf045d0ee7/Class_Schedule_Spring_26.png MATH 362 -- Operations Research II My Spring 2026 Schedule https://brooksemerick.com/math-140 daily 0.75 2026-04-02 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/4f92a5eb-1ae7-4e94-ab01-0b3999b67334/Class_Schedule_Spring_26.png MATH 140 -- Applied Stats - Make it stand out My Spring Weekly Schedule (above) 020 Final Exam — Friday, May 15, 8:00 — 10:00 AM, AF 202 050 Final Exam — Monday, May 11, 8:00 — 10:00 AM, AF 202 https://brooksemerick.com/math-210 daily 0.75 2026-04-03 https://images.squarespace-cdn.com/content/v1/55189425e4b017e0133cc3c2/1b30378f-bed9-4767-a2fb-f3dea04584e1/Class_Schedule_Spring_26.png MATH 210 -- Mathematical Typesetting and Computing - Make it stand out Spring 2026 Weekly Schedule