Mathematics and Computer Science Faculty/Staff Scholarship
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Combinatorial Identities for the Padovan Numbers
<p>We interpret the Padovan numbers combinatorially by having them count the number of tilings of an n-strip using dominoes and triominoes. Using this interpretation, we develop a collection of identities satisfied by the sequence of Padovan numbers.</p>

https://digitalcommons.misericordia.edu/mathcomp_facstaff/4
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Fibonacci Graphs
<p>By considering the Fibonacci numbers combinatorially, as counting the number of tilings of a strip of blocks with squares and dominoes, we introduce a graph that represents the sequence of Fibonacci numbers. Additionally, we consider individual graphs representing each Fibonacci number. Finally, we consider the graphic structure of these Fibonacci graphs and show how certain graphic properties relate to some wellknown identities of the Fibonacci numbers.</p>

https://digitalcommons.misericordia.edu/mathcomp_facstaff/3
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What Makes a Good Research Consultant?
<p>Statistical and research consulting is defined as the collaboration of a statistician or methodologist with another professional for devising solutions to research problems. An in-depth, interview qualitative approach was taken to answer the research question of what makes a good research consultant. The authors interviewed four faculty members in the field of statistics and research methods and two experienced graduate student consultants. In-depth, face-to-face interviews revealed common themes regarding consultancy skills, resourcefulness, communication and interpersonal skills. The participants discussed how to improve consulting sessions and deal with clients with different statistics levels and backgrounds. Participants felt there was no difference in how they approached a qualitative vs. quantitative consulting session. Finally, all the participants stated that the training of graduate student consultants can be improved by project-based coursework.</p>

https://digitalcommons.misericordia.edu/mathcomp_facstaff/2
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A Super-Gaussian Poissonâ€“Boltzmann Model for Electrostatic Free Energy Calculation: Smooth Dielectric Distribution for Protein Cavities and in Both Water and Vacuum States
<p>Calculations of electrostatic potential and solvation free energy of macromolecules are essential for understanding the mechanism of many biological processes. In the classical implicit solvent Poissonâ€“Boltzmann (PB) model, the macromolecule and water are modeled as two-dielectric media with a sharp border. However, the dielectric property of interior cavities and ion-channels is difficult to model realistically in a two-dielectric setting. In fact, the detection of water molecules in a protein cavity remains to be an experimental challenge. This introduces an uncertainty, which affects the subsequent solvation free energy calculation. In order to compensate this uncertainty, a novel super-Gaussian dielectric PB model is introduced in this work, which devices an inhomogeneous dielectric distribution to represent the compactness of atoms and characterizes empty cavities via a gap dielectric value. Moreover, the minimal molecular surface level set function is adopted so that the dielectric profile remains to be smooth when the protein is transferred from water phase to vacuum. An important feature of this new model is that as the order of super-Gaussian function approaches the infinity, the dielectric distribution reduces to a piecewise constant of the two-dielectric model. Mathematically, an effective dielectric constant analysis is introduced in this work to benchmark the dielectric model and select optimal parameter values. Computationally, a pseudo-time alternative direction implicit (ADI) algorithm is utilized for solving the super-Gaussian PB equation, which is found to be unconditionally stable in a smooth dielectric setting. Solvation free energy calculation of a Kirkwood sphere and various proteins is carried out to validate the super-Gaussian model and ADI algorithm. One macromolecule with both water filled and empty cavities is employed to demonstrate how the cavity uncertainty in protein structure can be bypassed through dielectric modeling in biomolecular electrostatic analysis.</p>

https://digitalcommons.misericordia.edu/mathcomp_facstaff/1
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