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Proportion of p-adic Polynomials Which Are Irreducible
Full text linkWe attempt to quantify the exact proportion of p-adic polynomials of degree n which are irreducible. We find an exact answer to this when n is prime and p != n, and also when n = 4 and p != 2. Our answers are rational functions in p. This relates to previous work done to find exact proportions of p-adic polynomials of degree n which have k roots
Tilings in the 3 dimensional lattice with L-tetrominoes
Full text linkWe consider three dimensional L-tetrominoes. We show that there exists at least one way to tile every three dimensional rectangle whose side lengths are at least and area is congruent to such that one square goes untiled. In addition, we show that every three dimensional rectangle is tileable provided one side has length at least and the other is a multiple of
Incremental Increases Between Successive Integers when Raised to the nth Power
Full text linkFor thousands of years, the beautiful field of number theory has captivated mathematicians with its elegant simplicity. Positive integers continue to reveal properties and relationships that are a joy to uncover, and in this paper, we investigate a pattern involving exponents and factorials while exploring some common notations in the field of number theory. Combinatorics, the field dealing with the mathematics of counting and arranging, also holds a presence in this paper. Pascal’s Triangle–the foundation of binomial expressions, also comes into play due to its tight relationship with combinatorics. Pascal’s Identity, the property that builds the triangle, becomes very useful as well. In our study, we explore the incremental increases between successive integers when raised to the nth power. For example, if we raise consecutive positive integers to the second power and enact two orders of differences on these values, we arrive at the constant increment of two, which is 2!. Further, if raise consecutive positive integers to the third power and enact three orders of differences, we obtain the constant increment of six–which is 3!. In this paper, we prove that if we raise consecutive positive integers to the nth power and take the nth difference, we always arrive at the constant increment of n!
Generative AI in CS Education: Navigating the Intersection of Student Usage and Faculty Vision
Full text linkGenerative AI tools are rapidly transforming both educational and professional landscapes, creating an imperative for computing education to adapt. This mixed-methods study investigates how Computer Science and Software Engineering students utilize these tools and how faculty perceive their impact within a small private STEM institution. Through surveys of 40 CS/SE students and 29 faculty members, complemented by in-depth interviews with 10 faculty, this dual perspective enables direct comparison between student practices and faculty perceptions, helping to identify areas where educational policies and practices may need adjustment.
Analysis identified six distinct categories of GenAI usage: Learning Enhancement (29.2%), Productivity and Efficiency (28.1%), Research and Information Processing (15.7%), Creativity and Ideation (13.9%), Problem-Solving (8.2%), and Accessibility Support (4.9%). These patterns demonstrate students\u27 strategic adoption of AI to enhance their educational experience, mirroring industry\u27s increasing integration of these tools into professional workflows.
Findings reveal tensions between AI\u27s efficiency benefits and concerns about fundamental skill development, suggesting institutions should consider structured AI learning opportunities, appropriate use guidelines, and balanced assessment approaches to prepare students for AI-augmented professional environments
Finite Posets as Prime Spectra of Commutative Noetherian Rings
Full text linkWe study finite partially ordered sets of prime ideals as found in commutative Noetherian rings. In doing so, we establish that these posets have a bipartite structure and devise a construction for finding ring spectra that are order-isomorphic to many such posets. Specifically, we prove that any finite complete bipartite graph is order-isomorphic to the spectrum of a ring of essentially finite type over the field of rational numbers. Furthermore, we prove that prime spectra of such rings can also depict any finite path or even cycle
Fixing the Potholes on the Road to Academic Success: A Curriculum for Engineering Educators to Create and Sustain Meaningful Change
No full textDegeneracies In a Weighted Sum Of Two Squares
No full textThis work is an attempt to classify and quantify instances when a weighted sum of two squares of positive integers, 3n2 1 +n2 2, can be realized in more than one way. Our project was inspired by a particular study of two-dimensional quantum billiards [S. G. Jackson, H. Perrin, G. E. Astrakharchik, and M. Olshanii, SciPost Phys. Core 7, 062 (2024)] where the weighted sums of interest represents an energy level with the two integers being the billiard’s quantum numbers; there, the 3-fold degeneracies seem to dominate the energy spectrum. Interestingly, contrary to the conventional paradigm, these degeneracies are not caused by some non-commuting symmetries of the system
2025 Rose-Hulman Institute of Technology : ONE HUNDRED AND FORTY-SEVENTH COMMENCEMENT MAY 31, 2025
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