Saturday – Oct 24
Location: Library Concourse
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- Time:
- 3:30 pm – 4:00 pm
- Title:
- Pattern avoidance in lattice and Yamanouchi words
- Speakers:
- Alanna Pellicane (SUNY Brockport), Rebecca Smith (SUNY Brockport)
Abstract
This research focuses on pattern avoidance in binary lattice and Yamanouchi words. A lattice word (sometimes referred to as a ballot sequence) when read from left to right always has at least as many entries of value $i$ as it has of value $i+1$. These words were noted by Mark Dukes as being ascent sequences. A Yamanouchi word is a reversed lattice word. For each pattern $w$ of size three, we determine the number of binary Yamanouchi words of size $n$ that avoid $w$.
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- Time:
- 3:30 pm – 4:00 pm
- Title:
- Generalizations of the Classical Gaussian Integral
- Speaker:
- Asim Nepal (SUNY Brockport)
Abstract
The classical Gaussian integral is a fundamental result in analysis, probability, and statistics, but it represents only one member of a broader family of improper integrals. This project investigates what happens when its fixed quadratic exponent is replaced by a positive parameter. Specifically, we study \[ I(n)=\int_0^\infty e^{-x^n}\,dx,\qquad n > 0, \] with the goal of determining when the integral converges, finding an exact formula for its value, and understanding how the area under $e^{-x^n}$ changes as $n$ varies. Using the substitution $t=x^n$ and properties of the Gamma function, we obtain \[ I(n)=\Gamma\left(1+\frac{1}{n}\right). \] We then analyze this expression with respect to $n$ and show that $I(n)$ converges for every $n > 0$. Rather than changing monotonically, the area decreases to a unique minimum at approximately $n=2.16623$ and then increases toward $1$ as $n\to\infty$, while $I(n)\to\infty$ as $n\to0^+$. We also extend the same Gamma-function approach to weighted and scaled Gaussian-type integrals. Finally, composite Trapezoidal and Simpson's Rules are used to compare numerical approximations with exact values and to distinguish quadrature error from truncation error. Overall, the project shows how varying a single exponent transforms the classical Gaussian integral into a broader family with an exact analytic structure and predictable numerical behavior.