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Full-Text Articles in Physical Sciences and Mathematics
Novel Theorems And Algorithms Relating To The Collatz Conjecture, Michael R. Schwob, Peter Shiue, Rama Venkat
Novel Theorems And Algorithms Relating To The Collatz Conjecture, Michael R. Schwob, Peter Shiue, Rama Venkat
Mathematical Sciences Faculty Research
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.
On The Solutions Of Three-Variable Frobenius-Related Problems Using Order Reduction Approach, Tian-Xiao He, Peter J.-S. Shiue, Rama Venkat
On The Solutions Of Three-Variable Frobenius-Related Problems Using Order Reduction Approach, Tian-Xiao He, Peter J.-S. Shiue, Rama Venkat
Mathematical Sciences Faculty Research
This paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius-related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three-variable Frobenius-related problems. The first method is used to reduce the equation of order three from a three-variable Frobenius-related problem to be a system of equations with two fixed variables. The second method reduces the equation of …
Impact Of Neoantigen Expression And T-Cell Activation On Breast Cancer Survival, Wenjing Li, Amei Amei, Francis Bui, Saba Norouzifar, Lingeng Lu, Zuoheng Wang
Impact Of Neoantigen Expression And T-Cell Activation On Breast Cancer Survival, Wenjing Li, Amei Amei, Francis Bui, Saba Norouzifar, Lingeng Lu, Zuoheng Wang
Mathematical Sciences Faculty Research
Neoantigens are derived from tumor-specific somatic mutations. Neoantigen-based syn-thesized peptides have been under clinical investigation to boost cancer immunotherapy efficacy. The promising results prompt us to further elucidate the effect of neoantigen expression on patient survival in breast cancer. We applied Kaplan–Meier survival and multivariable Cox regression models to evaluate the effect of neoantigen expression and its interaction with T-cell activation on overall survival in a cohort of 729 breast cancer patients. Pearson’s chi-squared tests were used to assess the relationships between neoantigen expression and clinical pathological variables. Spearman correlation analysis was conducted to identify correlations between neoantigen expression, mutation …