Discriminators of quadratic polynomials

Given $f \in \mathbb{Z}[x]$ and $n \in \mathbb{Z^{+}}$, the \emph{discriminator} $D_f(n)$ is the smallest positive integer $m$ such that $f(1), \ldots, f(n)$ are distinct mod $m$. In a recent paper, Z.-W. Sun proved that $D_f(n) = d^{\lceil \log_d n \rceil}$ if $f(x) = x(dx - 1)$ for $d \in \{2, 3\}$. We extend this result to $d = 2^r$ for any $r \in \mathbb{Z}^{+}$ and find that $D_f(n) = 2^{\lceil \log_2 n \rceil}$ in this case. We also provide more general statements for $d = p^r$, where $p$ is a prime. In addition, we present a potential method for generating prime numbers with discriminators of polynomials which do not always take prime values. Finally, we describe some general statements and possible topics for study about the discriminator of an arbitrary polynomial with integer coefficients.Comment: 8 page

Symmetries and intrinsic vs. extrinsic properties of M‾0,n\overline{\mathcal{M}}_{0, n}

We consider the following question: How much of the combinatorial structure determining properties of $\overline{\mathcal{M}_{0, n}}$ is ``intrinsic'' and how much new information do we obtain from using properties specific to this space? Our approach is to study the effect of the $S_n$-action. Apart from being a natural action to consider, it is known that this action does not extend to other wonderful compactifications associated to the $A_{n - 2}$ hyperplane arrangement. We find the differences in intersection patterns of faces on associahedra and permutohedra which characterize the failure to extend to other compactifications and show that this is reflected by most terms of degree $\ge 2$ of the cohomology/Chow ring. Even from a combinatorial perspective, terms of degree 1 are more naturally related to geometric properties. In particular, imposing $S_n$-invariance implies that many of the log concave sequences obtained from degree 1 Hodge--Riemann relations (and all of them for $n \le 2000$) on the Chow ring of $\overline{\mathcal{M}_{0, n}}$ can be restricted to those with a special recursive structure. A conjectural result implies that this is true for all $n$. Elements of these sequences can be expressed as polynomials in quantum Littlewood--Richardson coefficients multiplied by terms such as partition components, factorials, and multinomial coefficients. After dividing by binomial coefficients, polynomials with these numbers as coefficients can be interepreted in terms of volumes or resultants. Finally, we find a connection between the geometry of $\overline{\mathcal{M}_{0, n}}$ and higher degree Hodge--Riemann relations of other rings via Toeplitz matrices.Comment: 19 pages, Comments welcome

Characterizing cubic hypersurfaces via projective geometry

We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. Removing the conditions involving the middle Betti number from the numerical conditions used extends the possible $Y$ to cubic hypersurfaces, complete intersections of two quadric hypersurfaces, or complete intersections of two quartic hypersurfaces. The same method also gives a family of other cut and paste relations that can only possibly be satisfied by cubic hypersurfaces.Comment: More concise exposition; 20 page

Bounded gaps between primes in special sequences

We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor n\alpha\rfloor\}$, where $\alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.Comment: 14 page

Matroidal Cayley-Bacharach and independence/dependence of geometric properties of matroids

We consider the relationship between a matroidal analogue of the degree $a$ Cayley-Bacharach property (finite sets of points failing to impose independent conditions on degree $a$ hypersurfaces) and geometric properties of matroids. If the matroid polytopes in question are nestohedra, we show that the minimal degree matroidal Cayley-Bacharach property denoted $MCB(a)$ is determined by the structure of the building sets used to construct them. This analysis also applies for other degrees $a$. Also, it does not seem to affect the combinatorial equivalence class of the matroid polytope. However, there are close connections to minimal nontrivial degrees $a$ and the geometry of the matroids in question for paving matroids (which are conjecturally generic among matroids of a given rank) and matroids constructed out of supersolvable hyperplane arrangements. The case of paving matroids is still related to with properties of building sets since it is closely connected to (Hilbert series of) Chow rings of matroids, which are combinatorial models of the cohomology of wonderful compactifications. Finally, our analysis of supersolvable line and hyperplane arrangements give a family of matroids which are natrually related to independence conditions imposed by points one plane curves or can be analyzed recursively.Comment: 16 pages; Comments welcome

A Social Networks Approach to Interaction Patterns of BTS compared to Justin Bieber on Twitter

A Korean-pop boy band, BTS, has broken the cultural barrier to make changes in the nature of the global pop industry. This study examined the unique approach BTS has taken through social media to building its fan base and interacting with its fans. Twitter datasets were analyzed to explore the nature of BTS’s interaction with its fans on social media, as compared to another pop star, Justin Bieber. Findings and implications are discussed

Effect of Preferred Music on Agitation after Traumatic Brain Injury.

Background: Agitation after Traumatic brain injury (TBI) is a common behavioral problem which threatens the safety of patients and caregivers, disrupts the rehabilitation process, and becomes a major stress on caregivers. Preferred music as an environmental intervention can reduce agitation by inducing positive memories and emotions and relaxation in a patient. Purpose: The purpose of this study was to evaluate the effects of preferred music intervention on agitation for patients after TBI compared to those of classical “relaxation” music intervention. Methods: The study was a crossover design within subjects using a non-probability and purposive sampling method. The sample consisted of 14 patients from the acute rehabilitation unit in the University of Pittsburg Medical Center (UPMC) who suffered from cognitive impairments after severe brain injury and exhibited agitated behaviors. Subjects were divided into two groups. The first group listened to classical “relaxation” music first and then to preferred music. The second group listened to the music in reverse order. The scores of the Agitated Behavior Scale (ABS) were calculated in 1 hour increments (i.e., one hour before and immediately after listening and then one hour later). Data were analyzed using a mixed model ANOVA, a repeated measures ANOVA and a t-test. Results: There was a significantly greater effect of preferred music intervention compared to the effect of classical “relaxation” music (p= .046). The preferred music significantly reduced agitation (p= .01) while the classical “relaxation” music did not (p= .76). In particular, the patients’ physical agitation (p= .006) and cognitive agitation (p= .008) were significantly reduced after preferred music intervention. Conclusion: According to the findings, preferred music has a potential to provide therapeutic approaches for care of agitated patients after TBI. Additionally, positive effects of preferred music on agitation can give health care providers and patients’ family members motivation to explore more familiar environments for managing agitation.Ph.D.NursingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77884/1/soohyunp_2.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/77884/2/soohyunp_1.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/77884/3/soohyunp_3.pd