Symmetries on the Lattice of <Emphasis Type="Italic">k</Emphasis>-Bounded Partitions

In 2002, Suter [25] identified a dihedral symmetry on certain order ideals in Young’s lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result geometrically, the order ideals can be seen to be in bijection with the alcoves in a 2- fold dilation in the geometric realization of the affine symmetric group. By considering the m-fold dilation we observe a larger set of order ideals in the k-bounded partition lattice that was considered by Lapointe, Lascoux, and Morse [14] in the study of k-Schur functions. We identify the order ideal and the cyclic action on it explicitly in a geometric and combinatorial form.     

Shear thickening and shear-induced agglomeration of chemical mechanical polishing slurries using electrolytes

Chemical mechanical polishing is a fundamental technology used in the semiconductor manufacturing industry to polish and planarize a wide range of materials for the fabrication of microelectronic devices. During the high-shear (～1,000,000 s^?1) polishing process, it is hypothesized that individual slurry particles are driven together to form large agglomerates (≥0.5 µm). These agglomerates are believed to trigger a shear-induced thickening effect and cause defects during polishing. We examined how the addition of various monovalent salts (CsCl, KCl, LiCl, and NaCl) and electrostatic stabilizing bases (KOH, NaOH, or CsOH) influenced the slurry’s thickening behavior. Overall, as the added salt concentration was increased from 0.02 to 0.15 M, the shear rate at which the slurry thickened (i.e., the critical shear rate) decreased. Slurries with added CsCl, NaCl, and LiCl thickened at comparable shear rates (～20,000–70,000 s^?1) and, in general, followed ion hydration theory (poorly hydrated ions caused the slurry to thicken at lower shear rates). However, slurries with added KCl portrayed thickening behavior at higher critical shear rates (～35,000–100,000 s^?1) than other chloride salts. Also, slurries stabilized with CsOH thickened at higher shear rates (～90,000–140,000 s^?1), regardless of the added salt cation or concentration, than the slurries with KOH or NaOH. The NaOH-stabilized slurries displayed thickening at the lowest shear rates (～20,000 s^?1). The thickening dependence on slurry base cation indicates the existence of additional close-range structure forces that are not predicted by the Derjaguin–Landau–Verwey–Overbeek colloidal stability theory.     

Saturation of Berge hypergraphs

Given a graph $F$, a hypergraph is a Berge- $F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subhypergraph that is a Berge-$F$, but for any edge $e\in E\left(\overline{H}\right)$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars and matchings in $k$-uniform hypergraphs.     

Parameter choices and a better bound on the list size in the Guruswami–Sudan algorithm for algebraic geometry codes

Given an algebraic geometry code C_L(D, aP){C_{\mathcal L}(D, \alpha P)}, the Guruswami–Sudan algorithm produces a list of all codewords in C_L(D, aP){C_{\mathcal L}(D, \alpha P)} within a specified distance of a received word. The initialization step in the algorithm involves parameter choices that bound the degree of the interpolating polynomial and hence the length of the list of codewords generated. In this paper, we use simple properties of discriminants of polynomials over finite fields to provide improved parameter choices for the Guruswami–Sudan list decoding algorithm for algebraic geometry codes. As a consequence, we obtain a better bound on the list size as well as a lower degree interpolating polynomial.     

Nonsmooth bifurcations of equilibria in planar continuous systems

In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.     

The function of diorism in ancient Greek analysis

This paper is a contribution to our knowledge of Greek geometric analysis. In particular, we investigate the aspect of analysis know as diorism, which treats the conditions, arrangement, and totality of solutions to a given geometric problem, and we claim that diorism must be understood in a broader sense than historians of mathematics have generally admitted. In particular, we show that diorism was a type of mathematical investigation, not only of the limitation of a geometric solution, but also of the total number of solutions and of their arrangement. Because of the logical assumptions made in the analysis, the diorism was necessarily a separate investigation which could only be carried out after the analysis was complete.     

Counting arithmetical structures on paths and cycles

Let $G$ be a finite, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(diag\left(\mathbf{d}\right)?A)\mathbf{r}=0$, where $A$ is the adjacency matrix of $G$. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices $(diag\left(\mathbf{d}\right)?A)$). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients $\left(\genfrac{}{}{0ex}{}{2n?1}{n?1}\right)$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.     

Data-driven robust optimization

Mathematical Programming - The last decade witnessed an explosion in the availability of data for operations research applications. Motivated by this growing availability, we propose a novel schema...     

The Lang–Trotter conjecture for products of non-CM elliptic curves

The Ramanujan Journal - Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $${mathbb {Q}}$$ and by the subsequent...     

Distortion-free magnetic resonance imaging in the zero-field limit

MRI is a powerful technique for clinical diagnosis and materials characterization. Images are acquired in a homogeneous static magnetic field much higher than the fields generated across the field of view by the spatially encoding field gradients. Without such a high field, the concomitant components of the field gradient dictated by Maxwell’s equations lead to severe distortions that make imaging impossible with conventional MRI encoding. In this paper, we present a distortion-free image of a phantom acquired with a fundamentally different methodology in which the applied static field approaches zero. Our technique involves encoding with pulses of uniform and gradient field, and acquiring the magnetic field signals with a SQUID. The method can be extended to weak ambient fields, potentially enabling imaging in the Earth’s field without cancellation coils or shielding. Other potential applications include quantum information processing and fundamental studies of long-range ferromagnetic interactions.