as y′′ + t t2−16y ′ + 9 (t2−16)(t−3)y = 0. So p(t) = t t2−16 and q(t) = 9 (t2−16)(t−3) . Hence p(t) is continuous if t ∈ (−∞,−4) ∪ (4,∞) and q(t) is continuous if t ∈ (−∞,−4) ∪ (−4, 3) ∪ (3,∞). Therefore p(t) and q(t) are continuous if t ∈ (−∞,−4)∪(−4, 3)∪(3,∞). The initial conditions are y(2) = 1 and y′(2) = 2. We have 2 ∈ (−4, 3). Thus the solution exists on the interval (−4, 3). (b) The init...

We find all solutions to the parametrized family of norm-form equations $$ x^3-(t^3-1)y^3+3(t^3-1)xy+(t^3-1)^2 = \pm 1, where $t \gt 1$ is an integer, studied by Amoroso, Masser and Zannier. Our proof relies upon appeal lower bounds for linear fo

This paper presents a simple and robust method for computing the bisector of two planar rational curves. We represent the correspondence between the foot points on two planar rational curves C1ðtÞ and C2ðrÞ as an implicit curve F(t,r) 1⁄4 0, where F(t,r) is a bivariate polynomial B-spline function. Given two rational curves of degree m in the xy-plane, the curve F(t,r) 1⁄4 0 has degree 4m 1 2, ...

By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found...

Let μ be a rational-valued metric on a finite set T . We consider (a version of) the multifacility location problem: given a finite set V ⊇ T and a function c : V2 ) → Z+, attach each element x ∈ V − T to an element γ(x) ∈ T minimizing ∑(c(xy)μ(γ(x)γ(y)) : xy ∈ V2 ) ), letting γ(t) := t for each t ∈ T . Large classes of metrics μ have been known for which the problem is solvable in polynomial t...

CONTEXT Although a rare occurrence, previously undiagnosed disorders of sex development (DSD) with hyperandrogenism are sometimes detected by hormonal screening during the international sports competitions. Identifying the cause of XY,DSD raises medical and ethical concerns, especially with regard to issues of the eligibility to compete. OBJECTIVE The aim of this study was to determine whethe...

PURPOSE 45,XY,t(13;14)(q10;q10) karyotype can suggest infertility associated with more or less severe oligospermia in male adults. In addition, 45,XX,t(13;14)(q10;q10) karyotype carries reproductive risks such as miscarriage or infertility in female adults. However, reports on the phenotype of this karyotype in children are very rare. This study was done to observe various phenotypes of this ka...

We study a multiorbital model for the alkaline iron selenides K(1-x)Fe(2-y)Se(2) using a slave-spin method. With or without ordered vacancies, we identify a metal-to-Mott-insulator transition at the commensurate filling of six 3d electrons per iron ion. For Hund's couplings beyond a threshold value, this occurs via an intermediate orbital-selective Mott phase, in which the 3d xy orbital is Mott...

For phase transitions in dissipative quantum impurity models, the existence of a quantum-to-classical correspondence has been discussed extensively. We introduce a variational matrix product state approach involving an optimized boson basis, rendering possible high-accuracy numerical studies across the entire phase diagram. For the sub-Ohmic spin-boson model with a power-law bath spectrum ∝ω(s)...

The Hall resistivity xy , resistivity xx , and magnetization of three metallic ferromagnets are investigated as a function of magnetic field and temperature. The three ferromagnets, EuFe4Sb12 Tc 84 K , Yb14MnSb11 Tc 53 K , and Eu8Ga16Ge30 Tc 36 K are Zintl compounds with carrier concentrations between 1 1021 and 3.5 1021 cm−3. The relative decrease in xx below Tc xx Tc / xx 2 K is 28, 6.5, and ...