اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA Conjectural Inequality for Visible Points in Lattice Parallelograms
48
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $a,n in mathbb{Z}^+$, with $a<n$ and $gcd(a,n)=1$. Let $P_{a,n}$ denote the lattice parallelogram spanned by $(1,0)$ and $(a,n)$, that is, $$P_{a,n} = left{ t_1(1,0)+ t_2(a,n) , : , 0leq t_1,t_2 leq 1 right}, $$ and let $$V(a,n) = # textrm{ of visible lattice points in the interior of } P_{a,n}.$$ In this paper we prove some elementary (and straightforward) results for $V(a,n)$. The most interesting aspects of the paper are in Section 5 where we discuss some numerics and display some graphs of $V(a,n)/n$. (These graphs resemble an integral sign that has been rotated counter-clockwise by $90^circ$.) The numerics and graphs suggest the conjecture that for $a ot= 1, n-1$, $V(a,n)/n$ satisfies the inequality $$ 0.5 < V(a,n)/n< 0.75.$$
قيم البحث
اقرأ أيضاً
This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefull
y about both the visible points and the invisible points in the definition of previous research. We prove asymptotic formulas for the number of lattice points in different levels of visibility.
We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker vis
ible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger than a positive constant.
We present a generalization of Brouwers conjectural family of inequalities -- a popular family of inequalities in spectral graph theory bounding the partial sum of the Laplacian eigenvalues of graphs -- for the case of abstract simplicial complexes o
f any dimension. We prove that this family of inequalities holds for shifted simplicial complexes, which generalize threshold graphs, and give tighter bounds (linear in the dimension of the complexes) for simplicial trees. We prove that the conjecture holds for the the first, second, and last partial sums for all simplicial complexes, generalizing many known proofs for graphs to the case of simplicial complexes. We also show that the conjecture holds for the tth partial sum for all simplicial complexes with dimension at least t and matching number greater than $t$. Returning to the special case of graphs, we expand on a known proof to show that the Brouwers conjecture holds with equality for the tth partial sum where t is the maximum clique size of the graph minus one (or, equivalently, the number of cone vertices). Along the way, we develop machinery that may give further insights into related long-standing conjectures.
Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $mathscr B$-free numbers w
ith Sarnaks conjecture on the `randomness of the Mobius function, another the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.
A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice $mathbb{Z}^{2}$, gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are
said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a fractal structure. This complicated structure in some sense arises from prime powers - singularities do not occur for circles of radius $sqrt{n}$ if $n$ is square free.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
