ترغب بنشر مسار تعليمي؟ اضغط هنا

Optimal measures for p-frame energies on spheres

85   0   0.0 ( 0 )
 نشر من قبل Josiah Park
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the $600$-cell for several ranges of $p$ in different dimensions. Our methods apply to a much broader class of potential functions, those which are absolutely monotonic up to a particular order as functions of the cosine of the geodesic distance. In addition, a preliminary numerical study is presented which suggests optimality of several other highly symmetric configurations and weighted designs in low dimensions. In one case we improve the best known lower bounds on a minimal sized weighted design in $mathbb{CP}^4$. All these results point to the discreteness of minimizing measures for the $p$-frame energy with $p$ not an even integer.



قيم البحث

اقرأ أيضاً

For a collection of $N$ unit vectors $mathbf{X}={x_i}_{i=1}^N$, define the $p$-frame energy of $mathbf{X}$ as the quantity $sum_{i eq j} |langle x_i,x_j rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimizatio n problem, so giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-frac p 2} (2-p)^{frac {p-2} 2}$ which is sharp for $dleq Nleq 2d$ and $p=1$. We prove that for $1leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $pin[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies.
277 - Ruxi Shi 2020
A Borel probability measure $mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(mu)$. In this paper, we characterize all spectr al measures in the field $mathbb{Q}_p$ of $p$-adic numbers.
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analysts Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in $mathbb{R}^2$ (P. Jones, 1990), in $mathbb{R}^n$ (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones $beta$-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in $mathbb{R}^n$ that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.
To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X({ n_x : x in X })$. This is obtained from the blowup $X[{bf n}]$ - which contains $n_x$ copies of each point $x$ - by computing the determinant of the distance matrix of $X[{bf n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X({bf n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a delta-matroid to each metric space $X$ (and another delta-matroid to every tree), which also seem to be hitherto unexplored. We moreover show that the homogenization at $-1$ of $p_X({bf n})$ is Lorentzian (or strongly/completely log-concave), if and only if the normalization of $p_X(-{bf n})$ is strongly Rayleigh, if and only if a modification of the distance matrix of $X$ is positive semidefinite. We next specialize to the case of $X = G$ a connected unweighted graph - so $p_G$ is partially symmetric in ${ n_v : v in V(G) }$ - and show two further results: (a) We show that the univariate specialization $u_G(x) := p_G(x,dots,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of $G$ to the well-studied distance spectrum of $G$. (b) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph $G$ and its isometries, respectively.
94 - Alexey Glazyrin 2019
In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtain ed energy bounds are sharp for several infinite families of codes.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا