Orthogonal Dirichlet Polynomials
Abstract. Let fj g1 j=1 be a sequence of distinct positive numbers. Let w be a nonnegative function, integrable on the real line. One can form orthogonal Dirichlet polynomials fng from linear combinations of n  more » « less
Award ID(s):
NSF-PAR ID:
10408803
Author(s) / Creator(s):
Editor(s):
;
Date Published:
Journal Name:
Approximation and Computation in Science and Engineering
Page Range / eLocation ID:
573-588
Format(s):
Medium: X
National Science Foundation
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1. Abstract

Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$${a}_{j,i}\in {F}_{p}$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$${a}_{j,1}+\cdots +{a}_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m×m$minor of the$$m\times k$$$m×k$matrix$$(a_{j,i})_{j,i}$$${\left({a}_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq {F}_{p}^{n}$of size$$|A|> C\cdot \Gamma ^n$$$|A|>C·{\Gamma }^{n}$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$${x}_{1},\cdots ,{x}_{k}\in A$are all distinct. Here,Cand$$\Gamma$$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma . The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$in the solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. more » « less 2. Let F be a non-archimedean local field of residual characteristic p \neq 2 . Let G be a (connected) reductive group over F that splits over a tamely ramified field extension of F . We revisit Yu's construction of smooth complex representations of G(F) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [ Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference. more » « less 3. Let  Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-}. Here,  \lvert \varepsilon _I\rvert =1 , and  h_J is the Haar function defined on dyadic interval  J. We show that, for instance, \displaystyle \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} . Above, we use the one-sided  A_2 characteristic for the weight  w. This is an instance of a one-sided  A_2 conjecture. Our proof of this fact is difficult, as the very quick known proofs of the  A_2 theorem do not seem to apply in the one-sided setting. more » « less 4. Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by {(e_{(k,j)}^*)_j} , for k\in\N , let Z=\ell^\infty(X_k:k\kin\N) be their l ∞ -sum, and let T:Z\to Z be a bounded linear operator with a large diagonal, i.e. ,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*} Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive.
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5. Abstract Background

In a recent study, we reported beam quality correction factors,fQ, in carbon ion beams using Monte Carlo (MC) methods for a cylindrical and a parallel‐plate ionization chamber (IC). A non‐negligible perturbation effect was observed; however, the magnitude of the perturbation correction due to the specific IC subcomponents was not included. Furthermore, the stopping power data presented in the International Commission on Radiation Units and Measurements (ICRU) report 73 were used, whereas the latest stopping power data have been reported in the ICRU report 90.

Purpose

The aim of this study was to extend our previous work by computingfQcorrection factors using the ICRU 90 stopping power data and by reporting IC‐specific perturbation correction factors. Possible energy or linear energy transfer (LET) dependence of thefQcorrection factor was investigated by simulating both pristine beams and spread‐out Bragg peaks (SOBPs).

Methods

The TOol for PArticle Simulation (TOPAS)/GEANT4 MC code was used in this study. A 30 × 30 × 50 cm3water phantom was simulated with a uniform 10 × 10 cm2parallel beam incident on the surface. A Farmer‐type cylindrical IC (Exradin A12) and two parallel‐plate ICs (Exradin P11 and A11) were simulated in TOPAS using the manufacturer‐provided geometrical drawings. ThefQcorrection factor was calculated in pristine carbon ion beams in the 150–450 MeV/u energy range at 2 cm depth and in the middle of the flat region of four SOBPs. ThekQcorrection factor was calculated by simulating thefQocorrection factor in a60Co beam at 5 cm depth. The perturbation correction factors due to the presence of the individual IC subcomponents, such as the displacement effect in the air cavity, collecting electrode, chamber wall, and chamber stem, were calculated at 2 cm depth for monoenergetic beams only. Additionally, the mean dose‐averaged and track‐averaged LET was calculated at the depths at which thefQwas calculated.

Results

The ICRU 90fQcorrection factors were reported. Thepdiscorrection factor was found to be significant for the cylindrical IC with magnitudes up to 1.70%. The individual perturbation corrections for the parallel‐plate ICs were <1.0% except for the A11pcelcorrection at the lowest energy. ThefQcorrection for the P11 IC exhibited an energy dependence of >1.00% and displayed differences up to 0.87% between pristine beams and SOBPs. Conversely, thefQfor A11 and A12 displayed a minimal energy dependence of <0.50%. The energy dependence was found to manifest in the LET dependence for the P11 IC. A statistically significant LET dependence was found only for the P11 IC in pristine beams only with a magnitude of <1.10%.

Conclusions

The perturbation andkQcorrection factor should be calculated for the specific IC to be used in carbon ion beam reference dosimetry as a function of beam quality.

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