Analytic properties of spherical cusp forms on GL(n)

Let $\phi$ be an $L^2$-normalized spherical vector in an everywhere unramified cuspidal automorphic representation of $\mathrm{PGL}_n$ over $\mathbb{Q}$ with Laplace eigenvalue $\lambda_{\phi}$. We establish explicit estimates for various quantities related to $\phi$ that are uniform in $\lambda_{\phi}$. This includes uniforms bounds for spherical Whittaker functions on $\mathrm{GL}_n(\mathbb{R})$, uniform bounds for the global sup-norm of $\phi$, and uniform bounds for the "essential support" of $\phi$, i.e. the region outside which it decays exponentially. The proofs combine analytic and arithmetic tools.Comment: 18 pages, LaTeX2e, submitted; v2: revised version fixing mainly (45) and its consequences; v3: revised version incorporating suggestions by the referee, e.g. Theorem 2 is now more general than before; v4: final version to appear in Journal d'Analyse Math\'ematique; v5: small corrections added in proo

Generic Birkhoff Spectra

Suppose that $\Omega = \{0, 1\}^ {\mathbb {N}}$ and ${\sigma}$ is the one-sided shift. The Birkhoff spectrum $\displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},$ where $\dim_{H}$ is the Hausdorff dimension. It is well-known that the support of $S_{f}( {\alpha})$ is a bounded and closed interval $L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]$ and $S_{f}( {\alpha})$ on $L_{f}$ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $f\in C( {\Omega})$ in the sense of Baire category. For a dense set in $C( {\Omega})$ the spectrum is not continuous on ${\mathbb {R}}$, though for the generic $f\in C( {\Omega})$ the spectrum is continuous on ${\mathbb {R}}$, but has infinite one-sided derivatives at the endpoints of $L_{f}$. We give an example of a function which has continuous $S_{f}$ on ${\mathbb {R}}$, but with finite one-sided derivatives at the endpoints of $L_{f}$. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $f$ and $g$ are close in $C( {\Omega})$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from neighborhoods of the endpoints.Comment: Revised version after the referee's repor

Analisis Dampak Aktivitas Pertambangan Nikel PT. X Terhadap Pendapatan Petani Padi Sawah Di Kecamatan Tinanggea Kabupaten Konawe Selatan

Upaya peningkatan pertumbuhan ekonomi nasional maupun daerah akan memanfaatkan potensi sumberdaya alam (SDA). Meskipun demikian, dalam pengelolannya terkadang tidak mengkuti apa yang menjadi langkah-langkah pemanfaatan lingkungan lestari. Tidak jarang dibeberapa negara dapat kita saksikan terjadinya kerusakan lingkungan yang ditimbulkan akibat pengelolaan yang tidak ramah lingkungan. Aktifitas penambangan nikel PT. X di Kecamatan Tinanggea telah menimbulkan degradasi lingkungan. Degradasi lingkungan berdampak pula bagi masyarakat yang berada di sekitar lokasi penambangan nikel PT. X, yakni petani padi sawah. Dampak negatif akibat pebambangan nikel berupa menurunnya kesburan tanah, menurunnya kualitas air irigasi karena mengandung material lumpur yang berasal dari lokasi penambangan nikel, lumpur yang berasal dari lokasi penambangan nikel mengendap dan menimbulkan pendangkalan Bendungan Lapoa, serta menurnnya debit air irigasi akibat konversi lahan dihulu sungai. Degradasi lingkungan akibat penambangan nikel PT. X di Kecamatan Tinanggea menimbulkan dampak negatif berupa menurunnya tingkat pendapatan petani padi sawah. Tingkat pendapatan petani padi sawah menurun sebesar Rp 3.102.476/Ha per musim tanam jika dibandingkan tingkat pendapatan petani padi sawah di Kecamatan Lalembuu.Kata Kunci: Pertambangan nikel, pendapatan, petani, padi sawah Efforts to increase national and regional economic growth will utilize the potential of natural resources (SDA). However, in its management, sometimes it does not follow the steps for sustainable use of the environment. Not infrequently, in some countries we can see the occurrence of environmental damage caused by management that is not environmentally friendly. Nickel mining activities of PT. X in Tinanggea District have caused environmental degradation. Environmental degradation also has an impact on the community around the nickel mining site of PT. X, namely rice farmers. The negative impacts due to nickel mining include decreased soil fertility; decreased quality of irrigation water because it contains mud from nickel mining sites; mud from nickel mining sites settles and causes silting of the Lapoa Dam; and decreased irrigation water discharge due to land conversion upstream of the river. Environmental degradation due to nickel mining PT. X in Tinanggea District has a negative impact in the form of a decrease in the income level of lowland rice farmers. The income level of lowland rice farmers decreased by Rp. 3,102,476/ha per planting season when compared to the income level of lowland rice farmers in Lalembuu District.Keywords: Nickel mining, income, farmers, rice padd

Generic power series on subsets of the unit disk

In this paper, we examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense of Baire category. Notably, we prove that for any open set $U$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of $\mathbb{C}$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.Comment: Wrong grant numbers fixe

Big and little Lipschitz one sets

Given a continuous function $f: {{\mathbb R}}\to {{\mathbb R}}$ we denote the so-called "big Lip" and "little lip" functions by ${{\mathrm {Lip}}} f$ and ${{\mathrm {lip}}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$ or ${{\mathrm {Lip}}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_\delta$ and UDT then there exists a continuous function $f$ satisfying ${{\mathrm {Lip}}} f =\mathbf{1}_E$, that is, $E$ is a ${{\mathrm {Lip}}} 1$ set. In the other direction we show that every ${{\mathrm {Lip}}} 1$ set is $G_\delta$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{{\delta}}$ sets which are not ${{\mathrm {Lip}}} 1$. We say that a set $E\subset \mathbb{R}$ is ${{\mathrm {lip}}} 1$ if there is a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${{\mathrm {lip}}} 1$ set is a strongly one-sided dense $F_\sigma$ set.Comment: This is the final preprint version accepted to appear in European Journal of Mathematic

Effect Of Drilling Quality On Biomass Yield

The paper is focused on the study and evaluation of quality of the seeding of seeds and its effect on the biomass yield. The aim was to evaluate the space arrangement of the seeds by using of polygon method on one field with the repetition for different forward speeds of the drill machine - 10, 12, 14, 16 a 18 km.h-1. Field experiments have been conducted with the drill machine with continuous seeding TUME NOVA COMBI 3000 having a double-disc seeding unit. Within the experiments there was used winter wheat, variety Magisters, with seeding rate 220 kg per ha. For the evaluation there were used digital photographs, which were taken during repeated measurements of the each value of the forward speed after sprouting of crop. These images have been used in order to determine the shape and size of the surface area belonging to the plant. Own software TfPolyM was used for the image analysis. The shape of the polygons belonging to the individual plants was expressed by values of the shape factor Tf. This factor characterises the suitability the shape of polygon surface related to the individual plant. By comparing of the values of the shape factors for different forward speeds of the drill machine we can determine the optimal value of the forward speed from the point of seed placement uniformity in horizontal level. During harvest of the crop there was analysed the variability of the biomass yield in relation to values of the forward speed used during seeding. The following parameters of the crop yield were measured: total yield, grain yield, straw yield and chaff yield. The best values of the shape factor Tf  (0.856) were obtained  for the forward speed 14 km.h-1. When the forward speed was higher or smaller the values of the shape factor Tf were decreased. Average size of polygons was also studied. Increasing of the forward speed up to 14 km.h-1 has caused the increase of the average size of polygons. Further increase of the forward speed up to 18 km.h-1 has resulted in slight decrease of the average size of polygons. Such fact can be caused by imperfect filling of the metering unit on the drill machine.   As the maximal yield 392.74 g.m-2 was found for the case of forward speed 14 km.h-1. The highest values of the grain yield 222.32 g.m-2 was measured for the drill machine forward speed 14 km.h-1. Based on the results obtained it can be stated that forward speed of the drill machine effects the placement of the seeds in horizontal level and this fact subsequently effects the yield of biomass