The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin

We provide a near-complete classification of the Lorentz spaces $\Lambda_{\varphi}$ for which the sequence $\{S_{n}\}_{n\in \mathbb{N}}$ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function $\varphi$, we identify $\Lambda_{\varphi}:= L\log\log L\log\log\log\log L$ as the \emph{largest} Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{1,\infty}$. In particular, we disprove a conjecture stated by Konyagin in his 2006 ICM address. Our proof relies on a newly introduced concept of a "Cantor Multi-tower Embedding," a special geometric configuration of tiles that can arise within the time-frequency tile decomposition of the Carleson operator. This geometric structure plays an important role in the behavior of Fourier series near $L^1$, being responsible for the unboundedness of the weak-$L^1$ norm of a "grand maximal counting function" associated with the mass levels.Comment: 82 pages, no figures. We have added the following items: 1) Section 5 presenting a suggestive example; 2) Section 6 explaining the fundamental role of the so called grand maximal counting function; 3) Section 12 presenting a careful analysis of the Lacey-Thiele discretized Carleson model and of the Walsh-Carleson operator. Accepted for publication in J. Eur. Math. Soc. (JEMS

Aplikasi Teori Teknik Kimia dalam Mencari Model Matematis Pengurangan Berat Tomat Selama Penyimpanan

This article is explained how to make chemical engineering concept more applicable and interesting to students through their experience in research project. The title of their project is “The evaporation rate of stored the water content of fruits and vegetables”. This research aimed to study some factors influencing the evaporation rate, to develop mathematical model describing the evaporation process of stored fruits’s and vegetables’s water content. The experiment was simple and easy. The students kept fruits and vegetables in a storage room and observed the changed in weight of them. Then, they processed data and made mathematical model to explain the behavior of weight loss during storage. During guiding students, faculty concerned with improving the competence of students. Lecturer took students recognize their learning style. By knowing learning style, students would learn more concepts easily. Students learned material through reading journals, textbooks and discussion with lecturer too. The understanding in theory to formulate mathematical model and communication skill could improved through discussion. Students managed to achieve the goal of their research. They could communicate their ideas well and appear confident at the final project seminar

Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections

On one hand, for a general Calabi-Yau complete intersection X, we establish a decomposition, up to rational equivalence, of the small diagonal in X^3, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally equivalent to 0, up to torsion. On the other hand, we find a similar decomposition of the smallest diagonal in a higher power of a hypersurface, which provides us an analogous result on the multiplicative structure of its Chow ring.Comment: 33 pages. Comments are welcom

Receiver Multiuser Diversity Aided Multi-Stage MMSE Multiuser Detection for DS-CDMA and SDMA Systems Employing I-Q Modulation

The so-called receiver multiuser diversity aided multistage minimum mean-square error multiuser detector (RMD/MS-MMSE MUD), which was proposed previously by the author, is investigated in the context of the direct-sequence code-division multiple-access (DS- CDMA) and space-division multiple-access (SDMA) systems that employ in- and quadrature-phase (I-Q) modulation schemes. A detection scheme is studied, which is operated in real domain in the principles of successive interference cancellation (SIC). The concept of noise recognition factor (NRF) is proposed for explaining the efficiency of SIC-type detectors and also for motivating to design other high-efficiency detectors. The achievable bit error rate (BER) performance of the RMD/MS-MMSE MUD is investigated for DS-CDMA and SDMA systems of either full-load or overload, when communicating over Rayleigh fading channels for the SDMA and over either additive white Gaussian noise (AWGN) or Rayleigh fading channels for the DS-CDMA. The studies and performance results show that the RMD/MS-MMSE MUD is a highly promising MUD. It has low implementation complexity and good error performance. Furthermore, it is a high-flexibility detector suitable for various communication systems operated in different communication environments