On the Waring--Goldbach problem for eighth and higher powers

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We obtain new results for all exponents $k\ge 8$, and in particular establish that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement on the classical result of Hua from the 1940s

Blind Interference Alignment for Private Information Retrieval

Blind interference alignment (BIA) refers to interference alignment schemes that are designed only based on channel coherence pattern knowledge at the transmitters (the "blind" transmitters do not know the exact channel values). Private information retrieval (PIR) refers to the problem where a user retrieves one out of K messages from N non-communicating databases (each holds all K messages) without revealing anything about the identity of the desired message index to any individual database. In this paper, we identify an intriguing connection between PIR and BIA. Inspired by this connection, we characterize the information theoretic optimal download cost of PIR, when we have K = 2 messages and the number of databases, N, is arbitrary

Berge Sorting

In 1966, Claude Berge proposed the following sorting problem. Given a string of $n$ alternating white and black pegs on a one-dimensional board consisting of an unlimited number of empty holes, rearrange the pegs into a string consisting of $\lceil\frac{n}{2}\rceil$ white pegs followed immediately by $\lfloor\frac{n}{2}\rfloor$ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ such {\em Berge 2-moves} for $n\geq 5$. Extending Berge's original problem, we consider the same sorting problem using {\em Berge $k$-moves}, i.e., moves which take $k$ adjacent pegs to $k$ vacant adjacent holes. We prove that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge 3-moves for $n\not\equiv 0\pmod{4}$ and in $\lceil\frac{n}{2}\rceil+1$ Berge 3-moves for $n\equiv 0\pmod{4}$, for $n\geq 5$. In general, we conjecture that, for any $k$ and large enough $n$, the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge $k$-moves. This estimate is tight as $\lceil\frac{n}{2}\rceil$ is a lower bound for the minimum number of required Berge $k$-moves for $k\geq 2$ and $n\geq 5$.Comment: 10 pages, 2 figure

An assessment of the Hua Oranga outcome instrument and comparison to other outcome measures in an intervention study with Maori and Pacific people following stroke

The Hua Oranga instrument, developed for Maori people with mental illness, showed good responsiveness and adequate psychometric properties in Maori and Pacific people after stroke. Its simplicity, relative brevity, minimal cost and adequate psychometric properties should favour its use in future studies with both Maori and Pacific people. Suggestions are made for refinements to the measure. These should be tested in a new population before Hua Oranga is recommended for general use in a clinical setting. Abstract Aim Health outcomes research for Maori has been hampered by the lack of adequately validated instruments that directly address outcomes of importance to Maori, framed by a Maori perspective of health. Hua Oranga is an outcome instrument developed for Maori with mental illness that uses a holistic view of Maori health to determine improvements in physical, mental, spiritual and family domains of health. Basic psychometric work for Hua Oranga is lacking. We sought to explore the psychometric properties of the instrument and compare its responsiveness alongside other, more established tools in an intervention study involving Maori and Pacific people following acute stroke. Method Randomised 2x2 controlled trial of Maori and Pacific people following acute stroke with two interventions aimed at facilitating self-directed rehabilitation, and with follow-up at 12 months after randomisation. Primary outcome measures were the Physical Component Summary (PCS) and Mental Component Summary (MCS) of the Short Form 36 (SF36) at 12 months. Hua Oranga was used as a secondary outcome measure for participants at 12 months and for carers and whanau (extended family). Psychometric properties of Hua Oranga were explored using plots and correlation coefficients, principal factors analysis and scree plots. Results 172 participants were randomised, of whom 139 (80.8%) completed follow-up. Of these, 135 (97%) completed the Hua Oranga and 117 (84.2%) completed the PCS and MCS of the SF36. Eighty-nine carers completed the Hua Oranga. Total Hua Oranga scores and PCS improved significantly for one intervention group but not the other. Total Hua Oranga scores for carers improved significantly for both interventions. Total Hua Oranga score correlated moderately with the PCS (correlation coefficient 0.55, p<0.001). Factor analysis suggested that Hua Oranga measures two and not four factors; one 'physical-mental' and one 'spiritual-family'. Conclusion The Hua Oranga instrument, developed for Maori people with mental illness, showed good responsiveness and adequate psychometric properties in Maori and Pacific people after stroke. Its simplicity, relative brevity, minimal cost and adequate psychometric properties should favour its use in future studies with both Maori and Pacific people. Suggestions are made for refinements to the measure. These should be tested in a new population before Hua Oranga is recommended for general use in a clinical setting

Towards explaining the speed of kk-means

The $k$-means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the $k$-means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences

Microdata protection through approximate microaggregation

Microdata protection is a hot topic in the field of Statistical Disclosure Control, which has gained special interest after the disclosure of 658000 queries by the America Online (AOL) search engine in August 2006. Many algorithms, methods and properties have been proposed to deal with microdata disclosure. One of the emerging concepts in microdata protection is k-anonymity, introduced by Samarati and Sweeney. k-anonymity provides a simple and efficient approach to protect private individual information and is gaining increasing popularity. k-anonymity requires that every record in the microdata table released be indistinguishably related to no fewer than k respondents. In this paper, we apply the concept of entropy to propose a distance metric to evaluate the amount of mutual information among records in microdata, and propose a method of constructing dependency tree to find the key attributes, which we then use to process approximate microaggregation. Further, we adopt this new microaggregation technique to study $k$-anonymity problem, and an efficient algorithm is developed. Experimental results show that the proposed microaggregation technique is efficient and effective in the terms of running time and information loss

Britain welcomes the world: Dressing up London

Copyright @ 2013 Routledgehttps://www.taylorfrancis.com/chapters/dressing-london-ozlem-edizel-graeme-evans-hua-dong/e/10.4324/9780203126486-9?context=ubx&refId=546b057b-44c2-4d37-91ac-982df6ff2fa

An extremal problem on potentially Km−PkK_{m}-P_{k}-graphic sequences

A sequence $S$ is potentially $K_{m}-P_{k}$ graphical if it has a realization containing a $K_{m}-P_{k}$ as a subgraph. Let $\sigma(K_{m}-P_{k}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-P_{k}, n)$ is potentially $K_{m}-P_{k}$ graphical. In this paper, we prove that $\sigma (K_{m}-P_{k}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq k+1\geq 4.$ We conjecture that equality holds for $n \geq m \geq k+1\geq 4.$ We prove that this conjecture is true for $m=k+1=5$ and $m=k+2=5$.Comment: 5 page

顧愷之の魏晋勝流画賛について

The Biography of Ku K\u27ai-chih, contained in the fifth volume of the Li Tai Ming Hua Chi (Notes on Famous Artists of Respective Periods), states that he wrote the Wei Chin Ming Ch\u27ên Hua Tsan (Eulogies on Portraits of Eminent Persons of the Wei and Chin Dynasties) in which he discussed on the subject in great details, and that he also wrote the Lun Hua (Discussions on Painting) to explain how to copy old masterpieces; this Biography ends with a paragraph beginning with an introductory sentence: “K’ai-chih, in his Wei Chin Shêng Liu Hua Tsan, said as follows.” (Ming Ch\u27ên and Shêng Liu are the same in meaning.) The Lun Hua and the Wei Chin Shêng Liu Hua Tsan mentioned here seem confused in appearance, for the former lists and criticizes ancient paintings while the latter discribes about the attitude of mind, the materials and the techniques required in copying old works. An attempt was therefore made to interchangesettle the apparent contradiction between their titles and their contents (KIMBARA, Shōgo: “Studies on Art Criticism in Ancient China). However, the Wei Ching Shêng Liu Hua Tsan by Ku K\u27ai-chih existed separately, and was different from what was quoted in the Li Tai Ming Hua Chi. Portions of this Hua Tsan are found quoted in annotations on the Shih Shuo Hsin Yü (a collection of Chinese annecdotes) and in annotations by Li Shan on the Wên Hsuan (a collection of old Chinese writings). Judged from these scattered segments, the original form of the Hua Tsan by Ku K\u27ai-chih appears to have been modelled after the Hua Tsan written by Ts\u27ao Chih in the Wei Dynasty: that is to say, it probably was a versified writing consisting of four-character lines preceded by an introductory paragraph. This is the real Wei Chin Shêng Liu Hua Tsan, or the Wei Chin Ming Chên Hua Tsan “in which he discussed in great details” according to the Li Tai Ming Hua Chi. The discussions in the Wei Chin Shêng Liu Hua Tsan are not on the characteristics and value of the paintings as works of art, but are on the personalities of the figure subjects depicted therein. They are notes, not on the paintings themselves but on their subject matters. This was the case even with Ku K\u27aichih, who was an artist and art critic of a very creative mind. This fact may be understood to represent an aspect of the characteristic Chinese term of view on art. The portion entitled Wei Chin Shêng Liu Hua Tsan in the Li Tai Ming Hua Chih is nothing but a part of the Lun Hua. It explains the mental and material preparations necessary in copying old paintings, while the portion entitled Lun Hua comments on the styles of old masters in order to tell what are important in copying their works. The two are parts of the same writing, Lun Hua, giving instructions for copyists. The present writer is inclined to think that a careless editor in a later period gave the title Wei Chin Shêng Liu Hua Tsan to the second half of what had been recorded as Lun Hua in the Li Tai Ming Hua Chi, simply because a mention of the Hua Tsan is found in a previous paragraph of the Biography