Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $\epsilon$, getting the optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a $(\log(n/\epsilon^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $\epsilon$. This improves upon the $\log(n/\epsilon^3)+O(1)$ upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive $\log\log(1/\epsilon)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $\log(n/\epsilon)+4$, that uses only pure states and computes the $n$-bit Equality function to error $\epsilon$. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any $\epsilon$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $\log(n/\epsilon)-\log\log(1/\epsilon)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $\log(\sqrt{n}/\epsilon)+3$, that uses mixed states and computes the $n$-bit Equality function to error $\epsilon$. This is also tight up to an additive $\log\log(1/\epsilon)+O(1)$, which follows from Alon's result. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix. This also implies improved upper bounds on these measures for the distributed SINK function, which was recently used to refute the randomized and quantum versions of the log-rank conjecture.Comment: 16 page

Rainfall Trend Detection in Northern Nigeria over the Period of 1970-2012

This study examined the trends in variability and spatial distribution of annual rainfall over northern Nigeria during the period 1970-2012 with a view to understand the pattern of rainfall trend (significance and magnitude), by applying various statistical tools on the data obtained from 11 weather stations. The non-parametric Mann–Kendall test was used to determine the statistical signi?cance of trends while the magnitude of trends was derived from the Sen slope estimator of the linear trends using Kendall robust line ?tting. Map of rainfall trends was generated by applying a geo-statistical interpolation technique to visualize the detected tendencies. The findings revealed that a significant positive increase of 2.16mm in rainfall was recorded in the entire northern Nigeria within the period of 1970 to 2012. It further indicated that majority of the stations revealed an upward trend, with Bauchi, Borno, Kebbi and Sokoto stations showing significant positive trends of 8.13mm, 4.30mm, 4.76mm and 4.42mm respectively. It is concluded that there is high variability in rainfall in the northern Nigeria which signifies a clear evidence of climate change in the region. Keywords: Rainfall trend, Man-Kendall test, Northern Nigeria, Climate chang

EXAFS study of intermetallics of the type RGe<SUB>2</SUB> (R=La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er and Y) Part I: determination of Ge-Ge distances

A study of the EXAFS associated with the K x-ray absorption discontinuity of germanium in pure germanium and in the rare-earth germanides RGe2 (where R=La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er and Y) has been carried out. The Ge-Ge distances have been obtained in these compounds. Considering the phase to the RGe2 system, the bond lengths in these compounds have been determined. The values obtained by us for the RGe2 compounds (R=La, Ce, Pr, Nd, Sm, Gd, Dy and Y) agree with those obtained earlier by crystallographic methods. The bond lengths for the compounds TbGe2, HoGe2 and ErGe2 are also being reported

Tight Chang’s-lemma-type bounds for Boolean functions

Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f):= Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold: the Fourier sparsity of f, denoted k(f), the Fourier max-supp-entropy of f, denoted k′(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient, the Fourier max-rank-entropy of f, denoted k′′(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω (r(f)2/(k(f) log2 k(f))), subsumes and refines the previously best known upper bound r(f) = O(pk(f) log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(pk(f)δ(f) log k(f)). Another lower bound, δ(f) = Ω (r(f)/(k′′(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of F2-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f). Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications

Environmental Response and Genomic Regions Correlated with Rice Root Growth and Yield under Drought in the OryzaSNP Panel across Multiple Study Systems

Funding: This research was funded by the Generation Challenge Program (GCP) project G3008.06, “Targeting Drought-Avoidance Root Traits to Enhance Rice Productivity under Water-Limited Environments". The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Peer reviewedPublisher PD

Can Phlorotannins Purified Extracts Constitute a Novel Pharmacological Alternative for Microbial Infections with Associated Inflammatory Conditions?

Bacterial and fungal infections and the emerging multidrug resistance are driving interest in fighting these microorganisms with natural products, which have generally been considered complementary to pharmacological therapies. Phlorotannins are polyphenols restricted to brown seaweeds, recognized for their biological capacity. This study represents the first research on the antibacterial, antifungal, anti-inflammatory and antioxidant activity of phlorotannins purified extracts, which were obtained from ten dominant brown seaweeds of the occidental Portuguese coast

Structural analysis on mutation residues and interfacial water molecules for human TIM disease understanding

10.1186/1471-2105-14-S16-S11BMC Bioinformatics14SUPPL16-BBMI

Dynamic Changes in Protein Functional Linkage Networks Revealed by Integration with Gene Expression Data

Response of cells to changing environmental conditions is governed by the dynamics of intricate biomolecular interactions. It may be reasonable to assume, proteins being the dominant macromolecules that carry out routine cellular functions, that understanding the dynamics of protein∶protein interactions might yield useful insights into the cellular responses. The large-scale protein interaction data sets are, however, unable to capture the changes in the profile of protein∶protein interactions. In order to understand how these interactions change dynamically, we have constructed conditional protein linkages for Escherichia coli by integrating functional linkages and gene expression information. As a case study, we have chosen to analyze UV exposure in wild-type and SOS deficient E. coli at 20 minutes post irradiation. The conditional networks exhibit similar topological properties. Although the global topological properties of the networks are similar, many subtle local changes are observed, which are suggestive of the cellular response to the perturbations. Some such changes correspond to differences in the path lengths among the nodes of carbohydrate metabolism correlating with its loss in efficiency in the UV treated cells. Similarly, expression of hubs under unique conditions reflects the importance of these genes. Various centrality measures applied to the networks indicate increased importance for replication, repair, and other stress proteins for the cells under UV treatment, as anticipated. We thus propose a novel approach for studying an organism at the systems level by integrating genome-wide functional linkages and the gene expression data

Understanding Communication Signals during Mycobacterial Latency through Predicted Genome-Wide Protein Interactions and Boolean Modeling

About 90% of the people infected with Mycobacterium tuberculosis carry latent bacteria that are believed to get activated upon immune suppression. One of the fundamental challenges in the control of tuberculosis is therefore to understand molecular mechanisms involved in the onset of latency and/or reactivation. We have attempted to address this problem at the systems level by a combination of predicted functional protein∶protein interactions, integration of functional interactions with large scale gene expression studies, predicted transcription regulatory network and finally simulations with a Boolean model of the network. Initially a prediction for genome-wide protein functional linkages was obtained based on genome-context methods using a Support Vector Machine. This set of protein functional linkages along with gene expression data of the available models of latency was employed to identify proteins involved in mediating switch signals during dormancy. We show that genes that are up and down regulated during dormancy are not only coordinately regulated under dormancy-like conditions but also under a variety of other experimental conditions. Their synchronized regulation indicates that they form a tightly regulated gene cluster and might form a latency-regulon. Conservation of these genes across bacterial species suggests a unique evolutionary history that might be associated with M. tuberculosis dormancy. Finally, simulations with a Boolean model based on the regulatory network with logical relationships derived from gene expression data reveals a bistable switch suggesting alternating latent and actively growing states. Our analysis based on the interaction network therefore reveals a potential model of M. tuberculosis latency