On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

Optimum Small Optical Beam Displacement Measurement

We derive the quantum noise limit for the optical beam displacement of a TEM00 mode. Using a multimodal analysis, we show that the conventional split detection scheme for measuring beam displacement is non-optimal with 80% efficiency. We propose a new displacement measurement scheme that is optimal for small beam displacement. This scheme utilises a homodyne detection setup that has a TEM10 mode local oscillator. We show that although the quantum noise limit to displacement measurement can be surpassed using squeezed light in appropriate spatial modes for both schemes, the TEM10 homodyning scheme out-performs split detection for all values of squeezing.Comment: 13 pages, 7 figure

Restoration of photosystem II photochemistry and carbon assimilation and related changes in chlorophyll and protein contents during the rehydration of desiccated Xerophyta scabrida leaves

Recovery of photosynthesis in rehydrating desiccated leaves of the poikilochlorophyllous desiccation-tolerant plant Xerophyta scabrida was investigated. Detached leaves were remoistened under 12 h light/dark cycles for 96 h. Water, chlorophyll (Chl), and protein contents, Chl fluorescence, photosynthesis–CO2 concentration response, and the amount and activity of Rubisco were measured at intervals during the rehydration period. Leaf relative water contents reached 87% in 12 h and full turgor in 96 h. Chl synthesis was slower before than after 24 h, and Chla:Chlb ratios changed from 0.13 to 2.6 in 48 h. The maximum quantum efficiency recovered faster during rehydration than the photosystem II operating efficiency and the efficiency factor, which is known to depend mainly on the use of the electron transport chain products. From 24 h to 96 h of rehydration, net carbon fixation was Rubisco limited, rather than electron transport limited. Total Rubisco activity increased during rehydration more than the Rubisco protein content. Desiccated leaves contained, in a close to functional state, more than half the amount of the Rubisco protein present in rehydrated leaves. The results suggest that in X. scabrida leaves Rubisco adopts a special, protective conformation and recovers its activity during rehydration through modifications in redox status

The Turkey Ig-like receptor family: identification, expression and function.

The chicken leukocyte receptor complex located on microchromosome 31 encodes the chicken Ig-like receptors (CHIR), a vastly expanded gene family which can be further divided into three subgroups: activating CHIR-A, bifunctional CHIR-AB and inhibitory CHIR-B. Here, we investigated the presence of CHIR homologues in other bird species. The available genome databases of turkey, duck and zebra finch were screened with different strategies including BLAST searches employing various CHIR sequences, and keyword searches. We could not identify CHIR homologues in the distantly related zebra finch and duck, however, several partial and complete sequences of CHIR homologues were identified on chromosome 3 of the turkey genome. They were designated as turkey Ig-like receptors (TILR). Using cDNA derived from turkey blood and spleen RNA, six full length TILR could be amplified and further divided according to the typical sequence features into one activating TILR-A, one inhibitory TILR-B and four bifunctional TILR-AB. Since the TILR-AB sequences all displayed the critical residues shown to be involved in binding to IgY, we next confirmed the IgY binding using a soluble TILR-AB1-huIg fusion protein. This fusion protein reacted with IgY derived from various gallinaceous birds, but not with IgY from other bird species. Finally, we tested various mab directed against CHIR for their crossreactivity with either turkey or duck leukocytes. Whereas no staining was detectable with duck cells, the CHIR-AB1 specific mab 8D12 and the CHIR-A2 specific mab 13E2 both reacted with a leukocyte subpopulation that was further identified as thrombocytes by double immunofluorescence employing B-cell, T-cell and thrombocyte specific reagents. In summary, although the turkey harbors similar LRC genes as the chicken, their distribution seems to be distinct with predominance on thrombocytes rather than lymphocytes

Need Polynomial Systems Be Doubly-Exponential?

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of zeros of a zero-dimensional system is singly-exponential in the number of variables. How should this contradiction be reconciled? We first note that [Mayr and Ritscher, 2013] shows that the doubly exponential nature of Gr\"{o}bner bases is with respect to the dimension of the ideal, not the number of variables. This inspires us to consider what can be done for Cylindrical Algebraic Decomposition which produces a doubly-exponential number of polynomials of doubly-exponential degree. We review work from ISSAC 2015 which showed the number of polynomials could be restricted to doubly-exponential in the (complex) dimension using McCallum's theory of reduced projection in the presence of equational constraints. We then discuss preliminary results showing the same for the degree of those polynomials. The results are under primitivity assumptions whose importance we illustrate.Comment: Extended Abstract for ICMS 2016 Presentation. arXiv admin note: text overlap with arXiv:1605.0249

Machine Learning for Mathematical Software

While there has been some discussion on how Symbolic Computation could be used for AI there is little literature on applications in the other direction. However, recent results for quantifier elimination suggest that, given enough example problems, there is scope for machine learning tools like Support Vector Machines to improve the performance of Computer Algebra Systems. We survey the authors own work and similar applications for other mathematical software. It may seem that the inherently probabilistic nature of machine learning tools would invalidate the exact results prized by mathematical software. However, algorithms and implementations often come with a range of choices which have no effect on the mathematical correctness of the end result but a great effect on the resources required to find it, and thus here, machine learning can have a significant impact.Comment: To appear in Proc. ICMS 201

Aluminium-induced ion transport in Arabidopsis: the relationship between Al tolerance and root ion flux

Aluminium (Al) rhizotoxicity coincides with low pH; however, it is unclear whether plant tolerance to these two factors is controlled by the same mechanism. To address this question, the Al-resistant alr104 mutant, two Al-sensitive mutants (als3 and als5), and wild-type Arabidopsis thaliana were compared in long-term exposure (solution culture) and in short-term exposure experiments (H+ and K+ fluxes, rhizosphere pH, and plasma membrane potential, Em). Based on biomass accumulation, als5 and alr104 showed tolerance to low pH, whereas alr104 was tolerant to the combined low-pH/Al treatment. The sensitivity of the als5 and als3 mutants to the Al stress was similar. The Al-induced decrease in H+ influx at the distal elongation zone (DEZ) and Al-induced H+ efflux at the mature zone (MZ) were higher in the Al-sensitive mutants (als3 and als5) than in the wild type and the alr104 mutant. Under combined low-pH/Al treatment, alr104 and the wild type had depolarized plasma membranes for the entire 30 min measurement period, whereas in the Al-sensitive mutants (als3 and als5), initial depolarization to around –60 mV became hyperpolarization at –110 mV after 20 min. At the DEZ, the Em changes corresponded to the changes in K+ flux: K+ efflux was higher in alr104 and the wild type than in the als3 and als5 mutants. In conclusion, Al tolerance in the alr104 mutant correlated with Em depolarization, higher K+ efflux, and higher H+ influx, which led to a more alkaline rhizosphere under the combined low-pH/Al stress. Low-pH tolerance (als5) was linked to higher H+ uptake under low-pH stress, which was abolished by Al exposure

Multivariate modeling of chromium-induced oxidative stress and biochemical changes in plants of Pistia stratiotes L.

Biochemical changes in the plants of Pistia stratiotes L., a free floating macrophyte exposed to different concentrations of hexavalent chromium (0, 10, 40, 60, 80 and 160 μM) for 48, 96 and 144 h were studied. Chromium-induced oxidative stress in macrophyte was investigated using the multivariate modeling approaches. Cluster analysis rendered two fairly distinct clusters (roots and shoots) of similar characteristics in terms of their biochemical responses. Discriminant analysis identified ascorbate peroxidase (APX) as discriminating variable between the root and shoot tissues. Principal components analysis results suggested that malondialdehyde (MDA), superoxide dismutase (SOD), APX, non-protein thiols (NP-SH), cysteine, ascorbic acid, and Cr-accumulation are dominant in root tissues, whereas, protein and guaiacol peroxidase (GPX) in shoots of the plant. Discriminant partial least squares analysis results further confirmed that MDA, SOD, NP-SH, cysteine, GPX, APX, ascorbic acid and Cr-accumulation dominated in the root tissues, while protein in the shoot. Three-way analysis helped in visualizing simultaneous influence of metal concentration and exposure duration on biochemical variables in plant tissues. The multivariate approaches, thus, allowed for the interpretation of the induced biochemical changes in the plant tissues exposed to chromium, which otherwise using the conventional approaches is difficult