Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved ...lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.
New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited ...to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration of our methods. Our new equations and results about them are put into a larger context by introducing vector bundle techniques for finding equations of secant varieties in general. We include a few homogeneous examples of this method.
Toll-like receptor (TLR) signaling is a key innate immunity response to pathogens. Recruitment of signaling adapters such as MAL (TIRAP) and MyD88 to the TLRs requires Toll/interleukin-1 receptor ...(TIR)-domain interactions, which remain structurally elusive. Here we show that MAL TIR domains spontaneously and reversibly form filaments in vitro. They also form cofilaments with TLR4 TIR domains and induce formation of MyD88 assemblies. A 7-Å-resolution cryo-EM structure reveals a stable MAL protofilament consisting of two parallel strands of TIR-domain subunits in a BB-loop-mediated head-to-tail arrangement. Interface residues that are important for the interaction are conserved among different TIR domains. Although large filaments of TLR4, MAL or MyD88 are unlikely to form during cellular signaling, structure-guided mutagenesis, combined with in vivo interaction assays, demonstrated that the MAL interactions defined within the filament represent a template for a conserved mode of TIR-domain interaction involved in both TLR and interleukin-1 receptor signaling.
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent
ω
of matrix multiplication. The Kronecker square of the small
q
=
2
Coppersmith–Winograd ...tensor equals the
3
×
3
permanent, and could potentially be used to show
ω
=
2
. We prove the negative result for complexity theory that its border rank is 16, resolving a longstanding problem. Regarding its
q
=
4
skew cousin in
C
5
⊗
C
5
⊗
C
5
, which could potentially be used to prove
≤
2.11
, we show the border rank of its Kronecker square is at most 42, a remarkable sub-multiplicativity result, as the square of its border rank is 64. We also determine moduli spaces VSP for the small Coppersmith–Winograd tensors.
In August 2022, a novel henipavirus (HNV) named Langya virus (LayV) was isolated from patients with severe pneumonic disease in China. This virus is closely related to Mòjiāng virus (MojV), and both ...are divergent from the bat-borne HNV members, Nipah (NiV) and Hendra (HeV) viruses. The spillover of LayV is the first instance of a HNV zoonosis to humans outside of NiV and HeV, highlighting the continuing threat this genus poses to human health. In this work, we determine the prefusion structures of MojV and LayV F proteins via cryogenic electron microscopy to 2.66 and 3.37 Å, respectively. We show that despite sequence divergence from NiV, the F proteins adopt an overall similar structure but are antigenically distinct as they do not react to known antibodies or sera. Glycoproteomic analysis revealed that while LayV F is less glycosylated than NiV F, it contains a glycan that shields a site of vulnerability previously identified for NiV. These findings explain the distinct antigenic profile of LayV and MojV F, despite the extent to which they are otherwise structurally similar to NiV. Our results carry implications for broad-spectrum HNV vaccines and therapeutics, and indicate an antigenic, yet not structural, divergence from prototypical HNVs.
Concise tensors of minimal border rank Jelisiejew, Joachim; Landsberg, J. M.; Pal, Arpan
Mathematische annalen,
01/2024, Letnik:
388, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We determine defining equations for the set of concise tensors of minimal border rank in
C
m
⊗
C
m
⊗
C
m
when
m
=
5
and the set of concise minimal border rank
1
∗
-generic tensors when
m
=
5
,
6
. We ...solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case
m
=
5
. Our proofs utilize two recent developments: the 111-equations defined by Buczyńska–Buczyński and results of Jelisiejew–Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in
C
5
⊗
C
5
⊗
C
5
.
The water balance and growth of
Eucalyptus grandis hybrid plantations in Brazil are presented based on 6 years of intensive catchment hydrology, physiological and forest growth surveying, and ...modelling. The results show a balance between water supply by precipitation and output through evapotranspiration (considered as canopy interception, soil evaporation and trees transpiration) and runoff. The annual average precipitation was 1147
mm and average evapotranspiration was 1092
mm. The runoff was only 3% of the precipitation, because of high soil infiltration and the flat topography where the trees are planted. Evapotranspiration rates varied from 781
mm to 1334
mm during the years of the study and are strongly influenced by variations in annual precipitation and leaf area index. When the precipitation was close to the regional mean annual precipitation of 1350
mm there was enough water to supply the demands of the trees and produce some runoff. Biomass production was high and the peak annual growth rate was 95
m
3
ha
−1
year
−1. The UAPE model Soares, J.V., Almeida, A.C., 2001. Modeling the water balance and soil water fluxes in a fast-growing
Eucalyptus plantation in Brazil. J. Hydrol. 253, 130–147 was used to estimate the water balance and the widely used 3-PG model Landsberg, J.J., Waring, R.H., 1997. A generalised model of forest productivity using simplified concepts of radiation-use efficiency, carbon balance and partitioning. For. Ecol. Manage. 95, 209–228 was used to estimate forest growth and water-use efficiency.
A set of parameter values for the forest growth model 3-PG was determined. These permit the modelling of the time-course of growth and development of even-aged, intensively-managed, fertilised stands ...of
Eucalyptus globulus. The parameters were determined by fitting output from 3-PG to observed stem biomass data (i.e. total above-ground woody biomass) and canopy leaf area index (LAI). To fit these data, it was necessary to modify 3-PG by making specific leaf area and the fraction of stem biomass in branch and bark explicitly age-dependent, and to include effects of temperature on canopy quantum efficiency.
With these parameter values, 3-PG gave a good description of growth at seven of nine disparate sites in Tasmania and Western Australia. A partial explanation for its failure at one site in Western Australia is that this site was subject to a prolonged drought not reflected in the long-term mean climatic data used in this study. At the other, a high altitude site in Tasmania, 3-PG predicted significant growth although extensive frost damage had prevented establishment of a viable canopy. The model was also applied to non-fertilised stands in northern Tasmania for which only basic growth and site data were available. 3-PG adequately predicted the observed peak MAI of the better sites, but over-estimated peak MAI at poor or high-altitude sites.
The following conclusions are drawn: 3-PG can provide a good simulation of future growth of intensively-managed, fertilised stands of
E. globulus if the model is initialised with observed biomass data at some age around or following canopy closure. If the model is initialised with typical seedling biomass at planting, 3-PG adequately predicts stem growth rate but not canopy LAI. Further development of 3-PG should take into account possible environmental effects on litterfall, the effects of partial canopy closure during early canopy development, and the prediction of mortality prior to the onset of self-thinning.
As recently as a decade ago,
Karenia brevis red tides and their effects on animal resources in the Gulf of Mexico were principally perceived as acute blooms that caused massive fish kills. Although ...occasional mortalities of higher vertebrates were documented, it has only been in the past decade that conclusive evidence has unequivocally demonstrated that red tides and their brevetoxins are lethal to these organisms. Brevetoxins can be transferred through the food chain and are accumulated in or transferred by biota at many trophic levels. The trophic transfer of brevetoxins in the food web is a complex phenomenon, one that is far more complicated than originally conceived. Unexplained fish kills and other animal mortalities in areas where red tide is endemic are being increasingly linked with post-bloom exposures of biota to brevetoxins. Mass mortality events of endangered Florida manatees (
Trichechus manatus latirostris) follow a consistent spatial and temporal pattern, occurring primarily in the spring in southwestern Florida. Persistent blooms can also cause a cascade of environmental changes, affecting the ecosystem and causing widespread die-offs of benthic communities. Ongoing fish kills from sustained blooms can lead to short-term declines in local populations. Although animal populations in areas where red tide is endemic are unquestionably at risk, it remains to be determined to what extent populations can continue to recover from these sustained effects.
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