Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension

We consider the one-dimensional model of a spin glass with independent Gaussian-distributed random interactions, that have mean zero and variance $1/|i-j|^{2\sigma}$, between the spins at sites $i$ and $j$ for all $i\neq j$. It is known that, for $\sigma>1$, there is no phase transition at any non-zero temperature in this model. We prove rigorously that, for $\sigma>3/2$, any Newman-Stein metastate for the ground states (i.e.\ the frequencies with which distinct ground states are observed in finite size samples in the limit of infinite size, for given disorder) is trivial and unique. In other words, for given disorder and asymptotically at large sizes, the same ground state, or its global spin flip, is obtained (almost) always. The proof consists of two parts: one is a theorem (based on one by Newman and Stein for short-range two-dimensional models), valid for all $\sigma>1$, that establishes triviality under a convergence hypothesis on something similar to the energies of domain walls, and the other (based on older results for the one-dimensional model) establishes that the hypothesis is true for $\sigma>3/2$. In addition, we derive heuristic scaling arguments and rigorous exponent inequalities which tend to support the validity of the hypothesis under broader conditions. The constructions of various metastates are extended to all values $\sigma>1/2$. Triviality of the metastate in bond-diluted power-law models for $\sigma>1$ is proved directly.Comment: 18 pages. v2: subsection on bond-diluted models added, few extra references. 19 pages. v3: published version; a few changes; 20 page

Additional file 3: of Relationships between drought, heat and air humidity responses revealed by transcriptome-metabolome co-analysis

Result lists of condition-specific gene expression and metabolomic mass levels estimated by linear modeling across the whole dataset. Tables show mean log2 gene expression values and log2metabolomic mass levels for each of the six environmental conditions: control, drought (D), heat with low relative air humidity (H_LrH), heat with high relative air humidity (H_HrH), drought stress combined with low air humidity heat stress (DH_LrH) and drought stress combined with high air humidity heat stress (DH_HrH). Values of all individual samples are available in the processed data file deposited at the ArrayExpress and MetaboLights databases, respectively (Availability of data and materials). (XLSX 2264 kb

Image1.JPEG

<p>Whisky can be described as a complex matrix integrating the chemical history from the fermented cereals, the wooden barrels, the specific distillery processes, aging, and environmental factors. In this study, using Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) and liquid chromatography coupled with tandem mass spectrometry (LC-MS/MS), we analyzed 150 whisky samples from 49 different distilleries, 7 countries, and ranging from 1 day new make spirit to 43 years of maturation with different types of barrel. Chemometrics revealed the unexpected impact of the wood history on the distillate's composition during barrel aging, regardless of the whisky origin. Flavonols, oligolignols, and fatty acids are examples of important chemical signatures for Bourbon casks, whereas a high number of polyphenol glycosides, including for instance quercetin-glucuronide or myricetin-glucoside as potential candidates, and carbohydrates would discriminate Sherry casks. However, the comparison of barrel aged rums and whiskies revealed specific signatures, highlighting the importance of the initial composition of the distillate and the distillery processes.</p

Table1.DOCX

<p>Whisky can be described as a complex matrix integrating the chemical history from the fermented cereals, the wooden barrels, the specific distillery processes, aging, and environmental factors. In this study, using Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) and liquid chromatography coupled with tandem mass spectrometry (LC-MS/MS), we analyzed 150 whisky samples from 49 different distilleries, 7 countries, and ranging from 1 day new make spirit to 43 years of maturation with different types of barrel. Chemometrics revealed the unexpected impact of the wood history on the distillate's composition during barrel aging, regardless of the whisky origin. Flavonols, oligolignols, and fatty acids are examples of important chemical signatures for Bourbon casks, whereas a high number of polyphenol glycosides, including for instance quercetin-glucuronide or myricetin-glucoside as potential candidates, and carbohydrates would discriminate Sherry casks. However, the comparison of barrel aged rums and whiskies revealed specific signatures, highlighting the importance of the initial composition of the distillate and the distillery processes.</p

Table3.DOCX

<p>Whisky can be described as a complex matrix integrating the chemical history from the fermented cereals, the wooden barrels, the specific distillery processes, aging, and environmental factors. In this study, using Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) and liquid chromatography coupled with tandem mass spectrometry (LC-MS/MS), we analyzed 150 whisky samples from 49 different distilleries, 7 countries, and ranging from 1 day new make spirit to 43 years of maturation with different types of barrel. Chemometrics revealed the unexpected impact of the wood history on the distillate's composition during barrel aging, regardless of the whisky origin. Flavonols, oligolignols, and fatty acids are examples of important chemical signatures for Bourbon casks, whereas a high number of polyphenol glycosides, including for instance quercetin-glucuronide or myricetin-glucoside as potential candidates, and carbohydrates would discriminate Sherry casks. However, the comparison of barrel aged rums and whiskies revealed specific signatures, highlighting the importance of the initial composition of the distillate and the distillery processes.</p