Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

Proteome Profiling of Wheat Shoots from Different Cultivars

Wheat is a cereal grain and one of the world's major food crops. Recent advances in wheat genome sequencing are by now facilitating its genomic and proteomic analyses. However, little is known about possible differences in total protein levels of hexaploid versus tetraploid wheat cultivars, and also knowledge of phosphorylated wheat proteins is still limited. Here, we performed a detailed analysis of the proteome of seedling leaves from two hexaploid wheat cultivars (Triticum aestivum L. Pavon 76 and USU-Apogee) and one tetraploid wheat (T. turgidum ssp. durum cv. Senatore Cappelli). Our shotgun proteomics data revealed that, whereas we observed some significant differences, overall a high similarity between hexaploid and tetraploid varieties with respect to protein abundance was observed. In addition, already at the seedling stage, a small set of proteins was differential between the small (USU-Apogee) and larger hexaploid wheat cultivars (Pavon 76), which could potentially act as growth predictors. Finally, the phosphosites identified in this study can be retrieved from the in-house developed plant PTM-Viewer (bioinformatics.psb.ugent.be/webtools/ptm_viewer/), making this the first searchable repository for phosphorylated wheat proteins. This paves the way for further in depth, quantitative (phospho) proteome-wide differential analyses upon a specific trigger or environmental change

Up-to-date workflow for plant (phospho)proteomics identifies differential drought-responsive phosphorylation events in maize leaves

Protein phosphorylation is one of the most common post-translational modifications (PTMs), which can regulate protein activity and localization as well as proteinprotein interactions in numerous cellular processes. Phosphopeptide enrichment techniques enable plant researchers to acquire insight into phosphorylation-controlled signaling networks in various plant species. Most phosphoproteome analyses of plant samples still involve stable isotope labeling, peptide fractionation, and demand a lot of mass spectrometry (MS) time. Here, we present a simple workflow to probe, map, and catalogue plant phosphoproteomes, requiring relatively low amounts of starting material, no labeling, no fractionation, and no excessive analysis time. Following optimization of the different experimental steps on Arabidopsis thaliana samples, we transferred our workflow to maize, a major monocot crop, to study signaling upon drought stress. In addition, we included normalization to protein abundance to identify true phosphorylation changes. Overall, we identified a set of new phosphosites in both Arabidopsis thaliana and maize, some of which are differentially phosphorylated upon drought. All data are available via ProteomeXchange with identifier PXD003634, but to provide easy access to our model plant and crop data sets, we created an online database, Plant PTM Viewer (bioinformatics.psb.ugent.be/webtools/ptm_viewer/), where all phosphosites identified in our study can be consulted

The effect of electrical neurostimulation on collateral perfusion during acute coronary occlusion

<p>Abstract</p> <p>Background</p> <p>Electrical neurostimulation can be used to treat patients with refractory angina, it reduces angina and ischemia. Previous data have suggested that electrical neurostimulation may alleviate myocardial ischaemia through increased collateral perfusion. We investigated the effect of electrical neurostimulation on functional collateral perfusion, assessed by distal coronary pressure measurement during acute coronary occlusion. We sought to study the effect of electrical neurostimulation on collateral perfusion.</p> <p>Methods</p> <p>Sixty patients with stable angina and significant coronary artery disease planned for elective percutaneous coronary intervention were split in two groups. In all patients two balloon inflations of 60 seconds were performed, the first for balloon dilatation of the lesion (first episode), the second for stent delivery (second episode). The Pw/Pa ratio (wedge pressure/aortic pressure) was measured during both ischaemic episodes. Group 1 received 5 minutes of active neurostimulation before plus 1 minute during the first episode, group 2 received 5 minutes of active neurostimulation before plus 1 minute during the second episode.</p> <p>Results</p> <p>In group 1 the Pw/Pa ratio decreased by 10 ± 22% from 0.20 ± 0.09 to 0.19 ± 0.09 (p = 0.004) when electrical neurostimulation was deactivated. In group 2 the Pw/Pa ratio increased by 9 ± 15% from 0.22 ± 0.09 to 0.24 ± 0.10 (p = 0.001) when electrical neurostimulation was activated.</p> <p>Conclusion</p> <p>Electrical neurostimulation induces a significant improvement in the Pw/Pa ratio during acute coronary occlusion.</p

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Taha Toros Arşivi, Dosya No: 220-Kaptanzade Ali Rıza Be

A glimpse at polynomials with quantifiers

A prefixed polynomial equation, or a polynomial expression with a quantifier prefix, is an equation of the form P(x(1), x(2),..., x(n)) = 0, where P is a polynomial with integer coefficients whose variables x(1), x(2),..., x(n) range over natural numbers, that is preceded by some quantifiers over all of its variables x(1), x(2),..., x(n). Here is a typical, seemingly random, example of such an expression, Phi : for all m c there exists N for all a b there exists c d X A X for all x y there exists BCF there exists hijklnpqrst x (y+B-x) center dot (A+m+B-y) center dot((A+h-d)(2) + ((d+1) center dot i + A-c)(2) + (B+n-dx)(2) + (+(()dx+1) center dot j+B-c)(2) + (C+ r-dy)(2) + ((dy+1) center dot k+C-c)(2) + (B + s + 1 - c)(2) + (+)(c+t-N)(2) + (F+p-b(B+C)(2))(2) + (a -lb(B+C-2)-F- l)(2) + (X-F+eq)(2)) = 0 In this note we initiate the study of the collection of all possible such expressions ('the Atlas'), equipped with the equivalence relation of "being EFA-provably equivalent" on its members. The Atlas is partially ordered by EFA-implication. Here is the first abstract picture of the Atlas to have in mind: [GRAPHICS] Notice that the set of all prefixed polynomial equations is arithmetically complete, that is, every first-order arithmetical formula is EFA-equivalent to some prefixed polynomial expression. In this sense, the Atlas is just another way of talking about first-order arithmetical statements. Godel's Incompleteness theorems guarantee existence of many distinct equivalence classes. Our first task is to find examples. We start off with examples of distinct equivalence classes of metamathematical interest: the 1-consistency of I Sigma(1) (the expression phi above), the 1-consistency of I Sigma(2), the 1-consistency of full Peano Arithmetic, the highly unprovable Finite Kruskal's Theorem. [GRAPHICS] Then we give an example of the phase transition phenomenon. We produce a polynomial expression that has two free variables m and n such that whenever m/n is smaller or equal to Weiermann's constant w approximate to 0.63957768999472013311..., the expression is EFA-provable, otherwise it is unprovable in the theory ATR0 (so we witness a phase transition between EFA-provability and predicative unprovability). Then we give a crude example of a polynomial equation with quantifiers that is equivalent to the famous Graph Minor Theorem and, hence, is of the unknown high strength beyond that of Pi(1)(1)-CA(0). A 'seed' is a prefixed polynomial equation that is of minimal length in its EFA-equivalence class. We discuss seeds and notice that the seed of the 1-consistency of I Sigma(1) is smaller than 131. We discuss the role of the Atlas and its possible future partial implementation as a metamathematically-sensitive database of mathematical knowledge. The purpose of this note is to give definitions, to arrange the set-up, give nontrivial examples, introduce the right unprovability-sensitive notions, and ask some first questions about the Atlas. The current note omits all proofs. All proofs can be found in the grand manuscript [4]. Eventually, a bigger article, stemming from [4] will appear. Some of the material from this project became a chapter in the second author's doctoral thesis [6] at the University of Gent, Belgium. The authors genuinely and cordially thank the John Templeton Foundation for its interest and support for Unprovability research. The first author thanks the John Templeton Foundation for its financial support

Sharp thresholds for a phase transition related to weakly increasing sequences

Motivated by the classical Ramsey for pairs problem in reverse mathematics, we investigate the recursion-theoretic complexity of certain assertions which are related to the Erdos-Szekeres theorem. We show that resulting density principles give rise to Ackermannian growth. We then parameterize these assertions with respect to a number theoretic function f and investigate for which functions f Ackermannian growth is still preserved. We show that this is the case for f(i)=i(1/Ack-1(i)) but not for f(i)=i(1/Ad-1(i))

A miniaturisation of Ramsey's theorem

We approximate the strength of the infinite Ramsey Theorem by iterating a finitary version. This density principle, in the style of Paris, together with PA will give rise to a first-order theory which achieves a lot of the strength of ACA(0) and the original infinitary version. To prove our result, we use a generalisation of the results by Bigorajska and Kotlarski about partitioning alpha-large sets