Blow up in finite time and dynamics of blow up solutions for the -critical generalized KdV equation

In this paper, we describe the dynamics of blow up solutions for the critical generalized KdV equation such that the initial data is close to the soliton in and has decay in at the right. In particular, we prove that blow up occurs in finite time, and we obtain an upper bound on the blow up rate.

     

A study of the reaction of n-BuLi with Ti(Oi-Pr)4 as a method to generate titanacyclopropane and titanacyclopropene species

The use of the combination of reagents Ti(Oi-Pr)₄/n-BuLi, introduced by the group of J.J. Eisch in 2001, has only found a few applications so far, with sometimes conflicting observations. This article describes a study aimed at clarifying the nature, the stability and the reactivity of the active organometallic species involved. Reactions with CO₂ and other trapping reagents reveal that it is generated within a few minutes at 0 °C in THF, where it can be considered to be stable for 30 min. Most of our results are consistent with the expected titanacyclopropane nature of this reagent but some observations suggest that the chemistry at play may be more complicated.     

The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

On the Gibbs states of the noncritical Potts model on \mathbb Z ^2

We prove that all Gibbs states of the $q$ -state nearest neighbor Potts model on $\mathbb Z ^2$ below the critical temperature are convex combinations of the $q$ pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature $\beta >\beta _c (q) = \log \left(1+\sqrt{q}\right)$ .     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

Development and validation of an ion‐exchange chromatography method for heparin and its impurities in heparin products

An anion‐exchange liquid chromatography method for the determination of heparin and its impurities (dermatan sulfate and oversulfated chondroitin sulfate) was developed using chemometric‐assisted optimization, including multivariate experimental design and response surface methodology. The separation of heparin, dermatan sulfate, and oversulfated chondroitin sulfate (R_s above 2.0) was achieved on a Dionex RF IC IonPac AS22 column with a gradient elution of 10–70% of 2.5 M sodium chloride and 20 mM Tris phosphate buffer (pH 2.1) at a flow rate of 0.6 mL/min and UV detection at 215 nm. Method validation shows good linearity (r > 0.99), acceptable precision (%relative standard deviations <11.4%) and trueness (%recovery of 92.3–103.9%) for all analytes. The limits of detection for dermatan sulfate and oversulfated chondroitin sulfate are equivalent to 0.11% w/w (10.5 μg/mL) and 0.07% w/w (7.2 μg/mL), while the limits of quantification are 0.32% w/w (31.5 μg/mL) and 0.22% w/w (22.0 μg/mL) relative to heparin, respectively. The method is specific for heparin, dermatan sulfate, and oversulfated chondroitin sulfate without interference from mobile phase and sample matrices and could be used for accurate quantitation the drug and its impurities in a single run. Applications of the method reveal contents of heparin between 90.3 and 97.8%. Dermatan sulfate and oversulfated chondroitin sulfate were not detected in any of the real‐life samples.     

Selective Introduction of Sulfhydryl Groups into Recombinant Proteins for Study of Protein–Protein Interactions

In the present work, we proposed to create special sorbents for the study of protein–protein interactions, based on the fixation of cysteine-inserted beta-casein mutants with thiol-Sepharose resin. As a model system, we used bovine beta-casein, which belongs to the family of intrinsically unstructured proteins. Insertion of distal cysteines into the unfolded protein was not found to significantly change beta-casein properties. An amphiphilic beta-casein molecule has one hydrophilic domain and one hydrophobic domain placed on the N- and C-terminus, thus enabling one to exploit its capacity to engage in different types of intermolecular interactions. Two different casein-Sepharose sorbents incorporating either C-4 or C-208 beta-casein mutants bound to thiol-Sepharose were produced, exposing the hydrophobic domain in the case of the C-4 and the hydrophilic domain in the case of the C-208 mutant, respectively. The results obtained using the proposed sorbents with native beta-casein, another partially unfolded protein prion, and an oligomeric globular glyceraldehyde-3-phosphate dehydrogenase were found to be consistent with the data obtained by ELISA on free protein–protein complexes. Thus, Sepharose modified with various proteins is suitable for isolation of proteins interacting with the chromatographic phase bound partners from multicomponent systems such as milk. The obtained results allow the proposing of a fast and convenient method to be used for isolation of proteins, determination of protein-interacting partners, and the study of multi-protein complexes.     

Electrochemical oxidation of La2CuO4 thin films grown by molecular beam epitaxy

Thin insulating and c-axis oriented films of La₂CuO₄ are grown using a molecular beam epitaxy technique. Subsequently, these films are oxidized electrochemically using a 1N KOH solution. This approach is used to induce superconductivity, leading to a maximum T_c0 of 31 K,, measured both resistively and inductively. The surface morphology, lattice constants and the resistivity before and after the electrochemical treatment are compared.