A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter \(\alpha >0\). Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.     

The effect of two distinct fast time scales in the rotating,stratified Boussinesq equations: variations from quasi-geostrophy

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.     

High-Resolution Infrared-Vacuum Ultraviolet Photoion and Pulsed Field Ionization-Photoelectron Methods for Spectroscopic Studies of Neutrals and Cations

We show that by scanning the frequency of a single mode infrared (IR) optical parametric oscillator (IR-OPO) laser to excite the molecular species of interest and fixing the frequency of a vacuum ultraviolet (VUV) laser to photoionize the IR excited species, high-resolution IR spectra of polyatomic neutrals can be obtained with high sensitivity. The fact that this IR-VUV-photoion (IR-VUV-PI) method is based on VUV photoionization probe, and thus, allows the identification of the neutral IR absorber, makes it applicable for IR spectroscopy measurements of isotopemers, radicals, and clusters, which usually exist as impure samples. The highly resolved IR-VUV-PI measurements achieved using the single mode IR-OPO laser have made possible the selection of single rovibrational states of CH3X (X=Br and I), C2H4, and C3H4 for VUV-pulsed field ionization-photoelectron (VUV-PFI-PE) measurements, resulting in rovibrationally resolved photoelectron spectra for these polyatomic molecules. These experiments show that the signal-to-noise ratios of the IR-VUV-PI and IR-VUV-PFI-PE spectra obtained by employing the high-resolution IR-OPO laser are significantly higher than those observed in previous IR-VUV-PI and IR-VUV-PFI-PE studies using a low-resolution IR-OPO laser. Further improvement in sensitivity of IR-VUV-PI and IR-VUV-PFI-PE measurements by using the collinear arrangement of IR-VUV lasers and molecular beam is discussed.     

Sequential and localized grafting on aliphatic polyester diblock copolymers using alkyne deprotection and click cycloaddition

Diblock copolymers, in which both blocks are composed of aliphatic polyesters, were synthesized from two different alkyne‐functionalized δ‐valerolactone monomers by ring opening polymerization and subsequent click cycloaddition. Trimethylsilyl protection of the alkyne functionality of one block was instrumental to the success of the synthesis. These novel aliphatic polyester diblock copolymers were characterized by ¹H and ¹³C NMR spectroscopy, gel permeation chromatography (GPC), and infrared (IR) spectroscopy. Sequential functionalization of the diblock copolymers with hydrophobic groups on one block, and hydrophilic groups on the other block, provides access to amphiphilic structures. Micellar structures generated from these polyester amphiphiles were characterized by fluorescence spectroscopy and transition electron microscopy (TEM). © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2009     

Some Connections between Frames,Mutually Unbiased Bases,and POVM’s in Quantum Information Theory

This paper gives a very brief overview of some aspects of quantum information theory, then explains the relationship between a positive operator valued measure (POVM) and a frame. Some special types of POVM’s are defined and conjectures about them shown to be essentially conjectures about the existence of special types of frames. Finally, some connections with quantum channels and separable states are discussed.     

Confusion about Bohm

Conclusion  It is important to distinguish between physics, which is an experimental science, andphysicists, who are people. The latter are most certainlynot objective. Thus, Lipschütz-Yevick’s assertion that Ruskai says that the publication of Bohm’s controversial articles in thePhysical Review is evidence of the objectivity of the establishment towards [Bohm] is not supported by my statement It should be noted that even though studying the foundations of quantum mechanics has long been far from the mainstream, it has never been suppressed. The papers of Bohm, Bell,et al. were published in reputable journals, … Reasonable people may disagree on the significance of a particular theory or individual’s contribution. It is here, rather than in the physicsper se, that questions of social influence are likely to arise. I have commented elsewhere, e.g., [17], on the role that gender sometimes plays. In a subsequent article, I will also discuss the distinction between the effect of the social and political climate on the development of the careers of individuals and the development of physics. The articles by Cronin and Lipschütz-Yevick have stimulated me to think anew about a number of issues related to Bohmian mechanics, for which a full discussion requires clarification of some technical issues regarding the EPR experiment and non-locality. These will be discussed in a forthcoming article.     

Interpolymer complex formation between helical poly(L-proline) and random-coil poly(methacrylic acid) through hydrogen bonding

Interpolymer complex formation between poly(L -proline) (PLP) with helical structure and poly(methacrylic acid) (PMAA) with random-coil structure through hydrogen bonding in aqueous medium has been studied by several experimental techniques, e.g., viscometry, turbidimetry, potentiometry, conductometry, scanning electron microscopy, and x-ray diffraction methods. The decreases in reduced viscosity of the solution on addition of an increasing quantity of PLP to a constant amount of PMAA reveals the formation of a complex between PLP and PMAA. The minimum in reduced viscosity at a unit-mole ratio [PLP]/[PMAA] = 1.0 suggests a 1 : 1 complex formation. A distinct change in the curves for turbidity, pH, and conductance versus [PLP]/[PMAA] supports this conclusion. A scanning electron micrograph for the 1 : 1 PLP–PMAA complexes shows that the PLP/PMAA complex has the shape of entangled long fibers. An x-ray diffraction pattern for the PLP/PMAA complexes gives no diffraction patterns which appear in pure PLP, indicating the destruction of the helical structure of PLP due to the interpolymer complexation. Mixtures of PMAA with poly(γ-hydroxy-L -proline) (PHLP) which has a similar conformation as PLP, but involves intra- or intermolecular hydrogen bonds, has also been investigated by vicometry measurements. The reduced viscosity of a solution of the mixed polymers increases with increasing [PHLP], indicating no complex formation. All the results reveal that the magnitude and the nature of the forces acting in the polymers play an important role in interpolymer complexation.     

Quantitative small-angle light-scattering studies of the melting and crystallization of LLDPE/LDPE blends

Small-angle light-scattering (SALS) patterns were obtained during melting and crystallization of blends of linear low-density polyethylene (LLDPE) with conventional low-density polyethylene (LDPE). Quantitative measurements of these SALS patterns using a two-dimensional optical multichannel analyzer apparatus (OMA2) indicate that the LLDPE which is miscible with the LDPE component in the molten state crystallizes first, forming volume-filling spherulites. The LDPE then crystallizes within the preformed spherulites. These findings are supported by optical microscopy studies showing that the blend samples were volume filled with one kind of the spherulites having a radius comparable to that of the pure LLDPE. The SALS intensity curve changes with composition of the blends in a manner that may be interpreted by considering the orientation of crystals within spherulites. It has been observed that the spherulites in the blend have more diffuse boundaries as the LDPE content increases. The lattice spacing and long spacings in blends were obtained by wide-angle and small-angle x-ray scattering, respectively. The SALS technique along with differential scanning calorimetry (DSC) is shown to be useful for determining the crystallization behavior of a crystallizable polymer blend system.     

Subitising activity relative to units construction: a case study

Subitising, a quick apprehension of the numerosity of a small set of items, has been found to change from an individual's reliance on perceptual to conceptual processes. In this study, we utilised a constructivist teaching experiment methodology to investigate how the subitising activity of one preschool student, Amy, related to her construction of prenumerical units. Subitising and counting tasks were designed to assess and perturb Amy's thinking relative to her construction of units, and to observe changes in Amy's activity associated with the different tasks. Findings indicate that as Amy's subitising activity changed from perceptual to conceptual, she constructed subitised motor units and subitised figurative units. Implications of this study suggest that the construction of subitised units may support young children's later development of arithmetic units.