Unmixed Case Research Articles

Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on Rd. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.In the second part of the paper we study nonlinear m-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for d≥2, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for m-term approximation can still be derived.

Mixing and diffusion in protoplanetary disc chemistry

We develop a simple iterative scheme to include vertical turbulent mixing and diffusion in PRODIMO thermo-chemical models for protoplanetary discs. The models are carefully checked for convergence towards the time-independent solution of the reaction-diffusion equations, as, for example, used in exoplanet atmosphere models. A series of five TTauri disc models is presented where we vary the mixing parameter αmix from zero to 10−2 and take into account: (a) the radiative transfer feedback of the opacities of icy grains that are mixed upwards; and (b) the feedback of the changing molecular abundances on the gas temperature structure caused by exothermic reactions, and increased line heating and cooling. We see considerable changes in the molecular and ice concentrations in the disc. The most abundant species (H2, CH4, CO, the neutral atoms in higher layers, and the ices in the midplane) are transported both up and down, and at the locations where these abundant chemicals finally decompose, for example by photo processes, the release of reaction products has important consequences for all the other molecules. This generally creates a more active chemistry, with a richer mixture of ionised, atomic, molecular, and ice species, and new chemical pathways that are not relevant in the unmixed case. We discuss the impact on three spectral observations caused by mixing and find that: (i) icy grains can reach the observable disc surface where they cause ice absorption and emission features at IR to far-IR wavelengths; (ii) mixing increases the concentrations of certain neutral molecules observable by mid-IR spectroscopy, in particular OH, HCN, and C2H2; and (iii) mixing can change the optical appearance of CO in ALMA line images and channel maps, where strong mixing would cause the CO molecules to populate the distant midplane.

Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm -estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular coefficients

In this paper, we study non-divergence form elliptic and parabolic equations with singular coefficients. Weighted and mixed-norm L_p -estimates and solvability are established under suitable partially weighted BMO conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed norm case. For the proof, we explore and utilize the special structures of the equations to show both interior and boundary Lipschitz estimates for solutions and for higher-order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman–Stein sharp function theorem, the Hardy–Littlewood maximal function theorem, as well as a weighted Hardy’s inequality.

Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD

We revisit the $U(1)_A$ anomaly in the holographic model of low-energy QCD by Witten, Sakai, and Sugimoto, presenting a new and direct derivation of the Witten-Veneziano mechanism for generating the mass of the $\eta'$ through an anomalous mixing of the Ramond-Ramond $C_1$ field with the singlet component of the pseudoscalar mesons. The latter turns out to have a kinetic mixing with the normalizable modes of the $C_1$ field representing pseudoscalar glueballs, yielding additional vertices for their production and their decay that dominate over those of the unmixed case considered previously in the Witten-Sakai-Sugimoto model. The leading channel is predicted to be decay into two vector mesons, followed in importance by decay into three pseudoscalar mesons. The issue of production of pseudoscalar glueballs in radiative $J/\psi$ decays and in double diffractive processes is also discussed briefly.

Mixed monolayers of alkane thiols with polar terminal group on gold: Investigation of structure dependent surface properties

Adsorption of thiols with cationic or anionic terminal group on gold has been studied from mixed solutions of 11-Amino-1-undecanethiol (AUT) and 3-Mercaptopropionic acid (MPA) using Quartz Crystal Microbalance with Dissipation (QCM-D), X-ray Photoelectron Spectroscopy (XPS), atomic force microscopy (AFM) and contact angles. The goal is to probe the nature of such layers, and the additivity or otherwise of the pH responsiveness, with a view to evaluate their suitability as smart materials. For each of the two pure (unmixed) cases, ordered molecular monolayers are formed with sulfur binding to gold and the alkane chain pointing out from the surface as expected. Adsorption from the thiol mixtures, however, leads to a more complex behaviour. The surface concentration of thiols from the mixtures, as determined by QCM-D, is considerably lower than for the pure cases and it reaches a minimum at a 3:1 MPA/AUT relative concentration in the solution. The XPS results confirm a reduction in adsorbed amount in mixtures with the lowest overall intensity for the 3:1 ratio. Monolayers formed from mixtures display a wettability which is much lower and less pH sensitive. Collectively these results confirm that for adsorption from mixed systems, the configuration is completely different. Complex formation in the mixed solutions leads to the adsorption of molecules parallel to the surface in an axially in-plane configuration. This parallel layer of thiols is mechanically relatively robust to nano-shaving based on AFM measurements. These results will have a significant impact on the design of biomimetic surface coatings particularly when mixtures of oppositely charged molecules are present on the surface, as is commonly the case in biological, proteinaceous surfaces (e.g. hair and skin).

Particle trajectories on hillslopes: Implications for particle age and 10Be structure

AbstractMany geomorphic systems act as conveyor belts onto which material is loaded at a particular rate and is transported in one direction toward another system that serves as a sink. As the material travels, it ages, it changes in grain size, it accumulates cosmogenic radionuclides, it adsorbs or releases nutrients, and it weathers. Here I address the hillslope conveyor. As many geochemical processes are depth‐dependent, the depth history of a particle becomes important to know. I calculate soil particle trajectories in the horizontal‐depth plane and address three cases, one in which horizontal speeds decline exponentially with depth, a second in which they are uniform with depth, and a third in which horizontal speeds are also uniform but all profile values are vertically well‐mixed. Vertical speeds are governed by continuity in an incompressible medium and by the boundary condition of zero vertical particle speed at the soil surface. Particle trajectories must therefore become surface parallel at the surface. Knowledge of soil particle trajectories allows calculation of residence times and concentration profiles of 10Be in the soil. The results inform strategies for interpretation of nuclide concentrations in soils and stream sediments and for inference of transport rate profiles. In all steady cases, the particle age and 10Be structure are uniform with distance from the divide. When significant vertical gradients in horizontal speed occur, the patterns of particle age and of 10Be concentration are dominated by the depth scale of the transport process. In unmixed cases, the particle age and 10Be concentration in near‐surface samples can greatly exceed the vertically averaged values, reflecting the fact that the vertical speeds of particles slow dramatically as they near the surface. In cases in which horizontal speed varies significantly with depth, the vertically averaged concentration of 10Be within the soil can significantly underpredict the mean 10Be concentration of sediment delivered to the channel. Proper averaging requires weighting by particle speed. These results provide the backdrop for addressing chemical evolution of soil on a hillslope.

Multihomogeneous resultant formulae for systems with scaled support

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bézout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bézout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

Newton Polyhedra of Discriminants of Projections

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results (Gelfand et al. in Discriminants, resultants, and multidimensional determinants. Birkhauser, Boston, 1994; Sturmfels in J. Algebraic Comb. 32(2):207---236, 1994; McDonald in Discrete Comput. Geom. 27:501---529, 2002; Gonzalez-Perez in Can. J. Math. 52(2):348-368, 2000; Esterov and Khovanskii in Funct. Anal. Math. 2(1), 2008).

HYPERFINE MIXING AND THE SEMILEPTONIC DECAYS OF DOUBLE-HEAVY BARYONS IN A QUARK MODEL

The semileptonic decays of the lowest-lying double-heavy baryons are treated in a quark model. For the Ξbb, hyperfine mixing in the spin wave function leaves the total rate for decay into the lowest lying daughter baryons essentially unchanged, but changes the relative rates into the Ξbc and [Formula: see text]. The same pattern is obtained in the decays of the Ωbb. For the Ξbc, this mixing leads to factor of about 17 suppression in the decay rate to the [Formula: see text] when wave functions truncated to the largest components are used, but the total semileptonic decay rate of the parent baryon remains essentially unchanged. For the Ωbc, the decay to the [Formula: see text] is suppressed by a factor of more than 25 from the unmixed case. When the full wave functions are used, the large suppression of the decays to the [Formula: see text] and [Formula: see text] persists.

Ultrasonically enhanced delivery and degradation of PAHs in a polymer–liquid partitioning system by a microbial consortium

The current study examined the effects of ultrasonic irradiation on mass transfer and degradation of PAHs, by an enriched consortium, when delivered from polymeric matrices. Rates of release into methanol under sonicated conditions, relative to unmixed cases, for phenanthrene, fluoranthene, pyrene, and benzo[a]pyrene were increased approximately fivefold, when delivered from Desmopan 9370 A (polyurethane). Similar effects were observed in Hytrel and Kraton D4150 K polymers as well as recycled Bridgestone tires. Enhancements were also displayed as shifts to higher release equilibria under sonicated conditions, relative to non-sonicated cases, agreeing with current knowledge in sonochemistry and attributed to cavitation. Ultrasonic effects on microbial activity were also investigated and cell damage was found to be non- permanent with consortium re-growth being observed after sonic deactivation. Finally, the lumped effect of sonication on degradation of phenanthrene delivered from Desmopan was examined under the absence and presence of sonication. Rates of degradation were found to be increased by a factor of four demonstrating the possibility of using ultrasonic irradiation for improved mass transport in solid-liquid systems. Cellular inactivation effects were not evident, and this was attributed to the attenuation of sonic energy arising from the presence of solid polymer materials in the medium. The findings of the study demonstrate that sonication can be used to improve mass transport of poorly soluble compounds in microbial degradations, and alleviate limiting steps of soil remediation processes proposed in previous research.

A model for the infrared dust emission from forming galaxies

Dust plays various important roles in galaxy formation and evolution. In the early epoch of galaxy evolution, dust is only supplied by supernovae (SNe). With the aid of a new physical model of dust production by SNe developed by Nozawa et al. (N03), we constructed a model of dust emission from forming galaxies on the basis of the theoretical framework of Takeuchi et al. (T03). N03 showed that the produced dust species depends strongly on the mixing within SNe. We treated both unmixed and mixed cases and calculated the infrared (IR) spectral energy distribution (SED) of forming galaxies for both cases. Our model SED is less luminous than the SED of T03 model by a factor of 2–3. The difference is due to our improved treatment of the ultraviolet (UV) photon absorption cross-section, as well as the different grain size and species newly adopted in this work. The SED for the unmixed case is found to have an enhanced near to mid-IR (N-MIR) continuum radiation in its early phase of the evolution (age ≲ 107.25 yr) compared with that for the mixed case. The strong N-MIR continuum is due to the emission from silicon grains, which only exist in the species of the unmixed dust production. We also calculated the IR extinction curves for forming galaxies. N03 dust was found to yield a smaller extinction than that of T03 model. For the unmixed case, near-IR (NIR) extinction is dominated by large grains of silicon and amorphous carbon, and silicate features are less prominent compared to the curve given by T03. To the contrary, the extinction curve of the mixed case has a similar shape to that of T03. Then we calculated the SED of a local starbursting dwarf galaxy SBS 0335−052. Our present model SED naturally reproduced the strong N-MIR continuum and the lack of cold far-IR emission of SBS 0335−052. We found that only the SED of unmixed case can reproduce the NIR continuum of this galaxy. We then made a prediction for the SED of another typical star-forming dwarf, I Zw 18. The MIR continuum of I Zw 18 is expected to be much weaker than that of T03 SED. We also presented the evolution of the SED of Lyman-break galaxies. Finally, we discussed the possibility of observing forming galaxies at z≳ 5.

Enhancement of thermal performance in a sintered miniature heat pipe

The performance of a sintered miniature heat pipe is enhanced. With the capillary limitation, porosity takes priority over the wick structure parameters that would affect the heat transfer capacity. Since sintered dendritic copper powder has higher porosity, it is used to mix with pore former (Na2CO3) in experiments for increasing porosity, and hence enhancing the thermal performance. The results show that, for a heat pipe with a 3mm outer diameter and 200 mm effective length, the heat transfer rate is up to 16.5W and the thermal resistance is 0.9°C/W. In comparison with the unmixed case, the performance increases about 40%.

Dust in the Early Universe: Dust Formation in the Ejecta of Population III Supernovae

Dust grains play a crucial role in the formation and evolution history of stars and galaxies in the early universe. We investigate the formation of dust grains in the ejecta of Population III supernovae, including pair-instability supernovae, which are expected to occur in the early universe, applying a theory of non-steady state nucleation and grain growth. Dust formation calculations are performed for core-collapse supernovae with progenitor mass Mpr ranging from 13 to 30 M☉ and for pair-instability supernovae with Mpr = 170 and 200 M☉. In the calculations, the time evolution of gas temperature in the ejecta, which strongly affects the number density and size of newly formed grains, is calculated by solving the radiative transfer equation, taking account of the energy deposition of radioactive elements. Two extreme cases are considered for the elemental composition in the ejecta, unmixed and uniformly mixed cases within the He core, and formation of CO and SiO molecules is assumed to be complete. The results of calculations for core-collapse supernovae and pair-instability supernovae are summarized as follows: in the unmixed ejecta, a variety of grain species condense, reflecting the difference of the elemental composition at the formation site in the ejecta; otherwise only oxide grains condense in the uniformly mixed ejecta. The average size of newly formed grains spans a range of 3 orders of magnitude, depending on the grain species and the formation condition, and the maximum radius is limited to less than 1 μm, which does not depend on the progenitor mass. The size distribution function of each grain species is approximately lognormal, except for Mg silicates, MgO, Si, and FeS in the unmixed case and Al2O3 in both cases. The size distribution function summed up over all grain species is approximated by a power-law formula whose index is -3.5 for the larger radius and -2.5 for the smaller one; the radius at the crossover point ranges from 0.004 to 0.1 μm, depending on the model of supernovae. The fraction of mass locked into dust grains increases with increasing the progenitor mass: 2%-5% of the progenitor mass for core-collapse supernovae and 15%-30% for pair-instability supernovae whose progenitor mass ranges from 140 to 260 M☉. Thus, if very massive stars populated the first generation of stars (Population III stars), a large amount of dust grains would be produced in the early universe. We also discuss the dependence of the explosion energy and the amount of 56Ni in the ejecta, as well as the efficiency of formation of CO and SiO molecules, on the formation of dust grains in the ejecta of supernovae.

Derivation of supersaturation during precipitation from the mixing pattern of an inert tracer in the same device: case of partially premixed feed streams

The average supersaturation field is predicted for precipitations in the case of partially premixed feed streams by a simple mixing model from a couple of successive experiments with the same inert tracer by Planar Laser Induced Fluorescence and using a single camera. Then, the mathematical derivation is different from that one used for the unmixed feed case. The corresponding experiments have been achieved in a 90° impinging jets premixer and the supersaturation field was predicted. Due to the non-simultaneity of tracer experiments, a calculation of the averaged nucleation flux is not possible in the premixed feed case. Nevertheless, a comparison between different premixers can be done on the basis of the generated supersaturation levels and fouling risks.

Sparse Resultant of Composed Polynomials I Mixed–Unmixed Case

The main question of this paper is: What happens to sparse resultants under composition? More precisely, let f 1,⋯, f n be homogeneous sparse polynomials in the variables y 1,⋯, y n and g 1,⋯, g n be homogeneous sparse polynomials in the variables x 1,⋯, x n . Let f i ∘ ( g 1,⋯, g n ) be the sparse homogeneous polynomial obtained from f i by replacing y j by g j . Naturally a question arises: Is the sparse resultant of f 1 ∘ ( g 1,⋯, g n ), ⋯, f n ∘ ( g 1,⋯, g n )in any way related to the (sparse) resultants of f 1,⋯, f n and g 1,⋯, g n ? The main contribution of this paper is to provide an answer for the case when g 1,⋯, g n are unmixed, namely, Res C 1,⋯, C n ( f 1 ∘ ( g 1,⋯, g n ),⋯, f n ∘ ( g 1,⋯, g n )) = Res d 1,⋯, d n ( f 1,⋯, f n ) Vol( Q) Res B ( g 1,⋯, g n ) d 1⋯ d n , where Res d 1,..., d n stands for the dense (Macaulay) resultant with respect to the total degrees d i of the f i ’s, Res B stands for the unmixed sparse resultant with respect to the support B of the g j ’s, Res C 1,..., C n stands for the mixed sparse resultant with respect to the naturally induced supports C i of the f i ∘ ( g 1,⋯, g n )’s, and Vol( Q)for the normalized volume of the Newton polytope of the g j . The above expression can be applied to compute sparse resultants of composed polynomials with improved efficiency.

Radion and Higgs mixing at the LHC

We study the resonance production of radions and Higgs via gluon-gluon fusion in the Randall-Sundrum model with Higgs-curvature mixing at the LHC. We find that radion can be detected both in mixed (with Higgs boson) and unmixed case if the radion vacuum expectation value $\Lambda_\phi$ is around 1 TeV. The $\Lambda_\phi \sim 10$ TeV case is also promising for certain values of mixing parameters and radion masses. The mixing can affect the production and decay of Higgs boson in a significant way. Thus Higgs search strategies at the LHC may need refinements in case of radion-Higgs mixing in the Randall-Sundrum model.

Some Aspects of the Behavior of High Order Mode Internal Gravity Waves on Interfaces With Finite Thickness

The role of interface thickness on the propagation of high order internal wave modes is examined using simple analytical models for an unmixed, thick interface and one that has been vigorously mixed. It is shown that for the unmixed case, a local maximum in wave speed occurs as the interface grows from zero thickness, but that only for mode-one waves does the maximum point occur within areas of practical significance. When these models are applied to three wave resonant interactions, it is found that vigorous mixing will augment energy transfer rates when the interface thickness surpasses a certain value.

On linearizing algebraic torus actions

Certain results of Bialynicki-Birula as well as other pieces of evidence suggest that an algebraic torus action on an affine space is always linear with respect to a suitably chosen coordinate system (cf. [l, 51). With eventual proof of this conjecture in mind, we examine in this note torus actions on a smooth affine variety over an algebraically closed ground field. It turns out that, in what we have termed unmixed cases (see 2.3), the variety in question is a vector bundle over the fixed point variety and the torus action is linear along the fibers (see Theorem 2.5 below). We have noticed that this fact is already known in essence to Bialynicki-Biruala [2; Th. 2.51. But our proof is elementary and seems simpler than his, resting upon the smoothness of fixed point schemes (Fogarty [3]) and a version of Nakayama’s Lemma for semigroup-graded rings and modules (see 1.2 and 1.4). It is clear that definiteness of a torus action as defined by Bialynicki-Birula (see [2; p. 482)) implies unmixedness, but the converse is true as well (see 1.6). Since unmixedness is a quite intrinsic condition, it seems worthwhile to point this out. As an immediate consequence of the above result and the Quillen-Suslin Theorem [9, 121, a smooth affine variety with an unmixed torus action is actually an affine space with a linear torus action, provided the fixed point variety is isomorphic to some affine space (possibly a single point). By making use of recent results of Fujita, Miyanishi and Sugie [4, 81, we show this last to be the case indeed for any unmixed torus action on an affine space with a fixed point variety of dimension 12 (see Theorem 3.4). Hyman Bass told one of the authors how a graded ring R =@,,ro R, becomes isomorphic to a symmetric algebra over Ro under a certain homological hypothesis (see 2.6). His remark was largely responsible for getting us started on the present investigation. His suggestions were also responsible for an improvement in our treat-

Granuloma inguinale (venereum) in the female

Definition. GranuIoma inguinaIe (venereum) is a chronic specific infection chieff y of the skin and subcutaneous tissues, and in the femaIe primariIy invoIves the externaI genitalia or the inguina1 region. Less frequentIy the vagina or the cervix is the primary site of the disease. The causa1 organism is beIieved to be the Donovan body. The disease is probabIy venerea1 in origin and occurs most commonIy in the coIored race. Etiology. It is generaIIy accepted that the Donovan body is the causative agent. SeveraI investigators, however, beIieve it is due to a baciIIus beIonging to the mucosus capsuIatus group of FriedIander. McIntosh’ claims to have grown this organism on various ordinary cuIture media, but this couId not be substantiated by Dienst, GreenbIatt and Sanderson who have been unabIe to cuhivate the Donovan bodies in spite of the fact that on severa occasions they fortunateIy obtained the organism free from any type of bacteria from unruptured pseudo-buboes. They are convinced that the organism is not a bacterium. De Monbreun and Goodpasture recovered the Donovan bodies from artificiaIIy infected monkeys. One of us (R. B. G.) has succeeded in transmitting the disease from one patient to another with recovery of the organism in a pure state from the inoculated person one month and two months afterwards. The disease is thought to be venerea1 because the primary Iesion most often appears on the genitaha, and onIy during the period of active sex rife of its victims. Individual susceptibiIity, uncIeanIiness and race are important etioIogic factors. It is possibIe that the disease may be transmitted by some insect vector, but this has never been proved. Pathology. Pund and GreenbIatt* summarize the pathoIogy thus: “GranuIoma venereum reveaIs in the pure or unmixed cases a uniform histoIogic picture. The essentia1 features are (a) the massiveness of the celIuIar reaction in which the Iuxuriant granuIation tissue is surcharged with pIasma ceIIs; (b) the relative and conspicuous paucity of Iymphocytes; (c) the diffuse sprinkIing of poIymorphonucIear Ieucocytes, with foca1 coIIections in the superficies and papiIIae; (d) the pronounced margina epitheIia1 proIiferation simuIating earIy epithehomatous changes; and (e) the pathognomonic Iarge mononucIear celIs scattered in various numbers throughout the granuIation tissue. (Fig. I.) “The pathognomonic celI is specific for granuIoma venereum. The reIativeIy Iarge size of the ceI1, the diameter of which varies from 25 to go microns, and the many intracytopIasmic cysts flied with deepIy stained bodies are its cardina1 features. Donovan bodies are round or rod-Eke and are grouped peripheraIIy within the cysts