On Indirect Approach to the Solvability of Quasi-Linear Dirichlet Elliptic Boundary Value Problem with BMO-Anisotropic p-Laplacian

We study here Dirichlet boundary value problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and L^1-control in coefficient of the low-order term. As characteristic feature of such problem  is a specification of the matrix of anisotropy A=A^{sym}+A^{skew} in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space W^{1,p}_0(\Omega), we specify a suitable functional class in which we look for solutions and prove existence of weak solutions in the sense of Minty using a non standard approximation procedure and compactness arguments in variable spaces

A NOTE ON WEIGHTED SOBOLEV SPACES RELATED TO WEAKLY AND STRONGLY DEGENERATE DIFFERENTIAL OPERATORS

In this paper we discuss some issues related to Poincar´e’s inequality for aspecial class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties

On Increasing of Resolution of Satellite Images via Their Fusion with Imagery at Higher Resolution

In this paper we propose a new statement of the spatial increasing resolution problem of MODIS-like multi-spectral images via their fusion with Lansat-like imagery at higher resolution. We give a precise definition of the solution to the indicated problem, postulate assumptions that we impose at the initial data, establish existence and uniqueness result, and derive the corresponding necessary optimality conditions. For illustration, we supply the proposed approach by results of numerical simulations with real-life satellite images.In this paper we propose a new statement of the spatial increasing resolution problem of MODIS-like multi-spectral images via their fusion with Lansat-like imagery at higher resolution. We give a precise definition of the solution to the indicated problem, postulate assumptions that we impose at the initial data, establish existence and uniqueness result, and derive the corresponding necessary optimality conditions. For illustration, we supply the proposed approach by results of numerical simulations with real-life satellite images

Making Good Lawyers

Today, the criticism of law schools has become an industry. Detractors argue that legal education fails to effectively prepare students for the practice of law, that it is too theoretical and detached from the profession, that it dehumanizes and alienates students, too expensive and inapt in helping students develop a sense of professional identity, professional values, and professionalism. In this sea of criticisms it is hard to see the forest from the trees. “There is so much wrong with legal education today,” writes one commentator, “that it is hard to know where to begin.” This article argues that any reform agenda will fall short if it does not start by recognizing the dominant influence of the culture of autonomous self-interest in legal education. Law schools engage in a project of professional formation and instill a very particular brand of professional identity. They educate students to become autonomously self-interested lawyers who see their clients and themselves as pursuing self-interest as atomistic actors. As a result, they understand that their primary role is to serve as neutral partisans who promote the narrow self-interest of clients without regard to the interests of their families, neighbors, colleagues, or communities and to the exclusion of counseling clients on the implications of those interests. They view as marginal their roles as an officer of the legal system and as a public citizen and accordingly place a low priority on traditional professional values, such as the commitment to the public good, that conflict with their primary allegiance to autonomous self-interest. In this work of professional formation, law schools are reflecting the values and commitments of the autonomously self-interested culture that is dominant in the legal profession. Therefore, even if law schools sought to form a professional identity outside of the mold of autonomous self-interest, such a commitment would require much more than curricular reform. It would, at minimum, require the construction of a persuasive alternative understanding of the lawyer’s role. The article seeks to offer such an understanding grounded in a relational perspective on lawyers and clients. Part I offers workable definitions of professionalism and professional identity that enable an informed discussion of the formation of professional identity in and by law schools. Part II explores what and how legal education teaches students showing that both institutionally (at the law school level) and individually (at the law professor level) legal education is proactively engaged in the formation of a professional identity of autonomous self-interest. Part II further explains that its dominance in legal education notwithstanding, autonomous self-interest is but one, often unpersuasive, account of professionalism and professional identity. Part III turns to the competing vision of relationally self-interested professionalism and professional identity and develops an outline for legal education grounded in these conceptions. Because legal education reflects a deep commitment to the dominant culture of autonomous self-interest, it is unlikely that reform proposals that are inconsistent with that culture are likely to succeed in the near future. Yet proposing an alternative account of professional identity that exposes the assumptions of the dominant culture, explains their limitations, and develops a more persuasive understanding is a necessary step toward providing a workable framework for reformers committed to promoting professional values in the long term

ON APPROXIMATION OF STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM IN COEFFICIENTS FOR p-BIHARMONIC EQUATION

We study a Dirichlet-Navier optimal design problem for a quasi-linear mono-tone p-biharmonic equation with control and state constraints. The coecient of the p-biharmonic operator we take as a design variable in BV ( )\L1( ). In order to handle the inherent degeneracy of the p-Laplacian and the pointwise state constraints, we use regularization and relaxation approaches. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted p-biharmonic operator and Henig approximation of the ordering cone. Further we discuss the asymptotic behaviour of the solutions to regularized problem on each ("; k)-level as the parameters tend to zero and innity, respectively

Dolomite-IV : Candidate structure for a carbonate in the Earth's lower mantle

We report the crystal structure of dolomite-IV, a high-pressure polymorph of Fe-dolomite stabilized at 115 GPa and 2500 K. It is orthorhombic, space group Pnma, a =10.091(3), b = 8.090(7), c = 4.533(3) Å, V = 370.1(4) Å3 at 115.2 GPa and ambient temperature. The structure is based on the presence of threefold C3O9 carbonate rings, with carbon in tetrahedral coordination. The starting Fe-dolomite single crystal during compression up to 115 GPa transforms into dolomite-II (at 17 GPa) and dolomite-IIIb (at 36 GPa). The dolomite-IIIb, observed in this study, is rhombohedral, space group R3, a =11.956(3), c =13.626(5) Å, V =1686.9(5) Å3 at 39.4 GPa. It is different from a previously determined dolomite-III structure, but topologically similar. The density increase from dolomite-IIIb to dolomite IV is ca. 3%. The structure of dolomite-IV has not been predicted, but it presents similarities with the structural models proposed for the high-pressure polymorphs of magnesite, MgCO3. A ring-carbonate structure match with spectroscopic analysis of high-pressure forms of magnesite-siderite reported in the literature, and, therefore, is a likely candidate structure for a carbonate at the bottom of the Earth's mantle, at least for magnesitic and dolomitic compositions

Dolomite-IV : Candidate structure for a carbonate in the Earth's lower mantle

We report the crystal structure of dolomite-IV, a high-pressure polymorph of Fe-dolomite stabilized at 115 GPa and 2500 K. It is orthorhombic, space group Pnma, a =10.091(3), b = 8.090(7), c = 4.533(3) \uc5, V = 370.1(4) \uc53 at 115.2 GPa and ambient temperature. The structure is based on the presence of threefold C3O9 carbonate rings, with carbon in tetrahedral coordination. The starting Fe-dolomite single crystal during compression up to 115 GPa transforms into dolomite-II (at 17 GPa) and dolomite-IIIb (at 36 GPa). The dolomite-IIIb, observed in this study, is rhombohedral, space group R3, a =11.956(3), c =13.626(5) \uc5, V =1686.9(5) \uc53 at 39.4 GPa. It is different from a previously determined dolomite-III structure, but topologically similar. The density increase from dolomite-IIIb to dolomite IV is ca. 3%. The structure of dolomite-IV has not been predicted, but it presents similarities with the structural models proposed for the high-pressure polymorphs of magnesite, MgCO3. A ring-carbonate structure match with spectroscopic analysis of high-pressure forms of magnesite-siderite reported in the literature, and, therefore, is a likely candidate structure for a carbonate at the bottom of the Earth's mantle, at least for magnesitic and dolomitic compositions

First in-situ measurements of Fe3+/Fetot for oxides and silicates included in natural diamonds with Synchrotron M\uf6ssbauer Source

Diamond is the paramount phase to understand the evolution and the physico- chemical condition of the deep portions of the Earth\u2019s mantle, mainly because: (i) it is the stable phase through which carbon is stored in the deep mantle for long geologic time; (ii) it does contain and preserve different types of inclusions (fluid, mineral, etc.); (iii) it is the only material sampling the mantle to depths of 800 km (e.g. Harte, 2010), although the majority of the mined diamonds worldwide derive from shallower depth (150 to 250 km). The study of mineral inclusions trapped in diamonds allows the retrieval of different pieces of information about the Earth\u2019s interior and its active geodynamics, providing important clues on the initiation of subduction processes (Shirey & Richardson, 2011; Smart et al., 2016), tracking the transfer of material through the mantle transition zone (Stachel et al., 2005; Walter et al., 2011), recording the timing of ingress of fluids to the continental lithosphere (e.g. Shirey et al., 2004), preserving carbonatitic fluid that trigger deep mantle melting (e.g. Schrauder & Navon, 1994; Kopylova et al., 2010), providing samples of primordial noble gases (e.g. Ozima & Igarashi, 2000), and capturing the redox state of the mantle (e.g. Rohrbach & Schmidt, 2011). Unfortunately the majority of the techniques used so far to study the mineral inclusions are destructive. It is only in the last decade that the studies on inclusions in diamond have started to use non-destructive techniques, providing new information which would otherwise be lost using earlier destructive techniques. Such an example is the rim fluids around inclusions in diamonds. In this study we present details of the experimental setup on the determination of Fe3+/Fetot ratios of mineral inclusions whilst still within the diamonds by a non-destructive approach using the Synchrotron M\uf6ssbauer Source (SMS; Potapkin et al., 2012) at the Nuclear Resonance beamline SOURCE ID18 (R\ufcffer & Chumakov, 1996), European Synchrotron Radiation Facility (ESRF), Grenoble. The extremely small X-ray spot size (10 7 15 \u3bcm2) is perfectly suited for our purposes as some inclusions are smaller than 30-50 \u3bcm and the Fe3+/Fetot variation over the same inclusion cannot be performed by using standard laboratory radioactive sources because of the larger beam size. The average collection time for thicker inclusions (~ 200 \u3bcm) was 2 hours per spectrum, whilst the smallest inclusion (~ 30 730 730 \u3bcm3) required a collection time of approximately 10-12 hours in order to get a spectrum with nicely distinguishable features and a high signal-to-noise ratio. In general, application to a suite of silicate and oxide inclusions in diamonds produced comparable results with respect to those obtained using conventional M\uf6ssbauer sources (e.g. McCammon et al., 2004)

Synchrotron M\uf6ssbauer Source technique for in situ measurement of iron-bearing inclusions in natural diamonds

Natural diamonds containing silicate, oxide and sulfide inclusions are a popular focus of investigation as they uniquely provide a window into the conditions of the Earth\u2019s interior at extreme depths. Recent discoveries based on investigations of deep diamonds have considerably improved our knowledge of the Earth\u2019s deep carbon and water cycles and the oxygen fugacity of the Earth\u2019s interior. Super deep diamonds are those that are believed to have formed at depths of at least 300 km and some evidence suggests depths of at least 800 km. A common inclu- sion in these diamonds is ferropericlase, (Mg,Fe2+)O. Ferropericlase is the second most abundant mineral in the lower mantle, constituting up to about 20 mol% of its volume. The Fe3+/Fetot of ferropericlase is a strong func- tion of oxygen fugacity, and provides a measure of the most recent redox conditions under which it equilibrated. Conventional M\uf6ssbauer spectroscopy using a 57Co point source has been used in the past decades to study the Fe3+/Fetot content in inclusions still trapped in their diamond\u2019s host, however its limitations are the low spatial resolution (not below 3c100 \u3bcm2) and the long acquisition time. The Flank method was also proposed, it is fast, it has high spatial resolution (down to 3c20 \u3bcm2) but it measures the bulk value of Fe3+/Fetot since it cannot distinguish between different phases. An ideal method to measure Fe3+/Fetot values of ferropericlase would com- bine (1) the advantage of M\uf6ssbauer spectroscopy to distinguish Fe3+ in different phases and measure inclusions while still in the diamond, with (2) the advantage of the Flank method to conduct rapid measurements with high spatial resolution. The only method that offers the possibility to satisfy all these requirements is the Synchrotron M\uf6ssbauer Source (SMS). We used the SMS for the first time, to study the iron content and iron distribution in ferropericlase inclusion still contained within its diamond host from Juina (Brazil). This definitive non-destructive technique with extremely high spatial resolution ( 3c15 \u3bcm2) enabled measurements in multiple regions of the 150 7 150 \u3bcm2 inclusion to be sampled and showed that while Fe3+/Fetot values in ferropericlase were below the detection limit (0.02) overall, there was a magnetic component whose abundance varied systematically across the inclusion. Hyperfine parameters of the magnetic component are consistent with magnesioferrite, and the absence of superparamagnetism allows the minimum particle size to be estimated as 3c30 nm. Bulk Fe3+/Fetot values are similar to those reported for other ferropericlase inclusions from Juina. Their variation across the inclusion can provide constraints on its history, and ultimate on the deep carbon processes behind diamonds formation and their exhumation from the transition zone and shallow lower mantle regions