Symmetric Trace Research Articles

Specialized symmetric functions and trace identities

Specialized symmetric functions and trace identities

A locking-free stabilized embedded discontinuous Galerkin method for linear elasticity with strong symmetric stress and continuous displacement trace approximation

A locking-free stabilized embedded discontinuous Galerkin method for linear elasticity with strong symmetric stress and continuous displacement trace approximation

Recovering the mass distribution of an extended gravitational lens

ABSTRACT We investigate the possibility of determining the mass distribution of a gravitational lens via lensing observations. We consider an extended, compact gravitational lens, representing its static external gravitational potential via an infinite set of symmetric trace free (STF) multipole moments. Within the wave-optical treatment, we evaluate the caustics formed in the lens’s point spread function (PSF). We study the only quantity that is available in astronomical lensing observations: the image of that PSF formed by an imaging telescope. This observable may be used to recover some physical characteristics of the lens, including its shape, orientation, and composition. Illustrating this, we study exotic gravitational lenses formed by several well-known solids with uniform density. We show that when moments beyond the quadrupole are observed, some of the symmetry properties of the lens can be recovered. The presence of an octupole moment implies breaking the ‘north–south’ symmetry of the mass distribution in the lens. The presence of a rotated hexadecapole moment implies breaking axial symmetry. As such, if observations of lensed images allow the reconstruction of these moments, important information about the mass distribution and dynamics of the lens can be obtained. This may help with choosing the most appropriate mass profile that is used to characterize the mass distribution of astrophysical lenses, such as the dark matter haloes that are presumed to contain most of the mass of galaxies and clusters of galaxies. Our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems.

Multipole decomposition of gravitational lensing

We study gravitational lensing by a generic extended mass distribution. For that, we consider the diffraction of electromagnetic (EM) waves by an extended, weakly aspherical, gravitating object. We account for the static gravitational field of such a lens by representing its exterior potential in the most generic form, expressed via an infinite set of symmetric trace free (STF) tensor multipole mass moments. This yields the most general form of the gravitational phase shift, which allows for a comprehensive description of the optical properties of a generic gravitational lens. We found that at each order of the STF moments, the gravitational phase shift is characterized by only two parameters: a magnitude and a rotation angle that characterize the corresponding caustics, which form in the point spread function (PSF) of the lens. Both of these parameters are uniquely expressed in terms of the transverse-trace free (TT) projections of the multipole moments on the lens plane. Not only does this result simplify the development of physically consistent models of realistic lenses, it also drastically reduces the number of required parameters in the ultimate model. To help with the interpretation of the results, we established the correspondence of the gravitational phase shift expressed via the TT-projected STF multipole mass moments and its representation via spherical harmonics. For axisymmetric mass distributions, the new results are consistent with those that we obtained in previous studies. For arbitrary mass distributions, our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems. These findings are discussed in the context of ongoing astrophysical gravitational lensing investigations as well as observations that are planned with the solar gravitational lens (SGL).

Symmetric products, linear representations and trace identities

We give the equations of the n-th symmetric product X^n/S_n of a flat affine scheme X=mathrm {Spec},A over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces.

Magmatic evolution and crustal growth of the Mediterranean region: New geochemical and geochronological perspectives

Zarnitsa kimberlite pipe in Central Yakutia contains pyrope garnets with Cr2O3 ranging from 9 to 19.3 wt% derived from the asthenospheric mantle. They show mostly S-shaped, inflected rare earth element (REE) patterns for dunitic and harzburgitic, lherzolitic and harzburgitic varieties and all are rich in high field strength elements (HFSE) due to reaction with protokimberlite melts. Lithospheric garnets (<9 wt% Cr2O3) show a similar division into four groups but have more symmetric trace element patterns. Cr-diopsides suggest reactions with hydrous alkaline, protokimberlitic and primary (hydrous) partial melts. Cr-diopsides of metasomatic origin have inclined REE patterns and high LILE, U, Th and Zr concentrations. Four groups in REE of Ti-rich Cr-diopsides, and augites have asymmetric bell-like REE patterns and are HFSE-rich. Mg-ilmenites low in REE were formed within dunite conduits. Ilmenite derived from differentiated melts have inclined REE patterns with LREE ~ 100 × chondrite levels. Thermobarometry for dunites shows a 34 mWm−2 geotherm with a HT branch (>50 mWm−2) at 6–9 GPa, and a stepped HT geotherm with heated pyroxenite lenses at four levels from 6.5 to 3.5 GPa. Parental melts calculated with KDs suggest that augites and high-Cr garnets in the lithosphere base reacted with essentially carbonatitic melts while garnets from lower pressure show subduction peaks in U, Ba and Pb. The roots of the Zarnitsa pipe served to transfer large portions of deep (>9 GPa) protokimberlite melts to the lithosphere. Smaller diamonds were dissolved due to the elevated oxidation state but in peripheral zones large diamonds could grow.

Deep mantle roots of the Zarnitsa kimberlite pipe, Siberian craton, Russia: Evidence for multistage polybaric interaction with mantle melts

Zarnitsa kimberlite pipe in Central Yakutia contains pyrope garnets with Cr2O3 ranging from 9 to 19.3 wt% derived from the asthenospheric mantle. They show mostly S-shaped, inflected rare earth element (REE) patterns for dunitic and harzburgitic, lherzolitic and harzburgitic varieties and all are rich in high field strength elements (HFSE) due to reaction with protokimberlite melts. Lithospheric garnets (<9 wt% Cr2O3) show a similar division into four groups but have more symmetric trace element patterns. Cr-diopsides suggest reactions with hydrous alkaline, protokimberlitic and primary (hydrous) partial melts. Cr-diopsides of metasomatic origin have inclined REE patterns and high LILE, U, Th and Zr concentrations. Four groups in REE of Ti-rich Cr-diopsides, and augites have asymmetric bell-like REE patterns and are HFSE-rich. Mg-ilmenites low in REE were formed within dunite conduits. Ilmenite derived from differentiated melts have inclined REE patterns with LREE ~ 100 × chondrite levels. Thermobarometry for dunites shows a 34 mWm−2 geotherm with a HT branch (>50 mWm−2) at 6–9 GPa, and a stepped HT geotherm with heated pyroxenite lenses at four levels from 6.5 to 3.5 GPa. Parental melts calculated with KDs suggest that augites and high-Cr garnets in the lithosphere base reacted with essentially carbonatitic melts while garnets from lower pressure show subduction peaks in U, Ba and Pb. The roots of the Zarnitsa pipe served to transfer large portions of deep (>9 GPa) protokimberlite melts to the lithosphere. Smaller diamonds were dissolved due to the elevated oxidation state but in peripheral zones large diamonds could grow.

A note on symmetric linear forms and traces on the restricted quantum group $\bar U_q(\mathfrak{sl}(2))$

In this paper we prove two results about ${\rm SLF}(\bar U_q)$, the algebra of symmetric linear forms on the restricted quantum group $\bar U_q = \bar U_q\left(\mathfrak{sl}(2)\right)$. First, we express any trace on finite dimensional projective $\bar U_q$-modules as a linear combination in the basis of ${\rm SLF}(\bar U_q)$ constructed by Gainutdinov - Tipunin and also by Arike. In particular, this allows us to determine the symmetric linear form corresponding to the modified trace on projective $\bar U_q$-modules. Second, we give the explicit multiplication rules between symmetric linear forms in this basis.

26-Year-Old Woman With Shortness of Breath and Chest Pain

26-Year-Old Woman With Shortness of Breath and Chest Pain

Hölder inequality for symmetric operator spaces and trace property of K-cycles

Hölder inequality for symmetric operator spaces and trace property of K-cycles

Coincidence invariants and higher Reidemeister traces

The Lefschetz number and fixed point index can be thought of as two different descriptions of the same invariant. The Lefschetz number is algebraic and defined using homology. The index is defined more directly from the topology and is a stable homotopy class. Both the Lefschetz number and index admit generalizations to coincidences and the comparison of these invariants retains its central role. In this paper, we show that the identification of the Lefschetz number and index using formal properties of the symmetric monoidal trace extends to coincidence invariants. This perspective on the coincidence index and Lefschetz number also suggests difficulties for generalizations to a coincidence Reidemeister trace.

The two-body interaction potential in the STF tensor formalism: an application to binary asteroids

The symmetric trace free (STF) tensor formalism, developed by Hartmann et al. (Celest Mech Dyn Astron 60:139–159. doi: 10.1007/BF00693097 , 1994), is a nice tool, not much used in Celestial Mechanics. It is fully equivalent to the usual spherical harmonics but permits more elegant and compact formulations. The coupling between the gravitational fields of extended bodies with this formalism has been used in Mathis and Le Poncin-Lafitte (Astron Astrophys 497:889–910. doi: 10.1051/0004-6361/20079054 , 2009) for binary stars or planetary systems, but not yet applied to binary asteroids. However, binary asteroids are common in the Solar System and usually their study requires a full two rigid body approach. The formulation of the two-body interaction potential in the STF formalism in the full two rigid body problem is detailed and completed in this article. An application to the binary asteroid (66391) 1999 KW4 is presented with a comparison of our results with other results of the literature for validation.

Consistency conditions for gauge theory S matrices from requirements of generalized unitarity.

In the modern on-shell approach, the perturbative S matrix is constructed iteratively using on-shell building blocks with manifest unitarity. As only gauge invariant quantities enter in the intermediate steps, the notion of a gauge anomaly is absent. In this Letter, we rephrase the anomaly cancellation conditions in a purely on-shell language. We demonstrate that while the unitarity methods automatically lead to a unitary S matrix, the rational terms that are required to enforce locality, invariably give rise to inconsistent factorization channels in chiral theories. In four dimensions, the absence of such inconsistencies implies the vanishing of the symmetric trace of three generators. In six dimensions, if the symmetric trace of four generators does not vanish, the rational term develops a factorization channel revealing a new particle in the spectrum: the 2-form of the Green-Schwarz mechanism. Thus, in the purely on-shell construction, the notion of a gauge anomaly is replaced by the difficulty to consistently impose locality on the unitary S matrix.

Dixmier traces are weak⁎ dense in the set of all fully symmetric traces

We extend Dixmier's construction of singular traces (see [2]) to arbitrary fully symmetric operator ideals. In fact, we show that the set of Dixmier traces is weak⁎ dense in the set of all fully symmetric traces (that is, those traces which respect Hardy–Littlewood submajorization). Our results complement and extend earlier work of Wodzicki [22].

On D-brane anti D-brane effective actions and their corrections to all orders in alpha-prime

Based on a four point function, the S-matrix elements at disk level of the scattering amplitude of one closed string Ramond-Ramond field (C) and two tachyons and one scalar field, we find out new couplings in brane anti brane effective actions for p = n, p+2 = n cases. Using the infinite corrections of the vertex of one RR, one gauge and one scalar field and applying the correct expansion, it is investigated in detail how we produce the infinite gauge poles of the amplitude for p = n case. By discovering new higher derivative corrections of two tachyon-two scalar couplings in brane anti brane systems to all orders in α′, we also obtain the infinite scalar poles in (t′+s′+u)-channel in field theory. Working with the complete form of the amplitude with the closed form of the expansion and comparing all the infinite contact terms of this amplitude, we derive several new Wess-Zumino couplings with all their infinite higher derivative corrections in the world volume of brane anti brane systems. In particular, in producing all the infinite scalar poles of < VCVϕVTVT > , one has to consider the fact that scalar's vertex operator in (-1)-picture must carry the internal σ3 Chan-Paton matrix. The symmetric trace effective action has a non-zero coupling between Dϕ(1)i and Dϕ(2)i while this coupling does not exist in ordinary trace effective action.

Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs "noncommutative" traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a "shadow." In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate "cylindrical" type of string diagram, which we justify formally in an appendix.

Surface oxides of Ir(111) prepared by gas-phase oxygen atoms

The Ir(111) surface is oxidized with gas-phase oxygen atoms under vacuum condition to achieve an oxidation level beyond its saturation coverage for chemisorption. Two surface oxides, rutile IrO2 of (100) domain and corundum Ir2O3 of (001) domain, have been grown at 550K with different oxygen exposure of 3.6×105L and 7.2×105L respectively. The temperature programmed desorption (TPD) experiment of rutile IrO2(100) shows its desorption curve (at 4Ks−1) peaks at 750K, followed by a long tail of less pronounced desorption features. On the other hand, TPD of corundum Ir2O3(001) displays a symmetric trace, peaking at 880K. Carbon monoxide titration experiments show that adsorbed CO reduces corundum Ir2O3(001) at 400K, but CO does not adsorb on rutile IrO2(100) and no reduction reaction occurs. Evidently, among the two surface oxides, corundum Ir2O3(001) involves in catalysis of carbon monoxide oxidation, while rutile IrO2(100) does not. The formation of two surface oxides is also compared, we conclude that the atom arrangement favors Ir2O3(001) at the oxide/metal interface.

Multipole expansion at the level of the action

Sources of long wavelength radiation are naturally described by an effective field theory (EFT) which takes the form of a multipole expansion. Its action is given by a derivative expansion where higher order terms are suppressed by powers of the ratio of the size of the source over the wavelength. In order to determine the Wilson coefficients of the EFT, i.e. the multipole moments, one needs the mapping between a linear source term action and the multipole expansion form of the action of the EFT. In this paper we perform the multipole expansion to all orders by Taylor expanding the field in the source term and then decomposing the action into symmetric trace free tensors which form irreducible representations of the rotation group. We work at the level of the action, and we obtain the action to all orders in the multipole expansion and the exact expressions for the multipole moments for a scalar field, electromagnetism and linearized gravity. Our results for the latter two cases are manifestly gauge invariant. We also give expressions for the energy flux and the (gauge dependent) radiation field to all orders in the multipole expansion. The results for linearized gravity are a component of the EFT framework NRGR and will greatly simplify future calculations of gravitational wave observables in the radiation sector of NRGR.

Analysis of PWI footprint traces and material damage on the first walls of the spherical tokamak QUEST

After several non-inductive current startup experimental campaigns in the spherical tokamak QUEST, its metallic first walls have revealed various kinds of damages as a signature of strong plasma wall interaction (PWI). Several types of footprint traces, namely colored regions formed due to material erosion/redeposition, melting of plasma facing components (PFCs) and numerous arc tracks on the chamber walls are recognized. Analysis of the re-deposited materials on collector probes is carried out using X-ray photoelectron spectroscopy (XPS), scanning electron microscopy (SEM) and energy dispersive X-ray spectrometry (EDS). Redeposition of several impurity materials such as carbon, oxygen and tungsten is identified. The footprint traces are majorly formed on the lower side PFCs, showing a large up/down asymmetry. Both toroidally symmetric and asymmetric footprint traces are formed on the bottoms side divertor plate and the lower part of the outboard side walls, respectively. Localized melting occurred on the outboard side limiters is attributed to the loss of energetic electrons produced via electron cyclotron resonance (ECR) heating. The observed damages are discussed in view of localized PWI, loss of energetic electrons, particle drifts, sputtering, arcing and redeposition of eroded materials. Material analysis and numerically calculated guiding center orbits of the charge particles are used to discuss these damages.

Direct One-Pot Synthesis of Primary 4-Amino-2,3-diaryl-quinolines via Suzuki-Miyaura Cross-Coupling of 2-Aryl-4-azido-3-iodoquinolines with Arylboronic Acids

Palladium-catalyzed Suzuki-Miyaura cross-coupling of 2-aryl-4-azido-3-iodo-quinolines with arylboronic acids afforded the corresponding primary 4-amino-2,3-diarylquinolines in a single-pot operation along with symmetrical biaryls and traces of the 2,3-diaryl-4-azidoquinolines. A plausible mechanism, which implicates palladium hydride species in the reduction of the incipient 2,3-diaryl-4-azidoquinolines to afford the 4-amino-2,3-diarylquinolines is proposed.