On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain special cases of our general approac

On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard–Saias–Yor equality, and an equality established by one of the authors, are certain special cases of our general approach.Показано як за допомогою узагальненої теореми Лiттлвуда про контурний iнтеграл, що мiстить логарифм аналiтичної функцiї, можна отримати нескiнченну кiлькiсть iнтегральних рiвностей, що мiстять iнтеграли вiд логарифма ζ-функцiї Рiмана i є еквiвалентними гiпотезi Рiмана, i наведено кiлька таких рiвностей у якостi прикладу. Показано, що деякi вiдомi рiвностi такого типу, а саме, рiвностi Ванга, Волчкова, Балазарда – Сайаса – Йора та рiвнiсть, що встановлена одним iз авторiв, є частинними випадками нaшого загального пiдход

The stability and melting of aragonite: An experimental and thermodynamic model for carbonated eclogites in the mantle

Subduction of calcium carbonate, sequestered in the oceanic crust by hydrothermal metamorphism and biogenic action, accounts for a significant flux of carbon into the mantle, where it contributes to the genesis of carbonatitic and silica-undersaturated melts. However, the reported phase relations in the system CaCO3, notably the transition boundary from disordered calcite (calcite V, here ccv) to aragonite (ara), vary considerably among different studies. Moreover, the thermodynamic properties of ccv and of liquid CaCO3 (CaCO3L) remain to be determined. In order to address the dearth of experimental data on phase relations, and to determine a set of internally consistent thermodynamic properties for ara, ccv and CaCO3L, multi-anvil experiments were performed at 3\u20136 GPa and 1300\u20131750 \ub0C. By re-evaluating all experimental data, the transformation of ccv-ara fits the equation Tccv-ara = 397.6 + 320.17 7 P and the melting curve Tm = 1578.9 + 139.65 7 P 12 11.646 7 P2, where pressure is in GPa and temperature in K. Thermodynamic properties retrieved for calcite V and liquid CaCO3 are used to compute phase diagrams of relevance for chemical compositions representative of eclogite heterogeneities of the astenospheric mantle, and compared with experimentally derived phase relationships. Aragonite represents a carbonate of major abundance in carbonated eclogites at high temperature, close to the solidus; its ability to fractionate REE and Ba-Sr contributes to the peculiar geochemical signatures of silica undersaturated magmas. The relatively refractory nature of aragonite impacts on our understanding of the deep carbon cycle

On preconditioning electromagnetic integral equations in the high frequency regime via helmholtz operators and quasi-helmholtz projectors

Fast and accurate resolution of electromagnetic problems via the boundary element method (BEM) is oftentimes challenged by conditioning issues occurring in three distinct regimes: (i) when the frequency decreases and the discretization density remains constant, (ii) when the frequency is kept constant while the discretization is refined and (iii) when the frequency increases along with the discretization density. While satisfactory remedies to the problems arising in regimes (i) and (ii), respectively based on Helmholtz decompositions and Calderon-like techniques have been presented, the last regime is still challenging. In fact, this last regime is plagued by both spurious resonances and ill-conditioning, the former can be tackled via combined field strategies and is not the topic of this work. In this contribution new symmetric scalar and vectorial electric type formulations that remain well-conditioned in all of the aforementioned regimes and that do not require barycentric discretization of the dense electromagnetic potential operators are presented along with a spherical harmonics analysis illustrating their key properties

Intermediate scapolite: behavior at non-ambient conditions and unusual symmetry

The scapolite series of minerals represents a complex non-binary solid solution, which end members are: marialite [Na4Al3Si9O24Cl], meionite [Ca4Al6Si6O24CO3] and silvialite [Ca4Al6Si6O24SO4]. The members which composition falls on the marialite-meionite joint appears to be the most common in natural occurrences [1,2]. The members close to marialite on one side and to meionite on the other side, are usually reported to crystallize in the tetragonal I4/m space group, whereas intermediate scapolites are usually found in the primitive space group P42/n. In this study, we report a scapolite of intermediate composition (Na1.86Ca1.86K0.23Fe0.01)(Al4.36Si7.64)O24[Cl0.48(CO3)0.48(SO4)0.01], which, based on both X-ray and neutron single-crystal diffraction data, shows an anomalous I-centered lattice (Figure 1), possibly due to anti-phase domains too small to be detected by diffraction techniques. The behavior at non-ambient conditions of the same sample has been investigated at high-P (ambient-T) by single-crystal XRD at the former ID09 beamline of ESRF (Grenoble) and at high-T (ambient-P) by powder XRD at the MCX beamline of the Elettra synchrotron (Trieste), providing the following thermodynamic parameters: \uf020\uf062V0 = 0.0143(4) GPa-1 and \u3b1V0 = 1.87(4)\ub710-5 K-1, respectively, which confirm that compressibility and thermal expansivity increase, along the solid solution series, from meionite to marialite [3-6]. A P-induced phase transition towards a triclinic polymorph has been observed at 9.87 GPa at ambient-T. An in situ single-crystal XRD experiment at combined high P and T (using a resistive-heated DAC), performed at the P02.2 beamline of the Petra-III synchrotron (Hamburg), allowed to detect the occurrence of the same phase transition at 10.51 GPa at 650 \ub0C

Single-crystal diffraction at the high-pressure Indo-Italian beamline Xpress at Elettra, Trieste

In this study the first in situ high-pressure single-crystal X-ray diffraction experiments at Xpress, the Indo-Italian beamline of the Elettra synchrotron, Trieste (Italy), are reported. A description of the beamline experimental setup and of the procedures for single-crystal centring, data collection and processing, using diamond anvil cells, are provided. High-pressure experiments on a synthetic crystal of clinoenstatite (MgSiO3), CaCO3 polymorphs and a natural sample of leucophoenicite [Mn7Si3O12(OH)2] validated the suitability of the beamline experimental setup to: (i) locate and characterize pressure-induced phase transitions; (ii) solve ab initio the crystal structure of high-pressure polymorphs; (iii) perform fine structural analyses at the atomic scale as a function of pressure; (iv) disclose complex symmetry and structural features undetected using conventional X-ray sources

Allanite at high pressure : effect of REE on the elastic behaviour of epidote-group minerals

The compressional behaviour of a natural allanite from Lago della Vecchia (upper Cervo valley, Italy) metagranitoids [A1(Ca0.69Fe0.312+)\u3a31.00A2(Ca0.46Ce0.24La0.12Sm0.02Pr0.05Nd0.09Th0.02)\u3a31.00M1(Al0.65Fe0.343+Ti0.02)\u3a31.01M2(Al0.99)M3(Fe0.543+Fe0.362+Mg0.06Ti0.024+Al0.01)\u3a30.99Si1,Si2,Si3(Si2.80Al0.20)\u3a33.00O11(OH,O)] has been investigated up to 16 GPa (at 298 K) by means of in situ synchrotron single-crystal X-ray diffraction. Experiments have been conducted under hydrostatic conditions, using a diamond anvil cell and the mix methanol:ethanol:water = 16:3:1 (up to 10 GPa) and neon (up to 16 GPa) as pressure-transmitting media. No phase transition has been observed within the pressure-range investigated. Data collected in decompression prove that, at least up to 16 GPa (at 298 K), the deformation mechanisms are fully reversible. A third-order Birch\u2013Murnaghan Equation of State (BM-EoS) was fitted to the P\u2013V data (up to 10 GPa), giving: V0 = 470.2(2) \uc53, KP0,T0 = 131(4) GPa and K\u2032= 1.9(8). The evolution of the lattice parameters with pressure shows a slight anisotropic compression pattern, with KP0,T0(a):KP0,T0(b):KP0,T0(c) = 1.24:1.52:1. The monoclinic \u3b2-angle decreases monotonically with pressure, with: \u3b2P(\ub0) = \u3b2P0\u2013 0.0902(4)P (R2 = 0.997, with P in GPa). The main deformation mechanisms at the atomic scale are described based on a series of structure refinements at different pressures. A comparison between the compressional behavior of allanite, epidote and clinozoisite is carried out

On equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain special cases of our general approach

A New Refinement-Free Preconditioner for the Symmetric Formulation in Electroencephalography

Widely employed for the accurate solution of the electroencephalography forward problem, the symmetric formulation gives rise to a first kind, ill-conditioned operator illsuited for complex modelling scenarios. This work presents a novel preconditioning strategy based on an accurate spectral analysis of the operators involved which, differently from other Calderón-based approaches, does not necessitate the barycentric refinement of the primal mesh (i.e., no dual matrix is required). The discretization of the new formulation gives rise to a well-conditioned, symmetric, positive-definite system matrix, which can be efficiently solved via fast iterative techniques. Numerical results for both canonical and realistic head models validate the effectiveness of the proposed formulation