On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].

Properties of translation operator and the solution of the eigenvalue and boundary value problems of arbitrary space–time periodic metamaterials

There is a recent interest in understanding and exploiting the intriguing properties of space–time metamaterials. In the current manuscript, the time periodic circuit theory is exploited to introduce an appropriate translation operator that fully describes arbitrary space–time metamaterials. It is shown that the underlying mathematical machinery is identical to the one used in the analysis of linear time invariant periodic structures, where time and space eigen-decompositions are successively employed. We prove some useful properties the translation operator exhibits. The wave propagation inside the space time periodic metamaterial and the terminal characteristics can be rigorously determined via the expansion in the operators eigenvectors (space–time Bloch waves). Two examples are provided that demonstrate how to apply the framework. In the first, a space time modulated composite right left handed transmission line is studied and results are verified via time domain computations. Furthermore, we apply the theory to explain the non-reciprocal behaviour observed on a nonlinear transmission line manufactured in our lab. Bloch-waves are computed from the extracted circuit parameters. Results predicted using the developed machinery agree with both measurements and time domain analysis. Although the analysis was carried out for electric circuits, the approach is valid for different domains such as acoustic and elastic media.

Conformal partial waves in momentum space

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension d \geq 3d≥3 (including non-integer dd). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in d = 4d=4 dimensions.

Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type

<p style='text-indent:20px;'>We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (<i>SIAM J. Appl. Dyn. Syst. 18</i>, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The cytosolic diffusion is typically much larger than the lateral diffusion on the membrane. This motivates to a corresponding asymptotic reduction, which consists of a nonlocal system on the membrane. We prove the convergence of solutions of the full system towards unique solutions of the reduction.</p>

Scattering equations in AdS: scalar correlators in arbitrary dimensions

We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two cur- rent algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems

We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.

Noether generators and the Klein–Gordon potential on spaces with nonzero Weyl tensor

In this study, we investigate the potential functions [Formula: see text] that appear in the Lagrangian [Formula: see text], where [Formula: see text] is the metric of an arbitrary space. Our approach is based on a connection between the conformal Killing group associated with [Formula: see text] and the Noether symmetries of [Formula: see text]. To this effect, we select certain spaces that are characterized by a nonzero Weyl tensor.

Hypergeometric presentation for one-loop contributing to H → Z γ

In this paper, new analytic formulas for one-loop contributing to Higgs decay channel $H \rightarrow Z\gamma$ are presented in terms of hypergeometric functions. The calculations are performed by following the technique for tensor one-loop reduction developed in [A. I. Davydychev, Phys. Lett. B 263 (1991) 107]. For the first time, one-loop form factors for the decay process are shown which are valid at arbitrary space–time dimension $d$.

DEVELOPMENT OF A MATHEMATICAL MODEL OF BEHAVIOR OF AIR FLOWS IN THE TERRITORY OF OPEN PORT COAL WAREHOUSES

The article discusses the need to use mathematical modeling to study the behavior of air flows in the territory of a port open coal warehouse. To study these processes of the interaction of the movement of air masses on the territory of the port with coal particles, the movement of a point volume of air in an arbitrary space of the air flow, which is inextricable, was considered. This approach allows to take into account all meteorological features of blowing more accurately: wind speed, altitude distribution of speeds, flow turbulization, various geometric parameters of the warehouse. As a result of the numerical implementation of this model by the control volume method, we get the opportunity to analyze the structure of the stream flowing around the coal warehouse and draw conclusions about the nature of dust formation, separation of coal dust particles, displacement of coal dust particles.