Partial isometries, duality, and determinantal point processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space [Formula: see text] with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel [Formula: see text] called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces [Formula: see text], where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

The Classical and Quantum Photon Field for Non-compact Manifolds with Boundary and in Possibly Inhomogeneous Media

In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by zero-modes that appear in the presence of non-trivial topology of the manifold or the boundary. An important special case is $${\mathbb {R}}^3$$ R 3 with obstacles. In this case the zero modes have a direct interpretation in terms of the topology of the obstacle. I give a formula for the renormalised stress energy tensor in terms of an integral kernel of an operator defined by spectral calculus of the Laplace Beltrami operator on differential forms with relative boundary conditions.

Commutatively deformed general relativity: foundations, cosmology, and experimental tests

An integral kernel representation for the commutative $$\star $$ ⋆ -product on curved classical spacetime is introduced. Its convergence conditions and relationship to a Drin’feld differential twist are established. A $$\star $$ ⋆ -Einstein field equation can be obtained, provided the matter-based twist’s vector generators are fixed to self-consistent values during the variation in order to maintain $$\star $$ ⋆ -associativity. Variations not of this type are non-viable as classical field theories. $$\star $$ ⋆ -Gauge theory on such a spacetime can be developed using $$\star $$ ⋆ -Ehresmann connections. While the theory preserves Lorentz invariance and background independence, the standard ADM $$3+1$$ 3 + 1 decomposition of 4-diffs in general relativity breaks down, leading to different $$\star $$ ⋆ -constraints. No photon or graviton ghosts are found on $$\star $$ ⋆ -Minkowski spacetime. $$\star $$ ⋆ -Friedmann equations are derived and solved for $$\star $$ ⋆ -FLRW cosmologies. Big Bang Nucleosynthesis restricts expressions for the twist generators. Allowed generators can be constructed which account for dark matter as arising from a twist producing non-standard model matter field. The theory also provides a robust qualitative explanation for the matter-antimatter asymmetry of the observable Universe. Particle exchange quantum statistics encounters thresholded modifications due to violations of the cluster decomposition principle on the nonlocality length scale $$\sim 10^{3-5} \,L_P$$ ∼ 10 3 - 5 L P . Precision Hughes–Drever measurements of spacetime anisotropy appear as the most promising experimental route to test deformed general relativity.

Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt.

Time–Frequency Analysis in $$\mathbb {R}^n$$

A time–frequency transform is a sesquilinear mapping from a suitable family of test functions to functions on the time–frequency plane. The goal is to quantify the energy present in the signal at any given time and frequency. The transform is further specified by imposing conditions such as those stemming from basic transformations of signals and those which an energy density should satisfy. We present a systematic study on how properties of a time–frequency transform are reflected in the associated evaluation at time–frequency origin, integral kernel and quantization and discuss some examples of time–frequency transforms.

An epidemic model integrating direct and fomite transmission as well as household structure applied to COVID-19

This paper stresses its base contribution on a new SIR-type model including direct and fomite transmission as well as the effect of distinct household structures. The model derivation is modulated by several mechanistic processes inherent from typical airborne diseases. The notion of minimum contact radius is included in the direct transmission, facilitating the arguments on physical distancing. As fomite transmission heavily relates to former-trace of sneezes, the vector field of the system naturally contains an integral kernel with time delay indicating the contribution of undetected and non-quarantined asymptomatic cases in accumulating the historical contamination of surfaces. We then increase the complexity by including the different transmission routines within and between households. For airborne diseases, within-household interactions play a significant role in the propagation of the disease rendering countrywide effect. Two steps were taken to include the effect of household structure. The first step subdivides the entire compartments (susceptible, exposed, asymptomatic, symptomatic, recovered, death) into the household level and different infection rates for the direct transmission within and between households were distinguished. Under predefined conditions and assumptions, the governing system on household level can be raised to the community level. The second step then raises the governing system to the country level, where the final state variables estimate the total individuals from all compartments in the country. Two key attributes related to the household structure (number of local households and number of household members) effectively classify countries to be of low or high risk in terms of effective disease propagation. The basic reproductive number is calculated and its biological meaning is invoked properly. The numerical methods for solving the DIDE-system and the parameter estimation problem were mentioned. Our optimal model solutions are in quite good agreement with datasets of COVID-19 active cases and related deaths from Germany and Sri Lanka in early infection, allowing us to hypothesize several unobservable situations in the two countries. Focusing on extending minimum contact radius and reducing the intensity of individual activities, we were able to synthesize the key parameters telling what to practice.

Generalized Weyl Quantization and Time

This work presents quantization of time of arrival functions using generalized Stratonovich-Weyl quantization. We take into account the ordering problems involved, mainly the Born-Jordan and the symmetric ordering schemes. We call attention to the combination of the group theoretic methods usually employed in Weyl quantization with the implementation of different ordering schemes via integral kernel factors. It is possible to, and we do, apply the Pegg-Barnett method to the quantization of time to address physical issues such as boundedness and self-adjointness.

Variable order, directional ℋ2-matrices for Helmholtz problems with complex frequency

The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence, their sparse approximation is of outstanding importance. In our paper, we will generalize the directional $\mathcal{H}^{2}$-matrix techniques from the ‘pure’ Helmholtz operator $\mathcal{L}u=-\varDelta u+\zeta ^{2}u$ with $\zeta =-\operatorname *{i}k$, $k\in \mathbb{R}$ to general complex frequencies $\zeta \in \mathbb{C}$ with $\operatorname{Re}\zeta\geq0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition that contains $\operatorname{Re}\zeta $ in an explicit way, and introduces the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We develop an error analysis that is explicit with respect to the expansion order and with respect to $\operatorname{Re}\zeta $ and $\operatorname{Im}\zeta $. This allows for choosing the variable expansion order in a quasi-optimal way, depending on $\operatorname{Re}\zeta $, but independent of, possibly large, $\operatorname{Im}\zeta $. The complexity analysis is explicit with respect to $\operatorname{Re}\zeta $ and $\operatorname{Im}\zeta $, and shows how higher values of $\operatorname{Re} \zeta $ reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.

Integral operators, bispectrality and growth of Fourier algebras

In the mid 1980s it was conjectured that every bispectral meromorphic function {\psi(x,y)} gives rise to an integral operator {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions {\psi(x,y)} where the commuting differential operator is of order {\leq 6}. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.