Determinant Det Research Articles

The Ergo Region of the Kerr Black Hole in the Isotropic Coordinate

The isotropic coordinate is the more physically meaningful coordinate in the Schwarzschild black hole. Then we apply this isotropic coordinate to the Kerr black hole, and we have found the ergo region does not appear and all metrices gμν become regular even at the “event horizon” in this coordinate. But the determinant det gμν becomes zero at the “event horizon”, which means that gμν becomes singular at the “event horizon”.

Determinants of further development and management of modern global maritime containerization (Redefinition after Covid-19 and War in Ukraine)

Maritime containerized shipping has recently suffered from various problems and challenges caused by stochastic shocks such as the Covid -19 pandemic or the war in Ukraine. The pandemic and recent war have clearly indicated shortfalls and weaknesses of maritime containerization and have triggered various attempts at identifying the potential for its further development determined by the new reality (after global shocks and with the consideration of new ones). The paper presents a holistic analysis of factors that globally affect the potential for further development of containerized maritime transport. The entire research design comes as an attempt made at identifying and updating the determinants and predictions of prospects for development of containerization in the current geopolitical situation and global trade conditions in the post-pandemic era, also taking the aspects of the ongoing Russian invasion in Ukraine and the post-war recovery into consideration. The results of the research are presented as a matrix of the eight main determinants Det 1-8 (each supported by two key factors F1, F2) that have been originally identified. Each factor is described by an originally developed calculation methodology (as an Index). The indexes are calculated for the period of 2018-2020 and subsequently discussed. Moreover, there is a potential for further utilization of the presented matrix as a universal tool that will define the future aspects of management in the field of the expected architecture of global supply chains and maritime transport of containerized cargo after the absorption of global shocks.

On the deformed Pearcey determinant

In this paper, we are concerned with the deformed Pearcey determinant det⁡(I−γKs,ρPe), where 0≤γ<1 and Ks,ρPe stands for the trace class operator acting on L2(−s,s) with the classical Pearcey kernel arising from random matrix theory. This determinant corresponds to the gap probability for the Pearcey process after thinning, which means each particle in the Pearcey process is removed independently with probability 1−γ. We establish an integral representation of the deformed Pearcey determinant involving the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations. Together with some remarkable differential identities for the Hamiltonian, this allows us to obtain the large gap asymptotics, including the exact calculation of the constant term, which complements our previous work on the undeformed case (i.e., γ=1). It comes out that the deformed Pearcey determinant exhibits a significantly different asymptotic behavior from the undeformed case, which suggests a transition will occur as the parameter γ varies. As an application of our results, we obtain the asymptotics for the expectation and variance of the counting function for the Pearcey process, and a central limit theorem as well.

A double Sylvester determinant

Given two $\left( n+1\right) \times\left( n+1\right)$-matrices $A$ and $B$ over a commutative ring, and some $k\in\left\{ 0,1,\ldots,n\right\}$, we consider the $\dbinom{n}{k}\times\dbinom{n}{k}$-matrix $W$ whose entries are $\left( k+1\right) \times\left( k+1\right)$-minors of $A$ multiplied by corresponding $\left( k+1\right) \times\left( k+1\right)$-minors of $B$. Here we require the minors to use the last row and the last column (which is why we obtain an $\dbinom{n}{k}\times\dbinom{n}{k}$-matrix, not an $\dbinom{n+1}{k+1}\times\dbinom{n+1}{k+1}$-matrix). We prove that the determinant $\det W$ is a multiple of $\det A$ if the $\left( n+1,n+1\right)$-th entry of $B$ is $0$. Furthermore, if the $\left( n+1,n+1\right)$-th entries of both $A$ and $B$ are $0$, then $\det W$ is a multiple of $\left( \det A\right) \left( \det B\right)$. This extends a previous result of Olver and the author ( arXiv:1802.02900 ).

Quantum Gravity Microstates from Fredholm Determinants.

Various two dimensional quantum gravity theories of Jackiw-Teitelboim (JT) form have descriptions as random matrix models. Such models, treated nonperturbatively, can give an explicit and tractable description of the underlying "microstate" degrees of freedom, which play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool, a Fredholm determinant det(1-K), which can be defined explicitly for a wide variety of JT gravity theories. To illustrate the methods, the statistics of the first several energy levels of a nonperturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy F_{Q}(T) of the system is computed for the first time. These results are also of relevance to quantum properties of black holes in higher dimensions.

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields TXpoly on X, and is generated as a TXpoly-algebra by the sum of the determinants det⁡(NXg) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants det⁡(NXg), which requires (for the cup product) a nontrivial normalization missing in previous literature.

Twisted Lefschetz numbers of infra-solvmanifolds and algebraic groups

Twisted Lefschetz numbers are extensions of the ordinary Lefschetz numbers for cohomologies with values in flat bundles. As a generalization of linearization formula for the ordinary Lefschetz number of a diffeomorphism of a nilmanifold, we show that a twisted Lefschetz number of any diffeomorphism of any infra-solvmanifold is equal to the determinant det(I−A) for some matrix A by using the cohomology of algebraic groups.

Invariance properties of Wronskian type determinants of classical and classical discrete orthogonal polynomials

Given a finite set F={f1,⋯,fk} of nonnegative integers (written in increasing order of magnitude) and a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant det⁡(pfi(x+j−1))i,j=1,⋯,k. In this paper we prove an invariance property of this kind of Casorati determinants when the set F is substituted by the set I(F)={0,1,2,⋯,max⁡F}∖{max⁡F−f:f∈F}. Our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators (Krall discrete polynomials). These polynomials are orthogonal with respect to certain Christoffel transforms of the classical discrete measures. By passing to the limit, this invariance property is extended to Wronskian type determinants whose entries are Hermite, Laguerre and Jacobi polynomials.

On the number of integer points in a multidimensional domain

Abstract We provide a new upper estimate for the modulus of the difference |Λ ∩ 𝓢| − voln(𝓢)/det Λ, where 𝓢 ⊂ ℝn is a set of volume voln(𝓢) and Λ ⊂ ℝn is a complete lattice with determinant det Λ. This result has an important practical application, for example, in estimating the number of integer solutions of an arbitrary system of linear and nonlinear inequalities.

Addendum to “Estimates of the determinant of a perturbed identity matrix” [Linear Algebra Appl. 558 (2018) 101–107

Many bounds for the determinant det⁡(I+E) of a perturbed identity matrix are known. Mostly, the upper bound is inferior to the classical Hadamard bound. In this note we give simple and efficiently computable relative bounds differing by ‖E‖F3, where the upper bound is usually better than Hadamard's bound.

The Toverline{T} deformation of quantum field theory as random geometry

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant det T of the stress tensor, commonly referred to as Toverline{T} . Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.

Berkovich spectra of elements in Banach rings

Extending the notion of the spectrum Σf of an element f in an ultrametric Banach algebra over a complete valued field (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum σR,SBer(a) of an element a in a unital Banach algebra R over a commutative unital Banach ring S (in particular, when R is a Banach ring). This spectrum is a compact subset of the affine analytic line AS1 over S, and the latter can be identified with the suitable equivalence classes of all elements in all complete valued fields that are Banach S-algebras. If R is generated by a as a unital Banach S-algebra, then σR,SBer(a) coincides with the spectrum M(R) of R. If R is a unital complex Banach algebra but we regard it as a Banach ring only, then σR,Z1Ber(a) is the quotient of the ordinary spectrum σRC(a) by complex conjugation.For a non-Archimedean complete valued field k and an infinite dimensional ultrametric Banach k-space E with an orthogonal basis, if u∈L(E) is a completely continuous operator, we show that several different ways to define the spectrum of u give the same compact set σL(E),kBer(u). As an application, we give a relation between σL(E),kBer(u) and zeros of the Fredholm determinant det⁡(1−tu) (as defined by Serre) in complete valued field extensions of k.

Black Box Galois representations

We develop methods to study 2-dimensional 2-adic Galois representations ρ of the absolute Galois group of a number field K, unramified outside a known finite set of primes S of K, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on K and S, we show how to determine the determinant det⁡ρ, whether or not ρ is residually reducible, and further information about the size of the isogeny graph of ρ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for K=Q, and for K imaginary quadratic, ρ being the representation attached to a Bianchi modular form.These results form part of the first author's thesis [2].

Differentiating the pseudo determinant

A class of derivatives is defined for the pseudo determinant Det(A) of a Hermitian matrix A. This class is shown to be non-empty and to have a unique, canonical member ∇Det(A)=Det(A)A+, where A+ is the Moore–Penrose pseudo inverse. The classic identity for the gradient of the determinant is thus reproduced. Examples are provided, including the maximum likelihood problem for the rank-deficient covariance matrix of the degenerate multivariate Gaussian distribution.

Transition asymptotics for the Painlevé II transcendent

We consider real-valued solutions u=u(x|s), x∈R, of the second Painlevé equation uxx=xu+2u3 which are parameterized in terms of the monodromy data s≡(s1,s2,s3)⊂C3 of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as x→−∞, between the oscillatory power-like decay asymptotics for |s1|<1 (Ablowitz–Segur) to the power-like growth behavior for |s1|=1 (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for |s1|>1 (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant det(I−γKAi)|L2(x,∞) in a double scaling limit x→−∞, γ↑1, as well as asymptotics for the spectrum of KAi.

Fast Direct Methods for Gaussian Processes.

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the n-dimensional setting, however, it requires the inversion of an n ×n covariance matrix, C, as well as the evaluation of its determinant, det(C). In many cases, such as regression using Gaussian processes, the covariance matrix is of the form C = σ(2) I + K, where K is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix C is typically dense, causing standard direct methods for inversion and determinant evaluation to require O(n(3)) work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix C can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an O (n log(2) n) algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant det(C), permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining K. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.

The asymptotics of an eigenfunction-correlation determinant for Dirac-δ perturbations

We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a 3-dimensional ball BL of radius L in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-δ perturbation. More precisely, we show the algebraic decay of the correlation determinant det〈φjL,ψkL〉j,k=1,…,N2=L−ζ(E)+o(1), as N, L → ∞ and N/BL→ρ&amp;gt;0, where φjL and ψkL denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift ζ(E)≔1π2δ2(E). For an attractive Dirac-δ perturbation, we conclude that the decay exponent 1π2arcsinT(E)/2HS2 found in the work of Gebert et al. [J. Spect. Theory (to be published); e-print arXiv:1407.2512] does not provide a sharp upper bound on the decay of the correlation determinant.

Asymptotics of a cubic sine kernel determinant

We study the one parameter family of Fredholm determinants det(I Kcsin), 2 R of an integrable Fredholm operator Kcsin acting on the interval ( s,s) whose kernel is a cubic generalization of the sine kernel which appears in random matrix theory. This Fredholm determinant appears in the description of the Fermi distribution of semiclassical non-equilibrium Fermi states in condensed matter physics as well as in random matrix theory. Using the Riemann-Hilbert method, we calculate the large s-asymptotics of det(I Kcsin) for all values of the real parameter .

$$R$$ R -Matrix Realization of Two-Parameter Quantum Group $$U_{r,s}(\mathfrak {gl}_n)$$ U r , s ( gl n )

We provide a Faddeev-Reshetikhin-Takhtajan's RTT approach to the quantum group Fun(GL_{r,s}(n)) and the quantum enveloping algebra U_{r,s}(gl_n) corresponding to the two-parameter R-matrix. We prove that the quantum determinant det_{r,s}T is a quasi-central element in Fun(GL_{r,s}(n)) generalizing earlier results of Dipper-Donkin and Du-Parshall-Wang. The explicit formulation provides an interpretation of the deforming parameters, and the quantized algebra U_{r,s}(R) is identified to U_{r,s}(gl_n) as the dual algebra. We then construct n-1 quasi-central elements in U_{r,s}(R) which are analogues of higher Casimir elements in U_q(gl_n).

Inverses, determinants, eigenvalues, and eigenvectors of real symmetric Toeplitz matrices with linearly increasing entries

We explicitly determine the skew-symmetric eigenvectors and corresponding eigenvalues of the real symmetric Toeplitz matricesT=T(a,b,n):=(a+b|j−k|)1≤j,k≤n of order n≥3 where a,b∈R, b≠0. The matrix T is singular if and only if c:=ab=−n−12. In this case we also explicitly determine the symmetric eigenvectors and corresponding eigenvalues of T. If T is regular, we explicitly compute the inverse T−1, the determinant det⁡T, and the symmetric eigenvectors and corresponding eigenvalues of T are described in terms of the roots of the real self-inversive polynomial pn(δ;z):=(zn+1−δzn−δz+1)/(z+1) if n is even, and pn(δ;z):=zn+1−δzn−δz+1 if n is odd, δ:=1+2/(2c+n−1).