Asymptotics of the Airy-Kernel Determinant

The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.     

Periodicity and the Determinant Bundle

The infinite matrix ‘Schwartz’ group G ^−∞ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on G ^−∞. We show that while the higher (even, Schwartz) loop groups of G ^−∞, again classifying for odd K-theory, do not carry multiplicative determinants generating the first Chern class, ‘dressed’ extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding τ invariant is a determinant in this sense. The first author acknowledges the support of the National Science Foundation under grant DMS0408993, the second author acknowledges support of the Fonds québécois sur la nature et les technologies and NSERC while part of this work was conducted.     

The Uniqueness of the World

We follow some wild speculations in trying to understand the uniqueness of our physical world, from the field concept to F-TheoryDedicated to Emilio Santos on his 70th birthday. I have enjoyed discussions with Emilio on physics for the last 40 years; in spite of disagreements on many issues, I have an everlasting admiration for him and for his approach to science     

电象选择的唯一性

讨论了电象选择的唯一性并指出了有关献中能量表达式写法的不妥之处。     

Uniqueness of the BPST instanton

The uniqueness of the BPST instanton solution is established by solving the field equations subject to prescribed boundary conditions, but without imposing a priori restrictions of self-duality on the gauge fields.     

Higher-Degree Analogs of the Determinant Line Bundle

 In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the subspace of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology classes of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem. Received: 27 September 2001 / Accepted: 5 April 2002 Published online: 22 August 2002     

Calculation of a Certain Determinant

The n×n determinant det[(a+j−i)Γ(b+j+i)] is evaluated. This completes the calculation of the Mellin transform of the probability density of the determinant of a random quaternion self-dual matrix taken from the gaussian symplectic ensemble. The inverse Mellin transform then gives the later probability density itself. Received: 30 August 1999 / Accepted: 29 March 2000     

Uniqueness of the Newton-Wigner position operator

It is shown that the quantum position operator of Newton and Wigner for nonzero-mass systems is uniquely determined if one imposes a quantum manifest covariance condition of the same type as the similar condition of Currie, Jordan, and Sudarshan in the framework of the Hamiltonian formalism.     

On the Uniqueness of Gravitational Centre

The dual volume of order α of a convex body A in R ⁿ is a function which assigns to every a ∈ A the mean value of α-power of distances of a from the boundary of A with respect to all directions. We prove that this function is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n (for α = 0 and for α = n the function is constant). It implies that the dual volume of a convex body has the unique minimizer for α > n or α < 0 and has the unique maximizer for 0 < α < n. The gravitational centre of a convex body in R³ coincides with the maximizer of dual volume of order 2, thus it is unique.      

Spectral Determinant of the Two-Photon Quantum Rabi Model

The various generalized spectral determinants (G-functions) of the two-photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the G-function proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. This function is derived rigorously employing the ${\mathbb{Z}}_4$-symmetry of the model and shown that its zeros correspond to the complete regular spectrum.     

Invariance of Jarlskog Determinant Above the GUT Scale

An asymmetry between the probabilities P(ν _μ →ν _e ) and \(P(\bar {\nu _{\mu }}\rightarrow \bar {\nu _{e}})\) would be direct indication of CP violation at the fundamental level. Planck scale effects on neutrino mixing, we have derived the mixing angles of neutrino flavour due to Planck scale effects. It has been shown that Jarlskog determinant remains nearly invariant above the GUT scale.     

The Shapovalov Determinant for the Poisson Superalgebras

Abstract

Among simple ?-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C ₂: these are the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra of special Hamiltonian fields in 2k odd indeterminates, and the Kac–Moody version of . Using C ₂ we compute N. Shapovalov determinant for and , and for the Poisson superalgebras associated with . A. Shapovalov described irreducible finite dimensional representations of and ; we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over and      

Uniqueness of the solution of the Lippmann-Schwinger equation

A few consequences of the uniqueness of the scattering wavefunction in non-break up channel are discussed.     

On the Uniqueness of Ghost-Free Special Gravity

Special gravity refers to interacting theories of massless gravitons in Minkowski space-time which are invariant under the abelian gauge invariance h_(ab) → h_(ab) + ?(aχb)only. In this article we determine the most general form of special gravity free of Ostrogradski ghosts, meaning its equation of motion is of at most second order. Together with the recent works, this result could be helpful in formulating proofs of General Relativity as the unique physical theory of self-interacting massless gravitons. We also study how to construct gauge invariant couplings to matter fields.     

Uniqueness of the standard electroweak gauge model

It is proved that the standard SU(2) × U(1) electroweak gauge model is unique against any extension if the effective low-energy neutral-current interaction is to be precisely of the form $\text{(4G}{}_{\mathrm{F}}\text{/}\text{2}\text{) (j}{}_{\mathrm{\mu }}{}^{\mathrm{(3)}}\text{?}\text{sin}{}^2\text{θ}{}_{\mathrm{<mtext>W</mtext>}}\text{j}{}_{\mathrm{\mu }}{}^{\mathrm{<mtext>em</mtext>}}\text{)}{}^2$naturally.     

Determinant bundles and Virasoro algebras

We consider the interplay of infinite-dimensional Lie algebras of Virasoro type and moduli spaces of curves, suggested by string theory. We will see that the infinitesimal geometry of determinant bundles is governed by Virasoro symmetries. The Mumford forms are just invariants of these symmetries. The representations of Virasoro algebra define (twisted)D-modules on moduli spaces; theseD-modules are equations on correlators in conformal field theory.To the memory of Vadik Knizhnik (20. 2. 1962–25. 12. 1987)