Weighted Bergman Kernels and Quantization}

Let Ω be a bounded pseudoconvex domain in C ^N , φ, ψ two positive functions on Ω such that − log ψ, − log φ are plurisubharmonic, and z∈Ω a point at which − log φ is smooth and strictly plurisubharmonic. We show that as k→∞, the Bergman kernels with respect to the weights φ ^{k ψ} have an asymptotic expansion

Covariant Perturbation Expansion of Off-Diagonal Heat Kernel

Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.     

New Classes of Off-Diagonal Cosmological Solutions in Einstein Gravity

In this work, we apply the anholonomic deformation method for constructing new classes of anisotropic cosmological solutions in Einstein gravity and/or generalizations with nonholonomic variables. There are analyzed four types of, in general, inhomogeneous metrics, defined with respect to anholonomic frames and their main geometric properties. Such spacetimes contain as particular cases certain conformal and/or frame transforms of the well known Friedman-Robertson-Walker, Bianchi, Kasner and Gödel universes and define a great variety of cosmological models with generic off-diagonal metrics, local anisotropy and inhomogeneity. It is shown that certain nonholonomic gravitational configurations may mimic de Sitter like inflation scenarios and different anisotropic modifications without satisfying any classical false-vacuum equation of state. Finally, we speculate on perspectives when such off-diagonal solutions can be related to dark energy and dark matter problems in modern cosmology.     

Off-Diagonal Long-Range Order (ODLRO) Solitons and Superconductivity

The analogue between monopole-fermion interaction in the spontaneously broken gauge theory and phonon-electron interaction in the theory of superconductivity is studied. It is shown that both monopole-fermion interaction and phonon-electron interaction are characterized by the existence of the off-diagonal long-range order. Starting from the analogue between monopole-fermion interaction and phononelectron interaction the possible relevance of condensed solitons to superconductivity is also discussed.     

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.     

Absence of Localization in One-Dimensional Lattice with Off-Diagonal Nonlinearity Disorder

The time evolution of an initially localized mode in a one-dimensional lattice with off-diagonal nonlinearity disorder is studied by solving a modified nonlinear Schrodinger equation. Absence of localization is found when the nonlinearity parameter is beyond the selftrapping region for corresponding periodic httice. The propagation of electrons shows ballistic behavior even in the presence of disorder. When self-trapping occurs, the untrapped portion still escapes ballistically.     

Propositional Kernels

The pervasive presence of artificial intelligence (AI) in our everyday life has nourished the pursuit of explainable AI. Since the dawn of AI, logic has been widely used to express, in a human-friendly fashion, the internal process that led an (intelligent) system to deliver a specific output. In this paper, we take a step forward in this direction by introducing a novel family of kernels, called Propositional kernels, that construct feature spaces that are easy to interpret. Specifically, Propositional Kernel functions compute the similarity between two binary vectors in a feature space composed of logical propositions of a fixed form. The Propositional kernel framework improves upon the recent Boolean kernel framework by providing more expressive kernels. In addition to the theoretical definitions, we also provide an algorithm (and the source code) to efficiently construct any propositional kernel. An extensive empirical evaluation shows the effectiveness of Propositional kernels on several artificial and benchmark categorical data sets.     

Crystal Melting and Toric Calabi-Yau Manifolds

We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.     

Toric Self-Dual Einstein Metrics as Quotients

We use the quaternion Kähler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov–Gauduchon and Bryant.During the preparation of this work the first and third authors were supported by NSF grant DMS-0203219. The second author was supported by the Leverhulme Trust, the William Gordon Seggie Brown Trust and an EPSRC Advanced Fellowship. The fourth author was supported by the MIUR Project Proprietà Geometriche delle Varietà Reali e Complesse. The authors are also grateful for support from EDGE, Research Training Network HPRN-CT-2000-00101, funded by the European Human Potential Programme.Acknowledgement The first author thanks the Ecole Polytechnique, Palaiseau and the Università di Roma La Sapienza for hospitality and support. The third author would like to thank the Università di Roma La Sapienza, I.N.d.A.M, M.P.I-Bonn, and IHES as parts of this paper were written during his visits there. The fourth named author would like to thank University of New Mexico for hospitality and support. The authors are grateful to Paul Gauduchon, Michael Singer and Pavel Winternitz for invaluable discussions.     

The Nekrasov Conjecture for Toric Surfaces

The Nekrasov conjecture predicts a relation between the partition function for N = 2 supersymmetric Yang–Mills theory and the Seiberg-Witten prepotential. For instantons on \mathbbR⁴{\mathbb{R}^4}, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov and Nakajima-Yoshioka. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.     

On-Diagonal Estimates on Schrödinger Semigroup Kernels and Reduced Heat Kernels

We prove various estimates for the kernels of semigroups generated by Schr?dinger operators with magnetic field and potential of polynomial growth. We also investigate the reduced heat kernels. Received: 25 May 1996 / Accepted: 29 January 1997     

Decay of hypernuclei

We present a nonrelativistic transition potential for the weak strangeness-changing reaction ΛN → NN. The potential is based on a one meson exchange model (OME), where, in addition to the long-ranged pion, the exchange of the pseudoscalar K, η, as well as the vector ,ω, K^* mesons is considered. Results obtained for different hypernuclear decay observables are compared to the available experimental data.     

Toric complete intersections and weighted projective space

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi–Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as special cases of the toric construction. As compared to hypersurfaces, codimension two more than doubles the number of spectra with h¹¹=1. Altogether we find 87 new (mirror pairs of) Hodge data, mainly with h¹¹≤4.     

Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on for each positive integer k.     

SO(4) Symmetry and Off-Diagonal Long-Range Order in the Hubbard Bilayer

Yang's η pairing operator is generalized to explore off-diagonal long-range order in the Hubbard bilayer with an arbitrary chemical potential. With this operator and a constraint condition on annihilation and creation operators, we construct explicitly eigenstates which possess simultaneously three kinds of off-diagonal long-range order, i.e., the intralayer one and the interlayer one for on-site pairing, and that for interlayer nearest-neighbor pairing. As in the simple Hubbard model there is also an SO(4) symmetry, with the generators properly defined. A sufficient condition leads to at least one of the above three kinds of off-diagonal long-range order. A constraint relation among different kinds of off-diagonal long-range order is also given. There exists a triplet of collective modes if the U(1) symmetry of a subgroup is spontaneously broken.     

Representations of Hermitian Kernels¶by Means of Krein Spaces.¶II. Invariant Kernels

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kre?n space. This is motivated by the GNS representation of *-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C ^*-algebras, as well as others. We explain the key role played by the technique of induced Kre?n spaces and a lifting property associated to them. Received: 27 March 2000/ Accepted: 5 September 2000     

Bergman Kernel from Path Integral

We rederive the expansion of the Bergman kernel on Kähler manifolds developed by Tian, Yau, Zelditch, Lu and Catlin, using path integral and perturbation theory, and generalize it to supersymmetric quantum mechanics. One physics interpretation of this result is as an expansion of the projector of wave functions on the lowest Landau level, in the special case that the magnetic field is proportional to the Kähler form. This is relevant for the quantum Hall effect in curved space, and for its higher dimensional generalizations. Other applications include the theory of coherent states, the study of balanced metrics, noncommutative field theory, and a conjecture on metrics in black hole backgrounds discussed in [24]. We give a short overview of these various topics. From a conceptual point of view, this expansion is noteworthy as it is a geometric expansion, somewhat similar to the DeWitt-Seeley-Gilkey et al short time expansion for the heat kernel, but in this case describing the long time limit, without depending on supersymmetry.     

Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds

We present a proof of the mirror conjecture of Aganagic and Vafa (Mirror Symmetry, D-Branes and Counting Holomorphic Discs. http://arxiv.org/abs/hep-th/0012041v1, 2000) and Aganagic et al. (Z Naturforsch A 57(1–2):128, 2002) on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Graber-Zaslow, Contemp Math 310:107–121, 2002), (ii) an outer brane at arbitrary framing in the resolved conifold ${\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)}$ O P 1 ( - 1 ) ⊕ O P 1 ( - 1 ) (Zhou, Open string invariants and mirror curve of the resolved conifold. http://arxiv.org/abs/1001.0447v1 [math.AG], 2010), and (iii) an outer brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Brini, Open topological strings and integrable hierarchies: Remodeling the A-model. http://arxiv.org/abs/1102.0281 [hep-th], 2011).     

Decay properties of baryonia

Decay properties of baryonia are discussed and widths are estimated for L = 1 states. All presently known candidates for baryonia are shown to have properties consistent with those expected. Several novel characteristics which are to be anticipated are pointed out.