External Gravitational Field of a Nonstatic Spherically Symmetric Body in the Inertial Frame

We show that the external gravitational field of a nonstatic spherically symmetric source described by a diagonal metric tensor can only be static in the field theory of gravity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 425–432, December, 2005.     

A Centrally Symmetric Version of the Cyclic Polytope

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j<k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least for some c _j (d)>0 and at most as n grows. We show that c ₁(d)≥1−(d−1)⁻¹ and conjecture that the bound is best possible. Research of A. Barvinok partially supported by NSF grant DMS 0400617. Research of I. Novik partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748.     

The Sine Representation of Centrally Symmetric Convex Bodies

The problem of the sine representation for the support function of centrally symmetric convex bodies is studied. We describe a subclass of centrally symmetric convex bodies which is dense in the class of centrally symmetric convex bodies. Also, we obtain an inversion formula for the sine-transform.     

Approximation of Convex Bodies by Centrally Symmetric Bodies

We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let C be a convex body and let D be a centrally symmetric convex body in the Euclidean d-space. We prove that if D is an affine image of D of maximal possible volume contained in C, then C a subset of the homothetic copy of D with the ratio 2d-1 and the homothety center in the center of D. The ratio 2d-1 cannot be lessened as a simple example shows.     

Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates

For a pair of convex bodies $K$ and $K$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K$ are represented as direct sums $K = R \oplus P$ and $K = R \oplus P$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P$ are isothetic parallelotopes, both of dimension $d-m$.     

Quantum Gravitational Effects on the Boundary

Quantum gravitational effects might hold the key to some of the outstanding problems in theoretical physics. We analyze the perturbative quantum effects on the boundary of a gravitational system and the Dirichlet boundary condition imposed at the classical level. Our analysis reveals that for a black hole solution, there is a contradiction between the quantum effects and the Dirichlet boundary condition: the black hole solution of the one-particle-irreducible action no longer satisfies the Dirichlet boundary condition as would be expected without going into details. The analysis also suggests that the tension between the Dirichlet boundary condition and loop effects is connected with a certain mechanism of information storage on the boundary.     

复对称算子的一些等价性质

证明了在复对称算子的前提下,对数-亚正规算子与正规算子是等价的,并且给出了复对称算子的一些等价性质;最后通过给出例子来说明我们的结论.     

Unitary Equivalence of Proper Extensions of a Symmetric Operator and the Weyl Function

Let A be a densely defined simple symmetric operator in ${\mathfrak{H}}$ , let ${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$ be a boundary triplet for A ^* and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A ₀}, uniquely up to the unitary similarity. Here ${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric ${A \subset A^*}$ and a special boundary triplet ${\widetilde{\Pi}}$ for{A, A} such that the corresponding Weyl function is ${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$ , where B is a non-self-adjoint bounded operator in ${\mathcal{H}}$ . We are interested in the problem whether the result on the unitary similarity remains valid for ${\widetilde{M}(\cdot)}$ in place of M(·). We indicate some sufficient conditions in terms of the operators A ₀ and ${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in ${L^2(\mathbb{R}_+)}$ , we show that ${\widetilde{M}(\cdot)}$ defined in ${\Omega_+ \subset \mathbb{C}_+}$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function ${\widetilde{M}(\cdot)}$ does not determine A _B even up to the similarity.     

一类射影平坦的球对称芬斯勒度量

In this paper, we study spherically symmetric Finsler metrics. Analyzing the solution of the projectively flat equation, we construct a new class of projectively flat Finsler metrics.     

一类对偶平坦的球对称的芬斯勒度量

In this paper,we study spherically symmetric Finsler metrics.By analysing the solution of the spherically symmetric dually flat equation,we construct several new families of dually flat spherically symmetric Finsler metrics.     

Equivalence of Many-Gluon Green’s Functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock Statistical Quantum Field Theories

We prove the equivalence of many-gluon Green’s functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock statistical quantum field theories. The proof is based on the functional integral formulation for the statistical generating functional in a finite-temperature quantum field theory. As an illustration, we calculate one-loop polarization operators in both theories and show that their expressions indeed coincide.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 368–374, June, 2005.     

Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $2$ -neighborly $d$ -dimensional polytopes with about $3^{d/2}\approx (1.73)^d$ vertices and of centrally symmetric $k$ -neighborly $d$ -polytopes with about $2^{{3d}/{20k^2 2^k}}$ vertices. Using this result, we construct for a fixed $k\ge 2$ and arbitrarily large $d$ and $N$ , a centrally symmetric $d$ -polytope with $N$ vertices that has at least $\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$ faces of dimension $k-1$ , where $\gamma _2=1/\sqrt{3}\approx 0.58$ and $\gamma _k = 2^{-3/{20k^2 2^k}}$ for $k\ge 3$ . Another application is a construction of a set of $3^{\lfloor d/2 -1\rfloor }-1$ points in $\mathbb R ^d$ every two of which are strictly antipodal as well as a construction of an $n$ -point set (for an arbitrarily large $n$ ) in $\mathbb R ^d$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.     

Quantum Symmetric Pairs and the Reflection Equation

It is shown that central elements in G. Letzter’s quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter’s approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras. Presented by Susan Montgomery.     

Gauge Dependence of the Effective Gravitational Field

We analyze the gauge ambiguity problem for the effective gravitational field from the standpoint of the measurement process. The motion of a test point particle playing the role of a measuring device is investigated in the field of a point gravitating mass in the one-loop approximation. We show that the gravitational field value determined from the effective equations of motion of the device explicitly depends on the Feynman gauge parameter. This dependence is essential in the sense that a gauge variation cannot be interpreted as a deformation of the reference frame, which leads to a gauge ambiguity in the values of observed quantities. In particular, this result disproves the hypothesis that gauge dependence is canceled in the effective equations of motion of a classical point particle.     

Quantum Uncertainty and the Spectra of Symmetric Operators

In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=〈φ,S φ〉, through a function ΔS _t of t, i.e., ΔS[φ]≥ΔS _t , for all physical states φ with 〈φ,S φ〉=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions and the function ΔS _t . We also discuss potential applications in quantum and classical information theory.      

弱重力场中的平面活塞运动

本文分析了弱重力场中等速活塞运动驱动恒星大气的气体动力学过程.在活塞前面,气体被压缩.压缩气体利用其部分内能,以及有些情况下其动能,以克服外加重力.当逃逸速度与等离子体速度之比值为一小参数时,所有量可对小参数展开,基态解给出均匀流,如同没有重力场的气体动力学所讨论的那样.一阶关系给出外加引力场对流场的影响,即激波强度变化不大而气体内能不断耗散.对于强激波和活塞附近,近似得到的分析解有类似的特征.由于外加重力场在天体物理和大气物理过程中的重要性,这些结果对于恒星和行星大气中瞬变过程的机制会有启发.