Algebraic Bethe Ansatz for a seven-vertex model

The paper is devoted to construction of algebraic Bethe Ansatz for a seven-vertex model. R-matrix of the system is obtained by means of twist from the six-vertex model considered by us earlier. The presence of the seventh nonzero element in the R-matrix complicates the situation. In particular, commutation relations of elements of the monodromy matrix become more complicated in comparison with the six-vertex model. We construct algebraic Bethe Ansatz by introducing a new operator that is the difference of two operators on the main diagonal of the monodromy matrix. The eigenstates and spectrum of the system are found. This is the first step on the way of comparison of systems with six-and seven-vertex R-matrices, respectively. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 178–186.     

Algebraic bethe anzatz and the tavis-cummings model

The N-atom-radiation-field model is solved for its eigenvalues and eigenstates by the algebraic Bethe ansatz. The eigenenergies and the amplitudes of the wave functions are expressed in terms of the solutions of the Bethe equations. The determinant representations of the expectation values and the norms of the wave functions are obtained. The algebraic approach establishes a relationship between the model under consideration and the other exactly solvable models. Bibliography:12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 66–79. Translated by N. M. Bogolyubov.     

Estimate of the remainder of a cubature formula for a hypersphere

An estimate is given for the remainder term of a cubature formula of special type for calculating an integral over an n-dimensional sphere. The algebraic degree of precision of the formula is the highest among formulas of this type and is equal to 4p-1. Appearing in the estimate is an upper bound of the absolute values of all the partial derivatives of the integrand function of order 4p in the domain of integration.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 627–632, November, 1969.     

Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the announced proposition. We dually characterize the associated submodel-injection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs (Boolean algebras with operators). The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL.     

Spaces for which the first uncountable ordinal space is a remainder

A remainder of a locally compact, non-compact Hausdorff spaceX is anyX – X whereX is a Hausdorff compactification ofX. LetK(X) be the lattice of compactifications ofX. Conditions onK(X) and an internal condition are obtained which characterize when the first uncountable ordinal space is a remainder ofX.     

Estimate for the remainder in the Tate formula

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 149–151, November, 1991.     

Improvement of remainder term for the divisors problem

New bounds are obtained for the remainder term in the divisors problem.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 545–554, November, 1969.     

Sharp Hardy-Leray inequality for curl-free fields with a remainder term

In this paper, we give a new and a simpler approach to the result in [8] concerning the best constant of Hardy-Leray inequality for curl-free fields. As a by-product, we obtain an improved inequality with a remainder term. The non-attainability of the best constant is an easy consequence of the new inequality. The proof is based on a decomposition of curl-free fields into radial and spherical parts.     

The remainder method for sample percentiles

A method called the Remainder Method is proposed for the calculation of sample quantiles of a given order, for example, quartiles, hexatiles, octatiles, deciles and percentiles assuming that all the observations are distinct. Proof is given for a special case of deciles. The criterion ‘equisegmentation’ is proposed, namely that the number of observations below the first quantile, that between the consecutive quantiles, and that above the last quantile are the same. The formulae for quantiles offered by the proposed method satisfy the equisegmentation property, and more interestingly provide the number of quantiles having integer ranks. Some open problems are indicated.     

On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality

where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,

where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:

and the corresponding eigenfunction spaces are

     

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder R _n,m (f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r ₁, r ₂) defined by a generalized shift operator. Here, r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in the O-term depends only on ??.     

On a calculus for a generalized scalar operator

Let $\text{X}$ be a Banach space; S and T bounded scalar-type operators in $\text{X}$. Define Δ on the space of bounded operators on $\text{X}$ by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on $\text{X}$.     

一类算子矩阵特征值的代数指标和代数重数

自伴算子特征值的几何重数与代数重数相等,但对于非自伴算子不一定成立,这主要是特征值的代数指标起着决定性的作用.讨论了一类非自伴算子矩阵特征值的几何重数,代数指标与代数重数.