Optimal strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties

Locally finite, congruence meet-semidistributive varieties have been characterized by numerous Mal’cev conditions and, recently, by two strong Mal’cev conditions. We provide three new strong Mal’cev characterizations and a new Mal’cev characterization each of which improves the known ones in some way.     

A strong Mal’cev condition for locally finite varieties omitting the unary type

We show that a finite algebra has a Taylor operation if and only if it has an operation satisfying a particular set of 6-ary Taylor identities. As a consequence we get the first strong Mal’cev condition for the family of locally finite varieties omitting the unary type. This is of interest to combinatorialists, as it is conjectured that a Constraint Satisfaction Problem defined by a core relational structure is polynomial time solvable exactly when a certain associated variety omits the unary type. Our result implies that the problem of deciding if a core relational structure meets this characterisation is itself in NP.     

Strong Mal’tsev conditions in varieties of Abelian groups

Let Ab_d be a variety of Abelian groups of a finite exponent d≥1 and SC (Ab_d) be the set of all strong Mal’tsev conditions satisfied in Ab_d. We define the concept of a η-basis in SC(Ab_d) in terms of a basis w.r.t. a class η of varieties with commutative operations. The algorithm for constructing η-bases of any finite length in SC(Ab_d) is presented. For the variety Ab of all Abelian groups, we specify absolute bases of length 2 in SC(Ab) which are simultaneously η-bases. Bases of length 2 with similar properties are constructed also in SC(Ab_d), for any natural number d≥2. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 723–742, November–December, 1999.     

Congruence Distributivity in Goursat and Mal’cev Categories

We characterize the congruence distributive property for the Goursat and regular Malcev categories in terms of preservation of the intersection by direct images. This splits a previous characterization for the exact Malcev categories in two cases: plain congruence distributive property and weak congruence distributive property.Mathematics Subject Classifications (2000) Primary: 08B10, 08C05, 18C05; secondary: 08A30, 18D05.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

Optimal Sobolev estimates for ¯∂ on convex domains of finite type

We prove that solutions for ¯ get 1/M-derivatives more than the data in L^p-Sobolev spaces on a bounded convex domain of finite type M by means of the integral kernel method. Also we prove that the Bergman projection is invariant under the L^p-Sobolev spaces of fractional orders by different methods from McNeal-Stein's. By using these results, we can get L^p-Sobolev estimates of order 1/M for the canonical solution for ¯. The author was supported by grant No. R01-2000-000-00001-0 from the Basic Research Program of the Korea Science&Engineering Foundation.     

Local finiteness for Green’s relations in semigroup varieties

A semigroup variety is said to be locally 𝒦-finite, where 𝒦 stands for any of Green’s relations ?, ?, ?, 𝒟, or 𝒥, if every finitely generated semigroup in this variety has only finitely many 𝒦-classes. We characterize locally 𝒦-finite varieties of finite axiomatic rank in the language of “forbidden objects”.     

Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations

Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several conditions on u ₀ to prove the existence of a unique strong solution with u(0) = u ₀ in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given u ₀, q, s as above and the Stokes operator A ₂, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions.     

Optimal Tauberian constant in Ingham’s theorem for Laplace transforms

It is well known that there is an absolute constant \(\mathfrak{C}\) > 0 such that if the Laplace transform \(G(s) = \int_0^\infty {\rho (x)} {e^{ - sx}}dx\) of a bounded function ρ has analytic continuation through every point of the segment (?iλ, iλ) of the imaginary axis, then

$$G(s) = |\int_0^\infty {\rho (\mu )} du - G(0)| \leqslant \frac{{\text{C}}}{\lambda }\mathop {limsup}\limits_{x \to \infty } |\rho (x)|$$

The best known value of the constant \(\mathfrak{C}\) was so far \(\mathfrak{C}\) = 2. In this article we show that the inequality holds with \(\mathfrak{C}\) = π/2 and that this value is best possible. We also sharpen Tauberian constants in finite forms of other related complex Tauberian theorems for Laplace transforms.

     

Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings

We define closed subvarieties of some Deligne–Lusztig varieties for GL(2) over finite rings and study their ´etale cohomology. As a result, we show that cuspidal representations appear in it. Such closed varieties are studied in [Lus2] in a special case. We can do the same things for a Deligne–Lusztig variety associated to a quaternion division algebra over a non-archimedean local field. A product of such varieties can be regarded as an affine bundle over a curve. The base curve appears as an open subscheme of a union of irreducible components of the stable reduction of the Lubin–Tate curve in a special case. Finally, we state some conjecture on a part of the stable reduction using the above varieties. This is an attempt to understand bad reduction of Lubin–Tate curves via Deligne–Lusztig varieties.     

Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M, J) of real dimension four. We construct a local peak J-plurisubharmonic function at every point p ∈ bD of finite D’Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D’Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.     

Optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h~2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.     

Necessary and sufficient conditions for local strong solvability of the Navier–Stokes equations in exterior domains

We consider the instationary Navier–Stokes equations in a smooth exterior domain \({\Omega \subseteq \mathbb{R}^3}\) with initial value u ₀, external force f = div  F and viscosity ν. It is an important question to characterize the class of initial values \({u_0\in L^2_{\sigma}(\Omega)}\) that allow a strong solution \({u \in L^s(0,T; L^q(\Omega))}\) in some interval \({[0,T[ \, , 0 < T \leq \infty}\) where s, q with 3 < q < ∞ and \({\frac{2}{s} + \frac{3}{q} =1}\) are so-called Serrin exponents. In Farwig and Komo (Analysis (Munich) 33:101–119, 2013) it is proved that \({\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t < \infty}\) is necessary and sufficient for the existence of a strong solution \({u \in L^s(0,T ; L^q(\Omega)) \, , 0 < T \leq \infty}\) , if additionally 3 < q ≤ 8; here, A denotes the Stokes operator. In this paper, we will show that this result remains true if q > 8, and consequently, \({\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t < \infty}\) is the optimal initial value condition to obtain such a strong solution for all possible Serrin exponents s, q.     

Asymptotic solution of the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type

With the help of the dressing procedure the singular Riemann problem corresponding to the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type is reduced to a regular Riemann problem. From the asymptotic analysis of the regular Riemann problem we get the principal term of the asymptotic solution of the Cauchy problem in the domain of superpolynomial decrease, which is described in terms of the scattering data corresponding to the initial condition.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 181–195, 1986.The author sincerely thanks A. R. Its for many helpful discussions.