This paper proves that the three-dimensional Boussinesqsystem with a nonlinear damping term has a solution that is well-posed. We have established the existence of a weak solution to aregularized Boussinesq system in Sobolev spaces, even if the initialdata has minimal regularity. We have also demonstrated continuousdependence on initial data, which leads to uniqueness. Further-more, we have shown that the solution decays exponentially fastas time goes to infinity. To achieve these results, we used energymethods, the compactness method, the Poincare inequality, and someGro nwall type inequalities. Additionally, we e & curren;ectively used a time-dependent change of function to handle the long-time asymptotic behavior.
Let (G) over cap ((lambda)) be a formal group scheme which deforms (G) over cap (a) to (G) over cap (m). And let psi ((l)) : (G) over cap ((lambda)) -> (G) over cap ((lambda p l )) be the l-th Frobenius-type homomorphism determined by lambda. We show that the homomorphism (psi ((l)))* : H-0(2)((G) over cap(lambda(p l)) , (G) over cap (m)) -> H-0(2) ((G) over cap ((lambda)), (G) over cap (m)) induced by psi((l)) is injective over a Z((p))-algebra under a suitable restriction on lambda. In this situation, the Cartier dual of Ker(psi ((l)) ), which is a finite group scheme of order p (l) , is described over a Z/(p(n) )-algebra.
For the error functions of the form E(r)f(z) = root pi z/2 er f(root z) = z + Sigma+ (infinity)(n=2 ) (-1)(n-1)/ (2n-1)(n-1)! z(n), let ESH & eth;(k, lambda, gamma) represent the class of harmonic error functions E R F = E R H + ERG in the open unit disk U={z is an element of C : | z | < 1 }. The paper attempts to present some basic properties for functions in this class.
This paper is concerned with the estimation of upper bound of the third order Hankel determinant for a unified subclass of analytic functions in the open unit disc E = {z is an element of C : |z| < 1} Also the sharp upper bounds for the first four coefficients, Fekete-Szeg & odblac; inequality, Zalcman inequality and second Hankel determinant, are established. The results are further extended for two-fold and three-fold symmetric functions.
The arctangent of a real rational function may have jumpdiscontinuities (discontinuities of the first kind). By adding an ap-propriate piecewise constant function, we can obtain a real analyticfunction on the whole real line. This real analytic function can beexpressed as a sum of arctangents of real linear functions and aconstant. In other words, an arctangent of a real rational functioncan be decomposed into a real analytic part, expressed as a sum ofarctangents of real linear functions, and a piecewise constant func-tion which expresses pure discontinuities. We prove this statementand show the uniqueness of the expression. We also give severalexamples and another unique expression using a single arctangent ofa rational function and a piecewise constant function, although thissingle arctangent may have discontinuities.
In the present paper, we investigate some initial coefficient bounds on a subclass of bounded turning functions R-cos associated with cosine functions. Also, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these sub-class R-cos. Further, initial consecutive coefficients module difference estimates for the subclass R-cos is studied. Moreover, we determine the upper bound of Zalcman functional for the subclass R-cos for the case n=3, showing that the Zalcman conjecture holds for these values.
In the present paper, we classify SU(2)-equivariant harmonic maps of complex projective line CP & sup1; into complex Grass-mannians Gr(n)(Cn+2) of rank two. To classify them, we use gauge theoretic approach.
In [11] we considered the Cauchy problem for hyperbolic operators with triple characteristics whose coefficients depend only on the time variable. And we gave sufficient conditions for C infinity well-posedness. In this paper we shall show that the sufficient conditions given in [11] are also necessary under additional assumptions.
V. Kac showed that from a local Lie algebra, we canconstruct a graded Lie algebra whose local part is the given local Liealgebra. Later, the author defined the notion of standard pentads.Using this concept, we can construct a local Lie algebra from a givenLie algebra and its representation. Therefore, by using standardpentads, we can construct a graded Lie algebra that has a local partconsisting of a given Lie algebra and its representation under certainassumptions. As special cases of Lie algebras associated with standardpentads, we have the notion of PC Lie algebras. A PC Lie algebra isa graded Lie algebra that is constructed from a finite-dimensionalcommutative Lie algebra and its finite-dimensional completely reduc-ible representation. Our aim in this paper is to show that the class ofPC Lie algebras contains the class of Kac-Moody algebras.
Addressing a question of Shioya, we show that two-step iterations of the Laver collapse can force saturated ideals and Chang conjectures.
A close relationship between the coding theory and the design theory has been studied by many researchers. The principal concern is directed to the designs formed by minimal weight codewords or very small weight codewords. In the present article we study more designs. We extend the concept of the incidence relation, one of which is a classical one and the other is a dual one to the classical one in a certain sense. In the present article we focus on the binary Golay code of length 24. But the idea will be applied to a wide class of self-dual codes such as self-dual extremal binary codes or self-dual extremal ternary codes.
We give explicit parametrizations for all the homogeneous contact Riemannian structures on 3-dimensional Sasakian space forms.
Let V be an affine algebraic variety, and let p is an element of V be a singular point. For a regular function g on V such that g(p) = 0 and for a positive integer n, we consider the cyclic covering Phi(n) : V-n -> V of degree n branched along the hypersurface defined by g. We will prove that for sufficiently large n, the tangent cone of V(n )at Phi(-1)(n) (p) is, as an affine variety, the product of the tangent cone of the branch locus and the affine line. In particular, the multiplicity of the singularity Phi(-1)(n) (p) is an element of V-n, which is a function of n determined by V and g, remains constant for sufficiently large n. This result generalizes Tomaru's theorem for normal surface singularities.
Ollivier and Lin--Lu--Yau established the theory of graph Ricci curvature (LLY curvature) via optimal transport on graphs. Ikeda--Kitabeppu--Takai--Uehara introduced a new distance called the Kantorovich difference on hypergraphs and generalized the LLY curvature to hypergraphs (IKTU curvature). As the LLY curvature can be represented by the graph Laplacian by M\"unch--Wojciechowski, Ikeda--Kitabeppu--Takai--Uehara conjectured that the IKTU curvature has a similar expression in terms of the hypergraph Laplacian. In this paper, we introduce a variant of the Kantorovich difference inspired by the above conjecture and study the Ricci curvature associated with this distance ($\mathsf{wIKTU}$ curvature). Moreover, for hypergraphs with a specific structure, we analyze a quantity $\mathcal{C}(x,y)$ at two distinct vertices $x,y$ defined by using the hypergraph Laplacian. If the resolvent operator converges uniformly to the identity, then $\mathcal{C}(x,y)$ coincides with the $\mathsf{wIKTU}$ curvature along $x,y$.
In 2001, M.Rekos described the analytic behavior for a function $f(z)$ connected with the Euler totient function for Im$z > 0$ (see (1.2)) imitating the previous research of [1] and [3]. In the present paper, for Im$z > 0$ we describe the analytic behavior of the generalized function $f(z,F)$ (see (2.1)), where the function $F$ belongs to the subclass of the Selberg class which has a polynomial Euler product and satisfies some special conditions.
We calculate the Fenchel-Nielsen twist in the enhanced Teichm\"uller space of a marked surface by the cross ratio coordinates.
It is known that there exists an order isomorphism between the Weyl group orbit through a minuscule weight of a simply-laced finite-dimensional simple Lie algebra and the set of all order filters in a self-dual connected d-complete poset. In this paper, we try to extend this fact to the case of multiply-laced finite-dimensional simple Lie algebras by using the "folding" technique with respect to a Dynkin diagram automorphism.
Just as a residue field can be considered for a point of an algebraic variety, we can also consider a residue field for a point of a Berkovich analytic space. This residue field is a valuation field in the algebraic sense. Then we can consider its residue field as a valuation field. We call it the Berkovich double residue field at the point. In this paper, we consider a point x of the Berkovich analytification of an algebraic variety and identify the Berkovich double residue field at x with the union of the residue fields at the center of x in birational models. Besides, we concretely compute the Berkovich double residue field for any quasi monomial valuation.
The order topologies on ordered sets are well-known. For ordered groups or ordered rings, we investigate algebraic order topologies which are compatible with their operations. We consider these topologies on the products of ordered groups, the product extension rings of ordered rings, and also Archimedean ordered groups. We give examples related to algebraic order topologies, etc.
