THE DIFFERENTIAL SYSTEMS WITH THE SAME REFLECTING FUNCTION

Using reflecting function of Mironenko we construct some difFerential sy-stems which are equivalent to the given differential system.     

Dupin indicatrices and families of curve congruences

We study a number of natural families of binary differential equations (BDE's) on a smooth surface in . One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE.

More generally, we consider a natural class of BDE's on such a surface , and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

     

Fast convolutions meet Montgomery

Arithmetic in large ring and field extensions is an important problem of symbolic computation, and it consists essentially of the combination of one multiplication and one division in the underlying ring. Methods are known for replacing one division by two short multiplications in the underlying ring, which can be performed essentially by using convolutions.

However, while using school-book multiplication, modular multiplication may be grouped into operations (where denotes the number of operations of one multiplication in the underlying ring), the short multiplication problem is an important obstruction to convolution. It raises the costs in that case to . In this paper we give a method for understanding and bypassing this problem, thus reducing the costs of ring arithmetic to roughly when also using fast convolutions. The algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper.

     

Eisenstein Series in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.     

N.M.R. and I.R. studies of 1, 1, 1 trifluoroacetone in proton donor solvents

Interactions of 1, 1, 1 trifluoroacetone with proton donor solvents of the type RH [where R=HO, CH₃S, CH₃CH₂CH₂CH₂S, CH₃O, (CH₃)₂N, (C₂H₅)₂N] have been investigated using NMR and infra-red techniques. Evidence of the formation of addition products of the type CF₃.C(OH)R.CH₃ where OH group is hydrogen-bonded with the fluorine of the CF₃ group has been obtained.     

Arithmetical and semihereditary semigroup rings

In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ring S (with identity) are found which characterize those semigroup rings R[S] which are reduced or have weak global dimension at most one. Likewise, those semigroup rings R[S] which are semihereditary are completely determined in terms of R and S.     

Cyclic Maximal Ideals of Rings of Differential Operators over Power Series Rings

For an integral domain R, several necessary and sufficient conditions are given for R to be unibranched inside its absolute integral closure; one such condition is that R^pbe Henselian for each prime ideal P of R. Additional conditions are given in case R is a going-down domain. Unlike the situation in the Noetherian context, such going‐down domains R need not be quasilocal or of Krull dimension at most 1. A number of examples are given for the locally pseudo‐valuation domain case     

局部环上正交群中一类子群的扩群

定出了局部环上正交群中一类子群的扩群,得到了如下结果:设R是局部环,M是R的唯一极大理想,O(2m,R)为R上正交群.对R的任意理想S,G(2m,S)表示子群{A BC D∈O(2m,R)|B∈Sm×m}.如果char(R)≠2,m≥3,G(2m,0)≤X≤G(2m,M),那么存在R的理想S,使得X=G(2m,S).     

全局渐近稳定性的Jacobi猜想的证明

设 ｆ∈ Ｃ１（Ｒ２,Ｒ２）,ｊ（０）＝０．设 Ｄｆ（ｘ）为ｆ（ｘ）的 Ｊａｃｏｂｉ矩阵．Ｊａｃｏｂｉ猜想称：如果　ｘ∈Ｒ２,Ｄｆ（ｘ）的特征值都具有负实部,则微分方程ｘ＝ｆ（ｘ）的零解全局渐近稳定．本文证明此猜想成立．     

On the stability and boundedness of solutions of a kind of third order delay differential equations

In this paper, we study Eq. (1.1) for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when      

Universal perturbations of linear differential equations

Let be a fundamental solution of with and bounded on . We prove that there exist arbitrary small matrix functions with limit as such that has solutions with dense in .

     

Existence theorems for perturbed abstract measure differential equations

In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [B.C. Dhage, On some nonlinear alternatives of Leray-Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11-23] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage given in the above-mentioned reference, on ordered Banach spaces. Our existence results include the existence results of Sharma [R.R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972) 503-510], Joshi [S.R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979) 497-506] and Shendge and Joshi [G.R. Shendge, S.R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41 (1983) 53-54] as special cases under weaker continuity condition.     

Simply transitive geodesic ball packings to \mathbf{S}^2\!\times \!\mathbf{R} space groups generated by glide reflections

The \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry can be derived by the direct product of the spherical plane \(\mathbf{S}^2\) and the real line \(\mathbf{R}\) . In (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 42:235–250, 2001), Farkas has classified and given the complete list of the space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . The \(\mathbf{S}^2\!\times \!\mathbf{R}\) manifolds were classified by Molnár and Farkas in [2] by similarity and diffeomorphism. In Szirmai (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 52(2):413–430, 2011), we have studied the geodesic balls and their volumes in \(\mathbf{S}^2\!\times \!\mathbf{R}\) space; moreover, we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . In this paper, we study the locally optimal ball packings to the \(\mathbf{S}^2\!\times \!\mathbf{R}\) space groups having Coxeter point groups, and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The density of the densest packing is \(\approx 0.80407553\) , may be surprising enough in comparison with the Euclidean result \(\frac{\pi }{\sqrt{18}}\approx 0.74048\) . Molnár has shown in (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 38(2):261–288, 1997) that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere \(\mathcal PS ^3(\mathbf{V}^4,\varvec{V}_4,\mathbb R )\) . In our work, we shall use this projective model of \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry.     

S-Cohn–Jordan Extensions

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn–Jordan extension of R. A series of results relating properties of R and that of A(R; S) are presented. In particular it is shown that: (1) A(R; S) is semiprime (prime) iff R is semiprime (prime), provided R is left Noetherian; (2) if R is a semiprime left Goldie ring, then so is A(R; S), Q(A(R; S)) = A(Q(R); S) and udim R = udim A; (3) A(R; S) is semisimple iff R is semisimple, provided R is left Artinian. Some applications to the skew semigroup ring R#S are given.     

關於平均P葉函數

<正> §1.設函數f(z)=在單位圓|z|<1中是正則的;W表示w=f(z)將|z|>1照像到w平面上的黎曼面;以w(R)表示圓|w|≤R所掩蓋W的面積(重叠的黎曼面以重叠的次數計算)。若對任意的R>0,     

The ring of quotients of R(S)

A ring of quotients of the semigroup ring R(S) is discussed where R has a σ-set Σ and S has a σ-set Δ. In particular, we study the cases where (1) R is an integral domain and S is a commutative cancellative semigroup, (2) R is a commutative ring and S is a semilattice and (3) R is a commutative ring and S is a Rees matrix semigroup over a semigroup. Communicated by G. Lallement     

Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form where is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.

     

Measures invariant under the geodesic flow and their projections

Let be the -sphere of constant positive curvature. For , we will show that a measure on the unit tangent bundle of , which is even and invariant under the geodesic flow, is not uniquely determined by its projection to .

     

Gaussian estimates and regularized groups

We show that if a bounded analytic semigroup on satisfies a Gaussian estimate of order and is the generator of its consistent semigroup on , then generates a -regularized group on where . We obtain the estimate of () and the -independence of , and give applications to Schrödinger operators and elliptic operators of higher order.

     

Krull dimension of the enveloping algebra of a semisimple Lie algebra

Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .