Wreath Hurwitz numbers, colored cut-and-join equations, and 2-Toda hierarchy

Let G be arbitrary finite group,define H G· (t;p +,p) to be the generating function of G-wreath double Hurwitz numbers.We prove that H G· (t;p +,p) satisfies a differential equation called the colored cutand-join equation.Furthermore,H G·(t;p +,p) is a product of several copies of tau functions of the 2-Toda hierarchy,in independent variables.These generalize the corresponding results for ordinary Hurwitz numbers.     

Supercyclic sequences of differential operators

Summary Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff--MacLane--Godefroy--Shapiro's theorems. A new, general eigenvalue criterium for supercyclicity is also provided.     

A formula about W-operator and its application to Hurwitz number

$W$-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the $W$-operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the $W$-operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between $W$-operator $W\left(\left[d\right]\right)$ and the central elements $K_{1^{n?d}d}$ in $?S_n$. Based on the relation we find, we give another proof about a differential equation of the generating function of $d$-Hurwitz number.     

Invertibility of linear differential operators with closed range and poisson coefficients

The invertibility and injectivity properties of linear differential operators with closed range and Poisson coefficients are studied in the context of their equivalence in several spaces of vector functions defined on the axis. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 143–147, January, 1999.     

Commutators of generalized Hardy operators

LetH_a,b be the commutator generated by the generalized Hardy operator and the CMO function. The (L^p, L^p) boundedness of H_a,b is discussed in this paper. Furthermore, the authors consider the boundedness of H_a,b on the weighted homogeneous Herz spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fixed points of generalized e-concave (generalized e-convex) operators and their applications

In this paper, we present the definitions of generalized e-concave operators and generalized e-convex operators, which are the generalizations of e-concave operators and e-convex operators, respectively. Without compactness or continuity assumption of generalized e-concave operators and generalized e-convex operators, we have proved the existence, uniqueness and monotone iterative techniques of their fixed points. Our results are even new to e-concave operators and e-convex operators. Finally, we apply the results to the singular boundary value problems for second order differential equations.     

A hierarchy of generalized invariants for linear partial differential operators

We study invariants of linear partial differential operators in two variables under gauge transformations. Using the Beals-Kartashova factorization, we construct a hierarchy of generalized invariants for operators of an arbitrary order. We study the properties of these invariants and give some examples. We also show that the classic Laplace invariants correspond to some particular cases of generalized invariants. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 470–478, June, 2006.     

The nonquadratic imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic fields of -power degrees with class numbers equal to their genus class numbers.

     

Anisotropic torsional creep problems involving rapidly growing differential operators

The asymptotic behaviour of the sequence of positive solutions for a family of torsional creep-type problems involving anisotropic rapidly growing differential operators is studied in a bounded domain from the Euclidian space ${\mathbb{R}}^N$. We prove that the sequence of solutions converges uniformly on the domain to a certain distance function defined in accordance with the anisotropy of the problem.     

Unique continuation properties for generalized Baouendi-Grushin operators with singular weights

In this paper, properties of the spherical functions and Hardy-Sobolev inequalities of generalized Baouendi-Grushin vector fields are established, and then some unique continuation results for generalized Baouendi-Grushin operators with singular weights are given.     

Generating functions, Fibonacci numbers and rational knots

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots of given crossing number depending on genus and/or signature. This allows to determine the asymptotical average value of these invariants among rational knots. We give also an application to the enumeration of lens spaces.     

A criterion for the logarithmic differential operators to be generated by vector fields

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give

• a sufficient criterion for the property,
• a simple proof of F.J. Calderón-Moreno's theorem that free divisors have the property,
• a proof that divisors in dimension with only isolated quasi-homogeneous singularities have the property,
• an example of a nonfree divisor with nonisolated singularity having the property,
• an example of a divisor not having the property, and
• an algorithm to compute the V-filtration along a divisor up to a given order.

     

Asymptotic behaviours and generalized Toeplitz operators

We study some generalized Toeplitz operators associated to operators T on a Hilbert space H, for which there exists the limit of {‖Tnh‖} for every h∈H. We refer to the asymptotic limit ST of such a T, in the sense of [L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997) 571-605; L. Kerchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged) 68 (2002) 373-400; G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems, J. Operator Theory 53 (1) (2005) 101-140; C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997], and we give some conditions of ergodicity for T. Also, certain results of Douglas [R.G. Douglas, On the operator equation S^∗XT=X and related topics, Acta Sci. Math. (Szeged) 30 (1969) 19-32] involving generalized Toeplitz operators are extended in our more general setting, and we apply these results to ρ-contractions.     

Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making

Trapezoidal intuitionistic fuzzy numbers (TrIFNs) is a special intuitionistic fuzzy set on a real number set. TrIFNs are useful to deal with ill-known quantities in decision data and decision making problems themselves. The focus of this paper is on multi-attribute group decision making (MAGDM) problems in which the attribute values are expressed with TrIFNs, which are solved by developing a new decision method based on power average operators of TrIFNs. The new operation laws for TrIFNs are given. From a viewpoint of Hausdorff metric, the Hamming and Euclidean distances between TrIFNs are defined. Hereby the power average operator of real numbers is extended to four kinds of power average operators of TrIFNs, involving the power average operator of TrIFNs, the weighted power average operator of TrIFNs, the power ordered weighted average operator of TrIFNs, and the power hybrid average operator of TrIFNs. In the proposed group decision method, the individual overall evaluation values of alternatives are generated by using the power average operator of TrIFNs. Applying the hybrid average operator of TrIFNs, the individual overall evaluation values of alternatives are then integrated into the collective ones, which are used to rank the alternatives. The example analysis shows the practicality and effectiveness of the proposed method.     

Certain family of integral operators preserving subordination and superordination

We obtain subordination, superordination and sandwich-preserving new theorems for certain integral operators defined on the space of normalized analytic functions in the open unit disk. The sandwich-type theorem for these integral operators is also derived, and the results generalize some recently ones.     

Hyponormality of Toeplitz operators with generalized circulant symbols

In this paper we obtain a necessary and sufficient condition for Toeplitz operators with generalized circulant symbols to be hyponormal.     

Star-critical Ramsey numbers for large generalized fans and books

In this paper, we show that for any fixed integers $m\ge 2$ and $t\ge 2$, the star-critical Ramsey number $r_1(K_1+n{K}_t,K_{m+1})=(m?1)tn+t$ for all sufficiently large $n$. Furthermore, for any fixed integers $p\ge 2$ and $m\ge 2$, $r_1(K_p+n{K}_1,K_{m+1})=(m?1+o\left(1\right))n$ as $n\to \infty$.