differential operator

We propose a technique that we call HodgeRank for ranking data that may be incomplete and imbalanced, characteristics common in modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as

In the present paper we establish several fuzzy differential subordinations regardind the operator RD!,"
m , given
by RD!,"
m : A # A, RD!,"
m f (z) = (1#")Rm f (z)+"D!
m f (z), where Rm f (z) denote the Ruscheweyh derivative, D!
m f (z) is the
generalized S !
a l
!
a gean operator and A = { f !H(U), f (z) = z + a2z
2 +…, z !U} is the class of normalized analytic
functions. A certain fuzzy class, denoted by RDm
F (!,",# ), of analytic functions in the open unit disc is introduced by
means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties
and characteristics of the class RDm
F (!,",# ). Also, several fuzzy differential subordinations are established regarding
the operator RD!,"
m .

This paper deals with the numerical solution of nonuniform Transmission Lines (TL) with nonlinear loads. The method presented here is based on the combination of wavelet expansion and Finite Elements (FE) in space and time domain. By expanding voltages, currents and operators by means of wavelet functions, the TL equations are transformed into algebraic equations where the differential operator is

We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced -- characteristics almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in score or rating-based cardinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method uses the graph Helmholtzian, the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an analogue of the Laplace operator or scalar Laplacian. We study the graph Helmholtzian using combinatorial Hodge theory: we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the L2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained -- if this is large, then the data does...

We discuss the issues that arise in developing quasi-distributions for arbitrary operators in contrast to the usual case of time and frequency. The arbitrary operator case has many mathematical and physical challenges that have not been solved. We also discuss the connection with differential equations and pseudo-differential operators. In regard to differential equations we argue that the proper generalization of the constant coefficient case is not the variable coefficients case but an equation where the coefficients are kept constant and the differential operator is replaced by a Hermitian operator.

Let L L be an ample line bundle over a complex abelian variety A A . We show that the space of all global sections over A A of Diff A n ⁡ ( L , L ) \operatorname {Diff}^{n}_A(L,L) and S n ( Diff A 1 ⁡ ( L , L ) ) S^n(\operatorname {Diff}^1_A(L,L)) are both of dimension one. Using this it is shown that the moduli space M X M_X of rank one holomorphic connections on a compact Riemann surface X X does not admit any nonconstant algebraic function. On the other hand, M X M_X is biholomorphic to the moduli space of characters of X X , which is an affine variety. So M X M_X is algebraically distinct from the character variety if X X is of genus at least one.