Size and momentum of an infalling particle in the black hole interior

The future interior of black holes in AdS/CFT can be described in terms of a quantum circuit. We investigate boundary quantities detecting properties of this quantum circuit. We discuss relations between operator size, quantum complexity, and the momentum of an infalling particle in the black hole interior. We argue that the trajectory of the infalling particle in the interior close to the horizon is related to the growth of operator size. The notion of size here differs slightly from the size which has previously been related to momentum of exterior particles and provides an interesting generalization. The fact that both exterior and interior momentum are related to operator size growth is a manifestation of complementarity.

k-ORDER FIBONACCI QUATERNIONS

In this paper, we define and study another interesting generalization of the Fibonacci quaternions is called k-order Fibonacci quaternions. Then we obtain for Fibonacci quaternions, for Tribonacci quaternions and for Tetranacci quaternions. We give generating function, the summation formula and some properties about k-order Fibonacci quaternions. Also, we identify and prove the matrix representation for k-order Fibonacci quaternions. The matrix given for k-order Fibonacci numbers is defined for k-order Fibonacci quaternions and after the matrices with the k-order Fibonacci quaternions is obtained with help of auxiliary matrices, important relationships and identities are established.

Note on the Labelled tree graphs

In the CHY-frame for the tree-level amplitudes, the bi-adjoint scalar theory has played a fundamental role because it gives the on-shell Feynman diagrams for all other theories. Recently, an interesting generalization of the bi-adjoint scalar theory has been given in [1] by the “Labelled tree graphs”, which carries a lot of similarity comparing to the bi-adjoint scalar theory. In this note, we have investigated the Labelled tree graphs from two different angels. In the first part of the note, we have shown that we can organize all cubic Feynman diagrams produces by the Labelled tree graphs to the “effective Feynman diagrams”. In the new picture, the pole structure of the whole theory is more manifest. In the second part, we have generalized the action of “picking pole” in the bi-adjoint scalar theory to general CHY-integrands which produce only simple poles.

On Some Inequalities for the Generalized Euclidean Operator Radius

There are many criterion to generalize the concept of numerical radius; one of the most recent interesting generalization is what so-called the generalized Euclidean operator radius. Simply, it is the numerical radius of multivariable operators. In this work, several new inequalities, refinements, and generalizations are established for this kind of numerical radius.

A 2-DIMENSIONAL ALGORITHM RELATED TO THE FAREY–BROCOT SEQUENCE

Moshchevitin and Vielhaber gave an interesting generalization of the Farey–Brocot sequence for dimension d ≥ 2 (see [N. Moshchevitin and M. Vielhaber, Moments for generalized Farey–Brocot partitions, Funct. Approx. Comment. Math.38 (2008), part 2, 131–157]). For dimension d = 2 they investigate two special cases called algorithm [Formula: see text] and algorithm [Formula: see text]. Algorithm [Formula: see text] is related to a proposal of Mönkemeyer and to Selmer algorithm (see [G. Panti, Multidimensional continued fractions and a Minkowski function, Monatsh. Math.154 (2008) 247–264]). However, algorithm [Formula: see text] seems to be related to a new type of 2-dimensional continued fractions. The content of this paper is first to describe such an algorithm and to give some of its ergodic properties. In the second part the dual algorithm is considered which behaves similar to the Parry–Daniels map.

A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

On ideal convergence in probabilistic normed spaces

An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

Mine Rule

Mining of association rules is one of the most adopted techniques for data mining in the most widespread application domains. A great deal of work has been carried out in the last years on the development of efficient algorithms for association rules extraction. Indeed, this problem is a computationally difficult task, known as NP-hard (Calders, 2004), which has been augmented by the fact that normally association rules are being extracted from very large databases. Moreover, in order to increase the relevance and interestingness of obtained results and to reduce the volume of the overall result, constraints on association rules are introduced and must be evaluated (Ng et al.,1998; Srikant et al., 1997). However, in this contribution, we do not focus on the problem of developing efficient algorithms but on the semantic problem behind the extraction of association rules (see Tsur et al. [1998] for an interesting generalization of this problem).

On certain methods of solving a class of integral equations of Fredholm type

The authors begin by presenting a brief survey of the various useful methods of solving certain integral equations of Fredholm type. In particular, they apply the reduction techniques with a view to inverting a class of generalized hypergeometric integral transforms. This is observed to lead to an interesting generalization of the work of E. R. Love [9]. The Mellin transform technique for solving a general Fredholm type integral equation with the familiar H-function in the kernel is also considered.