Experimental calculations of number,energy and dose albedos for various materials using 662 keV gamma rays

ABSTRACT

Number, energy and dose albedos are measured at a scattering angle of 180° for a broad beam of 662 keV gamma rays obtained from a radioactive source of ¹³⁷Cs (having strength in µCi; 1 Ci?=?3.7?×?10¹⁰ disintegrations per second). The gamma beam is incident on semi-infinite thick targets of variable atomic numbers. The scattering media is divided into three sets, which are pure elements, alloys and composite materials. Experiments are carried out using a 3^″?×?3^″ NaI(Tl) scintillation detector. To obtain precision in data, the response unfolding of a scintillation detector is used, which converts the observed pulse-height distribution to a true photon spectrum over the energy range of 2.5 to 640 keV. The detector response unfolding results in the true intensity of back-scattered gamma flux by shifting the events resulting from partial absorption of photons to the full energy peak of the spectrum. In the present study, albedo factors are studied as a function of target thickness and their atomic number. The experimentally calculated numbers of back-scattered gamma photon are in good agreement with theoretically generated numbers of multiple back-scattered counts by using a Monte Carlo simulation code. The experimental data on energy and intensity of 662 keV gamma photons are used to evaluate the number, energy and dose albedos for different materials under investigation.     

An averaging method applied to a duffing equation

We approximate a Duffing equation by an averaged system. We solve the system explicitely and draw bifurcation diagrams in dependence of the forcing term. We discuss the goodness of the averaging method, relative to the behavior of the solutions in dependence of involved parameters, by comparing with results obtained in [6,10] for the original Duffing equation.     

ERM learning algorithm for multi-class classification

The multi-class classification problem is considered by an empirical risk minimization (ERM) approach. The hypothesis space for the learning algorithm is taken to be a ball of a Banach space of continuous functions. When the regression function lies in some interpolation space, satisfactory learning rates for the excess misclassification error are provided in terms of covering numbers of the unit ball of the Banach space. A comparison theorem is proved and is used to bound the excess misclassification error by means of the excess generalization error.     

A new parametric method for ranking fuzzy numbers

Ranking fuzzy numbers is important in decision-making, data analysis, artificial intelligence, economic systems and operations research. In this paper, to overcome the limitations of the existing studies and simplify the computational procedures an approach to ranking fuzzy numbers based on α$\alpha$-cuts is proposed. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of existing fuzzy ranking approaches.     

Classroom note: The number c in Cauchy's average value theorem

The location of the number c arising from Cauchy's Average Value Theorem is described when the size of the interval is small.     

Sometimes binary is better

This note uses material involving perfect numbers and Zeno's paradoxes to show that although most students prefer to use base 10 when working with mathematical concepts there are times when the binary system is best.     

Generic Initial Ideals,Graded Betti Numbers,and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are:

1. Let I be an ideal of R = K[x ₁, x ₂,…, x _n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.

2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.     

On the Frobenius Problem of Numerical Semigroups

We define the minimal transversal of a numerical semigroup with respect to one of its elements and use it to calculate the Frobenius number, genus, and the Hilbert function of the semigroup. We give various examples for the use of this method.     

Imaginary Quadratic Fields with Ono Number 3

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ?/2? ⊕ ?/2? if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.