A special but common type of scenario is one in which a company has a promotion that is designed to make the customer purchase more of their product than they otherwise might. Although this can be aimed specifically at children, it really applies to all persons. The basic premise is that the company issues a "set" of different items or…

The unified approximations reduce the conceptual complexity by combining solutions for a relatively large number of different situations into just two similar sets of processes. Processes used to solve problems by either the unified or classical approximations require similar degrees of understanding of the underlying chemical processes.

Given a set of urns, each filled with a mix of black chips and white chips, what is the probability of drawing a black chip from the last urn after some sequential random shifts of chips among the urns? The Total Probability Formula (TPF) is the common tool to solve such a problem. However, when the number of urns is more than two and the number…

While waiting for his meal to arrive, a truck driver was using his straw to move water from one glass to another when he was struck by this question: If I keep doing this, will the two glasses ever have exactly the same amount of water? This article looks at various problems related to that question.

A lattice is a (rectangular) grid of points, usually pictured as occurring at the intersections of two orthogonal sets of parallel, equally spaced lines. Polygons that have lattice points as vertices are called lattice polygons. It is clear that lattice polygons come in various shapes and sizes. A very small lattice triangle may cover just 3…

It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-1)])" have the same particular solutions in a suitable region where "f" and "g" are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical…

A factorisation of a general second order ordinary differential equation is introduced from which the full solution to the equation can be obtained by performing two integrations. The method is compared with traditional methods for solving these type of equations. It is shown how the Green's function can be derived directly from the factorisation…

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this…

A method for the numerical evaluation of the error term in Gaussian quadrature rules is derived by means of Chebyshev polynomials of the first kind.

An essential element in successful psychotherapy involves helping clients become more creative in their approach to their problems. While Otto Rank, one of the pioneers of psychoanalysis, loosely alluded to this as a fitting goal, it was the inventor of psychodrama, Jacob L. Moreno, MD (1889-1974), who made this an explicit objective of the…

Solutions-focused role play provides students with structured opportunities to transfer what they have learned to authentic situations they are likely to encounter as peer helpers. Rather than coming up with a single way to solve a problem, students are encouraged to explore multiple possible solutions. Peer training programs have traditionally…

The matrix diagonalization method is used to solve a limit problem.

Biotechnological processes (BTP) are characterized by a complicated structure of organization and interdependent characteristics. Partial differential equations or systems of partial differential equations are used for their behavioural description as objects with distributed parameters. Modelling of substrate without regard to dispersion…

Crisis intervention is a role that fits exceedingly well with counseling psychologists' interests and skills. This article provides an overview of a new crisis intervention model, the Integrated Problem-Solving Model (IPSM), and demonstrates its application to a specific crisis, sexual assault. It is hoped that this article will encourage…

No matter how skilled students may be, those skills will not lead to a job if potential employers do not know the applicant has them. Job seekers--using a strategy of age-old rhetoric--must provide concrete, detailed support to demonstrate the skills they possess. Because many recruiters feel that the best predictor of future performance is past…