Reducing a class of two-dimensional integrals to one-dimension with application to Gaussian Transforms

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers times an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions

An entrepreneurial model of economic and environmental co-evolution

A basic tenet of ecological economics is that economic growth and development are ultimately constrained by environmental carrying capacities. It is from this basis that notions of a sustainable economy and of sustainable economic development emerge to undergird the ‘standard model’ of ecological economics. However, the belief in ‘hard’ environmental constraints may be obscuring the important role of the entrepreneur in the coevolution of economic and environmental relations, and hence limiting or distorting the analytic focus of ecological economics and the range of policy options that are considered for sustainable economic development. This paper outlines a co-evolutionary model of the dynamics of economic and ecological systems as connected by entrepreneurial behaviour. We then discuss some of the key analytic and policy implications.

Analytically Continued Hypergeometric Expression of the Incomplete Beta Function

The Incomplete Beta Function is rewritten as a Hypergeometric Function that is the analytic continuation of the conventional form, a generalization of the finite series, which simpifies the Stieltjes transform of powers of a monomial divided by powers of a binomial

Reduced-mass Fock-Tani Representations for a+ + (b+c-) --\u3e (a+c-) + b+ and First-order Results for {abc} = {ppe, epe, μpμ, μdμ, and μtμ}

The Fock-Tani transformation in the Jacobi three ⟶ two-body reduced-mass system is carried out and the first-order T matrix is found to be identical to that for the full three-body transformation. The Fock-Tani transformation in the reduced-mass system in which particle b is fixed at the origin is found to give a first-order T matrix with an error of mc /mb in the initial momentum wave function. First-order differential and total cross sections are calculated for a+ + (b+c-)⟶(a+c-) + b+ where |abc|= {ppe, epe, μpμ, μdμ, and μtμ}

An Evaluation of Stream Flow Characteristics and Fecal Coliform Loads in Sayler\u27s Creek Watershed, South Central Virginia

The Sayler\u27s Creek watershed is located within Prince Edward County, Nottoway, and Amelia Counties of the south central region of Virginia. The Sayler\u27s Creek Watershed consists of two small creeks: Big Sayler\u27s Creek and Little Sayler\u27s Creek. The Environmental Protection Agency has Big Sayler\u27s Creek listed as impaired and not Little Sayler\u27s Creek. Based upon visual inspections of the Sayler\u27s Creek watershed throughout the year, Little Sayler\u27s Creek is inlpaired for fecal coliform instead of Big Sayler\u27s Creek. Another hypothesis of this study is that fecal coliform levels are directly related to runoff from cattle ranches in the immediate floodplain of this river basin. The objectives of this study was to attempt to prove or support that there is a positive relationship between discharge and fecal coliform concentration and to analyze hydrologic conditions and fecal coliform concentrations in order to assist in an accurate determination of fecal coliform loads in Sayler\u27s Creek. Five sites were chosen for the study due to their accessibility. Cross sectional profiles, depths, and current velocities were measured at each site. Hydrologic and Microbiological data was regressed and graphically analyzed to determine the significance of relationships. Within the watershed as a whole.the analyses show a statistically significant relationship between discharge and fecal coliform concentration (p-value = 0.004). Statistical analysis for each site was performed as well to examine the results obtained for the overall watershed. Fecal coliform loads are comparatively higher in Little Sayler\u27s Creek than in Big Sayler\u27s Creek. Within the watershed as a whole, there is evidence to support that fecal coliform concentration is related to stream discharge. Comparatively speaking, Big Sayler\u27s Creek has higher discharge values overall in the four flow stages. However, Little Sayler\u27s Creek has a higher fecal coliform load. The results from this study indicate that both Big Sayler\u27s Creek and Little Sayler\u27s Creek are impaired with respect to fecal coliform standards. More research is needed to accurately determine the level and potential causes of the fecal coliform impairment inthe Sayler\u27s Creek watershed. To determine the extent of impairment, hydrologic characteristics need to be determined. Loads of fecal coliform, instead of concentration, is a necessary determination to better assess the actual amounts of fecal coliform that is being delivered by the Sayler\u27s Creek System. This data can be used to set a priority ranking for the Sayler\u27s Creek Watershed. This method would incorporate hydrologic characteristics into the TMDL decision-making process

Reduced Form for the General-state Multicenter Integral from an Integro-differential Transform

In a previous paper Gaussian transforms were utilized to obtain the analytically reduced form for the class of multicenter integrals containing a product of hydrogenic orbitals for s states, Yukawa or Coulomb potentials, and plane waves. In the present paper a related transformation is developed for nonspherical functions, leading to the reduced form for multicenter integrals that include hydrogenic orbitals representing states of arbitrary angular momentum

Analytically Reduced Form Of Multicenter Integrals From Gaussian Transforms

In a previous paper the analytically reduced form was found for the general class of integrals containing multicenter products of ls hydrogenic orbitals, Coulomb or Yukawa potentials, and plane waves. The method consisted of combining all angular dependence within a single quadratic form by means of a three-dimensional Fourier transform and a one-dimensional Feynman transform for each term in the product and an additional integral transformation to move the resulting denominator into an exponential to be summed with the vector products in the plane waves. This quadratic form was then diagonalized with respect to the (introduced) momentum integrals and diagonalized again with respect to the (original) spatial integrals. In the present paper the four-dimensional Fourier-Feynman transformations are replaced by the one-dimensional Gaussian transformation so that only one diagonalization is required, yielding a simpler reduced form for the integral. The present work also extends the result to include all s states and pairs of states with I f\u27=O summed over the m quantum number

Fourier Transform of the Multicenter Product of 1s Hydrogenic Orbitals and Coulomb or Yukawa Potentials and the Analytically Reduced Form for Subsequent Integrals that Include Plane Waves

The Fourier transform of the multicenter product of N 1s hydrogenic orbitals and M Coulomb or Yukawa potentials is given as an (M+N-1)-dimensional Feynman integral with external momenta and shifted coordinates. This is accomplished through the introduction of an integral transformation, in addition to the standard Feynman transformation for the denominators of the momentum representation of the terms in the product, which moves the resulting denominator into an exponential. This allows the angular dependence of the denominator to be combined with the angular dependence in the plane waves

An Integral Transform for quantum amplitudes

The central impediment to reducing multidimensional integrals of transition amplitudes to analytic form, or at least to a fewer number of integral dimensions, is the presence of magnitudes of coordinate vector differences (square roots of polynomials) |x1−x2|2=x21−2x1x2cosθ+x2 √ in disjoint products of functions. Fourier transforms circumvent this by introducing a three-dimensional momentum integral for each of those products, followed in many cases by another set of integral transforms to move all of the resulting denominators into a single quadratic form in one denominator whose square my be completed. Gaussian transforms introduce a one-dimensional integral for each such product while squaring the square roots of coordinate vector differences and moving them into an exponential. Addition theorems may also be used for this purpose, and sometimes direct integration is even possible. Each method has its strengths and weaknesses. An alternative integral transform to Fourier transforms and Gaussian transforms is derived herein and utilized. A number of consequent integrals of Macdonald functions, hypergeometric functions, and Meijer G-functions with complicated arguments is given

Reasons for Patronage of Traditional Bone Setting as an Alternative to Orthodox Fracture Treatment A case of Muleba District, Kagera Tanzania

The study examined the factors for the preference of Traditional Bone Setting (TBS) in the treatment of fractures among Tanzanians. It sought to unfold other reasons for consulting TBS practitioners besides poverty, ignorance and inaccessibility to modern orthopedic services which are commonly associated with the pull factors. From the available literature, though very popular, TBS is associated with complications like malunion, non-union of the fractured bones, and limb gangrene. In order to find out why there is a paradox, the investigation was mainly done in Muleba, a district of Kagera Region where the treatment is most common according to the Institute of Traditional and Alternative Medicine, at Muhimbili University of Health and Allied Sciences. The study revealed that the therapy management groups were often more vocal than their fractured individuals in deciding the model of treatment. And, the fractured people who are financially able, formally educated and geographically closer to orthopedic services are among the adherents of TBS. Besides, the respondents unanimously expressed their dislike of orthopedic amputation, Plaster of Paris (POP), internal and external fixation let alone the length of time spent in hospital for treatment. All these have significant implications including continued use of TBS by rural and urban people for themselves and livestock. Combining X-ray reading and alternative medicine makes TBS sustainable. Thus, in future, it is suggested TBS services be integrated to orthodox treatment so as to control its negative aspects while harnessing its positive aspects