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A three-dimensional integral containing the kernel g(x, y, z)su,v(z) is derived. The function g(x, y, z) is a generalized function containing the logarithmic and exponential functions and su,v(z) is the Lommel function and the integral is taken over the cube 0 ≤ y ≤ ∞, 0 ≤ x ≤∞, 0 ≤ z ≤ ∞. A representation in terms of the Lerch function is derived, from which special cases can be evaluated. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distribution. All the results in this work are new.

A triple integral involving the Bessel-integral function Jiv(z) is derived and evaluated for certain real numbers of the parameters. The derived integral allows a representation in terms of the product of the Hurwitz-Lerch zeta and Gamma functions with seven parameters. All the results in this work are new.

A closed form expression of a quadruple integral involving the Legerndre polynomial Pn(x) is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.

The known exact expression for an octuple integral relating to research in the fields of mathematics and physics is summarized. A new closed form expression for this integral is given in terms of the Lerch function.

The main focus of the present paper is to establish a triple integral involving the Struve function in terms of the Hurwitz-Lerch zeta function by using our contour integral method. We also consider some useful examples as special cases of this integral involving the Struve function to give the applications of our main results.

The aim of the present document is to evaluate a triple integral involving a general class of logarithmic, special function and exponential function. Importance of our resultss lies in the fact that they involve the Bessel function of the First Kind, whichis used in a wide range of areas spanning Science and Engineering. Further we establish some special cases

The aim of the present document is to evaluate a quadruple integral involving the product of the generalized Parabolic Cylinder functions D v ( β x ) D u ( α z ) D_{v}(\beta x)D_{u}(\alpha z) expressed in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

Bone loss, commonly seen in osteoporosis, is a condition that entails a progressive decline of bone mineral density and microarchitecture, often seen in post-menopausal women. Bone loss has been widely reported in astronauts exposed to a plethora of stressors and in patients with osteoporosis following radiotherapy for cancer. Studies on mechanisms are well documented but the causal connectivity of events to bone loss development remains incompletely understood. Herein, the adverse outcome pathway (AOP) framework was used to organize data and develop a qualitative AOP beginning from deposition of energy (the molecular initiating event) to bone loss (the adverse outcome). A literature review was conducted to compile and evaluate the state of knowledge based on the modified Bradford Hill criteria. Following review of 1865 studies, an empirically supported AOP was developed, showing the progression to bone loss through many factors affecting the activities of bone-forming osteoblasts and bone-resorbing osteoclasts. The structural, functional, and quantitative basis of each proposed relationship was defined, for inference of causal changes between key events. Current knowledge and its gaps relating to dose-, time- and incidence-concordance across the key events were identified, as well as modulating factors that influence linkages. The new priorities for research informed by the AOP highlight areas for improvement to enable development of a quantitative AOP used to support risk assessment strategies for space travel or cancer radiotherapy.

A closed form expression of a quadruple integral involving the Hermite polynomial $H_{n}(x)$ is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.

We derive the infinite sum of the Lerch function in terms of the incomplete gamma function and the Lerch function. Special cases are evaluated in terms of fundamentals constants. All the results in this work are new.

Closed expressions using the Lerch function for a definite integral are derived and evaluated. Some of these closed expressions are given in Gradshteyn and Ryzhik. Some special cases of the integral are derived and discussed. The majority of the results in this work are new.

A quadruple integral involving the logarithmic, exponential, polynomial and Gamma functions is derived in terms of the Hurwitz-Lerch zeta function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

Carl Johan Malmsten (1846) and David Beirens de Haan (1847) published work containing some interesting integrals. While no formal derivations of the integrals in his book Nouvelles Tables d'Integrales Defines are available in current literature deriving and evaluating such formulae are useful in all aspects of science and engineering whenever such formulae are used. Formulae in the book of Bierens de Haan are used in connection with certain potential problems where there is the need to determine the vector potential of two parallel, infinitely long, tubular rectangular conductors carrying cur-rents in opposite directions. In this current work we supply formal derivations for some of these integrals along with deriving some special cases as new integrals in order to expand upon the book of Bierens de haan to aid in potential research where these formulae are applicable. Updating book of integrals is always a useful exercise as it keeps the volume accurate and more useful for potential readers and researchers. Formal derivations are also useful as they help in verifying the correctness of integrals in such volumes. The definite integral we derived in this work is given by (1) in terms of the Lerch function, where the parameters a; k; m; and p are general complex numbers subject to their restrictions. This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.

The aim of this paper is to provide a table of definite integrals which includes both known and new integrals. This work is important because we provide a formal derivation for integrals in [7] not currently present in literature along with new integrals. By deriving new integrals we hope to expand the current list of integral formulae which could assist in research where applicable. The authors apply their contour integral method [9] to an integral in [8] to achieve this new integral formula in terms of the Lerch function. In this present work, the authors provide a formal derivation for an interesting Exponential Fourier transform and express it in terms of the Lerch function. The Exponential Fourier transform has many real world applications namely, in the field of Electrical engineering, in the work of electrical transients by [10] and in the field of Civil engineering, in the work of stress analysis of boundary load on soil by [11]. The definite integral we derived in this work is given by (1) where the variables . This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.

This article addresses the emergence of children and young people as simultaneously sexual/gender non-normative and political subjects during the 2016 Safe Schools controversy. What appeared as a s...

It is always useful to improve the catalogue of definite integrals available in tables. In this paper we use our previous work on Lobachevsky integrals to derive entries in the tables by Bierens De Haan and Anatolli Prudnikov featuring errata results and new integral formula for interested readers. In this work we derive a definite integral given by (1) in terms of the Lerch function. The importance of this work lies in the derivation of known and new results not presently found in current literature. We used our contour integral method and applied it to an integral in Prudnikov and used it to derive a closed form solution in terms of a special function. The advantage of using a special function is the added benefit of analytic continuation which widens the range of computation of the parameters. Special functions have significance in mathematical analysis, functional analysis, geometry, physics, and other applications. Special functions are used in the solutions of differential equations or integrals of elementary functions. Special functions are linked to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.

Objectives. The purpose of this study is to present data from the clinical commissioning of an Xstrahl 150 x-ray unit used for superficial radiotherapy, Methods. Commissioning tasks included vendor acceptance tests, timer reproducibility, linearity and end-effect measurements, half-value layer (HVL) measurements, inverse square law verification, head-leakage measurements, and beam output calibration. In addition, percent depth dose (PDD) curves were determined for different combinations of filter/kV settings and applicators. Automated PDD water phantom scans were performed utilizing four contemporary detectors: a microDiamond detector, a microSilicon detector, an EDGE detector, and a PinPoint ionization chamber. The measured PDD data were compared to the published values in BJR Supplement 25, Results. The x-ray unit’s mechanical, safety, and radiation characteristics were within vendor-stated specifications. Across sixty commissioned x-ray beams, the PDDs determined in water using solid state detectors were in excellent agreement with the BJR 25 data. For the lower (2 collimator was (−0.7 ± 1.0)%, Conclusions. This study presents machine characterization data collected for clinical use of a superficial x-ray unit. Special focus was placed on utilizing contemporary detectors and techniques for the relative PDD measurements using a motorized water phantom. The results in this study confirm that the aggregate values published in the BJR 25 report still serve as a valid benchmark when comparing data from site-specific measurements, or the reference data for clinical utilization without such measurements, Advances in knowledge. This paper presents comprehensive data from the acceptance and commissioning of a modern kilovoltage superficial x-ray radiotherapy machine. Comparisons between the PDD data measured in this study using different detectors and BJR 25 data are highlighted.

In 1974 Fritz Oberhettinger published the Tables of Mellin Transforms. We derive the correct result for one of these integrals along with some interesting special cases.

Closed expressions for the integral \begin{document}$ \begin{equation*} \int_{0}^{\infty}\frac{x^{m-1} \log ^k(a x)}{\left(x^{2 u}+1\right) \left(x^{3 u}+1\right)}dx \end{equation*} $\end{document} are given where the variables $ a $, $ k $, $ m $ and $ u $ are general complex numbers. Some of these closed expressions are given in [4]. Some special cases of the integral are derived and discussed.

A new genus and species of rhinatrematid caecilian, Amazops amazops gen. et sp. nov., is described based on a single specimen from Orellana, Ecuador collected in 1990. Among other features the new taxon differs from all other rhinatrematid caecilians in having less than four annular grooves interrupted in the region of the vent and in the squamosal contributing to the bony margin of the orbit. A consideration of its distinctive morphology suggests that it is plausible that the new taxon is the sister taxon of all other rhinatrematid caecilians. That the genus is known from a single specimen, and that this is the first new rhinatrematid species from the Andes described for more than 50 years, highlights the poor sampling (collecting) of rhinatrematid caecilians and limited knowledge of their diversity.

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

We apply experimental data to identify contexts in which Russian aspect is categorically determined (‘categorical contexts’) and contexts in which both aspects can appear (‘overlapping contexts’). Traditional analyses suggest that these contexts are distinct and that certain features are typical for each type of context. Such analyses tend to rely on constructed examples typically consisting of a single sentence. Our experiment, by contrast, is based on extensive (over 1100 words each) authentic texts (created by and for native Russians), and the identification of contexts is not a priori, but emerges from contexts normed by over 500 native speakers. We present two main findings: 1) the distinction between categorical contexts and overlapping contexts is scalar, not discrete; and 2) a multitude of factors distinguish the two types of contexts, modality in particular is prominent in overlapping contexts, a factor that has received little prior attention. Our data both confirms and significantly extends previous analyses by presenting empirically justified factors that distinguish categorical from overlapping contexts.

The relationship between construal and redundancy has not been previously explored empirically. Russian aspect allows speakers to construe situations as either Perfective or Imperfective, but it is not clear to what extent aspect is determined by context and therefore redundant. We investigate the relationship between redundancy and open construal by surveying 501 native Russian speakers who rated the acceptability of both Perfective and Imperfective verb forms in complete extensive authentic contexts. We find that aspect is largely redundant in 81% of uses, and in 17% of contexts aspect is relatively open to construal. We contend that anchoring in redundant contexts likely facilitates the independence of construal in contexts with less redundancy. However further research is needed to discover what makes contexts redundant since known cues for aspect are absent in the majority of such contexts. Native speakers are fairly consistent in giving the original aspect high ratings, but less consistent in rating the non-original aspect, indicating potential problems in testing the reactions of speakers to non-authentic data.

Previous descriptions of the composition and stability of children’s households have focused on the presence of parents and the stability of mothers’ marital and cohabiting relationships. We use data available in the 2008 Survey of Income and Program Participation to expand the description of children’s household composition and stability. We find that one in five children lives with nonnuclear household members. These other household members are a source of substantial household instability. In addition, during the period of observation (2008–2013), children experienced considerable residential instability. Thus, children’s experience of household instability is much more common and frequent than previously documented. Moreover, levels of both residential and compositional instability are higher for children with less-educated mothers and for racial/ethnic minorities.

The aim of this study was to report a single‐institution experience and commissioning data for Elekta VersaHD linear accelerators (LINACs) for photon beams in the Eclipse treatment planning system (TPS). Two VersaHD LINACs equipped with 160‐leaf collimators were commissioned. For each energy, the percent‐depth‐dose (PDD) curves, beam profiles, output factors, leaf transmission factors and dosimetric leaf gaps (DLGs) were acquired in accordance with the AAPM task group reports No. 45 and No. 106 and the vendor‐supplied documents. The measured data were imported into Eclipse TPS to build a VersaHD beam model. The model was validated by creating treatment plans spanning over the full‐spectrum of treatment sites and techniques used in our clinic. The quality assurance measurements were performed using MatriXX, ionization chamber, and radiochromic film. The DLG values were iteratively adjusted to optimize the agreement between planned and measured doses. Mobius, an independent LINAC logfile‐based quality assurance tool, was also commissioned both for routine intensity‐modulated radiation therapy (IMRT) QA and as a secondary check for the Eclipse VersaHD model. The Eclipse‐generated VersaHD model was in excellent agreement with the measured PDD curves and beam profiles. The measured leaf transmission factors were less than 0.5% for all energies. The model validation study yielded absolute point dose agreement between ionization chamber measurements and Eclipse within ±4% for all cases. The comparison between Mobius and Eclipse, and between Mobius and ionization chamber measurements lead to absolute point dose agreement within ±5%. The corresponding 3D dose distributions evaluated with 3%global/2mm gamma criteria resulted in larger than 90% passing rates for all plans. The Eclipse TPS can model VersaHD LINACs with clinically acceptable accuracy. The model validation study and comparisons with Mobius demonstrated that the modeling of VersaHD in Eclipse necessitates further improvement to provide dosimetric accuracy on par with Varian LINACs.

BACKGROUND: Pre-treatment liver tumor localization remains a challenging task for radiation therapy, mostly due to the limited tumor contrast against normal liver tissues, and the respiration-induced liver tumor motion. Recently, we developed a biomechanical modeling-based, deformation-driven cone-beam CT estimation technique (Bio-CBCT), which achieved substantially improved accuracy on low-contrast liver tumor localization. However, the accuracy of Bio-CBCT is still affected by the limited tissue contrast around the caudal liver boundary, which reduces the accuracy of the boundary condition that is fed into the biomechanical modeling process. In this study, we developed a motion modeling and biomechanical modeling-guided CBCT estimation technique (MM-Bio-CBCT), to further improve the liver tumor localization accuracy by incorporating a motion model into the CBCT estimation process. METHODS: MM-Bio-CBCT estimates new CBCT images through deforming a prior high-quality CT or CBCT volume. The deformation vector field (DVF) is solved by iteratively matching the digitally-reconstructed-radiographs (DRRs) of the deformed prior image to the acquired 2D cone-beam projections. Using the same solved DVF, the liver tumor volume contoured on the prior image can be transferred onto the new CBCT image for automatic tumor localization. To maximize the accuracy of the solved DVF, MM-Bio-CBCT employs two strategies for additional DVF optimization: (I) prior-knowledge-guided liver boundary motion modeling with motion patterns extracted from a prior 4D imaging set like 4D-CTs/4D-CBCTs, to improve the liver boundary DVF accuracy; and (II) finite-element-analysis-based biomechanical modeling of the liver volume to improve the intra-liver DVF accuracy. We evaluated the accuracy of MM-Bio-CBCT on both the digital extended-cardiac-torso (XCAT) phantom images and real liver patient images. The liver tumor localization accuracy of MM-Bio-CBCT was evaluated and compared with that of the purely intensity-driven 2D-3D deformation technique, the 2D-3D deformation technique with motion modeling, and the Bio-CBCT technique. Metrics including the DICE coefficient and the center-of-mass-error (COME) were assessed for quantitative evaluation. RESULTS: Using limited-view 20 projections for CBCT estimation, the average (± SD) DICE coefficients between the estimated and the ‘gold-standard’ liver tumors of the XCAT study were 0.57±0.31, 0.78±0.26, 0.83±0.21, and 0.89±0.11 for 2D-3D deformation, 2D-3D deformation with motion modeling, Bio-CBCT and MM-Bio-CBCT techniques, respectively. Using 20 projections for estimation, the patient study yielded average DICE results of 0.63±0.21, 0.73±0.13 and 0.78±0.12, and 0.83±0.09, correspondingly. The MM-Bio-CBCT localized the liver tumor to an average COME of ~2 mm for both the XCAT and the liver patient studies. CONCLUSIONS: Compared to Bio-CBCT, MM-Bio-CBCT further improves the accuracy of liver tumor localization. MM-Bio-CBCT can potentially be used towards pre-treatment liver tumor localization and intra-treatment liver tumor location verification to achieve substantial radiotherapy margin reduction.

Total body irradiation is an important part of the conditioning regimens frequently used to prepare patients for allogeneic hematopoietic stem cell transplantation (SCT). Volumetric-modulated arc therapy enabled total body irradiation (VMAT-TBI), an alternative to conventional TBI (cTBI), is a novel radiotherapy treatment technique that has been implemented and investigated in our institution. The purpose of this study is to (1) report our six-year clinical experience in terms of treatment planning strategy and delivery time and (2) evaluate the clinical outcomes and toxicities in our cohort of patients treated with VMAT-TBI. This is a retrospective single center study. Forty-four patients at our institution received VMAT-TBI and chemotherapy conditioning followed by allogeneic SCT between 2014 and 2020. Thirty-two patients (73%) received standard-dose TBI (12-13.2 Gy in 6-8 fractions twice daily), whereas 12 (27%) received low-dose TBI (2-4 Gy in one fraction). Treatment planning, delivery, and treatment outcome data including overall survival (OS), relapse-free survival (RFS), and toxicities were analyzed. The developed VMAT-TBI planning strategy consistently generated plans satisfying our dose constraints, with planning target volume coverage90%, mean lung dose ∼50% to 75% of prescription dose, and minimal hotspots in critical organs. Most of the treatment deliveries were100 minutes (range 33-147, mean 72). The median follow-up was 26 months. At the last follow-up, 34 of 44 (77%) of patients were alive, with 1- and 2-year OS of 90% and 79% and RFS of 88% and 71%, respectively. The most common grade 3+ toxicities observed were mucositis (31 patients [71%]) and nephrotoxicity (6 patients [13%]), both of which were deemed multifactorial in cause. Four patients (9%) in standard-dose cohort developed grade 3+ pneumonitis, with 3 cases in the setting of documented respiratory infection and only 1 (2%) deemed likely related to radiation alone. VMAT-TBI provides a safe alternative to cTBI. The dose modulation capability of VMAT-TBI may lead to new treatment strategies, such as simultaneous boost and further critical organ sparing, for better malignant cell eradication, immune suppression, and lower toxicities.

This is a collection of definite integrals involving the logarithmic and polynomial functions in terms of special functions and fundamental constants. All the results in this work are new.