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In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions.

The Moreno-García cosine product is extended to evaluate an extensive number of trigonometric products previously published. The products are taken over finite and infinite domains defined in terms of the Hurwitz-Lerch Zeta function, which can be simplified to composite functions in special cases of integer values of the parameters involved. The results obtained include generalizations of finite and infinite products cosine functions, in certain cases raised to a complex number power.

A Prudnikov sum is extended to derive the finite sum of the Hurwitz-Lerch Zeta function in terms of the Hurwitz-Lerch Zeta function. This formula is then used to evaluate a number trigonometric sums and products in terms of other trigonometric functions. These sums and products are taken over positive integers which can be simplified in certain circumstances. The results obtained include generalizations of linear combinations of the Hurwitz-Lerch Zeta functions and involving powers of 2 evaluated in terms of sums of Hurwitz-Lerch Zeta functions. Some of these derivations are in the form of a new recurrence identity and finite products of trigonometric functions.

Precision radial velocity spectrographs that use adaptive optics (AO) show promise to advance telescope observing capabilities beyond those of seeing-limited designs. We are building a spectrograph for the Large Binocular Telescope (LBT) named iLocater that uses AO to inject starlight directly into single mode fibers. iLocater's first acquisition camera system (the SX camera), which receives light from one of the 8.4 m diameter primary mirrors of the LBT, was initially installed in summer 2019 and has since been used for several commissioning runs. We present results from first-light observations that include on-sky measurements as part of commissioning activities. Imaging measurements of the bright B3IV star 2 Cygni ( V = 4.98) resulted in the direct detection of a candidate companion star at an angular separation of only θ = 70 mas. Follow-up AO measurements using Keck/NIRC2 recover the candidate companion in multiple filters. An R ≈ 1500 miniature spectrograph recently installed at the LBT named Lili provides spatially resolved spectra of each binary component, indicating similar spectral types and strengthening the case for companionship. Studying the multiplicity of young runaway star systems like 2 Cygni (36.6 ± 0.5 Myr) can help to understand formation mechanisms for stars that exhibit anomalous velocities through the Galaxy. This on-sky demonstration illustrates the spatial resolution of the iLocater SX acquisition camera working in tandem with the LBT AO system; it further derisks a number of technical hurdles involved in combining AO with Doppler spectroscopy.

In this paper, we apply usage-based linguistic analysis to systematize the inventory of orthographic errors observed in the writing of non-native users of Russian. The data comes from a longitudinal corpus (560K tokens) of non-native academic writing. Traditional spellcheckers mark errors and suggest corrections, but do not attempt to model why errors are made. Our approach makes it possible to recognize not only the errors themselves, but also the conceptual causes of these errors, which lie in misunderstandings of Russian phonotactics and morphophonology and the way they are represented by orthographic conventions. With this linguistically-based system in place, we can propose targeted grammar explanations that improve users’ command of Russian morphophonology rather than merely correcting errors. Based on errors attested in the non-native academic writing corpus, we introduce a taxonomy of errors, organized by pedagogical domains. Then, on the basis of this taxonomy, we create a set of mal-rules to expand an existing finite-state analyzer of Russian. The resulting morphological analyzer tags wordforms that fit our taxonomy with specific error tags. For each error tag, we also develop an accompanying grammar explanation to help users understand why and how to correct the diagnosed errors. Using our augmented analyzer, we build a webapp to allow users to type or paste a text and receive detailed feedback and correction on common Russian morphophonological and orthographic errors.

Background and Purpose: A recently developed biology-guided radiotherapy platform, equipped with positron emission tomography (PET) and computed tomography (CT), provides both anatomical and functional image guidance for radiotherapy. This study aimed to characterize performance of the kilovoltage CT (kVCT) system on this platform using standard quality metrics measured on phantom and patient images, using CT simulator images as reference. Materials and Methods: Image quality metrics, including spatial resolution/modular transfer function (MTF), slice sensitivity profile (SSP), noise performance and image uniformity, contrast-noise ratio (CNR) and low-contrast resolution, geometric accuracy, and CT number (HU) accuracy, were evaluated on phantom images. Patient images were evaluated mainly qualitatively. Results: On phantom images the MTF10% is about 0.68 lp/mm for kVCT in PET/CT Linac. The SSP agreed with nominal slice thickness within 0.7 mm. The diameter of the smallest visible target (1% contrast) is about 5 mm using medium dose mode. The image uniformity is within 2.0 HU. The geometric accuracy tests passed within 0.5 mm. Relative to CT simulator images, the noise is generally higher and the CNR is lower in PET/CT Linac kVCT images. The CT number accuracy is comparable between the two systems with maximum deviation from the phantom manufacturer range within 25 HU. On patient images, higher spatial resolution and image noise are observed on PET/CT Linac kVCT images. Conclusions: Major image quality metrics of the PET/CT Linac kVCT were within vendor-recommended tolerances. Better spatial resolution but higher noise and better/comparable low contrast visibility were observed as compared to a CT simulator when images were acquired with clinical protocols.

In this paper, we derive an integral transform involving the product of Hermite polynomial $ H_{n}(\beta x) $ and parabolic cylinder function $ D_{v}(\alpha t) $. These integral transforms will be evaluated in terms of Lerch function. Various formulae are also evaluated in terms of special functions to complete this paper. All the results in this paper are new.

In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.

In this work we present derivations of the formula listed in entry 4.113 in the sixth edition of Gradshteyn and Rhyzik's table of integrals. We evaluate two definite integrals of the formandin terms of the Lerch function where $ k $, $ a $, $ z $ and $ b $ are arbitrary complex numbers. The entries in the table(s) are obtained as special cases in the paper below.

We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik [5].

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.

With a possible connection to integrals used in General Relativity, we used our contour integral method to write a closed form solution for a quadruple integral involving exponential functions and logarithm of quotient radicals. Almost all Hurwitz–Lerch Zeta functions have an asymmetrical zero distribution. All the results in this work are new.

The western pond turtle (WPT) was formerly considered a single species (Actinemys or Emys marmorata) that ranged from southern British Columbia, Canada to Baja California, México. More recently it was divided into a northern and a southern species. WPTs are found primarily in streams that drain into the Pacific Ocean, although scattered populations exist in endorheic drainages of the Great Basin and Mojave deserts. Populations in the Mojave Desert were long thought to be restricted to the Mojave River, but recently another population was documented in Piute Ponds, a terminal wetland complex associated with Amargosa Creek on Edwards Air Force Base. WPT fossils in the Mojave Desert are known from the Miocene to the Pleistocene. Recently, Pleistocene fossils have been found as far into the desert as Salt Springs, just south of Death Valley. The oldest fossil records suggest that WPTs were present in wetlands and drainages of the geological feature known as the Mojave block prior to the uplift of the Sierra Nevada Range about 8 Ma and prior to the ~ 3 Ma uplift of the Transverse Ranges. Archaeological records document use of turtles by Native Americans for food and cultural purposes 1,000 or more years ago at the Cronese Lakes on the lower Mojave River and Oro Grande on the upper river. The first modern publication documenting their presence in the Mojave River was 1861. Museum specimens were collected as early as 1937. These fossil and early literature records support the indigenous status of WPTs to the Mojave River. However, mtDNA-based genetic evidence shows that Mojave River turtles share an identical haplotype with turtles on the California coast. Limited nuclear data show some minor differences. Overdraft of water from the Mojave River for municipal and agricultural uses, urban development, and saltcedar expansion are threats to the continued survival of WPTs in the Mojave River.

Intelligent Language Tutoring Systems typically attempt to automatically diagnose learner errors in order to provide individualized feedback. One common approach is the use of mal-rules to extend norma- tive grammars by licensing specific types of learner errors. In finite-state morphologies, mal-rules can be implemented as two-level rules or replace rules. However, unlike the phonological rules of natural languages, mal-rules do not necessarily behave as a coherent system, especially with respect to feed- ing interactions. Using examples from learner errors attested in the RULEC corpus of Russian learner texts, we illustrate the problem of cyclic feeding interactions that can occur between mal-rules. We then describe a formal algorithm for identifying an optimal ordering for mal-rules to be applied to a transducer.

Closed expressions for the integralare 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.

In this paper by means of contour integration we will evaluate definite integrals of the form \begin{equation*} \int_{0}^{1}\left(\ln^k(ay)-\ln^k\left(\frac{a}{y}\right)\right)R(y)dy \end{equation*} in terms of a special function, where $R(y)$ is a general function and $k$ and $a$ are arbitrary complex numbers. These evaluations can be expressed in terms of famous mathematical constants such as the Euler's constant $\gamma$ and Catalan's constant $C$. Using derivatives, we will also derive new integral representations for some Polygamma functions such as the Digamma and Trigamma functions along with others.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form $\int_{0}^{\infty}\ln^k(\alpha y)\ln(R(y))dy$ in terms of a special function, where $R(y)$ is a general function and $k$ and $\alpha$ are arbitrary complex numbers.

The Wang sum involving the exponential sums of Lerch's Zeta functions is extended to the finite sum of the Huwitz-Lerch Zeta function to derive sums and products involving cosine and tangent trigonometric functions. The general theorem used to derive these sums and products is in the form of the finite sum over positive integers of the Hurwitz-Lerch Zeta function where the associated parameters are general complex numbers. New Hurwitz-Lerch Zeta function recurrence identities with consecutive neighbours are derived. Some finite sum and product formulae examples involving cosine, tangent and the product of cosine and tangent functions are also derived and evaluated.

A two-dimensional integral containing Sx is derived. Sx is the Fresnel integral function, and the double integral is taken over the range 0

We consider a Hurwitz-Lerch zeta function Φs,z,a sum over the natural numbers. We provide an analytically continued closed form solution for this sum in terms of the addition of Hurwitz-Lerch zeta functions. A new recurrence identity with consecutive neighbours and the product of trigonometric functions is derived.

Using a contour integral method, the de Montmort-Prudnikov sum is extended to derive a new series representation involving the incomplete gamma function. The series is uniformly convergent and completely analytical, which can be evaluated for general complex ranges of the parameters involved. Applications and evaluations of this formula are discussed.

This paper employs a contour integral method to derive and evaluate the infinite sum of the Euler polynomial expressed in terms of the Hurwitz Zeta function. We provide formulae for several classes of infinite sums of the Euler polynomial in terms of the Riemann Zeta function and fundamental mathematical constants, including Catalan’s constant. This representation of Catalan’s constant suggests it could be rational.

A family of generalized definite logarithmic integrals given by $$ \int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$ built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [6] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions.

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by ∫0∞xm−1logkax/β2+x2γ+xdx, where the parameters a,k,β, and γ are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

We derive a new formula for the Hurwitz–Lerch zeta function in terms of the infinite sum of the incomplete gamma function. Special cases are derived in terms of fundamental constants.

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation, which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants, such as Catalan’s constant C and π.

An indexed rotational immobilization system was developed for supine total body irradiation (TBI). Treatment plans had multi-isocentric volumetric modulated arc therapy (VMAT) beams to the upper body and parallel-opposed fields to the lower body, with a 12 Gy prescription dose to >90% of the body and mean lung dose ∼8 Gy. In the end-to-end test, point dose measurements had

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.

This paper gives new integrals related to a class of special functions. This paper also showcases the derivation of definite integrals involving the quotient of functions with powers and the exponential function expressed in terms of the Lerch function and special cases involving fundamental constants. The goal of this paper is to expand upon current tables of definite integrals with the aim of assisting researchers in need of new integral formulae.

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.

In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.

In this article we derive some entries and errata for the book of Gradshteyn and Ryzhik which were originally published by Bierens de Haan. We summarize our results using tables for easy reading and referencing.

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form ∫ 0 ∞ log ( 1 ± e − α y ) R ( k , a , y ) d y in terms of a special function, where R ( k , a , y ) is a general function and k, a and α are arbitrary complex numbers, where R e ( α ) > 0 .

The habitat in the vicinity of Kurupukari, on the Essequibo River in central Guyana, is tall evergreen lowland forest. The area has suffered some human disturbance from agriculture, road construction and ferry activity. The area was sampled for 10 days in 1990 and 12 days in 1997; seven days in rainy season and 15 in dry season. During this sampling 23 anuran and 17 reptile species were collected. Some differences exist between species collected on either side of the river. Comparisons are made with collections from other locations in Guyana.

Background: Drug adverse event (AE) signal detection using the Gamma Poisson Shrinker (GPS) is commonly applied in spontaneous reporting. AE signal detection using large observational health plan databases can expand medication safety surveillance. Methods: Using data from nine health plans, we conducted a pilot study to evaluate the implementation and findings of the GPS approach for two antifungal drugs, terbinafine and itraconazole, and two diabetes drugs, pioglitazone and rosiglitazone. We evaluated 1676 diagnosis codes grouped into 183 different clinical concepts and four levels of granularity. Several signaling thresholds were assessed. GPS results were compared to findings from a companion study using the identical analytic dataset but an alternative statistical method—the tree-based scan statistic (TreeScan). Results: We identified 71 statistical signals across two signaling thresholds and two methods, including closely-related signals of overlapping diagnosis definitions. Initial review found that most signals represented known adverse drug reactions or confounding. About 31% of signals met the highest signaling threshold. Conclusions: The GPS method was successfully applied to observational health plan data in a distributed data environment as a drug safety data mining method. There was substantial concordance between the GPS and TreeScan approaches. Key method implementation decisions relate to defining exposures and outcomes and informed choice of signaling thresholds.

We report the herpetofauna of two neighboring upland locations in west-central Guyana. Twenty amphibian and 24 reptile species were collected. Only 40% of amphibians and 12.5% of reptiles were collected in both locations. This is one of the few collections made at upland (750–800 m) locations in the Guiana Shield.

We provide an initial checklist of the herpetofauna of Baramita, a lowland rainforest site in the Northwest Region of Guyana. Twenty-five amphibian and 28 reptile species were collected during two separate dry-season visits. New country records for two species of snakes are documented, contributing to the knowledge on the incompletely known herpetofauna of Guyana.

Respiratory drugs are widely used in children to treat labeled and non-labeled indications but only some data are available quantifying comprehensively off-label usage. Thus, we aim to analyse drug utilisation and off-label prescribing of respiratory drugs focusing on age- and indication-related off-label use. Patients aged ≤18 years documented in the Bavarian Association of Statutory Health Insurance Physicians database (approx. 2 million children) between 2004 and 2008 were included in our study. Annual period prevalence rates (PPRs) per 10,000 children and the proportion of age- and indication-related off-label prescriptions were calculated and stratified by age and gender. Within the study period, highest PPRs were found for the fixed combination of clenbuterol/ambroxol (between 374-575 per 10,000 children) and the inhaled short acting beta-2-agonist salbutamol (between 378-527 per 10,000 children). Highest relative PPR increase was found for oral salbutamol (approx. 39-fold) whereas the most distinct decrease was found for oral long-acting beta-2-agonist clenbuterol (-97%). Compound classes most frequently involved in off-label prescribing were inhaled bronchodilative compounds (91,402; 37.3%) and oral beta-2-agonists (26,850; 22.5%). The highest absolute number of off-label prescriptions were found for inhaled salbutamol (n = 67,084; 42.0%) and oral clenbuterol/ambroxol (fixed combination, n = 18,897; 20.7%). Off-label prescribing due to indication was of much greater relevance than age-related off-label use. Most frequently, bronchodilative compounds were used off-label to treat respiratory tract infections. Highest off-label prescription rates were found in the youngest patients without relevant gender-related differences. Off-label prescribing of respiratory drugs is common especially in young children. Bronchodilative drugs were most frequently used off-label for treating acute bronchitis or upper respiratory tract infections underlining the essential need for a more rational prescribing in this area.