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This article investigates the sufficient conditions for the rearrangement of multiple iterated series into a single-level series through the utilization of fundamental proof methods. These conditions were subsequently applied to establish reduction formulas for a series of the form ∞∑x1=0∞ ∑x2=0 ∞∑xk=0(x1x2･･xk)t・f(x1+x2+･･･+xk) where f: N→R and t∈ {0,1,2}, employing techniques derived from generating functions. [ABSTRACT FROM AUTHOR]

Background: Dose evaluation at longitudinal positions of body computed tomography (CT) scans is useful for CT quality assurance programs and patient organ dose evaluation. Accurate estimates depend on both patient size and scan length. Purpose: To propose practical evaluation of the average dose to the transverse slab of an axial image slice for adult body CT examinations, considering not only patient size but also scan length, and to compare the results with those of Monte Carlo (Geant4) simulation [Dsim(z)] and size-specific dose estimates at longitudinal positions of scans [SSDE(z)] from international standards (IEC publication no. 62985). Methods: In a scan series, the total dose at each z-axis location was calculated using the input information identical to the SSDE(z) evaluation. Each axial image slice (slice thickness, 2.5 or 5 mm) was first considered independently. Its z-axis coverage and CTDIvol (from the DICOM headers) were used to directly calculate a z-axis dose profile for the average dose over the cross-section of a water phantom, using the approach to equilibrium function. The phantom diameter was taken to be equal to the patient water equivalent diameter at that slice. The above was repeated at all slices and the dose at each z-axis location was accumulated from all profiles, referred to as Dcalc(z). For validation, we considered a cohort of 65 patients, who underwent chest and abdominopelvic examinations. The resultant Dcalc(z) was compared with Dsim(z) and SSDE(z), both available in a previous paper. Results: Dcalc(z) evaluation could be used to accurately assess the scan range average dose, with an accuracy of 7.1%-8.7% for 65 patients in two examinations. On individual image slices, the maximum difference in magnitude between Dcalc(z) and Dsim(z) [and between SSDE(z) and Dsim(z) in parentheses] was 37.5% (85%) [two edges (2 × 5 cm) of chest scan range], 17.8% (35.2%) (the remaining central region of chest scan),26.8% (74.1%) [two edges (2 × 5 cm) of abdominopelvic scan range], and 14.2% (22.5%) (the remaining central region of abdominopelvic scan). Conclusions: Identical input data are used for Dcalc(z) and SSDE(z) evaluations. The latter is limited to the z-axis locations within scan range. At each image slice, SSDE(z) is equivalent to the midpoint dose of a fixed-mA scan of 15-30 cm (scan length). In contrast, Dcalc(z) considers dose accumulation from varying scan length (from sub-centimeter to about 1 m) and tube current, and dose profile is also computed outside scan range. Besides greatly improving dose evaluation for individual image slices, Dcalc(z) allows for evaluating dose accumulation from multiple series, which typically span different scan ranges. Our proposal may assist CT manufacturers and dose index monitoring software in assessing dose at longitudinal positions of body CT scans. [ABSTRACT FROM AUTHOR]

It is proved that if a multiple series in the Franklin system converges in the sense of Pringsheim everywhere, except, perhaps, on a set that is a Cartesian product of sets of measure zero, to an everywhere finite integrable function, then it is the Fourier–Franklin series of this function. A uniqueness theorem is also proved for multiple Franklin series whose rectangular partial sums at each point have a sequential limit. [ABSTRACT FROM AUTHOR]

In this paper we prove that the quadratic partial sums of a multiple Franklin series with indices , , cannot converge to on a set of positive measure. [ABSTRACT FROM AUTHOR]

Purpose: Multi‐series CT examination is common in the clinic, but no metric is agreed upon to report the overall dose from such an examination. This work proposes a relevant metric for tracking patient dose from multi‐series examinations and illustrates the evaluation method through explanatory examples. Materials and methods: In each acquisition series, a previously reported method was used to evaluate the cross‐sectional average dose along the z‐axis of a water phantom, with inputs of CTDIvol, scan length, tube current, and patient water‐equivalent diameter. With a multi‐series examination, the dose at each z‐location was accumulated over all acquisition series. This method was applied to four clinical CT examinations. In three abdominal/pelvic examinations (patient weight, 107, 79, 79 kg), tube current modulation was applied in five acquisition series with scan lengths of 30–41.8 cm, while tube current was fixed in other series with short scan lengths (1.0, 7.9 cm). In another CT‐guided liver ablation procedure (patient weight, 114 kg), 22 series were acquired with constant mA and scan lengths of 1–30 cm. The maximum value of the overall dose profile of each examination was compared to five dose quantities, including CTDIvol,sum and SSDEsum by the ACR CT Dose Index Registry, scan length‐weighted CTDIvol and SSDE by a CT dose monitoring platform, and "max z location CTDIvol" by a CT manufacturer. Results: A simple graphic display of dose as a function of the z‐axis location was presented for each acquisition series and for the whole examination. Differences up to 43.4% and 42.8%, or down to −93.5%, −93.5%, and −49.0%, were observed between the maximum value of the overall dose profile and five dose quantities (in the above order), respectively. Conclusion: The overall dose profile gives a complete description of z‐axis dose distribution for the studied CT examinations under a wide range of patient variables and acquisition conditions, including multiple acquisition series. Simple visualization of the doses across and beyond the scan ranges may provide a new tool for CT dose optimization. [ABSTRACT FROM AUTHOR]

Srivastava and Panda have studied the simple and multiple generating relations concerning the multivariable H-function. The aim of this paper is to derive the various classes of simple and multiple generating relations involving the multivariable Aleph-function. The generating function is used in the theory of numbers, in physics and other fields of mathematics. We see the particular cases concerning the multivariable I-function, the Aleph-function of two variables and the I-function of two variables. [ABSTRACT FROM AUTHOR]

In this paper a method of summation of double absolutely convergent numerical series was obtained. This method was named the expansion of double numerical series in infinite diagonals. The form of the expansion for symmetric and skew-symmetric general term of double series was established. This expansion was generalized to the case of multiple series. It was called the expansion of multiple numerical series in infinite diagonals for two indices. The expansion of triple series in infinite diagonals for two indices was generalized to the case of three indices.

We consider a semi-parametric approach to perform the joint segmentation of multiple series sharing a common functional part. We propose an iterative procedure based on Dynamic Programming for the segmentation part and Lasso estimators for the functional part. Our Lasso procedure, based on the dictionary approach, allows us to both estimate smooth functions and functions with local irregularity, which permits more flexibility than previous proposed methods. This yields to a better estimation of the functional part and improvements in the segmentation. The performance of our method is assessed using simulated data and real data from agriculture and geodetic studies. Our estimation procedure results to be a reliable tool to detect changes and to obtain an interpretable estimation of the functional part of the model in terms of known functions. [ABSTRACT FROM AUTHOR]

The paper presents the proof of the uniqueness theorem formultiple series in the Franklin systemthat converge inmeasure and whosemajorant of cubic partial sums with numbers 2 satisfies a certain necessary condition. This result is new in the one-dimensional case as well. [ABSTRACT FROM AUTHOR]

In this paper we solve three open problems and a conjecture related to the calculations of some classes of multiple series posed by Furdui in [1].

Background: Dose evaluation at longitudinal positions of body computed tomography (CT) scans is useful for CT quality assurance programs and patient organ dose evaluation. Accurate estimates depend on both patient size and scan length., Purpose: To propose practical evaluation of the average dose to the transverse slab of an axial image slice for adult body CT examinations, considering not only patient size but also scan length, and to compare the results with those of Monte Carlo (Geant4) simulation [D sim (z)] and size-specific dose estimates at longitudinal positions of scans [SSDE(z)] from international standards (IEC publication no. 62985)., Methods: In a scan series, the total dose at each z-axis location was calculated using the input information identical to the SSDE(z) evaluation. Each axial image slice (slice thickness, 2.5 or 5 mm) was first considered independently. Its z-axis coverage and CTDI vol (from the DICOM headers) were used to directly calculate a z-axis dose profile for the average dose over the cross-section of a water phantom, using the approach to equilibrium function. The phantom diameter was taken to be equal to the patient water equivalent diameter at that slice. The above was repeated at all slices and the dose at each z-axis location was accumulated from all profiles, referred to as D calc (z). For validation, we considered a cohort of 65 patients, who underwent chest and abdominopelvic examinations. The resultant D calc (z) was compared with D sim (z) and SSDE(z), both available in a previous paper., Results: D calc (z) evaluation could be used to accurately assess the scan range average dose, with an accuracy of 7.1%-8.7% for 65 patients in two examinations. On individual image slices, the maximum difference in magnitude between D calc (z) and D sim (z) [and between SSDE(z) and D sim (z) in parentheses] was 37.5% (85%) [two edges (2 × 5 cm) of chest scan range], 17.8% (35.2%) (the remaining central region of chest scan), 26.8% (74.1%) [two edges (2 × 5 cm) of abdominopelvic scan range], and 14.2% (22.5%) (the remaining central region of abdominopelvic scan)., Conclusions: Identical input data are used for D calc (z) and SSDE(z) evaluations. The latter is limited to the z-axis locations within scan range. At each image slice, SSDE(z) is equivalent to the midpoint dose of a fixed-mA scan of 15-30 cm (scan length). In contrast, D calc (z) considers dose accumulation from varying scan length (from sub-centimeter to about 1 m) and tube current, and dose profile is also computed outside scan range. Besides greatly improving dose evaluation for individual image slices, D calc (z) allows for evaluating dose accumulation from multiple series, which typically span different scan ranges. Our proposal may assist CT manufacturers and dose index monitoring software in assessing dose at longitudinal positions of body CT scans., (© 2022 American Association of Physicists in Medicine.)

The subject of this paper is the recovery of an additive function defined on P-adic parallelepipeds from its derivative with respect to P-adic parallelepipeds. The resulting theorem is applied to the study of the uniqueness problem for multiple series with respect to the Haar and Price systems. [ABSTRACT FROM AUTHOR]

The class of functions of s variables is defined. For the functions of this class, Sász's theorem for s-dimensional Fourier-Haar series is proved. When , it is shown that the obtained theorem is best possible for general complete orthonormal systems. [ABSTRACT FROM AUTHOR]

We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions.

Battery applications, such as electric vehicles, electric propulsion ships, and energy storage systems, are developing rapidly, and battery management issues are gaining attention. In this application field, a battery system with a high capacity and high power in which numerous battery cells are connected in series and parallel is used. Therefore, research on a battery management system (BMS) to which various algorithms are applied for efficient use and safe operation of batteries is being conducted. In general, maintenance/replacement of multi-series/multiple parallel battery systems is only possible when there is no load current, or the entire system is shut down. However, if the circulating current generated by the voltage difference between the newly added battery and the existing battery pack is less than the allowable current of the system, the new battery can be connected while the system is running, which is called hot swapping. The circulating current generated during the hot-swap operation is determined by the battery's state of charge (SOC), the parallel configuration of the battery system, temperature, aging, operating point, and differences in the load current. Therefore, since there is a limit to formulating a circulating current that changes in size according to these various conditions, this paper presents a circulating current estimation method, using an artificial neural network (ANN). The ANN model for estimating the hot-swap circulating current is designed for a 1S4P lithium battery pack system, consisting of one series and four parallel cells. The circulating current of the ANN model proposed in this paper is experimentally verified to be able to estimate the actual value within a 6% error range. [ABSTRACT FROM AUTHOR]