The present study explores the application of hypergeometric functions in evaluating fractional integrals, providing a comprehensive framework to bridge fractional calculus and special functions. As a generalization of classical integrals, fractional integrals have gained prominence due to their wide applicability in modeling anomalous diffusion, viscoelastic systems, and other non-local phenomena. Hypergeometric functions, renowned for their rich analytical properties and ability to represent solutions to differential equations, offer an elegant and versatile tool for solving fractional integrals. In this paper, we evaluate a new class of fractional integrals, presenting results that contribute significantly to the study of generalized hypergeometric functions, particularly 3F2(1). The results reveal previously unexplored connections within these functions, providing new insights and extending their applicability. Furthermore, evaluating these fractional integrals holds promise for advancing the theoretical understanding and practical applications of fractional differential equations.
Citation: Abdelhamid Zaidi, Saleh Almuthaybiri. Explicit evaluations of subfamilies of the hypergeometric function 3F2(1) along with specific fractional integrals[J]. AIMS Mathematics, 2025, 10(3): 5731-5761. doi: 10.3934/math.2025264
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| [8] |
Dongyi Liu, Genqi Xu .
Input-output |
| [9] | Tong Li . Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8(3): 773-781. doi: 10.3934/nhm.2013.8.773 |
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The present study explores the application of hypergeometric functions in evaluating fractional integrals, providing a comprehensive framework to bridge fractional calculus and special functions. As a generalization of classical integrals, fractional integrals have gained prominence due to their wide applicability in modeling anomalous diffusion, viscoelastic systems, and other non-local phenomena. Hypergeometric functions, renowned for their rich analytical properties and ability to represent solutions to differential equations, offer an elegant and versatile tool for solving fractional integrals. In this paper, we evaluate a new class of fractional integrals, presenting results that contribute significantly to the study of generalized hypergeometric functions, particularly 3F2(1). The results reveal previously unexplored connections within these functions, providing new insights and extending their applicability. Furthermore, evaluating these fractional integrals holds promise for advancing the theoretical understanding and practical applications of fractional differential equations.
Determination of cell viability is essential for in vitro studies. Approaches for the evaluation of live/dead cells such as flow cytometry and microscopy use fluorescent dyes, but flow cytometry generates numeric values immediately. A wide range of dyes are available, and their ability to stain depends on vital properties such as cell membrane integrity (e.g. PI, SYTOX, TOTO), cell membrane potential (e.g. Rh123, DiBAC4), membrane pump activity (e.g. CFDA, Rh123), and enzymatic activity (e.g. CFDA, Calcein-AM, ChemChrome). In addition, permanent DNA stains (e.g. SYTO, DAPI, SYBR) are also available [1]. Eukaryotic cells generally require the use of only one dye when assays are performed on a flow cytometer. Using side and forward scatter signals, an operator can determine the population of whole homogeneous cells with similar size and granularity. DNA-binding dyes such as SYTO and DAPI are commonly used to differentiate between live and dead cells, and PI is commonly used for the evaluation of cell membrane integrity. These dyes are usually applied separately, and other parameters are calculated by the operator or automatically (e.g. with commercial kits) because the count of the general cell population is already known.
Determination of prokaryotic cell viability by flow cytometry is more difficult. If the cytometer detector is poor, it is impossible to count cells using side and forward scatter signals. Therefore, it is necessary to apply a combination of dyes with emissions of different wavelengths to count live and dead cells separately. Combining DNA-binding dyes with those used to evaluate cell membrane integrity is the most common method (e.g. the combination of SYTO 9 and PI). While SYTO 9, with a green emission, enters all cells and binds to their DNA, PI, which fluoresces red after binding to DNA, only enters cells with damaged membranes [2],[3]. Nevertheless, Stiefel et al., 2012 noted that in addition to background fluorescence and the cross-signaling of these dyes, SYTO 9 causes bleaching effects and has different binding affinities to live and dead cells [4]. Therefore, there is a need for studies on alternative dyes. Moreover, the quantitative assessment of prokaryotic viability is essential, especially for the confirmation of activity of novel antimicrobial substances. Traditional microbiological methods have long been used successfully for these purposes. However, disk and agar well diffusion methods, agar and broth dilution tests for minimum inhibitory concentration (MIC) determination, Etests and Time-kill tests [5]–[8], bioautography [9],[10], and qPCR [11] mainly provide descriptive information. Therefore, the development of an effective, alternative approach using flow cytometry for the evaluation of prokaryotic cell viability is a necessity. Some dyes used in PCR analysis have also been used for the assessment of bacterial viability. It has also been shown that a combination of SYBR green and PI was more successful than with SYTO 9 [12]. Therefore, the aim of this study was to assess the feasibility of EvaGreen in determining the cell viability of Listeria monocytogenes АТСС 13932 and Staphylococcus aureus АТСС 25923 using flow cytometry.
Meat samples were ground with a Retsch GM200 blade homogenizer (Retsch, Germany). A total of 50 ± 1.0 mg was collected for DNA isolation. DNA was isolated using a Sorb-GMO-B kit (Syntol CJSC, Russia) according to the instructions. Samples were mixed with 800 µL of lysis buffer and 15 µL of proteinase K, incubated for 60 min at 60 °C, and centrifuged on MiniSpin columns (Eppеndorf, Germany) at 13000 rpm for 5 min. Supernatants were transferred to new tubes, and 500 µL of extraction solution was added. Then, mixtures were intensively mixed and centrifuged on MiniSpin columns (Eppеndorf, Germany) at 13000 rpm for 10 min. A total of 300 µL of supernatant was transferred to new tubes and mixed with 625 µL of precipitating solution and sorbent (24:1). Then, mixtures were intensively mixed for 10 min and centrifuged on MiniSpin columns (Eppеndorf, Germany) at 7000 rpm for 1 min. Precipitates were washed three times by the addition of 500 µL of wash solution, intensively mixed for 2 min, and centrifuged on MiniSpin columns (Eppеndorf, Germany) at 7000 rpm for 0.5 min. Precipitates were dried at 60 °C for 5 min; then, 200 µL of buffer was added, samples were mixed, incubated at 60 °C for 10 min, and centrifuged on MiniSpin columns (Eppеndorf, Germany) at 13000 rpm for 2 min. A total of 150 µL of supernatant was taken for further testing.
The Escherichia coli ATCC 25922 strain was obtained from the State Research Center for Applied Biotechnology and Microbiology (Obolensk, Moscow region, Russia). RNA from E. coli ATCC 25922 was isolated using a MagNA Pure LC RNA Isolation Kit-High Performance (Roche, Germany) according to the instructions. Cells were lysed by incubation with a special buffer containing chaotropic salt, and proteinase K digestion destroyed the remaining proteins and nucleases. Then, magnetic glass particles (MGPs) were added to bind RNA, and DNA was degraded with DNase. Unbound substances were removed by several wash steps, and the purified RNA was eluted using elution buffer. Cells were centrifuged for 10 minutes at 1500 rpm on MiniSpin columns (Eppеndorf, Germany), resuspended in 200 µL of phosphate buffered saline (PBS), and transferred into a well of the sample cartridge. Sample cartridges were placed on the reagent/sample stage, and the “RNA HP Cells” protocol was started. The isolation was performed on a MagNA Pure LC® 2.0 (Roche, Switzerland).
L. monocytogenes АТСС 13932 and Staphylococcus aureus АТСС 25923 strains were obtained from the State Research Center for Applied Biotechnology and Microbiology (Obolensk, Moscow region, Russia). Cultures were grown on slanted Trypticase soy agar (TSA, Liofilchem) at 37 °C for 24 h. Cultures from the agar surface were removed with normal saline solution and adjusted to a concentration of 1 × 106 cells/mL according to the standard of McFarland turbidity. A total of 100 µL of the suspensions was transferred to Eppendorf tubes and mixed with 2 mL of Trypticase soy broth (TSB, Liofilchem) and incubated on a thermoshaker TS-100 (BioSan, Latvia) at 30 °C for 4 h. Suspensions were centrifuged on MiniSpin columns (Eppеndorf, Germany) at 5000 rpm for 5 min. Cell precipitates were washed three times with normal saline solution followed by centrifugation. Cell suspensions at an approximate concentration of 1 × 106 cells/mL were prepared in normal saline solution according to the McFarland turbidity standard. Suspensions with an approximate concentration of 1 × 106 cells/mL were used as positive controls. To obtain negative control-1 of L. monocytogenes АТСС 13932 and S. aureus АТСС 25923, the resulting suspensions were heated at 100 °C for 10 min. To obtain negative control-2 of L. monocytogenes АТСС 13932, cell pellets were washed with 70% isopropyl alcohol and resuspended at an approximate concentration of 1 × 106 cells/mL.
A total of 5 µL of DNA/RNA was mixed with 5 µL EvaGreen (Synthol, Russia), 380 µL of deionized water, and 10 µL of DMSO (Biolot, Russia); then, samples were incubated in the dark for 15 min and green and red fluorescence signals were measured on a Guava EasyCyte flow cytometer (Merck Millipore, Germany) up to 5000 events.
(1) Protocol 1: A total of 20 µL of L. monocytogenes АТСС 13932 cells or S. aureus АТСС 25923 cells was mixed with 5 µL of EvaGreen (Synthol, Russia), 365 µL of deionized water, and 10 µL of DMSO (Biolot, Russia); then, samples were incubated in the dark for 15 min and green and red fluorescence signals were measured on a Guava EasyCyte flow cytometer (Merck Millipore, Germany) up to 5000 events. To prepare mixed samples, a 10-µL positive control (live cells) and a 10-µL negative control-1 (dead cells killed by heating) were used.
(2) Protocol 2: A total of 20 µL of L. monocytogenes АТСС 13932 cells was mixed with 5 µL of EvaGreen (Synthol, Russia) and 375 µL of 0.9% NaCl solution (Panreac, Spain); then, the samples were incubated in the dark for 15 min and green and red fluorescence signals were measured on a Guava EasyCyte flow cytometer (Merck Millipore, Germany) up to 5000 events.
Parameters of analysis are presented in Table 1. Low gain on the green channel was used for chicken DNA analysis due to the primary intense fluorescence when bound to EvaGreen. Low gain on the red channel was used for E. coli RNA measurement. Compensation coefficients were applied to obtain better separation between live and dead cells in L. monocytogenes samples.
| Sample ID | Gains |
Compensation |
||||
| FSC | SSC | GRN | YEL | RED | GRN-%RED | |
| Chicken DNA | 5.2 | 1.9 | 8.0 | 12.3 | 6.7 | 0.0 |
| Escherichia coli RNA | 1.7 | 3.1 | 20.7 | 12.3 | 3.5 | 0.0 |
| Listeria monocytogenes АТСС 13932 (positive control) | 5.2 | 1.9 | 23.6 | 12.3 | 8.7 | 3.6 |
| Listeria monocytogenes АТСС 13932 (negative control-1) | 5.2 | 1.9 | 23.6 | 12.3 | 8.7 | 5.1 |
| Listeria monocytogenes АТСС 13932 (negative control-2) | 5.2 | 1.9 | 15.3 | 12.3 | 8.7 | 2.2 |
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Results of the flow cytometry analysis of DNA, RNA, and L. monocytogenes АТСС 13932 according to protocol 1 are presented in Figure 1. Both green and red fluorescence were observed in chicken DNA stained with EvaGreen; corresponding counted events are located in the upper right square of the plot. RNA isolated from E. coli and stained with EvaGreen displayed only red fluorescence; all counted events are located in the lower right square of the plot. The positive control of L. monocytogenes АТСС 13932 displayed a localization similar to chicken DNA; a large number of live cells are located in the upper right square of the plot. Negative controls (1 and 2) of L. monocytogenes АТСС 13932 showed a similar localization to E. coli RNA; a large number of dead cells are located in the lower right square of the plot. The positive control of L. monocytogenes АТСС 13932 displayed a concentration of 2.26 × 106 live cells/mL, negative control 1 displayed a concentration of 2.35 × 106 dead cells/mL, and negative control 2 displayed a concentration of 2.77 × 106 dead cells/mL. To examine the discriminatory ability of EvaGreen, a mixed sample of L. monocytogenes АТСС 13932 was prepared according to protocol 1 using 10 µL of positive control and 10 µL of negative control. Both green and red fluorescence were observed in live cells, and corresponding counted events are located in the upper right square of the plot. Only red fluorescence was observed in dead cells, and all counted events are located in the lower right square of the plot. However, the separation was not distinct enough.
According to the safety report for EvaGreen® dye [13] and Chiaraviglio & Kirby, 2014 [14], EvaGreen is a non-permeable, nontoxic (class III) dye and appears to be membrane-impermeable; this is evident from the absence of cell nuclear staining with HeLa cells over a 30 min incubation period. Nevertheless, we observed fluorescence of live L. monocytogenes АТСС 13932 cells in the positive control stained with EvaGreen. Presumably, the observed phenomenon can be linked to the solution composition; a mixture of DMSO and deionized water was used in the protocol. When a 0.9% NaCl solution was used, no fluorescence was observed in the positive control of L. monocytogenes АТСС 13932 stained with EvaGreen (Figure 2).
However, eukaryotic mammalian cells were used for the study of EvaGreen permeability in both reports. Prokaryotic cells have different cell wall structures depending on whether they are Gram-positive or Gram-negative [15]. The type of cell envelope could influence EvaGreen permeability; however, in our study, we only considered the staining of Gram-positive bacteria.
We also examined S. aureus АТСС 25923 staining by EvaGreen (Figure 3). The positive control of S. aureus АТСС 25923 displayed a localization similar to L. monocytogenes АТСС 13932; a large number of live cells are located in the upper right square of the plot. Mixed samples of S. aureus were prepared according to protocol 1 using 10 µL of the positive control and 10 µL of the negative control. Live cells displayed both green and red fluorescence, and corresponding counted events are located in the upper right square of the plot. Dead cells displayed only red fluorescence, and all counted events are located in the lower right square of the plot. The separation was more distinct than in the case of L. monocytogenes АТСС 13932.
We also noticed red fluorescence of dead L. monocytogenes АТСС 13932 cells in the negative control stained with EvaGreen that was similar to E. coli RNA bound to EvaGreen. Presumably, the heat treatment of L. monocytogenes АТСС 13932 destroyed cell integrity and led to the conversion of double-stranded DNA to single-stranded DNA [16]. RNA and ssDNA were located in the buffer solution where ssDNA could be degraded by nucleases [17]; therefore, both degraded ssDNA and RNA stained with EvaGreen could demonstrate red fluorescence.
It was observed that EvaGreen dye, which is commonly used in PCR analysis, stained live cells of L. monocytogenes АТСС 13932 and fluoresced in green and red spectra; the dye also stained dead cells and only demonstrated red fluorescence. However, the solvent composition and cell type influenced EvaGreen permeability. Therefore, it is necessary to repeat this analysis on a wide range of both Gram-positive and Gram-negative bacteria. In conclusion, EvaGreen dye has prospective applications in flow cytometry.
| [1] | E. Özergin, Some properties of hypergeometric functions, Eastern Mediterranean University, 2011. Available from: http://hdl.handle.net/11129/217 |
| [2] | F. Ouimet, Central and noncentral moments of the multivariate hypergeometric distribution, arXiv, 2024. https://doi.org/10.48550/arXiv.2404.09118 |
| [3] | W. Briggs, R. Zaretzki, A new look at inference for the hypergeometric distribution, 2009. |
| [4] |
P. Sheridan, M. Onsjö, The hypergeometric test performs comparably to TF-IDF on standard text analysis tasks, Multimed. Tools Appl., 83 (2024), 28875–28890. https://doi.org/10.1007/s11042-023-16615-z doi: 10.1007/s11042-023-16615-z
|
| [5] |
H. Alzer, K. Richards, Combinatorial identities and hypergeometric functions, Rocky Mountain J. Math., 52 (2022), 1921–1928. https://doi.org/10.1216/rmj.2022.52.1921 doi: 10.1216/rmj.2022.52.1921
|
| [6] | H. Gould, Combinatorial identities: A standardized set of tables listing 500 binomial coefficient summations, Morgantown, 1972. |
| [7] | E. Diekema, Combinatorial identities and hypergeometric series, arXiv, 2022. https://doi.org/10.48550/arXiv.2204.05647 |
| [8] |
J. Borwein, A. Straub, C. Vignat, Densities of short uniform random walks in higher dimensions, J. Math. Anal. Appl., 437 (2016), 668–707. https://doi.org/10.1016/j.jmaa.2016.01.017 doi: 10.1016/j.jmaa.2016.01.017
|
| [9] | J. Borwein, D. Nuyens, A. Straub, J. Wan, Random walks in the plane, Discrete Math. Theor. Comput. Sci., 2010,191–202. |
| [10] |
J. McCrorie, Moments in Pearson's four-step uniform random walk problem and other applications of very well-poised generalized hypergeometric series, Sankhya B, 83 (2021), 244–281. https://doi.org/10.1007/s13571-020-00230-1 doi: 10.1007/s13571-020-00230-1
|
| [11] |
S. Janson, D. Knuth, T. Łuczak, B. Pittel, The birth of the giant component, Random Struct. Algor., 4 (1993), 233–358. https://doi.org/10.1002/rsa.3240040303 doi: 10.1002/rsa.3240040303
|
| [12] | E. P. Wigner, On the matrices which reduce the Kronecker products of representations of S. R. groups, In: The collected works of eugene paul wigner, Berlin, Heidelberg: Springer, 1993,608–654. https://doi.org/10.1007/978-3-662-02781-3_42 |
| [13] | K. S. Rao, Symmetries of 3n-j coefficients and generalized hypergeometric functions, In: Symmetries in science X, Boston: Springer, 1998,383–399. https://doi.org/10.1007/978-1-4899-1537-5_24 |
| [14] |
M. Harmer, Note on the Schwarz triangle functions, Bull. Aust. Math. Soc., 72 (2005), 38–389. https://doi.org/10.1017/S0004972700035218 doi: 10.1017/S0004972700035218
|
| [15] |
A. Ali, M. Islam, A. Noreen, Z. Nisa, Solution of fractional k-hypergeometric differential equation, Int. J. Math. Anal., 14 (2020), 125–132. https://doi.org/10.12988/ijma.2020.91287 doi: 10.12988/ijma.2020.91287
|
| [16] |
M. Abul-Ez, M. Zayed, A. Youssef, Further study on the conformable fractional Gauss hypergeometric function, AIMS Mathematics, 6 (2021), 10130–10163. https://doi.org/10.3934/math.2021588 doi: 10.3934/math.2021588
|
| [17] |
M. Chen, W. Chu, Yabu's formulae for hypergeometric 3F2-series through Whipple's quadratic transformations, AIMS Mathematics, 9 (2024), 21799–21815. https://doi.org/10.3934/math.20241060 doi: 10.3934/math.20241060
|
| [18] |
M. Atia, M. Alkilayh, Extension of Chu–Vandermonde identity and quadratic transformation conditions, Axioms, 13 (2024), 825. https://doi.org/10.3390/axioms13120825 doi: 10.3390/axioms13120825
|
| [19] |
M. Atia, On the inverse of the linearization coefficients of Bessel polynomials, Symmetry, 16 (2024), 737. https://doi.org/10.3390/sym16060737 doi: 10.3390/sym16060737
|
| [20] |
M. Atia, A. Rathie, On a generalization of the Kummer's quadratic transformation and a resolution of an isolated case, Axioms, 12 (2023), 821. https://doi.org/10.3390/axioms12090821 doi: 10.3390/axioms12090821
|
| [21] |
M. Atia, A. Al-Mohaimeed, On a resolution of another isolated case of a Kummer's quadratic transformation for 2F1, Axioms, 12 (2023), 221. https://doi.org/10.3390/axioms12020221 doi: 10.3390/axioms12020221
|
| [22] |
A. Shehata, S. Moustafa, Some new results for Horn's hypergeometric functions Γ1 and Γ2, J. Math. Comput. Sci., 23 (2020), 26–35. https://doi.org/10.22436/jmcs.023.01.03 doi: 10.22436/jmcs.023.01.03
|
| [23] |
W. Mohammed, C. Cesarano, F. Al-Askar, Solutions to the (4+1)-dimensional time-fractional Fokas equation with M-truncated derivative, Mathematics, 11 (2023), 194. https://doi.org/10.3390/math11010194 doi: 10.3390/math11010194
|
| [24] |
F. He, A. Bakhet, M. Hidan, H. Abd-Elmageed, On the construction of (p, k)-hypergeometric function and applications, Fractals, 30 (2022), 2240261. https://doi.org/10.1142/S0218348X22402617 doi: 10.1142/S0218348X22402617
|
| [25] |
H. Exton, A new two-term relation for the 3F2 hypergeometric function of unit argument, J. Comput. Appl. Math., 106 (1999), 395–397. https://doi.org/10.1016/S0377-0427(99)00077-1 doi: 10.1016/S0377-0427(99)00077-1
|
| [26] | M. Milgram, On hypergeometrics 3F2(1)-A review, arXiv, 2010. https://doi.org/10.48550/arXiv.1011.4546 |
| [27] |
Y. Kim, J. Choi, A. Rathie, Two results for the terminating 3F2(2) with applications, Bull. Korean Math. Soc., 49 (2012), 621–633. https://doi.org/10.4134/BKMS.2012.49.3.621 doi: 10.4134/BKMS.2012.49.3.621
|
| [28] |
K. Chen, Explicit formulas for some infinite 3F2(1)-series, Axioms, 10 (2021), 125. https://doi.org/10.3390/axioms10020125 doi: 10.3390/axioms10020125
|
| [29] |
M. Chen, W. Chu, Bisection series approach for exotic 3F2(1)-series, Mathematics, 12 (2024), 1915. https://doi.org/10.3390/math12121915 doi: 10.3390/math12121915
|
| [30] | A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006). |
| [31] | E. D. Rainville, Special functions, New York: Macmillan, 1960. |
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| 2. | Martin Gugat, Jan Giesselmann, G. Buttazzo, E. Casas, L. de Teresa, R. Glowinski, G. Leugering, E. Trélat, X. Zhang, Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks, 2021, 27, 1292-8119, 67, 10.1051/cocv/2021061 | |
| 3. | Martin Gugat, Stephan Gerster, On the limits of stabilizability for networks of strings, 2019, 131, 01676911, 104494, 10.1016/j.sysconle.2019.104494 | |
| 4. | Charlotte Rodriguez, Networks of geometrically exact beams: Well-posedness and stabilization, 2022, 12, 2156-8472, 49, 10.3934/mcrf.2021002 | |
| 5. | Martin Gugat, Günter Leugering, Ke Wang, Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-lyapunov function, 2017, 7, 2156-8499, 419, 10.3934/mcrf.2017015 | |
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| 7. | Enrique Zuazua, 2013, Chapter 9, 978-3-642-32159-7, 463, 10.1007/978-3-642-32160-3_9 | |
| 8. | Martin Gugat, Boundary feedback stabilization of the telegraph equation: Decay rates for vanishing damping term, 2014, 66, 01676911, 72, 10.1016/j.sysconle.2014.01.007 | |
| 9. | Zhong-Jie Han, Gen-Qi Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, 2011, 84, 0020-7179, 458, 10.1080/00207179.2011.561441 | |
| 10. | Farhat Shel, Exponential stability of a network of elastic and thermoelastic materials, 2013, 36, 01704214, 869, 10.1002/mma.2644 | |
| 11. | Abdessamad El Alami, Imad El Harraki, Ali Boutoulout, Regional Feedback Stabilization for Infinite Semilinear Systems, 2018, 24, 1079-2724, 343, 10.1007/s10883-017-9386-3 | |
| 12. | Kaïs Ammari, Farhat Shel, 2022, Chapter 2, 978-3-030-86350-0, 15, 10.1007/978-3-030-86351-7_2 | |
| 13. | Martin Gugat, Markus Dick, Günter Leugering, 2013, Chapter 26, 978-3-642-36061-9, 255, 10.1007/978-3-642-36062-6_26 | |
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| 15. | Abdessamad El Alami, Ali Boutoulout, 2020, Chapter 6, 978-3-030-26148-1, 67, 10.1007/978-3-030-26149-8_6 | |
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| 18. | Markus Dick, Martin Gugat, Gunter Leugering, 2012, Feedback stabilization of quasilinear hyperbolic systems with varying delays, 978-1-4673-2124-2, 125, 10.1109/MMAR.2012.6347931 | |
| 19. | Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe, Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks, 2020, 170, 0167-8019, 823, 10.1007/s10440-020-00360-8 | |
| 20. | Yacine Chitour, Guilherme Mazanti, Mario Sigalotti, Persistently damped transport on a network of circles, 2016, 369, 0002-9947, 3841, 10.1090/tran/6778 | |
| 21. | Markus Dick, Martin Gugat, Günter Leugering, A strict H1-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction, 2011, 1, 2155-3297, 225, 10.3934/naco.2011.1.225 | |
| 22. | Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier, Finite-time stabilization of a network of strings, 2015, 5, 2156-8499, 721, 10.3934/mcrf.2015.5.721 | |
| 23. | Martin Gugat, Sonja Steffensen, Dynamic boundary control games with networks of strings, 2018, 24, 1292-8119, 1789, 10.1051/cocv/2017082 | |
| 24. | Martin Gugat, Markus Dick, Günter Leugering, Gas Flow in Fan-Shaped Networks: Classical Solutions and Feedback Stabilization, 2011, 49, 0363-0129, 2101, 10.1137/100799824 | |
| 25. | Mohamed Ouzahra, Global stabilization of semilinear systems using switching controls, 2012, 48, 00051098, 837, 10.1016/j.automatica.2012.02.018 | |
| 26. | Pierre-Olivier Lamare, Antoine Girard, Christophe Prieur, 2013, Lyapunov techniques for stabilization of switched linear systems of conservation laws, 978-1-4673-5717-3, 448, 10.1109/CDC.2013.6759922 | |
| 27. | Martin Gugat, Markus Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction, 2011, 1, 2156-8499, 469, 10.3934/mcrf.2011.1.469 | |
| 28. | Markus Dick, Martin Gugat, Michael Herty, Günter Leugering, Sonja Steffensen, Ke Wang, 2014, Chapter 31, 978-3-319-05082-9, 487, 10.1007/978-3-319-05083-6_31 | |
| 29. | Markus Dick, Martin Gugat, Günter Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, 2010, 5, 1556-181X, 691, 10.3934/nhm.2010.5.691 | |
| 30. | Jon Asier Bárcena-Petisco, Márcio Cavalcante, Giuseppe Maria Coclite, Nicola De Nitti, Enrique Zuazua, Control of hyperbolic and parabolic equations on networks and singular limits, 2024, 0, 2156-8472, 0, 10.3934/mcrf.2024015 | |
| 31. | Martin Gugat, Stabilization of a cyclic network of strings by nodal control, 2025, 25, 1424-3199, 10.1007/s00028-024-01030-0 | |
| 32. | Yacine Chitour, Paolo Mason, Mario Sigalotti, 2025, Chapter 7, 978-3-031-82981-9, 145, 10.1007/978-3-031-82982-6_7 | |
| 33. | Felipe Gonçalves Netto, Yacine Chitour, Guilherme Mazanti, Strong Stability of Linear Delay-Difference Equations, 2025, 9, 2475-1456, 1664, 10.1109/LCSYS.2025.3577929 | |
| 34. | Giuseppe Maria Coclite, Nicola De Nitti, Mauro Garavello, Francesca Marcellini, Feedback stabilization for entropy solutions of a 2 × 2 hyperbolic system of conservation laws at a junction, 2025, 00217824, 103774, 10.1016/j.matpur.2025.103774 |
Abdelhamid Zaidi, Saleh Almuthaybiri. Explicit evaluations of subfamilies of the hypergeometric function 3F2(1) along with specific fractional integrals[J]. AIMS Mathematics, 2025, 10(3): 5731-5761. doi: 10.3934/math.2025264
| Sample ID | Gains |
Compensation |
||||
| FSC | SSC | GRN | YEL | RED | GRN-%RED | |
| Chicken DNA | 5.2 | 1.9 | 8.0 | 12.3 | 6.7 | 0.0 |
| Escherichia coli RNA | 1.7 | 3.1 | 20.7 | 12.3 | 3.5 | 0.0 |
| Listeria monocytogenes АТСС 13932 (positive control) | 5.2 | 1.9 | 23.6 | 12.3 | 8.7 | 3.6 |
| Listeria monocytogenes АТСС 13932 (negative control-1) | 5.2 | 1.9 | 23.6 | 12.3 | 8.7 | 5.1 |
| Listeria monocytogenes АТСС 13932 (negative control-2) | 5.2 | 1.9 | 15.3 | 12.3 | 8.7 | 2.2 |
DownLoad:
CSV
| Sample ID | Gains |
Compensation |
||||
| FSC | SSC | GRN | YEL | RED | GRN-%RED | |
| Chicken DNA | 5.2 | 1.9 | 8.0 | 12.3 | 6.7 | 0.0 |
| Escherichia coli RNA | 1.7 | 3.1 | 20.7 | 12.3 | 3.5 | 0.0 |
| Listeria monocytogenes АТСС 13932 (positive control) | 5.2 | 1.9 | 23.6 | 12.3 | 8.7 | 3.6 |
| Listeria monocytogenes АТСС 13932 (negative control-1) | 5.2 | 1.9 | 23.6 | 12.3 | 8.7 | 5.1 |
| Listeria monocytogenes АТСС 13932 (negative control-2) | 5.2 | 1.9 | 15.3 | 12.3 | 8.7 | 2.2 |
