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Published: 18 May 2024

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1

Sarkhel, D. N. "A wide Perron integral." Bulletin of the Australian Mathematical Society 34, no. 2 (October 1986): 233–51. http://dx.doi.org/10.1017/s0004972700010108.

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In terms of an arbitrary limit process T, defined abstractly for real functions, we define in a novel way a T-continuous integral of Perron type, admitting mean value theorems, integration by parts and the analogue of the Marcinkiewicz theorem for the ordinary Perron integral. The integral is shown to include, as particular cases, the various known continuous, approximately continuous, cesàro-continuous, mean-continuous and proximally Cesàro-continuous integrals of Perron and Denjoy types. An interesting generalization of the classical Lebesgue decomposition theorem is also obtained.

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2

Kim, Joo Bong, Deok Ho Lee, Woo Youl Lee, Chun-Gil Park, and Jae Myung Park. "The s-Perron, sap-Perron and ap-McShane Integrals." Czechoslovak Mathematical Journal 54, no. 3 (September 2004): 545–57. http://dx.doi.org/10.1007/s10587-004-6407-7.

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3

SIUDUT, STANISLAW. "ON THE DENJOY-PERRON-BOCHNER INTEGRAL." Tamkang Journal of Mathematics 25, no. 4 (December 1, 1994): 295–300. http://dx.doi.org/10.5556/j.tkjm.25.1994.4457.

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The notion of Denjoy integrals of abstract functions was first intro- duced by A. Alexiewicz [1]. His descriptive definitions are based upon a concept of the approximate derivative. In this paper we present another descriptive definition for the Denjoy-Perron integral of abstract functions - via the parametric derivative of Tolstov [8]. Some properties of this integral are examined.

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4

Jurkat, W. B., and R. W. Knizia. "A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem." Canadian Journal of Mathematics 43, no. 3 (June 1, 1991): 526–39. http://dx.doi.org/10.4153/cjm-1991-032-8.

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AbstractIn this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.

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5

Ene. "INTEGRALS OF LUSIN AND PERRON TYPE." Real Analysis Exchange 14, no. 1 (1988): 115. http://dx.doi.org/10.2307/44153633.

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6

Tulone, Francesco. "Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space." Tatra Mountains Mathematical Publications 49, no. 1 (December 1, 2011): 81–88. http://dx.doi.org/10.2478/v10127-011-0027-z.

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ABSTRACT A Henstock-Kurzweil type integral on a compact zero-dimensional metric space is investigated. It is compared with two Perron type integrals. It is also proved that it covers the Lebesgue integral.

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7

Henstock. "LIMITS OF ÇESARO-PERRON AND SIMILAR INTEGRALS." Real Analysis Exchange 24, no. 1 (1998): 95. http://dx.doi.org/10.2307/44152934.

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8

YEE. "GENERALIZED CONVERGENCE THEOREMS FOR DENJOY–PERRON INTEGRALS." Real Analysis Exchange 14, no. 1 (1988): 48. http://dx.doi.org/10.2307/44153618.

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9

Bongiorno, B., W. F. Pfeffer, and B. S. Thomson. "A Full Descriptive Definition of the Gage Integral." Canadian Mathematical Bulletin 39, no. 4 (September 1, 1995): 395–401. http://dx.doi.org/10.4153/cmb-1996-047-x.

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AbstractWe consider a specific Riemann type integral, called the gage integral. Using variational measures, we characterize all additive functions of intervals that are indefinite gage integrals. The characterization generalizes the descriptive definition of the classical Denjoy-Perron integral to all dimensions.

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10

Yee, Lee Peng, and Chew Tuan Seng. "On Convergence theorems for nonabsolute integrals." Bulletin of the Australian Mathematical Society 34, no. 1 (August 1986): 133–40. http://dx.doi.org/10.1017/s0004972700004585.

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11

Ahmed, S. I., and W. F. Pfeffer. "A Riemann integral in a locally compact Hausdorff space." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 1 (August 1986): 115–37. http://dx.doi.org/10.1017/s1446788700028123.

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AbstractWe present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

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12

Bullen, P. S. "A survey of intergration by parts for Perron integrals." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 3 (June 1986): 343–63. http://dx.doi.org/10.1017/s1446788700027555.

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13

De Sarkar, S., and A. G. Das. "Riemann derivatives and general integrals." Bulletin of the Australian Mathematical Society 35, no. 2 (April 1987): 187–211. http://dx.doi.org/10.1017/s0004972700013174.

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Sargent and later Bullen and Mukhopadhyay obtained a definition of absolutely continuous functions, functions, that is related to kth Peano derivatives. The generalised notions of ACkG*, [ACkG*], ACkG* above, etcetera functions led Bullen and Mukhopadhyay to define certain general integrals of the kth order.The present work is concerned with a further simplification of the definitions of such functions by the use of divided differences but still retaining similar fundamental properties. These concepts lead to the introduction of Denjoy and Ridder type integrals which are shown to be equivalent to a Perron type integral that corresponds to kth Riemann* derivatives. All three of these integrals are shown to be equivalent to the three integrals of Bullen and Mukhopadhyay.

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14

Schwabik, Štefan. "A note on integration by parts for abstract Perron-Stieltjes integrals." Mathematica Bohemica 126, no. 3 (2001): 613–29. http://dx.doi.org/10.21136/mb.2001.134198.

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15

Dergachev, A. V. "Generalized derivatives and integrals of Cesàro-Perron Type. I." Moscow University Mathematics Bulletin 69, no. 2 (March 2014): 56–66. http://dx.doi.org/10.3103/s002713221402003x.

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16

Dergachev, A. V. "Generalized derivatives and integrals of Cesàro-Perron type. II." Moscow University Mathematics Bulletin 69, no. 6 (November 2014): 229–36. http://dx.doi.org/10.3103/s0027132214060011.

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17

Seng, Chew Tuan. "On the generalised dominated convergence theorem." Bulletin of the Australian Mathematical Society 37, no. 2 (April 1988): 165–71. http://dx.doi.org/10.1017/s0004972700026691.

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In this paper we give another version of the generalised dominated convergence theorem, which is better than other convergence theorems for Perron integrals in the sense that it can be applied more easily.

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18

Boccuto, A., A. R. Sambucini, and V. A. Skvortsov. "Integration by Parts for Perron Type Integrals of Order 1 and 2 in Riesz Spaces." Results in Mathematics 51, no. 1-2 (November 16, 2007): 5–27. http://dx.doi.org/10.1007/s00025-007-0254-4.

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19

Skvortsov, Valentin, and Francesco Tulone. "Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals in the theory of Haar and Walsh series." Journal of Mathematical Analysis and Applications 421, no. 2 (January 2015): 1502–18. http://dx.doi.org/10.1016/j.jmaa.2014.08.002.

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20

A. Boccuto and V. A. Skvortsov. "The Ward, Perron and Henstock-Kurzweil Integrals with Respect to Abstract Derivation Bases in Riesz Spaces." Real Analysis Exchange 31, no. 2 (2006): 431. http://dx.doi.org/10.14321/realanalexch.31.2.0431.

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21

Bardadyn, Krzysztof, Bartosz Kosma Kwasniewski, Kirill S. Kurnosenko, and Andrei V. Lebedev. "t-Entropy formulae for concrete classes of transfer operators." Journal of the Belarusian State University. Mathematics and Informatics, no. 3 (November 29, 2019): 122–28. http://dx.doi.org/10.33581/2520-6508-2019-3-122-128.

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t-Entropy is a principal object of the spectral theory of operators, generated by dynamical systems, namely, weighted shift operators and transfer operators. In essence t-entropy is the Fenchel – Legendre transform of the spectral potential of an operator in question and derivation of explicit formulae for its calculation is a rather nontrivial problem. In the article explicit formulae for t-entropy for two the most exploited in applications classes of transfer operators are obtained. Namely, we consider transfer operators generated by reversible mappings (i. e. weighted shift operators) and transfer operators generated by local homeomorphisms (i. e. Perron – Frobenius operators). In the first case t-entropy is computed by means of integrals with respect to invariant measures, while in the second case it is computed in terms of integrals with respect to invariant measures and Kolmogorov – Sinai entropy.

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22

Boccuto, A., A. R. Sambucini, and V. A. Skvortsov. "Erratum to: Integration by Parts for Perron Type Integrals of Order 1 and 2 in Riesz Spaces." Results in Mathematics 57, no. 3-4 (March 19, 2010): 393–96. http://dx.doi.org/10.1007/s00025-010-0023-7.

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23

Choi, In. "Residual-Based Tests for the Null of Stationarity with Applications to U.S. Macroeconomic Time Series." Econometric Theory 10, no. 3-4 (August 1994): 720–46. http://dx.doi.org/10.1017/s0266466600008744.

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This paper proposes residual-based tests for the null of level- and trend-stationarity, which are analogs of the LM test for an MA unit root. Asymptotic distributions of the tests are nonstandard, but they are expressed in a unified manner by expressing stochastic integrals. In addition, the tests are shown to be consistent. By expressing the distributions expressed as a function of a chi-square variable with one degree of freedom, the exact limiting probability density and cumulative distribution functions are obtained, and the exact limiting cumulative distribution functions are tabulated. Finite sample performance of the proposed tests is studied by simulation. The tests display stable size when the lag truncation number for the long-run variance estimation is chosen appropriately. But the power of the tests is generally not high at selected sample sizes. The test for the null of trend-stationarity is applied to the U.S. macroeconomic time series along with the Phillips-Perron Z(⋯) test. For some monthly and annual series, the two tests provide consistent inferential results. But for most series, the two contradictory nulls of trend-stationarity and a unit root cannot be rejected at the conventional significance levels.

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24

Schwabik, Štefan. "Abstract Perron-Stieltjes integral." Mathematica Bohemica 121, no. 4 (1996): 425–47. http://dx.doi.org/10.21136/mb.1996.126036.

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25

Benedicks, M., and W. F. Pfeffer. "The Dirichlet Problem With Denjoy-Perron Integrable Boundary Condition." Canadian Mathematical Bulletin 28, no. 1 (March 1, 1985): 113–19. http://dx.doi.org/10.4153/cmb-1985-013-1.

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AbstractThe Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.

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26

Král, Josef. "Note on generalized multiple Perron integral." Časopis pro pěstování matematiky 110, no. 4 (1985): 371–74. http://dx.doi.org/10.21136/cpm.1985.118252.

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27

Tvrdý, Milan. "Regulated functions and the Perron-Stieltjes integral." Časopis pro pěstování matematiky 114, no. 2 (1989): 187–209. http://dx.doi.org/10.21136/cpm.1989.108713.

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28

Kurzweil, Jaroslav, and Jiří Jarník. "Equivalent definitions of regular generalized Perron integral." Czechoslovak Mathematical Journal 42, no. 2 (1992): 365–78. http://dx.doi.org/10.21136/cmj.1992.128325.

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29

Morales, Pedro. "The Perron product integral in Lie groups." Czechoslovak Mathematical Journal 43, no. 2 (1993): 349–66. http://dx.doi.org/10.21136/cmj.1993.128392.

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30

Kim, Byung-Moo, Young-Kuk Kim, and Jae-Myung Park. "THE STRONG PERRON INTEGRAL." Bulletin of the Korean Mathematical Society 40, no. 1 (February 1, 2003): 77–83. http://dx.doi.org/10.4134/bkms.2003.40.1.077.

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31

Schwabik, Štefan. "The Perron product integral and generalized linear differential equations." Časopis pro pěstování matematiky 115, no. 4 (1990): 368–404. http://dx.doi.org/10.21136/cpm.1990.118415.

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32

Bose, M. K., and B. Ghosh. "On the Cesàro-Perron integral." Bulletin of the Australian Mathematical Society 43, no. 3 (June 1991): 483–89. http://dx.doi.org/10.1017/s0004972700029336.

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33

Navarro and Skvortsov. "ON THE N-DIMENSIONAL PERRON INTEGRAL." Real Analysis Exchange 21, no. 1 (1995): 69. http://dx.doi.org/10.2307/44153880.

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34

Skvortsov, Valentin A., and Piotr Sworowski. "THE STRONG PERRON INTEGRAL IN ℝnREVISITED." Communications of the Korean Mathematical Society 22, no. 1 (January 31, 2007): 15–18. http://dx.doi.org/10.4134/ckms.2007.22.1.015.

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35

Bullen, P. S., and R. Vyborny. "Some Applications of a Theorem of Marcinkiewicz." Canadian Mathematical Bulletin 34, no. 2 (June 1, 1991): 165–74. http://dx.doi.org/10.4153/cmb-1991-027-x.

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AbstractA classical theorem of Marcinkiewicz states that a function is Perron integrable iff it has one continuous major and one continuous minor function. Using an elaboration of this remarkable theorem three applications are made; to obtain a new proof of a recent characterization of the Perron integral, to proofs of some theorems on interchange of limits and integration and to extend classical existence theorems for ordinary differential equations.

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36

Cross. "INTEGRATION BY PARTS FOR THE PERRON INTEGRAL." Real Analysis Exchange 16, no. 2 (1990): 546. http://dx.doi.org/10.2307/44153734.

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37

Schurle, Arlo W. "Perron Integrability Versus Lebesgue Integrability." Canadian Mathematical Bulletin 28, no. 4 (December 1, 1985): 463–68. http://dx.doi.org/10.4153/cmb-1985-055-1.

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AbstractThe paper investigates the relationship between Perron - Stieltjes integrability and Lebesgue-Stieltjes integrability within the generalized Riemann approach. The main result states that with certain restrictions a Perron-Stieltjes integrable function is locally Lebesgue-Stieltjes integrable on an open dense set. This is then applied to show that a nonnegative Perron-Stieltjes integrable function is Lebesgue-Stieltjes integrable. Finally, measure theory is invoked to remove the restrictions in the main result.

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38

Skvortsov. "A PERRON TYPE INTEGRAL IN AN ABSTRACT SPACE." Real Analysis Exchange 13, no. 1 (1987): 76. http://dx.doi.org/10.2307/44151850.

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39

Shchepin, E. V. "The Leibniz Differential and the Perron–Stieltjes Integral." Journal of Mathematical Sciences 233, no. 1 (July 2, 2018): 157–71. http://dx.doi.org/10.1007/s10958-018-3932-8.

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40

Thomson. "THE SPACE OF DENJOY-PERRON INTEGRABLE FUNCTIONS." Real Analysis Exchange 25, no. 2 (1999): 711. http://dx.doi.org/10.2307/44154028.

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41

Montenegro Carrasco, Wilfredo. "La educación del siglo XXI: un proceso de formación integral de la persona humana." Cultura, no. 35 (December 30, 2021): 107–31. http://dx.doi.org/10.24265/cultura.2021.v35.07.

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La educación del hombre es un tema que no deja de ser cuestionado. A pesar de los esfuerzos y de las diversas innovaciones en la materia, la crisis educativa de la «sociedad líquida» es cada vez más profunda y al mismo tiempo más relativa (Bauman, 2004). El hombre y sus actitudes, hoy fuertemente controvertidas por sus manifestaciones de violencia, irracionalidad, corrupción y rechazo de los valores, revelan la cruda realidad educativa. Frente a la situación descrita, la presente investigación se propone analizar las raíces de esta problemática y a la vez proponer reflexiones y alternativas innovadoras que ayuden a superar dicha crisis. Los resultados encontrados son reveladores y esperanzadores. Varios de los planteamientos de este trabajo ya están contemplados en la legislación y en la teoría educativa desde hace varias décadas; sin embargo, han quedado en el olvido por efectos del individualismo, la anomia, el escepticismo y el nihilismo propios de la sociedad posmoderna. El aporte factible y necesario es la educación sistémica de la persona que, superando los diversos reduccionismos, se centre en el «desarrollo humano integral» (Benedicto XVI, 2009); dejando en claro, asimismo, los roles de la familia, del Estado, de la escuela, de los medios de comunicación social, de la sociedad y del mismo educando frente al desafío de la educación actual.

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42

Bendová, Hana, and Jan Malý. "An elementary way to introduce a Perron-like integral." Annales Academiae Scientiarum Fennicae Mathematica 36 (2011): 153–64. http://dx.doi.org/10.5186/aasfm.2011.3609.

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43

Bongiorno, Di Piazza, and Skvortsov. "A NEW FULL DESCRIPTIVE CHARACTERIZATION OF DENJOY-PERRON INTEGRAL." Real Analysis Exchange 21, no. 2 (1995): 656. http://dx.doi.org/10.2307/44152676.

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44

Zeng-Tai. "ON A PROBLEM OF SKVORTSOV INVOLVING THE PERRON INTEGRAL." Real Analysis Exchange 17, no. 2 (1991): 748. http://dx.doi.org/10.2307/44153766.

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45

Skvortsov, V. A., and F. Tulone. "Perron type integral on compact zero-dimensional Abelian groups." Moscow University Mathematics Bulletin 63, no. 3 (June 2008): 119–24. http://dx.doi.org/10.3103/s002713220803008x.

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46

Kurzweil, J., and J. Jarnik. "Generalized Multidimensional Perron Integral Involving a New Regularity Condition." Results in Mathematics 23, no. 3-4 (May 1993): 363–73. http://dx.doi.org/10.1007/bf03322308.

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47

Kwapisz, Jarosław. "Inflations of self-affine tilings are integral algebraic Perron." Inventiones mathematicae 205, no. 1 (November 27, 2015): 173–220. http://dx.doi.org/10.1007/s00222-015-0633-5.

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48

Boccuto, Antonio, and Valentin A. Skvortsov. "The Perron integral of order k in Riesz spaces." Positivity 14, no. 4 (April 9, 2010): 595–612. http://dx.doi.org/10.1007/s11117-010-0054-z.

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49

Dong, Jingcheng, and Henry Tucker. "Integral Modular Categories of Frobenius-Perron Dimension pq n." Algebras and Representation Theory 19, no. 1 (July 29, 2015): 33–46. http://dx.doi.org/10.1007/s10468-015-9560-9.

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50

Skvortsov, V. A. "On the Marcinkiewicz theorem for the binary Perron integral." Mathematical Notes 59, no. 2 (February 1996): 189–95. http://dx.doi.org/10.1007/bf02310959.

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